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Riso and Hudson have identified 'lost messages' that we needed to hear as children but didn't. The absence of these words may be at the heart of our most basic fear. Too often as children we instead hear admonishments such as "You're useless!!", or "Do as I tell you to". Other times we hear absolutes like : "You're always a nuisance!" or "You never show gratitude".
Our inner child however does not want to hear such words, and he is hurt spiritually by them. This hurt resides deep inside and manifests itself in adulthood as feelings of insecurity, low self-worth or social withdrawal.
Which one do you most need to hear to heal the hurt from all those years ago? And when you have identified one, use it as a daily affirmation to yourself. You may repeat the phrase when you are alone, while having a drink or even while walking. Allow it to sink into you….feel the warmth within as it does. And as it sinks it, all the old beliefs and behaviour will recede as the new positive and desirable feelings about yourself rises to the foreground of your being.
Alternatively, do take advantage of my services to quickly resolve the hurt and heal the child within. Call 97597683 NOW! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,347 |
Kings of the Sun is unique for the time in that it doesn't feature a single "white man" character. There's no invading Europeans with their fancy coats, silver tongues and daggers behind their backs. But hang about you cry, neither did Apocalypto (excluding the end). Maybe so, but Kings of the Sun takes it to the extreme by having not one, not two, but three different tribes of natives all fighting each other. It is the Gangs of New York of Mesoamerican inter-tribal gang warfare movies. But unlike Apocalypto there's controversially no actual Native American actor on set here. And the lead is a Russian?! Something sure is afoot. | {
"redpajama_set_name": "RedPajamaC4"
} | 23 |
{"url":"https:\/\/www.physicsforums.com\/threads\/sum-to-product-product-to-sum.89313\/","text":"# Sum to Product \/ Product to Sum\n\n1. Sep 16, 2005\n\n### amcavoy\n\nIs there any reliable way to convert a series to a product, or the opposite? I was looking at the following and wanted to know more:\n\n$$\\sum_{n=1}^{\\infty}\\frac{1}{n^{s}}=\\prod_{p}\\left(1-p^{-s}\\right)^{-1}$$\n\n2. Sep 16, 2005\n\n### shmoe\n\nIf the coefficients of your Dirichlet series is a multiplicative function f, that is\n\n$$\\sum_{n=1}^\\infty f(n)n^{-s}$$\n\nthen you can write this as an Euler product\n\n$$\\prod_{p}(1+f(p)p^{-s}+f(p^2)p^{-2s}+\\ldots)$$\n\nwhere the product is over the primes (this is assuming you have absolute convergence of both product and sum). You can think of this as the fundamental theorem of arithmetic in an analytic form. There are plenty of interesting examples of this, powers of Zeta, Dirichlet L-functions, and anything that gets the name \"L-function\" is usually assumed to satisfy some form of this (as well as many other properties).\n\nFor more general sums and products you can still use exponentiation and logarithms to convert from one to another, again being careful with convergence issues if any.\n\n3. Sep 16, 2005\n\n### SGT\n\nExponentials turn sums into products, while logarithms turn products into sums. So:\n$$exp(\\sum_{n=1}^{\\infty}\\frac{1}{n^{s}})=\\prod_{n=1}^{\\infty}exp(\\frac{1}{n^{s}})$$\nYou must now find $$p$$ such that\n$$\\left(1-p^{-s}\\right)^{-1} = exp(\\frac{1}{n^{s}})$$\n\n4. Sep 16, 2005\n\n### shmoe\n\nAlthough it wasn't mentioned, the product in the orignal post is almost surely a product over all the primes (it's the Euler product form of the Riemann Zeta function. The terms won't match up via exponentiation like this.","date":"2016-12-08 09:56:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8529849052429199, \"perplexity\": 481.69486341682654}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698542520.47\/warc\/CC-MAIN-20161202170902-00084-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:0}
Besides the well-known quantum bifurcation in the ground state, known from the Dicke model~\cite{Emary2003} and the bosonic Josephson
junction~\cite{Chuchem2010}, quantum criticality also takes place in excited states~\cite{Caprio2008}.
A key feature of the so-called excited-state quantum phase transitions (ESQPTs),
is a level clustering at critical energies, which results in a logarithmic singularity in the density of states (DOS)~\cite{Caprio2008,Ribeiro2008,Brandes2013}.
This singularity is induced by a saddle point of the semiclassical energy surface~\cite{Caprio2008}.
Accordingly, in the thermodynamic limit the eigenstates of the quantum system
also experience symmetry breaking, in the sense of being degenerate or not, depending on their energy~\cite{Puebla2013}.
As a consequence of the strong relation to the classical dynamics, the underlying classical dynamics drives the ESQPT, which can be
considered to be a quantum manifestation of a separatrix~\cite{Scharf1992}.
Among the quantum signatures of the separatrix~\cite{Aubry1996,Chuchem2010,Nissen2010,Caballero-Benitez2010,Juli'a-D'iaz2010,Juli'a-D'iaz2010a,Kawaguchi2012,
Bernstein1993,Franzosi2000,Franzosi2001,Karkuszewski2002,Buonsante2004,Graefe2007},
intriguing relations to spin squeezing and entanglement~\cite{Juli'a-D'iaz2012,Juli'a-D'iaz2012a,Hennig2012} have been explored.
Previous works study a plethora of dynamical consequences of ESQPTs, i.e.,
their influence on quantum quenches \cite{P'erez-Fern'andez2011a}, the
relation to chaos~\cite{P'erez-Fern'andez2011}, and decoherence of a qubit coupled to a system possessing an ESQPT~\cite{P'erez-Fern'andez2009}.
In the context of driven quantum systems, singular behavior resembling ESQPTs appears in the quasienergy
spectrum of the kicked top~\cite{Bastidas2014}.
Experimental signatures of ESQPTs have been found in molecular
systems~\cite{Winnewisser2005}, in the diverging period of a spinor Bose-Einstein condensate~\cite{Zhao2014}, and in microwave billiards~\cite{Dietz2013}.
In pioneering experiments, Oberthaler and collaborators realized the anisotropic Lipkin-Meshkov-Glick (LMG) model by coupling the internal
degrees of freedom of a $^{87}$Rb Bose-Einstein condensate~\cite{Zibold2010,Gross2010,Steel1998}. Additionally,
the striking experimental observation of Dicke superradiance in Bose gases loaded in a cavity~\cite{Baumann2010,Baumann2011},
opens the possibility to explore the physics of excited states, due to the high degree of control of the system.
Recently, there has been a renewed interest in the experimental investigation of the Tavis-Cummings model, which
is realized using cavity-assisted Raman transitions \cite{Baden2014}.
Motivated by these experimental highlights, in this paper we explore
the signatures of the ESQPT that appear in the periodic dynamics of the LMG model and the Tavis-Cummings (TC) model.
In doing so, we use the fact that
expectation values in eigenstates can be calculated by a semiclassical temporal average~\cite{Ribeiro2008,Paul1993}.
In contrast to the LMG model, the classical dynamics of the TC model do not occur on a two-dimensional manifold. Therefore, it is
interesting to investigate
if our approach is also applicable for models with regular dynamics and a high-dimensional phase space such as the TC model.
The rest of the paper is organized as follows. In Sec.~\ref{sec:I}, we introduce the models that will be discussed
along the paper, their corresponding semiclassical limits and the method to calculate time-averaged expectation values.
In Sec.~\ref{sec:II}, we discuss our results for the LMG and TC models. In particular, we perform numerical calculations
for a finite-size system to compare with the semiclassical results. Finally, in Sec.~\ref{sec:III} we discuss the application
of our method to the LMG model for realistic experimental parameters. The conclusions in Sec.~\ref{sec:III} are followed
by the Appendix, where we discuss in detail the semiclassical calculations for the Tavis-Cummings model.
\section{Models and Methods}
\label{sec:I}
\subsection{Hamiltonians}
The LMG model describes a set of $N$ two-level systems in a transverse field in the $z$-direction which have anisotropic
interactions~\cite{Lipkin1965,Meshkov1965,Glick1965}. The Hamiltonian reads
\begin{equation}
H_{\text{LMG}} = -h J_z -\frac 1 N \left(\gamma_x J_x^2 + \gamma_y J_y^2 \right),
\label{LMGHamiltonian}
\end{equation}
where $J_\alpha= \frac 12 \sum_{i}^N \sigma_i^\alpha$ and $\sigma_i^{\alpha}$, with $\alpha \in \{x,y,z\}$ denote the Pauli
matrices.
The parameters $\gamma_x$ and $\gamma_y$ describe the interaction strength in the $x$ and $y$ directions, respectively.
To get a well-defined thermodynamic limit, we restrict the spin Hilbert space to the subspace with maximal total angular
momentum $j= N/2$ as in Refs.~\cite{Emary2003,Ribeiro2008,Brandes2013}.
This allows to uniquely describe the atomic Hilbert space by using Dicke states $\left|j,m\right\rangle$, which are the eigenstates of $J_z$~\cite{Dicke1954}.
In the experimental realization of the LMG model in Ref.~\cite{Zibold2010,Gross2010} by coupling the internal
degrees of freedom of a Bose-Einstein condensate, one is naturally
restricted to $j=N/2$ in the case of a system consisting of $N$ indistinguishable
bosonic particles \cite{Steel1998}.
The underlying reason is that the spatial wave functions of all bosons are identical
due to condensation. This implies that the internal degrees of freedom have to be symmetric under permutation.
As the Dicke states with $j=N/2$ satisfy this condition~\cite{Dicke1954}, these are the physically relevant states.
The TC Hamiltonian describes an ensemble of two-level systems with level splitting $\omega_{0}$, which are collectively
coupled with strength $\lambda$ to a cavity field of frequency $\omega$
\begin{equation}
\label{TCHamiltonian}
H_{\text{TC}} = \omega \hat a^\dagger \hat a + \omega_0 J_z + \frac{\lambda}{\sqrt{N}} \left( \hat a J_+ + \hat a^\dagger J_- \right),
\end{equation}
where $\hat a$ and $\hat a ^{\dagger}$ are bosonic operators and $J_{\pm}= J_x\pm i J_y$~\cite{Emary2003,Keeling2009,Brandes2013,Tavis1968,Narducci1973}.
The bosonic mode
is described in terms of Fock states $\left|n \right\rangle$. The TC Hamiltonian commutes with the operator
$
\hat{\mathcal{M}} = \hat a^\dagger \hat a + J_z.
$
This allows to restrict the basis of the Hilbert space $\left\{|n\rangle\otimes|j,m\rangle \right\}$ to the symmetry-adapted
basis $\left\{|M-m\rangle\otimes|j,m\rangle \right\}$, where $M=n+m$ is the eigenvalue of $\hat{\mathcal{M}}$~\cite{Tavis1968}.
\subsection{The semiclassical energy landscapes}
\begin{figure}[t]
\centering
\begin{minipage}{0.95\linewidth}
\begin{overpic}[width=1.\linewidth]{JxQuadOverJz1}
\put(-3,75){ \textbf a)}
\end{overpic}
\end{minipage}
\begin{minipage}{0.95\linewidth}
\begin{overpic}[width=\linewidth]{JxQuadOverJz2}
\put(-3,75){ \textbf b)}
\end{overpic}
\end{minipage}
\caption{(Color online)
(a) Expectation values of $J_z$ and $J_x^2$ for $\gamma_x/h =2$, $\gamma_y/h=0$ and $j=30$ for the LMG model.
(b) Same as in (a) but for $\gamma_x/h =3$ and $\gamma_y/h=2$. For
each eigenstate $\left| E_i \right>$ we calculate $\langle E_i|J_{z}/j|E_i\rangle$ and $\langle E_i|J^{2}_{x}/j^{2}|E_i\rangle$
and place a dot in the diagram. Black (gray) dots correspond to states with energy less (greater) than the saddle point energy. The solid orange
line depicts the semiclassical calculation Eq.~\eqref{eq:ClassicalExp}. The other symbols (triangles, squares, circles and
double triangles) correspond to time-averaged expectation values Eq.~\eqref{temporalAverage} for $j=30$ and $\tau = 20/h$.
The initial conditions are depicted on the Bloch sphere with the same shape as the symbols in the plot.
We mark states
centered at the global minimum, saddle point and local maximum with $m_g$,
$S$ and $M_l$, respectively, and emphasize the points $p$, $p_1$ and $p_2$. Additionally, we show a density plot of the
participation ratio $P_r^{-1}$ on the Bloch sphere.
The black curves mark the classical trajectories with constant energy.
}
\label{fig:ExpectaionValuesC}
\end{figure}
To obtain the semiclassical energy landscape $E_{\text{LMG}}$ of the LMG model, we scale the Hamiltonian Eq.~\eqref{LMGHamiltonian}
with $j$ and consider the thermodynamic limit $j\gg 1$.
In this limit, quantum fluctuations are negligible and we can define classical variables $j_{\alpha}=J_{\alpha}/j$ with Poisson
bracket $\{j_{x},j_{y}\}=j_{z}$ as in Ref.~\cite{Dusuel2005}.
By using these classical variables we define the energy landscape
\begin{equation}
\label{LMGSemLandscape}
E_{\text{LMG}}(\textbf{j})=-h j_z -\frac{1}{2} \left(\gamma_x j_x^2 + \gamma_y j_y^2 \right)
\ ,
\end{equation}
where $\textbf{j}=(j_x,j_y,j_z)$. In this paper we consider the parametrization $\textbf{j}=(\sin\theta \cos\phi,\sin\theta \sin\phi,\cos \theta)$
of the Bloch sphere in terms of the local coordinates $(\theta,\phi)$.
In the insets of Fig.~\ref{fig:ExpectaionValuesC}, we depict the level sets of the classical energy $E_{\text{LMG}}$ on the Bloch sphere.
In these figures we mark different kinds of fixed points, namely local maxima $M_l$, global minima $m_{g}$ and saddle points $S$.
References~\cite{Ribeiro2008,Castanos2006} perform a detailed analysis of the classical energy of the LMG model with respect to the fixed points and present
a phase diagram of the system.
Reference~\cite{Ribeiro2008} finds an exact expression for the DOS in the thermodynamic limit,
and shows that the fixed points of $E_{\text{LMG}}$ are related to non-analyticities in the DOS. Therefore, a saddle point of the energy
surface corresponds to a logarithmic singularity and a local maximum to a jump in the DOS.
Due to the continuous symmetry $\hat{\mathcal{M}}$ of the TC model Eq.~\eqref{TCHamiltonian}, we can derive an effective
semiclassical energy landscape $E_{\text{TC}}$ for the atomic ensemble.
Like for the LMG model, to obtain the semiclassical energy landscape, we scale the Hamiltonian Eq.~\eqref{TCHamiltonian} with $j$. This allows us to use
the classical variables $\textbf{j}=(j_x,j_y,j_z)$ for the atomic ensemble as in the LMG model. Furthermore, we use the classical variables $a, \vartheta \in \mathbb{R}$ for the bosonic mode
defined by $a e^{i\vartheta}=\hat a /\sqrt{j}$. In addition, we define the classical conserved quantity
$\mathcal{M}=\hat{\mathcal M}/j=j_z+ a^2$, which enables us to obtain the semiclassical energy landscape
\begin{align}
\label{TCRedSemLandscape}
E_{\text{TC}}(\vartheta,\textbf{j})&= \omega \mathcal{M} + \left(\omega_0 - \omega \right) j_{z}
\nonumber \\&
+ \lambda \sqrt{2\left(\mathcal{M}- j_{z} \right)}(j_x \cos \vartheta-j_y\sin \vartheta)
\ .
\end{align}
Finally, we introduce rotated spin variables $\tilde j_x = j_x \cos \vartheta - j_y \sin \vartheta$,
$\tilde j_y = j_x \sin \vartheta + j_y \cos \vartheta$ and $\tilde j_z = j_z$. In terms of the new coordinates
$\tilde{\textbf{j}}=(\tilde{j}_x,\tilde{j}_y,\tilde{j}_z)$, the energy landscape
\begin{equation}
\label{TCRedSemLandscapeRot}
E_{\text{TC}}( \tilde{\textbf{ j}})= \omega \mathcal{M} + \left(\omega_0 - \omega \right) \tilde j_{z}
+ \lambda \sqrt{2\left(\mathcal{M}- \tilde j_{z} \right)} \tilde j_x
\end{equation}
represents an effective energy landscape for the atomic ensemble.
In the Appendix we show that the equations of motion of the rotated spins are given by
$\frac d{dt} \tilde j_\alpha = \left\lbrace \tilde j_\alpha, E_{\text{TC}} ( \tilde{\textbf{ j}}) \right\rbrace$. The time evolution of
the bosonic variables follows
\begin{equation}
a^2 = \mathcal {M} - \tilde j_z \qquad \qquad \frac d {dt} \vartheta = -\omega -\frac \lambda {a\sqrt 2} \tilde j_{x}.
\end{equation}
Consequently, the time evolution of $a$,$\vartheta$ depends on the rotated spins but not the other way around.
In the inset of Fig.~\ref{fig:JcPlot} we depict the level sets of the energy landscape $E_{\text{TC}}$ on the Bloch sphere.
For $\mathcal M =1$, the energy surface $E_{\text{TC}}( \tilde{ \textbf{ j}})$ possesses a saddle point, giving rise to an
ESQPT~\cite{Narducci1973,Bastarrachea-Magnani2014a,P'erez-Fern'andez2011}.
\begin{figure}[t]
\begin{overpic}[width=1.\linewidth]{JcPlot}
\end{overpic}
\caption{(Color online)
Expectation values in eigenstates (black dots)
for the TC model~\eqref{TCHamiltonian} for the parameters $\omega / \omega_0=\lambda/\omega_0=2$, $j=30$
and $ M= 30$.
Time-averaged expectation values~\eqref{temporalAverage} depicted as circles and triangles fulfill $\left< \hat {\mathcal M}\right>=30$.
The solid orange
line depicts the semiclassical calculation~\eqref{eq:ClassicalExp}.
On the Bloch spheres the black lines depict the level sets of the energy landscape $E_{\text{TC}}\left(\tilde {\mathbf{j}} \right)$.
The coloring shows the participation ratio $P_r^{-1}$ of the product of coherent states.
The triangles and balls mark the positions of the initial states of our simulations.
The light blue and
green curve on the Bloch sphere mark the sections $\phi=0$ and $\phi=\pi/2$, respectively. In the lower right inset
we depict $E_{\text{TC}}$ as a function of the local coordinates of the Bloch sphere $(\theta,\phi)$.
In particular, we depict the dependence of the energy as a function of
the polar angle $\theta$ for the azimuthal angles $\phi=0$ and
$\phi=\pi/2$. Here one can clearly see the saddle point
at $\theta=0$.}
\label{fig:JcPlot}
\end{figure}
\subsection{Time-averaged expectation values}
According to Refs.~\cite{Ribeiro2008,Paul1993,Ribeiro2009},
the expectation value $\langle\hat{O}\rangle_{E}$ of an operator $\hat O$ in an eigenstate of the system can be related
to the dynamics of the corresponding classical observable $o(t)$. At leading order in $1/j$, such a relation reads
\begin{equation}
\label{eq:ClassicalExp}
\langle\hat{O}\rangle_{E} = \frac 1 L \sum_{l=1}^L \frac 1 {T_l(E)} \int_0^{T_l(E)} o^{(l)}(t) dt
\ ,
\end{equation}
where $L$ denotes the number of connected trajectories defined by the relations $E=j E_{\text{LMG}}(\textbf{j})$ for the LMG
model and $E=j E_{\text{TC}}(\vartheta,\textbf{j})$ for the TC model. In Eq.~\eqref{eq:ClassicalExp} we consider the representation
$o^{(l)}(t)$ of the observable $o(t)$ restricted
to the $l$-th trajectory with period $T_l(E)$. Furthermore, one has to sum over the trajectories $l$ in such a way that their union
has the symmetry of the underlying system, i.e., the symmetries of Hamiltonians~\eqref{LMGHamiltonian} and~\eqref{TCHamiltonian}.
References~\cite{Brandes2013,Ribeiro2008} show how to express the expectation values of observables in eigenstates in terms of the DOS.
As a consequence, the expectation values inherit the singularities of the DOS.
Based on Eq.~\eqref{eq:ClassicalExp}, we now suggest a method which could be used to experimentally detect a signature of the ESQPT.
The measurement of the energy of the system is often not
experimentally accessible, which motivates us to employ an alternative representation of the observables. In order to resolve the
singularities of measurable quantities, we choose two observables $\hat O_1$ and $\hat O_2$. In the numerical simulations we
take $(\hat O_1,\hat O_2)= \left(\frac{J_z}{j},\frac{J_x^2}{j^{2}}\right)$ for the LMG model and
$(\hat O_1,\hat O_2)=\left(\frac{J_z}{j},\frac{\hat a J_+ + \hat a^\dagger J_-}{j^{3/2}}\right)$ for the TC model.
In contrast to the LMG model, the classical dynamics of the TC model is not restricted to a two-dimensional manifold.
For this reason, we discuss the applicability
of our method to this more complicated model below.
As spin-coherent states $|\theta,\phi\rangle$
are the closest ones to classical states~\cite{Arecchi1972}, we take
these as initial conditions for our quantum-mechanical simulations of the finite-size LMG model.
A spin-coherent state is obtained by a rotation of the Dicke-State $\left| j, j \right>$ so that its mean is located at the
Bloch sphere coordinates $ (\theta,\phi) $.
More precisely, one can show that $|\theta,\phi\rangle= (1+\left|\rho \right|)^{-j}e^{\rho J_-}\left| j, j \right>$,
where $\rho= e^{i \phi} \tan \frac\theta 2$.
In the basis of Dicke states, a spin-coherent state reads
$\sum_{m=-j}^{j} t_m \left| j,m \right> $, where \cite{Zhang1990}
\begin{equation}
t_m = \sqrt{\left( \begin{array}{c}
2j \\
j+m
\end{array} \right) } \left(\sin\frac{\theta}{2} \right)^{j-m} \left(\cos\frac{\theta}{2 } \right) ^{j+m} e^{-i (j+m) \phi}.
\label{eq:coherentStateCoeff}
\end{equation}
For the TC model, one can use a product of coherent states of both the spin system as well as the bosonic mode
(see below for additional information).
We initially prepare the system in a (product of) coherent state(s) $\left| \psi(0) \right>$.
Afterwards, one measures the expectation values $\langle\hat O_1\rangle$ and $\langle\hat O_2\rangle$ in the state $\left| \psi(t) \right>$,
which allows us to define the temporal average
\begin{equation}
\overline{\left< O_i\right>} = \frac 1 \tau \int_0^{\tau} \left<\psi(t)\right|\hat O_i \left|\psi(t) \right> dt ,
\label{temporalAverage}
\end{equation}
where $\tau$ is the evolution time.
This sequence is repeated for a set of different initial states.
For a notational reason we define $\boldsymbol \chi =\left(\left< O_1\right>, \left< O_2 \right>\right)$. For the LMG model we
specify $\boldsymbol \chi_{\left(\theta,\phi \right)} $ at which the expectation values correspond to Eq.~\eqref{temporalAverage}
with the initial state $\left| \theta,\phi \right>$.
\section{Results}
\label{sec:II}
From an experimental point of view, an energy-independent representation of the expectation values has the big advantage that it is not necessary to measure the
energy. Furthermore, such a representation would be useful to study systems in which energy is
not a conserved quantity like in dissipative~\cite{Morrison2008a,Kopylov2013} or in feed-back systems~\cite{Kopylov2015}.
Thus, in doing so one can examine the implications of
an ESQPT under nonequilibrium conditions.
\subsection{LMG model}
In Fig.~\ref{fig:ExpectaionValuesC} we compare the expectation values of observables in eigenstates,
the semiclassical calculation found by using Eq.~\eqref{eq:ClassicalExp}, and the quantum-mechanical
averaging method Eq.~\eqref{temporalAverage}.
In the thermodynamic limit, the expectation values $\left< J_z\right>$ and $\left< J_x^2\right>$ calculated using
Eq.~\eqref{eq:ClassicalExp} describe a parametric curve as a function of
energy, which we denote with $\boldsymbol \chi_{cl} (E)$. Accordingly, we denote the expectation values in eigenstates
with $\boldsymbol \chi_{es} (E)$.
In Fig.~\ref{fig:ExpectaionValuesC}~(a) there is a cusp of $\boldsymbol \chi_{cl}(E)$ at $\boldsymbol \chi =(1,0) $.
This is a result of the ESQPT~\cite{Ribeiro2008}, as both expectation values exhibit a singular behavior there in
the thermodynamic limit as a function of energy.
The critical energy $E_{S}$ corresponds to the energy of the separatrix, which is $E_S=j E_{\text{LMG}}(\textbf{j})$ in
Fig.~\ref{fig:ExpectaionValuesC}~(a).
The expectation values of observables in eigenstates approximately agree with the semiclassical calculation.
However, for finite sizes the expectation values in eigenstates are not directly located at $\boldsymbol \chi =(1,0) $,
because these points can be achieved only in the thermodynamic limit.
In Fig.~\ref{fig:ExpectaionValuesC}(b) the curve $\boldsymbol \chi_{cl} (E)$ representing the
semiclassical calculation exhibits a qualitatively
different behavior.
In contrast to Fig.~\ref{fig:ExpectaionValuesC}(a) there is no cusp at the saddle point.
Starting from the global minimum $m_g$ and increasing the energy $E$, the curve $\boldsymbol \chi _{cl} (E)$
exhibits a bifurcation at the
saddle point energy $E_S$. One branch continues to expectation values corresponding to the global maximum $M_g$ and the
other one continues to expectation values corresponding to the local maximum $M_l$. Thus, this
bifurcation is due to the
emergence of a local maximum of the energy surface~\cite{Ribeiro2008} which can be seen in the inset.
Most of the expectation values in eigenstates
are well described by the classical calculation, but there are significant deviations for states close to the saddle point.
States with energies less than the saddle point energy are nearly degenerate. Therefore, two states that are nearly degenerate have similar
expectation values for the chosen observables. We color these expectation values black in Fig.~\ref{fig:ExpectaionValuesC}
, while the others are depicted in gray.
For the simulations of Eq.~\eqref{temporalAverage} we use the
initial states depicted on the Bloch spheres in the insets of Fig.~\ref{fig:ExpectaionValuesC}(a) and (b). These are the
most interesting initial states containing all fixed points of the energy landscape Eq.~\eqref{LMGSemLandscape}.
Therefore, trajectories close to every possible energy in the system can be probed \cite{Bastidas2014}.
In Fig.~\ref{fig:ExpectaionValuesC}(a),
the finite-size simulation agrees with the semiclassical and
eigenstate calculations.
The curve $\boldsymbol \chi_{\left(\theta,\phi\right)}$ exhibits a cusp close to $\boldsymbol \chi = (1,0)$, although the
energy of the spin-coherent state is smooth as a function of $\left( \theta,\phi \right)$. For this reason
the path $\boldsymbol \chi_{\left(\theta,\phi \right)}$ shows a signature of the non-analytic character of the ESQPT.
There is no point exactly located at the cusp at $\boldsymbol \chi = (1,0)$, although the system is
initialized at the saddle point.
This is a consequence of the quantum-mechanical deformation of the wave packet being prepared in the vicinity of the saddle point.
The enhanced quantum fluctuations due to the influence of the saddle point cause a collapse and revival behavior in the time evolution
of observables as it is discussed in Ref.~\cite{Tonel2005}. This also contributes to the deviation from the semiclassical calculation visible
in Fig.~\ref{fig:ExpectaionValuesC}.
The deformation of the wave packet has been investigated experimentally in Refs.~\cite{Zibold2010,Gross2010,Gerving2012}.
In contrast, wave packets
remaining essentially Gaussian resemble the semiclassical calculation much more. Detailed investigations of the deviations of semiclassical and quantum dynamics
can be found in Refs.~\cite{Nissen2010,Chuchem2010,Caballero-Benitez2010,Hennig2012,Buchleitner2002}.
This argumentation also applies to Fig.~\ref{fig:ExpectaionValuesC}(b), where the finite-size simulation is unable to resolve the bifurcation point
$\boldsymbol \chi = (1/2,0)$.
The initial states located at points $p$, $p_1$ and $p_2$ marked in the insets of Fig.~\ref{fig:ExpectaionValuesC} lying close to the separatrix also
experience a strong deformation. Consequently, they considerably deviate from the semiclassical limit.
Deviations of the quantum-mechanical calculation
from the semiclassical results are strongly related to the participation ratio $P_r^{-1}$ of the initial
states $\left| \psi(0) \right>=\left|\theta, \phi \right>$, with $\left|\theta, \phi \right>$ being a spin-coherent state.
Following Refs.~\cite{Scharf1992,Weaire1977}, we define the ``inverse participation ratio'' as
$
P_r= \sum_{i} \left| \left<\psi(0) \right. \left| E_i\right> \right|^4,
$
where $\left| E_i \right>$ denotes an eigenstate of the system.
The participation ratio is a measure of the number of eigenstates needed to construct a specific state. Therefore, a high
participation ratio means that our initial state is a superposition of a lot of eigenstates.
In the insets of Fig.~\ref{fig:ExpectaionValuesC} we depict a density plot of $P_r^{-1}$ on the Bloch sphere.
In particular, the initial points
$p$, $p_1$ and $p_2$ in Fig.~\ref{fig:ExpectaionValuesC} which exhibit quite a strong deviation from the semiclassical calculation,
have a high participation ratio.
Consequently, the temporal average of these initial states is influenced by many eigenstates, so that it deviates strongly from
the semiclassical calculation. The relation between the participation ratio and deviations of quantum dynamics is addressed in Refs.
\cite{Chuchem2010,Hennig2012,Khripkov2013,Caballero-Benitez2010}.
To improve the simulation close to $\boldsymbol{\chi}=(1/2,0)$ in Fig.~\ref{fig:ExpectaionValuesC}(b), we suggest new initial conditions, which we depict with purple
double triangles in the inset.
We choose therefore the points, at which the classical velocity is minimal~\cite{Berry1972} which possess a very low
participation ratio.
The time-averaged expectation values $\overline{\left< O_i\right>}_{(\theta,\phi)}$ converge to the corresponding
semiclassical ones $\langle\hat{O}_{i}\rangle$ of
Eq.~\eqref{eq:ClassicalExp}.
Given a finite size $j$, the scaling
$|\langle\hat{O}_{i}\rangle-\overline{\left< O_i\right>}_{(\theta,\phi)}| \propto 1/\log j$ for an initial
state at a saddle point has been discussed in Ref. \cite{Chuchem2010} for $\gamma_y=0$. We also checked this scaling
numerically for $\gamma_y\neq 0$ and for other initial states at the separatrix .
\subsection{TC model}
For the TC model, Fig.~\ref{fig:JcPlot} shows the expectation values of the observables
$\hat{O}_1=J_{z}/j$ and $\hat{O}_2=(\hat{a} J_{+}+ \hat{a}^{\dagger} J_{-})/j^{3/2}$ using the different calculation techniques.
As explained before, the classical energy surface exhibits a saddle point for $M=j$.
For this reason, black dots depict the observables in eigenstates $\boldsymbol \chi_{es} (E)$ for a finite-size system with $j = M =30$.
Although the classical dynamics is not restricted to a two-dimensional manifold for the TC model, the expectation values
in eigenstates can be calculated semiclassically with a high accuracy. The ESQPT cusp is located at $\boldsymbol \chi = (1,0)$. However, as for the
LMG model, $\boldsymbol \chi_{es} (E)$ reaches this point only in the thermodynamic limit.
As stated before we suggest a product of spin and bosonic coherent states
\begin{equation}\label{TC_coh_state}
\left|\psi(0)\right> = \left|\alpha \right>\otimes\left|\theta,\phi\right>
\end{equation}
as initial state, where $\left| \alpha \right>$ denotes a coherent state of the bosonic mode~\cite{Glauber1963}.
Its mean photon number is given by $\left< \hat a^\dagger \hat a \right> = \left| \alpha \right|^2 $.
In order to satisfy the condition $M=j$,
the initial state $\left|\psi(0)\right>$ shall fulfill
\begin{equation}
\left< \hat{\mathcal M} \right> = j \cos \theta + \left| \alpha \right|^2 = j.
\label{eq:initialConstrain}
\end{equation}
This constrain is also fulfilled by the expectation value of the time-evolved state as $\hat {\mathcal M}$ commutes
with the Hamiltonian.
Due to its definition, the initial state~\eqref{TC_coh_state} is not restricted to the subspace $M=j$ and needs the whole Hilbert
space to be defined.
The variance of $\hat{\mathcal M}$ in the proposed initial state is $\text{Var}\; \hat{\mathcal M} = j \left(1- \cos \theta + \frac 12\sin^2\theta \right)$.
Thus,
in the thermodynamic limit the variance of the scaled quantity $\hat{\mathcal M}/j$
scales as $1/j$.
We can use the symmetry $\hat{\mathcal M}$ to reduce the numerical effort.
To this end, we decompose the initial state in a sum of states with different quantum numbers $M$.
Therefore, we use the representation of spin and bosonic coherent states in terms of Fock and Dicke states, respectively \cite{Zhang1990}.
We write the initial state in Eq.~\eqref{TC_coh_state}
\begin{equation}
\left|\psi(0) \right>
= \sum_{M=M_{min}}^{M_{max}} \sum_{m=-j}^{min(j,M)} a_{M-m} t_m \left| M-m \right> \otimes \left| j,m \right> , \\
\label{eq:prodCoherentStates}
\end{equation}
where
$(M_{min}, M_{max})=\left(-j,\infty\right)$,
\begin{align}
a_n &= e^{-\frac{\left|\alpha \right|^2}{2}} \frac{\alpha^n}{\sqrt{n!}}
\end{align}
and $t_m$ is given in~\eqref{eq:coherentStateCoeff}
in terms of the symmetry-adapted basis discussed in Sec.~\ref{sec:I}.
Therefore, $a_n$ and $t_m$ denote the coefficients of the bosonic and spin-coherent states, respectively.
The amplitude of $\alpha$ is fixed due to the constrain~\eqref{eq:initialConstrain}. We choose the phase of $\alpha$ to be
$\vartheta = \arg \alpha = 0 $, thus $\alpha = \sqrt{j-j \cos \theta}$.
The time evolution for different $M$ for that initial state decouples due to the symmetry $\hat{\mathcal M}$, which
reduces the numerical effort.
In the numerical calculation we can truncate the state at
$(M_{min},M_{max}) = (j-\Delta M, j+\Delta M)$ with $\Delta M$ chosen in such a way that the time evolution
of the expectation values converges.
Figure~\ref{fig:JcPlot} depicts the results of the time-averaged quantum simulations.
The initial conditions are
sketched on the Bloch sphere with blue balls and red triangles located along the paths
$\theta \in \left(0, \pi\right)$ for $\phi=0$ and $\phi=\pi$, respectively. The time-averaged expectation values
are depicted with corresponding symbols.
At this point we recall that the energy
surface on the Bloch sphere is depicted in a rotated frame for which the rotation angle is given by $\vartheta$. As we choose
$\vartheta=0$ for our initial conditions, the rotated frame is equivalent to the laboratory frame.
The result of the finite-size simulations resembles the findings
for the LMG model in Fig.~\ref{fig:ExpectaionValuesC} (a). The initial conditions close to the saddle point of the classical
energy surface reproduce the expectation values of the semiclassical calculation with a high accuracy. However,
for an initial condition located on the separatrix but away from the saddle point, there are also significant
deviations from the semiclassical calculation. In Fig.~\ref{fig:JcPlot} we denote this point with $p_3$. We also
find that this initial condition is characterized by a high participation ratio $P_r^{-1}$,
which smoothens the signature of the ESQPT.
\section{Applications}
\label{sec:III}
Finally, we discuss the experimental applicability of our method for the LMG model. Here, we refer to the
experimental realization of the LMG model in Refs.~\cite{Zibold2010,Gross2010}.
This experimental realization allows us to prepare the system in a spin-coherent state on arbitrary
positions on the Bloch sphere~\cite{Oberthaler}. The measurement of the expectation values of $J_z$ and $J_x^2$
is performed
by repeating the time evolution for the same initial state up to a given time $t$. Based on the
single measurements one can then calculate the desired expectation value at time $t$.
The maximum feasible time $\tau$ should be long enough to observe recurrences in the time evolution of observables for all initial
states~\cite{Oberthaler}.
In Fig.~\ref{fig:expTest} we simulate these experimental circumstances for an experimentally feasible
particle number $N=300$ and $\gamma_x/h=2$. Based on the realization of the LMG model in Ref.~\cite{Zibold2010,Gross2010}, experimentally feasible parameters
are $h = 2\pi \times 9.45\ \mathrm{Hz}$ and $\gamma_x= 2\pi \times 18.9\ \mathrm{Hz} $.
From the time evolution of
$\left< J_z(t)\right>$ and $\left< J_x^2(t)\right>$ we estimate the duration
of one period $\tau$ for each initial preparation.
As examples we depict in Fig.~\ref{fig:expTest}(a) the time evolution of an initial state located at the saddle point and
of an initial state located close to the global minimum. We also mark the respective estimated period $\tau$.
In the inset of Fig.~\ref{fig:expTest}(b) we show the chosen $\tau$ for the initial states
depicted in the inset of Fig.~\ref{fig:ExpectaionValuesC}(a). As continuous measurements are not possible, we consider a
discretized version of Eq.~\eqref{temporalAverage}, namely
\begin{equation}
\overline{\left< O_i\right>}_{\left(\theta,\phi \right)} = \frac 1 \tau \sum_{k=0}^{n_{(\theta,\phi)}-1} \left<\psi( t_k)\right|\hat O_i \left|\psi( t_k) \right> \Delta t ,
\label{temporalAverageDiscrete}
\end{equation}
where the time step $\Delta t $ and $n_{(\theta,\phi)}$ fulfill $\Delta t n_{(\theta,\phi)} = \tau $
and $t_k = k \Delta t$.
To minimize the experimental effort it will be convenient to take $\Delta t$ as large as possible.
In Fig.~\ref{fig:expTest} we choose $\Delta t=\frac{1}{4h} $, which still enables a sufficient precision.
For the parameters given above this means that $\tau \leq 134\ \mathrm{ms}$ and $\Delta t = 4.2\ \mathrm{ms}$, so that at most
$n_{(\theta,\phi)}=32$ for an initial state at the saddle point.
As a demonstration, in Fig.~\ref{fig:expTest}(a) we mark the points used in the average~\eqref{temporalAverageDiscrete} with dots. Due to our
choice of $\Delta t$ these points are dense in relation to the temporal variation of the observables. As the chosen
$\tau$ are quite small, the average does not suffer from the complex collapse and revival behavior appearing for longer
evolution times observed in Ref.~\cite{Tonel2005}.
\begin{figure}[t]
\begin{overpic}[width=1.\linewidth]{timeEvolution}
\put(-3,40){ \textbf a)}
\end{overpic}
\begin{overpic}[width=1.\linewidth]{experimentalFigure}
\put(-3,75){ \textbf b)}
\end{overpic}
\caption{(Color online)
(a) Example of the time evolution of $\hat O_1=J_z/j$ (solid lines) and $\hat O_2=J_x^2/j^2$ (dashed lines) in the LMG model for initial states located at the saddle point [pink (dark gray)]
and close to a global minimum [yellow (light gray)]. Parameters are the same as in Fig.~\ref{fig:ExpectaionValuesC}(a), but $j=150$.
The black lines depict
the estimated periods $\tau_{S,m_g}$ for the chosen initial states, respectively.
The dots mark the times $t_k$ used to evaluate Eq.~\eqref{temporalAverageDiscrete} at which the time step is chosen to be $\Delta t = \frac 1{4h} $.
(b) Realistic simulation of the method for the LMG model. The colored symbols and the orange line depict the same as in Fig.~\ref{fig:ExpectaionValuesC}(a).
The period $\tau$ is
always estimated by the time evolution of the observables and depicted in the top inset. Here we also used $\Delta t = \frac 1{4h} $.
The bars depict the time-averaged variance of the observables which is defined in \eqref{eq:temporalAverageVar}.
In the bottom inset we depict the time-averaged variance of $J_z$. }
\label{fig:expTest}
\end{figure}
As explained before, depending on the initial condition the state can be strongly deformed. This hinders the
measurement of the expectation values as one has to repeat the measurement more often to obtain the required
precision. To estimate this effort, we also included bars for each point in Fig.~\ref{fig:expTest}(b) depicting
the corresponding time-averaged variance of the time evolution,
\begin{equation}
\overline{\text{Var} \;\hat O_i} = \frac 1 \tau \sum_{k=0}^{n_{(\theta,\phi)}-1} \text{Var} \; \hat O_i(t_k) \Delta t
\label{eq:temporalAverageVar}
\end{equation}
where
\begin{equation}
\text{Var} \;\hat O_i(t) = \left<\psi( t )\right|\left( \hat O_i - \left<\hat O_i(t) \right> \right)^2 \left|\psi( t) \right>
\label{eq:VarO}
\end{equation}
is the variance of the observable $\hat O_i$ at time $t$.
We depict the time-averaged variance of $\hat O_i=J_z$ in the inset of Fig.~\ref{fig:expTest}(b).
We see that for initial states away from the saddle point the variance is vanishing small, as the states
remain essentially Gaussian, but for states close to the separatrix one has to repeat the time evolution for
a given time $t$ quite often.
The time-averaged variance defined in Eq.~\eqref{eq:temporalAverageVar} does account only for the quantum fluctuations of the system.
However, the actual uncertainties appearing in our method depend on the experimental realization.
For example, one could also consider other influences, e.g., uncertainties in preparing the initial
state. However, in the context of the experimental realization of Ref.~\cite{Gross2010,Zibold2010} we assume this to have a
minor influence on the measured results, as there is a high degree of control of the preparation of the initial state.
Referring to Ref.~\cite{Gross2010}, we assume that about $n_{\text{EM}}=60$ experimental measurements are sufficient to determine adequately
the expectation values of $\hat O_i$ for each time step in Fig.~\ref{fig:expTest}(a). Consequently, to obtain the point in Fig.~\ref{fig:expTest}(b)
corresponding to the saddle point, one has to perform $n_{\text{SEM}} =2 n_{\text{EM}} n_{(\theta,\phi)} = 3840$ single experimental measurements,
where $n_{(\theta,\phi)}=32$ for the saddle point as in Eq.~\eqref{temporalAverageDiscrete}. We note that in the experimental realization presented
in Ref.~\cite{Muessel2014} up to $n_{\text{PM}}=30$ measurements can be performed in parallel in one experimental run so that one
needs $n_{\text{SEM}}/n_{\text{PM}}=128$ experimental runs. For the other points in Fig.~\ref{fig:expTest}(b) fewer experimental runs are necessary.
\section{Conclusions}
\label{sec:IV}
Expectation values of quantum-mechanical operators in eigenstates can be calculated using the classical dynamics of the system.
This motivated us to
suggest a method which opens a new avenue to experimentally detect signatures of the ESQPT in systems in which energy is not experimentally accessible or not conserved.
The temporal averaging of a finite-size system resembles the expectation values
in eigenstates and the semiclassical calculations.
However, there are
partial deviations from the semiclassical calculations for energies near an ESQPT. These deviations
are bigger for initial states obeying a high participation ratio.
A point to be addressed in the future is the application of our method to more complicated mean-field type models such as spinor
Bose-Einstein condensates \cite{Hamley2012,Gerving2012} and the Dicke model \cite{Baumann2010,Baumann2011}.
In particular,
our representation of observables might be
interesting in the description of
nonequilibrium systems, in which the energy is not a conserved quantity, such as driven~\cite{Bastidas2012b,Engelhardt2013},
dissipative~\cite{Morrison2008a,Kopylov2013} or feedback~\cite{Kopylov2015} mean
field-type systems.
\begin{acknowledgments}
The authors gratefully acknowledge financial support from the DFG Grants BR 1528/7, BR 1528/8, BR 1528/9, SFB 910 and GRK 1558 and inspiring
conversations with P. P\'erez-Fern\'andez, M. Vogl and M. Oberthaler and the members of his group.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 388 |
Since interviewing IMAscore a couple of weeks ago I've massively gotten back into listening to theme park music! It really gets me in the mood for the season and can instantly take you back to that happy place where you're queuing for your favourite ride and just taking in the sights, and more importantly the sounds of the attractions around you. There's something truly unique and magical about theme park soundtracks. They take us on a journey and their sole purpose is to evoke feelings of excitement and anticipation as well as communicate a narrative and set a mood for the attraction you're about to experience. I don't think there's any other music in the world whose job it is to do all of those things and more, so I wanted to take the time today to pay homage to some of my all time favourites!
Just for clarification, this is by no means a definitive list of what is the 'best'. It's just my opinion and of course we're bound to disagree in some instances! So bear that in mind when you're reading through this and getting infuriated that I didn't mention your personal fav haha! Let's get to it.
Kicking things off with one of the most epic and...slightly terrifying soundtracks in the UK: Valhalla. This soundtrack perfectly sets the mood for the utter carnage you're about to embark upon when you board your viking longboat. It really puts you on edge, communicates the Norse god theme perfectly but is also slightly camp when you listen to it, giving it that Blackpool Pleasure Beach edge. Some theme park soundtracks are interchangeable, but the epic chanting combined with those drums mean this one is undeniably Valhalla and I couldn't imagine it anywhere else.
Anyone who knows me knows I'm not the biggest fan of Walibi Belgium. I think the place is tacky, I hate the 'WAB' theme and I think pretty much all of the rides are terrible. And honestly, I think Psyke Underground wouldn't be that amazing if it weren't for this soundtrack. It's over the top, ridiculous EuroPop and wouldn't be out of place as a Eurovision entry. Catchy, annoying and yet actually really enjoyable and really pumps you up ready for the launch! Paired with the fact that the inside of the station looks like a gay nightclub and you're on for a winner with this one.
Oh god, OK. I'm NOT that emotional a person. It takes a LOT to make me cry, but listening to the Soarin' soundtrack is a surefire way to set off the waterworks with me. It's just so inspiring! So epic! I honestly think my love for EPCOT is tied directly to this bloody soundtrack. I'm listening to it now as I type this and I'm getting goosebumps. IT'S JUST SO GOOD! I get that it's meant to inspire the sensation of flight and blah blah, but for me it just transports me straight to sunny Orlando, enjoying a fab Disney World holiday with my family and I'll always cherish this soundtrack for its ability to do that. Oh and Soarin' is fab, haters gonna hate.
How did we get this far into this article without mentioning an IMAscore soundtrack yet? No idea, but here we go. Y'all know I love IMAscore to death, and the guys are excellent at executing a creepy as epic soundtrack, but in my opinion where they really come into their own is with a fantasy theme. And they absolutely NAIL this with Dweverlwind. The swooping sounds of the wind blowing and fairydust sparkling alongside the epic drums and brass, it all comes together perfectly and instantly transports me to a magical fairy land whenever I listen to it. Weirdly, because I first rode this attraction in October this soundtrack also has deep connections with Autumn for me, and as its my favourite month it just makes me love it even more!
OK, so, I am a huge Disney fan, and in particular the music. The Disneyland Soundsational parade is essentially a celebration of Disney's best songs, brought to life with colourful floats and characters and tied together with a Disney theme parkified melody. I know it's kind of cheating because technically the songs in the parade music are just songs from the films I grew up with and love, but there's something incredible exciting and upbeat about the Soundsational chorus and how the theme links them all together. It also reminds me of my first visit to Disneyland with my best friend which is always a special memory for me!
So, Klugheim is basically one of the greatest things to happen in my life in recent years theme park wise, and one of the best things about this land is the way IMAscore scored the area. I'm a huge Lord of the Rings fan and this area always feels really Tolkien-esque to me, and I think that's largely to do with the music. It wouldn't feel out of place in Middle Earth: it's epic, it gives a sense of adventure, the way the chorus of voices cuts in gives a sense of battle and fighting and just epicness. It's stunning, it perfectly sets the scene and even though I'm still not 100% clear on what Taron or Klugheim are themed to exactly there's no denying the heroic connotations of this score!
You know earlier how I was talking about how ride soundtracks set the scene for the journey we're about to take? Well I feel like the music for De Vliegender Hollander does this expertly. It's incredibly cinematic, it feels like the first scene of a movie where our hero is being introduced before they go up against certain death. Anybody who has ridden De Vliegender Hollander knows that things get real dark, real quick, but the soundtrack doesn't let onto that at ALL, meaning when it does happen it takes us all the more by surprise! It's also got that beautiful Efteling fairytale element to it, magical but with a hint of a dark side. I guess you could say it's Efteling's answer to Yo Ho A Pirate's Life For Me!
Of course, I love the original Grim Grinning Ghosts Haunted Mansion soundtrack from Disneyland, but I think the Phantom Manor soundtrack at Disneyland Paris takes that one step further. Not only does it more explicitly tell the story of the Phantom Manor with the tragic singing bride included in the music itself, there's also something entirely more elegant about the whole thing that really ties in with the theme of a wedding night gone wrong. There's also more of an edge of beauty to this soundtrack, tying it perfectly in with the Disneyland Paris aesthetic. I also love that there are smidges of Grim Grinning Ghosts mixed in there too to tie the whole thing together! Perfect classic horror soundtrack.
This is my all time favourite theme park soundtrack, and I'm gutted because it's unlikely that this attraction will ever return! For those who don't know, The Mystery of Hocus Pocus Hall was a seasonal overlay for Hocus Pocus Hall at Chessington World of Adventures for Halloween. The entire attraction looked completely different and incorporated some very clever projection mapping and special effects. It was fab, but the best part by far was this magnificent soundtrack by IMAscore. Similarly to the Dwervelwind soundtrack, I think this one fills me with so much joy because it transports my brain straight to the Halloween season, which as we all know is the best. On top of that it's magical, it's fun, it's whimsical and it's got just the right dash of fantasy. It's stunning, and I continue to listen to it throughout the year.
There we have it, an insight into some of my favourite theme park soundtracks! This is by no means all of my favourites but if I were to list them all we'd be here all day. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,802 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/116960\/how-much-energy-in-form-of-heat-does-a-human-body-emit","text":"# How much energy in form of heat does a human body emit?\n\nHow much energy in form of heat does a human body emit at rest level?\n\nThat's easy. The energy requirement of an average man is 2,500 Calories per day, and one Calorie is 4184J. Therefore he emits about 10.5MJ\/day or about 120W.\n\nAn average woman requires 2,000 Calories per day, so she emits about 97W.\n\n\u2022 Does the human body emit as heat, all the energy it got from food? If yes, in typing this answer, you spent some energy - what is its energy source? \u2013\u00a0299792458 Jun 5 '14 at 10:09\n\u2022 @New_new_newbie: assuming your weight stays the same you are emitting all the energy from your food as heat. If you're getting fatter it means you're retaining some of the energy and converting it to chemical energy (i.e. fat). Conversely if you're getting thinner it means you're converting chemical energy in fat to heat, so you're emitting more heat than you get from your food. I got the energy to type this from my body reacting glucose with oxygen to produce water, carbon dioxide and energy. \u2013\u00a0John Rennie Jun 5 '14 at 10:14\n\u2022 Wrong. Energy gets stored in human body as ATPs (en.wikipedia.org\/wiki\/Adenosine_triphosphate). The chemical energy stored as a P-O bond gets released. Of course, some part of that will be lost as heat, but if picking up a 400g stone to 1m height required 4J of work, that 4J does come from the food you have consumed. Otherwise energy conservation would stand violated, if your input = output as heat'' conjecture was true. So, again, what is the energy source of typing this post if all the energy you got from food got EMITTED out as heat? \u2013\u00a0299792458 Jun 5 '14 at 10:32\n\u2022 @New_new_newbie: yes, if I lift a weight some of the energy I get from food has been converted to potential energy of the weight. So I suppose it would be more precise to say the energy from food ends as heat + any potential energy gain. In real life we are unlikely to make any long term potential energy changes to our environment (I suppose builders do, though their buildings will fall down eventually so the energy ends up as heat eventually). \u2013\u00a0John Rennie Jun 5 '14 at 10:42\n\u2022 @jameslarge Of course it's not exactly true, but you're missing the point - this is a great back-of-the-hand calculation that gives you something very close to the real value for the average Joe. Even if you did nothing all day but eat and put heavy objects up, almost all the energy you got in would be released as heat. Mammals are extremely inefficient heat engines - depending on the amount of work you do, the effective \"food -> work done\" efficiency is somewhere around 20% to as low as 1%. And the upper limit is the \"cycle all day\" kind, not \"workout a bit once a day\". \u2013\u00a0Luaan Jan 11 '16 at 14:11\n\nI hesitate to contradict John, but: it's simplistic to assume caloric input equals caloric output, or that caloric output is purely heat, as opposed to moving from one place to another, lifting boxes, etc. A far better model IMHO is to set up the human body as a black-body source with $\\epsilon = 0.98$ (emissivity), temperature = 310K, and some reasonable estimate as to total body area. Then you compare the absorption of heat from, say an ambient environment of 294K to see the net outflow of heat.\n\nThat does ignore conductive and convective heat flow :-) .\n\nSee, for example, the excellent calculator at hyperphysics\n\n\u2022 Yes, I recall that the result using Stefan-Boltzmann's law is close to that of John Rennie. In space, the power output is close to $1 kW$, quite impressive! \u2013\u00a0auxsvr Jun 5 '14 at 12:26\n\u2022 I expect that black body radiation losses are small compared to other heat losses of the human body in most ordinary circumstances. I haven't done the numbers, but I betcha the heat loss from inhaling gas at ambient temperature and exhaling it at body temperature is much greater than black body radiation when sitting behind a desk in a typical office. Add to that evaporative cooling since exhaled breath usually contains more moisture. \u2013\u00a0Olin Lathrop Jun 5 '14 at 12:47\n\u2022 Unless you're building a ziggurat or otherwise storing lots of gravitational potential, don't you think all of that mechanical energy also ends up as heat? \u2013\u00a0rob Jun 5 '14 at 12:55\n\u2022 @Carl Well, if I commit a terabyte of information to memory (seems unreasonably large), the minimum entropy change is $S=k\\ln2^{40}\\approx 1200\\,\\mathrm{K\/eV}$. At my body temperature the minimum energy release is only about 1\/4 eV. Even if the ratio of physical to information entropy is a billion to one I still have much less than a joule of heat associated with learning. I don't think it's a big correction. \u2013\u00a0rob Jun 5 '14 at 13:51\n\u2022 This answer means that, during a hot summer week in my really hot appartment, I emit no heat, and I will directly convert all my food calories into work at 100% efficiency. During summer, I regularly break the 2nd law of thermodynamics. \u2013\u00a0Sanchises Sep 15 '15 at 18:55\n\nAs a rough planning figure for building design etc, 1 human = 100 watt.\n\nJohn Rennie's answer is correct +\/- 1% or so.\n\nSure, you can lift something up, doing work instead of emitting heat. But you equally often lift something down, converting its potential energy to heat that your body emits. Besides, most of the loads you lift are small compared to the energy \"cost\" of running the chemical factory that is your body.\n\nOr say you climb up the stairs in a 20-story building. Assuming 1 floor = 10', that's 200 ft-lb. of work, or around 270J. Let's say that takes you 2 minutes, during which your body emits 120*100 = 12,000J of heat. So the work you did amounts to only about 2% of the total energy you consumed during that stair climb. And a modern human doesn't do physical work for very much of the day, so the conversion to work amounts to less than 1% error.\n\nBesides, you (usually) climb down about as much as you climb up -- again converting your potential energy to heat -- so that pretty much cancels out.\n\nJust assume that it takes around 100W to run your body at idle, more if you have to do \"work\" (running, etc.)\n\n\u2022 Umm... you forgot to include the weight of the person in your 200 ft stair climb calculation! A typical person can provide over 100W of sustained mechanical power, elite athletes can do closer to 500W. Sprint exertion is perhaps double. Tempted to edit this answer... \u2013\u00a0Michael Jan 11 '17 at 20:13\n\u2022 @Michael and bgold: we're not very efficient in converting chemically stored energy into mechanical work. IIRC, around 20 % for running, up to 25 % for biking. So when our output is 100 W mechanical, we have at the same time an additional output of 300 - 400 W in form of heat. When lifting something down, we're not only emitting heat due to the loss in potential energy of the weight but also additional head from our muscles (lifting things down makes hungry as well!) \u2013\u00a0cbeleites Jun 11 '18 at 17:28\n\n## protected by Qmechanic\u2666Nov 11 '15 at 15:24\n\nThank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).","date":"2019-05-19 07:17:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.5821163654327393, \"perplexity\": 1227.0737459352872}, \"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-2019-22\/segments\/1558232254253.31\/warc\/CC-MAIN-20190519061520-20190519083520-00448.warc.gz\"}"} | null | null |
{"url":"https:\/\/web2.0calc.com\/questions\/reasoning_1","text":"+0\n\n# reasoning\n\n0\n39\n1\n\nYou can paint 75 square feet of a surface every 45 minutes. Determine how long it takes you to paint a wall with the given dimensions.8ft X 5ft\n\nMar 16, 2021\n\n#1\n+507\n0\n\n$$\\frac{75}{45} = \\frac{100}{60}$$. This means you can paint 100 square feet every 60 minutes. 8ft x 5ft = 40 square feet\n\n$$\\frac{40}{x} = \\frac{100}{60}$$\n\n$$2400 = 100x$$\n\n$$x = 24$$\n\nThis means it will take you 24 minutes to paint an 8ft by 5ft wall :)\n\nMar 16, 2021","date":"2021-04-22 00:36:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.9107373356819153, \"perplexity\": 1388.7428009205928}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039554437.90\/warc\/CC-MAIN-20210421222632-20210422012632-00023.warc.gz\"}"} | null | null |
Frayser community rallies at latest homicide scene
Yolanda Jones
yolanda.jones@commercialappeal.com
Over the last three years, Pastor DeAndre Brown and members of his mentoring group have gone to homicide scenes in Frayser to reclaim the location from violence.
"For the past three years we have been responding actively to murder scenes. We go out and have an instant block party, if possible, just to let the community know that it is safe. That it is not normal behavior, Brown said. "We refuse to accept this as normal."
Brown said his Lifeline to Success group stopped going to the crime scenes last year to see if anyone else in the Frayser community would step up and help.
When no one did, Wednesday Brown and his group were at the scene of the latest homicide in the city.
Brown organized an anti-violence rally outside Creative Cuts and Design, a Frayser barbershop where a 25-year-old man was shot and killed Tuesday when two men opened fire on the business shortly after 8 p.m.
Friends identified the man killed as Devario Burks.
Police said another man was also shot, but his condition was non-critical. Two other people were also in the business, but were not hit by gunfire.
No arrests have been made, and as police continue to search for the suspects, Brown implored the community to do more.
"The only thing necessary for the triumph of evil is for good men to do nothing," Brown said.
He urged residents to be visible in their communities, mentor youth and "if you see something, say something."
"Good citizens are good neighbors and we watch out for each other," Brown said. "Stop harboring evil. Stop being a part of the madness. Stop being a victim. Let's stop it."
Shelby County School Board Member Stephanie Love showed up to the rally to show support.
"When I heard about the shooting, I said 'not again' and 'why,' " Love said. "My son lives in this community, and I worry about him and this community."
About 40 people turned out for the 11:30 a.m. rally in the parking lot of the barber shop at 2127 Frayser Blvd.
People left stuffed animals and cards on the porch of the barber shop in remembrance of the victim, Burks.
"He loved his family and was a good dude," said friend Gabrielle Cole. | {
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{"url":"http:\/\/mathoverflow.net\/revisions\/69436\/list","text":"5 edited body\n\n(EDIT 07\/06\/11: although the question has not been settled definitely, S\u00e1ndor's excellent answer and the comments by Algelo Angelo and ulrich have highlighted many potential obstructions to the constructions I wanted. Thank you all! I am still very interested in any leads, so please keep them coming if you have some more.)\n\nI would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:\n\nFix a number $c$, for what $n$ can one construct an affine variety $X \\subseteq \\mathbb A^n_{\\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? Given $c$, can we construct such $X$ for all $n$ big enough?\n\nThe case $c=1$ is easy, there are no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.\n\nMotivation\/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.\n\n4 added 322 characters in body\n\n(EDIT 07\/06\/11: although the question has not been settled definitely, S\u00e1ndor's excellent answer and the comments by Algelo and ulrich have highlighted many potential obstructions to the constructions I wanted. Thank you all! I am still very interested in any leads, so please keep them coming if you have some more.)\n\nI would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:\n\nFix a number $c$, for what $n$ can one construct an affine variety $X \\subseteq \\mathbb A^n_{\\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? Given $c$, can we construct such $X$ for all $n$ big enough?\n\nThe case $c=1$ is easy, there are no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.\n\nMotivation\/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.\n\n3 added 7 characters in body\n\nI would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:\n\nGiven\n\nFix a number $c$, for what $n$ can one construct an affine variety $X \\subseteq \\mathbb A^n_{\\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? If we fix Given $c$, can we construct such $X$ for all $n$ big enough?\n\nThe case $c=1$ is easy, there are a lot of log-canonical no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.\n\nMotivation\/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.\n\n2 added 43 characters in body\n1","date":"2013-05-22 02:14:34","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\": 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\": 0.7719340920448303, \"perplexity\": 520.647600473741}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368701153213\/warc\/CC-MAIN-20130516104553-00029-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Q: DDD, CQRS - Identity service and user avatar (image) handling I'm designing an Identity Service using DDD and CQRS pattern. So far I have one aggregate User that has all the logic for changing user's state (creating, resetting password, confirming email, etc.). In my understanding, user's avatar is also part of his identity and should be handled in the context.
I've decided that such small pictures can be saved directly to the database, but I'm not sure how to design it inside the domain (or whether it should belong to the domain at all).
Avatar could in fact be modelled as an Entity and referenced in User aggregate by its ID, like that:
public class Avatar: Entity<AvatarId>
{
AvatarId Id { get; } // Strongly typed ID
byte[] Picture { get; private set; }
AvatarExtension Extension { get; private set; } // Enumeration: JPG, PNG, GIF, etc.
public void ChangePicture(byte[] newPicture, AvatarExtension newExtension)
{
// Ommited parameters validation for brevity
Picture = newPicture;
Extension = newExtension;
}
}
public class User: IAggregateRoot, Entity<UserId>
{
...
AvatarId AvatarId { get; } //Avatar referenced by ID
public void SetAvatar(AvatarId avatarId)
{
AvatarId = avatarId;
}
}
I'm afraid it leaks some infrastructure concerns, Avatar seems more like a table definition in EF Core (code-first). In addition, Avatar would require to has its own repository, because it has to be handled separately (added to DdContext etc. before creating User), but this is not an aggregate root and my gut feeling is that it shouldn't be. Maybe referencing Avatar directly as an object in User and keeping it as one-to-one relationship would be a good idea?
A: The purpose of an aggregate (especially in CQRS) is validating commands, so the only state in an aggregate should be that for which there exists a command which would be rejected (resp. accepted) but would be accepted (resp. rejected) if the state changed.
It's hard to see how the bytes of an avatar would affect whether a future command against a user is accepted or rejected. That a user has an avatar? Sure, but that doesn't depend on the content of the avatar (a reference, e.g. a URL, to the avatar would suffice). That no two users can have the same avatar? That's not something the user aggregate modeling a single user can enforce (setting aside what exactly is meant by "no two can have the same avatar").
| {
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namespace base {
class FilePath;
}
class AppRegistrationData;
class GoogleChromeDistribution : public BrowserDistribution {
public:
// Opens the Google Chrome uninstall survey window.
// version refers to the version of Chrome being uninstalled.
// local_data_path is the path of the file containing json metrics that
// will be parsed. If this file indicates that the user has opted in to
// providing anonymous usage data, then some additional statistics will
// be added to the survey url.
// distribution_data contains Google Update related data that will be
// concatenated to the survey url if the file in local_data_path indicates
// the user has opted in to providing anonymous usage data.
void DoPostUninstallOperations(
const Version& version,
const base::FilePath& local_data_path,
const base::string16& distribution_data) override;
base::string16 GetActiveSetupGuid() override;
base::string16 GetShortcutName(ShortcutType shortcut_type) override;
base::string16 GetIconFilename() override;
int GetIconIndex(ShortcutType shortcut_type) override;
base::string16 GetBaseAppName() override;
base::string16 GetBaseAppId() override;
base::string16 GetBrowserProgIdPrefix() override;
base::string16 GetBrowserProgIdDesc() override;
base::string16 GetInstallSubDir() override;
base::string16 GetPublisherName() override;
base::string16 GetAppDescription() override;
std::string GetSafeBrowsingName() override;
std::string GetNetworkStatsServer() const override;
// This method reads data from the Google Update ClientState key for
// potential use in the uninstall survey. It must be called before the
// key returned by GetVersionKey() is deleted.
base::string16 GetDistributionData(HKEY root_key) override;
base::string16 GetUninstallLinkName() override;
base::string16 GetUninstallRegPath() override;
bool GetCommandExecuteImplClsid(base::string16* handler_class_uuid) override;
void UpdateInstallStatus(
bool system_install,
installer::ArchiveType archive_type,
installer::InstallStatus install_status) override;
bool ShouldSetExperimentLabels() override;
bool HasUserExperiments() override;
protected:
// Disallow construction from others.
GoogleChromeDistribution();
explicit GoogleChromeDistribution(
scoped_ptr<AppRegistrationData> app_reg_data);
private:
friend class BrowserDistribution;
};
#endif // CHROME_INSTALLER_UTIL_GOOGLE_CHROME_DISTRIBUTION_H_
| {
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} | 1,071 |
Head of a Bearded Man
Arts of the Ancient Mediterranean and Byzantium
5th century BCE
Roman Intaglio of Mars
This large gemstone featuring the Roman god of war, a prime example of an ancient carving technique, makes quite the impression.
Conversation: Joseph Cornell | December 17, 2022
Lecture: Photographing the Bundu Helmet Masks of the Mende Peoples in Sierra Leone | December 10, 2022
This limestone head has features that are commonly found on Greek sculptures carved about one hundred years earlier (during the Archaic period), such as almond-shaped eyes, face-framing ringlets, and a faint smile. It comes from the island of Cyprus, a major center for the exchange of goods and ideas throughout the ancient world. The sculptor may have visited Greece or otherwise known about the earlier Greek art that appears to have influenced this work.
Ancient Cypriot
Cyprus (Object made in)
500 BCE–401 BCE
29.2 × 20.3 × 26 cm (11 1/2 × 8 × 10 1/4 in.)
Robert A. Waller Fund
H. F. M., "A Limestone Cypriote Head," Bulletin of the Art Institute of Chicago 20, 7 (October 1926), p. 99.
Art Institute of Chicago, "Accessions and Loans," Bulletin of the Art Institute of Chicago 20, 6 (September 1926), p. 85
Cornelius Vermeule, Art and Archeology of Antiquity, vol. III (London: Pindar Press, 2003), pp. 32-34.
Karen B. Alexander, "From Plaster to Stone: Ancient Art at the Art Institute of Chicago," in Karen Manchester, Recasting the Past: Collecting and Presenting Antiquities at the Art Institute of Chicago (Chicago: Art Institute of Chicago; New Haven: Yale University Press, 2012), p. 31.
University of Minnesota Gallery, Space in Sculpture, November 15-December 31, 1948.
Art Institute of Chicago, The Human Figure in Early Greek Art, A Preview Part I, Gallery 101A, September 1, 1988-September 24, 1989.
Art Institute of Chicago, Ancient Art Galleries, Gallery 155, April 20, 1994-February 6, 2012.
Art Institute of Chicago, Of Gods and Glamour: The Mary and Michael Jaharis Galleries of Greek, Roman, and Byzantine Art, Gallery 151, November 11, 2012-September 13, 2016.
The Metropolitan Museum of Art, New York; sold to the Art Institute of Chicago, 1926.
Early cypriote
1000 BCE–1 CE
Buddha Shakyamuni Seated in Meditation (Dhyanamudra), Chola period (c. 855-1279), about 12th century
Coronation Stone of Motecuhzoma II (Stone of the Five Suns), 1503
Aztec (Mexica)
Stela of Amenemhat and Hemet, Middle Kingdom, early Dynasty 12, about 1956–1877 BCE
Karttikeya, Commander of the Divine Army, Seated on a Peacock, Ganga Period, about 12th century
Cow Suckling a Calf, About 9th century
Kneeling Figure, late 2nd millennium B.C.
Head of a Male Deity (Deva), Angkor period, late 12th–early 13th century
Statue of the Aphrodite of Knidos, 2nd century
Ancient Roman
Amulet of the Goddess Isis Nursing the God Horus, Third Intermediate Period, Dynasty 25 (about 747–656 BCE)
Portrait Head of Emperor Hadrian, 130-138
Relief of a Falling Warrior, 2nd century
Portrait Head of a Woman, about 140
Buddha, Tang dynasty (A.D. 618–907), c. 725/50 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,455 |
{"url":"https:\/\/scaron.info\/blog\/getting-started-with-awesome.html","text":"# Getting started with awesome\n\nTiling window managers are useful companions. They manage windows for you, so that you don't have to drag-and-click windows around all the time. They avoid window overlaps and unused screen space. Yet, they are less intuitive for beginners. In this post, I would like to introduce you to the basics of awesome, which is the tiling window manager I use. If you already know the basics, you can jump directly to the keyboard cheat sheet.\n\n## First contact\u00b6\n\nGetting awesome to run on Debian\/Ubuntu systems is easy: run sudo apt-get install awesome, then select \"awesome\" as your window manager when you log-in.\n\nThe first time you start awesome, it won't be very engaging: nothing more than a default background and a desktop bar at the top of the screen. When you click on the left end of this bar, a small menu pops out, with basic options such as spawning a terminal, reading the manual (to get the list of keyboard shortcuts), restart or exit.\n\n### Terminal\u00b6\n\nFirst, launch a terminal by pressing Mod4 + Enter.\n\n\u2022 Mod4 is the \"special\" key (typically, the Windows key...) located between Ctrl and Alt on your keyboard.\n\nFrom there, you can run your usual software, for instance nautilus (the Gnome file manager), firefox, gvim or thunderbird.\n\n### Run prompt\u00b6\n\nAs you don't want to open terminals all the time, awesome lets you run software directly: type Mod4 + r and a \"Run:\" prompt will appear in your desktop bar. You can then type your command as if you were in a terminal, for example: firefox -new-tab https:\/\/wikipedia.org\/.\n\n## Tags, layouts and windows\u00b6\n\n### Tags\u00b6\n\nBy default, you will have nine desktops, which awesome calls \"tags\" if you read the manual. You can switch between them by using Mod4 + tag number, or Mod4 + left and Mod4 + right. New windows are spawned in the current tag.\n\n### Layouts\u00b6\n\nIn awesome, windows are organised on your desktop following a desired layout. At first, you will be in the \"floating\" layout, where awesome does not reorganize your windows. Press Mod4 + Space to switch to the next layout, which should be \"tile\". The icon at the right end of your desktop bar represents your current layout:\n\nAs depicted by its icon, the tiling layout consists of two columns: one for master windows and the other for \"non-master\" windows. By default, there is only one master window occupying half of the screen, while all other non-master windows are stacked vertically.\n\nThere are twelve default layouts in awesome, which you can configure in the layouts variable of your configuration file ~\/.config\/awesome\/rc.lua. In my case, I only use two of the four tiling layouts: tile and tile.bottom.\n\nlayouts =\n{\nawful.layout.suit.tile,\nawful.layout.suit.tile.bottom,\n-- awful.layout.suit.max,\n-- awful.layout.suit.floating,\n-- awful.layout.suit.tile.left,\n-- awful.layout.suit.tile.top,\n-- awful.layout.suit.fair,\n-- awful.layout.suit.fair.horizontal,\n-- awful.layout.suit.spiral,\n-- awful.layout.suit.spiral.dwindle,\n-- awful.layout.suit.max.fullscreen,\n-- awful.layout.suit.magnifier\n}\n\n\n### Windows\u00b6\n\nFor beginners, the easiest way to focus a window is to use the mouse: when your pointer hovers a window, it becomes active. You can also move windows around in your layout by Mod4 + drag-and-clicking it around.\n\nWhen your workspace gets cluttered, you can make your active window fullscreen with Mod4 + f, or minimize it with Mod4 + n. The best way to close a program is to quit it from inside, using its menu (FileQuit in Firefox) or sending it a command like exit in a terminal. If the program is frozen or no such option is available, you can send it a (rather tough) kill signal by doing Mod4 + Shift + c on its window. This is usually terminal ;-)\n\n### Example: four windows on one screen\u00b6\n\nHere is a simple example: imagine you want four windows of equal dimensions on your screen. One way to realize this is through the default tiling layout:\n\n\u2022 Go to the tiling layout (the layout icon is the rightmost icon you see on the image above). Initially, you should see one window maximized to the left, and three windows stacked to the right half of your screen.\n\u2022 Press Mod4 + h to increase the number of windows in your main column (left part of the screen).\n\u2022 Finally, you can use Mod4 + drag-and-clicking to move your windows around in the order you like.\n\n## Beginner's Cheat Sheet\u00b6\n\nNow we have touched all the basics to survive the first steps in awesome. Here is the summary of keyboard shortcuts we have seen:\n\n Mod4 + Enter spawn a terminal Mod4 + r spawn the run prompt Mod4 + n minimize window Mod4 + f fullscreen window Mod4 + Left go to left desktop Mod4 + Right go to right desktop Mod4 + Space go to the next layout Mod4 + Shift + c kill focused window Mod4 + 1-9 switch to desktop 1-9 Mod4 + Shift + q quit awesome\n\nThe list is not exhaustive, check out man awesome to learn about other shortcuts and additional features. In awesome terminology, you will find that a \"client\" is a window and a \"tag\" is what we commonly think of as a \"desktop\".\n\n## Setting up Awesome with GNOME\u00b6\n\nSome features provided by GNOME, for instance the screensaver of password keyring, will not be available when you run Awesome by itself. See the post configuring Awesome with GNOME on Ubuntu 14.04 for instructions.\n\n## Discussion \u00b6\n\nYou can use Markdown with $\\LaTeX$ formulas in your comment.","date":"2023-01-28 12:55:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19611380994319916, \"perplexity\": 3859.1333914554307}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499634.11\/warc\/CC-MAIN-20230128121809-20230128151809-00318.warc.gz\"}"} | null | null |
How do I benefit from a fixed rate car loan?
When it comes to IMB Bank's fixed rate car loans, you can have your cake (as the competitively low rate, you so dearly like, won't increase) and eat it too (with no early repayment penalties). Fixed rate car loans can also help you plan your budget well ahead as there'll be no hikes to surprise you.
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For the lowest rate IMB Bank can offer you, buy a new ride (or one that's under 2 years old) and secure it to the car loan as collateral. To be eligible for security at all, the car you buy must be under 6 years old. Keep in mind, IMB Bank will have the power to sell your car if for some reason you forfeit the loan. Alternatively, if you don't want to use your new wheels as collateral, go for IMB Bank's unsecured car loan with a steeper interest rate but less paperwork.
The most expensive set of wheels you can buy with an IMB Bank car loan is $75k, but that's only if IMB Bank assesses you'll be able to reasonably afford the ongoing repayments without stretching your budget. Looking for a more budget-friendly ride? The smallest loan amount is $2k.
Say you bought a brand new $30k car with an IMB Bank secured loan. Using Mozo's personal loan repayment calculator, based on the comparison rate at the time of writing at 6.77%, each monthly repayment would be $591.
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IMB Bank car loan terms range from one to 7 years, depending on the age of your car and if you've secured it. The length of your contract will also be based on how much you've borrowed and your personal preference. Remember, it's okay to have a lengthy loan term as you have the opportunity to repay the loan early fee free.
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Not at all. There are no fees charged for making extra repayments, but you can't go back once you've made them, as you don't have the flexibility of a redraw facility. So be sure you can afford to exceed those regular monthly repayments and won't need to dip into them later on.
Can I repay my IMB Bank car loan off sooner rather than later?
Sure can! Even though fixed rate loans usually spell early loan repayment penalties, IMB Bank car loans are an exception. This means you can wipe off that debt sooner and pay less in interest if it suits.
Just to clarify...so I can't redraw from additional car loan repayments?
Correct, IMB Bank does not have a redraw facility, meaning you can't dip into any extra car loan repayments you have made. If this feature is a deal breaker for you, visit our comparison tables and select a different provider.
I don't want a new car, but a second-hand one. Can I still use an IMB Bank loan to help finance it?
There's nothing wrong with choosing an old wagon, but it needs to have been in the world under 6 years to qualify for loan security. If you want a vintage Cadillac, you could opt for the IMB Bank unsecured car loan with a higher interest rate instead.
What happens if I don't make a car loan repayment on time?
Here's the catch - IMB Bank charges higher late repayment fees, compared to many other car loan providers in the market. So before you take out a car loan with IMB Bank, ensure you can afford your ongoing repayments to avoid the steep late payment fees or even worse defaulting on your loan and damaging your credit score.
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Tag Archives: drama
My Top TV Moments (Part 2)
Posted on November 30, 2019 by Steve Higgins
As I was away last week enjoying the delights of Southport I was a little rushed when it came to putting together a new post for this week. A couple of weeks back I wrote about my Top TV moments and looking back at my notes I see I had a few 'moments' left over so rather than consign them to the waste paper bin I think it's time to welcome you to my Top TV Moments, Part 2!
24 was an action/espionage series which was shot in 'real' time, the 24 hour long episodes of each series covering a full 24 hour day. Kiefer Sutherland stars as special agent Jack Bauer of the CTU, Counter Terrorist Unit. Jack and his colleagues have to deal with various terrorist threats including in the opening season, plans to assassinate presidential candidate David Palmer. The show is full of twists and turns and other plots and villains emerge and unfold. Events are shown in real time and to emphasise this a digital clock is frequently shown with split screens depicting the various elements happening in the same time scale.
Bauer is a tough hombre who stands no messing and is perhaps similar to the Bruce Willis Die Hard character. A meme I saw on the Internet went like this 'Jack Bauer threw a grenade and killed 50 terrorists. Then the grenade went off . . .'
Carrie Matheson, a CIA agent who also suffers from a bipolar disorder has information that Al-Quaeda are planning a strike against the US using one of their own people. When Nicholas Brody, an army officer who has until recently been a captive in Iraq is rescued and returned to the US, Carrie believes this may be the man in question and he could have been programmed or brainwashed to act against his own country. The series builds the tension quietly and is a psychological drama rather than an action series like 24. Claire Danes as Carrie produces an outstanding performance as does Mandy Patinkin as Saul Berenson, Carrie's mentor in the agency.
A few years ago Liz and I spent part of the winter months in sunny and warm Lanzarote and to keep us occupied on those winter nights we took along the box set of Happy Valley. I have to say I wasn't that interested at first. Sarah Lancashire who has long since moved on from the scatty part of Raquel in TV soap Coronation Street, plays Catherine Cawood a police sergeant in a small West Yorkshire town. She is divorced from her husband and the two of them are scarred by the suicide of their daughter Becky 8 years earlier. Becky had been raped and gave birth to a son Ryan who lives with Catherine and her sister, a recovering alcoholic and heroin addict. Plenty of drama in that set up alone but a kidnapping occurs and it turns out that Tommy Lee Royce, the man who raped Becky is involved. All in all, an outstanding production.
Not the comic book superhero Avengers but the 1960's TV show about secret agents John Steed and Mrs Emma Peel. Steed was played throughout all the various incarnations of the Avengers by Patrick MacNee and Mrs Peel was portrayed by Diana Rigg. Mrs Peel was the leather jumpsuit wearing judo expert and together she and the charming bowler hatted Steed foiled various villains. The series was not in the same action packed mould as 24 or Homeland but had a slightly camp and comic edge to it. Mrs Peel drove a Lotus Elan as I remember while Steed preferred a vintage Bentley. When Mr Peel returned from being lost in the jungle Mrs Peel left the series to join him, handing over to Tara King, Steed's new assistant. The two passed on the stairs to Steed's apartment with Mrs Peel advising Tara to always stir Steed's tea anti clockwise!
Department S
Department S was about an Interpol department that tries to solve cases that are particularly baffling. In the very first episode the team investigate an aircraft that lands at Heathrow having been missing for 6 days, although the passengers and crew have no recollection of what has happened. Department S consisted of three investigators, Stuart Sullivan, novelist Jason King and computer expert Annabelle Hurst. Jason King played by the flamboyant Peter Wyngarde was the real star and his stylish clothes preempted the fashion trends of the early 70's. Wyngarde loved the part and I read somewhere that he even invented Mark Caine the hero of Jason King's novels. Wyngarde later starred in a spin off series Jason King.
Across the Lake was a BBC film made in 1988. It starred Anthony Hopkins as speed king Donald Campbell in the final days of his life as he tried to raise the water speed record to over 300 miles per hour. Hopkins gives a lovely performance as Donald Campbell, a man who believed himself to be living in the shadow of his father, record breaker Sir Malcolm Campbell. He decided to take his old Bluebird boat, update her and try to break the 300 mph mark on Coniston water in the lake district. The jet boat flipped over and Campbell was killed. His body was not found until 2001.
The film shows the unglamorous side to record-breaking. Waiting in poor weather, the endless delays, the mechanical issues, the press waiting for something to happen. Something drove Campbell onwards in his pursuit of records. He was short of money and had sold all sorts of rights to his name, his films of record-breaking and so on. This was all before the days of big time sponsorship in the speed and motor racing industry and Hopkins shows us a Donald Campbell undefeated, perhaps even a little desperate but still with considerable style.
The record-breaking team disperse for Christmas and then return after the holidays. They begin their preparations again until a fine January morning appeared. Campbell powered up his speedboat and did a run of 297 mph but lost his life on his second run.
Spend, Spend, Spend
Vivian Nicholson was a british woman who became famous after telling the press she was going to spend spend spend when her husband won £152,000 on the pools in 1961. Lavish spending depleted their fortune quickly and after her husband was killed in a car crash Viv was declared bankrupt. Nicholson wrote her life story with author Stephen Smith and a copy of the book was given to TV writer Jack Rosenthal who dramatised the work for the BBC's Play for Today. The episode was broadcast in March of 1977 and stars Susan Littler as pools winner Viv Nicholson. The film tells the story of a hard working class life in Yorkshire that is transformed when she and husband Keith, played by John Duttine, win the huge amount. Three years later Keith was killed in a car accident and Viv was declared bankrupt. The film tells the story of their early life together and their inability to deal with their huge fortune.
The Magic Boomerang
There are a series of TV adverts on at the moment for 'Quick Quid', a loan company which invites you to apply for a quick loan (as long as you don't mind paying their incredible interest rates that is!) There are various versions of the ad but they all go a similar way; the boiler has conked out or the car has broken down and some hapless member of the public has no money to pay to get it sorted. Suddenly that's the clue for time to freeze while the person calls up 'Quick Quid' and arranges a loan. In the Magic Boomerang, a 1960's black and white show from Australia, a young lad comes across a magic boomerang and finds that time freezes for everyone except him, just like those aforementioned adverts, while the boomerang is in the air. I remember running home from school years ago just to watch it.
Whatever Happened to The Likely Lads
The Likely lads was a TV sitcom from the 1960's about two young Geordie lads. The follow up colour version, Whatever Happened to The Likely Lads, aired in the 1970's and followed the antics of those same two lads. Rodney Bewes played Bob who is now happily married to Thelma and James Bolam played Terry, still footloose and fancy free. Each is jealous of the other in their own way and together they comment on the changing nature of life from pubs closing down to high rise flats but in particular their working class roots. Bob is constantly tormented by Terry as he is keen to become part of the middle class; he has a white collar job and a new house on a brand new housing estate. Terry however constantly laments the changing attitudes of the 1970's.
Writers Dick Clement and Ian La Frenais had planned a new series meeting up with the pair in their later years but James Bolam declined to be involved. The two actors apparently fell out after making the feature film version in 1976. After the death of Rodney Bewes in 2017 James Bolam denied rumours of a rift between him and Bewes saying "I think that Rodney wanted to do some more Likely Lads and I never did . ." Such a pity, I would have loved to see the pair together in later life.
After the success of the Monty Python series and before the appearance of the Python films, the various members of the Python team set about various other personal projects. John Cleese began writing the sitcom Fawlty Towers based on his experiences staying in a small hotel, actually the Gleneagles Hotel in Torquay, where he stayed while filming for Monty Python. He co-wrote the sitcom with his then wife Connie Booth although they had divorced by the time of the second series. The series is about hotel owner Basil Fawlty played by Cleese and his wife Sybil played by Prunella Scales. Other characters are the waitress played by Connie Booth and Manuel, a spanish waiter played by Andrew Sachs.
Only two series of six episodes each were made and the initial reception was only lukewarm but as the series gained popularity, critical acclaim began to follow. The show has won many plaudits including being ranked first on the BFI's list of the top 100 British television Programmes and was named the greatest ever sitcom by a panel of comedy experts for the Radio Times magazine.
Floating in Space is a novel by Steve Higgins set in Manchester, 1977. Click the links at the top of the page to buy or for more information.
Posted in television | Tagged comedy, drama, Television, TV, TV Reviews, TV shows | 2 Comments
A Completely Inaccurate and Unreliable History of TV and Film Time Travel!
I think I'll start this post with that fabulous time travel film Back to the Future. If you have never seen it, (shame on you) the film concerns young Marty McFly who helps his friend Doc Brown with a time travel experiment. The time machine is a 1985 DeLorean sports car which morphs through time when it reaches the magical speed of 88 mph. The experiment goes horribly wrong when a gang of Libyan terrorists whom the Doc has double crossed in order to get some vital nuclear supplies for his car/time machine arrive and shoot the Doc apparently dead.
Marty escapes in the DeLorean, accidentally hits the time travel switch (the flux capacitor) and zooms back to 1955 when he speeds up to 88 mph.
There Marty has to enlist the help of Doc Brown's younger self, get his parents' romance back on course after accidentally knocking that off target, and get himself back to 1985 in time to prevent the Doc being murdered by terrorists.
In Back to the Future 2, Crispin Glover, who played Marty's dad, failed to agree terms with the producers so was not in films 2 or 3. Instead a look-alike actor was used prompting Glover to sue the production company. His legal challenge failed but next time you watch 2 and 3, that's why Marty's dad isn't in the film much!
There are some great little touches to the film too, some you may not even have noticed, for instance, when Marty leaves 1985 and goes back to the past, he departs from the Twin Pines Mall. There, in 1955 he hits a pine tree and later arrives back in 1985 at the Lone Pine Mall!
In the final film Doc Brown gets accidentally flipped back to 1885 and Marty has to time travel back there to save the Doc. How does he know the Doc is in 1885? Well, the Doc arranges for a courier service to deliver a message to Marty at the exact time and spot where he disappears and flips back to 1885. Of course as Marty rescued him and brought him back to the future, then he wouldn't have been there to write the message for the courier, well wouldn't he? Maybe he wrote the message before departing!
Funnily enough, that is a similar situation in one of my favourite of the rebooted Doctor Who episodes, Blink. The doctor, played by David Tennant, gets stuck in the past courtesy of the Weeping Angels, aliens who appear frozen when watched but otherwise move in the blink of an eye. However he manages to add some special video links to a future DVD so people in the present can pick up his messages and help him.
Blink is possibly the most beloved episode of the modern Doctor Who era, with an oddball mystery that is intriguing and a slightly off beat approach to the show's usual format, focusing mainly on a new character named Sally Sparrow instead of David Tennant's Doctor. Sally is played by the now famous Carey Mulligan, and her portrayal of a normal young woman who has to solve a crazy time mystery when her friend is transported to the past by a living angel statue won her plenty of fans among the Doctor Who faithful.
Really though, the best Doctor Who episodes were back in the 1980's with Tom Baker as the Doctor and Elizabeth Sladen as assistant Sarah Jane Smith. A great episode was Pyramids of Mars, which combines sci- fi with the mystery of Egyptian tombs and artefacts. The Tardis materialises back at Unit headquarters in the UK. (UNIT by the way, is a military organisation the Doctor worked with in the 1980's.) However, they are not at the current time period but have arrived earlier, in 1911. A great country house is there and it appears that a mysterious Egyptian has taken over the late Professor Scarman's estate and there are many strange goings on. Later the Doctor finds that an ancient alien called Sutekh, imprisoned thousands of years ago by another alien, Horus is trying to escape captivity and wreak death and destruction on the universe. The Doctor, naturally, foils his plans.
Doctor Who is the world's longest running sci-fi show having been first broadcast on the 23rd November 1963. Not many people watched the show that day as most people were desperate to find out more about the Kennedy assassination which had happened the previous day and so it was repeated again the following week. William Hartnell, the original doctor, left the show in 1966 and the role passed to Patrick Troughton. The producers came up with the ingenious idea of having the doctor, an alien from the planet Gallifrey, regenerating into another body making it easy to reboot the series every time the lead actor left.
Star Trek, although not really a time travel programme actually had quite a few episodes which involved time travel. The fans' firm favourite, an episode voted the best ever Star Trek episode, was City on the Edge of Forever. The crew of the Enterprise arrive at a distant planet searching for the source of some time displacement. The source is a time portal, left among the ruins of an ancient civilisation which although abandoned, still emits waves of time displacement. In the meantime, Doctor McCoy is suffering from paranoia brought on by an accidental overdose of the wonder drug cordrazine which any Star Trek fan will tell you can cure any known Galactic ailment. McCoy in his crazed state bumbles through the time portal, back to 1930's America (handy for that old 1930's set on the Paramount back lot) and changes history. Kirk and Spock are forced to also go back in time, stop McCoy from changing history and restore things to as they were. Joan Collins plays a charity worker at the core of events; does she have to die in order to restore things to as they were?
In the Star Trek movie world there was another great time travel film, Star Trek 4 in which earth is threatened by a space vehicle causing havoc with the world's weather. It turns out that the aliens are sending signals in whale-speak so the crew travel back to the 1980's in order to find a hump back whale which can respond to the aliens. Sounds a bit mad when I put it like that but actually Star Trek 4 was one of the best Trek films. Highlights included Mr Spock diving into a giant pool and mind melding with a whale and later asking a punk rocker to turn down his ghetto blaster.
I must of course mention the sixties show the Time Tunnel the Irwin Allen show about two American scientists 'lost in the swirling maze of past and future ages, during the first experiments on America's greatest and most secret project, the Time Tunnel. Tony Newman and Doug Phillips now tumble helplessly toward a new fantastic adventure, somewhere along the infinite corridors of time' as the opening blurb used to go.
The Time Tunnel starts off with a Congressman coming to investigate the growing budget of the time tunnel complex and threatens to close things down unless he sees results. Scientist Tony Newman decides he must therefore travel back in time to prove that the tunnel really works and save the project. Tony ends up on the ill-fated liner Titanic. His colleague Doug follows him back to 1912 and the control room struggle to shift the two in time before the ship sinks. Unable to return the duo to the present, the technicians struggle every week to shift the duo to somewhere new just in the nick of time. One episode that I particularly remember was when the pair land in Pearl Harbour, just before the Japanese attack in 1941. Tony meets himself as a young boy and finally solves the mystery of the disappearance of his father in the attack.
I'm at the point of running out of time travel TV shows but here is one great time travel movie, 12 Monkeys. I didn't like it the first time I saw it then some time later I saw part of the film again and thought, hey, maybe this film isn't so bad. The third time I saw it all the way through and did enjoy it although it can be a little hard to follow. 12 Monkeys was inspired by the French short film La Jetée made in 1962 and it goes something like this; Bruce Willis plays prisoner James Cole who lives in a post apocalyptic society in the year 2035 where people are forced to live underground after a deadly virus was released in 1996 which wiped out most of humanity. The virus was released by a group known only as the 12 Monkeys and Cole is selected to be sent back into the past to try and find the original virus so scientists can perfect a cure. Cole is plagued by a vision which constantly returns to him in which a man is shot dead on a railway station and as the film reaches its final moments, this tragic vision is finally explained.
I've tried to keep this post pretty much research free so no doubt numerous errors and omissions will be evident. What was your favourite time travel film or TV episode?
Floating in Space is a novel set in Manchester 1977. Click the links at the top of the page to buy or for more information.
Posted in motion pictures, television | Tagged cinema, classic tv, drama, film, Sci Fi, time travel, TV | Leave a comment | {
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{"url":"http:\/\/mathoverflow.net\/questions\/48067\/is-square-of-delta-function-defined-somewhere\/93905","text":"# Is square of Delta function defined somewhere?\n\nHello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where?\n\nIn the beginning, this question might look strange. But by restricting the space of the test functions, I think it is still possible. For example, in order to make sense of $\\delta_0^2$, we can think that it is the limit of $\\frac{e^{-x^2\/t}}{2\\pi t}$ as $t\\rightarrow 0_+$. Now choose the test function $f(x)=x^2$. It is clear that $$\\int_{-\\infty}^{\\infty} x^2 \\frac{e^{-x^2\/t}}{2\\pi t} d x = \\frac{1}{2\\sqrt{\\pi t}} \\int_{-\\infty}^{\\infty} x^2 \\frac{e^{-x^2\/t}}{\\sqrt{\\pi t}} d x = \\frac{1}{2\\sqrt{\\pi t}} \\cdot \\frac{t}{2} = \\frac{\\sqrt{t}}{4\\sqrt{\\pi}}\\;.$$ Then let $t$ tend to $0$, we get $<\\delta_0^2,f>=0$ in this case. So we can restrict, for example, all test functions tend to 0 at the speed no less than $x^2$.\n\nI don't want to invent the whole stuff if it already exists. Otherwise, I might take care of the every details. Thank you in advance for any hints.\n\nEDIT: Here are some references that I found to be useful. 1: Mikusi\u0144ski, J. On the square of the Dirac delta-distribution. (Russian summary) Bull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astronom. Phys. 14 1966 511\u2013513. 44.40 (46.40)\n\n2:Ta Ngoc Tri, The Colombeau theory of generalized functions Master thesis, 2005\n\n-\nThere is the <a href=\"en.wikipedia.org\/wiki\/\u2026; Colombeau algebra <\/a>. Though I have no idea what applications it has. \u2013\u00a0 Rbega Dec 2 '10 at 17:29\nI am no expert but my understanding is that the Dirac delta function is not a function but is a distribution. Furthermore multiplication of distributions is not defined and this is the cause of much frustration in quantum field theory. \u2013\u00a0 Bruce Westbury Dec 2 '10 at 17:35\n@Bruce the multiplication of distributions is sometimes defined. If you are interested you should look up wavefront sets, one good reference is Chapter 5 of math.mit.edu\/~rbm\/18.157-F09\/18.157-F09.html, see e.g. Prop 5.12. For example, this would make rigorous the notion that the product of a delta function at $0$ and a delta function at $x\\neq 0$ should be $0$. However, you are right that the product of two delta functions is not well defined as a distribution. However, I think that @Anand is asking about restricting the domain of distributions to allow it to exist. \u2013\u00a0 Otis Chodosh Dec 2 '10 at 17:47\n@Otis. Thank you for your reference. I will have a look. Yes. As you said, I intend to restrict the space of the test functions, in order to give a rigorous definition of $\\delta_0^2$. \u2013\u00a0 Anand Dec 2 '10 at 17:54\n@Anad: I think you have to clarify what kind of object you want your square of $\\delta$ to be. E.g. restricting the test funtions does not give you something like $\\delta^2$ as long as you are looking for a linear functional (which you probably don't). \u2013\u00a0 Dirk Dec 2 '10 at 18:58\n\nWhen L. Schwartz \"invented\" distributions (actually, he only invented the mathematical theory as a part of functional analysis, because distributions were already used by physicists), he proved incidentally that it is impossible to define a product in such a way that distributions form an algebra with acceptable topological properties. What is possible is to define the product of distributions when their wave front sets do not meet. This is why $fT$ makes sense if $T$ is a distribution and $f$ is $C^\\infty$, for instance, because the front set of $f$ is void. But you can also multiply that way genuine distributions; for instance in $\\mathbb R^2$, $\\delta_{x=0}=\\delta_{x_1=0}\\delta_{x_2=0}$.\n\nJ.-F. Colombeau invented in the 70's an algebra of generalized functions, which has something to do with distributions. But each distribution has infinitely many representatives in the algebra, and you have to play with the equality and a \"weak equality\" (or \"association\"). I don't know of an example where this tool solved an open problem. In Colombeau's algebra, the square of $\\delta_0$ makes sense, but is highly non unique.\n\n-\n@Prof. Denis Serre, Thank you very much. Your explanation is very clear. Especially the case $\\delta_{x=0}=\\delta_{x_1=0}\\delta_{x_2=0}$ in $R^2$. I took this results for granted before. Now I know the reason. By the way, do you have a good reference on the topic of Wave front set? I encountered this topic in Hormander's books. His books seem too difficult. Thank you very much! :-) \u2013\u00a0 Anand Dec 3 '10 at 14:08\n\n$\\delta_0$ vanishes identically on the space of test functions you've defined. So it's not surprising that its square is well-defined: $0\\cdot 0 = 0$.\n\nI suspect you'll have a much harder time defining $\\delta_0^2$ on test functions which don't vanish at $0$.\n\n-\nMy intention is just to make $\\delta_0^2$ rigorous by restricting the test functions to vanish at 0. :-) \u2013\u00a0 Anand Dec 2 '10 at 17:58\nYes, but you've thrown the baby out with the bathwater. \u2013\u00a0 userN Dec 2 '10 at 17:59\nI think that you can restrict to functions which approach their value at $0$ like $x^2$, instead of functions which are zero at $0$, e.g. $x^2 + 3$. I don't think this leads to anything interesting, but @A.J.'s objection might not be fatal. \u2013\u00a0 Otis Chodosh Dec 2 '10 at 18:08\n@Otis, Thanks for your comments. Since I meet this problem in my research, I need something rigorous, even it is not very interesting. :-) \u2013\u00a0 Anand Dec 2 '10 at 18:11\n@Anand I think that you should edit your question to describe the properties of \\delta_0^1 you would like. A.J.'s argument basically shows that if you want to define $\\delta^2 (f)$ to be $\\lim_{t\\to 0} \\int_{-\\infty}^\\infty f e^{-x^2\/t}\/\\sqrt{2\\pi t}$ for some class of $f$, then basically the only thing you can do is to define it to be zero for functions which vanish faster than $x^2$. Perhaps you should explain why you would like to define $\\delta_0^2$ and what properties you would like from such a definition. Otherwise, I doubt that anyone can say anything more useful than A.J's answer. \u2013\u00a0 Otis Chodosh Dec 2 '10 at 18:19\n\nThere are whole theories in microlocal analysis that deal with the issues here, I believe. Some heuristics are that the \"singular support\" of a distribution controls what it can be multiplied by in a naive sense (distributions with a disjoint singular support). So squaring the delta function is the first bad case - whatever the singular support means, it must be the set containing 0 for the delta function. Need more heuristics.\n\nOne insight is that one dimension may be too few to show the real picture. \"Microlocal\" tends to mean localising in (co)tangential directions, and one dimension offers only two. Hyperfunctions in the case of one dimension make something of this by considering the real line as the boundary of the upper half complex plane. I.e up is not the same as down. Boundary values of functions holomorphic in the upper half plane have a candidate for the delta function analogue: take 1\/z. No problem squaring that. More of a problem saying what this analogy means that is worth anything. Mikio Sato did that. Now I shall be quiet, because this is probably already wrong enough.\n\n-\n@Charles, thank you very much for your comments. It is very interesting. But it is too hard for me to understand. I wish I could understand something about Microlocal analysis one day. :-) \u2013\u00a0 Anand Dec 2 '10 at 21:38\nThe process of squaring holomorphic functions on half-planes does not yield hyperfunctions that behave like squares, even when restricting to the regular case. \u2013\u00a0 S. Carnahan Dec 3 '10 at 4:38\n\nThe extent to which multiplication of distributions is defined was examined by Richards & Youn and some of the results are in their short and fairly elementary joint book on distributions. One can multiply something fairly exotic like the third derivative of the delta function by a very well-behaved function; that much everybody knows. But I think they had a result that as one factor becomes progressively less well-behaved the other must become more well-behaved in order to make multiplication possible. I don't recall the details. But I'm pretty sure theirs is not the last word on the subject.\n\n-\nYes. As you said that as one factor becomes progressively less well-behaved, the other should become more well-behaved. The assumption here might be that you consider the same space of test functions. In current case, both factor becomes ill-behaved (both of them are $\\delta_0$). So we might overcome this difficulty by restricting the test functions, requiring that they should vanish at 0 at certain speed. Thanks. :-) \u2013\u00a0 Anand Dec 2 '10 at 18:05\nBy the way, could you please give me some more details about the reference of Richards & Youn? I learnt this topic by Roudin's book, I am not an expert at this domain. Thank you very much! :-) \u2013\u00a0 Anand Dec 2 '10 at 18:13\nGoogle \"Richards Youn Distributions\" and it pops up in google books. \u2013\u00a0 B R Dec 2 '10 at 18:44\nThe Theory of Distributions: A Nontechnical Introduction by J. Ian Richards and Heekyung Youn, published in 1990 by Cambridge University Press. \"Nontechnical\" appears to mean that the reader is not assumed to know functional analysis, topology, or measure theory. But everything is rigorously proved. Including the fact that the \"tempered\" distributions---those that don't grow too fast at $\\pm\\infty$---are closed under the Fourier transform. \u2013\u00a0 Michael Hardy Dec 2 '10 at 20:51\nThanks BR and Michael Hardy. :-) \u2013\u00a0 Anand Dec 2 '10 at 21:31\n\nI'd like to point out that several of the concepts mentioned here are explained on the nLab:\n\nwhile several are missing and the parts on microlocal analysis and hyperfunctions could use some help .\n\n-\n@Tim van Beek. Thank you very much for the reference. It looks like what I am looking for...:-) \u2013\u00a0 Anand Dec 3 '10 at 14:03\n\nThe theory of distributions and operations on them are generally only useful in so far as they extend the operations on smooth functions. If you look in H\u00f6rmander, there is a criterion in terms of wavefront sets which is very useful (mentioned by others), and you'll also notice that the wavefront sets of $\\delta$ and $\\delta$ collide. The reason you can't square the delta-function is that when you approximate it by smooth functions, there is no unique limit. If you wanted to restrict to a smaller space of test functions, you would clearly have to consider test functions which vanish at the origin in some way. But do you have a particular purpose in mind for this question?\n\nEDIT: Sorry -- this was supposed to be a comment, not an answer.\n\n-\n@Phil Isett, I am doing stochastic heat equation. If the initial data is a Delta distribution, then the second moments of the solution is well defined for all $t>0$, but for time $t=0$, the second moment will be formally certain Delta square. This is why I encounter this problem. Thanks you for your comment. :-) \u2013\u00a0 Anand Jul 21 '11 at 9:10\n\nDenis Serre's answer is just perfect. Let me add a couple of examples of distributions that can be squared:\n\n(1) With $H$ the Heaviside function, define $Log(x+i0)=\\ln(\\vert x\\vert)+i\\pi H(-x)$ and $$T_1=\\frac{1}{x+i0}=\\frac{d}{dx}(Log(x+i0))=pv\\frac 1{x}-i\\pi \\delta_0(x).$$ It is easy to see that $WF T_1=[0]\\times (0,+\\infty),$ so that $WF T_1+WF T_1$ does not meet 0. Then there is no difficulty to define $T^2$ say as $$\\langle T^2,\\phi\\rangle=\\lim_{\\epsilon\\rightarrow 0_+}\\int\\frac{\\phi(x) dx}{(x+i\\epsilon)^2}.$$\n\n(2) Let us consider a smooth hypersurface $\\Sigma$ of $\\mathbf R^d$ defined by the equation $f(x)=0$ with a smooth $f$ such that $df\\not=0$ at $f=0$ and let $\\delta_\\Sigma$ be the Euclidean measure on $\\Sigma$. Then $$T_2=pv\\frac{1}{f}-i\\delta_\\Sigma$$ can be squared. The reason is the same than for the previous example, since $WF T_2$ is the positive conormal of $\\Sigma$. A point $(x,\\xi)\\in WF T_2$ iff $$x\\in \\Sigma\\quad \\xi =\\lambda df(x) \\text{ with \\lambda >0}.$$ Then of course, if $(x,\\xi_j)$, $j=1,2$ are both in $WF T_2$ then $$\\xi_1+\\xi_2\\not=0.$$\n\n-\n\nI've seen the idea of it used in image processing for denoising; the total variation energy\n\n$E_{TV}(f) = \\displaystyle\\int\\left(|\\nabla f| +(f-u)^2\\right)$\n\nis generally used instead of Tikhonov regularization\n\n$E_{Tikhonov}(f) = \\displaystyle\\int\\left(|\\nabla f|^2 +(f-u)^2\\right)$\n\nas the latter never has a discontinuous solution (since the integral would be infinite).\n\nI don't remember how rigorously this idea was developed - \"Mathematical Problems in Image Processing\" by Aubert and Kornprobst was the textbook I used at the time, but there are probably some more recent references in the field.\n\n-\n\nCheck out the papers by Accardi and Boukas\n\n[added by S. Carnahan: The relevant part of their first ArXiv paper is that they regularize powers of $\\delta$, not by setting $\\delta^n(x) = c_n \\delta(x)$ for some real $c_n$ (which they find to work poorly for their purposes), but by using two variables and setting $\\delta(t-s)^n = \\delta(s)\\delta(t-s)$ for all $n \\geq 2$. This definition allows them to define certain representations of Lie algebras by Fock space methods. As far as I can tell, this does not yield a workable definition of $\\delta^2$ for analysis on the real line.]\n\n-\nI was more or less ready to delete this non-answer, but I might as well summarize what I found by Googling. \u2013\u00a0 S. 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Tag: Bosworth
21st April, 1509: The king is Dead; Long Live the King!
At 11pm on 21st April 1509, (though some historians believe it was 22nd), King Henry VII died at Richmond Palace. His death wasn't a surprise to anybody because he'd been ill on and off for the past two years, and had locked himself away at Richmond due to the decline of his condition. Since Henry had been away from the public eye, it was able to be kept secret for the following two days until arrangements for succession had been made.
Henry VII's death bed. Apparently the 2 men on the right are holding piss pots to collect his urine…and I thought my job was wank.
Henry had had a turbulent life, for starters he was born to a fierce as fuck, 13 year old Lancastrian noble girl and his father had died before his birth. Then, for a long time after that, his life was blighted by The House of York who thought Henry was a threat to their throne, (and to be fair they had a point). Prior to his ascent to the throne his life went like this: He lived in exile because of the Yorkist's, He rallied against the Yorkist's, he scrapped with the Yorkist's, he killed a Yorkist king and then married a Yorkist princess.
His marriage to Elizabeth of York was said to be a happy one, despite the initial circumstances that led to their union. The idea was that when Henry kicked arse at Bosworth, and killed King Richard, the young Lancastrian king would then marry the beautiful York princess in a bid to unite the houses and end the wars…and it worked. I like to think of it like a bit of a Romeo and Juliet story, (and in my head its set to a West Side Story backdrop but with mincing Tudors instead of not very threatening New Yorkers).
Henry VII, the Miser King and his Smokin' hot wife, Elizabeth of York. Punched above his weight, that one.
His life settled somewhat after he had secured the throne… well I say that but there were a couple of hiccups, and a couple of kids claiming to be Elizabeth's long lost brothers / royal heirs to the throne which must have been awkward, but nothing he didn't handle. He ruled successfully for 23 years, but wasn't particularly popular with his subjects who saw him as a tight arse and a miser, but to his credit he took the country out of bankruptcy and got shit done so it goes to show you can't have it all.
His death was announced to the Kings Garter at their annual feast of St. George on 23rd April, then publically on the 24th. The throne was left to his 17 year old son, Henry, who, despite being the polar opposite to his father and a big fat misogynistic tool, also did a pretty good job of keeping the country ticking over*.
Henry is buried next to his wife in Westminster Abbey, in a chapel of his own making. He placed the Tudor's on the throne and started a dynasty that lasted for 118 years…Good work I'd say.
Elizabeth of York and Henry Tudor's tomb: they even look good in all gold.
*I had a hard time writing that sentence. I initially wanted to remain unbiased, but that was never going to happen. Then I was going to put 'also did a pretty good job of upholding the peace', but then I though about the break from Rome, the reformation and the total annihilation of the monasteries, let alone scrapping in Scotland and France. So I settled with 'ticking over'…I basically just see all of them as keeping it warm until Elizabeth came to sort shit out.
October 30th 1485: Henry Tudor, What A Genius…(oh and his coronation!)
October 30, 2015 October 28, 2015 1 Comment
Henry VII: Looking as smug as he should do
Henry VII was crowned king on 30th October, 1484, after kicking the shit out of his predecessor, Richard III, a few weeks earlier at the battle of Bosworth (which you can read about here). Now let's not be under any pretence: Henry had about as much claim to the English throne as Richard did, actually less of a claim, and there were many people who would see the throne return back to the York's if they had their way. Henry was nobody's fool though and did everything within his power to make sure that this didn't happen. Actually this is more of a story about a very clever man, than the coronation of a King.
As you probably already know, Henry married Elizabeth of York, the daughter of Edward IV and niece of Richard III. This was an attempt to unite the houses of Lancaster and York in order to seal his claim to the throne and suggest that the recent wars and battles were over. Although the pair were betrothed, henry didn't marry Liz until the following January (nearly 3 months after his coronation). This was to ensure nobody could claim that henry only had the throne through his wife's claim TO IT. He managed to delay the marriage by writing to the Pope to ask for special permission for the marriage to happen – the couple were distant relatives, though that didn't usually stop folks back then. Henry knew however that it would take fucking ages for the letter to get to the Pope and for a reply to be sent, buying him a bit of time to squeeze his coronation in.
His next genius move was to set the date of his assentation to the throne to the day before the battle of Bosworth so that he could claim anybody supporting Richard was a traitor and seize their lands. By seizing their lands he was not only showing them that they really shouldn't fuck with him, but also making himself incredibly wealthy in the process. I think the whole wealth thing would've come as a bit of an alien concept to Henry. He had been so used to moving around and living in relative poverty in France, (I say 'poverty'…he was poorer than his birth right would suggest, don't feel too bad for him, he wasn't a council estate in Tory Britain type of poor, more of a Kate Middleton after forgetting her purse kind of poor), then suddenly he finds himself rich with a whole army, a treasury and a shit tonne of land to his name.
Henry also learned from Bosworth that nobody could be trusted, (his step Dad had given him the run around at Bosworth and a few of the other noble men had shown themselves to be a bunch of fickle dick heads). Henry's answer to this problem was to make a law that no man should have his own army. This stopped anybody rising up against the King and reduced the power the noblemen had. Henry wasn't thick.
His next act of pure genius was to be crowned before the first meeting of parliament, so that nobody could argue the legitimacy of his claim to the throne. After all who is going to tell the King that he is not king? Especially if that King has just seen to it that the last man who pissed him off has an axe put through his head and his knob and bollock paraded about on the back of a horse for all to see before being shoved under a future car park?!
His actual coronation itself took place at Westminster Abbey. It must have been an emotional day for not only Henry, but for his Mum, Margaret Beaufort. Margaret had not seen her boy for 17 years. She had Henry when she was 13 and childbirth near killed her. She never had another child, and despite being scary as balls, I think she loved him very much. She sent him into exile for his own protection: being an heir to the throne of the house Lancaster at a time when the throne was occupied by the York family was pretty dangerous, (think Montague and Capulet if you need a perspective), and had Margaret not sent henry away he would've almost certainly been killed as a child.
Of course, Henry's 'unofficial' coronation took place on Bosworth battle field, when Lord Stanley dragged Richard III's crown out from under a bush and placed it upon Henry head. Henry knew that he had to have a proper coronation, one that 'was under the eyes of God' (i.e. in a church and not on Gods actual face), in order to cross it off the 'reasons to kick henry off the throne' list. By holding a coronation at Westminster Abbey and presenting his standards at St. Pauls cathedral, Henry was saying to the world 'Look God chose me so I must be King…I've put my flag up and everything'. It worked. A couple of years late and nobody even questioned henry's claim (well, apart from the pesky Lambert Simnel and Perkin Warbeck posse's that is, but that's a story for another Tudorial).
Henrys 'unofficial' coronation at Bosworth. Apparently this picture is based upon a tapestry, and not a crap colouring in book bought from a National Trust property.
Henry went on to reign for 23 years, 7 months and 28 days. His reign brought about peace to what had been a really shit past few decades, and also marks the birth of the Tudor reign. Henry always strikes me as an amazing bloke and the more I read about him, the more he becomes a contender for the 'my favourite Tudor Sovereign' spot.
Kenilworth Castle September 26, 2019
How the Protestant Reformation Rapidly Changed Life For people With Disabilities in England in the Sixteenth Century. June 10, 2019
13th February, 1542: The Execution of Katherine Howard February 13, 2019
February 7th, 1527: The Vicar of Hell loses an eye. February 7, 2019
Haddon Hall, Henry Vernon and the Runaway Bride. September 14, 2018
thetudorials@gmail.com | {
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Camer.be ist eine kamerunische Online-Zeitung, die hauptsächlich französischsprachig ist. Gegründet wurde sie 2005 durch die Camer-Group mit Sitz in Brüssel (Belgien). Die Abkürzung camer steht für Cameroun und .be für Belgien.
Die Zeitung liefert täglich Nachrichten über Politik, Wirtschaft, Gesellschaft, Kultur und Sport in Kamerun und Afrika. Der Herausgeber ist Jean-Pierre Bouneck und der Chefredakteur ist Hugues Seumo.
Eine Parallelseite, Camer-sport.be, beschäftigt sich ausschließlich mit Sportnachrichten. Der Chefredakteur ist Hermann Oswald G´nowa.
Mittlerweile zählt Camer.be zu den 10 meistbesuchten kamerunischen Webseiten.
Am 13. Juli 2012 erhielt diese Online-Zeitung die Auszeichnung "Le Njawe Prize Du Meilleur Journal En Ligne" in Erinnerung an den verstorbenen kamerunischen Journalisten, Menschen- und Presserechtler Pius Njawe.
Weblinks
www.camer.be
www.camer-sport.be
Einzelnachweise
Onlinezeitung
Nachrichtenwebsite
Französischsprachiges Medium
Medien (Kamerun) | {
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Leopold Van Esbroeck (Londerzeel, 14 december 1911 – Brasschaat, 27 juli 2010) was een Belgisch beeldhouwer en schilder.
Biografie
Leopold "Pol" studeerde aan de Koninklijke Academie voor Schone Kunsten van Antwerpen en won er verscheidene prijzen. Na zijn studies werd hij professor aan de Academie, de jongste ooit.
Tijdens de oorlogsjaren stond Leopold Van Esbroeck in voor de restauratie van het Rubenshuis te Antwerpen en in 1964 – 1969 restaureerde hij de beelden van de Calvarieberg nabij de Sint-Pauluskerk te Antwerpen.
Leopold was getrouwd met de Vlaamse beeldhouwster Mariëtte Coppens. Samen sculpteerden ze de Lindeboom in Zoersel.
Hij beeldhouwde verschillende oorlogsmonumenten, een groot aantal borstbeelden, maakte praalwagens en schilderde. Zijn stijl van werken evolueerde van barok naar soberheid.
Leopold en Mariëtte liggen begraven in het ereperk van de kunstenaars in het Schoonselhof te Antwerpen. Op de deksteen staat een werk in arduin van hem, genaamd Het Zonnebloemmeisje (1973), en een werk van haar in vaurion claire, genaamd Gezin (1975).
Dochters Mieke en Diana (Coppens) openden in 2011 Het beeldenhuis Van Esbroeck-Coppens VZW, een museum in Zoersel, gewijd aan de werken van hun ouders.
Werken
Een groot aantal van zijn beeldhouwwerken zijn te vinden over heel Vlaanderen en Brussel.
In Antwerpen vindt men onder andere het standbeeld moeder Netje, hoofdfiguur uit de roman Moeder, waarom leven wij? van Lode Zielens, naast de Sint-Andrieskerk en de Madonna op de Grote Markt.
Zoersel: Gesculptuurde Lindeboom. De 800 jaar oude lindeboom van Zoersel werd in 1974 geveld en bewerkt door Leopold en Mariette. Leopold kerfde in de stam de geschiedenis van Zoersel in vijfentachtig figuren terwijl Mariette het verhaal van De Loteling van Hendrik Conscience in de takken beeldhouwde.
In de Calvarieberg (Antwerpen) maakte hij de volgende beelden : de pelikaan, Adam, een haan, het fronton met fakkelhond, Christus op het kruis, de twee watermannen, twee houten putti in de grot, het vagevuur en de pauw.
Onderscheidingen
Prijs Van Rome (beeldhouwkunst)
Theodoor Van Lerius-prijs
Ereprofessor aan de Koninklijke Academie voor Schone Kunsten van Antwerpen
Officier in de Leopoldsorde
Officier in de Kroonorde
Ereburger van Klein-Brabant, Zoersel en Londerzeel
Trivia
Pol was de nonkel van zanger Dirk Van Esbroeck.
Literatuur
Sint-Paulus-Info. Wetenschappelijk tijdschrift van de Sint-Paulusvrienden. Bouwstoffen voor de geschiedenis van de Antwerpse Sint-Pauluskerk. 72 nummers (1982 tot 2009).
Sirjacobs Raymond. Sint-Pauluskerk Antwerpen. Historische Gids (tweede volledig herwerkte druk 2001).
Zie ook
Sint-Pauluskerk (Antwerpen)
Externe link
Werken Pol Van Esbroeck
Website over het Beeldenhuis in Zoersel
Toerisme Zoersel
Belgisch beeldhouwer
Belgisch kunstschilder | {
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Sarah Leah Whitson (@sarahleah1), executive director of Human Rights Watch's Middle East and North Africa Division, oversees the work of the division in 19 countries, with staff located in 10 countries. She has led dozens of advocacy and investigative missions throughout the region, focusing on issues of armed conflict, accountability, legal reform, migrant workers, and political rights. She has published widely on human rights issues in the Middle East in international and regional media, including The New York Times, Foreign Policy, The Los Angeles Times, and CNN. She appears regularly on Al-Jazeera, BBC, NPR, and CNN. Before joining Human Rights Watch, Whitson worked in New York for Goldman, Sachs & Co. and Cleary, Gottlieb, Steen & Hamilton. She graduated from the University of California, Berkeley and Harvard Law School. Whitson is a member of the Council on Foreign Relations. She speaks Armenian and Arabic. | {
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\section{#1} \setcounter{subsection}{0}}
\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\thesubsection}{\thesection.\arabic{subsection}}
\setcounter{secnumdepth}{1}
\begin{document}
\title{Gravitational properties of light - The emission of counter-propagating laser pulses from an atom}
\author{Dennis R\"atzel, Martin Wilkens, Ralf Menzel}
\address{University of Potsdam, Institute for Physics and Astronomy\\
Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany}
\begin{abstract}
The gravitational field of a laser pulse, although not detectable at the moment, comes with a peculiar feature which continues to attract attention; cause and effect propagate with the same speed, that of light. A particular result of this feature is that the gravitational field of an emitted laser pulse and the gravitational effect of the emitter's energy-momentum change are intimately entangled. In this article, a specific example of an emission process is considered - an atom, modeled as a point mass, emits two counter-propagating pulses. The corresponding curvature and the effect on massive and massless test particles is discussed. A comparison is made with the metric corresponding to a spherically symmetric massive object that isotropically emits radiation - the Vaidya metric.
\noindent{\it Keywords\/}: gravity, general relativity, laser pulses, electromagnetic radiation, radiation, star, black hole
\noindent PACS numbers: 04.20.-q, 42.55.-f, 42.60.Jf, 42.62.-b, 42.55.Ah
\end{abstract}
\pacs{04.20.-q, 42.55.-f, 42.60.Jf, 42.62.-b, 42.55.Ah}
\maketitle
\section{Introduction}
\label{sec:introduction}
In a seminal paper \cite{Tolman1931} by Tolman et al., a single light pulse of finite lifetime was modeled and the corresponding gravitational field was derived as a perturbation of Minkowski spacetime in the framework of linearized gravity. In \cite{Bonnor2009} and \cite{Raetzel2016pulse}, it was argued that the gravitational field of the pulse is due only to its emission and absorption.
In \cite{Raetzel2016pulse}, it was shown that the exact process in which a laser pulse is emitted can be disregarded, as long as one is only interested in the gravitational field close to the pulse trajectory. However, in general, the change in the emitter's energy-momentum accompanying the emission of the pulse induces a gravitational effect that is intimately entangled with the gravitational effect of the creation of the pulse. In particular, like the laser pulse and the gravitational effect of its emission, the gravitational effect of the emitter's energy-momentum change propagates with the speed of light.
In this article, a specific example of an emission process will be considered - an atom, modeled as a point mass, emits two counter-propagating pulses. This model respects the continuity equation of general relativity $\partial_\mu T^{\mu\nu}=0$ and the results are valid for all spacetime points. The model will be introduced and the corresponding gravitational field will be derived in Section \ref{sec:pulse}. In Section \ref{sec:curvature}, the curvature and the tidal forces will be given. In Section \ref{sec:acc} and \ref{sec:masslesstest}, the acceleration of test particles in the bi-pulse metric will be discussed.
The situation of two laser pulses emitted from a point particle shares certain features with the spacetime corresponding to a spherically symmetrical massive object that emits radiation isotropically. The corresponding solution of the full Einstein equations is called the Vaidya metric. In Section \ref{sec:compvaidya}, the relation to the Vaidya metric will be clarified and the acceleration of a massive test particle in this spacetime will be discussed.
\section{Two laser pulses emitted by a point mass}
\label{sec:pulse}
In this section, we will present the model for the emission of two counter-propagating laser pulses from a point particle. The situation is illustrated in Figure \ref{fig:pulse}. We start with the separation of the metric into the background Minkowski metric $\eta_{\mu\nu}=\mathrm{diag}(-,+,+,+)$ and a metric perturbation $h_{\mu\nu}$
\begin{equation}\label{eq:metricsplit}
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\,,
\end{equation}
where $|h_{\mu\nu}|\ll 1$ is assumed to hold. We assume the Lorentz gauge condition
\begin{equation}\label{eq:lorentzgauge}
\partial^\mu \left(h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu} {h_\alpha}^\alpha\right)=0\,,
\end{equation}
and the Einstein field equations become, in first order in the metric perturbation (see \cite{Misner1973} page 438)
\begin{equation}\label{eq:linearizedeinstein}
\left[\frac{1}{c^2}\frac{\partial^2}{\partial t^2} -\frac{\partial^2}{\partial x^2}
- \frac{\partial^2}{\partial y^2}
- \frac{\partial^2}{\partial z^2}
\right] h_{\mu\nu} = \frac{16\pi G}{c^4}
\left(T_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu} {T_\alpha}^\alpha\right)\,,
\end{equation}
where $T_{\mu\nu}$ is the energy-momentum tensor of the two pulses and the point mass.
\begin{figure}[h]
\includegraphics[width=12cm,angle=0]{laser_pulse_box_shaped_emitter_decay.pdf}
\caption{The laser pulses are modeled as pulses of electromagnetic radiation of length $L$, traveling from the emitter along the $z$-axis in the negative and positive $z$-direction. The extension of the pulses in the transverse $x/y$-directions is assumed to be negligible in comparison to their length.
\label{fig:pulse}}
\end{figure}
We assume the point mass to be at rest in the lab frame and the emission to happen at $z=0$, to start at $t=0$ and to end at $t=L/c$, where $L$ the length of the laser pulses. The only non-zero component of the energy momentum tensor for the emitting point particle is given as
\begin{equation}\label{eq:energymomentum}
\nonumber T^s_{00}=\left\{
\begin{array}{ccc}
mc^2\delta^{(3)}(\vec r) &:& ct\le 0\\
\left(mc^2-2\epsilon\frac{ct}{L}\right)\delta^{(3)}(\vec r) &:& 0 < ct < L\\
\left(mc^2-2\epsilon\right)\delta^{(3)}(\vec r) &:& L \le ct\,.
\end{array}
\right.
\end{equation}
where $m$ is the mass of the point particle before the emission and $\epsilon$ is the energy, $\hbar \omega$, of each laser pulse.
We model each pulse as a pulse of electromagnetic radiation, traveling from the emitter along the $z$-axis, with finite extension (pulse length) $L$ in the direction of propagation, but negligible extension $\Delta(z)$ in the transverse $x/y$-directions, $\Delta(z)\ll L$ (see figure \ref{fig:pulse}) such that we can well approximate the energy density as proportional to $\delta(x)\delta(y)$. All measures refer to a laboratory frame where the emitting system is at rest before emission of the pulse. The first pulse moves in the positive $z$-direction and the second pulse moves in the negative $z$-direction. Hence, in the coordinates $(ct,x,y,z)$, the terms in the energy momentum tensor due to the pulses have the non-zero components $T^{+}_{00}=T^{+}_{zz}=-T^{+}_{z0}=-T^{+}_{0z}=T^+$ and $T^{-}_{00}=T^{-}_{zz}=T^{-}_{z0}=T^{-}_{0z}=T^-$. We consider only circularly polarized laser pulses, which means that the energy momentum tensor is constant over the full extension of the pulses along the $z$-axis. Then, the functions $T^{+}$ and $T^{-}$ have the following form:
\begin{equation}\label{eq:Tmunufunctions}
\begin{array}{llll}
\textrm{For }ct\le 0: &
T^+=0\,, &T^-=0\\
\textrm{For } ct > 0 \,\, \mathrm{and} \,\, z>0: &
T^+= \frac{\epsilon}{L} \delta(x)\delta(y)\chi(ct-z)\, &
T^-= 0\\
\textrm{For }ct > 0 \,\, \mathrm{and} \,\, z<0: &
T^+= 0\,&
T^-= \frac{\epsilon}{L} \delta(x)\delta(y)\chi(ct+z)
\end{array}
\end{equation}
where $\chi$ is a characteristic function (normalized $\chi^2=\chi$) which -- for any given time $t$ -- encodes the momentary extension and location of the laser pulses on the $z$-axis. It is explicitly given as
\begin{equation}
\nonumber \chi(x)=\left\{
\begin{array}{ccc}
0 &:& x\le 0\\
1 &:& 0 < x < L\\
0 &:& L \le x\,.
\end{array}
\right.
\end{equation}
The linearized Einstein equation (\ref{eq:linearizedeinstein}) can be solved by the retarded potential as (see \cite{Misner1973} page 445)
\begin{equation}\label{eq:retarded}
h_{\mu\nu}(x,y,z,t)
=
\frac{4 G}{c^4} \int \frac{
\left(T_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}{T_\alpha}^\alpha\right)(x',y',z',t_\mathrm{ret})}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}} dx'dy'dz'\,,
\end{equation}
with $t_\mathrm{ret}$ the retarded time, $t_\mathrm{ret}=t-\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}/c$. The retarded potential for the energy momentum tensor of the point particle is easily calculated. The pulses define world sheets restricted to the $t$-$z$-plane starting from the emission points.
\begin{figure}[h]
\includegraphics[width=10cm,angle=0]{Regions_decay.pdf}
\caption{ The pulses define world sheets restricted to the $t$-$z$-plane. The world sheets touch along the world line of the emitter between the points A and B which correspond, respectively, to the start of the pulse emission and the end of the emission. The future directed light cones of A and B define the spacetime regions $I$-$III$ with qualitatively different metric perturbations. Region $I$ is the pre-emission cone, region $II$ is the emission cone and region $III$ is the post-emission cone.
\label{fig:Regions}}
\end{figure}
The retarded solution (\ref{eq:retarded}) for the pulses is given by the intersection of the backward light cone of the spacetime point $x$ with the world sheets. Hence, there are three different regions defined by the future directed light cones of the emission process. A spacetime diagram illustrating this situation is given in Figure \ref{fig:Regions}. We call these regions the pre-emission cone, the emission cone and the post-emission cone, respectively.
From the retarded potential (\ref{eq:retarded}), we obtain the metric perturbation
\begin{equation}\label{eq:gmunudecay}
h_{\mu\nu}=h^{s}_{\mu\nu}+h^{+}_{\mu\nu}+h^{-}_{\mu\nu}
\end{equation}
where the three terms $h^{s}_{\mu\nu}$, $h^{+}_{\mu\nu}$ and $h^{-}_{\mu\nu}$ correspond to the point particle, the laser pulse propagating in positive $z$-direction and the laser pulse propagating in negative $z$-direction, respectively. In the coordinates $(ct,x,y,z)$, they have the non-zero components $h^{s}_{00}=h^{s}_{xx}=h^{s}_{yy}=h^{s}_{zz}=h^s$, $h^{+}_{00}=h^{+}_{zz}=-h^{+}_{z0}=-h^{+}_{0z}=h^+$ and $h^{-}_{00}=h^{-}_{zz}=h^{-}_{z0}=h^{-}_{0z}=h^-$.
The function $h^{s}$, $h^{+}$ and $h^{-}$ have the following form:
\begin{equation}\label{eq:gfunctions}
\begin{array}{lllll}
\textrm{For }ct-r\le 0: & h^s=\frac{mc^2}{2c^4}\frac{4G}{r}\,, &
h^+=0\,, &h^-=0\\
\textrm{For }0 < ct-r < L: & h^s=\frac{1}{2c^4} \left(mc^2-2\epsilon\frac{ct-r}{L}\right)\frac{4G}{r}\,, &
h^+= \frac{\epsilon}{c^4}\frac{4G}{L} \ln\frac{ct-z}{r-z}\, &
h^-= \frac{\epsilon}{c^4}\frac{4G}{L} \ln\frac{ct+z}{r+z}\\
\textrm{For }L \le ct-r: & h^s= \frac{1}{2c^4} \left(mc^2-2\epsilon\right)\frac{4G}{r}\, &
h^+= \frac{\epsilon}{c^4}\frac{4G}{L} \ln\frac{ct-z}{ct-L-z}\,&
h^-= \frac{\epsilon}{c^4}\frac{4G}{L} \ln\frac{ct+z}{ct-L+z}
\end{array}
\end{equation}
with $r=(x^2+y^2+z^2)^{\frac{1}{2}}$.
We find that $h_{\mu\nu}$ coincides with the Schwarzschild metric, linearized in Schwarzschild coordinates, until the start of the emission at $t=0$.
\begin{figure}[h]
\includegraphics[width=6cm,angle=0]{atom_field_-Lhalf_Greys.png}
\includegraphics[width=7.2cm,angle=0]{atom_field_Lover1point3_Greys.png}
\caption{\label{fig:hval} The plots show one component of the metric perturbation $h_{00}$ for a point particle of mass $mc^2=3\epsilon$ that emits two laser pulses of energy $\epsilon$ and length $L$ in the coordinates $(ct,x,y,z)$ in the $(x,y)$-plane for different times $t$. $h_{00}$ is normalized to units of $\kappa$ and then the logarithm of the logarithm is taken. The $mc^2$ of the emitter is chosen close to the energy of the photons $\epsilon$ such that the change in the gravitational field of the emitter due to the emission becomes visible. The model allows arbitrary masses as long as they are small enough to allow for the use of linearized gravity. In a), we see the metric spherical symmetrical perturbation due to the massive point particle at rest at $(z,x)=(0,0)$. In b), we see the effect of the two light pulses on the metric expanding from the point of their emission at $(z,x)=(0,0)$ after the emission.
}
\end{figure}
By direct calculation, it can be checked easily that (\ref{eq:gmunudecay}) fulfills the Lorentz gauge condition (\ref{eq:lorentzgauge}).
In the next section, we will investigate the physical effect due to the emission we modeled.
\section{Curvature and tidal forces}
\label{sec:curvature}
It is convenient to investigate the Riemann curvature tensor to find out if and where a physical effect of the metric perturbation (\ref{eq:gmunudecay}), in principle, could be measured. If the Riemann curvature tensor vanishes in a spacetime region, a coordinate transformation can be found in which the metric $g$ looks like the Minkowski metric $\mathrm{diag}(-1,1,1,1)$ (see Section 13.9 in \cite{Straumann2004} and proposition 2.11 of \cite{Atiyah1973}). This means that there is no measurable physical effect.
In first order in the metric perturbation $h_{\mu\nu}$, the Riemann curvature tensor takes the form
\begin{equation}\label{eq:defriemann}
R_{\nu\rho\sigma\alpha}=\frac{1}{2}\left(\partial_{\rho}\partial_{\sigma}h_{\nu\alpha}-\partial_{\nu}\partial_{\sigma}h_{\rho\alpha}-\partial_{\rho}\partial_{\alpha}h_{\nu\sigma}+\partial_{\alpha}\partial_{\nu}h_{\rho\sigma}\right)\,.
\end{equation}
$R_{\nu\rho\sigma\alpha}$ has only $20$ independent components. This can be seen from its symmetries $R_{\nu\rho\sigma\alpha}=-R_{\rho\nu\sigma\alpha}=-R_{\nu\rho\alpha\sigma}$ and $R_{\nu\rho\sigma\alpha}=R_{\sigma\alpha\nu\rho}$ and the Bianchi identity $R_{\nu\rho\sigma\alpha}+R_{\nu\sigma\alpha\rho}+R_{\nu\alpha\rho\sigma}=0$ it fulfills. Due to these symmetries, the Riemann tensor is invariant under linearized coordinate transformations $x^\mu\rightarrow x^\mu+\xi^\mu$, where $\xi^\mu$ is assumed to be small, and terms of higher than linear order in $\xi^\mu$ are neglected.
The Riemann tensor ${R^{\mu}}_{\rho\sigma\alpha}$ has a direct physical and geometrical interpretation. It appears naturally in the geodesic deviation equation for the relative acceleration between two infinitesimally close geodesics $\gamma(\lambda)$ and $\gamma'(\lambda)=\gamma(\lambda)+s(\lambda)$ parameterized by $\lambda$:
\begin{equation}\label{eq:geodesicdev}
a^\mu=\frac{D^2s^\mu}{d\lambda^2}={R^{\mu}}_{\rho\sigma\alpha}(x)\dot{\gamma}^\rho \dot{\gamma}^\sigma s^\alpha\,,
\end{equation}
where $s$ is the separation vector between the geodesics and $D/d\lambda=\dot\gamma^\mu \nabla_\mu$ is the covariant derivative along the geodesic $\gamma(\lambda)$. Equation (\ref{eq:geodesicdev}) can be interpreted as the effect of tidal forces on neighboring test particles.
The linearized Riemann curvature tensor (\ref{eq:defriemann}) corresponding to a linear combination of metric perturbations can be written as a linear combination of curvature tensors. Therefore, we obtain
\begin{equation}\label{eq:curvaturesep}
R_{\mu\nu\rho\sigma}= R^s_{\mu\nu\rho\sigma} + R^+_{\mu\nu\rho\sigma} + R^-_{\mu\nu\rho\sigma}
\end{equation}
with the curvature tensors corresponding to $h^s_{\mu\nu}$, $h^+_{\mu\nu}$ and $h^-_{\mu\nu}$, respectively. Expressions for the non-vanishing, independent components of the three terms in equation (\ref{eq:curvaturesep}) are given in Appendix A. Using the Lorentz gauge condition (\ref{eq:lorentzgauge}), we can write the only non-vanishing, independent components of the whole curvature as
\begin{equation}
\begin{array}{lllllll}
R_{aiaj}&=&-\frac{1}{2}\partial_i\partial_j(h^s + h^- + h^+)-\frac{1}{2}\delta_{ij}\partial_a^2 h^s&\quad & R_{xyxy}&=&-\frac{1}{2}\,\left(\partial_x^2+\partial_y^2 \right)h^s\\
R_{0izj}&=&-\frac{1}{2}\partial_i\partial_j(h^- - h^+)-\frac{1}{2}\,\delta_{ij}\partial_z\partial_0 h^s &\quad & R_{0zai}&=& -\frac{1}{2} \partial_i\partial_a h^s \\
& & & & R_{0z0z}&=&\frac{1}{2}\,(\partial_0^2-\partial_z^2) h^s
\end{array}
\end{equation}
where $a=0,z$ and $i,j\in \{x,y\}$.
We see that $R_{0z0z}$, $R_{0z0i}$ and $R_{0zzi}$ are of the order of derivatives of $h^s$. In order to evaluate them we must specify the creation process. This means, for example, that we cannot predict the deflection of a test particle due to one of the laser pulses along the $z$-axis in general, without first specifying the context of the laser pulse emission. The necessity to specify the context can be seen by investigating the geodesic deviation equation (\ref{eq:geodesicdev}) \cite{Raetzel2016pulse}. For expressions for the geodesic deviation due to the emitting particle and the two laser pulses see Appendix B.
\begin{figure}[h]
\includegraphics[width=6.5cm,angle=0]{atom_field_Lover1point3_gradgrad_0101.png}
\includegraphics[width=6.5cm,angle=0]{atom_field_Lover1point3_gradgrad_0131.png}
\includegraphics[width=6.5cm,angle=0]{atom_field_Lover1point3_gradgrad_0303.png}
\includegraphics[width=6.5cm,angle=0]{atom_field_Lover1point3_gradgrad_3131.png}
\caption{\label{fig:curve} The plots show the curvature components $R_{0x0x}$, $R_{0xzx}$, $R_{0z0z}$ and $R_{zxzx}$ for the metric perturbation $h_{\mu\nu}$ induced by a point particle of mass $mc^2=3\epsilon$ that emits two laser pulses of energy $\epsilon$ in the coordinates $(ct,x,y,z)$ in the $(x,z)$-plane for different times $t$. The logarithm of value of the curvature components is encoded in the opacity of the color. Red is a negative value of curvature component and blue a positive value. White stands for zero.}
\end{figure}
Plots for some of the curvature components are given in Figure \ref{fig:curve}. The components presented in the plots as well as all other components share the following features: Inside a sphere, which is expanding with the speed of light, the curvature is that of a point particle with the reduced mass $m-2\epsilon/c^2$. This sphere corresponds to the end of the emission process, when the point particle hast lost the mass $2\epsilon/c^2$, due to the emission of the laser pulses. The sphere is the intersection of the spatial plane of constant $ct$ with the post-emission cone (see Figure \ref{fig:Regions}).
We can distinguish two further regions. The first is the intersection of the $ct=const.$-plane with the pre-emission cone. Here, the gravitational field is that of the emitter with mass $m$. The spherical shell between the first and the second spherical regions is the intersection of the $ct=const.$-plane with the emission cone. It is only here that we find a gravitational effect of the laser pulses since only here they contribute to the curvature. We see that the gravitational field of light (corresponding to our model and the framework of linearized gravity) is only due to its emission. In the next section, we will investigate the acceleration of a test particle due to the metric perturbation (\ref{eq:gmunudecay}) in the emission cone.
\section{Acceleration of a test particle at rest}
\label{sec:acc}
In this section, we investigate the acceleration experienced by a test particle at rest with respect to the emitter. In the coordinates $(ct,x,y,z)$, the geodesic equation governing the trajectories $\gamma$ of freely falling test particles is given in first order in the metric perturbation as
\begin{equation}\label{eq:geodesic}
\ddot{\gamma}^\mu=-\left(\eta^{\mu\nu}\left(\partial_\rho h_{\nu\sigma} -\frac{1}{2}\partial_\nu h_{\rho\sigma}\right)-h^{\mu\nu}\left(\partial_\rho \eta_{\nu\sigma} -\frac{1}{2}\partial_\nu \eta_{\rho\sigma}\right)\right)\dot{\gamma}^\rho \dot{\gamma}^\sigma\,.
\end{equation}
We will only consider terms in the metric perturbation that contribute to the curvature. All other terms can be canceled by a linearized coordinate transformation which changes the coordinates only by terms that are of the order $\kappa=\frac{4 G\epsilon}{c^4L}$. Hence, the effect of this coordinate transformation on equation (\ref{eq:geodesic}) is of order $\kappa^2$ and can be neglected as a contribution of higher order. Then, in the emission cone, the functions $h^s$, $h^+$ and $h^-$ are given as
\begin{equation}\label{eq:hfunccurv}
h^s=\kappa \frac{\mu - ct + r}{r}\,,\quad h^+= -\kappa(\ln r +\ln(1-cos\theta))\,, \quad h^-= -\kappa(\ln r + \ln(1+cos\theta))\,,
\end{equation}
expressed in terms of the spherical coordinates $z=r\cos\theta$, $x=r\cos\phi\sin\theta$ and $y=r\sin\phi\sin\theta$ and using the definition $\mu=\frac{Lmc^2}{2\epsilon}$. In these coordinates, the Minkowski metric has the form $\eta=\mathrm{diag}(-1,1,r^2,r^2\sin^2\theta)$.
For a test particle at rest we have $\dot\gamma=(c,0,0,0)$, and the geodesic equation (\ref{eq:geodesic}) becomes
\begin{equation}\label{eq:geodesicrest}
\ddot{\gamma}^\mu=-c^2\eta^{\mu\nu}\left(\partial_0 h_{\nu0} -\frac{1}{2}\partial_\nu h_{00}\right)\,.
\end{equation}
This means that only $\partial_a h_{00}$ and $\partial_0 h_{a0}$ with $a=x,y,z$ contribute to the spatial part of $\ddot{\gamma}$. From the equations (\ref{eq:hfunccurv}), we see that the metric perturbation is time independent, and the contributions of $\partial_0 h_{a0}$ vanish.
It is interesting to consider the contributions of the massive particle and the laser pulses separately. We will start with the contribution of the massive point particle. We find from equation (\ref{eq:geodesicrest}) that a test particle at rest experiences a radial acceleration
\begin{equation}\label{eq:ddotr}
\ddot r^s = \frac{c^2}{2}\partial_r h^s = -\frac{c^2}{2}\kappa \frac{\mu-ct}{r^2}
\end{equation}
inside the emission cone due to the massive point particle. Interestingly, it contains an acceleration away from the origin as we will see in the following. In the emission cone, $ct$ lies between $r$ and $r+L$. We can define the retarded time $t_\mathrm{ret}=t - r/c$ with $0\le ct_\mathrm{ret}\le L/c$ in the emission cone, and we find
\begin{equation}\label{eq:pointpartacc}
\ddot r^s = -\frac{c^2}{2}\kappa \left( \frac{\mu-ct_\mathrm{ret}}{r^2} - \frac{1}{r} \right) = - \frac{G}{r^2}\left(m - \frac{2\epsilon}{c^2} \frac{ct_\mathrm{ret}}{ L}\right) + \frac{2G\epsilon}{c^2L}\frac{1}{r}
\end{equation}
The first term in (\ref{eq:pointpartacc}) is an acceleration proportional to the mass of the point particle $m - \frac{2\epsilon}{c^2}\frac{ct_\mathrm{ret}}{L}$ at the retarded time $t_\mathrm{ret}$. This acceleration is proportional to $1/r^2$ and corresponds to the Newtonian attraction of the massive point particle. It gradually decreases as the emission cone passes the position of the test particle in the retarded time $t_\mathrm{ret}=ct-r$ corresponding to the position of the test particle.
The second part gives rise to an acceleration away from the origin. This acceleration is proportional to 1/r, where r is the distance from the point particle. We can call it a gravitational induction force as it is associated with the change of the gravitational field due to the mass loss of the emitter.
For the contribution of the two laser pulses, we find from equation (\ref{eq:geodesicrest}) for the spatial part of the acceleration
\begin{equation}\label{eq:laseracc}
\left(\begin{matrix} \ddot r^p \\ r\ddot \theta^p \end{matrix}\right) =\frac{c^2}{2} \left(\begin{matrix} \partial_r (h^+ + h^-) \\ \frac{1}{r}\partial_\theta (h^+ + h^-) \end{matrix}\right) = -\frac{4G\epsilon}{c^2 L } \frac{1}{\rho}\left(\begin{matrix} \sin\theta \\ \cos\theta \end{matrix}\right)\,,
\end{equation}
where $\rho=\sqrt{x^2+y^2}=r\sin\theta$. This acceleration always points towards the $z$-axis.
For long lifetimes after the pulse emission or close to its trajectory, the acceleration in equation (\ref{eq:laseracc}) coincides with the attraction experienced by a massive test particle at rest due to a single laser pulse.
Note that the absolute value of the radial part of the acceleration due to the laser pulses in equation (\ref{eq:laseracc}) is exactly twice as large as the induction force (\ref{eq:pointpartacc}) due to the mass change of the point particle, the second term in equation (\ref{eq:pointpartacc}). Since they are of opposite sign, the half of the radial acceleration due to the laser pulses is canceled and the total radial acceleration is
\begin{equation}\label{eq:radacc}
\ddot r =\ddot r^s + \ddot r^p =- \frac{G}{r^2}\left(m - \frac{2\epsilon}{c^2} \frac{ct_\mathrm{ret}}{ L}\right)- \frac{2G\epsilon}{c^2L}\frac{1}{r}\,.
\end{equation}
Hence, the non-Newtonian part of the radial acceleration, proportional to $\frac{1}{r}$, is attractive. Since the emission is not isotropic, there is an acceleration in the $\theta$-direction. It is only due to the pulses and given as
\begin{equation}\label{eq:rddottheta}
r\ddot \theta = r\ddot \theta^p = \frac{c^2}{2}\frac{1}{r}\partial_\theta h_{00} =-\frac{c^2\kappa}{2r} \left(\frac{\sin{\theta}}{1-\cos\theta}-\frac{\sin{\theta}}{1+\cos\theta}\right) = -\frac{4G\epsilon}{c^2 L }\frac{1}{r} \cot\theta\,.
\end{equation}
In the next section, we want to investigate the acceleration of a massless test particle witnessing the emission process.
\section{Acceleration of massless test particles}
\label{sec:masslesstest}
For a massless test particle traveling in the -$z$-direction, the situation is very different than for a particle at rest. The $4$-velocity vector is $\dot\gamma=(c,0,0,-c)$. Hence, we find with the geodesic equation (\ref{eq:geodesic})
\begin{equation}\label{eq:vecxhsh+}
\ddot{\vec{\gamma}}=c^2\left[\left(\begin{matrix} \partial_\rho h^s \\ \partial_0 h^s \end{matrix}\right) +2 \left(\begin{matrix}
\partial_\rho h^+ \\ \partial_0 h^+
\end{matrix} \right) \right]\,,
\end{equation}
where we defined the acceleration vector as $\ddot{\vec{\gamma}}:=(\ddot \rho,\ddot z)$ in the cylindrical coordinates $(ct,\rho,\phi,z)$ with $x=\rho\cos\phi$ and $y=\rho\sin\phi$. $\ddot \rho$ expresses the transversal acceleration and $\ddot z$ the longitudinal acceleration with respect to the $z$-axis. The second laser pulse does not contribute to the acceleration since the massless test particle propagates parallel to the second laser pulse and into the same direction.
Again neglecting all terms that do not contribute to the curvature, we have
\begin{eqnarray}
h^s &=& \kappa \frac{\mu - ct + r}{r}\\
h^+ &=& -\kappa \ln(r-z)\,
\end{eqnarray}
in the emission cone. Then, evaluation of the expressions in (\ref{eq:vecxhsh+}) gives
\begin{eqnarray}
\ddot{\vec{\gamma}}&=&-\frac{\kappa c^2}{r}\left(\begin{matrix} \left(\frac{\mu - ct}{r^2} + 2\frac{1}{r-z}\right)\rho \\ 1\end{matrix}\right)
\end{eqnarray}
Note, that the contribution of the laser pulse to the acceleration is only transversal. With the parameterization $ct_\mathrm{ret}=ct-r$ in the emission cone we find
\begin{eqnarray}\label{eq:restacc}
\ddot{\vec{\gamma}}&=&-\frac{2G}{r^2}\left(m - \frac{2\epsilon}{c^2} \frac{ct_\mathrm{ret}}{L}\right)\left(\begin{matrix} \frac{\rho}{r} \\ 0 \end{matrix}\right)-\frac{4G\epsilon}{c^2L }\frac{1}{r}\left(\begin{matrix}\frac{r+z}{r-z}\frac{\rho}{r} \\ 1 \end{matrix}\right)\,.
\end{eqnarray}
In equation (\ref{eq:restacc}), there are two terms with differing dependence on the radial distance $r$. The term that is proportional to $1/r$ is due to the emission of radiation and the change of mass of the emitter. The first term in (\ref{eq:restacc}) is the gravitational acceleration of light by a massive object of mass $\left(m - \frac{2\epsilon}{c^2} \frac{ct_\mathrm{ret}}{L}\right)$. The factor $2$ in comparison to the Newtonian force experienced by a massive particle at rest given in equation (\ref{eq:radacc}) is
known from general relativistic effects like gravitational lensing.
For very small $\rho/r$ and positive $z$, we find that only the gravitational effect of the laser pulse contributes, and we obtain
\begin{eqnarray}\label{eq:restaccclose}
\ddot{\vec{\gamma}}
&=&-\frac{16G\epsilon}{c^2L}\frac{1}{\rho} \left(\begin{matrix}
1 \\ 0
\end{matrix} \right)
\end{eqnarray}
\section{Isotropic emission and the relation to the Vaidya metric}
\label{sec:compvaidya}
In the case of the emission of two counter propagating laser pulses, light is emitted with directional preference. In this section, we will discuss the isotropical case and its relation to the metric due to a spherically symmetrical, isotropically radiating massive object; the Vaidya metric.
In the limit of infinitely many laser pulses of infinitely small energy that are emitted isotropically in all spatial directions, the energy momentum tensor becomes
\begin{equation}\label{eq:energymomentumlinvaidya}
T_{\mu\nu}(x)= \left\{\begin{array}{ccc}
0 &:& ct-r\le ct\\
\frac{P}{ 4\pi r^2c}l_\mu(x) l_\nu(x) &:& 0 < ct-r < L\\
0 &:& L \le ct-r
\end{array}\right.
\end{equation}
where $P$ is the total radiation power and $l_\mu(x)$ is the four momentum of a light ray that is moving radially outwards at the spacetime point $x$ and that fulfills the normalization condition $l_0=-1$. We define the retarded time $u=ct-r$ and the advanced time $v=ct+r$. In the set of coordinates $(u,v,\theta,\phi)$, the only non-vanishing component of the energy momentum tensor (\ref{eq:energymomentumlinvaidya}) is $T_{uu}=\frac{P}{4\pi r^2c}$. We define the mass of the emitter as $M(u)=m-Pu/c^3$. Then, we can identify $P$ with the rate of mass loss of the central particle $-c^3\partial_u M(u)$, and we find
\begin{equation}\label{eq:emvaidya}
T_{uu}=-\frac{c^2\partial_u M(u)}{4\pi r^2}\,.
\end{equation}
A solution of the full Einstein equations to this energy momentum tensor for general monotonously decreasing functions $M(u)$ is the retarded Vaidya metric. It was derived as the spacetime induced by a radiating spherically symmetric massive object in\cite{Vaidya1951} by Vaidya. It is given by the line element (see \cite{Lindquist1965})
\begin{equation}\label{eq:vaidya}
ds^2=-\left(1-\frac{2GM(u)}{c^2r}\right)du^2-2dudr+r^2d\Omega^2=\frac{2GM(u)}{c^2r}du^2+ds^2_{\mathbb{M}}\,,
\end{equation}
where $d\Omega^2=d\theta^2+\sin^2\theta d\varphi^2$ and $ds^2_{\mathbb{M}}$ is the line element of Minkowski space. Note that the Vaidya metric becomes the Schwarzschild metric for constant mass, which takes its standard form $ds^2=-(1-\frac{2GM}{c^2r})dc\bar{t}^2+(1-\frac{2GM}{c^2r})^{-1}dr^2+r^2d\Omega^2$ using the time coordinate $c\bar t=u+r+2M\ln(r/2M-1)$.
\begin{figure}[h]
\includegraphics[width=10cm,angle=0]{vaidya_arrows_axes_exterior.pdf}
\caption{The Vaidya metric can be interpreted as a Schwarzschild metric with changing mass. The dependence of the mass term on the retarded time leads to an everywhere non-zero energy momentum tensor which can be interpreted as a radially outward directed flux of electromagnetic radiation moving in the Schwarzschild background (represented by radial arrows in the picture).
\label{fig:vaidya}}
\end{figure}
From the second expression for the line element in equation (\ref{eq:vaidya}) we see that a linearized version of the Vaidya metric is given by the metric perturbation $h^V_{\mu\nu}$ with the only non-vanishing component
\begin{equation}\label{eq:vaidyalin}
h^V_{uu}=\frac{2GM(u)}{c^2r}
\end{equation}
where $\chi(u)$ is the characteristic function encoding the envelope of the pulse. The metric perturbation $h^V_{\mu\nu}$ presented in equation (\ref{eq:vaidyalin}) does not fulfill the Lorentz gauge condition. Therefore, we are not able to derive it using the method of retarded potentials which we employed in Section \ref{sec:pulse}. However, the discrepancy is only due to a different choice of coordinates. Thus, the linearized Vaidya metric (\ref{eq:vaidyalin}) can really be seen as the isotropic version of the pulse emission metric $\eta_{\mu\nu}+h_{\mu\nu}$ with $h_{\mu\nu}$ given in equation (\ref{eq:gmunudecay}).
The equations of motion for a test particle in the full Vaidya metric (\ref{eq:vaidya}) can be derived from the metric $g_{\mu\nu}$ encoded in the line element $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ using the geodesic equation
\begin{equation}\label{eq:geodesiceq}
\ddot{\gamma}^\mu = -{\Gamma^\mu}_{\rho\sigma}\dot{\gamma}^\rho\dot{\gamma}^\sigma
\end{equation}
with subsidiary conditions $g_{\mu\nu}\dot{\gamma}^\mu\dot{\gamma}^\nu=-1$ for massive test particles, and $g_{\mu\nu}\dot{\gamma}^\mu\dot{\gamma}^\nu = 0$ for massless test particles. In equation (\ref{eq:geodesiceq}), the dot indicates the derivative with respect to the curve parameter, which is proper time $\tau$ for the time-like geodesics of massive particles. For the null-geodesics of massless particles, we will use coordinate time $t$. ${\Gamma^\mu}_{\rho\sigma}$ are the Christoffel symbols, ${\Gamma^{\mu}}_{\rho\sigma} = \frac{1}{2} g^{\mu\nu}\left(\partial_\rho g_{\sigma\nu} + \partial_{\sigma} g_{\nu\rho} -\partial_\nu g_{\rho\sigma}\right)$.
From the line element (\ref{eq:vaidya}), we obtain the equations of motion in spherical coordinates
\begin{eqnarray}\label{eq:geodesicdevexpl}
\nonumber \ddot{\gamma^r} & = & -\frac{GM(u)}{c^2r^2}\left(\left(1-\frac{2GM(u)}{c^2r}\right)-r\partial_u\ln M(u)\right)\left(\dot\gamma^u\right)^2-\frac{2GM(u)}{c^2r^2}\dot\gamma^u\dot\gamma^r +\\
\nonumber && + \,r\left(1-\frac{2GM(u)}{c^2r}\right)\left(\left(\dot\gamma^\theta\right)^2+\sin^2\theta\left(\dot\gamma^\phi\right)^2\right)
\\
\ddot{\gamma^\theta} & = & -\frac{2}{r}\dot\gamma^\theta\dot\gamma^r + \sin\theta\cos\theta \left(\dot\gamma^\phi\right)^2
\\
\nonumber \ddot{\gamma^\phi} & = & -\left(\frac{2}{r}\dot\gamma^r + \cot\theta\dot\gamma^\theta \right)\dot\gamma^\phi
\end{eqnarray}
and the subsidiary conditions
\begin{equation}\label{eq:subsed}
-\left(1-\frac{2GM(u)}{c^2r}\right) \left(\dot\gamma^u\right)^2 - 2\dot\gamma^u\dot\gamma^r + r^2 \left(\left(\dot\gamma^\theta\right)^2+\sin^2\theta\left(\dot\gamma^\phi\right)^2\right) =
\left\{\begin{array}{ccl}
-1 & \quad & \mbox{for time-like geodesics}
\\
0 & \quad & \mbox{for null-geodesics}
\end{array}\right.\,.
\end{equation}
For a massive particle initially at rest, we obtain from the subsidiary condition $\left(\dot\gamma^u\right)^2=\left(1-\frac{2GM}{c^2r}\right)^{-1}$ and, hence,
\begin{equation}\label{eq:restaccvaidya}
\ddot{\gamma^r} = -\frac{GM(u)}{c^2r^2} + \left(1-\frac{2GM(u)}{c^2r^2}\right)^{-1}\frac{G\partial_u M(u) }{c^2r}
\end{equation}
For $M(u)=m-Pu/c^3$ and small mass $m$, equation (\ref{eq:restaccvaidya}) becomes equation (\ref{eq:radacc}) which we derived for the emission of two counter-propagating laser pulses. The first term in equation (\ref{eq:restaccvaidya}) is the Newtonian gravitational acceleration due to a spherically symmetric massive object whose mass decreases as a function of the retarded time $u$. The second term decreases only as $1/r$ and is proportional to the radiation power of the spherical object $P=-c^3\partial_u M$. The term "gravitational induction force" for the second term was coined in \cite{Lindquist1965}. As we explained in Section \ref{sec:acc}, it contains contributions from the massive body and from the radiation that cancel partially. The contribution due to the mass loss of the massive body leads to an acceleration away from the body. The contribution of the radial radiation leads to an inward acceleration that is twice as strong as the acceleration due to the mass loss.
Due to the different dependence of the first and second term in equation (\ref{eq:restaccvaidya}) on the distance to the radiating object $r$, the second term of equation (\ref{eq:restaccvaidya}) becomes dominant for distances larger than $r_0=M/\partial_u M$. At this distance, the induction force is $G(\partial_u M)^2/Mc^2$. Hence, there needs to be a very high emission rate for the induction force to be significant. In \cite{Lindquist1965}, it is argued that the gravitational induction plays a role for radiating stars, when at all, only in catastrophic phases of gravitational collapse. For a decaying particle, however, $r_0$ is $L=cT$, where $T$ is the duration of the emission process. If the decay process takes a femtosecond, $r_0$ is only of the order $100\mathrm{nm}$. Hence, already very close to the decaying particle, the induction force becomes the dominant force during the decay. For a rubidium atom emitting a femtosecond light pulse at $780\mathrm{nm}$, we find that $r_0=15\mathrm{km}$. Hence, the radial induction force is much weaker than the Newtonian gravitational force for all relevant distances to the atom. In the an-isotropic case of the emission of two counter-propagating laser pulses, there is a non-spherical-symmetrical part of the force, which is only due to the laser pulses. As can be seen in the equations (\ref{eq:radacc}), (\ref{eq:rddottheta}) and (\ref{eq:restaccclose}), this is the only significant force at points far from the emitter but close to the trajectory of one of the pulses.
\section{Conclusions}
\label{sec:conclusions}
In this article, the situation of two counter-propagating laser pulses emitted from a massive point particle was considered. The corresponding metric perturbation in the framework of linearized gravity and the corresponding curvature were derived. It was shown that the curvature is that of a massive point particle at all spacetime points lying in the causal future of the end of the emission process and in the causal past of the beginning of the emission process. It was concluded that the laser pulses only contribute to the curvature during their emission and their absorption. This is in agreement with the results presented in \cite{Raetzel2016pulse}, where only one pulse was considered and the gravitational effect of the emitter was neglected. In contrast to the model presented in the former article, in the model presented in this article, the emitter itself is taken into account, and the continuity equation of general relativity is fulfilled.
The acceleration of test particles at rest and massless test particles propagating parallel to the trajectories of the two laser pulses in the gravitational field of the emitter and the two light pulses was calculated. It was found that the acceleration due the emitter can be separated into two qualitatively different terms. One is proportional to the inverse square of the distance to the emitter. This term is the Newtonian attraction due to the emitter and it decreases with its mass loss. The second part of the acceleration is proportional to the inverse of the distance to the emitter. It is the gravitational induction force due the change of mass of the emitter.
The effect of the laser pulses on the test particles is an acceleration which is proportional to the inverse of the distance to the axis on which the two pulses propagate. It always points in the direction of this axis.
In the model presented in this article, the only condition on the mass of the emitter is that it is small enough to allow for the application of the linearized Einstein equations. Hence, the emitter could be, for example, an atom or a laser device emitting in two opposite directions. In that case, the energy corresponding to the rest mass of the point particle is much larger than the energy of the two light pulses.
Another particular case included in the model presented in this article is the equivalence of the total energy of the emitted laser pulses $2\epsilon$ and the energy contained in the rest mass of the point mass $mc^2$. This is the case of a complete decay of the massive point particle. The resulting energy momentum tensor and the corresponding metric perturbation were already presented in \cite{Voronov1973} by Voronov and Kobzarev. The authors modeled the creation of massless particles by the decay of a massive scalar particle. This can be seen as a classical model for the decay of a Higgs boson into two photons.
The case of the isotropic emission of laser pulses from a massive point particle was discussed in Section \ref{sec:compvaidya}. It was argued that this situation corresponds to a particular Vaidya metric with a pulsed mass loss. The corresponding acceleration of massive test particles was investigated. Again, two terms with different dependence on the distance to the emitter were obtained: a Newtonian gravitational force proportional to the mass of the emitter that decreases with the distance square and a force proportional to the radiation power that decreases with the distance to the emitter. The latter was called the gravitational induction force. Since the induction force decreases more slowly than the Newtonian force, it becomes larger than the latter at some distance from the emitter. However, for emitting atoms it is much smaller than the Newtonian force for all experimentally relevant distances. For a decaying particle it becomes relevant already when very close to the particle. This agrees with the observation of \cite{Lindquist1965} that the induction force of a radiating star is only significant in catastrophic situations of gravitational collapse.
As explained above, the situation is different for the an-isotropic situation of two counter-propagating laser pulses emitted from a massive point particle which was considered in this article; there is a non-radial acceleration of test particles that is due only to the laser pulses.
\section*{Appendix A}
The only non-vanishing, independent components of the three curvature terms are
\begin{eqnarray}
\nonumber R^+_{0z0z}&=&-\frac{1}{2}\,(\partial_0+\partial_z)^2 h^+\\
R^+_{0z0i}=-R^+_{0zzi}&=&-\frac{1}{2} \partial_i(\partial_0+\partial_z)h^+\\
\nonumber R^+_{0i0j}=R_{+\,zizj}=-R^+_{0izj}&=&-\frac{1}{2} \partial_i \partial_j h^+\,,
\end{eqnarray}
and
\begin{eqnarray}
\nonumber R^-_{0z0z}&=&-\frac{1}{2}\,(\partial_0-\partial_z)^2 h^-\\
R^-_{0z0i}=R^-_{0zzi}&=&\frac{1}{2} \partial_i(\partial_0-\partial_z)h^-\\
\nonumber R^-_{0i0j}=R^-_{zizj}=R^-_{0izj}&=&-\frac{1}{2} \partial_i \partial_j h^-\,,
\end{eqnarray}
and
\begin{equation}
\begin{array}{lllllll}
R^s_{aiaj}&=&-\frac{1}{2}\,\left(\partial_a^2\delta_{ij}+\partial_i\partial_j \right)h^s &\quad & R^s_{0z0i}&=&-\frac{1}{2}\,\partial_i\partial_z h^s \\
R^s_{0z0z}&=&-\frac{1}{2}\,\left(\partial_0^2+\partial_z^2 \right)h^s &\quad & R^s_{0zzi}&=&\frac{1}{2}\,\partial_i\partial_0 h^s \\
R^s_{xyxy}&=&-\frac{1}{2}\,\left(\partial_x^2+\partial_y^2 \right)h^s & & R^s_{0izj}&=&-\frac{1}{2}\,\delta_{ij}\partial_z\partial_0 h^s
\end{array}
\end{equation}
where $a=0,z$ and $i,j=x,y$.
\section*{Appendix B}
For example, consider the case of two geodesics that start at the same time and move initially with $\dot \gamma=(\dot \gamma^0,0,\dot \gamma^y,\dot \gamma^z)$ and an infinitesimal separation $s=(0,-ds,0,0)$. We find from equation (\ref{eq:geodesicdev})
\begin{eqnarray}
a^0&=&ds((R_{0y0x}\dot \gamma^0+R_{0yyx}\dot \gamma^y)\dot \gamma^y+(R_{0zyx}+R_{0yzx})\dot \gamma^z\dot \gamma^y+(R_{0z0x}\dot \gamma^0+R_{0zzx}\dot \gamma^z)\dot \gamma^z)\\
a^x&=&-ds((R_{x00x}\dot \gamma^0+2R_{x0yx}\dot \gamma^y + 2R_{x0zx}\dot \gamma^z)\dot \gamma^0+2R_{xyzx}\dot \gamma^y \dot \gamma^z + R_{xyyx}(\dot \gamma^y)^2 + R_{xzzx}(\dot \gamma^z)^2)\\
a^y&=&-ds((R_{y00x}\dot \gamma^0 + R_{y0yx}\dot \gamma^y+(R_{y0zx}+R_{yz0x})\dot \gamma^z)\dot \gamma^0 + (R_{yzyx}\dot \gamma^y + R_{yzzx}\dot \gamma^z)\dot \gamma^z)\\
a^z&=&-ds((R_{z00x}\dot \gamma^0+(R_{z0yx}+R_{zy0x})\dot \gamma^y+R_{z0zx}\dot \gamma^z)\dot \gamma^0+ ( R_{zyyx} \dot \gamma^y + R_{zyzx}\dot \gamma^z) \dot \gamma^y)\,.
\end{eqnarray}
Only $\frac{D^2s^i}{d\tau^2}$ with $i=x,y$ is independent of the curvature terms $R_{z00j}$ and $R_{z0zj}$ for which the contributions of the metric perturbation due to the point mass cannot be neglected in general.
\section*{References}
\bibliographystyle{ieeetr}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,044 |
package com.teamwizardry.refraction.common.block;
import com.teamwizardry.librarianlib.features.base.block.tile.BlockModContainer;
import com.teamwizardry.refraction.common.tile.TileInvisibleRedstone;
import net.minecraft.block.Block;
import net.minecraft.block.material.Material;
import net.minecraft.block.properties.IProperty;
import net.minecraft.block.properties.PropertyInteger;
import net.minecraft.block.state.BlockStateContainer;
import net.minecraft.block.state.IBlockState;
import net.minecraft.entity.Entity;
import net.minecraft.item.ItemBlock;
import net.minecraft.tileentity.TileEntity;
import net.minecraft.util.EnumBlockRenderType;
import net.minecraft.util.EnumFacing;
import net.minecraft.util.math.AxisAlignedBB;
import net.minecraft.util.math.BlockPos;
import net.minecraft.world.IBlockAccess;
import net.minecraft.world.World;
import javax.annotation.Nonnull;
import javax.annotation.Nullable;
import java.util.List;
/**
* Created by Demoniaque.
*/
public class BlockInvisibleRedstone extends BlockModContainer {
public static final PropertyInteger POWER = PropertyInteger.create("power", 0, 15);
public BlockInvisibleRedstone() {
super("invisible_redstone", Material.AIR);
setDefaultState(getDefaultState().withProperty(POWER, 0));
setLightLevel(15);
setLightOpacity(15);
}
@Override
public boolean canConnectRedstone(IBlockState state, IBlockAccess world, BlockPos pos, EnumFacing side) {
return false;
}
@Nullable
@Override
public ItemBlock createItemForm() {
return null;
}
@SuppressWarnings("deprecation")
@Override
public @Nonnull
IBlockState getStateFromMeta(int meta) {
return getDefaultState().withProperty(POWER, meta);
}
@Override
public int getMetaFromState(IBlockState state) {
return state.getValue(POWER);
}
@Override
protected @Nonnull BlockStateContainer createBlockState() {
return new BlockStateContainer(this, POWER);
}
@Nullable
@Override
public IProperty<?>[] getIgnoredProperties() {
return new IProperty[]{POWER};
}
@Override
public boolean getWeakChanges(IBlockAccess world, BlockPos pos) {
return true;
}
@Override
@Deprecated
public boolean canProvidePower(IBlockState state) {
return true;
}
@Override
@Deprecated
public int getWeakPower(IBlockState blockState, IBlockAccess blockAccess, BlockPos pos, EnumFacing side) {
return blockState.getValue(POWER);
}
@Override
public boolean isCollidable() {
return false;
}
@Nullable
@Override
@Deprecated
public AxisAlignedBB getCollisionBoundingBox(IBlockState blockState, @Nonnull IBlockAccess worldIn, @Nonnull BlockPos pos) {
return Block.NULL_AABB;
}
@Override
@Deprecated
public void addCollisionBoxToList(IBlockState state, @Nonnull World worldIn, @Nonnull BlockPos pos, @Nonnull AxisAlignedBB entityBox, @Nonnull List<AxisAlignedBB> collidingBoxes, @Nullable Entity entityIn, boolean p_185477_7_) {
}
@Override
public boolean isReplaceable(IBlockAccess worldIn, BlockPos pos) {
return true;
}
@Override
public int getLightOpacity(IBlockState state, IBlockAccess world, BlockPos pos) {
return 0;
}
@Override
@Deprecated
public @Nonnull EnumBlockRenderType getRenderType(IBlockState state) {
return EnumBlockRenderType.INVISIBLE;
}
@Override
@Deprecated
public boolean isFullBlock(IBlockState state) {
return false;
}
@Override
@Deprecated
public boolean isOpaqueCube(IBlockState state) {
return false;
}
@Nullable
@Override
public TileEntity createTileEntity(@Nonnull World world, @Nonnull IBlockState iBlockState) {
return new TileInvisibleRedstone();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,338 |
class BaseEvent(object):
"""
Generic event class
"""
def __init__(self, value):
self.value = value
def __unicode__(self):
return "<{}: '{}'>".format(self.__class__, self.value)
class PlayNextSongEvent(BaseEvent):
"""
Play Next Song
"""
pass
class GetNextSongEvent(BaseEvent):
"""
Get next song to play
"""
pass
class UpdateMetadataEvent(BaseEvent):
"""
Update metadata event
"""
pass
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,287 |
\section{Introduction}
Cooperative control of multi-agent systems generally focuses on designing local control laws to achieve a global goal, such as reference-tracking~\cite{6595620}, consensus~\cite{Ren05}, or formation~\cite{ji2007distributed}. In addition to these objectives, various relative-motion constraints are often imposed to achieve stability, safety and integrity of the overall system, such as collision avoidance~\cite{dimarogonas2006feedback}, network connectivity~\cite{ji2007distributed,5948318}, or relative velocity constraint~\cite{6595620}.
This work is motivated by the desire to specify and achieve more structured and complex team behaviors than the listed ones. Particularly, following a recent trend, we consider Linear Temporal Logic (LTL) formulas as suitable descriptions of desired high-level goals. LTL allows to rigorously specify various temporal tasks, including periodic surveillance, sequencing, request-response, and their combinations. Furthermore, with the use of formal verification-inspired methods, a discrete plan that guarantees the specification satisfaction can be automatically synthesized, while various abstraction techniques bridge the continuous control problem and the discrete planning one. As a result, a generic hierarchical approach that allows for correct-by-design control with respect to the given LTL specification has been formulated and largely employed during the last decade or so in single-agent as well as multi-agent settings \cite{Din11, Fil12, Guo-icra14, Klo11, loizou-cdc2005, Lyg13, saha14, Jana14, Ulu13}.
In temporal logic-based multi-agent control, two different points of view can be taken: a top-down and a bottom-up. In the former one, a global specification captures requirements on the overall team behavior. Typically, the focus of decentralization is on decomposing the specification into tasks to be executed by the individual agents in a synchronized~\cite{Din11} or partially synchronized~\cite{Klo11, Ulu13} manner. A central monitoring unit then ensures that the composition of the local plans yields the satisfaction of the global goal.
In contrast, in bottom-up approach, each agent is assigned its own local task. The tasks can be independent ~\cite{Fil12, Guo-icra14} or partially dependent, involving requests for collaboration with the others~\cite{Jana14}. A major research interest here is the decentralization of planning and control procedures. For instance, in~\cite{Guo-icra14}, a decentralized revision scheme is suggested for a team in a partially-known workspace. In~\cite{Fil12}, gradual verification is employed to ensure that independent LTL tasks remain mutually satisfiable while avoiding collisions. In~\cite{Jana14}, a receding horizon approach is employed to achieve partially decentralized planning for collaborative tasks. In~\cite{saha14}, the authors propose a compositional motion planning
framework for multi-robot systems under safe LTL specifications.
In this work, we tackle the multi-agent control problem under local LTL tasks from the bottom-up perspective. Even though the local tasks are mutually independent, the agents within a multi-agent group are often more than a collection of stand-alone systems. Instead, they are subject to dynamic constraints with their neighboring agents and in such a case, integration of the continuous motion control with the high-level discrete network structure control is essential~\cite{6595620,5948318}. Here, the agents are subject to relative-distance constraints that need to be satisfied at all times. This coupling constraints make the team of agents {competitive} as each agent has to satisfy its local task and at the same time {cooperative} as they have to maintain the relative distance within the team.
We addressed a version of this problem in~\cite{Guo-cdc14}, where we proposed a dynamic leader-follower coordination and control scheme as a solution. In this work, however, we aim for a fully decentralized and communication-free solution that is applicable, e.g., to low-cost robotic systems equipped with range and angle sensors, but without communication units.
Our solution consists of three ingredients: a~standard discrete plan synthesis algorithm, a decentralized, hybrid, potential-field-based motion controller with two different control modes and a switching strategy between the two different continuous control modes.
In summary, we propose a fully communication-free decentralized hybrid control scheme for multi-agent systems under both complex high-level local LTL tasks and low-level relative-distance constraints. Specifically, our main contribution is the design of a two-mode communication-free control law that brings a group of agents to a region of interest.
The rest of the paper is organized as follows. Sec.~\ref{sec:prelims} introduces preliminaries. Sec.~\ref{sec:pf} formalizes the considered problem. Sec.~\ref{sec:solution} presents our solution in details. Sec.~\ref{sec:example} demonstrates the results in simulations. We conclude in~Sec.~\ref{sec:conc}.
\section{Preliminaries}\label{sec:prelims}
{\emph{Linear Temporal Logic (LTL) formula} over a set of \emph{atomic propositions} $\Sigma$ that can be evaluated as true or false is defined inductively according to the following rules:}
{
\begin{itemize}\itemsep-0.5ex
\item an atomic proposition $\sigma \in \Sigma$ is an LTL formula;
\item if $\varphi$ and $\psi$ are LTL formulas, then also $\neg \varphi$, $\varphi \wedge \psi$, $\bigcirc \varphi$, $\varphi \, \textsf U \, \psi$, $\Diamond \, \varphi$, and $\square \, \varphi$ are LTL formulas,
\end{itemize}
where $\neg $ (\emph{negation}), $\wedge$ (\emph{conjunction}) are standard Boolean connectives and $\bigcirc$ (\emph{next}), $\textsf U$ (\emph{until}), $\Diamond$ (\emph{eventually}), and $\square$~(\emph{always}) are temporal operators.}
The semantics of LTL is defined over the infinite words over~$2^\Sigma$. Informally, $\sigma \in \Sigma$ is satisfied on a word $w = w(1)w(2)\ldots$ if $\sigma \in w(1)$. Formula $\bigcirc \, \varphi$ holds true if $\varphi$ is satisfied on the word suffix that begins in the next position $w(2)$, whereas $\varphi_1 \, \textsf{U}\, \varphi_2$ states that $\varphi_1$ has to be true until $\varphi_2$ becomes true. Finally, $\Diamond \, \varphi$ and $\square \, \varphi$ are true if $\varphi$ holds on $w$ eventually and always, respectively. {For full details, see e.g.,~\cite{Bai08}.
{\emph{Syntactically co-safe LTL (sc-LTL)} is a subclass of LTL built without the \emph{always} operator $\Box$ and with the restriction that the \emph{negation} $\neg$ can be applied to atomic propositions only~\cite{Kup01}. In contrast to general LTL formulas, the satisfaction of an sc-LTL time can be achieved in a finite time, i.e., each word satisfying an sc-LTL formula $\varphi$ consists of a \emph{satisfying prefix} that can be followed by an arbitrary suffix.}{}
\medskip
{\emph{An undirected weighted graph}
is a tuple $G=(\mathcal{N},{E}, {W})$}, where
$\mathcal{N}=\{1,\ldots,N\}$ is a set of {nodes};
${E} \subseteq \mathcal{N}\times \mathcal{N}$ is a set of \emph{edges}; and
${W}:{E}\rightarrow \mathbb{R}^+$ is the weight {function}.
Each node~{$i$} has a {set of \emph{neighbors}} $\mathcal{N}_i=\{j\in \mathcal{N}\,|\,(i,j)\in {E}\}$.
A path from node $i$ to $j$ is a sequence of nodes starting with $i$ and ending with $j$ such that the consecutive nodes are neighbors.
$G$ is \emph{connected} if there is a path between any two nodes and $G$ is \emph{complete} if $E=\mathcal{N}\times \mathcal{N}$.
The Laplacian matrix $\textbf{H}$ of $G$ is an $N\times N$ positive semidefinite matrix:
$\textbf{H}(i,i)=\sum_{j\in \mathcal{N}_i}W(i,j), \forall i\in \mathcal{N}$;
$\textbf{H}(i,j)=W(i,j)$, $\forall (i,j)\in {E}$, and $\textbf{H}(i,j)=0$ otherwise.
{For a connected graph $G$,} $\textbf{H}$ has nonnegative eigenvalues~\cite{Godsil} and a single zero eigenvalue with the eigenvector $\textbf{1}_N$, where $\textbf{1}_N=[1,\ldots,1]^T$.
\medskip
{In this paper, each vector norm over $\mathbb{R}^n$ is the Euclidean norm~\cite{horn2012matrix}}.
{We use $|S|$ to denote the cardinality of a set $S$ and $v[i]$ to denote the $i$-{th} element of a vector $v$.}
\section{Problem Formulation}\label{sec:pf}
\subsection{Agent Dynamics and Network Structure}\label{system}
We consider a team of $N$ autonomous agents with unique identities (IDs) {$i\in\mathcal{N}=\{1,\ldots,N\}$}. They all satisfy the single-integrator dynamics:
\begin{equation}\label{dynamics}
\dot{x}_i(t) \triangleq u_i(t), \qquad i\in \mathcal{N}
\end{equation}
where $x_i(t), \, u_i(t) \in \mathbb{R}^2$ are the {respective} state and the control input of agent~$i$ at time $t> 0${, and} $x_i(0)$ is the given {initial} state. The agents are modeled as point masses without volume, {i.e.,} inter-agent collisions are not considered
Each agent has a sensing radius $r>0$, which is assumed to be identical for all agents. Namely, each agent can only observe
another agent' state if their relative distance is less than $r$.
Thus, given $\{x_i(0), i\in \mathcal{N}\}$, we
define the undirected graph
$G_0\triangleq(\mathcal{N}, E_0)$, where
$(i,\,j)\in E_0$ if $\|x_i(0)-x_j(0)\|<r$.
{We assume that the initial graph $G_0$ is connected.}
\subsection{{Task Specifications}}\label{taskspec}
Within the 2D workspace, each agent $i\in \mathcal{N}$ has a set of {$M_i$} regions of interest: $\Pi_i\triangleq\{\varpi_{i1},\ldots,\varpi_{iM_i}\}$.
These regions can be {of} different shapes, such as spheres, triangles,
or polygons. For simplicity of presentation, $\varpi_{i\ell}\in \Pi_i$ is here {represented by} a circular area around a point of interest:
\begin{equation}\label{regions}
\varpi_{i\ell} = \mathcal{B}(c_{i\ell}, r_{i\ell})=\{y\in \mathbb{R}^2 \mid \|y-c_{i\ell}\|\leq r_{i\ell}\},
\end{equation}
where $c_{i\ell}\in \mathbb{R}^2$ is the center;
$r_{i\ell}\geq r_{\min}$ is the radius, and $r_{\min} > 0$ is a given minimal radius for all regions.
{We assume that the regions of interest do not intersect and that the workspace is bounded, which {imply the following assumption} necessary for the design of the agents' controllers:}
\begin{assumption}\label{region-assump}
(I) $\|c_{i\ell_i}-c_{j\ell_j}\|>2\,r_{\min}$, $\forall i,\, j\in \mathcal{N}$, $\forall \varpi_{i\ell_i} \in \Pi_i $ and $\forall \varpi_{j\ell_j} \in \Pi_j$.
(II) $\|c_{i\ell}\|<c_{\max}$, $\forall i\in \mathcal{N}$ and $\forall \varpi_{i\ell}\in \Pi_i$, where $c_{\max}>0$ is a given constant.
\end{assumption}
{Each region of interest is associated with a subset of atomic propositions $\Sigma_i$ through
the labeling function $L_i:\Pi_i\rightarrow 2^{\Sigma_i}$.
Without loss of generality, we assume that $\Sigma_i \cap \Sigma_j = \emptyset$, for all $i,j \in \mathcal N$ such that $i \neq j$.
{We view the atomic propositions $L_i(\varpi_{i\ell})$ as \emph{services} that the agent $i$ can provide when being present in region $\varpi_{i\ell} \in \Pi_i$. Hence,}
{upon the visit to} $\varpi_{i\ell}$, the agent~$i$ {chooses among $L_i(\varpi_{i\ell})$ the subset of atomic propositions to be evaluated as true (i.e., the subset of services it provides among the available ones).
{We denote} by $\mathbf{x}_i(T)$ the \emph{trajectory} of agent $i$ during the time interval {$[0,T)$}, where $T>0$ and $T$ can be infinity.
{The trajectory $\mathbf{x}_i(T)$ is associated with a unique {finite or infinite} sequence $\mathbf{p}_i(T) \triangleq \pi_{i1}\pi_{i2}\dots$ of regions in $\Pi_i$ that the agent $i$ crosses, and with a finite or infinite sequence of time instants $t_{i0}'t_{i1}t_{i1}' t_{i2} t_{i2}'\cdots$ when $i$ enters/leaves the respective regions. {Formally,} for all $k \geq 1$:
$0 = t_{i0}' \leq t_{ik}\leq t_{ik}'< t_{ik+1} < T$, $x_i(t) \in \pi_{ik}$, $\pi_{ik}\in\Pi_i$, $\forall t\in [t_{ik},\,t_{ik}']$, and $x_i(t) \notin \varpi_{i\ell}$, $\forall \varpi_{i\ell} \in \Pi_i$ and $\forall t\in (t_{ik-1}',\,t_{ik})$.}
{The \emph{trace} corresponding to $\mathbf{x}_i(T)$ is a sequence of labels of the visited regions
$\textup{\texttt{trace}}_i(T) \triangleq L_i(\pi_{i1})L_i(\pi_{i2})\cdots, $
which represents the sequence of atomic propositions that \emph{can} be true (i.e., the services that can be provided) by the agent $i$ following $\mathbf{x}_i(T)$.}
{The \emph{word} the agent $i$ produces is a sequence of atomic propositions that actually \emph{are} evaluated as true (i.e., the actually provided services). Note that the agent's word and trajectory have to comply: if $\textup{\texttt{trace}}_i(T)$ is as above, then $\textup{\texttt{word}}_i(T) = w_{\ell_1}w_{\ell_2}\cdots$, where $w_{\ell_k} \subseteq L_i(\pi_{\ell_k})$, for all $k \geq 1$.}
\medskip
{The specification of the local task for
each agent $i\in \mathcal{N}$
is given as a general LTL or an sc-LTL formula} $\varphi_i$ over $\Sigma_i$ and captures requirements on the services to be provided. In this work, we do not focus on how the service providing is executed by an agent; we only aim at controlling an agent's motion to reach regions where these services are available.
{Formally, an infinite trajectory $\mathbf{x}_i(T)$ of an agent $i$ satisfies a given formula $\varphi_i$ if and only if there exists an infinite word $\textup{\texttt{word}}_i(T)$ that complies with $\mathbf{x}_i(T)$ and satisfies $\varphi_i$.}
\subsection{Problem Statement}
\begin{problem}\label{main-prob}
{Given a team of $N$ agents as in Sec.~{\ref{system}}, and their task specifications as in Sec.~\ref{taskspec}, design distributed control laws $u_i$, $\forall i\in \mathcal{N}$, {such that for $T \rightarrow \infty$}:
\begin{itemize} \itemsep-0.5ex
\item[(1)] $\mathbf{x}_i(T)$ satisfies $\varphi_i$; and
\item[(2)] $\|x_i(t)-x_j(t)\|<r$, $\forall (i,\,j)\in E_0$, $\forall t\in [0,\, T)$.
\end{itemize}}
\end{problem}
\section{Solution}\label{sec:solution}
{The proposed solution consists of three layers:
(i)
an offline synthesis of an discrete plan, i.e., a sequence of progressive goal regions for each agent;
(ii)
a distributed continuous control scheme guaranteeing that one of the agents reaches its progressive goal region in finite time while the relative-distance constraints are fulfilled at all time;
(iii)
a hybrid control layer, which monitors the discrete plan execution and switches between different continuous control modes to achieve the satisfaction of each LTL task. }
\subsection{Discrete Plan Synthesis}\label{synthesis}
{The discrete plan can be generated using standard techniques leveraging ideas from automata-based formal verification. Loosely speaking, an LTL or an sc-LTL formula $\varphi_i$ is first translated into a B\"uchi or a finite automaton, respectively. The automaton is viewed as a graph and analyzed using graph search algorithms. As a result, a word that satisfies $\varphi_i$ is obtained and mapped onto the sequence of regions to be visited. Current temporal logic-based discrete plan synthesis algorithms can accommodate various environmental constraints and advanced plan optimality criteria. We refer the interested reader to related literature, e.g.,~\cite{Bai08,Din11}.
}
{It can be shown that without loss of generality, the derived plan of an agent $i$ is in a prefix-suffix form}
$\tau_i = \tau_{i,\textup{pre}}(\tau_{i,\textup{suf}})^\omega$,
where {$\tau_{i,\textup{pre}}=(\pi_{i1},w_{i1}) \cdots(\pi_{i{k_i}},w_{ik_i})$ is the plan prefix, and $\tau_{i,\textup{suf}}=(\pi_{i{{k_i}+1}},w_{i{k_i}+1}) \cdots(\pi_{i{K_i}},w_{iK_i})$ is the periodical plan suffix};
$\pi_{ik}\in \Pi_i$ and $w_{ik}\subseteq L_i(\pi_{ik})$, $\forall k=1,\cdots, K_i$.
Simply speaking, $\tau_i$ represents the sequence of \emph{progressive goal regions} $\pi_{i1}\pi_{i2}\cdots$ and the {word, i.e., the sequence of services to be provided there $w_{i1}w_{i2}\cdots$ that satisfies $\varphi_i$}. {If $\{\varphi_i, i\in \mathcal{N}\}$ are all sc-LTL formulas, $\tau_{i,\textup{pre}}$ represents the satisfying prefix and the suffix $\tau_{i,\textup{suf}}$ can be disregarded.}
\subsection{Continuous Controller Design}\label{continuous-design}
Before stating the proposed control scheme, let us first introduce the notion of connectivity graph, which will allow us to handle the relative-distance constraints.
Recall that each agent has a limited sensing radius $r>0$ as mentioned in Sec.~\ref{system}. Let $\delta \in (0,\,r)$ be a given constant.
Then we define the connectivity graph $G(t)$ as follows:
\begin{definition}\label{edge}
Let $G(t)\triangleq(\mathcal{N}, E(t))$ denote the undirected time-varying connectivity graph at time $t\geq 0$, where $E(t)\subseteq \mathcal{N}\times \mathcal{N}$is the set of edges.
(I) $G(0)=G_0$;
(II)~At time $t > 0$, $(i,\, j)\in E(t)$ iff one of the following conditions hold: (1) $\|x_i(t)-x_j(t)\|\leq r -\delta $; or (2)~$r -\delta<\|x_i(t)-x_j(t)\|\leq r $ and $(i,j) \in E(t^-)$, where $t^-<t$ and $|t-t^-|\rightarrow 0$.
\end{definition}
Note that the condition (II) above guarantees that a new edge will only be added when the distance between two unconnected agents decreases below $r -\delta$.
In other words, there is a hysteresis effect when adding new edges to the connectivity graph.
Each agent $i\in \mathcal{N}$ has a time-varying set of neighbors ${\mathcal{N}}_i(t)=\{j\in \mathcal{N}\,|\,(i,\,j)\in E(t)\}$.
Note that the graph $G_0$ defined in Sec.~\ref{system} is assumed to be connected.
{Given that the progressive goal region at time $t$ is $\pi_{ig}= \mathcal{B}(c_{i\text{g}}, r_{i\text{g}})\in \Pi_i $}, we propose the following two control modes:
\noindent (1) the \emph{active} mode:
\begin{equation}\label{law1}
\textbf{C}_{act}: \quad u_i(t)\triangleq -\displaystyle d_i\, p_i -\sum_{j\in \mathcal{N}_i(t)}h_{ij}\,x_{ij},
\end{equation}
(2) the \emph{passive} mode:
\begin{equation}\label{law2}
\textbf{C}_{pas}: \quad u_i(t)\triangleq -\sum_{j\in \mathcal{N}_i(t)}h_{ij}\,x_{ij},
\end{equation}
where $x_{ij}\triangleq x_i-x_j$; $p_i\triangleq x_i-c_{i\text{g}}$;
and the coefficients $d_{i}$ and $h_{ij}$ are given by
\begin{equation}\label{d}
d_i \triangleq \frac{{\varepsilon^3}}{(\|p_i\|^2+\varepsilon)^2}+\,\frac{\varepsilon^2}{2\,(\|p_i\|^2+\varepsilon)};
h_{ij}\triangleq \frac{r^2}{(r^2-\| x_{ij}\|^2)^2},
\end{equation}
where $\varepsilon>0$ is a design parameter to be appropriately tuned.
We show in detail how to choose $\varepsilon$ in the sequel.
Note that both controllers in~\eqref{law1} and~\eqref{law2} are nonlinear and rely on only locally-available states: $x_i(t)$ and $\{x_j(t),j\in \mathcal{N}_i(t)\}$.
Assume that $G(T_s)$ is connected at time $T_s>0$. Moreover, assume that there are $N_a\geq 1$ agents within $\mathcal{N}$ that are in the \emph{active} mode obeying~\eqref{law1} with its goal region as $\pi_{i\text{g}}=\mathcal{B}(c_{i\text{g}},\,r_{i\text{g}})\in \Pi_i$; and the rest $N_p=N-N_a$ agents that are in the \emph{passive} mode obeying~\eqref{law2}.
For simplicity, denote by the group of active and passive agents $\mathcal{N}_a, \mathcal{N}_p \subseteq \mathcal{N}$ respectively.
Note that it is allowed that $N_a=N$ and $N_p=0$ when all agents are in the active mode.
In the rest of this section, we show that for \emph{any} allowed combination of $N_a > 1$ and $N_p < N$, by following the control laws~\eqref{law1} and~\eqref{law2}, \emph{one} active agent reaches its goal region within finite time $T_f\in (T_s,\,+\infty)$, while $\|x_i(t)-x_j(t)\|<r$, $\forall (i,\,j)\in E(T_s)$ and $\forall t\in [T_s,\,T_f]$.
\subsubsection{Relative-Distance Maintenance}\label{subsubsec:distance}
In this part, we show that the relative-distance constraints are always satisfied by following the control laws~\eqref{law1} and~\eqref{law2} for \emph{any} number of active and passive agents within the system following a potential-field based analysis.
We propose the following potential-field function:
\begin{equation}\label{Lyapunov}
V(t)\triangleq\frac{1}{2}\sum_{i\in \mathcal{N}}\sum_{j\in \mathcal{N}_i(t)} \phi_c (x_{ij} ) +b_i\sum_{i\in \mathcal{N}}\phi_g (x_{i} )
\end{equation}
where $\phi_c(\cdot)$ stands for an attractive potential to agent $i$'s neighbors and is given by:
\begin{equation}\label{potentialc}
\phi_c (x_{ij} )\triangleq\frac{1}{2}\,\frac{\| x_{ij}\|^2}{r^2-\| x_{ij}\|^2}, \qquad \|x_{ij}\|\in [0,\, r-\delta);
\end{equation}
while $\phi_g(\cdot)$ is an attractive force to agent $i$'s goal defined by:
\begin{equation}\label{potentialg}
\phi_g (x_{i} )\triangleq\frac{\varepsilon^2}{2}\,\frac{{\| p_i\|^2}}{\| p_i\|^2+\varepsilon} + \frac{\varepsilon^2}{4} \, \ln(\|p_i\|^2+\varepsilon),
\end{equation}
where function $\ln(\cdot)$ is the natural logarithm; $b_i\in \mathbb{B}$ indicates the agent $i$'s control mode. Namely, $b_i=1$, $\forall i\in \mathcal{N}_a$ and $b_i = 0$, $\forall i\in \mathcal{N}_p$.
It can be verified that the gradient of $V(t)$ from~\eqref{Lyapunov} with respect to $x_i$ is given by
\begin{equation}\label{gradient}
\begin{split}
\nabla_{x_i} V=\frac{\partial V}{\partial x_i} &= \nabla_{x_i}\phi_g (x_i )+\sum_{j\in \mathcal{N}_i} \nabla_{x_i}\phi_c (x_{ij} )\\
&= b_i\, d_i\, p_i+\sum_{j\in \mathcal{N}_i(t)} h_{ij}\,x_{ij}=-u_i.
\end{split}
\end{equation}
\begin{theorem}\label{connectivity}
$G(t)$ remains connected and no existing edges within $E(T_s)$ will be lost, namely $E(T_s)\subseteq E(t)$, $\forall t\geq T_s$.
\end{theorem}
\begin{proof}
Assume that the network $G(t)$ remains \emph{invariant} during the time period $[t_1,\, t_2)\subseteq [T_s,\,\infty)$.
Thus the neighboring sets $\{\mathcal{N}_i, i\in \mathcal{N}\}$ also remain invariant and $V(t)$ is differentiable for $t\in [t_1,\, t_2)$.
Then the time derivative of $V(t)$ is given by
\begin{equation}\label{derivative}
\begin{split}
\dot{V}(t)&=\sum_{i\in \mathcal{N}} \big( \nabla_{x_i} V\big)^T \, \dot{x}_i=\sum_{i\in \mathcal{N}} \big( \nabla_{x_i} V\big)^T \, u_i\\
&=-\sum_{i\in \mathcal{N}} \big{\|}b_i\,d_i\, p_i +\sum_{j\in \mathcal{N}_i(t)}h_{ij}\, x_{ij} \big{\|}^2\leq 0,
\end{split}
\end{equation}
meaning that $V(t)$ is non-increasing, $\forall t\geq T_s$. Thus $V(t)\leq V(T_s)<+\infty$ for $t\geq T_s$.
On the other hand, assume a \emph{new} edge $(p,\,q)$ is added to $G(t)$ at $t=t_2$, where $p,\, q\in \mathcal{N}$.
By Def.~\ref{edge}, $\|x_{pq}(t_2)\|\leq r-\delta$ and $\phi_c(x_{pq}(t_2))=\frac{(r-\delta)^2}{\delta(2r-\delta)}<+\infty$ since $0<\varepsilon<r$.
Denote by $\widehat{E}\subset \mathcal{N}\times \mathcal{N}$ the set of newly-added edges at $t=t_2$. Let $V(t_2^+)$ and $V(t_2^-)$ be the value of $V(t)$ before and after adding the set of new edges to $G(t)$ at $t=t_2$. We get
$
V(t_2^+) = V(t_2^-) + \sum_{(p,\,q)\in \widehat{E}}\phi_c(x_{pq}(t_2))
\leq V(t_2^-) + |\widehat{E}|\, \frac{(r-\delta)^2}{\varepsilon(2r-\delta)}<+\infty,
$
where we use the fact that $|\widehat{E}|$ is bounded as $\widehat{E}\subset \mathcal{N}\times \mathcal{N}$. Thus $V(t)<+\infty$ also holds when new edges are added.
Similar analysis can be found in~\cite{ji2007distributed}.
As a result, $V(t)<+\infty$ for $t\in [T_s,\, \infty)$. By Def.~\ref{edge}, one existing edge $(i,\,j)\in E(t)$ will be lost only if $x_{ij}(t)=r$. It implies that $\phi_c(x_{ij})\rightarrow +\infty$, i.e., $V(t)\rightarrow +\infty$ by~\eqref{Lyapunov}. By contradiction, we can conclude that new edges might be added but no existing edges will be lost, namely $E(T_s)\subseteq E(t)$, $\forall t\geq T_s$. If $G(T_s)$ is connected, then $G(t)$ remains connected for $\forall t\geq T_s$. It completes the proof.
\end{proof}
Note that Theorem~\ref{connectivity} holds also when $N_a=0$, i.e., there are no active agents, as \eqref{derivative} still holds when $b_i=0$, $\forall i\in \mathcal{N}$.
\subsubsection{Convergence Analysis}\label{subsubsec:convergence}
In this part, we aim at analyzing the convergence properties of the closed-loop system.
Since we have shown that $V(t)$ is non-increasing for all $t>T_s$ by Theorem~\ref{connectivity} above, by LaSalle's invariance principle~\cite{khalil2002nonlinear} we only need to find out the largest invariant set that $\dot{V}(t)=0$, which implies:
\begin{equation}\label{equi}
b_i\, d_i\,p_i +\sum_{j\in \mathcal{N}_i(t)}h_{ij}\,x_{ij} =0, \quad \forall i\in \mathcal{N}.
\end{equation}
Then we can construct one $N\times N$ diagonal matrix $\textbf{D}$ that $\textbf{D}(i,i)=b_i\,d_i$, $\forall i\in\mathcal{N}$ and $\textbf{D}(i,j)=0$, $i\neq j$ and $i,j\in \mathcal{N}$.
and another $N\times N$ matrix $\textbf{H}$ that $\textbf{H}(i,i)=\sum_{j\in \mathcal{N}_i}h_{ij}$, $\forall i\in \mathcal{N}$ and $\textbf{H}(i,j)=-h_{ij}$, $i\neq j$ and $\forall(i,\,j)\in E(t)$ while $\textbf{H}(i,j)=0$, $\forall (i,\,j)\notin E(t)$. Note that $h_{ij} > 0$ as $\|x_{ij}\|\in [0,\,r)$ by~\eqref{gradient}, $\forall (i,\,j)\in E(t)$.
As a result, $\textbf{H}$ is the Laplacian matrix of the graph $G(t)=(\mathcal{N}, E(t), W)$, where $W(i,\,j)=h_{ij}$, $\forall (i,\,j)\in E(t)$.
Then~\eqref{equi} can be written in vector form:
\begin{equation}\label{equi2}
\textbf{H}\otimes \textbf{I}_2 \cdot \mathbf{x} + \textbf{D} \otimes \textbf{I}_2 \cdot (\mathbf{x}-\mathbf{c})=0,
\end{equation}
where $\otimes$ is the Kronecker product~\cite{horn2012matrix}; $\mathbf{x}$ is the stack vector for $x_i$, $i\in \mathcal{N}$ and $\mathbf{x}[i]=x_i$;
$\textbf{I}_2$ is the identity matrix; $\mathbf{c}$ is the stack vector that $\mathbf{c}[i]=c_{i\text{g}}$ if $i\in \mathcal{N}_a$ and $\mathbf{c}[i]=\textbf{0}_2$ if $i\in \mathcal{N}_p$. Let $\mathcal{C}$ be the set of critical points satisfying~\eqref{equi2}, i.e.,
\begin{equation}\label{equiset}
\mathcal{C}\triangleq\{x\in \mathbb{R}^{2N}\,|\, \textbf{H}\otimes \textbf{I}_2 \cdot \mathbf{x} + \textbf{D} \otimes \textbf{I}_2 \cdot (\mathbf{x}-\mathbf{c})=0\}.
\end{equation}
Now we show that at the critical points all agent relative distances can be made arbitrarily small by reducing $\varepsilon$ and the corresponding connectivity graph is a complete graph.
\begin{lemma}\label{difference}
For all critical points $\mathbf{x}_c\in \mathcal{C}$, (I) $\|x_{ij}\|$ can be made arbitrarily small by reducing $\varepsilon$, $\forall (i,\,j)\in E(t)$;
(II) there exists $\varepsilon_0>0$ such that if $\varepsilon<\varepsilon_0$, then the connectivity graph $G(t)$ is complete.
\end{lemma}
\begin{proof}
(I) Consider the following equation for $\mathbf{x}_c\in \mathcal{C}$
$$
\sum_{(i,j)\in {E}(t)} h_{ij} \|x_{ij}\|^2
= \mathbf{x}^T_c \cdot (\textbf{H}\otimes \textbf{I}_2) \cdot \mathbf{x}_c.
$$
Combining the above equation with~\eqref{equi2}, we get
\begin{equation}\label{relativebound}
\begin{split}
&\sum_{(i,j)\in {E}(t)} h_{ij} \|x_{ij}\|^2 =-\mathbf{x}_c^T \cdot (\textbf{D}\otimes \textbf{I}_2)\cdot (\mathbf{x}_c-\mathbf{c})\\
& =-(\mathbf{x}_c-\mathbf{c})^T \cdot (\textbf{D}\otimes \textbf{I}_2)\cdot (\mathbf{x}_c-\mathbf{c})\\
&\quad \; \; -\mathbf{c}^T \cdot (\textbf{D}\otimes \textbf{I}_2)\cdot (\mathbf{x}_c-\mathbf{c})\\
& =-\sum_{i\in \mathcal{N}}b_i\,d_i\, \big(\|p_i\|^2 + c_{i\ell}^T \, p_i\big) \leq \sum_{i\in \mathcal{N}} b_i\, \|c_{i\ell}\|\,d_i\|p_i\|.
\end{split}
\end{equation}
Since $d_i\,\|p_i\|<\varepsilon\sqrt{\varepsilon}$ for $\|p_i\|\geq 0$, we get
\begin{equation}\label{relativebound2}
\sum_{(i,j)\in {E}(t)} h_{ij} \|x_{ij}\|^2 <N_a\,c_{\max}\,\varepsilon \sqrt{\varepsilon}\leq N\,c_{\max}\,\varepsilon \sqrt{\varepsilon},
\end{equation}
where $\|c_{i\ell}\|<c_{\max}$ is given in Assump.~\ref{region-assump}.
Thus $\forall (i,\,j)\in E(t)$, it holds that $h_{ij}\|x_{ij}\|^2<N\,c_{\max}\,\varepsilon \sqrt{\varepsilon}\triangleq \varsigma$. It can be verified that $h_{ij}\|x_{ij}\|^2$ is monotonically increasing as a function of $\|x_{ij}\|$.
This implies that $\forall (i,\,j)\in E(t)$,
$\|x_{ij}\|^2\leq r^2\,\varsigma$, or equivalently $\|x_{ij}\|^2\leq \varepsilon \sqrt{\varepsilon}\,\xi$,
where
\begin{equation}\label{xi}
\xi \triangleq r^2\, N\, c_{\max}.
\end{equation}
Thus $\|x_{ij}\|$ can be made arbitrarily small by reducing $\varepsilon$.
(II) Moreover, let $\varepsilon_0$ satisfy
\begin{equation}\label{v0}
(N-1)\sqrt{\varepsilon_0\sqrt{\varepsilon_0}\, \xi} <r-\delta.
\end{equation}
If $\varepsilon<\varepsilon_0$, then for {any} pair $(p,\,q)\in \mathcal{N}\times \mathcal{N}$, $\|x_{pq}\|$ satisfies
$$
\|x_{pq}\|= |x_p-x_1+x_1-x_2+\ldots -x_q|\leq (N-1) \sqrt{\varepsilon\sqrt{\varepsilon}\,\xi} <r-\delta,
$$
where we use two facts: there exists a path in $G(t)$ of maximal length $N$ from any node $p\in \mathcal{N}$ to another node $q$ as $G(t)$ remains connected for $t>T_s$ by Lemma~\ref{connectivity}; and $\|x_{ij}\|\leq \varepsilon \sqrt{\varepsilon}\,\xi$ from above, $\forall (i,\,j)\in E(t)$.
By Def.~\ref{edge} this implies $(p,\,q)\in E(t)$. Thus $G(t)$ is a complete graph.
\end{proof}
Before stating the convergence property, we need to define the following sets for all $i\in \mathcal{N}_a$:
\begin{equation}\label{si}
\mathcal{S}_i\triangleq \{\mathbf{x}\in \mathbb{R}^{2N}\,|\,\|\mathbf{x}-\mathbf{1}_N\otimes c_{i\ell}\|\leq r_S(\varepsilon)\},
\end{equation}
where $r_S(\varepsilon) \triangleq \sqrt{3N\,\varepsilon}+\sqrt{(N-1)\varepsilon\sqrt{\varepsilon}\, \xi}$ and $\xi$ is defined in~\eqref{xi}. Loosely speaking, $\mathcal{S}_i$ represents the neighbourhood around the goal region center of the active agent $i\in \mathcal{N}_a$. Furthermore, let $\mathcal{S}\triangleq\cup_{i\in \mathcal{N}_a} \mathcal{S}_i$ and $\mathcal{S}^{\neg}\triangleq\mathbb{R}^{2N} \setminus\mathcal{S}$.
In the following, we analyze the properties of the critical points within $\mathcal{S}$ and $\mathcal{S}^{\neg}$. More specifically:
by Lemma~\ref{away} there are no local minimal but saddle points within $\mathcal{S}^{\neg}$; by Lemma~\ref{isolated} these saddle points are non-degenerate,
(II) by Lemmas~\ref{distance}-\ref{localmini} all critical points within $\mathcal{S}$ are local minima.
To explore these properties, we compute the second partial derivatives of $V(t)$ with respect to $x_i$, which are given by
\begin{equation}\label{2gradient}
\begin{split}
\frac{\partial^2 V}{\partial x_i\partial x_i }&= b_i\, d_i \otimes \textbf{I}_2 + b_i\, d_i'\, p_i\cdot p_i^T \\
&+\sum_{j\in \mathcal{N}_i(t)} \big(h_{ij}\otimes \textbf{I}_2 + h_{ij}'\,x_{ij}\cdot x_{ij}^T\big)
\end{split}
\end{equation}
and
\begin{equation}\label{2gradient2}
\begin{split}
\frac{\partial^2 V}{\partial x_i\partial x_j }&= - h_{ij}\otimes \textbf{I}_2 - h_{ij}'\,x_{ij}\cdot x_{ij}^T, \qquad \forall j\neq i,
\end{split}
\end{equation}
where
\begin{equation}\label{hp}
d_i' =\frac{{-4\,\varepsilon^3}}{(\|p_i\|^2+\varepsilon)^3}+\frac{{-\,\varepsilon^2}}{(\|p_i\|^2+\varepsilon)^2},
\text{ and }
h_{ij}'= \frac{4\,r^2}{(r^2-\| x_{ij}\|^2)^3}.
\end{equation}
\begin{lemma}\label{away}
There are no local minima of $V$ within $\mathcal{S}^\neg$.
\end{lemma}
\begin{proof}
We prove this by showing that if a critical point $\mathbf{x}_c\in \mathcal{S}^{\neg}$ there always exists a direction $\mathbf{z}\in \mathbb{R}^{2N}$ at $\mathbf{x}_c$ such that the {quadratic form} $\mathbf{z}^T \nabla^2V \mathbf{z}$ is negative semi-definite.
Given a critical point $\mathbf{x}_c\in \mathcal{C}$ and $\mathbf{x}_c\in \mathcal{S}^\neg$, then by definition $\|\mathbf{x}-\mathbf{1}_N\otimes c_{i\ell}\|>r_S,\, \forall i \in \mathcal{N}_a$. On the other hand, for any $i\in \mathcal{N}_a$, we can bound $\|\mathbf{x}-\mathbf{1}_N\otimes c_{i\ell}\|$ as follows:
\begin{equation*}
\begin{split}
&\|\mathbf{x}-\mathbf{1}_N\otimes c_{i\ell}\|=\|\mathbf{x}-\mathbf{1}_N\otimes x_{i}+\mathbf{1}_N\otimes (x_{i}- c_{i\text{g}})\|\\
&\leq \sqrt{\sum_{j\in \mathcal{N}} \|x_{ij}\|^2} + \sqrt{N}\, \|p_i\|\leq \sqrt{(N-1)\varepsilon\sqrt{\varepsilon}\, \xi} + \sqrt{N} \,\|p_i\|,
\end{split}
\end{equation*}
where $\|x_{ij}\|^2\leq \varepsilon\, \sqrt{\varepsilon}\,\xi$ at $\mathbf{x}_c$, $\forall (i,\,j)\in E(t)$ by Lemma~\ref{difference}.
By comparing it with $r_S(\varepsilon)$, we get
$\|p_i\|\geq \sqrt{3\,\varepsilon}$, $\forall i \in \mathcal{N}_a$.
Choose $\mathbf{z} \triangleq \mathbf{1}_N\otimes {z}$, where $z\in \mathbb{R}^2$ and $\|z\|\triangleq 1$. Then $\mathbf{z}^T \, \nabla^2V\, \mathbf{z}$ is evaluated by using~\eqref{2gradient}-\eqref{hp}:
\begin{equation*}
\begin{split}
\mathbf{z}^T \nabla^2V \mathbf{z}&=\sum_{i\in \mathcal{N}} b_i\, d_{i}\,z^T z + b_i\, d_{i}' \, z^T p_i\, p_{i}^Tz\triangleq z^T M z,
\end{split}
\end{equation*}
where $M\triangleq \sum_{i\in \mathcal{N}_a}(d_i\otimes \textbf{I}_2 + d_i'\, p_i\, p_{i}^T)$ is a $2\times 2$ Hermitian matrix.
The trace of $M$ is computed as
\begin{equation}\label{trace}
\begin{split}
\textbf{trace}(M)&= \sum_{i\in \mathcal{N}_a} 2\,d_i + d_i'\, \|p_i\|^2\\
&=\varepsilon^3\sum_{i\in \mathcal{N}_a}\frac{{3\varepsilon-\|p_i\|^2}}{(\|p_i\|^2+\varepsilon)^3}<0,
\end{split}
\end{equation}
as we have shown that $\|p_i\|\geq \sqrt{3\varepsilon}, \forall i \in \mathcal{N}_a$ if $\mathbf{x}_c \in \mathcal{S}^{\neg}$. On the other hand, denote by $p_i=[p_{i,x},\,p_{i,y}]$ the coordinates of~$p_i$. The determinant of $M$ is given by
\begin{equation}\label{determinant}
\begin{split}
&\textbf{det}(M)=
-(\sum_{i\in \mathcal{N}_a} d_i' \, p_{i,x}\,p_{i,y})^2\\
&\qquad +(\sum_{i\in \mathcal{N}_a} d_i + d_i'\, p_{i,x}^2)(\sum_{i\in \mathcal{N}_a} d_i + d_i'\, p_{i,y}^2)\\
&\geq \frac{1}{2}\sum_{i,\,j\in \mathcal{N}_a} \big[(d_i+d_i'\|p_{i}\|^2)(d_j+d_j'\|p_{j}\|^2)\big]>0,
\end{split}
\end{equation}
since $d_i'\|p_i\|^2<-d_i$ for $\|p_i\|>\sqrt{3\varepsilon}$, $\forall i\in \mathcal{N}_a$; and $(p_{i,x}p_{i,y}-p_{j,x}p_{j,y})^2\leq \|p_i\|^2\|p_j\|^2$ by Cauchy-Schwarz inequality~\cite{horn2012matrix}.
Denote by $\lambda_1$ and $\lambda_2$ the eigenvalues of $M$, where $\lambda_{1},\lambda_2\in \mathbb{R}$ as $M$ is Hermitian.
Since $\textbf{trace}(M)<0$ and $\textbf{det}(M)>0$, then $M$ is negative definite and both eigenvalues are negative~\cite{horn2012matrix}, i.e., $\lambda_1, \lambda_2<0$. Thus for any vector $v= \textbf{1}_N\otimes z$ where $z\in \mathbb{R}^2$, $v^T\nabla^2 V v<0$.
To conclude, for any critical point $\mathbf{x}_c\in \mathcal{C}$, if $\mathbf{x}_c\in \mathcal{S}^\neg$ then $\mathbf{x}_c$ is not a local minimum.
\end{proof}
\begin{lemma}\label{isolated}
There exists $\varepsilon_1>0$ such that if $\varepsilon<\varepsilon_1$, all critical points of $V$ in $\mathcal{S}^{\neg}$ are non-degenerate saddle points.
\end{lemma}
\begin{proof}
To show that $V$ is Morse we use Lemma 3.8 from~\cite{rimon1988exact}, which states that the non-singularity of a linear operator follows from the fact that its associated quadratic form is sign definite on complementary subspaces.
Let $\mathcal{Q}=\{v\in \mathbb{R}^{2N}\,|\, v=\mathbf{1}_N\otimes z, \, z \in\mathbb{R}^2\}$.
In Lemma~\ref{away}, we have shown that for any vector $v\in \mathcal{Q}$, $v^T\nabla^2 V v<0$.
Let $\mathcal{P}=\{v\in \mathbb{R}^{2N}\,|\, v=\mathbf{e}_N\otimes z,\; \mathbf{e}_N\perp \mathbf{1}_N,\, \mathbf{e}_N\in \mathbb{R}^N,\, z \in \mathbb{R}^2\}$.
Firstly, it can be easily verified that
$\mathcal{P}$ is the orthogonal complement of $\mathcal{Q}$.
In the following, we show that $\nabla^2 V$ is positive definite in $\mathcal{P}$.
Let $\mathbf{z} \in \mathcal{P}$, i.e., $\mathbf{z}\triangleq \mathbf{e}_N \otimes z \triangleq [z_1^T\;z_2^T\ldots z_n^T]^T$, where $z\in \mathbb{R}^2$, $\mathbf{e}_N\in \mathbb{R}^N$, $\mathbf{e}_N^T \perp \mathbf{1}_N$, $z_i\in \mathbb{R}^2$, $\forall i\in \mathcal{N}$.
The quadratic form $\mathbf{z}^T \, \nabla^2V\, \mathbf{z}$ at $\mathbf{x}_c$ is computed using~\eqref{2gradient}-\eqref{hp}:
\begin{equation*}
\begin{split}
&\mathbf{z}^T \nabla^2V \mathbf{z}=\sum_{i\in \mathcal{N}_a} \big(d_{i}\,\|z_i\|^2 + d_{i}' \, |p_{i}^Tz_i|^2\big) \\
&+\sum_{(i,\,j)\in E(t)}\big( h_{ij} \,\|z_i-z_j\|^2 +2\,h'_{ij}\,|(x_i-x_j)^T(z_i-z_j)|^2\big)\\
&\geq \sum_{i\in \mathcal{N}_a} \big(d_{i}\,\|z_i\|^2 + d_{i}' \, |p_{i}^Tz_i|^2)+\sum_{(i,\,j)\in E(t)} h_{ij} \,\|z_i-z_j\|^2 \\
&\geq \sum_{i\in \mathcal{N}_a} \big(d_{i} + d_i'\|p_i\|^2\big)\|z_i\|^2+
\mathbf{z}^T (\mathbf{H}\otimes \textbf{I}_2) \mathbf{z},
\end{split}
\end{equation*}
where we use the fact that $h_{ij}'>0$, $d_i'<0$ and $|p_{i}^Tz_i|\leq \|p_i\|\|z_i\|$. It can be verified that $d_{i} + d_i'\|p_i\|^2>-0.1\varepsilon$ for $\|p_i\|\geq \sqrt{3\varepsilon}$, $\forall i\in \mathcal{N}_a$. Moreover,
\begin{equation}\label{laplacian}
\begin{split}
\mathbf{z}^T (\mathbf{H}\otimes \textbf{I}_2) \mathbf{z}&=(\mathbf{e}_N \otimes z)^T \cdot (\mathbf{H}\otimes \textbf{I}_2)\cdot (\mathbf{e}_N \otimes z)\\
&= (\mathbf{e}^T_N\cdot \mathbf{H}\cdot \mathbf{e}_N) \|z\|^2\geq \lambda_2(\mathbf{H})\|z\|^2,
\end{split}
\end{equation}
where we apply the Courant-Fischer Theorem~\cite{horn2012matrix}:
$$\min_{\mathbf{e}_N\perp \textbf{1}_N}\{\mathbf{e}^T_N\cdot \mathbf{H}\cdot \mathbf{e}_N\}=\lambda_2(\textbf{H})>0,$$
since $\textbf{H}$ is the Laplacian matrix defined in~\eqref{equi2}, which is positive semidefinite with $\lambda_1(\mathbf{H})=0$, of which the corresponding eigenvector is $\mathbf{1}_N$; and the second smallest eigenvalue $\lambda_2(\mathbf{H})>0$.
In addition, since $h_{ij}>1/{r^2}$ and $G(t)$ is a complete graph at $\mathbf{x}_c$ by Lemma~\ref{difference}, it holds that $\lambda_2(\mathbf{H})>N/{r^2}$ by~\cite{Godsil}.
This implies that
\begin{equation}\label{quadratic5}
\begin{split}
\mathbf{z}^T \nabla^2V \mathbf{z}
&\geq \sum_{i\in \mathcal{N}_a} \big(\frac{N}{r^2}+d_{i} + d_i'\|p_i\|^2\big)\|z_i\|^2\\
&\geq \sum_{i\in \mathcal{N}_a} \big(\frac{N}{r^2}-0.1\varepsilon\big)\|z_i\|^2.
\end{split}
\end{equation}
Thus if
$\varepsilon < {N}/({0.1 r^2})$,
it holds that $\mathbf{z}^T \nabla^2V \mathbf{z}>0$, $\forall \mathbf{z}=\mathbf{e}_N \otimes z$ where $\mathbf{e}_N\perp \mathbf{1}_N$, $z\in \mathbb{R}^2$.
To conclude, $\nabla^2V|_{\mathcal{Q}}$ is negative definite by Lemma~\ref{away} and $\nabla^2V|_{\mathcal{P}}$ is positive definite by the analysis above.
By applying Lemma 3.8 from~\cite{rimon1988exact}, we can conclude that $\nabla^2V$ is non-singular at the saddle points $\mathbf{x}_c\in \mathcal{S}^{\neg}$, if
\begin{equation}\label{v1}
\varepsilon<\min\{\varepsilon_0,\, \frac{N}{0.1 r^2}\}\triangleq \varepsilon_1.
\end{equation}
In other words, all critical points within $\mathcal{S}^{\neg}$ are non-degenerate saddle points if $\varepsilon<\varepsilon_1$.
\end{proof}
Now we focus on proving that all critical points within $\mathcal{S}$ are stable local minima. First of all, we need the following two lemmas to show that when a critical point belongs to $\mathcal{S}_i$ corresponding to one active agent $i\in \mathcal{N}_a$, then all the other agents are within its goal region $\pi_{i\text{g}}$ and away from their own goal region center by at least distance $r_{\min}$.
\begin{lemma}\label{distance}
There exists $\varepsilon_2>0$, such that if $\varepsilon<\varepsilon_2$, the following hold:
(I)
$\mathcal{S}_i\cap \mathcal{S}_j=\emptyset$, $\forall i\neq j$ and $i,\,j \in \mathcal{N}_a$;
(II)
If $\mathbf{x}_c\in \mathcal{S}_{i^\star}$ for some $i^\star \in \mathcal{N}_a$, then $x_j\in \pi_{i^\star\textup{g}}$, $\forall j\in \mathcal{N}$ and $\|x_j-c_{j\textup{g}}\|>r_{\min}$, $j\neq i^\star$, $\forall j\in \mathcal{N}_a$.
\end{lemma}
\begin{proof}
Let $\varepsilon_2$ be given as the solution of
\begin{equation}\label{v2}
r_S(\varepsilon_2)=\sqrt{3N\,\varepsilon_2}+\sqrt{(N-1)\varepsilon_2\sqrt{\varepsilon_2}\, \xi}\triangleq r_{\min},
\end{equation}
where $r_{\min}$ is given in Assump.~\ref{region-assump}. Note that~\eqref{v2} has an unique solution as the left-hand side is a function of $\varepsilon_2$ that monotonically increases and has the range $[0,\;\infty)$.
Assume that $\mathbf{x}_c \in \mathcal{S}_{i^\star}$ for $i^\star\in \mathcal{N}_a$, i.e., $\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{i^\star\text{g}}\|\leq r_S(\varepsilon_2)$.
Then $\forall j\neq i^\star$, $j\in \mathcal{N}_a$, it holds that
(I)
$\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{j\text{g}}\|=\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{i\text{g}}+\mathbf{1}_N\otimes c_{i \text{g}}-\mathbf{1}_N\otimes c_{j\text{g}}\|
\geq \sqrt{N}\, \|c_{i\text{g}}-c_{j\text{g}}\|-\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{i\text{g}}\|
\geq 2\sqrt{N} \,{r}_{\min} -r_S(\varepsilon),$
due to that $\|c_{i\text{g}}-c_{j\text{g}}\|>2r_{\min}$ by Assump.~\ref{region-assump}.
Since $\varepsilon<\varepsilon_2$, then $r_S(\varepsilon)<r_S(\varepsilon_2)=r_{\min}$.
Thus $\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{j\text{g}}\|>2\sqrt{N} \,{r}_{\min}-r_{\min}>r_{\min}=r_S(\varepsilon_2)$, implying that $\mathbf{x}_c \notin \mathcal{S}_j$.
(II)
$\|x_j-c_{i^\star\text{g}}\|<\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{i^\star\text{g}}\|<r_{\min}<r_{i^\star\text{g}}$, meaning that $x_j\in \pi_{i^\star\text{g}}$, $\forall j\in \mathcal{N}$.
Moreover, $
\|x_j- c_{j\text{g}}\|=\|x_j- c_{i^\star\text{g}}+c_{i^\star\text{g}}-c_{j\text{g}}\|\geq \|c_{i^\star\text{g}}-c_{j\text{g}}\|-\|x_j- c_{i^\star\text{g}}\|\geq 2r_{\min}-r_{\min}>r_{\min}.
$
\end{proof}
\begin{lemma}\label{furtherbound}
There exists $\varepsilon_6>0$ such that if $\varepsilon<\varepsilon_6$, then for a critical point $\mathbf{x}_c \in \mathcal{S}_{i^\star}$, $i^\star\in \mathcal{N}_a$, then it holds that $\|p_{i^\star}\|<\sqrt{0.4\varepsilon}$.
\end{lemma}
\begin{proof}
By summing~\eqref{equi} for all $i\in \mathcal{N}$, we get
\begin{equation}\label{distancecond}
d_{i^\star}\, p_{i^\star}=-\sum_{j\neq i^\star, j\in \mathcal{N}_a} d_j\, p_j.
\end{equation}
Consider the scalar function $f(\|p_j\|)=d_j(\|p_j\|)\|p_j\|$ for $\|p_j\|\geq 0$.
It is monotonically increasing for $\|p_j\|\in [0, \, 3.2\sqrt{\varepsilon})$ and decreasing for $\|p_j\|\in [3.2\sqrt{\varepsilon},\, \infty)$.
If $\mathbf{x}_c \in \mathcal{S}_{i^\star}$ for $i^\star\in \mathcal{N}_a$, then $\|\mathbf{x}_c-\mathbf{1}_N\otimes c_{i^\star\text{g}}\|\leq r_S(\varepsilon_2)$.
Moreover,
$\|\mathbf{x}-\mathbf{1}_N\otimes c_{i^\star \ell}\|\geq
\|\mathbf{1}_N\otimes x_{i^\star}-\mathbf{1}_N\otimes c_{i^\star\text{g}}\|-
\|\mathbf{x}-\mathbf{1}_N\otimes x_{i^\star}\|
\geq \sqrt{N} \,\|p_{i^\star}\|- \sqrt{(N-1)\varepsilon\sqrt{\varepsilon}\, \xi}.$
This implies
$\|p_{i^\star}\|\leq \sqrt{3\,\varepsilon}+2\sqrt{\varepsilon\sqrt{\varepsilon}\, \xi}.$
Moreover by Lemma~\ref{distance}, $\|p_j\|>r_{\min}$ , $\forall j\neq i^\star$, $j\in \mathcal{N}_a$.
Thus if $r_{\min}>3.2\sqrt{\varepsilon}$, namely
\begin{equation}\label{v3}
\varepsilon<0.07\, r^2_{\min}\triangleq \varepsilon_3,
\end{equation}
it holds that $d_j\,\|p_j\|<0.5\varepsilon^2/r_{\min}$, $\forall j\neq i^\star$, $j\in \mathcal{N}_a$.
Thus $d_{i^\star}\, \|p_{i^\star}\|<0.5(N_a-1)\varepsilon^2/r_{\min}$ by~\eqref{distancecond}.
If the following two conditions hold:
(i) $\sqrt{3\,\varepsilon}+2\sqrt{\varepsilon\sqrt{\varepsilon}\, \xi}<3.2\sqrt{\varepsilon}$;
(ii)
$0.5(N_a-1)\varepsilon^2/r_{\min}< d_j(\sqrt{0.4\varepsilon})\sqrt{0.4\varepsilon}$,
then $\|p_{i^\star}\|<\sqrt{0.4\varepsilon}$ since it is shown earlier that function $d_j(\|p_j\|)\|p_j\|$ is monotonically increasing for $\|p_j\|\in [0,\, 3.2\sqrt{\varepsilon})$. Condition~(i) above implies that
$
\varepsilon<4.1/\xi^2\triangleq \varepsilon_4
$
and condition~(ii) holds for all $N_a\leq N$ if
$
\varepsilon<0.8\,r_{\min}^2/(N-1)^2\triangleq \varepsilon_5.
$
To conclude, if $\varepsilon<\varepsilon_6$, where
\begin{equation}\label{v6}
\varepsilon_6\triangleq \min\{\varepsilon_3, \varepsilon_4, \varepsilon_5\},
\end{equation}
then $\mathbf{x}_c\in \mathcal{S}_{i^\star}$ implies $\|p_{i^\star}\|<\sqrt{0.4\varepsilon}$.
\end{proof}
With the above two lemmas, we can now show that all critical points within $\mathcal{S}$ are stable local minima.
\begin{lemma}\label{localmini}
There exists $\varepsilon_{\min}>0$ such that if $\varepsilon<\varepsilon_{\min}$,
all critical points of $V$ within $\mathcal{S}$ are local minima.
\end{lemma}
\begin{proof}
A critical point $\mathbf{x}_c\in \mathcal{S}$ can only belong to one set $\mathcal{S}_i$ for $i\in \mathcal{N}_a$ by Lemma~\ref{distance}. Without loss of generality, let $\mathbf{x}_c \in S_{i^{\star}}$, where $i^\star \in \mathcal{N}_a$. In the following we show that $\mathbf{x}_c$ is a local minimum.
Let $\mathbf{z}\in \mathbb{R}^{2N}$ and $\|\mathbf{z}\|=1$. Set $\mathbf{z}=[z_1^T\;z_2^T\ldots z_n^T]^T$, where $z_i\in \mathbb{R}^2$, $\forall i\in \mathcal{N}$. Then $\mathbf{z}^T \, \nabla^2V\, \mathbf{z}$ at $\mathbf{x}_c$ is computed as:
\begin{equation}\label{quadratic3}
\begin{split}
&\mathbf{z}^T \nabla^2V \mathbf{z}=\sum_{i\in \mathcal{N}_a} \big(d_{i}\,\|z_i\|^2 + d_{i}' \, |p_{i}^Tz_i|^2\big)+ \\
&\sum_{(i,\,j)\in E(t)}\big( h_{ij}\|z_{ij}\|^2 +2\,h'_{ij}\,|x_{ij}^T\,z_{ij}|^2\big).
\end{split}
\end{equation}
where $z_{ij}\triangleq z_i-z_j$.
Since $|p_i^Tz_i|\leq \|p_i\|\|z_i\|$, $d_i>0$ and $d_i'<0$, it holds that
$
d_{i}\,\|z_i\|^2 + d_{i}' \, |p_{i}^T\,z_i|^2\geq (d_i +d_{i}' \|p_i\|^2)\|z_i\|^2, \forall i\in \mathcal{N}_a.
$
It can be verified that for $j\neq i^\star$ and $\forall j\in \mathcal{N}_a$, $d_j+d_j'\|p_j\|^2>\varepsilon^2\hat{g}$ where $\hat{g} \triangleq -2/r^2_{\min}$, since $\|p_j\|>r_{\min}$ by Lemma~\ref{distance}; and $d_{i^\star}+d_{i^\star}'\|p_{i^\star}\|^2>0.08\varepsilon $ since $\|p_{i^\star}\|>\sqrt{0.4\varepsilon}$ by Lemma~\ref{furtherbound}. Regarding the second term of~\eqref{quadratic3}, since Lemma~\ref{difference} shows that $G(t)$ is a complete graph at $\mathbf{x}_c$ with $h_{ij}>1/{r^2}$ and $h_{ij}'>0$, we get
\begin{equation}\label{hbound}
\begin{split}
&\sum_{(i,\,j)\in E}\big( h_{ij} \,\|z_{ij}\|^2 +2\,h'_{ij}\,|x_{ij}^T\, z_{ij}|^2\big)\\
&\geq \sum_{j\in \mathcal{N}} h_{i^\star j} \,\|z_{i^\star j}\|^2 \geq \frac{1}{r^2}\sum_{j\in \mathcal{N}}\,\|z_{i^\star j}\|^2.
\end{split}
\end{equation}
Thus ~\eqref{quadratic3} can be bounded by
\begin{equation*}\label{quadratic4}
\begin{split}
&\mathbf{z}^T \nabla^2V \mathbf{z}
\geq \sum_{i\in \mathcal{N}_a} \big(d_{i} + d_{i}' \, \|p_{i}\|^2\big)\|z_i\|^2 +\sum_{j\in \mathcal{N}} h_{i^\star j} \,\|z_{i^\star j}\|^2\\
&\geq 0.08\,\varepsilon \|z_{i^\star}\|^2 -\varepsilon^2 \sum_{j\neq i^\star,j\in \mathcal{N}_a}|\hat{g}|\|z_j\|^2+\frac{1}{r^2} \sum_{j\in \mathcal{N}} \,\|z_{i^\star j}\|^2\\
&\geq \sum_{j\in \mathcal{N}_a}\big(\frac{1}{r^2}+ \frac{0.08\varepsilon}{N}\big) \|z_{i^\star}\|^2 +\big(\frac{1}{r^2}-\varepsilon^2|\hat{g}|\big)\|z_j\|^2-\frac{2}{r^2} z_{i^\star}^T\, z_j,
\end{split}
\end{equation*}
as $1\leq N_a\leq N$. If the following condition holds:
\begin{equation}\label{condition11}
\begin{split}
\big(\frac{1}{r^2}+ \frac{0.08\varepsilon}{N}\big) \big(\frac{1}{r^2}-\varepsilon^2|\hat{g}|\big)> \big(\frac{1}{r^2}\big)^2,
\end{split}
\end{equation}
it implies
$
\mathbf{z}^T \nabla^2V \mathbf{z}> (|z_{i^\star}^T\, z_j|-z_{i^\star}^T\, z_j)/r^2\geq 0,
$
$\forall \mathbf{z}\in \mathbb{R}^{2N}$.
Namely, $\nabla^2V$ is positive definite at critical points $\mathbf{x}_c \in \mathcal{S}$.
Condition~\eqref{condition11} is equivalent to
$$
\varepsilon^2+\frac{N}{0.08\,r^2}\varepsilon -\frac{1}{r^2|\hat{g}|}<0.
$$
Since $\varepsilon>0$, this implies that
\begin{equation}\label{v7}
0<\varepsilon <\frac{\sqrt{(\frac{N}{0.08\,r^2})^2+\frac{4}{r^2|\hat{g}|}}-\frac{N}{0.08\,r^2}}{2}\triangleq \varepsilon_7,
\end{equation}
To conclude, if
\begin{equation}\label{v8}
\varepsilon<\min\{\varepsilon_1, \varepsilon_2,\, \varepsilon_6,\varepsilon_7\}\triangleq \varepsilon_{\min},
\end{equation}
where $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_6$ and $\varepsilon_7$ are defined in~\eqref{v1},~\eqref{v2},~\eqref{v6} and~\eqref{v7}, then all local minima within $\mathcal{S}$ are stable. \end{proof}
By summarizing Lemmas~\ref{away}-\ref{localmini}, we can derive the following convergence result:
\begin{theorem}\label{convergence}
Assume that $G(T_s)$ is connected and $\varepsilon<\varepsilon_{\min}$ by~\eqref{v8}.
Then starting from anywhere in the workspace except a set of measure zero, there exists a finite time $T_f\in [T_s,\infty)$ and one agent $i^\star\in \mathcal{N}_a$, such that $x_j(T_f)\in \pi_{i^\star\text{g}}$, $\forall j\in \mathcal{N}$, while at the same time $\|x_i(t)-x_j(t)\|<r$, $\forall (i,\,j)\in E(T_s)$ and $\forall t\in [T_s,\, T_f]$.
\end{theorem}
\begin{proof}
First of all, the second part follows directly from Theorem~\ref{connectivity} which guarantees that all edges within $E(T_s)$ will be reserved for all $t>T_s$. Secondly, we have shown that $V(t)$ by~\eqref{Lyapunov} is non-increasing for all $t>T_s$ by Theorem~\ref{connectivity}. By LaSalle's invariance principle~\cite{khalil2002nonlinear} we only need to find out the largest invariant set within $\dot{V}(t)=0$.
Theorems~\ref{away} and~\ref{localmini} ensure that the potential function $V(t)$ has only local minima inside $\mathcal{S}$ and saddle points outside $\mathcal{S}$. These saddle points have attractors of measure zero by Lemma~\ref{isolated}.
Thus starting from anywhere in the workspace except a set of measure zero, the system converges to the set of local minima.
Part~(I) of Lemma~\ref{distance} shows that a local minimum can not belong to two different $\mathcal{S}_i$ simultaneously.
Thus the system converges to the set of local minima within $\mathcal{S}_{i^\star}$ for one active agent $i^\star\in \mathcal{N}_a$.
By part~(II) of Lemma~\ref{distance}, all agents would enter $\pi_{i^\star\text{g}}$, i.e., $x_j\in \pi_{i^\star\text{g}}$, $\forall j\in \mathcal{N}$.
Consequently, there exists $T_f<\infty$ that $x_j(T_f)\in \pi_{i^\star\text{g}}$, $\forall j\in \mathcal{N}$, for exactly one active agent $i^\star\in \mathcal{N}_a$.
\end{proof}
\begin{remark}\label{general}
Note that Theorem~\ref{convergence} holds for any number of active agents with $1\leq N_a\leq N$. In other words, independent of the number of active agents within the team, one active agent will reach its goal region first within finite time, while fulfilling the relative-distance constraints.
\end{remark}
\subsection{Hybrid Control Structure}
In {Sec.~\ref{synthesis}, we have generated a sequence of progressive goal regions for each agent and in} Sec.~\ref{continuous-design}, we have shown that under the proposed control laws all agents converge to one active agent's {progressive} goal region. In this part, we {propose a local procedure for each agent to decide on its own activity/passivity. Thus we integrate the discrete plans and the continuous control laws into
a hybrid control scheme to guarantee that every agent's local task is fulfilled. }
\subsubsection{Switching Protocol for sc-LTL}\label{switch}
{Let us first focus on the case when each task $\varphi_i$, $i \in \mathcal N$ is an sc-LTL formula.} As {introduced} in Sec.~\ref{synthesis}, the {discrete} plan $\tau_i $ for agent $i$ {can be represented} by a finite {satisfying prefix of progressive} goal regions in $\Pi_i$ of length $k_i >0$:
{
$$
\tau_{i,\textup{pre}} = (\pi_{i1},w_{i1})\cdots(\pi_{i{k_i}},w_{ik_i})
$$}
{We propose the following \emph{activity switching protocol} for each agent $i\in \mathcal{N}$}:
\begin{itemize}\itemsep-0.5ex
\item[(I)] At time $t=0$, agent $i$ {sets $\varkappa_i:= 1$ and itself as} active and sets {$\pi_{i\text{g}}:= \pi_{i\varkappa_i}$}, namely the first goal region in $\tau_i$. The \emph{active} controller~\eqref{law1} is applied, where the progressive goal region is $\pi_{i\text{g}}$
\item[(II)] Whenever agent $i$ reaches its current {progressive} goal region $\pi_{ig} = \pi_{i\varkappa_i}$ and $\varkappa_i<k_i$, it provides the {prescribed set of services $w_{i\varkappa_i}$} and it sets $\varkappa_i:= \varkappa_i+1$ and $\pi_{i\text{g}}:= \pi_{i\varkappa_i}$. {The controller~\eqref{law1} for agent~$i$ is updated while
the other agents' controllers remain unchanged. }
\item[(III)] Whenever agent $i$ reaches {its last progressive} goal region $\pi_{ig}= \pi_{i{k_i}}${, it provides the set of services $w_{ik_i}$ by which it finishes the execution of its discrete plan.}
Afterwards it remains \emph{passive} and controller~\eqref{law2} applies.
\end{itemize}
\begin{theorem}\label{satisfyall}
{By following the protocol above, it is guaranteed that $\forall i \in \mathcal N$, $\varphi_i$ is satisfied by $\mathbf{x}_i(T)$, and $\|x_i(t)-x_j(t)\|<r$, $\forall (i,\,j)\in E(0)$ and $\forall t \geq 0$, where $T\rightarrow\infty$.}
\end{theorem}
\begin{proof}
At $t=0$, all agents are active and following the controller~\eqref{law1}. By Theorem~\ref{convergence}, all agents converge to one agent's goal region at a finite time $t_1>0$. Denote by $i \in \mathcal{N}$ this agent. Then either by step~(II) of the protocol, the agent $i$ updates its active control law by setting $\pi_{i\text{g}}=\pi_{i2}$, or by step~(III) the agent $i$ has completed its plan $\tau_{i,\text{pre}}$ and becomes passive.
{Since all agents' plans are finite and Theorem~\ref{convergence} holds for any number of active agents, we obtain that there exists a finite time instant $T_{f_j}$, at which one of the agents $j \in \mathcal N_a$ finishes executing its plan $\tau_{j,\mathrm{pre}}$, i.e., such that $\varphi_{j}$ becomes satisfied.}
Then by step~(III), this agent is passive and following the controller~\eqref{law2} for all times $t \in [T_{f_j}, \infty)$.
{Inductively, we conclude that there exists a time instant $T_f$, by which all agents complete their plans and all formulas are satisfied. All agents are passive for all $t \in (T_{f},\infty)$
and by controller~\eqref{law2} they all converge to one point.}
The second part of the theorem follows directly from Theorem~\ref{convergence}.
\end{proof}
Note that this protocol is fully decentralized as the decisions on an agents' activity/passivity are local and do not depend on any relative-state measurements.
\subsubsection{Switching Protocol for full LTL}\label{switch-infinite}
As {introduced} in Sec.~\ref{synthesis}, if the {task specification $\varphi_i$ is given as a general LTL formula, then the plan $\tau_i$ is represented} by an infinite sequence of {progressive goal regions in a prefix-suffix form}
\begin{align*}
\tau_{i} = \tau_{i,\text{pre}}(\tau_{i,\text{suf}})^{\omega} & = (\pi_{i1},w_{i1})(\pi_{i2},w_{i2})\ldots, \text{where} \\
\tau_{i,\text{pre}} & = (\pi_{i},w_{i1})\ldots(\pi_{i{k_i}},w_{ik_i}), k_i>0 \text{ and} \\
\tau_{i,\text{suf}} & = (\pi_{i{k_i+1}},w_{ik_i+1})\ldots(\pi_{i{K_i}},w_{iK_i}), K_i>0.
\end{align*}
The main challenge in this case is to ensure that each agent visits its progressive goal region infinitely often. The activity switching protocol from Sec.~\ref{switch} could not be applied here since all agents would remain active at all times. As a result, the team may repetitively converge to $\pi_{ig}$ for some $i \in \mathcal N$ while never visiting the other agents' progressive goal regions
(see Sec.~\ref{sec:example} for an example). Hence, we aim to design a ``fair'' activity switching protocol that enforces a progress towards each agent's task.}
{Thereto, we first introduce a communication-free reaching-event detector that enables an agent to monitor its neighbors' plan executions.}
\medskip
\noindent{\emph{Reaching-Event Detector}}.
Agent $i \in \mathcal{N}$ can detect when it reaches its own {progressive} goal region $\pi_{i\text{g}}$ by checking if $x_i(t)\in \pi_{i\text{g}}$.
For our switching protocol presented below, it is also essential that it can detect when another agent
$j\in \mathcal{N}$ reaches $\pi_{jg}$. Note that by Lemma~\ref{difference}, the connectivity graph is complete since the first time any agent $i \in \mathcal N$ reaches its progressive goal region $\pi_{ig}$, hence it is sufficient to detect when a neighboring agent $j \in \mathcal N_i(t)$ reaches $\pi_{jg}$.
Given that the agents satisfy the dynamics by~\eqref{dynamics} and that each agent $i\in \mathcal{N}$ can measure $x_i(t)-x_j(t)$ for all its neighbors $j\in \mathcal{N}_i(t)$ in real time, we assume that the agent~$i$ can measure or estimate $u_j(t)$, for all $j\in \mathcal{N}_i(t)$~\cite{Fran08}.
{Let $\Omega_i(j,\,t)\in \mathbb{B}$ be a Boolean variable indicating that agent~$i$ detects its neighboring agent $j\in \mathcal{N}_i(t)$ reaching the goal region $\pi_{j\text{g}}$ at time $t>0$. We propose the following reaching-event detector inspired by~\cite{Tabu11}. Simply speaking, the detector checks if within a short time period $[t-\Delta_t,\, t]$, there exists $j \in \mathcal N_i(t)$, such that $u_j(t)$ has changed from a relatively small value (below a given $\Delta_u$) by a difference larger than certain $\Delta_d$. If so, it means that the agent $j$ has reached its progressive goal region $\pi_{j\text{g}}$.
The choice of this reaching-event detector is motivated by the following facts:
By~\eqref{equi}, all control inputs $u_i(t)$ are close to zero when the system is close to a local minimal, $\forall i\in \mathcal{N}$.
Afterwards, our switching protocol introduced below guarantees that
\emph{only} agent $j$ switches its control law either to~\eqref{law1} in order to navigate to the next progressive goal region or to~\eqref{law2} in order to become passive. This change is lower-bounded by constant $\Delta_d$ derived using~control law \eqref{law1} and Lemmas~\ref{distance},~\ref{furtherbound} as
$\Delta_d\triangleq |f(r_{\min})-f(\sqrt{0.4\varepsilon})|$, where $f(\|p_j\|)=d_j(\|p_j\|)\|p_j\|$ is a scalar function and $d_j(\|p_j\|)$ is defined by~\eqref{d}.
In contrast, for the other agents $i\neq j$, $i \in \mathcal{N}$, the control input $u_i(t)$ remains unchanged and close to zero. Hence, agent $j$ is identified as the only one who has reached its progressive goal region.
Formally,
\begin{definition}
$\Omega_i(j,\,t) \triangleq \textup{\texttt{True}}$
if and only if there exists $ t'\in [t-\Delta_t,\, t]$, where $|u_j(t')|<\Delta_u$ and $|u_j(t)-u_j(t')|>\Delta_d$.
\end{definition}
\medskip
\noindent\emph{Activity Switching Protocol}.
Loosely speaking, in the proposed protocol, an agent $i \in \mathcal N$ becomes passive if it has made a certain progress towards the satisfaction of its specification, hence giving the other agents an opportunity to advance in execution of their plans. However, once each agent has achieved certain progress, the agent $i$ becomes active again to proceed with its infinite plan.
We define a \emph{round} as the time period during which each agent has reached at least one of its goal regions according to their plans.
\begin{definition}
For all $m \geq 1$, the \emph{$m$-th round} is defined as the time interval $[T_{{\circlearrowleft}_{m-1}},T_{{\circlearrowleft}_{m}})$, where $T_{{\circlearrowleft}_{0}} = 0$, $T_{{\circlearrowleft}_{m-1}} < T_{{\circlearrowleft}_{m}}$ and for all $m \geq 1$, $T_{\circlearrowleft_m}$ is the smallest time satisfying the following conditions $\forall i \in \mathcal N$: $\mathtt{word}_i(T_{\circlearrowleft_{m}}) = w_{i1}\cdots w_{i\ell}$ for some $\ell \geq 1$ and
$\mathtt{word}_i(T_{\circlearrowleft_{m}}) \neq \mathtt{word}_i(T_{\circlearrowleft_{m-1}})$.
\end{definition}
The notion of a round is crucial to the design of the activity switching protocol. To recognize a round completion, we introduce the following variables:
$\chi_i\geq 0$ indicates the starting time of the current round, and
$\Upsilon_i \in \mathbb{Z}^N$ is a vector to record how many progressive goal regions each agent has reached within one round since $\chi_i$.
Although these variables are locally maintained by each agent.
By Lemma~\ref{difference}, the connectivity graph is complete since the first time one active agent reaches its goal region $\pi_{ig}$ and under the assumption of unbiased measurements, it holds that at the same time instant $\chi_i = \chi_j$, and $\Upsilon_i = \Upsilon_j$, $\forall i,j \in \mathcal N$.
We propose the following \emph{activity switching protocol} for each agent $i\in \mathcal{N}$:
\begin{itemize}\itemsep-0.5ex
\item[(I)] At time $t=0$, $\Upsilon_i := \textbf{0}_{N}$, $\chi_i : =0$, $\varkappa_i := 1$.
The agent $i$ is active and follows control law~\eqref{law1}, where $\pi_{i\text{g}} := \pi_{i\varkappa_i}$.
\item[(II)]
Whenever the agent $i$ reaches its current progressive goal region $\pi_{ig} = \pi_{i\varkappa_i}$ and waits until $|u_i(t)|<\Delta_u$, it provides the prescribed set of services $w_{i\varkappa_i}$ and updates the current progressive goal region accordingly: If $\varkappa_i < K_i$ then $\varkappa_i := \varkappa_i+1$, and if $\varkappa_i = K_i$ then $\varkappa_i := k_i+1$. Furthermore, $\pi_{ig} := \pi_{i\varkappa_i}$, and finally $\Upsilon_i[i] := \Upsilon_i[i]+1$.
Generally speaking, the agent $i$ decides to stay {active} or to become {passive} based on the probability function:
$$
\textbf{Pr}(b_i=1)= \begin{cases}
f_{\mathrm{prob}}(\cdot) & \text{if } f_{\mathrm{cond}}(\cdot) = \texttt{True},\\
0& \text{otherwise,}
\end{cases}
$$
where $f_\mathrm{prob}(\cdot) \in [0,1]$ and $f_\mathrm{cond}(\cdot) \in \{\texttt{True},\texttt{False}\}$ are functions of time $t$ and the local variables $\Upsilon_i$ and~$\chi_i$, subject to the following: given that the current round is the $m$-th one, there exists a time $T \in (T_{\circlearrowleft_{m-1}},\, T_{\circlearrowleft_{m}})$, such that $f_\mathrm{cond}(\cdot)= \texttt{False}$ for all $t \in [T,\, T_{\circlearrowleft_m})$.
Whenever $b_i=1$, the agent $i$ keeps following the control law~\eqref{law1} with the updated $\pi_{ig}$. Otherwise, it becomes passive and the control law~\eqref{law2} is applied.
\item[(III)] Whenever agent $i$ detects that $\Omega_i(j,t)=\texttt{True}$, for some $j \neq i \in \mathcal N$, it sets $\Upsilon_i[j]=\Upsilon_i[j]+1$.
\item[(IV)] Whenever for all $j \in \mathcal N$ it holds that $\Upsilon_i[j]>0$ the agent $i$ sets $\Upsilon_i:=\textbf{0}_{N}$, $\chi_i:=t$ and follows the active control law~\eqref{law1}.
\end{itemize}
{A straightforward example of the functions choice in (II) is $f_\mathrm{cond} = \texttt{False}$, for all $t \geq 0$. Then the agent $i$ always becomes passive once it visits $\pi_{ig}$. Note that it becomes active after the current round is completed by step (IV). However, a different selection of the functions may allow for trading the fairness of activity switching for the increased efficiency of plan executions.
The switching to passive control mode may be temporarily postponed and as a result, the visits to progressive goal regions may become more frequent. An example of such a case is given in Sec.~\ref{sec:example}. }
\begin{lemma}\label{finitetime}
The round $[T_{\circlearrowleft_{m-1}},T_{\circlearrowleft_m})$ is finite, $\forall m \geq 1$.
\end{lemma}
\begin{proof}
Let $t=T_{\circlearrowleft_{m-1}} = 0$, and thus $\Upsilon_i[j]=0$, for all $i,j \in \mathcal N$ by step (I). By Theorem~\ref{convergence}, one of the agents reaches its progressive goal region in finite time at $t_1 \geq T_{\circlearrowleft_{j-1}}$. Since there are only finite number of agents and due to the required properties of $f_\mathrm{cond}$, there exists a finite time $T_{f_j} \geq 0$, when either the step (IV) applies or when one of the agents $j \in \mathcal N_a$ necessarily becomes passive by the function $\textbf{Pr}(\cdot)$ in step (II) and remains passive till the end of the round. In the former case $T_{\circlearrowleft_m}=T_{f_j}$, i.e., we directly obtain that the $m$-th round is finite. In the latter case, by inductive reasoning we obtain that there exists a finite time instant $T_f$, such that step (IV) applies, i.e., such that $T_{\circlearrowleft_m}=T_{f}$. Again, we have that $m$-th round is finite.
Inductively, let $m >1$, $t = T_{\circlearrowleft_{m-1}}$, and $\Upsilon_i[j]=0$, for all $i,j \in \mathcal N$ by step (IV). Using analogous arguments as above, we obtain the existence of a finite $ T{\circlearrowleft_{m}}$.
\end{proof}
\begin{theorem}
By following the protocol above, it is guaranteed that $\forall i\in \mathcal{N}$, $\varphi_i$ is satisfied by $\mathbf{x}_i(T)$ and $\|x_i(t)-x_j(t)\|<r$, $\forall (i,\,j)\in E(0)$ and $\forall t>0$, where $T \rightarrow \infty$.
\end{theorem}
\begin{proof}
The satisfaction of $\varphi_i$ follows directly from the correctness of each agent's discrete plan and the fact that each round is finite by Lemma~\ref{finitetime}.
At last, the distance constraints between neighbouring agents are always maintained
as shown in Theorem~\ref{convergence}.
\end{proof}
\section{Simulation}\label{sec:example}
In the following case study, we simulate a team of four autonomous robots $\mathcal N = \{\mathfrak{R}_1, \cdots, \mathfrak{R}_4\}$ {subject to the dynamics~\eqref{dynamics} in a bounded, obstacle-free workspace of $40 \times 40$ meters ($m$)}. {Each robot $\mathfrak{R}_i$ is given a local sc-LTL or LTL task $\varphi_i$}. All algorithms and modules {were} implemented in Python 2.7. {Simulations were} carried out on a desktop computer (3.06 GHz Duo CPU and 8GB of RAM) with a simulation stepsize set to $1ms$.
\begin{figure}[t!]
\begin{minipage}[t!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=1\textwidth]{safe_traj_static.pdf}
\end{minipage}
\begin{minipage}[ht!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=0.54\textwidth]{safe_distance_static.pdf}
\includegraphics[width =0.95\textwidth, height=0.44\textwidth]{safe_reach_static.pdf}
\end{minipage}
\caption{Left: agents' respective regions of interest in red, green, blue and cyan respectively and their trajectories after execution of switching policy from Sec.~\ref{switch}. All agents accomplish their sc-LTL tasks after $9s$. Top-right: the evolution of pair-wise distances $\|x_{12}\|,\|x_{23}\|, \|x_{34}\|$, which all stay below $7.5m$. Bottom-right: the time instants when the agents reach their goal regions.}
\label{safe_static}
\end{figure}
\medskip
As shown Fig.~\ref{safe_static}, several sphere regions of interest for each agent are placed in top-left, top-right, bottom-right, and bottom-left corners of the workspace and they all satisfy Assump.~\ref{region-assump} with $c_{\max}=40$ and $r_{min}=2$:
\begin{itemize}\itemsep-0.5ex
\item $\Pi_1 = \{\varpi_{1\texttt{tl}},\varpi_{1\texttt{tr}},\varpi_{1\texttt{br}},\varpi_{1\texttt{bl}}\}$ shown in red;
\item $\Pi_2 = \{\varpi_{2\texttt{tl}}, \varpi_{2\texttt{tr}}, \varpi_{2\texttt{bl}}\}$ shown in green;
\item $\Pi_3 = \{\varpi_{3\texttt{tr}}, \varpi_{3\texttt{br}}, \varpi_{3\texttt{bl}}\}$ shown in blue;
\item $\Pi_4 = \{\varpi_{4\texttt{tl}}, \varpi_{4\texttt{tr}}, \varpi_{4\texttt{br}}, \varpi_{4\texttt{bl}}\}$ shown in cyan.
\end{itemize}
The respective sets of atomic propositions (services) are $\Sigma_1 = \{\sigma_{11},\sigma_{12}\}$; $\Sigma_2 = \{\sigma_{21},\sigma_{22}, \sigma_{23}\}$; $\Sigma_3 = \{\sigma_{31},\sigma_{32}, \sigma_{33}\}$; and $\Sigma_4 = \{\sigma_{41},\sigma_{42}\}$. The regions are labeled as follows:
$L_1(\varpi_{1\texttt{tl}})=L_1(\varpi_{1\texttt{br}}) = \{\sigma_{11}\}$, $L_1(\varpi_{1\texttt{tr}})=L_1(\varpi_{1\texttt{bl}}) = \{\sigma_{12}\}$;
$L_2(\varpi_{2\texttt{tl}})= \{\sigma_{21}\}$, $L_2(\sigma_{2\texttt{tr}})=\{\sigma_{22}\}$, $L_2(\varpi_{2\texttt{bl}})=\{\sigma_{23}\}$;
$L_3(\varpi_{3\texttt{tr}})= \{\sigma_{31}\}$, $L_3(\varpi_{3\texttt{br}})=\{\sigma_{32}\}$, $L_3(\varpi_{3\texttt{bl}})=\{\sigma_{33}\}$; and finally
$L_4(\varpi_{4\texttt{tl}})=L_4(\varpi_{4\texttt{tr}}) = \{\sigma_{41}\}$, $L_4(\varpi_{4\texttt{bl}})=L_4(\varpi_{4\texttt{br}}) = \{\sigma_{42}\}$.
The agents start from $[25,15]$, $[20,15]$, $[15,20]$, and $[20,25]$, respectively.
The uniform neighboring radius is $r=8m$ and the design parameter needed in Def.~\ref{edge} is $\delta=0.5m$.
The edge set of $G(0)$ is hence $E(0)=\{(\mathfrak{R}_1,\mathfrak{R}_2),(\mathfrak{R}_2,\mathfrak{R}_3)$, $(\mathfrak{R}_3,\mathfrak{R}_4)\}$.
The upper bound by~\eqref{v8} is $\varepsilon < \varepsilon_{\textrm{min}} \approx 0.031$ and we choose $\varepsilon=0.03$.
We consider two cases of the agent task specifications: one {with} sc-LTL formulas and one {with} general LTL formulas.
\medskip
\noindent
\emph{sc-LTL Task Specifications.}
The local task of agent $\mathfrak{R}_1$ to provide service $\sigma_{12}$, then $\sigma_{11}$ and at last again $\sigma_{12}$. The corresponding LTL formula is $\varphi_1 = \Diamond(\sigma_{12} \wedge \Diamond (\sigma_{11} \wedge \Diamond \sigma_{12}))$.
Agent $\mathfrak{R}_2$ is asked to provide service $\sigma_{21}$ or $\sigma_{22}$ and service $\sigma_{23}$ in any order, formalized as $\varphi_2^s= \Diamond(\sigma_{21} \vee \sigma_{22})\wedge \Diamond \sigma_{23}$.
The task of agent $\mathfrak{R}_3$ is to provide service $\sigma_{31}$ or $\sigma_{32}$ and service $\sigma_{33}$ in any order, formalized as $\varphi_2^s= \Diamond(\sigma_{31} \vee \sigma_{32})\wedge \Diamond \sigma_{33}$.
Finally, agent $\mathfrak{R}_4$ is required to provide service $\sigma_{42}$, then $\sigma_{41}$ and at last again service $\sigma_{42}$, represented by the LTL formula $\varphi_4 =\Diamond(\sigma_{42} \wedge \Diamond (\sigma_{41} \wedge \Diamond \sigma_{42}))$.
The synthesized discrete plans are as follows:
\begin{itemize}\itemsep-0.5ex
\item $\tau_{1} = (\varpi_{1\texttt{bl}},\{\sigma_{12}\})(\varpi_{1\texttt{tl}},\{\sigma_{11}\})(\varpi_{1\texttt{bl}},\{\sigma_{12}\})$
\item $\tau_{2} = (\varpi_{2\texttt{tl}},\{\sigma_{21}\})(\varpi_{2\texttt{bl}},\{\sigma_{23}\})$
\item $\tau_{3} = (\varpi_{3\texttt{tr}},\{\sigma_{31}\})(\varpi_{3\texttt{br}},\{\sigma_{33}\})$
\item $\tau_{4} = (\varpi_{4\texttt{br}},\{\sigma_{41}\})(\varpi_{4\texttt{tr}},\{\sigma_{42}\})(\varpi_{4\texttt{tl}},\{\sigma_{41}\})$
\end{itemize}
At $t=0$, the switching policy from Sec.~\ref{switch} is applied.
The agent trajectories are shown in Fig.~\ref{safe_static}, where the distances between the neighboring agents along with times of reaching the agents' progressive goal regions are plotted, too.
\medskip
\begin{figure}[t]
\begin{minipage}[t!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=1\textwidth]{live_traj_stuck.pdf}
\end{minipage}
\begin{minipage}[ht!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=0.54\textwidth]{live_distance_stuck.pdf}
\includegraphics[width =0.95\textwidth, height=0.44\textwidth]{live_reach_stuck.pdf}
\end{minipage}
\caption{The system with general LTL formulas after application of switching policy from Sec.~\ref{switch}. All agent converge to $\varpi_{2\texttt{tl}}$ and stop there due to the fact that $\mathfrak{R}_2$ remains active with an infinite plan to stay at $\varpi_{2\texttt{tl}}$.}
\label{live_stuck}
\end{figure}
\noindent \emph{General LTL task specifications}.
The task of agent $\mathfrak{R}_1$ to periodically provide both services $\sigma_{11}$ and $\sigma_{12}$, represented by $\phi_1 = \square \Diamond \sigma_{11} \wedge \square \Diamond \sigma_{12}$.
The task of agent $\mathfrak{R}_2$ is to periodically provide one of the services $\sigma_{21}$ or $\sigma_{22}$ or $\sigma_{23}$, formalized as $\phi_2= \square \Diamond(\sigma_{21} \vee \sigma_{22} \vee \sigma_{23})$. Finally, the tasks of agents $\mathfrak{R}_3$ and $\mathfrak{R}_4$ are
$\phi_3= \square \Diamond(\sigma_{31} \vee \sigma_{32} \vee \sigma_{33})$, and $\phi_4 = \square \Diamond \sigma_{41} \wedge \square \Diamond \sigma_{42}$.
The synthesized discrete plans are as follows:
\begin{itemize}\itemsep-0.5ex
\item $\tau_{1} = \big((\varpi_{1\texttt{bl}},\{\sigma_{12}\})\big((\varpi_{1\texttt{tl}},\{\sigma_{11}\})\big)^\omega$
\item $\tau_{2} = (\varpi_{2\texttt{tl}},\{\sigma_{21}\})^\omega$
\item $\tau_{3} = (\varpi_{3\texttt{bl}},\{\sigma_{33}\})^\omega$
\item $\tau_{4} = \big((\varpi_{4\texttt{br}},\{\sigma_{41}\})(\varpi_{4\texttt{tr}},\{\sigma_{42}\})\big)^\omega$
\end{itemize}
{
First, we simulated the scenario where we applied the activity switching protocol for sc-LTL formulas proposed in Sec.~\ref{switch}. Fig.~\ref{live_stuck} shows that the first progressive goal visited is $\varpi_{2\texttt{tl}}$. Since the agent $\mathfrak{R}_2$ stays active by the protocol, and its next progressive goal region is again $\varpi_{2\texttt{tl}}$, the whole system has reached its stable local minimum. Hence, all agents converge very close to one point and stop.
In contrast, the activity switching protocol from Sec.~\ref{switch-infinite} avoids such an unwanted behavior.
The simulation results for the activity switching protocol from Sec.~\ref{switch-infinite} are illustrated in Fig.~\ref{live_static}.
The functions $f_{\mathrm{prob}}$ and $f_{\mathrm{cond}}$ were chosen in a way that allows to partially trade fairness of activity switching for increased efficiency of plan executions measured in terms of the distance traveled between consecutive visits to progressive goal regions. More specifically, an agent is not switched to passive immediately after it reaches one of its goal region. Rather than that, it has the following probability of remaining active:
$$
\textbf{Pr}(b_i=1)= \begin{cases}
e^{-\alpha_i\Upsilon_i[i](t-\chi_i)},&\quad \text{if}\quad \Upsilon_i[i]\cdot (t-\chi_i)<\bar{\chi}_i,\\
0,&\quad \text{if}\quad \Upsilon_i[i]\cdot (t-\chi_i)\geq \bar{\chi}_i,
\end{cases}
$$
where $\bar{\chi}_i = 5$, and $\alpha_i = 1$. The probability of remaining active decreases with the increasing time elapsed since the current round started and with the increasing number agent $\mathfrak{R}_i$'s own progressive goal region was visited. Note that there exists a finite $T \in (T_{\circlearrowleft_{m-1}},T_{\circlearrowleft_{m}})$, such that $\Upsilon_i[i]\cdot (t-\chi_i)\geq \bar{\chi}_i$ for all $t \in [T, T_{\circlearrowleft_m}]$, hence each agent $\mathfrak{R}_i$ is guaranteed to be switched to passive control mode eventually.
The selected function does not necessarily yield a monotonic decrease of the total number of active agents in the team and is particularly useful when one agent has a set of goal regions whose locations are close.
\begin{figure}[t]
\begin{minipage}[t!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=1\textwidth]{live_traj_static.pdf}
\end{minipage}
\begin{minipage}[ht!]{0.49\linewidth}
\centering
\includegraphics[width =1\textwidth, height=0.54\textwidth]{live_distance_static.pdf}
\includegraphics[width =0.95\textwidth, height=0.44\textwidth]{live_reach_static.pdf}
\end{minipage}
\caption{Left: agents' respective regions of interest in red, green, blue and cyan respectively and their trajectories after execution of switching policy from Sec.~\ref{switch-infinite} for $20$s. Top-right: the evolution of pair-wise distances $\|x_{12}\|,\|x_{23}\|, \|x_{34}\|$, which all stay below $7.5m$. Bottom-right: the time instants when the agents reach their goal regions according to their plan.}
\label{live_static}
\end{figure}
\section{Conclusion and Future Work}\label{sec:conc}
We proposed a distributed communication-free hybrid control scheme for multi-agent systems to fulfil locally-assigned tasks as general or sc-LTL formulas, while at the same time subject to relative-distance constraints.
Future work plans include handling uncertainties in the relative state measurements and considering more complex agent dynamics. We also plan to relax the requirement on the completness of the graph $G(t)$.
\section{Acknowledgements}
{This work was supported by EU STREP RECONFIG: FP7-ICT-2011-9-600825 and the Swedish Research Council.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,889 |
Q: How can I sort a Map>? I have done a lot of research for this question, but I have not found a way to sort a map of custom object lists (Map<String, List<CustomObj>>), basing the comparison on CustomObj attributes (as SORT_BY_NAME, SORT_BY_DATE, etc).
A motivating example of my question is:
*
*I have a custom object: Person (with attribute as Name, DateOfBith, etc ...);
*I have a Map of Person object List as: Map<String, List<Person>>. The map key is a String used for other purposes;
*I would like to create a comparator and a sorting method that sorts the map in ascending order based on comparisons between the attributes of the Person object (name, date, etc ..)
For simplicity I report the real code but adapted to a simplified case of Person object, because it would already represent the concept of entity.
Person.java -> Custom object
public class Person {
private String name;
private Date dateOfBirth;
...
// Empty and Full attrs Constructors
...
// Getter and Setter
...
// Comparator by name
public static Comparator<Person> COMPARE_BY_NAME = Comparator.comparing(one -> one.name);
// Comparator by date
public static Comparator<Person> COMPARE_BY_DATE = Comparator.comparing(one -> one.dateOfBirth);
}
Sorter.java -> Sorter object
public class Sorter {
// List Comparator of Person by Date
public static final Comparator<? super List<Person>> COMPARATOR_BY_DATE = (Comparator<List<Person>>) (p1, p2) -> {
for (Persontab person1: p1) {
for (Person person2: p2) {
return Person.COMPARE_BY_DATE.compare(person1, person2);
}
}
return 0;
};
// List Comparator of Person by Name
public static final Comparator<? super List<Person>> COMPARATOR_BY_NAME = (Comparator<List<Person>>) (p1, p2) -> {
for (Persontab person1: p1) {
for (Person person2: p2) {
return Person.COMPARE_BY_NAME.compare(person1, person2);
}
}
return 0;
};
// Sorting method
public Map<String, List<Person>> sort(Map<String, List<Person>> map, Comparator<? super List<Person>> comparator) {
return map.entrySet()
.stream()
.sorted(Map.Entry.comparingByValue(comparator))
.collect(Collectors.toMap(Map.Entry::getKey, Map.Entry::getValue, (v1, v2) -> v1, LinkedHashMap::new));
}
}
Main.java -> Start code
public class MainApp {
public static void main(String[] args) {
Map<String, List<Person>> exampleMap = new HashMap<>();
List<Person> personList = new ArrayList<>();
personList.add(new Person("name1", new Date("2022-01-01")));
personList.add(new Person("name12", new Date("2022-01-05")));
personList.add(new Person("name13", new Date("2022-01-03")));
map.put("2022-01", personList);
personList.clear();
personList.add(new Person("name14", new Date("2021-02-01")));
personList.add(new Person("name3", new Date("2021-02-05")));
personList.add(new Person("name4", new Date("2021-02-03")));
map.put("2021-02", personList);
Sorter sorter = new Sorter();
// Example of sorting by date
map = sorter.sort(exampleMap, Sorter.COMPARATOR_BY_DATE);
// In this case the sorting works correctly, or rather it sorts the items by date as I expect
// Example of sorting by name
map = sorter.sort(exampleMap, Sorter.COMPARATOR_BY_NAME);
// In this case, I don't think sorting works correctly. Sort each list of elements for each key in ascending order. But it doesn't sort the map elements.
/* I expect to have the following map when sort by date:
"2021-02": [
Person("name14", new Date("2021-02-01")),
Person("name4", new Date("2021-02-03")),
Person("name3", new Date("2021-02-05"))
],
"2022-01": [
Person("name14", new Date("2021-02-01")),
Person("name13", new Date("2022-01-03")),
Person("name12", new Date("2022-01-05"))
]
}
}
A: First, let's reiterate: a HashMap is unordered so you need something else. Your Sorter.sort() method actually collects the values into a LinkedHashMap which provides an iteration order based on insert order and would be ok for your use case. Just to be clear (also for the sake of others): this doesn't sort the map itself but creates a new LinkedHashMap.
Now to your comparators: if you want to compare 2 lists you probably want to compare elements at equal indices. Thus your comparator needs to be something like this:
Comparator<List<Person>> = (l1, l2) -> {
Iterator<Person> itr1 = l1.iterator();
Iterator<Person> itr2 = l2.iterator();
while( itr1.hasNext() && itr2.hasNext() ) {
Person p1 = itr1.next();
Person p2 = itr1.next();
int result = Person.COMPARE_BY_DATE.compare(p1, p2);
if( result != 0 ) {
return result;
}
}
return 0;
};
However, the lists might have different lengths as well so you might want to handle that too:
Comparator<List<Person>> = (l1, l2) -> {
//iterators and loop here
//after the loop it seems all elements at equal indices are equal too
//now compare the sizes
return Integer.compare(l1.size(), l2.size());
}
A: By changing the type of Map used in my code above, as suggested by @Thomas, in TreeMap<>, I found the solution to the problem as follows:
*
*I first merged all the Person object lists into one list. This was then sorted by the chosen criterion, for example Person.COMPARE_BY_NAME;
*I created an algorithm that would re-group the sorted lists, in according to the criteria of my project, in a map. The key of this map corresponds to the concatenation of the month + year of the Person object.
The algorithm is reported at the bottom of the comment;
*I sort the map based on the chosen attribute, for example Sorter.COMPARATOR_BY_NAME;
Di seguito il codice è come segue:
Merge all List<Person> in one -> main or somewhere before the Map was created
...
//
List<Person> newPersonList = new ArrayList<>();
newPersonList.addAll(oldPersonList1);
newPersonList.addAll(oldPersonList2);
...
Main or somewhere before the Map was created
...
groupList(Person.COMPARE_BY_NAME, Sorter.COMPARATOR_BY_NAME);
...
GroupPerson -> method to group the merged List<Person> in a TreeMap<String, List<Person>>
public Map<String, List<Person>> groupList(final Comparator<? super Person> itemComparator, final Comparator<? super List<Person>> listComparator)
// Sort Person list by comparator before create TreeSet
newPersonList.sort(itemComparator);
Map<String, List<Person>> personMapGrouped = new TreeMap<>();
// Here, create a Map of list
for (Person person: newPersonList) {
final SimpleDateFormat dateFormat = new SimpleDateFormat("yyyy MM", Locale.getDefault());
final String groupKey = dateFormat.format(person.getDateOfBirth());
if (personMapGrouped.containsKey(groupKey)) {
// The key is already in the TreeMap; add the Person object against the existing key.
final List<Person> personListGrouped = personMapGrouped.get(groupKey);
if (personListGrouped!= null) {
personListGrouped.add(person);
}
} else {
// The key is not there in the TreeMap; create a new key-value pair
final List<Person> personListGrouped = new ArrayList<>();
personListGrouped.add(person);
personMapGrouped.put(groupKey, personListGrouped);
}
}
// Here sort the Map by params passed
final TabPersonSorter sorter = new TabPersonSorter();
personMapGrouped = sorter.sort(personMapGrouped, listComparator);
}
In this case, using the lists created in the main above, the results obtained are:
"List<Person> mergedList": [
Person("name1", new Date("2022-01-01")),
Person("name3", new Date("2021-02-05")),
Person("name4", new Date("2021-02-03")),
Person("name12", new Date("2022-01-05")),
Person("name13", new Date("2022-01-03")),
Person("name14", new Date("2021-02-01"))
]
"Map<String, List<Person>> mergedMap": {
"2022-01": [
Person("name1", new Date("2022-01-01")),
Person("name12", new Date("2022-01-05")),
Person("name13", new Date("2022-01-03"))
],
"2021-02": [
Person("name3", new Date("2021-02-05")),
Person("name4", new Date("2021-02-03"))
],
"2022-02": [
Person("name14", new Date("2021-02-01"))
]
}
Obviously if the grouping in the map were not bound by such a restrictive date as only year + month, the sorting would have the desired effect in distinct groups.
In fact, in the case of the sorting by date, this is respected very well.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,475 |
{"url":"https:\/\/peeterjoot.wordpress.com\/2013\/02\/","text":"Peeter Joot's (OLD) Blog.\n\nMath, physics, perl, and programming obscurity.\n\n papasu on PHY450H1S. Relativistic Electr\u2026 papasu on Energy term of the Lorentz for\u2026 lidiodu on PHY450H1S. Relativistic Electr\u2026 lidiodu on PHY450H1S. Relativistic Electr\u2026 lidiodu on bivector form of Stokes\u00a0t\u2026\n\n\u2022 310,899\n\nArchive for February, 2013\n\nRotation of diatomic\u00a0molecules\n\nPosted by peeterjoot on February 28, 2013\n\n[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]\n\nQuestion: Rotation of diatomic molecules ([2] problem 3.6)\n\nIn our first look at the ideal gas we considered only the translational energy of the particles. But molecules can rotate, with kinetic energy. The rotation motion is quantized; and the energy levels of a diatomic molecule are of the form\n\n\\begin{aligned}\\epsilon(j) = j(j + 1) \\epsilon_0\\end{aligned} \\hspace{\\stretch{1}}(1.0.1)\n\nwhere $j$ is any positive integer including zero: $j = 0, 1, 2, \\cdots$. The multiplicity of each rotation level is $g(j) = 2 j + 1$.\n\na\n\nFind the partition function $Z_R(\\tau)$ for the rotational states of one molecule. Remember that $Z$ is a sum over all states, not over all levels \u2014 this makes a difference.\n\nb\n\nEvaluate $Z_R(\\tau)$ approximately for $\\tau \\gg \\epsilon_0$, by converting the sum to an integral.\n\nc\n\nDo the same for $\\tau \\ll \\epsilon_0$, by truncating the sum after the second term.\n\nd\n\nGive expressions for the energy $U$ and the heat capacity $C$, as functions of $\\tau$, in both limits. Observe that the rotational contribution to the heat capacity of a diatomic molecule approaches 1 (or, in conventional units, $k_{\\mathrm{B}}$) when $\\tau \\gg \\epsilon_0$.\n\ne\n\nSketch the behavior of $U(\\tau)$ and $C(\\tau)$, showing the limiting behaviors for $\\tau \\rightarrow \\infty$ and $\\tau \\rightarrow 0$.\n\na. Partition function $Z_R(\\tau)$\n\nTo understand the reference to multiplicity recall (section 4.13 [1]) that the rotational Hamiltonian was of the form\n\n\\begin{aligned}H = \\frac{\\mathbf{L}^2}{2 M r^2},\\end{aligned} \\hspace{\\stretch{1}}(1.0.2)\n\nwhere the $\\mathbf{L}^2$ eigenvectors satisfied\n\n\\begin{subequations}\n\n\\begin{aligned}\\mathbf{L}^2 {\\left\\lvert {l m} \\right\\rangle} = l (l + 1) \\hbar^2 {\\left\\lvert {l m} \\right\\rangle}\\end{aligned} \\hspace{\\stretch{1}}(1.0.3a)\n\n\\begin{aligned}L_z {\\left\\lvert {l m} \\right\\rangle} = m \\hbar {\\left\\lvert {l m} \\right\\rangle}\\end{aligned} \\hspace{\\stretch{1}}(1.0.3b)\n\n\\end{subequations}\n\nand $-l \\le m \\le l$, where $l \\ge 0$ is a positive integer. We see that $\\epsilon_0$ is of the form\n\n\\begin{aligned}\\epsilon_0 = \\frac{\\hbar^2}{2 M R_l(r)},\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nand our partition function is\n\n\\begin{aligned}Z_R(\\tau) = \\sum_{l = 0}^\\infty \\sum_{m = -l}^l e^{-l (l + 1)\\epsilon_0\/\\tau}= \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nWe have no dependence on $m$ in the sum, and just have to sum terms like fig 1, and are able to sum over $m$ trivially, which is where the multiplicity comes from.\n\nFig 1: Summation over m\n\nTo get a feel for how many terms are significant in these sums, we refer to the plot of fig 2. We plot the partition function itself in, truncation at $l = 30$ terms in fig 3.\n\nFig 2: Plotting the partition function summand\n\nFig 3: Z_R(tau) truncated after 30 terms in log plot\n\nb. Evaluate partition function for large temperatures\n\nIf $\\tau \\gg \\epsilon_0$, so that $\\epsilon_0\/\\tau \\ll 1$, all our exponentials are close to unity. Employing an integral approximation of the partition function, we can somewhat miraculously integrate this directly\n\n\\begin{aligned}Z_R(\\tau) &\\approx \\int_0^\\infty dl (2 l + 1) e^{-l(l+1)\\epsilon_0\/\\tau} \\\\ &= \\int_0^\\infty dl \\frac{d}{dl} \\left( -\\frac{\\tau}{\\epsilon_0} e^{-l(l+1)\\epsilon_0\/\\tau} \\right) \\\\ &= \\frac{\\tau}{\\epsilon_0}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\nc. Evaluate partition function for small temperatures\n\nWhen $\\tau \\ll \\epsilon_0$, so that $\\epsilon_0\/\\tau \\gg 1$, all our exponentials are increasingly close to zero as $l$ increases. Dropping all the second and higher order terms we have\n\n\\begin{aligned}Z_R(\\tau) \\approx 1 + 3 e^{-2 \\epsilon_0\/\\tau}\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nd. Energy and heat capacity\n\nIn the large $\\epsilon_0\/\\tau$ domain (small temperatures) we have\n\n\\begin{aligned}U &= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\ln Z \\\\ &= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\ln \\left( 1 + 3 e^{-2 \\epsilon_0\/\\tau} \\right) \\\\ &= \\tau^2 \\frac{3 (-2\\epsilon_0)(-1\/\\tau^2)}{1 + 3 e^{-2 \\epsilon_0\/\\tau}} \\\\ &= \\frac{6 \\epsilon_0}{1 + 3 e^{-2 \\epsilon_0\/\\tau}} \\\\ &\\approx 6 \\epsilon_0.\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nThe specific heat in this domain is\n\n\\begin{aligned}C_{\\mathrm{V}} = \\frac{\\partial {U}}{\\partial {\\tau}}=\\left( \\frac{6 \\epsilon_0\/\\tau}{1 + 3 e^{-2 \\epsilon_0\/\\tau}} \\right)^2\\approx \\left( \\frac{6 \\epsilon_0}{\\tau} \\right)^2\\end{aligned} \\hspace{\\stretch{1}}(1.0.9)\n\nFor the small $\\epsilon_0\/\\tau$ (large temperatures) case we have\n\n\\begin{aligned}U = \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\ln Z= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\ln \\frac{\\tau}{\\epsilon_0}= \\tau^2 \\frac{1}{{\\tau}}= \\tau\\end{aligned} \\hspace{\\stretch{1}}(1.0.10)\n\nThe heat capacity in this large temperature region is\n\n\\begin{aligned}C_{\\mathrm{V}} = \\frac{\\partial {U}}{\\partial {\\tau}} = 1,\\end{aligned} \\hspace{\\stretch{1}}(1.0.11)\n\nwhich is unity as described in the problem.\n\ne. Sketch\n\nThe energy and heat capacities are roughly sketched in fig 4.\n\nFig 4: Energy and heat capacity\n\nIt\u2019s somewhat odd seeming that we have a zero point energy at zero temperature. Plotting the energy (truncating the sums to 30 terms) in fig 5, we don\u2019t see such a zero point energy.\n\nFig 5: Exact plot of the energy for a range of temperatures (30 terms of the sums retained)\n\nThat plotted energy is as follows, computed without first dropping any terms of the partition function\n\n\\begin{aligned}U &= \\tau^2 \\frac{\\partial}{\\partial \\tau} \\ln\\left( \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right) \\\\ &= \\epsilon_0\\frac{\\left( \\sum_{l = 1}^\\infty l (l + 1)(2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{\\left( \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)} \\\\ &= \\epsilon_0\\frac{\\left( \\sum_{l = 1}^\\infty l (l + 1)(2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{Z}\\end{aligned} \\hspace{\\stretch{1}}(1.0.12)\n\nTo avoid the zero point energy, we have to use this and not the truncated partition function to do the integral approximation. Doing that calculation (which isn\u2019t as convenient, so I cheated and used Mathematica). We obtain\n\n\\begin{aligned}U \\approx \\frac{\\int_1^\\infty l (l + 1)(2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau}}{\\int_0^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau}}=\\epsilon_0 e^{2 \\epsilon_0\/\\tau} \\left( 2 + \\frac{\\tau}{\\epsilon_0} \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.13)\n\nThis approximation, which has taken the sums to infinity, is plotted in fig 6.\n\nFig 6: Low temperature approximation of the energy\n\nFrom eq. 1.0.12, we can take one more derivative to calculate the exact specific heat\n\n\\begin{aligned}C_{\\mathrm{V}} &= \\epsilon_0\\frac{\\partial {}}{\\partial {\\tau}}\\left(\\frac{\\left( \\sum_{l = 1}^\\infty l (l + 1)(2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{\\left( \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}\\right) \\\\ &= \\left( \\frac{\\epsilon_0}{\\tau} \\right)^2\\left(\\frac{\\left( \\sum_{l = 1}^\\infty l^2 (l + 1)^2 (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{\\left( \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}+\\frac{\\left( \\sum_{l = 1}^\\infty l (l + 1)(2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)^2}{\\left( \\sum_{l = 0}^\\infty (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)^2}\\right) \\\\ &= \\left( \\frac{\\epsilon_0}{\\tau} \\right)^2\\left(\\frac{\\left( \\sum_{l = 1}^\\infty l^2 (l + 1)^2 (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{Z}+ \\frac{U^2}{\\epsilon_0^2}\\right) \\\\ &= \\frac{U^2}{\\epsilon_0^2}+\\left( \\frac{\\epsilon_0}{\\tau} \\right)^2\\frac{\\left( \\sum_{l = 1}^\\infty l^2 (l + 1)^2 (2 l + 1) e^{-l (l + 1)\\epsilon_0\/\\tau} \\right)}{Z}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.14)\n\nThis is plotted to 30 terms in fig 7.\n\nFig 7: Specific heat to 30 terms\n\nReferences\n\n[1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.\n\n[2] C. Kittel and H. Kroemer. Thermal physics. WH Freeman, 1980.\n\nPHY452H1S Basic Statistical Mechanics. Lecture 12: Helmholtz free energy. Taught by Prof. Arun\u00a0Paramekanti\n\nPosted by peeterjoot on February 28, 2013\n\n[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]\n\nDisclaimer\n\nPeeter\u2019s lecture notes from class. May not be entirely coherent.\n\nCanonical partition\n\nWe found\n\n\\begin{subequations}\n\n\\begin{aligned}\\frac{\\sigma_{\\mathrm{E}}}{E} \\propto \\frac{T \\sqrt{C_V}}{E} k_{\\mathrm{B}}^2\\end{aligned} \\hspace{\\stretch{1}}(1.0.1a)\n\n\\begin{aligned}Z = \\sum_{\\{c\\}} e^{-\\beta E(c)}\\end{aligned} \\hspace{\\stretch{1}}(1.0.1b)\n\n\\begin{aligned}C_V \\sim N\\end{aligned} \\hspace{\\stretch{1}}(1.0.1c)\n\n\\begin{aligned}E \\sim N\\end{aligned} \\hspace{\\stretch{1}}(1.0.1d)\n\n\\end{subequations}\n\nwhere the partition function \\index{partition function} acts as a probability distribution so that we can define an average as\n\n\\begin{aligned}\\left\\langle{{A}}\\right\\rangle = \\frac{\\sum_{\\{c\\}} A(c) e^{-\\beta E(c)}}{Z}\\end{aligned} \\hspace{\\stretch{1}}(1.0.2)\n\nIf we suppose that the energy is typically close to the average energy as in fig. 1.1.\n\nFig 1.1: Peaked energy distribution\n\n, then we can approximate the partition function as\n\n\\begin{aligned}Z \\approx e^{-\\beta \\left\\langle{{E}}\\right\\rangle} \\sum_{\\{c\\}} \\delta_{E, \\bar{E}}= e^{-\\beta \\left\\langle{{E}}\\right\\rangle} e^S\/k_{\\mathrm{B}},\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nwhere we\u2019ve used $S = k_{\\mathrm{B}} \\ln \\Omega$ to express the number of states where the energy matches the average energy $\\Omega = \\sum \\delta_{E, \\bar{E}}$.\n\nThis gives us\n\n\\begin{aligned}Z = e^{-\\beta (\\left\\langle{{E}}\\right\\rangle - k_{\\mathrm{B}} T S\/k_{\\mathrm{B}}) } = e^{-\\beta (\\left\\langle{{E}}\\right\\rangle - T S) } \\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nor\n\n\\begin{aligned}\\boxed{Z = e^{-\\beta F},}\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nwhere we define the Helmholtz free energy $F$ as\n\n\\begin{aligned}\\boxed{F = \\left\\langle{{E}}\\right\\rangle - T S.}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\nEquivalently, the log of the partition function provides us with the partition function\n\n\\begin{aligned}F = - k_{\\mathrm{B}} T \\ln Z.\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nRecalling our expression for the average energy, we can now write that in terms of the free energy\n\n\\begin{aligned}\\left\\langle{{E}}\\right\\rangle = \\frac{\\sum_{\\{c\\}} E(c) e^{-\\beta E(c)}}{\\sum_{\\{c\\}} e^{-\\beta E(c)}}= -\\frac{\\partial {}}{\\partial {\\beta}}\\ln Z=\\frac{\\partial {(\\beta F)}}{\\partial {\\beta}}\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nQuantum mechanical picture\n\nConsider a subsystem as in fig. 1.2 where we have states of the form\n\nFig 1.2: subsystem in heat bath\n\n\\begin{aligned}{\\left\\lvert {\\Psi_{\\text{full}}} \\right\\rangle} = {\\left\\lvert {\\chi_{\\text{subsystem}}} \\right\\rangle} {\\left\\lvert {\\phi_{\\text{bath}}} \\right\\rangle}\\end{aligned} \\hspace{\\stretch{1}}(1.0.9)\n\nand a total Hamiltonian operator of the form\n\n\\begin{aligned}H_{\\text{full}} = H_{\\text{subsystem}} + H_{\\text{bath}} (+ H_{\\text{coupling}})\\end{aligned} \\hspace{\\stretch{1}}(1.0.10)\n\nwhere the total energy of the state, given energy eigenvalues $\\mathcal{E}_n$ and $\\lambda_n$ for the states ${\\left\\lvert {\\chi_{\\text{subsystem}}} \\right\\rangle}$ and ${\\left\\lvert {\\phi_{\\text{bath}}} \\right\\rangle}$ respectively, is given by the sum\n\n\\begin{aligned}E = \\mathcal{E}_m + \\lambda_n.\\end{aligned} \\hspace{\\stretch{1}}(1.0.11)\n\nHere $\\mathcal{E}_m, \\lambda_n$ are many body energies, so that $\\delta E \\sim \\#e^{-\\#N}$.\n\nWe can now write the total number of states as\n\n\\begin{aligned}\\Omega(E) &= \\underbrace{\\sum_m}_{\\text{subsystem}}\\underbrace{\\sum_n}_{\\text{bath}}\\delta(E - \\mathcal{E}_m -\\lambda_n)\\\\ &= \\sum_m e^{\\frac{1}{{k_{\\mathrm{B}}}} S(E - \\mathcal{E}_m)} \\\\ &\\approx \\sum_m e^{\\frac{1}{{k_{\\mathrm{B}}}} S(E)}e^{\\beta \\mathcal{E}_m}\\end{aligned} \\hspace{\\stretch{1}}(1.0.12)\n\n\\begin{aligned}Z = \\sum_m e^{-\\beta \\mathcal{E}_m} = \\text{Tr} \\left( e^{-\\beta \\hat{H}_{\\text{subsystem}}} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.13)\n\nWe\u2019ve ignored the coupling term in eq. 1.0.10. This is actually a problem in quantum mechanics since we require this coupling to introduce state changes.\n\nExample: Spins\n\nGiven $N$ spin $1\/2$ objects $\\uparrow$, $\\downarrow$, satisfying\n\n\\begin{aligned}S_z = \\pm \\frac{1}{{2}} \\hbar\\end{aligned} \\hspace{\\stretch{1}}(1.0.14)\n\nDropping $\\hbar$ we have\n\n\\begin{aligned}S_z \\rightarrow \\pm \\frac{1}{{2}} \\sigma\\end{aligned} \\hspace{\\stretch{1}}(1.0.15)\n\nOur system has a state ${\\left\\lvert {\\sigma_1, \\sigma_2, \\cdots \\sigma_N} \\right\\rangle}$ where $\\sigma_i = \\pm 1$. The total number of states is $2^N$.\n\nOur Hamiltonian is\n\n\\begin{aligned}\\hat{H} = - B \\sum_i \\hat{S}_{z_i}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.16)\n\nThis is the associated with the Zeeman effect, where states can be split by a magnetic field, as in fig. 1.3.\n\nFig 1.3: Zeeman splitting\n\nOur minimum and maximum energies are\n\n\\begin{subequations}\n\n\\begin{aligned}E_{\\mathrm{min}} = -\\frac{B}{2} N\\end{aligned} \\hspace{\\stretch{1}}(1.0.17a)\n\n\\begin{aligned}E_{\\mathrm{max}} = -\\frac{B}{2} N\\end{aligned} \\hspace{\\stretch{1}}(1.0.17b)\n\n\\end{subequations}\n\nThe total energy difference is\n\n\\begin{aligned}\\Delta E = B N,\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nand the energy differences are\n\n\\begin{aligned}\\delta E \\sim \\frac{B N}{2^N} \\sim \\# e^{-\\# N}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nThis is a measure of the average energy difference between two adjacent energy levels. In a real system we cannot assume that we have non-interacting spins. Any weak interaction will split our degenerate energy levels as in fig. 1.4.\n\nFig 1.4: Interaction splitting\n\nWe can now express the partition function\n\n\\begin{aligned}Z &= \\sum_{\\{\\sigma\\}} e^{-\\beta \\left( -\\frac{B}{2} \\sum_i \\sigma_i \\right)} \\\\ &= \\left( \\sum_{\\{\\sigma_1\\}} \\exp \\left( -\\frac{\\beta B}{2} \\sigma_i \\right) \\right)\\left( \\sum_{\\{\\sigma_2\\}} \\exp \\left( -\\frac{\\beta B}{2} \\sigma_i \\right) \\right)\\cdots \\\\ &= \\left( \\exp \\left( -\\frac{\\beta B}{2} (1) \\right) + \\exp \\left( -\\frac{\\beta B}{2} (-1) \\right) \\right)^N \\\\ &= \\left( 2 \\cosh\\left( \\frac{B}{2 k_B T} \\right) \\right)^N\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nOur free energy is\n\n\\begin{aligned}F = - k_B T N \\ln \\left( 2 \\cosh \\left( \\frac{B}{2 k_B T} \\right) \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nFor the expected value of the spin we find\n\n\\begin{aligned}\\left\\langle{{S_z}}\\right\\rangle = \\sum_i \\left\\langle{{S_{z_i}}}\\right\\rangle\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\n\\begin{aligned}\\left\\langle{{S_{z_i}}}\\right\\rangle=\\frac{1}{{2}} \\frac{\\sum_\\sigma \\sigma e^{\\beta B \\sigma\/2}}{\\sum_\\sigma e^{\\beta B \\sigma\/2}}= \\frac{1}{{2}} \\tanh \\left( \\frac{B}{2 k_B T} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nPHY452H1S Basic Statistical Mechanics. Lecture 11: Statistical and thermodynamic connection. Taught by Prof.\\ Arun\u00a0Paramekanti\n\nPosted by peeterjoot on February 27, 2013\n\nDisclaimer\n\nPeeter\u2019s lecture notes from class. May not be entirely coherent.\n\nConnections between statistical and thermodynamic views\n\n\u2022 \u201cHeat\u201d. Disorganized energy.\n\u2022 $S_{\\text{Statistical entropy}}$. This is the thermodynamic entropy introduced by Boltzmann (microscopic).\n\nIdeal gas\n\n\\begin{aligned}H = \\sum_{i = 1}^N \\frac{\\mathbf{p}_i^2}{2m}\\end{aligned} \\hspace{\\stretch{1}}(1.3.1)\n\n\\begin{aligned}\\Omega(E) = \\frac{1}{{h^{3N} N!}}\\int d\\mathbf{x}_1 d\\mathbf{x}_2 \\cdots d\\mathbf{x}_Nd\\mathbf{p}_1 d\\mathbf{p}_2 \\cdots d\\mathbf{p}_N\\delta( E - H )\\end{aligned} \\hspace{\\stretch{1}}(1.3.2)\n\nLet\u2019s isolate the contribution of the Hamiltonian from a single particle and all the rest\n\n\\begin{aligned}H = \\frac{\\mathbf{p}_1^2}{2m}+\\sum_{i \\ne 1}^N \\frac{\\mathbf{p}_i^2}{2m}=\\frac{\\mathbf{p}_1^2}{2m}+H'\\end{aligned} \\hspace{\\stretch{1}}(1.3.3)\n\nso that the number of states in the phase space volume in the phase space region associated with the energy is\n\n\\begin{aligned}\\Omega(N, E) &= \\frac{V^N}{h^{3N} N!}\\int d\\mathbf{p}_1\\int d\\mathbf{p}_2 d\\mathbf{p}_3 \\cdots d\\mathbf{p}_N\\delta( E - H' - H_1) \\\\ &= \\frac{V^{N-1}}{h^{3(N-1)} (N-1)!} \\frac{V}{h^3 N}\\int d\\mathbf{p}_1\\int d\\mathbf{p}_2 d\\mathbf{p}_3 \\cdots d\\mathbf{p}_N\\delta( E - H' - H_1) \\\\ &= \\frac{ V }{ h^3 N} \\int d\\mathbf{p}_1 \\Omega( N-1, E - H_1 )\\end{aligned} \\hspace{\\stretch{1}}(1.3.4)\n\nWith entropy defined by\n\n\\begin{aligned}S = k_{\\mathrm{B}} \\ln \\Omega,\\end{aligned} \\hspace{\\stretch{1}}(1.3.5)\n\nwe have\n\n\\begin{aligned}\\Omega( N-1, E - H_1 ) = \\exp\\left( \\frac{1}{k_{\\mathrm{B}}} S \\left( N-1, E - \\frac{\\mathbf{p}_1^2}{2m} \\right) \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.3.6)\n\nso that\n\n\\begin{aligned}\\Omega(N, E) =\\frac{ V }{ h^3 N} \\int d\\mathbf{p}_1 \\exp\\left( \\frac{1}{k_{\\mathrm{B}}} S \\left( N-1, E - \\frac{\\mathbf{p}_1^2}{2m} \\right) \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.3.7)\n\nFor $N \\gg 1$ and $E \\gg \\mathbf{p}_1^2\/2m$, the exponential can be approximated by\n\n\\begin{aligned}\\exp\\left( \\frac{1}{k_{\\mathrm{B}}} S \\left( N-1, E - \\frac{\\mathbf{p}_1^2}{2m} \\right) \\right)= \\exp\\left( \\frac{1}{k_{\\mathrm{B}}} \\left( S(N, E) - \\left( \\frac{\\partial {S}}{\\partial {N}} \\right)_{E, V} - \\frac{\\mathbf{p}_1^2}{2m} \\left( \\frac{\\partial {S}}{\\partial {E}} \\right)_{N, V} \\right) \\right),\\end{aligned} \\hspace{\\stretch{1}}(1.3.8)\n\nso that\n\n\\begin{aligned}\\Omega(N, E) = \\underbrace{\\frac{ V }{ h^3 N} \\int d\\mathbf{p}_1 e^{\\frac{S}{k_{\\mathrm{B}}}(N, E)}e^{-\\frac{1}{{k_{\\mathrm{B}}}}\\left( \\frac{\\partial {S}}{\\partial {N}} \\right)_{E, V}}}_{B}\\int d\\mathbf{p}_1 e^{-\\frac{\\mathbf{p}_1^2}{2m k_{\\mathrm{B}}}\\left( \\frac{\\partial {S}}{\\partial {E}} \\right)_{N, V}}.\\end{aligned} \\hspace{\\stretch{1}}(1.3.9)\n\nor\n\n\\begin{aligned}\\Omega(N, E) = B\\int d\\mathbf{p}_1 e^{-\\frac{\\mathbf{p}_1^2}{2m k_{\\mathrm{B}}}\\left( \\frac{\\partial {S}}{\\partial {E}} \\right)_{N, V}}.\\end{aligned} \\hspace{\\stretch{1}}(1.3.10)\n\n\\begin{aligned}\\mathcal{P}(\\mathbf{p}_1) \\propto e^{-\\frac{\\mathbf{p}_1^2}{2m k_{\\mathrm{B}} T}}.\\end{aligned} \\hspace{\\stretch{1}}(1.3.11)\n\nThis is the Maxwell distribution.\n\nNon-ideal gas. General classical system\n\nFig 1: Partitioning out a subset of a larger system\n\nBreaking the system into a subsystem $1$ and the reservoir $2$ so that with\n\n\\begin{aligned}H = H_1 + H_2\\end{aligned} \\hspace{\\stretch{1}}(1.4.12)\n\nwe have\n\n\\begin{aligned}\\Omega(N, V, E) &= \\int d\\{x_1\\}d\\{p_1\\}d\\{x_2\\}d\\{p_2\\}\\delta( E - H_1 - H_2 ) \\frac{1}{{ h^{3N_1} N_1! h^{3 N_2} N_2!}} \\\\ &\\propto \\int d\\{x_1\\}d\\{p_1\\}e^{\\frac{1}{{k_{\\mathrm{B}}}} S(E - H_1, N - N_1)}\\end{aligned} \\hspace{\\stretch{1}}(1.4.13)\n\n\\begin{aligned}\\Omega(N, V, E) \\sim \\int d\\{x_1\\}d\\{p_1\\}\\underbrace{e^{\\frac{1}{{k_{\\mathrm{B}}}}S(E, N)}e^{-\\frac{N_1 }{k_{\\mathrm{B}}}\\left( \\frac{\\partial {S}}{\\partial {N}} \\right)_{E, V}}}_{\\text{environment'', or heat bath''}}e^{-\\frac{H_1 }{k_{\\mathrm{B}}}\\left( \\frac{\\partial {S}}{\\partial {E}} \\right)_{N, V}}\\end{aligned} \\hspace{\\stretch{1}}(1.4.14)\n\n\\begin{aligned}H_1 = \\sum_{i \\in 1} \\frac{\\mathbf{p}_i}{2m}+\\sum_{i \\in j} V(\\mathbf{x}_i - \\mathbf{x}_j)+ \\sum_{i \\in 1} \\Phi(\\mathbf{x}_i)\\end{aligned} \\hspace{\\stretch{1}}(1.4.15)\n\n\\begin{aligned}\\mathcal{P} \\propto e^{-\\frac{H( \\{x_1\\} \\{p_1\\} ) }{k_{\\mathrm{B}} T} }\\end{aligned} \\hspace{\\stretch{1}}(1.4.16)\n\nand for the subsystem\n\n\\begin{aligned}\\mathcal{P}_1 =\\frac{e^{-\\frac{H_1}{k_{\\mathrm{B}} T} }}{\\int d\\{x_1\\}d\\{p_1\\}e^{-\\frac{H_1}{k_{\\mathrm{B}} T} }}\\end{aligned} \\hspace{\\stretch{1}}(1.4.17)\n\nCanonical ensemble\n\nCan we use results for this subvolume, can we use this to infer results for the entire system? Suppose we break the system into a number of smaller subsystems as in fig. 1.2.\n\nFig 2: Larger system partitioned into many small subsystems\n\n\\begin{aligned}\\underbrace{(N, V, E)}_{\\text{microcanonical}}\\rightarrow (N, V, T)\\end{aligned} \\hspace{\\stretch{1}}(1.5.18)\n\nWe\u2019d have to understand how large the differences between the energy fluctuations of the different subsystems are. We\u2019ve already assumed that we have minimal long range interactions since we\u2019ve treated the subsystem $1$ above in isolation. With $\\beta = 1\/(k_{\\mathrm{B}} T)$ the average energy is\n\n\\begin{aligned}\\left\\langle{{E}}\\right\\rangle = \\frac{\\int d\\{x_1\\}d\\{p_1\\}He^{- \\beta H }}{\\int d\\{x_1\\}d\\{p_1\\}e^{- \\beta H }}\\end{aligned} \\hspace{\\stretch{1}}(1.5.19)\n\n\\begin{aligned}\\left\\langle{{E^2}}\\right\\rangle = \\frac{\\int d\\{x_1\\}d\\{p_1\\}H^2e^{- \\beta H }}{\\int d\\{x_1\\}d\\{p_1\\}e^{- \\beta H }}\\end{aligned} \\hspace{\\stretch{1}}(1.5.20)\n\nWe define the partition function\n\n\\begin{aligned}Z \\equiv \\frac{1}{{h^{3N} N!}}\\int d\\{x_1\\}d\\{p_1\\}e^{- \\beta H }.\\end{aligned} \\hspace{\\stretch{1}}(1.5.21)\n\nObserve that the derivative of $Z$ is\n\n\\begin{aligned}\\frac{\\partial {Z}}{\\partial {\\beta}} = -\\frac{1}{{h^{3N} N!}}\\int d\\{x_1\\}d\\{p_1\\}He^{- \\beta H },\\end{aligned} \\hspace{\\stretch{1}}(1.5.22)\n\nallowing us to express the average energy compactly in terms of the partition function\n\n\\begin{aligned}\\left\\langle{{E}}\\right\\rangle = -\\frac{1}{{Z}} \\frac{\\partial {Z}}{\\partial {\\beta}} = - \\frac{\\partial {\\ln Z}}{\\partial {\\beta}}.\\end{aligned} \\hspace{\\stretch{1}}(1.5.23)\n\nTaking second derivatives we find the variance of the energy\n\n\\begin{aligned}\\frac{\\partial^2 {{\\ln Z}}}{\\partial {{\\beta}}^2} &=\\frac{\\partial {}}{\\partial {\\beta}}\\frac{\\int d\\{x_1\\}d\\{p_1\\}(-H)e^{- \\beta H }}{\\int d\\{x_1\\}d\\{p_1\\}e^{- \\beta H }} \\\\ &= \\frac{\\int d\\{x_1\\}d\\{p_1\\}(-H)^2e^{- \\beta H }}{\\int d\\{x_1\\}d\\{p_1\\}e^{- \\beta H }}-\\frac{\\left( \\int d\\{x_1\\} d\\{p_1\\} (-H) e^{- \\beta H } \\right)^2}{\\left( \\int d\\{x_1\\} d\\{p_1\\} e^{- \\beta H } \\right)^2} \\\\ &= \\left\\langle{{E^2}}\\right\\rangle - \\left\\langle{{E}}\\right\\rangle^2 \\\\ &= \\sigma_{\\mathrm{E}}^2\\end{aligned} \\hspace{\\stretch{1}}(1.5.24)\n\nWe also have\n\n\\begin{aligned}\\sigma_{\\mathrm{E}}^2 &= -\\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {\\beta}} \\\\ &= \\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {T}} \\frac{\\partial {T}}{\\partial {\\beta}} \\\\ &= -\\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {T}} \\frac{\\partial {}}{\\partial {\\beta}} \\frac{1}{{k_{\\mathrm{B}} \\beta}} \\\\ &= \\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {T}} \\frac{1}{{k_{\\mathrm{B}} \\beta^2}} \\\\ &= k_{\\mathrm{B}} T^2 \\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {T}}\\end{aligned} \\hspace{\\stretch{1}}(1.5.25)\n\nRecalling that the heat capacity was defined by\n\n\\begin{aligned}C_V = \\frac{\\partial {\\left\\langle{{E}}\\right\\rangle}}{\\partial {T}},\\end{aligned} \\hspace{\\stretch{1}}(1.5.26)\n\nwe have\n\n\\begin{aligned}\\sigma_{\\mathrm{E}}^2 = k_{\\mathrm{B}} T^2 C_V \\propto N\\end{aligned} \\hspace{\\stretch{1}}(1.5.27)\n\n\\begin{aligned}\\frac{\\sigma_{\\mathrm{E}}}{\\left\\langle{{E}}\\right\\rangle} \\propto \\frac{1}{{\\sqrt{N}}}\\end{aligned} \\hspace{\\stretch{1}}(1.5.28)\n\nOne dimensional well problem from Pathria chapter\u00a0II\n\nPosted by peeterjoot on February 16, 2013\n\nProblem 2.5 [2] asks to show that\n\n\\begin{aligned}\\oint p dq = \\left( { n + \\frac{1}{{2}} } \\right) h,\\end{aligned} \\hspace{\\stretch{1}}(1.0.1)\n\nprovided the particle\u2019s potential is such that\n\n\\begin{aligned}m \\hbar \\left\\lvert { \\frac{dV}{dq} } \\right\\rvert \\ll \\left( { m ( E - V ) } \\right)^{3\/2}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.2)\n\nI took a guess that this was actually the WKB condition\n\n\\begin{aligned}\\frac{k'}{k^2} \\ll 1,\\end{aligned} \\hspace{\\stretch{1}}(1.0.3)\n\nwhere the WKB solution was of the form\n\n\\begin{aligned}k^2(q) = 2 m (E - V(q))\/\\hbar^2\\end{aligned} \\hspace{\\stretch{1}}(1.0.4a)\n\n\\begin{aligned}\\psi(q) = \\frac{1}{{\\sqrt{k}}} e^{\\pm i \\int k(q) dq}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.4b)\n\nThe WKB validity condition is\n\n\\begin{aligned}1 \\gg \\frac{-2 m V'}{\\hbar} \\frac{1}{{2}} \\frac{1}{{\\sqrt{2 m (E - V)}}} \\frac{\\hbar^2}{2 m(E - V)}\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nor\n\n\\begin{aligned}m \\hbar \\left\\lvert {V'} \\right\\rvert \\ll \\left( {2 m (E - V)} \\right)^{3\/2}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\nThis differs by a factor of $2 \\sqrt{2}$ from the constraint specified in the problem, but I\u2019m guessing that constant factors of that sort have just been dropped.\n\nEven after figuring out that this question was referring to WKB, I didn\u2019t know what to make of the oriented integral $\\int p dq$. With $p$ being an operator in the QM context, what did this even mean. I found the answer in [1] section 12.12. Here $p$ just means\n\n\\begin{aligned}p(q) = \\hbar k(q),\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nwhere $k(q)$ is given by eq. 1.0.4a. The rest of the problem can also be found there and relies on the WKB connection formulas, which aren\u2019t derived in any text that I own. Quoting results based on other results that I don\u2019t know the origin of it\u2019s worthwhile, so that\u2019s as far as I\u2019ll attempt this question (but do plan to eventually look up and understand those WKB connection formulas, and then see how they can be applied in a problem like this).\n\nReferences\n\n[1] D. Bohm. Quantum Theory. Courier Dover Publications, 1989.\n\n[2] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.\n\n1D pendulum problem in phase\u00a0space\n\nPosted by peeterjoot on February 15, 2013\n\nProblem 2.6 in [1] asks for some analysis of the (presumably small angle) pendulum problem in phase space, including an integration of the phase space volume energy and period of the system to the area $A$ included within a phase space trajectory. With coordinates as in fig. 1.1, our Lagrangian is\n\nFig 1.1: 1d pendulum\n\n\\begin{aligned}\\mathcal{L} = \\frac{1}{{2}} m l^2 \\dot{\\theta}^2 - g m l ( 1 - \\cos\\theta ).\\end{aligned} \\hspace{\\stretch{1}}(1.0.1)\n\nAs a sign check we find for small $\\theta$ from the Euler-Lagrange equations $\\dot{d}{\\theta} = -(g\/l) \\theta$ as expected. For the Hamiltonian, we need the canonical momentum\n\n\\begin{aligned}p_\\theta = \\frac{\\partial {\\mathcal{L}}}{\\partial {\\dot{\\theta}}} = m l^2 \\dot{\\theta}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.2)\n\nObserve that this canonical momentum does not have dimensions of momentum, but that of angular momentum ($m l \\dot{\\theta} \\times l$).\n\nOur Hamiltonian is\n\n\\begin{aligned}H = \\frac{1}{{2 m l^2}} p_\\theta^2 + g m l ( 1 - \\cos\\theta ).\\end{aligned} \\hspace{\\stretch{1}}(1.0.3)\n\nHamilton\u2019s equations for this system, in matrix form are\n\n\\begin{aligned}\\frac{d{{}}}{dt}\\begin{bmatrix}\\theta \\\\ p_\\theta\\end{bmatrix}=\\begin{bmatrix}\\frac{\\partial {H}}{\\partial {p_\\theta}} \\\\ -\\frac{\\partial {H}}{\\partial {\\theta}} \\end{bmatrix}=\\begin{bmatrix}p_\\theta\/m l^2 \\\\ - g m l \\sin\\theta\\end{bmatrix}\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nWith $\\omega = g\/l$, it is convient to non-dimensionalize this\n\n\\begin{aligned}\\frac{d{{}}}{dt}\\begin{bmatrix}\\theta \\\\ p_\\theta\/ \\omega m l^2\\end{bmatrix}=\\omega\\begin{bmatrix}p_\\theta\/\\omega m l^2 \\\\ - \\sin\\theta\\end{bmatrix}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nNow we can make the small angle approximation. Writing\n\n\\begin{aligned}\\mathbf{u} = \\begin{bmatrix}\\theta \\\\ p_\\theta\/ \\omega m l^2\\end{bmatrix}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6a)\n\n\\begin{aligned}i = \\begin{bmatrix}0 & 1 \\\\ -1 & 0\\end{bmatrix}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6b)\n\nOur pendulum equation is reduced to\n\n\\begin{aligned}\\mathbf{u}' = i \\omega \\mathbf{u},\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nWith a solution that we can read off by inspection\n\n\\begin{aligned}\\mathbf{u} = e^{i \\omega t} \\mathbf{u}_0=\\begin{bmatrix}\\cos\\omega t & \\sin\\omega t \\\\ -\\sin\\omega t & \\cos \\omega t\\end{bmatrix}\\mathbf{u}_0\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nLet\u2019s put the initial phase space point into polar form\n\n\\begin{aligned}\\mathbf{u}_0^2= \\theta_0^2 + \\frac{p_0^2}{\\omega^2 m^2 l^4}= \\frac{2}{\\omega^2 m l^2}\\left( { \\frac{p_0^2}{2 m l^2} + \\frac{1}{{2}} \\omega^2 m l^2 \\theta_0^2 } \\right)=\\frac{2}{g m l}\\left( { \\frac{p_0^2}{2 m l^2} + \\frac{1}{{2}} g m l \\theta_0^2 } \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.9)\n\nThis doesn\u2019t appear to be an exact match for eq. 1.0.3, but we can write for small $\\theta_0$\n\n\\begin{aligned}1 - \\cos\\theta_0=2 \\sin^2 \\left( { \\frac{\\theta_0}{2} } \\right)\\approx2 \\left( { \\frac{\\theta_0}{2} } \\right)^2=\\frac{\\theta_0^2}{2}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.10)\n\nThis shows that we can rewrite our initial conditions as\n\n\\begin{aligned}\\mathbf{u}_0 = \\sqrt{ \\frac{2 E}{g m l} }e^{i \\phi }\\begin{bmatrix}1 \\\\ 0\\end{bmatrix},\\end{aligned} \\hspace{\\stretch{1}}(1.0.11)\n\nwhere\n\n\\begin{aligned}\\tan \\phi =\\left( { \\omega m l^2 \\theta_0\/ p_0 } \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.12)\n\nOur time evolution in phase space is given by\n\n\\begin{aligned}\\begin{bmatrix}\\theta(t) \\\\ p_\\theta(t)\\end{bmatrix}=\\sqrt{ \\frac{2 E}{g m l} }\\begin{bmatrix}\\cos(\\omega t + \\phi) \\\\ - \\omega m l^2\\sin(\\omega t + \\phi)\\end{bmatrix},\\end{aligned} \\hspace{\\stretch{1}}(1.0.14)\n\nor\n\n\\begin{aligned}\\boxed{\\begin{bmatrix}\\theta(t) \\\\ p_\\theta(t)\\end{bmatrix}=\\frac{1}{{\\omega l}}\\sqrt{ \\frac{2 E}{m} }\\begin{bmatrix}\\cos(\\omega t + \\phi) \\\\ - \\omega m l^2\\sin(\\omega t + \\phi)\\end{bmatrix}.}\\end{aligned} \\hspace{\\stretch{1}}(1.0.14)\n\nThis is plotted in fig. 1.2.\n\nFig 1.2: Phase space trajectory for small angle pendulum\n\nThe area of this ellipse is\n\n\\begin{aligned}A = \\pi \\frac{1}{{\\omega^2 l^2}} \\frac{2 E}{m} \\omega m l^2 = \\frac{2 \\pi}{\\omega} E.\\end{aligned} \\hspace{\\stretch{1}}(1.0.15)\n\nWith $\\tau$ for the period of the trajectory, this is\n\n\\begin{aligned}A = \\tau E.\\end{aligned} \\hspace{\\stretch{1}}(1.0.16)\n\nAs a final note, observe that the oriented integral from problem 2.5 of the text $\\oint p_\\theta d\\theta$, is also this area. This is a general property, which can be seen geometrically in fig. 1.3, where we see that the counterclockwise oriented integral of $\\oint p dq$ would give the negative area. The integrals along the $c_4, c_1$ paths give the area under the blob, whereas the integrals along the other paths where the sense is opposite, give the complete area under the top boundary. Since they are oppositely sensed, adding them gives just the area of the blob.\n\nFig 1.3: Area from oriented integral along path\n\nLet\u2019s do this $\\oint p_\\theta d\\theta$ integral for the pendulum phase trajectories. With\n\n\\begin{aligned}\\theta = \\frac{1}{{\\omega l}} \\sqrt{\\frac{2 E}{m}} \\cos(\\omega t + \\phi)\\end{aligned} \\hspace{\\stretch{1}}(1.0.17a)\n\n\\begin{aligned}p_\\theta = -m l \\sqrt{\\frac{2 E}{m}} \\sin(\\omega t + \\phi)\\end{aligned} \\hspace{\\stretch{1}}(1.0.17b)\n\nWe have\n\n\\begin{aligned}\\oint p_\\theta d\\theta = \\frac{m l}{\\omega l} \\frac{2 E}{m} \\int_0^{2\\pi\/\\omega} \\sin^2( \\omega t + \\phi) \\omega dt= 2 E \\int_0^{2\\pi\/\\omega} \\frac{ 1 - \\cos\\left( { 2(\\omega t + \\phi) } \\right) }{2} dt= E \\frac{2 \\pi}{\\omega} = E \\tau.\\end{aligned} \\hspace{\\stretch{1}}(1.0.18)\n\nReferences\n\n[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.\n\nPHY452H1S Basic Statistical Mechanics. Lecture 10: Continuing review of thermodynamics. Taught by Prof. Arun\u00a0Paramekanti\n\nPosted by peeterjoot on February 14, 2013\n\n[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]\n\nDisclaimer\n\nPeeter\u2019s lecture notes from class. May not be entirely coherent.\n\nContinuing review of thermodynamics\n\nWe have energy conservation split into two types of energy\n\n\\begin{aligned}dE = \\underbrace{d W }_{\\text{Organized macroscopic variable X}}+ \\underbrace{d Q}_{\\text{disorganized}}\\end{aligned} \\hspace{\\stretch{1}}(1.2.1)\n\nIn fig. 1 we plot changes that are adiabatic processes ($d Q = 0$) and heating and cooling processes (with $d W = 0$).\n\nGiven a dimensionality of $d_w + 1$, a cyclic change is that for which we have\n\n\\begin{aligned}\\{ X_{\\mathrm{initial}} \\} \\rightarrow \\{ X_{\\mathrm{final}} \\} \\end{aligned} \\hspace{\\stretch{1}}(1.0.2a)\n\n\\begin{aligned}E_{\\mathrm{initial}} \\rightarrow E_{\\mathrm{final}}\\end{aligned} \\hspace{\\stretch{1}}(1.0.2b)\n\n\\begin{aligned}\\Delta W \\ne 0\\end{aligned} \\hspace{\\stretch{1}}(1.0.2c)\n\n\\begin{aligned}\\Delta Q \\ne 0\\end{aligned} \\hspace{\\stretch{1}}(1.0.2d)\n\n\\begin{aligned}\\Delta E = 0\\end{aligned} \\hspace{\\stretch{1}}(1.0.2e)\n\nSuch a cyclic process could be represented as in fig. 2.\n\nFig2: Cyclic process\n\nHere we\u2019ve labeled the level curves with a parameter $\\sigma$, as yet undefined. We call $\\sigma$ the thermodynamic entropy, and say that\n\n\\begin{aligned}\\left( {\\sigma, \\{x_i\\}} \\right),\\end{aligned} \\hspace{\\stretch{1}}(1.0.3)\n\nspecifies the state of the system.\n\nExample: Pushing a block against a surface with friction.\n\nEquilibrium\n\nConsidering two systems in contact as in fig. 3.\n\nFig3: Two systems in contact\n\nWe require\n\n\u2022 Mechanical equilibrium.\n\nrequires balance of the forces $f_i$\n\n\\begin{aligned}\\frac{\\partial {E}}{\\partial {x_i}} = f_i,\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\n(Note the neglect of the sign here, the direction of the force isn\u2019t really of interest).\n\nand\n\n\\begin{aligned}\\frac{\\partial {E_1}}{\\partial {x_i}} = \\frac{\\partial {E_2}}{\\partial {x_i}} \\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\n\u2022 Thermal stability\n\n\\begin{aligned}\\frac{\\partial {E_1}}{\\partial {\\sigma}} = \\frac{\\partial {E_2}}{\\partial {\\sigma}} \\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\nWe must have some quantity that characterizes the state of the system in a non-macroscopic fashion. The identity eq. 1.0.6 is a statement that we have equal temperatures.\n\nWe define temperature as\n\n\\begin{aligned}T \\equiv \\frac{\\partial {E}}{\\partial {\\sigma}}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nWe could potentially define different sorts of temperature, for example, perhaps $T^3 \\equiv {\\partial {E}}\/{\\partial {\\sigma}}$. Should we do this, we effectively also define $\\sigma$ in a specific way. The definition eq. 1.0.7 effectively defines this non-macroscopic parameter $\\sigma$ (the entropy) in the simplest possible way.\n\nCyclic state variable verses non-state variables\n\n\\begin{aligned}\\{x_i\\}, \\sigma \\rightarrow \\text{state variable''}\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nA non-cyclic process changes these, whereas a cyclic process takes $\\sigma, \\{x_i\\}$ back to the initial values. This is characterized by\n\n\\begin{aligned}\\oint d\\sigma = 0\\end{aligned} \\hspace{\\stretch{1}}(1.0.9a)\n\n\\begin{aligned}\\oint dx_i = 0.\\end{aligned} \\hspace{\\stretch{1}}(1.0.9b)\n\nThis doesn\u2019t mean that the closed loop integral of other qualities, such as temperature are necessarily zero\n\n\\begin{aligned}\\oint T d\\sigma = \\oint d Q \\ne 0\\end{aligned} \\hspace{\\stretch{1}}(1.0.10a)\n\n\\begin{aligned}\\oint f_i dx_i = \\oint d W \\ne 0.\\end{aligned} \\hspace{\\stretch{1}}(1.0.10b)\n\nNote that the identification of $d Q = T d\\sigma$ follows from our definition\n\n\\begin{aligned}\\left( { \\frac{\\partial {E}}{\\partial {\\sigma}} } \\right)_x = T\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nso that with $d W = 0$ we have\n\n\\begin{aligned}dE = T d \\sigma\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nGraphically we have for a cyclic process fig. 4.\n\nFig4: Cyclic process\n\nWe have\n\n\\begin{aligned}d W_{\\rightarrow} = -d W_{\\leftarrow} \\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\n\\begin{aligned}d Q_{\\rightarrow} = -d Q_{\\leftarrow} \\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nso that\n\n\\begin{aligned}\\Delta Q_{12}^{(\\mathrm{A})} + \\Delta Q_{21}^{(\\mathrm{B})} \\ne 0,\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nor\n\n\\begin{aligned}\\Delta Q_{12}^{(\\mathrm{A})} \\ne \\Delta Q_{12}^{(\\mathrm{B})}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nIrreversible and reversible processes\n\nReversible means that an undoing of the macroscopic quantities brings us back to the initial state. A counter example is a block on a spring as illustrated in fig. 5.\n\nFig 5: Heat loss and irreversibility\n\nIn such a system the block will hit gas atoms as it moves. It\u2019s hard to imagine that such gas particles will somehow spontaneously reorganize itself so that they return to their initial positions and velocities. This is the jist of the Second law of thermodynamics. Real processes introduce a degree of irreversibility with\n\n\\begin{aligned}\\text{Energy}_1 \\rightarrow \\text{Energy}_2\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\n\\begin{aligned}\\text{Work} \\rightarrow \\text{Heat}\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nbut not all\n\n\\begin{aligned}\\text{Heat} \\rightarrow \\text{Work}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nPHY452H1S Basic Statistical Mechanics. Lecture 9: Lightning review of thermodynamics. Taught by Mr. (Eric) Kin-Ho\u00a0Lee\n\nPosted by peeterjoot on February 13, 2013\n\nDisclaimer\n\nPeeter\u2019s lecture notes from class. May not be entirely coherent.\n\nLightning review of thermodynamics\n\nFirst law\n\nEnergy conservation.\n\n\u2022 Work. Macroscopic control\n\u2022 heat. Uncontrollable (microscopically)\n\nThis is summarized by the differential relationship\n\n\\begin{aligned}dE = d W + d Q.\\end{aligned} \\hspace{\\stretch{1}}(1.2.1)\n\nExamples of work\n\nWe have many types of work (in contrast to only one type of heat). Exmaples\n\n1. $-P dV = d W$\n2. $q\\mathbf{E} \\cdot dl$\n3. $k x dx$\n4. $H dm$\n\nHomework: verify the signs of these.\n\nWe put these into a general form, to first order, of\n\n\\begin{aligned}d W_i = f_i dx_i,\\end{aligned} \\hspace{\\stretch{1}}(1.2.2)\n\nwhere we assume that higher order terms are not significant.\n\n\\begin{aligned}d W = \\sum_i d W_i = \\sum_i f_i dx_i.\\end{aligned} \\hspace{\\stretch{1}}(1.2.3)\n\nHeat\n\nWe have only one type of heat, which we loosely describe as something imbued by contact with a \u201chotter\u201d system, as in (Fig 1)\n\nFig1: System in contact with heat source\n\nThis is defined as the condition where we have no heat exchange with the environment, or\n\n\\begin{aligned}d Q = 0.\\end{aligned} \\hspace{\\stretch{1}}(1.2.4)\n\nWe contrast this with heating processes for which we have\n\n\\begin{aligned}d W = 0.\\end{aligned} \\hspace{\\stretch{1}}(1.2.5)\n\nSince we have $N$ coordinates ($d W = \\sum_{i = 1}^N f_i dx_i$). We can think about an $n + 1$ dimensional space, where\n\n1. $N$-dimensions are $x_i$\n2. 1 dimension that characterizes heat exchange.\n\nn = 1\n\nGiven work on gas\n\n\\begin{aligned}d W = -P dV\\end{aligned} \\hspace{\\stretch{1}}(1.2.6)\n\nWe have a coordinate, not yet precisely defined, for which fixed levels indicate that there is no heat exchange occuring, as in (Fig 2)\n\nFig2: Adiabatic and heat exchange processes\n\nWe\u2019ll call this axis $\\sigma$, the thermodynamic entropy.\n\nWe\u2019ve been introduced to statistical entropy\n\n\\begin{aligned}S = k_B \\ln \\Omega.\\end{aligned} \\hspace{\\stretch{1}}(1.2.7)\n\nWe\u2019ll assume for now that these are not related and will eventually figure out the connection between these two concepts.\n\nn = 2\n\nA representation of a adiabatic, or constant $\\sigma$-hypersurface process is given in (Fig 3), a heating\/cooling process with transition between $\\sigma$-hypersurfaces in (Fig 4), and a cyclic process, in (Fig 5)\n\nFig4: Heat exchange process\n\nFig 5: Cyclic process\n\nThe cyclic process is one for which $dE = d W + d Q = 0$, however, this does not imply $d W = 0$ and $d Q = 0$ since we only require that the sum of the two is zero. In this whole process, we can have for example a net change in heat. Example: the engine of a car. Work is done, and heat is generated, but a car that was initially stopped and returns to its final destination, stops and cools down again, has still had significant internal action in the process.\n\nReversible processes\n\nWhat do we mean by reversible? We mean that any of the changes in the system have been done so slowly that we could reverse the direction of the processes at any point, and should we do so, both the system and the environment will be returned to its initial state. This is an idealization that is, most of the time, a good approximation, but gives us an excellent idea of the limits of what we can theoretically describe.\n\nQuestion: Why does the speed of the process make a difference?\n\nIf we are making changes to the system quickly, imagine that we are compressing a gas as in (Fig 6)\n\nFig 6: Fast gas compression by a piston\n\nDoing work slowly means that the whole system can react to the change imposed. If we compressed the gas quickly, then changes to the system start only at the contact point with the piston. This can\u2019t be reversed. If we pull the piston out at this point, none of the non-front gas particles will be able to react. The system will not be in thermal equalibrium for fast changes.\n\nCartesian to spherical change of variables in 3d phase\u00a0space\n\nPosted by peeterjoot on February 11, 2013\n\nQuestion: Cartesian to spherical change of variables in 3d phase space\n\n[1] problem 2.2 (a). Try a spherical change of vars to verify explicitly that phase space volume is preserved.\n\nOur kinetic Lagrangian in spherical coordinates is\n\n\\begin{aligned}\\mathcal{L} &= \\frac{1}{{2}} m \\left( \\dot{r} \\hat{\\mathbf{r}} + r \\sin\\theta \\dot{\\phi} \\hat{\\boldsymbol{\\phi}} + r \\dot{\\theta} \\hat{\\boldsymbol{\\theta}} \\right)^2 \\\\ &= \\frac{1}{{2}} m \\left( \\dot{r}^2 + r^2 \\sin^2\\theta \\dot{\\phi}^2 + r^2 \\dot{\\theta}^2 \\right)^2\\end{aligned} \\hspace{\\stretch{1}}(1.0.1)\n\nWe read off our canonical momentum\n\n\\begin{aligned}p_r &= \\frac{\\partial {\\mathcal{L}}}{\\partial {r}} \\\\ &= m \\dot{r}\\end{aligned} \\hspace{\\stretch{1}}(1.0.2a)\n\n\\begin{aligned}p_\\theta &= \\frac{\\partial {\\mathcal{L}}}{\\partial {\\theta}} \\\\ &= m r^2 \\dot{\\theta}\\end{aligned} \\hspace{\\stretch{1}}(1.0.2b)\n\n\\begin{aligned}p_\\phi &= \\frac{\\partial {\\mathcal{L}}}{\\partial {\\phi}} \\\\ &= m r^2 \\sin^2\\theta \\dot{\\phi},\\end{aligned} \\hspace{\\stretch{1}}(1.0.2c)\n\nand can now express the Hamiltonian in spherical coordinates\n\n\\begin{aligned}H &= \\frac{1}{{2}} m \\left(\\left( \\frac{p_r}{m} \\right)^2+ r^2 \\sin^2\\theta \\left( \\frac{p_\\phi}{m r^2 \\sin^2\\theta} \\right)+ r^2 \\left( \\frac{p_\\theta}{m r^2} \\right)\\right) \\\\ &= \\frac{p_r^2}{2m} + \\frac{p_\\phi^2}{2 m r^2 \\sin^2\\theta} + \\frac{p_\\theta^2}{2 m r^2}\\end{aligned} \\hspace{\\stretch{1}}(1.0.3)\n\nNow we want to do a change of variables. The coordinates transform as\n\n\\begin{aligned}x = r \\sin\\theta \\cos\\phi\\end{aligned} \\hspace{\\stretch{1}}(1.0.4a)\n\n\\begin{aligned}y = r \\sin\\theta \\sin\\phi\\end{aligned} \\hspace{\\stretch{1}}(1.0.4b)\n\n\\begin{aligned}z = r \\cos\\theta,\\end{aligned} \\hspace{\\stretch{1}}(1.0.4c)\n\nor\n\n\\begin{aligned}r = \\sqrt{x^2 + y^2 + z^2}\\end{aligned} \\hspace{\\stretch{1}}(1.0.5a)\n\n\\begin{aligned}\\theta = \\arccos(z\/r)\\end{aligned} \\hspace{\\stretch{1}}(1.0.5b)\n\n\\begin{aligned}\\phi = \\arctan(y\/x).\\end{aligned} \\hspace{\\stretch{1}}(1.0.5c)\n\nIt\u2019s not too hard to calculate the change of variables for the momenta (verified in sphericalPhaseSpaceChangeOfVars.nb). We have\n\n\\begin{aligned}p_r = \\frac{x p_x + y p_y + z p_z}{\\sqrt{x^2 + y^2 + z^2}}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6a)\n\n\\begin{aligned}p_\\theta = \\frac{(p_x x + p_y y) z - p_z (x^2 + y^2)}{\\sqrt{x^2 + y^2}}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6b)\n\n\\begin{aligned}p_\\phi = x p_y - y p_x\\end{aligned} \\hspace{\\stretch{1}}(1.0.6c)\n\nNow let\u2019s compute the volume element in spherical coordinates. This is\n\n\\begin{aligned}d\\omega &= dr d\\theta d\\phi p_r p_\\theta p_\\phi \\\\ &= \\frac{\\partial(r, \\theta, \\phi, p_r, p_\\theta, p_\\phi)}{\\partial(x, y, z, p_x, p_y, p_z)}dx dy dz dp_x dp_y dp_z \\\\ &= \\begin{vmatrix} \\frac{x}{\\sqrt{x^2+y^2+z^2}} & \\frac{y}{\\sqrt{x^2+y^2+z^2}} & \\frac{z}{\\sqrt{x^2+y^2+z^2}} & 0 & 0 & 0 \\\\ \\frac{x z}{\\sqrt{x^2+y^2} \\left(x^2+y^2+z^2\\right)} & \\frac{y z}{\\sqrt{x^2+y^2} \\left(x^2+y^2+z^2\\right)} & -\\frac{\\sqrt{x^2+y^2}}{x^2+y^2+z^2} & 0 & 0 & 0 \\\\ -\\frac{y}{x^2+y^2} & \\frac{x}{x^2+y^2} & 0 & 0 & 0 & 0 \\\\ \\frac{\\left(y^2+z^2\\right) p_x-x y p_y-x z p_z}{\\left(x^2+y^2+z^2\\right)^{3\/2}} & \\frac{\\left(x^2+z^2\\right) p_y-y \\left(x p_x+z p_z\\right)}{\\left(x^2+y^2+z^2\\right)^{3\/2}} & \\frac{\\left(x^2+y^2\\right) p_z-z \\left(x p_x+y p_y\\right)}{\\left(x^2+y^2+z^2\\right)^{3\/2}} & \\frac{x}{\\sqrt{x^2+y^2+z^2}} & \\frac{y}{\\sqrt{x^2+y^2+z^2}} & \\frac{z}{\\sqrt{x^2+y^2+z^2}} \\\\ \\frac{y z \\left(y p_x-x p_y\\right)-x \\left(x^2+y^2\\right) p_z}{\\left(x^2+y^2\\right)^{3\/2}} & \\frac{x z \\left(x p_y-y p_x\\right)-y \\left(x^2+y^2\\right) p_z}{\\left(x^2+y^2\\right)^{3\/2}} & \\frac{x p_x+y p_y}{\\sqrt{x^2+y^2}} & \\frac{x z}{\\sqrt{x^2+y^2}} & \\frac{y z}{\\sqrt{x^2+y^2}} & -\\sqrt{x^2+y^2} \\\\ p_y & -p_x & 0 & -y & x & 0 \\\\ \\end{vmatrix}dx dy dz dp_x dp_y dp_z \\\\ &= dx dy dz dp_x dp_y dp_z\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nThis also has a unit determinant, as we found in the similar cylindrical change of phase space variables.\n\nReferences\n\n[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.\n\nChange of variables in 2d phase\u00a0space\n\nPosted by peeterjoot on February 10, 2013\n\nMotivation\n\nIn [1] problem 2.2, it\u2019s suggested to try a spherical change of vars to verify explicitly that phase space volume is preserved, and to explore some related ideas. As a first step let\u2019s try a similar, but presumably easier change of variables, going from Cartesian to cylindrical phase spaces.\n\nCanonical momenta and Hamiltonian\n\nOur cylindrical velocity is\n\n\\begin{aligned}\\mathbf{v} = \\dot{r} \\hat{\\mathbf{r}} + r \\dot{\\theta} \\hat{\\boldsymbol{\\theta}},\\end{aligned} \\hspace{\\stretch{1}}(1.2.1)\n\nso a purely kinetic Lagrangian would be\n\n\\begin{aligned}\\mathcal{L} &= \\frac{1}{{2}} m \\mathbf{v}^2 \\\\ &= \\frac{1}{{2}} m \\left( \\dot{r}^2 + r^2 \\dot{\\theta}^2 \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.2.2)\n\nOur canonical momenta are\n\n\\begin{subequations}\n\n\\begin{aligned}p_r &= \\frac{\\partial {\\mathcal{L}}}{\\partial {\\dot{r}}} \\\\ &= m \\dot{r}\\end{aligned} \\hspace{\\stretch{1}}(1.0.3a)\n\n\\begin{aligned}p_\\theta &= \\frac{\\partial {\\mathcal{L}}}{\\partial {\\dot{\\theta}}} \\\\ &= m r^2 \\dot{\\theta}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.3b)\n\n\\end{subequations}\n\nand our kinetic energy is\n\n\\begin{aligned}H &= \\mathcal{L} \\\\ &= \\frac{1}{{2m}} p_r^2 + \\frac{1}{{2 m r^2}} p_\\theta^2.\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nNow we need to express our momenta in terms of the Cartesian coordinates. We have for the radial momentum\n\n\\begin{aligned}p_r &= m \\dot{r} \\\\ &= m \\frac{d{{}}}{dt} \\sqrt{x^2 + y^2} \\\\ &= \\frac{1}{{2}} \\frac{2 m}{r} \\left( x \\dot{x} + y \\dot{y} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nor\n\n\\begin{aligned}p_r = \\frac{1}{{r}} \\left( x p_x + y p_y \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\n\\begin{aligned}p_\\theta &= m r^2 \\frac{d{{\\theta}}}{dt} \\\\ &= m r^2 \\frac{d{{}}}{dt} \\arctan \\left( \\frac{y}{x} \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nAfter some reduction (cyclindrialMomenta.nb), we find\n\n\\begin{aligned}p_\\theta = p_y x - p_x y.\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nWe can assemble these into a complete set of change of variable equations\n\n\\begin{subequations}\n\n\\begin{aligned}r = \\sqrt{x^2 + y^2}\\end{aligned} \\hspace{\\stretch{1}}(1.0.9a)\n\n\\begin{aligned}\\theta = \\arctan\\left( \\frac{y}{x} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.9b)\n\n\\begin{aligned}p_r = \\frac{1}{{\\sqrt{x^2 + y^2}}} \\left( x p_x + y p_y \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.9c)\n\n\\begin{aligned}p_\\theta = p_y x - p_x y.\\end{aligned} \\hspace{\\stretch{1}}(1.0.9d)\n\n\\end{subequations}\n\nOur phase space volume element change of variables is\n\n\\begin{aligned}dr d\\theta dp_r dp_\\theta &= \\frac{\\partial(r, \\theta, p_r, p_\\theta)}{\\partial(x, y, p_x, p_y)}dx dy dp_x dp_y \\\\ &= \\begin{vmatrix} \\frac{x}{\\sqrt{x^2+y^2}} & \\frac{y}{\\sqrt{x^2+y^2}} & 0 & 0 \\\\ -\\frac{y}{x^2+y^2} & \\frac{x}{x^2+y^2} & 0 & 0 \\\\ \\frac{y \\left(y p_x-x p_y\\right)}{\\left(x^2+y^2\\right)^{3\/2}} & \\frac{x \\left(x p_y-y p_x\\right)}{\\left(x^2+y^2\\right)^{3\/2}} & \\frac{x}{\\sqrt{x^2+y^2}} & \\frac{y}{\\sqrt{x^2+y^2}} \\\\ p_y & -p_x & -y & x \\end{vmatrix}dx dy dp_x dp_y \\\\ &= \\frac{x^2 + y^2}{\\left(x^2 + y^2\\right)^{3\/2}}\\frac{x^2 + y^2}{\\left(x^2 + y^2\\right)^{1\/2}} \\\\ &= dx dy dp_x dp_y.\\end{aligned} \\hspace{\\stretch{1}}(1.0.10)\n\nWe see explicitly that this point transformation has a unit Jacobian, preserving area.\n\nReferences\n\n[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.\n\nSome problems from Kittel chapter\u00a03\n\nPosted by peeterjoot on February 10, 2013\n\n[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]\n\nQuestion: Classical gas partition function\n\n[1] expresses the classical gas partition function (3.77) as\n\n\\begin{aligned}Z_1 \\propto \\int \\exp\\left( - \\frac{p_x^2 + p_y^2 + p_z^2 }{2 M \\tau}\\right) dp_x dp_y dp_z\\end{aligned} \\hspace{\\stretch{1}}(1.0.1)\n\nShow that this leads to the expected $3 \\tau\/2$ result for the thermal average energy.\n\nLet\u2019s use the adjustment technique from the text for the $N$ partition case and write\n\n\\begin{aligned}Z_N = \\frac{1}{{N!}} Z_1^N,\\end{aligned} \\hspace{\\stretch{1}}(1.0.2)\n\nwith $Z_1$ as above. This gives us\n\n\\begin{aligned}U &= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\ln Z_N \\\\ &= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\left(N \\ln Z_1 - \\ln N!\\right) \\\\ &= N \\tau^2 \\frac{\\partial {\\ln Z_1 }}{\\partial {\\tau}} \\\\ &= N \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}} \\sum_{k = 1}^{3} \\ln\\int \\exp\\left( - \\frac{p_k^2 }{2 M \\tau}\\right) dp_k \\\\ &= N \\tau^2\\sum_{k = 1}^{3}\\frac{\\frac{\\partial {}}{\\partial {\\tau}} \\int \\exp\\left( - \\frac{p_k^2 }{2 M \\tau} \\right) dp_k}{\\int \\exp\\left( - \\frac{p_k^2 }{2 M \\tau} \\right) dp_k} \\\\ &= N \\tau^2\\sum_{k = 1}^{3}\\frac{\\frac{\\partial {}}{\\partial {\\tau}} \\sqrt{ 2 \\pi M \\tau }}{\\sqrt{ 2 \\pi M \\tau}} \\\\ &= 3 N \\tau^2\\frac{\\frac{1}{{2}} \\tau^{-1\/2}}{\\sqrt{ \\tau}} \\\\ &= \\frac{3}{2} N \\tau \\\\ &= \\frac{3}{2} N k_{\\mathrm{B}} T\\end{aligned} \\hspace{\\stretch{1}}(1.0.3)\n\nQuestion: Two state system\n\n[1] problem 3.1.\n\nFind an expression for the free energy as a function of $\\tau$ of a system with two states, one at energy $0$ and one at energy $\\epsilon$. From the free energy, find expressions for the energy and entropy of the system.\n\nOur partition function is\n\n\\begin{aligned}Z = 1 + e^{-\\epsilon \/\\tau}\\end{aligned} \\hspace{\\stretch{1}}(1.0.4)\n\nThe free energy is just\n\n\\begin{aligned}F = -\\tau \\ln Z = -\\tau \\ln (1 + e^{-\\epsilon\/\\tau})\\end{aligned} \\hspace{\\stretch{1}}(1.0.5)\n\nThe entropy follows immediately\n\n\\begin{aligned}\\sigma \\\\ &= -\\frac{\\partial {F}}{\\partial {\\tau}} \\\\ &= \\frac{\\partial {}}{\\partial {\\tau}}\\left( \\tau \\ln \\left( 1 + e^{-\\epsilon\/\\tau} \\right) \\right) \\\\ &= \\ln \\left( 1 + e^{-\\epsilon\/\\tau} \\right)-\\tau \\epsilon \\frac{-1}{\\tau^2} \\frac{1}{{1 + e^{-\\epsilon\/\\tau}}} \\\\ &= \\ln \\left( 1 + e^{-\\epsilon\/\\tau} \\right)+\\frac{\\epsilon}{\\tau} \\frac{e^{-\\epsilon\/\\tau}}{1 + e^{-\\epsilon\/\\tau}}\\end{aligned} \\hspace{\\stretch{1}}(1.0.6)\n\nThe energy is\n\n\\begin{aligned}U \\\\ &= F + \\tau \\sigma \\\\ &= -\\tau \\ln (1 + e^{-\\epsilon\/\\tau}) + \\tau \\sigma \\\\ &= \\tau\\left( \\not{{\\ln \\left( 1 + e^{-\\epsilon\/\\tau} \\right)}} + \\frac{\\epsilon}{\\tau} \\frac{e^{-\\epsilon\/\\tau}}{1 + e^{-\\epsilon\/\\tau}} -\\not{{\\ln (1 + e^{-\\epsilon\/\\tau}) }} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.7)\n\nThis is\n\n\\begin{aligned}U=\\frac{\\epsilon e^{-\\epsilon\/\\tau}}{1 + e^{-\\epsilon\/\\tau}}=\\frac{\\epsilon}{1 + e^{\\epsilon\/\\tau}}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.8)\n\nThese are all plotted in (Fig 1).\n\nFig1: Plots for two state system\n\n\\imageFigure{kittelCh3Problem1PlotsFig1}{Plots for two state system}{fig:kittelCh3Problem1Plots:kittelCh3Problem1PlotsFig1}{0.2}\n\nQuestion: Magnetic susceptibility\n\n[1] problem 3.2.\n\nUse the partition function to find an exact expression for the magnetization $M$ and the susceptibility $\\chi = dM\/dB$ as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization, found by other means, was $M = n m \\tanh( m B\/\\tau)$, where $n$ is the particle concentration. Find the free energy and express the result as a function only of $\\tau$ and the parameter $x = M\/nm$. Show that the susceptibility is $\\chi = n m^2\/\\tau$ in the limit $m B \\ll \\tau$.\n\nOur partition function for a unit volume containing $n$ spins is\n\n\\begin{aligned}Z=\\frac{\\left( e^{-m B\/\\tau} +e^{m B\/\\tau} \\right)^n}{n!}=2 \\frac{\\left( \\cosh\\left( m B\/\\tau \\right) \\right)^n}{n!},\\end{aligned} \\hspace{\\stretch{1}}(1.0.9)\n\nso that the Free energy is\n\n\\begin{aligned}F = -\\tau\\left( \\ln 2 - \\ln n! + n \\ln \\cosh\\left( m B\/\\tau \\right) \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.15)\n\nThe energy, magnetization and magnetic field were interrelated by\n\n\\begin{aligned}- M B &= U \\\\ &= \\tau^2 \\frac{\\partial {}}{\\partial {\\tau}}\\left( -\\frac{F}{\\tau} \\right) \\\\ &= \\tau^2 n\\frac{\\partial {}}{\\partial {\\tau}}\\ln \\cosh\\left( m B\/\\tau \\right) \\\\ &= \\tau^2 n \\frac{ -m B\/\\tau^2\\sinh\\left( m B\/\\tau \\right)}{\\cosh\\left( m B\/\\tau \\right)} \\\\ &= - m B n \\tanh \\left( m B\/\\tau \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.11)\n\nThis gives us\n\n\\begin{aligned}M = m n \\tanh \\left( m B\/\\tau \\right),\\end{aligned} \\hspace{\\stretch{1}}(1.0.12)\n\nso that\n\n\\begin{aligned}\\chi = \\frac{dM}{dB}= \\frac{m^2 n}{\\tau \\cosh^2 \\left( m B\/\\tau \\right)}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.13)\n\nFor $m B\/\\tau \\ll 1$, the cosh term goes to unity, so we have\n\n\\begin{aligned}\\chi \\approx= \\frac{m^2 n}{\\tau},\\end{aligned} \\hspace{\\stretch{1}}(1.0.14)\n\nas desired.\n\nWith $x = M\/nm$, or $m = M\/nx$, the free energy is\n\n\\begin{aligned}F =-\\tau\\left( \\ln 2\/n! + n \\ln \\cosh\\left( \\frac{M B}{n x \\tau} \\right) \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.15)\n\nThat last expression isn\u2019t particularly illuminating. What was the point of that substitution?\n\nQuestion: Free energy of a harmonic oscillator\n\n[1] problem 3.3.\n\nA one dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\\epsilon_s = s \\hbar \\omega$, where $s$ is a positive integer or zero, and $\\omega$ is the classical frequency of the oscillator. We have chosen the zero of energy at the state $s = 0$. Show that for a harmonic oscillator the free energy is\n\n\\begin{aligned}F = \\tau \\ln\\left( 1 - e^{-\\hbar \\omega\/\\tau} \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.16)\n\nNote that at high temperatures such that $\\tau \\gg \\hbar \\omega$ we may expand the argument of the logarithm to obtain $F \\approx \\tau \\ln (\\hbar \\omega\/\\tau)$. From 1.0.16 show that the entropy is\n\n\\begin{aligned}\\sigma = \\frac{\\hbar\\omega\/\\tau}{e^{\\hbar \\omega\/\\tau} - 1} -\\ln\\left( 1 - e^{-\\hbar \\omega\/\\tau} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.17)\n\nI found it curious that this problem dropped the factor of $\\hbar\\omega\/2$ from the energy. Including it we have\n\n\\begin{aligned}\\epsilon_s = \\left( s + \\frac{1}{{2}} \\right) \\hbar \\omega,\\end{aligned} \\hspace{\\stretch{1}}(1.0.18)\n\nSo that the partition function is\n\n\\begin{aligned}Z= \\sum_{s = 0}^\\infty e^{-\\left( s + \\frac{1}{{2}} \\right) \\hbar \\omega\/\\tau }=e^{-\\hbar \\omega\/2\\tau}\\sum_{s = 0}^\\infty e^{-s \\hbar \\omega\/\\tau}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.19)\n\nThe free energy is\n\n\\begin{aligned}F &= -\\tau \\ln Z \\\\ &= -\\tau\\left( -\\frac{\\hbar \\omega}{2\\tau} + \\ln \\left( \\sum_{s = 0}^\\infty e^{-s \\hbar \\omega\/\\tau} \\right) \\right) \\\\ &= \\frac{\\hbar \\omega}{2} +\\ln\\left( \\sum_{s = 0}^\\infty e^{-s \\hbar \\omega\/\\tau} \\right)\\end{aligned} \\hspace{\\stretch{1}}(1.0.20)\n\nWe see that the contribution of the $\\hbar \\omega\/2$ in the energy of each state just adds a constant factor to the free energy. This will drop out when we compute the entropy. Dropping that factor now that we know why it doesn\u2019t contribute, we can complete the summation, so have, by inspection\n\n\\begin{aligned}F = -\\tau \\ln Z=\\tau \\ln\\left( 1 - e^{-\\hbar \\omega\/\\tau} \\right).\\end{aligned} \\hspace{\\stretch{1}}(1.0.21)\n\nTaking derivatives for the entropy we have\n\n\\begin{aligned}\\sigma &= -\\frac{\\partial {F}}{\\partial {\\tau}} \\\\ &= -\\ln\\left( 1 - e^{-\\hbar \\omega\/\\tau} \\right)+\\tau\\frac{\\hbar \\omega}{\\tau^2} \\frac{e^{-\\hbar \\omega\/\\tau}}{1 - e^{-\\hbar \\omega\/\\tau}} \\\\ &= -\\ln\\left( 1 - e^{-\\hbar \\omega\/\\tau} \\right)+\\frac{\\frac{\\hbar \\omega}{\\tau}}{e^{\\hbar \\omega\/\\tau} - 1}\\end{aligned} \\hspace{\\stretch{1}}(1.0.22)\n\nQuestion: Energy fluctuation\n\n[1] problem 3.4.\n\nConsider a system of fixed volume in thermal contact with a reservoir. Show that the mean square fluctuation in the energy of the system is\n\n\\begin{aligned}\\left\\langle{{ (\\epsilon - \\left\\langle{\\epsilon}\\right\\rangle)^2 }}\\right\\rangle = \\tau^2\\left( \\frac{\\partial {U}}{\\partial {\\tau}} \\right)_V\\end{aligned} \\hspace{\\stretch{1}}(1.0.23)\n\nHere $U$ is the conventional symbol for $\\left\\langle{{\\epsilon}}\\right\\rangle$. Hint: Use the partition function $Z$ to relate ${\\partial {U}}\/{\\partial {t}}$ to the mean square fluctuation. Also, multiply out the term $(\\cdots)^2$.\n\nWith a probability of finding the system in state $s$ of\n\n\\begin{aligned}P_s = \\frac{e^{-\\epsilon_s\/\\tau}}{Z}\\end{aligned} \\hspace{\\stretch{1}}(1.0.24)\n\nthe average energy is\n\n\\begin{aligned}U &= \\left\\langle{{\\epsilon}}\\right\\rangle \\\\ &= \\sum_s P_s \\epsilon_s \\\\ &= \\sum_s \\epsilon_s \\frac{e^{-\\epsilon_s\/\\tau}}{Z} \\\\ &= \\frac{1}{{Z}} \\sum_s \\epsilon_s e^{-\\epsilon_s\/\\tau}\\end{aligned} \\hspace{\\stretch{1}}(1.0.25)\n\nSo we have\n\n\\begin{aligned}\\tau^2 \\frac{\\partial {U}}{\\partial {\\tau}} \\\\ &= -\\frac{\\tau^2}{Z^2} \\frac{dZ}{d\\tau}\\sum_s \\epsilon_s e^{-\\epsilon_s\/\\tau}+ \\frac{\\tau^2}{Z}\\sum_s \\frac{\\epsilon_s^2}{\\tau^2} e^{-\\epsilon_s\/\\tau} \\\\ &= -\\frac{\\tau^2}{Z^2} \\frac{dZ}{d\\tau}\\sum_s \\epsilon_s e^{-\\epsilon_s\/\\tau}+ \\frac{1}{{Z}}\\sum_s \\epsilon_s^2 e^{-\\epsilon_s\/\\tau}.\\end{aligned} \\hspace{\\stretch{1}}(1.0.26)\n\nBut\n\n\\begin{aligned}\\frac{dZ}{d\\tau}=\\frac{d}{d\\tau} \\sum_s e^{-\\epsilon_s\/\\tau}=\\sum_s \\frac{\\epsilon_s}{\\tau^2} e^{-\\epsilon_s\/\\tau},\\end{aligned} \\hspace{\\stretch{1}}(1.0.27)\n\ngiving\n\n\\begin{aligned}\\tau^2 \\frac{\\partial {U}}{\\partial {\\tau}} &= \\frac{1}{Z^2}\\sum_s \\epsilon_s e^{-\\epsilon_s\/\\tau} \\sum_s \\epsilon_s e^{-\\epsilon_s\/\\tau}+ \\frac{1}{{Z}}\\sum_s \\epsilon_s^2 e^{-\\epsilon_s\/\\tau} \\\\ &= -\\left\\langle{{ \\epsilon }}\\right\\rangle^2 + \\left\\langle{{\\epsilon^2}}\\right\\rangle,\\end{aligned} \\hspace{\\stretch{1}}(1.0.28)\n\nwhich shows 1.0.23 as desired.\n\nReferences\n\n[1] C. Kittel and H. Kroemer. Thermal physics. 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\section{Introduction}
In Nature we often meet phenomena with a~large number of variables where the few-body approach breaks down. In these cases the standard procedure is to apply tools from statistical physics and inspect thermodynamical quantities of the system, instead of treating all degrees of freedom one-by-one. A certain generalization of the standard Boltzmann--Gibbs entropy is promoted by Constantino Tsallis, introducing the $q$-entropy formula, central to non-extensive statistical theory~\cite{book:ts, book:ts2}. Despite its unconventional form, in~the last two decades the Tsallis-entropy was found to be a~very general and descriptive notion. Numerous physical observations were successfully explained using non-extensive statistical physics~\cite{book:nonextcollection, TsallisBibHtml, artic:tsappl1, artic:tsappl2, artic:tsappl3}. From our perspective the field of high-energy physics is especially important, since that community uses efficiently these tools to describe the results of high-energy particle and nuclear collisions. It is an~experimental finding that the distributions derived from Tsallis-entropy fit the spectrum of high-energy particles, produced by many systems starting from electron-positron collisions up to the cosmic rays. In this paper we focus on identified hadron spectra, measured in proton-proton collisions. We put emphasis on the investigation of the center-of-mass energy ($\sqrt{s}$) dependence of the Tsallis parameters $q$ and $T$, assuming a~Quantum Chromodynamics (QCD) inspired logarithmic scaling of these parameters. We use units in which $\hbar=c=k_B=1$.
The~outline of the paper is the following: in the next section we enlist our motivation from high energy physics and the goals of our analysis. In Section~\ref{sec:second} we briefly introduce the mathematical apparatus we used during our investigation. In Section~\ref{sec:third} we show experimental results, while in Section~\ref{sec:fourth} we compare them to state-of-the-art theoretical models. Finally, in~Section \ref{sec:fifth} we summarize our work and give a~discussion, including our future plans.
\section{Connection with High Energy Physics}
\label{sec:first}
One of the main goals in high-energy heavy-ion physics is to understand the properties of the so-called Quark Gluon Plasma (QGP), a~particular form of the strongly interacting matter which existed shortly after the Big Bang. With today's high-energy particle accelerators we are able to reach the energy range where this superdense matter of the early Universe can be formed for a~short, $\mathcal{O}$(fm/$c$)$\sim10^{-23}$ s time. The~properties of the QGP can be studied in ultra-relativistic heavy-ion collisions indirectly. Due to the nature of the strong interaction there is no way for direct observation, only signatures stemming from the final state allow us to draw conclusions.
On the other hand, the~reactions occur during a~very short time and our information about their nature is very limited. This is a~strong restraint in our possibilities, especially we cannot treat properly the description at the microscopical level. Nevertheless it is essential to understand the processes in proton-proton collisions, the~baseline for heavy-ion measurements.
To date we still do not have a~well established, detailed, and~throughout probed theory of the {\em hadronization}, the~process where the color degrees of freedom confine into hadrons. This is related to the {\em Yang--Mills Mass Gap}, one of the so-called ``Millennium problems'' of the Clay Mathematical Institute~\cite{artic:millenium}. Recent hadronization models are phenomenological, and~it is quite typical that their parameters lack of any clear physical meaning.
Recently, complex detector systems, like ALICE at the Large Hadron Collider (CERN LHC) or STAR and PHENIX at the Relativistic Heavy Ion Collider (BNL RHIC) are able to measure with high accuracy the final state particles.
The~hadron spectra, measured in high-energy collisions, are one of the most fundamental characteristics of these events and involve both microscopic and collective effects in high-energy collisions. Identifying their most crucial problems is a~key task for understanding hadronization. A remarkable phenomenon is that these properties occur not only in heavy-ion collisions, but even for small colliding systems like proton-proton or electron-positron collisions~\cite{artic:flowing0, artic:flowing1, artic:flowing2, artic:flowing3, artic:gbiro}.
The~aim of our study is to find the common source of these similarities and to recognize the driving mechanisms behind the observations. For this aim, we~built a~consistent non-extensive approach, in~which fit parameters carry important physical information about the observed system of high-energy and strongly interacting particles including reactions and collective effects among them.
\section{Non-Extensive Statistics in High-Energy Physics}
\label{sec:second}
High-energy physics, and~in particular high-energy heavy ion physics is an~interdisciplinary topic. It uses the theory of relativistic quantum fields, statistical physics, thermo- and hydrodynamics, and~even the theory of curved space-times. Earlier studies show that non-extensive statistical physics provides a~useful tool to describe particle-particle collisions, where ``{\it particle}'' now stands either for electron/positron, or for proton or a~heavy nucleus. The~non-extensivity in high-energy physics manifests itself both in the non-exponential energy- and non-Poissonian multiplicity distributions.
The~hadron spectra can be characterized with Tsallis--Pareto-like distributions, both at low-, and~high transverse momenta very well~\cite{artic:gbiro, artic:tsbeurphys2, artic:cleymansphyslett, artic:cleymansjphys, artic:wilk1, artic:wilk2, artic:wilk3, artic:wilk4, artic:tsorig, artic:wilkentr, artic:actaphys, artic:chaossolitions}.
The~origin of these distributions lies in the assumption that subsystems are not independent of each other. It makes them a~good candidate to investigate the QGP in heavy ion collisions or the baseline in smaller colliding systems. Recently a~number of systematic analyses have been made in order to find the best form for these distributions~\cite{artic:lilin1, artic:lilin2, artic:cleymansTS, artic:cleymansqspecies, artic:flowing0, artic:flowing1, artic:flowing2, artic:flowing3}, but in this present study we compare theoretical and experimental results in a~wide energy range comprehensively.
\subsection{The~Description of the Inclusive Hadron Production}
\label{sec:hadronprod}
The~Quantum Chromodynamics is the fundamental theory of the strong interaction. Due to the energy-scale dependent behavior of the strong coupling, the~perturbative QCD (pQCD) based parton model--initiated by Bjorken and Feynman--works extremely well at high energies~\cite{artic:bjorken}. In the framework of the pQCD-based parton model, all hadrons are made up from partons (bare, nearly massless quarks and gluons), therefore the inner structure of the initial colliding and the finally produced hadrons are described by the parton distribution functions (PDF) and by the fragmentation functions (FF), respectively. These~non-perturbative distribution functions are defined in the momentum space and can be parametrized by a~polynomial {\em ansatz}. The~PDF,
\begin{equation}
f_{a/h}(x_a, Q^2) \ \ ,
\end{equation}
gives the distribution of parton $a$ inside the hadron $h$ at the energy scale, $Q^2$, while $x_a=p_a/p_h$ is the momentum fraction carried by that parton. On the other hand, the~confinement of the parton $c$ into the final state hadron $h$ with the momentum fraction $z_c=p_h/p_c$ can be described at scale, $\tilde{Q}^2$ with the help of fragmentation functions
\begin{equation}
D_c^h(z_c,\tilde{Q}^2) .
\end{equation}
In this framework the inclusive cross-section of a~given hadron $h$ produced in proton-proton collisions can be calculated by the following convolution:
\begin{equation}
\label{eq:hadronprod}
E_h\frac{ {\textrm d}^3 \sigma_h^{pp}}{ {\textrm d} p_h^3}\sim \sum\limits_{a,b,c}f_{a/p}(x_a,Q^2)\otimes f_{b/p}(x_b,Q^2) \otimes \frac{ {\textrm d} \sigma^{ab\rightarrow cd}}{ {\textrm d} \hat{t}} \otimes \frac{D_c^h(z_c,\tilde{Q^2})}{\pi z_c^2} \ \ \ .
\end{equation}
Here the parton distribution function of a~proton is denoted by $f_{a/p}(x_a, Q^2)$ and $\frac{ {\textrm d} \sigma}{ {\textrm d} \hat{t}}$ is the differential cross-section of the $ab\rightarrow cd$ partonic process, the~variable $\hat{t}$ is related to the 4-momentum exchange of the particles.
The~hadronization is described within the parton model by the above phenomenological fragmentation functions, for which several forms of parametrization exist in the literature. These~parametrizations are usually fitted to lepton scattering data, therefore they describe existing experimental results in a~broad-range in the parameter space. In Section \ref{sec:fourth}, after investigating the energy dependence, we~show the latest results of a~new fragmentation function parametrization based on non-extensive phenomena.
\subsection{Hadronization Using Non-Extensive Statistics}
\label{sec:non-exthadron}
As we have already mentioned, the~transverse energy distribution of the measured hadrons---the particle yield measured in the $y \in [-0.5,+0.5] $ midrapidity region--is an~important quantity accessible to measurement. In practice, the~low-energy regime is described by exponential-like functions, as~a~thermalized system, while the high $p_T$ regime behaves like a~power-law, $p_T^{-n}$. The~Tsallis--Pareto-like distributions handle these two regimes simultaneously.
The~technical apparatus in the high-energy physics shows a~great advancement, nowadays the statistics of these spectra is larger than ever. It is no surprise that the Tsallis--Pareto distributions are widely used by the high-energy community to describe hadron spectra. The~STAR and PHENIX collaborations at RHIC BNL (USA) and the European CERN's ALICE, ATLAS, and~CMS collaborations at the LHC are using the following form to characterize the particle yield~\cite{artic:phenixts, artic:starts, artic:62gevphenix, artic:097tevalice, artic:cmsts, artic:atlasts}:
\begin{equation}
\frac{1}{2\pi p_T}\frac{ {\textrm d}^2N}{ {\textrm d} y {\textrm d} p_T} = \frac{ {\textrm d} N}{ {\textrm d} y}\frac{(n-1)(n-2)}{2\pi n C\left(nC+m(n-2)\right)}\left(1+\frac{m_T-m}{nC}\right)^{-n} \ \ \ ,
\label{eq:tsexp}
\end{equation}
where $n$ and $C$ are fit parameters and $m_T=\sqrt{p_T^2+m^2}$ is the transverse mass, including the rest mass $m$ of the given identified hadron species. We note that this formula is based on the QCD-Hagedorn formula~\cite{artic:qcdhagedorn, artic:ua1hagedorn1, artic:ua1hagedorn2, artic:mpprod1, artic:mpprod2}. This and other variations of the distribution are exhaustively tested e.g.,~in~\cite{artic:gbiro, artic:tsbeurphys2, artic:flowing0, artic:flowing1, artic:flowing2, artic:flowing3, artic:cleymansphyslett, artic:lilin1, artic:lilin2}. Below we theoretize over the origin of such Tsallis-type formulas. Contrary to the fixed fit parameters of the Tsallis--Pareto distributions as in Equation~(\ref{eq:tsexp}), we~assume that the identified hadron spectra are characterized with a~scaling Tsallis-distribution, where an~energy scaling of the Tsallis-parameters is also present. In the following we refer to these as Tsallis-like~distributions.
In extensive systems the entropy is finite in the thermodynamical limit, $\lim\limits_{N\rightarrow\infty}\frac{S_N}{N}<\infty$. This is the case with the Boltzmann--Gibbs--Shanon entropy formula, $S=-\sum\limits_i P_i \ln{P_i}$, where $P_i$ is the probability of being in state $i$. In strongly correlated systems, it turns out that the total entropy of the system is not the sum of the entropy of the subsystems:
\begin{equation}
S_{12}\neq S_1 + S_2 \ \ .
\end{equation}
For our generalization, we~use the well established terminology of the thermodynamics, since we expect to include the classical Boltzmann--Gibbs case too. Let us consider a~monotonic, transformed entropy, $L(S)$, which satisfies additivity ,
\begin{equation}
L(S_{12})=L(S_1)+L(S_2) \ .
\end{equation}
Note, on general terms $L(S)$ is the logarithm of the formal group of phase space factors $\Omega(S)=e^S$.
Due to this assumption, applied recursively in ensembles, we~arrive at the following general class of entropies~\cite{artic:tsbentr}:
\begin{equation}
L(S)=\sum\limits_i P_i L(- \ln{P_i}) \ \ \ .
\label{eq:entrdef}
\end{equation}
Because $L(S)$ is by definition a~monotonic function, the~most likely state of a~heat reservoir and its subsystem at maximum entropy is equivalently at
\begin{equation}
\label{eq:l-maxent}
L\left( S_1(E_1^* \right) )+L \left( S_2(E_2^*) \right)= \mathrm{max} \ \ \ ,
\end{equation}
where $E_1$ is the energy of the subsystem, $E_2=E-E_1$ is the energy of the reservoir. While keeping $E~=~$const in the entropy maximum Equation~(\ref{eq:l-maxent}) we obtain
\begin{equation}
0= \left.\frac{\partial L}{\partial S_1}\right|_{S_1(E_1^*)} \times \left.\frac{\partial S_1}{\partial E_1}\right|_{E_1^*}
-\left.\frac{\partial L}{\partial S_2}\right|_{S_2(E_2^*)} \times \left.\frac{\partial S_2}{\partial E_2}\right|_{E_2^*} \ \ \ .
\end{equation}
It makes the use of the usual definition of thermodynamical temperature expedient,
\begin{equation}
\beta_1 :=\left.\frac{\partial L}{\partial S_1}\right|_{S_1(E_1^*)} \times \left.\frac{\partial S_1}{\partial E_1}\right|_{E_1^*} =
\left.\frac{\partial L}{\partial S_2}\right|_{S_2(E_2^*)} \times \left.\frac{\partial S_2}{\partial E_2}\right|_{E_2^*} = \beta_2 \ \ \ .
\label{eq:beta1}
\end{equation}
Assuming now $E_1\ll E$ in high-energy collisions, we~consider: $E\rightarrow\sqrt{s}$ and $E_1\rightarrow (m_T-m)\approx p_T$ of the particle,
\vspace{12pt}
\begin{equation}
L(S_2(E-E_1))\approx L\left(S_2(E)-\left.\frac{\partial S_2}{\partial E_2}\right|_E \times E_1\right) \approx L(S_2(E)) -
\left.\frac{\partial L}{\partial S_2}\right|_{S_2(E)} \times \left.\frac{\partial S_2}{\partial E_2}\right|_{E} \times E_1 \ \ .
\end{equation}
Inserting this into Equation~(\ref{eq:beta1}), we~arrive at the formula,
\begin{equation}
\beta_1 \approx \left.\frac{\partial L}{\partial S_2}\right|_{S_2(E)} \times \left.\frac{\partial S_2}{\partial E_2}\right|_{E}
- \left[
\left.\frac{\partial^2 L}{\partial S^2_2}\right|_{S_2(E)} \times \left(\left.\frac{\partial S_2}{\partial E_2}\right|_{E} \right)^2 +
\left.\frac{\partial L}{\partial S_2}\right|_{S_2(E)} \times \left.\frac{\partial^2 S_2}{\partial E^2_2}\right|_{E} \right] \times E_1 + \dots \ \ .
\label{eq:beta1_2}
\end{equation}
By looking for an~{\em universal termostat}, lending to $\beta_1$ an~absolute temperature interpretation, we~assume that the energy of the subsystem is independent from the energy of the reservoir, i.e.,~we~require the term linear in $E_1$ to vanish. After ordering we obtain:
\begin{equation}
\left.\left.\frac{\partial^2 L}{\partial S^2_2}\right|_{S_{2}(E)}\right/ \left.\frac{\partial L}{\partial S_2}\right|_{S_2(E)} =
- \left.\left.\frac{\partial^2 S_2}{\partial E^2_2}\right|_{E}\right/ \left(\left.\frac{\partial S_2}{\partial E_2}\right|_{E}\right)^2 \ \ \ .
\end{equation}
This equality among general functions, $L(S)$ and $S(E)$ is possible only if both are equal with a~constant,
\begin{equation}
\frac{L''(S)}{L'(S)} = - \frac{S''(E)}{S'(E)^2} =\frac{1}{C} := 1-q \ ,
\end{equation}
where $C$ is the heat capacity of the reservoir.
The~solution of this differential equation has all desired features:
\begin{equation}
L(S)=\frac{ {\textrm e}^{(1-q)S}-1}{1-q} \ \ \ .
\end{equation}
Replacing it into the Equation~(\ref{eq:entrdef}),
\begin{equation}
L(S)= \sum\limits_i P_i \frac{ {\textrm e}^{-(1-q) \ln P_i}-1}{1-q} = \frac{1}{1-q} \sum\limits_i
P_i \left( P_i^{q-1} -1\right) = \frac{1}{1-q} \sum\limits_i
\left( P_i^{q} -P_i \right)
\end{equation}
is the (now additive) Tsallis entropy, while
\begin{equation}
S= \frac{1}{1-q} \ln \left( 1+ (1-q) L(S) \right) = \frac{1}{1-q} \ln \sum\limits_i P_i^q \ \ ,
\end{equation}
turns out to be the R\'enyi entropy.
This argumentation can be used also for microcanonical systems, with $S_1=- \ln{P_1}$ and $P_1$ being the distribution of the subsystem's states. Using the previously-defined generalized entropy, $L(S)$, one~arrives at the following energy distribution, which maximizes the $q$-entropy:
\begin{equation}
P_i = \left( Z^{1-q} + (1-q)\frac{E_i}{T}\right)^{-\frac{1}{1-q}} \ \ \ .
\label{eq:ts1}
\end{equation}
It is a~Tsallis--Pareto distribution with the individual energy, $E_i$ and $Z$ is calculated form $\sum p_i=1$.
In high-energy collisions we also have to deal with fluctuations event by event. Following the calculations in references~\cite{artic:tsbentr, artic:tsbphysica}, one may assume that the multiplicity of the created hadrons follow a~negative-binomial distribution for bosons, a~binomial one for fermions. Due to such general reservoir fluctuations, the~$q$ non-extensivity parameter receives a~correction~\cite{artic:tsbentr, artic:tsbphysica}:
\begin{equation}
\label{eq:qwithCandBeta}
q=1-\frac{1}{C}+\frac{\Delta \beta^2}{\left<\beta\right>^2} \ \ \ .
\end{equation}
As the average number of created particles can vary in a~wide range depending on the studied system--typically $\mathcal{O}(10^2)$ in proton-proton collisions, while $\mathcal{O}(10^3-10^5)$ in nucleus-nucleus collisions. We~expect this fluctuation effect to overcome the finite heat capacity condition, therefore one observes $q>1$. It is also straightforward to see that enlarging the system results in $C \to \infty $, and~if fluctuations become sufficiently suppressed, we~get back the Boltzmann--Gibbs case with exponential distribution.
On the other hand, the~assumption $q\rightarrow 1$ leads to a~Gaussian distribution for the $\beta$ values, which can be an~approximation, but never the complete truth. This parameter is used to call the {\it entropic index} or the {\it non-extensivity} parameter, present as a~measure of the deviation from the Boltzmann--Gibbs case, $q=1$. We note, in~high-energy nuclear collisions this value is in the range $1.0<q<1.5$, which~suggests fluctuations override the system size effects, related to the heat capacity of the reservoir.
The~distributions Equations~(\ref{eq:tsexp}) and~(\ref{eq:ts1}) behave similarly, they both can be regarded as Tsallis--Pareto-type distributions. The~authors in \cite{artic:lilin1, artic:lilin2} investigated how the different types fit the experimental data.
Many further useful readings regarding the thermodynamically consistent non-extensive approach can be found in the literature. The~first possibility is presented in \cite{artic:wilkentr,artic:actaphys,artic:chaossolitions}, representing the case where the power is proportional to $\frac{1}{q-1}$. An~another kind of approach where the power is $\frac{q}{1-q}$, as discussed in references~\cite{artic:tsthermo1,artic:tsthermo2,artic:tsthermo3,artic:tsthermo4}.
For our analysis the chosen form is the following~\cite{book:tsb}:
\begin{equation}
\left.\frac{1}{N_{ev}}\frac{ {\textrm d}^2N}{2\pi p_T {\textrm d} p_T {\textrm d} y}\right|_{y\approx0}=A\times\left[1+\frac{q-1}{T} (m_T-m) \right]^{-\frac{1}{q-1}} \ \ \
\label{eq:TS}
\end{equation}
As it was shown in \cite{artic:tsbphysica, artic:tsbphysica2} the parameters $q$ and $T$ for an~ideal case are connected to the mean multiplicity and its variance:
\begin{equation}
T=\frac{E}{\left<N\right>}, \ \ \ \ \ q=\frac{\left< N(N+1) \right>}{\left<N \right>^2} \ \ \ .
\label{eq:variance}
\end{equation}
They also may depend on each other. Since in case of $q \to 1$ one has $T \to T_{BG}$, the~parameter $q$ is a~measure of non-extensitivity (i.e.,~non-Gaussivity in $\beta$ fluctuations, non-Poissonity in the multiplicity distribution $P(N)$). $T$ is like the kinetic temperature.
Based on Equation (\ref{eq:variance}), for fixed $\Delta N^2/\left<N\right>^2=\sigma^2$ one obtains:
\begin{equation}
\frac{T}{E}=\sigma^2+(1-q) \ \ \ .
\end{equation}
On the other hand for an~NBD (Negative Binomial Distribution) with fixed $\left<N\right>/k=f$ one gets
\begin{equation}
T=E\times f \times(q-1) \ \ \ .
\end{equation}
Our aim in the followings is to explore the center-of-mass energy evolution of the parameters $q$ and $T$, especially keeping in our mind their physical meaning.
Based on the definition of the PDF and FF of the pQCD-based parton model, we~expect a~logarithmic scaling.
Since this was observed even in fits of electron-positron data~\cite{artic:flowing0}, where PDFs do not appear, we~connect the non-extensive features with the hadronization (fragmentation) processes only.
The~argumentation behind this will be explained in the next subsection.
\subsection{Motivation for Qcd-Like Energy Scaling of the Parameters}
\label{sec:scaling}
Partons, the~elementary momentum carriers in the strongly interacting matter, are tagged with a~quantum number named \emph{color},
which property is not observable directly.
The~quarks, antiquarks~and gluons together confine into color singlet hadrons during the hadronization process.
Hadron formation can happen at any energy scale, $Q^2$. The~dependency on it can be factorized into the running nature of the strong interaction's coupling constant, $\alpha(Q^2)$.
Since any observable quantity should be independent of the arbitrary fixing of the energy scale appearing in perturbative QCD calculations, the~mathematical description ought to be (energy-)scale independent at any fixed order. To satisfy this request is not an~easy task, due to the non-perturbative nature of non-Abelian fields at low-energy.
As we presented the hadron production within the pQCD-based parton model in Section~\ref{sec:hadronprod}, the~convolution in Equation (\ref{eq:hadronprod}) includes the scale parameter ($Q^2$) in its kernel.
However, the~cross-section---being an~observable quantity---should be independent of this inner parameter.
Technically, this is achieved with the following method: while calculating the partonic (color) cross sections at a~given order, the~scale can be factorized out and merged into the non-perturbative phenomenological functions.
These are the parton distribution and fragmentation functions, and~they must satisfy a~proper scale evolution equation for avoiding scale-dependent hadronic yields.
To obtain the scale invariance, the~following formula should be fulfilled at any fixed order for any generic form of such phenomenological functions~\cite{artic:dglap1, artic:dglap2, artic:dglap3}:
\begin{equation}
\frac{\partial}{\partial \ln{Q^2}} R(z, Q^2) = 0 \ \ \ ,
\end{equation}
where $R(z, Q^2)$ can be either the PDF or FF, typically at the momentum fraction of the mother and daughter particles, $z$.
In general, this evolution equation determines the possible form of quantity $R$, which naively depends on the current energy scale at a~given fixed order.
Solving this Callan--Symanzik equation one can obtain the energy scale dependence of the running coupling at a~given order in any theory~\cite{artic:pdg}.
In the perturbative QCD based parton model, the~parametrized parton distribution and fragmentation {\em ansatz} functions are typically given in a~power-law form~\cite{artic:ellis}.
One can get then the proper scale evolution by solving the Doksitzer--Gribov--Lipatov--Altarelly--Parisi (DGLAP) equations~\cite{artic:dglap1, artic:dglap2, artic:dglap3}.
Due to its polynomial power-law form, the~predictive power of the calculations gets weaker at low-$z$. We expect that a~Tsallis-like distribution with the appropriate parameter evolution can resolve this problem, providing a~better description.
This motivates us to fit $q$ and $T$ parameters as a~function of the center-of-mass energy $Q^2\sim\sqrt{s}$ in a~similar fashion, as it was done in reference~\cite{artic:gribov80}.
Here our aim is to test the validity of this approach via investigating the energy-evolution dependence of the parameters.
\subsection{The~Improved Quark-Coalescence Model}
\label{sec:coal}
Another description of the hadron formation is based on the constituent quark scaling.
In the \emph{quark coalescence model} the usual underlying assumption is that the hadronization takes place in a~thermal system, where all the participating partons emerge at the same temperature~\cite{artic:tsbeurphys, artic:coalescence1}.
This idea was developed for the description of hadron production in high-energy heavy-ion collisions, where the bulk of the hadron yield closely follows the exponential shape.
In larger colliding systems, like in central collisions of large nuclei, this idea worked well, especially for the low transverse momentum regime, $p_T$ < 3--5 GeV/$c$, with a~single temperature parameter.
\textls[-15]{In the original approach the energy distribution of the partons follow the Boltzmann--Gibbs statistics. Then, one approximates the formation rate as the multiplication of $k$ such Boltzmann--Gibbs distributions:}
\begin{equation}
P_k=\left[f_{BG}(E/k)\right]^k= A' e^{-\beta E} \ \ \ .
\label{eq:Pk}
\end{equation}
In the present non-extensive framework we still assume that the partons are part of a~simple ensemble, but we replace the Boltzmann--Gibbs exponentials by Tsallis--Pareto distributions.
Now the rate is the following:
\begin{equation}
P_k=\left[f_{Ts}(E/k)\right]^k=A^k \left[ 1+ \frac{q-1}{T} E_k \right] ^{-\frac{k}{q-1}} = A' \left[ 1+ \frac{q'-1}{T} E \right] ^{-\frac{1}{q'-1}} \ \ \ ,
\label{eq:Pknonext}
\end{equation}
with $(q'-1)=\frac{(q-1)}{k}$.
In order to test this model, we~check the fitted $q$ parameters of the identified hadrons.
If the equal-energy quark-coalescence theory is correct for both the quark-antiquark containing mesons and triple (anti)quark (anti)baryons, then one observes:
\begin{equation}
(q_{quark}-1)=2(q_{meson}-1)=3(q_{baryon}-1) \ \ \ .
\end{equation}
The~meson-baryon ratio for the Tsallis parameters should be around 3/2:
\begin{equation}
\label{qmperqb}
\frac{q_{meson}-1}{q_{baryon}-1}=\frac{3}{2} \ .
\end{equation}
This idea formulated by Equation~(\ref{qmperqb}) can be tested as fitting the hadron spectra and investigating the ratio of the parameter $(q_i-1)$ for different identified hadron species. As the constituent quark scaling is getting stronger at larger energies, we~expect to reach this theoretical value only~asymptotically.
\section{Fitted Parameters}
\label{sec:third}
For our analysis we use identified hadron spectra datasets measured in proton-proton collisions in recent years~\cite{artic:62gevphenix, artic:62gevphenix2, artic:200gevphenix, artic:500gevphenix, artic:200gevstar,artic:097tevalice, artic:09tevalice,artic:276tevalice,artic:276tevpi0alice,artic:7tevalice, artic:7tevalice2}.
The~numerical fits of the various datasets were made utilizing the CERN Root analysis software (\myurl{https://root.cern.ch/}, version 6.06/00).
Although we conducted a~comprehensive study, it is worth to note that it is not necessarily meaningful to compare the fit-parameter values for all existing data, because kinematical ranges may vary and the multiplicity-classes are also not defined evenly. This especially holds for kaons and protons at high $p_T$; this generates some uncertainty in those fits. To circumvent these difficulties, we~fixed a~recipe for the fit procedure making it consistently insensitive to the fit program(s), the~size of the fit parameter space and input parameters.
In order to counterbalance the effect of varying ranges we perform the fit procedure in multipl~steps:
\begin{enumerate}[leftmargin=*,labelsep=5mm]
\item fit of the high-$p_T$ part by fixing $T$ and changing $q$;
\item fit of the low-$p_T$ part by fixing $q$ and changing $T$;
\item fit of the whole $p_T$ range with both parameters, starting from the above obtained $q$ and $T$.
\end{enumerate}
We define ``low-$p_T$'' as $p_T< 4$ GeV/$c$ and the ``high-$p_T$'' part as $p_T\geq 2$ GeV/$c$, respectively. The~overlap is intended. In the function defined in Equation~(\ref{eq:TS}) the parameter $T$ sets the characteristic $p_T$ scale. In fact, for $q\longrightarrow 1$ one obtains $f_1(m_T)=Ae^{-(m_T-m)/T}$. The~parameter $q$ on the other hand is linked to the 'power-law like' tail at high $p_T$.
The~procedure was evaluated by comparing the $\chi^2/NDF$~values.
The~investigated spectra and the fitted Tsallis--Pareto functions are shown in Figure~\ref{fig:dataperfit}. {\em Upper panels} of the plots present the fits of experimental data measured in proton-proton collisions at $\sqrt{s}=62.4$~GeV, 200 GeV, 500 GeV, 900 GeV, 2.76 TeV, and~7 TeV center-of-mass energies. We considered various neutral, charged and charge-averaged hadron species, $\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $p$, and~$\bar{p}$. Identified hadron $m_T$ spectra are scaled by constant factors ($2^n$) for better visibility, as indicated in the panels.
In the {\em lower panels} ``Data/Fit'' plots are presented for each case. One can observe how well the distribution~(\ref{eq:TS}) describes the yields in the whole 62.4 GeV $\leq \sqrt{s} \leq 7$ TeV center-of-mass energy range in the $m_T \lesssim 20$ GeV region. Within the mid $m_T$-regime the overlap with data is excellent, while at the highest $m_T$ values or for heavier hadrons the deviation is somewhat larger.
\begin{figure}[H]
\centering
\centerline{\includegraphics[width=0.98\textwidth]{dataperfit_all.pdf}}
\caption{\textit{Upper panels} are the identified hadron spectra as the function of the transverse mass, $m_T$ is plotted, measured by the PHENIX~\citep{artic:200gevphenix, artic:62gevphenix, artic:62gevphenix2, artic:500gevphenix}, STAR~\citep{artic:200gevstar}, and~ALICE~\citep{artic:097tevalice,artic:09tevalice,artic:276tevalice,artic:276tevpi0alice,artic:7tevalice,artic:7tevalice2} at different center-of-mass energies from 62.4 GeV $\leq \sqrt{s} \leq 7$ TeV. Experimental data is in comparison with the fitted Tsallis--Pareto functions is indicated as solid lines. The~$m_T$ spectra were scaled by constant factors ($2^m$) for the better visibility as indicated on the graphs. \textit{Lower panels} are present the Data/Fit ratio plots including the estimated fit errors.}
\label{fig:dataperfit}
\end{figure}
To determine the center-of-mass energy dependence, we~review the $\sqrt{s}$ evolution of the fitted $q_i$ and $T_i$ parameters for each hadron species, $i \in ~\left\{ \pi^{\pm}, \pi^{0}, K^{\pm}, p, \textrm{and } \bar{p} \right\} $. According to the formula~(\ref{eq:TS}), the~parameters of the Tsallis--Pareto distribution are plotted in Figure~\ref{fig:q,T} as a~function of $\sqrt{s}$. Based on the motivation presented in Section~\ref{sec:scaling}, we~assumed an~energy-evolution for each hadron type $i$ as follows~\cite{artic:karesz}:
\begin{equation}
\label{eq:q-evol}
q_i(\sqrt{s})=q_{1i}+ q_{2i} \log\left( \sqrt{s}/m_i\right),
\end{equation}
\begin{equation}
\label{eq:T-evol}
T_i(\sqrt{s})=T_{1i}+ T_{2i} \log\left( \sqrt{s}/m_i\right).
\end{equation}
In these formulae, the~mass $m_i$ of the identified hadron $i$ is used to set the physically relevant energy scale.
In Equation (\ref{eq:T-evol}) the parameter $T_{1,i}$ is fixed at $T_{1,i}=50$ MeV, as suggested by reference~\cite{artic:tsbeurphys2}. Hence, the~function in Equation (\ref{eq:T-evol}) is parametrized by $T_{2,i}$.
In Figure~\ref{fig:q,T}a, the~fitted $q_i$ values are plotted for each $\sqrt{s}$, for given identified hadron spectra summarized in Table~\ref{tab:qT}. One can see in the graphs, that the $q_i(\sqrt{s})$ values are close to each other and all curves slightly increase with $\sqrt{s}$, following nicely the formula~(\ref{eq:q-evol}). For pions and kaons the increase is very similar and their evolutions are alike. However, the~precise increase seems to be larger for the~kaons.
We note, that charged and neutral pion results should be consistent. Taking $\pi^0$ and $\pi^{\pm}$ together for the fits, the~mesonic components overlap more. This alternative behavior is thought to be the effect of different kinematical ranges. See more on fitted parameters and $\chi^2/NDF$ in Table~\ref{tab:fitparams} and on kinematical ranges in Table~\ref{tab:kine} in the Appendixes~\ref{appendix:a} and~\ref{appendix:b}.
\begin{table}[H]
\centering
\begin{tabular}{cccccc}
\toprule
\textbf{Hadron,} \boldmath{$i$} & \boldmath{$m_i$} & \boldmath{$q_{1i}$} & \boldmath{$q_{2i}$} & \boldmath{$T_{1i}$} & \boldmath{$T_{2i}$} \\
\midrule
$\pi^{0}$ & 135.0 MeV & $1.03\pm0.002$ & $0.011\pm 0.002 $ & 50 MeV & $0.006\pm 0.001$ MeV \\
$\pi^{\pm}$ & 140.0 MeV & $1.04\pm0.01$ & $0.009\pm 0.002 $ & 50 MeV & $0.009\pm 0.001 $ MeV \\
$K^{\pm} $ & 493.0 MeV & $1.00\pm0.01 $ & $0.016\pm 0.001 $ & 50 MeV & $0.018\pm 0.001 $ MeV \\
$p(\bar{p})$ & 938.0 MeV & $1.09\pm0.01 $ & $0.004\pm 0.001 $ & 50 MeV & $0.021\pm 0.001 $ MeV \\
\bottomrule
\end{tabular}
\caption{The~$\sqrt{s}$-evolution of the parameters of the fitted Tsallis--Pareto distributions for hadrons, \mbox{$i \in$ $\pi^{\pm}$}, $\pi^{0}$, $K^{\pm}$, $p$, and~$\bar{p}$ in the $62.4$ GeV $\leq \sqrt{s} \leq 7$ TeV c.m. energy range.}
\label{tab:qT}
\end{table}
\unskip
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\includegraphics[width=70mm,scale=0.8,angle=0]{q_TS_mT.pdf}
& \includegraphics[width=70mm,scale=0.8,angle=0]{T_TS_mT.pdf}\\
({\bf a})&({\bf b})\\
\end{tabular}
\caption{The~fitted $q_i$ (\textbf{a}) and $T_i$ (\textbf{b}) as a~function of $\sqrt{s}$ for hadron species, $i$ marked as points. Only the species of the particles are indicated. The~solid color lines are fitted to the pion, kaon, and~(anti)proton points. Vertical lines indicate the place of the $\sqrt{s}=13 $ TeV and $14$ TeV data.}
\label{fig:q,T}
\end{figure}
Figure~\ref{fig:q,T}b presents the evolution of the parameter $T_i$ with the center of mass energy. We applied an~evolution according to $\sim \log(\sqrt{s})$ in the Figure~\ref{fig:q,T} using Formula~(\ref{eq:T-evol}) with the evolution parameters listed in Table~\ref{tab:qT}.
Here the energy evolution of the parameter $T(\sqrt{s})$ shows an~increasing trend with the mass of the hadron species. The~obtained parameter values supports the idea of a~mass hierarchy effect: the higher the mass, $m_i$, the~larger $T_{2i}$.
One can realize from Figure~\ref{fig:q,T}a,b, by comparing them in the $\sqrt{s}<3$ TeV regime, that massive protons and kaons present the smaller change in $q_i$, and~their masses are closer to the lattice QCD crossover temperature $T\approx 170$ MeV~\cite{Katz}. Light pions deviate more as increasing the energy, and~the obtained $T_{\pi}$ is smaller, around $\approx 100$ MeV. It is consistent with our picture that, lighter particle can suffer larger fluctuations, which increases the parameter $q_i$ following Equation~(\ref{eq:qwithCandBeta}).
Based on the $\sqrt{s}$ evolution of the experimental fit curves in Figure~\ref{fig:q,T}, we~could predict the parameter values for the soon-to-be available LHC-energy collisions at $\sqrt{s}= 13$ TeV and $14$ TeV. These~energies are indicated on both panels with vertical lines. According to the the assumptions given by the {\em ansatz} Formulae~(\ref{eq:q-evol}) and~(\ref{eq:T-evol}) and the fit parameters from Table~\ref{tab:qT} we summarized these values in Table~\ref{tab:qT-for13TeV} for $\sqrt{s}=13$ TeV. These~data were used to plot the 13 TeV center-of-mass energy prediction on the {\em bottom right panel} of Figure~\ref{fig:dataperfit}. Note, $\sqrt{s}=14$ TeV data is expected to have very similar values within errors.
In Figure~\ref{fig:qspecies} we show the fitted $q_i$ and $T_i$ values for different hadron species at the center-of-mass energy values listed above. In agreement with reference~\cite{artic:cleymansqspecies}, we~observe that the non-extensivity parameter $q_i$ on Figure~\ref{fig:qspecies}a is less sensitive to the hadron mass, however the importance of the center-of-mass energy of the colliding system is remarkable. As we have seen already on Figure~\ref{fig:q,T}, pions have somewhat larger non-extensitivity than more massive hadrons:
\begin{equation}
q_{\pi}(\sqrt{s}) \ \gtrsim \ q_{K}(\sqrt{s}) \ \gtrsim \ q_{p}(\sqrt{s}) \ \ \ .
\label{eq:q-order}
\end{equation}
Figure~\ref{fig:qspecies} also highlights that protons present weaker c.m. energy dependence than mesons.
The~parameter $T_i$ reflects an~opposite mass-hierarchy ordering, on Figure~\ref{fig:qspecies}b. The~more massive hadron, the~larger $T_i$ value:
\begin{equation}
T_{\pi}(\sqrt{s}) \ < \ T_{K}(\sqrt{s}) \ < \ T_{p}(\sqrt{s})\ \ \ .
\label{eq:T-order}
\end{equation}
Nevertheless, we~observe only a~weak center-of-mass energy dependence.
\begin{table}[H]
\centering
\begin{tabular}{cccc}
\toprule
\textbf{Hadron,} \boldmath{$i$} & \boldmath{$m_i$} & \boldmath{$q_{i}$} & \boldmath{$T_{i}$} \textbf{(MeV)}\\
\midrule
$\pi^{0}$ & 135.0 MeV & $1.156\pm0.001$/$1.157\pm0.001$ &
$119.0\pm 2.0$/$119.0\pm 2.0$ \\
$\pi^{\pm}$ & 140.0 MeV & $1.143\pm0.001$/$1.144\pm0.001$ &
$153.0\pm 2.0$/$154.0\pm 2.0$ \\
$K^{\pm} $ & 493.0 MeV & $1.163\pm0.002$/$1.164\pm0.002$ &
$233.0\pm 2.0$/$234.0\pm 2.0$ \\
$p(\bar{p})$ & 938.0 MeV & $1.128\pm 2.0.003$/$1.128\pm0.003$ &
$250.0\pm 2.0$/$252.0\pm 2.0$ \\
\bottomrule
\end{tabular}
\caption{Predictions for the Tsallis--Pareto parameters for
$\sqrt{s}=13$ TeV ({\it left}) and $14$ TeV ({\it right}), for~hadrons $i \in$ $\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $p$, and~$\bar{p}$, based on the Formulae~(\ref{eq:q-evol}) and~(\ref{eq:T-evol}).}
\label{tab:qT-for13TeV}
\end{table}
\unskip
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\includegraphics[width=70mm,scale=0.8,angle=0]{qspecies.pdf}
& \includegraphics[width=70mm,scale=0.8,angle=0]{Tspecies.pdf}\\
({\bf a})&({\bf b})\\
\end{tabular}
\caption{The~fitted $q_i$ ({\bf a}) and $T_i$ ({\bf b}) values for each hadron type, $i \in $ $\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $K_s^0$, $p$, and~$\bar{p}$, c.f.~\cite{artic:cleymansqspecies}}.
%
\label{fig:qspecies}
\end{figure}
\subsection{The~$(T,q)$ Parameter Space for Identified Hadrons}
\label{sec:qT}
Summarizing our obserations we can conclude that the center-of-mass energy evolution of the fit parameters works well with our logarithmic evolution {\em ansatz}.
\begin{itemize}[leftmargin=*,labelsep=6mm]
\item
The~$q_{2i}$ and $T_{2i}$ parameters are getting slightly larger with the larger hadron mass, $m_i$, and~applying Formulae~(\ref{eq:q-evol}) and~(\ref{eq:T-evol}) the evolution is described nicely in the whole tested energy range, \mbox{$62.4$~GeV $\leq \sqrt{s} \leq 7$ TeV}.
\item The~obtained $q_i(\sqrt{s})$ function increases with $\sqrt{s}$ in the range 1.07--1.17 indicating the deviation from the Boltzmann--Gibbs case where $q=1$. The~deviation from this ``{\em thermodynamical limit}'' case grows as the center-of-mass energy gets higher values. However large statistical errorbars correspond to the lack of statistics in specific particle identification methods of the measurements. (See more in Appendix~\ref{appendix:b}.)
\item The~$T_i(\sqrt{s})$ kinetical temperature parameters almost keep constant values, with the following hadron (mass) hierarchy: $T_{\pi}$ = 120--140 MeV, $T_{K}$ = 120--200 MeV, and~$T_p$ = 70--240 MeV.
\end{itemize}
We plot the parameters $q_i$ and $T_i$ on the Figure~\ref{fig:T-q2}. The~fitted parameters gather in the $T_{i}\in [70,240]$ MeV and $q_i(\sqrt{s}) \in [1.07,1.17]$ parameter space, which is indicated by the shaded area.
In Figure~\ref{fig:T-q2}b, while keeping the shaded area, we~included the fit-result of theoretical calculations as well. We used two model to get the identified hadron spectra series, namely~PYTHIA8~\cite{artic:pythia6,artic:pythia8} and kTpQCD\_v20~\cite{artic:ktpqcd}. We chose several c.m. energy values and the pseudorapidity region, $|\eta|< 0.5$ for our calculations, and~finally we applied the same fit procedure as described in Section~\ref{sec:fourth}. We plotted the parameters $q_i$ and $T_i$ together with the experimental data-fitted {\em shaded} region.
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\includegraphics[width=70mm,scale=0.8,angle=0]{Tvsq_woA.pdf}
& \includegraphics[width=70mm,scale=0.8,angle=0]{Tvsq_wA.pdf}\\
({\bf a})&({\bf b})\\
\end{tabular}
\caption{(\textbf{a}) The~parameter space $T_i-q_i$ extracted from experimental $p_T$ spectra for hadron types $i \in $ $\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $p$, and~$\bar{p}$; (\textbf{b}) The~PYTHIA and kTpQCD\_20 calculated spectra fit results on the same hadron types added to the measurement fit points.}
\label{fig:T-q2}
\end{figure}
\vspace{-6pt}
The~calculated points are denoted by {\em empty symbols} for each identified hadron at several center-of-mass energies. Theoretical data fits partially overlap with the {\em shaded} area defined by the experimental fits, although several points are deviating.
\vspace{6pt}
\noindent \textbf{PYTHIA8}
\vspace{6pt}
We generated 10M events using PYTHIA8~\cite{artic:pythia6,artic:pythia8} Lund's high-energy Monte Carlo event generator to simulate the identified hadron spectra. Points for (charge-averaged) pion, kaon and protons spread in the $q_i \in [1.08,1,23]$ range, wider than the experimental points. In contrast to that, the~$T_i \in [80,150]$ MeV corresponding to the experimental values on the {\em left panel}. Deviating points of the PYTHIA8 results are those which lack sufficient statistics at the highest transverse momenta. Here, the~tail of the distribution is indefinite, thus $q_i$ values fall outside of the experimental $q_i(\sqrt{s}) \in [1.07,1.17]$ parameter space. One recognizes that pions deviate less, since they have the highest statistics among all, followed by kaons and protons. We note that the deviance of $q_i$ disappear as we exclude the low-statistic data at the highest momenta at each energy value, while the consistency with $T_i$ remains.
\vspace{6pt}
\noindent \textbf{kTpQCD\_v20}
\vspace{6pt}
Hadron spectra calculated within the framework of perturbative QCD were also used utilizing kTpQCD\_v20~\cite{artic:ktpqcd}. These~calculations deliver similar results for all hadron species, because of the similar (polynomial) fragmentation parametrizations. Concerning the correspondence between perturbative QCD results and experimental data fits, both $T_i$ and $q_i$
are running out of the experimental regime in a~similar way. Deviation from the measurement-based data is most remarkable at low c.m. energies, where the $p_T$ range of the spectra is too short due to the limited phase-space. The~domain of pQCD is $p_T > 1.5$ GeV/$c$ and the maximal energy is typically $p_T<\sqrt{s}/2$. This limited range makes the fits more doubtful.
\vspace{6pt}
We also investigated the center-of-mass energy dependence of the fit parameters calculated theoretically from the the PYTHIA8~\cite{artic:pythia6,artic:pythia8} and kTpQCD\_v20~\cite{artic:ktpqcd} models. In Figure~\ref{fig:qth}, we~compare the experimentally observed $\sqrt{s}$-dependence from Figure~\ref{fig:q,T} ({\em solid lines}) to these theoretical model results ({\em data points}). Figure~\ref{fig:qth}a is for parameter $q_i$.
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\includegraphics[width=70mm,scale=0.8,angle=0]{q_TS_mT_theory_cut.pdf}
& \includegraphics[width=70mm,scale=0.8,angle=0]{T_TS_mT_theory_cut.pdf}\\
({\bf a})&({\bf b})\\
\end{tabular}
\caption{The~fitted $q_i$ (\textbf{a}) and $T_i$ (\textbf{b}) as a~function of $\sqrt{s}$ for hadron species, $i$ marked as points. Only~the species of the particles are indicated by points. The~solid color lines are fitted to the experimentally measured pion, kaon, and~(anti)proton points.}
\label{fig:qth}
\end{figure}
\noindent \textbf{Non-extensivity,} $\boldsymbol{q_i}$:
\vspace{6pt}
According to the above observations, the~perturbative QCD points are close to each other, due to the similar fragmentation function parametrization for the hardon species, mostly at the tail of the distributions at the highest energies and momenta---where experimental and theoretical data meet each other. With kTpQCD\_v20 pions and kaons have similar slopes in $\log(\sqrt{s})$. PYTHIA8 results violate the inequality~(\ref{eq:q-order}) and are larger: $q_{PYTHIA,i} > q_{EXP,i}$. However, the~$\log(\sqrt{s})$ evolution has the same trend and similar slopes---except for protons.
\vspace{6pt}
\noindent \textbf{Temperature-like,} $\boldsymbol{T_i}$:
\vspace{6pt}
The~kTpQCD\_v20 points for each hadron species are close to each other and meet the temperature values only at the highest-energy regime. In this case the formula~(\ref{eq:T-order}) represents a~trend opposite to the perturbative QCD calculations. Theoretical fit parameters deviate here appreciably. We count this for the non-applicability of the pQCD at the low-momentum regime, $p_T<1.5$ GeV/$c$, where the spectra are more thermal-like. On the other hand, PYTHIA8 works well for both the {\em soft} and {\em hard} regimes for the light hadrons. The~$\sqrt{s}$ evolution follows the experimentally observed trend, only~a~small offset is present for kaons and~protons.
We conclude that these model calculations proved their validity in several ways~\cite{artic:predict1, artic:predict2}. Nevertheless, these pictures are not fully consistent with the fit parameters obtained from the data. In~other words, comparison of theoretical models should be made with care within their region of~validity.
\section{Comparison with the Improved Quark-Coalescence Model}
\label{sec:fourth}
As we have explained in Section~\ref{sec:coal}, the~quark-coalescence model was developed for heavy-ion collisions originally, but it was improved by the previously introduced Tsallis distribution. In the followings we endeavor to extend this idea also for smaller systems, such as proton-proton collisions.
\subsection{Connecting Non-Extensivity with the Quark-Coalescence Model}
\label{sec:coal-q}
According to the coalescence picture, the~observed of $(q_{meson}-1)/(q_{baryon}-1)$ should be $3/2$~\cite{artic:quarkscaling}.
In Figure~\ref{fig:qratios} we plot the ratio $\chi_{ij}=(q_{i}-1)/(q_{j}-1)$ as a~function of the center-of-mass energy, $\sqrt{s}$. Figure~\ref{fig:qratios}a presents the ratios of experimental {\em data points} compared to the fit curve ratios of $\pi/p$, $K/p$ and $K/p$ listed in Table~\ref{tab:qT}. A monotonic, increasing trend of the ratios is clearly seen at the lower energies and saturation can be expected in the most energetic reactions. There are two important observations that is worth note:
\begin{enumerate}[leftmargin=*,labelsep=5mm]
\item the $(q_{meson}-1)/(q_{baryon}-1)$ fit curves lie below the {\em dashed line} with the value of $3/2$ within the $\sqrt{s} \in [62.4~\textrm{GeV},10~\textrm{TeV}]$ c.m. energy range;
\item the $\chi_{K\pi}$ kaon-pion ratio shoots over the expected value, 1, a~bit.
\end{enumerate}
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\includegraphics[width=70mm,scale=0.8,angle=0]{qmesonbarion_woA.pdf}
& \includegraphics[width=70mm,scale=0.8,angle=0]{qmesonbarion_wA_cut.pdf}\\
({\bf a})&({\bf b})\\
\end{tabular}
\caption{ ({\bf a}) Ratio of $(q_{meson}-1) / (q_{baryon}-1)$ plotted in the function of $\sqrt{s}$ as data {\em points} and fit {\em lines}. ({\bf b}) The~PYTHIA and kTpQCD\_20 calculated spectra fit results on the same hadron types added to the measurement fit lines. }
\label{fig:qratios}
\end{figure}
On Figure~\ref{fig:qratios}b we plotted the theoretically calculated \emph{points} along with the experimentally fitted {\em solid lines}. It is not surprising, that theoretical curves present quite flat functions for all combination of the ratios, since both the PYTHIA8's Lund fragmentation and the fragmentation function parametrizations in the kTpQCD\_v20 are based on the constituent quark model and the infinite momentum frame is assumed as well. Thus, there is no room for the evolution apart from a~constant hadron-mass effect. This can result only in a~shift of the $q_i$ values as we have seen on the {\em left~panel} of Figure~\ref{fig:qth}. Deviation from the constancy appears only at the lowest energies, but as we have seen earlier in Section~\ref{sec:qT}, at low energies both PYTHIA8 and kTpQCD\_v20 have limited phase space. The~average values of the ratios
$\chi_{ij}=(q_{i}-1)/(q_{j}-1)$ are summarized in Table~\ref{tab:qij} for $i,j \in \{\pi, K, p\}$.
\begin{table}[H]
\caption{The~average values of the hadron spectra parameter ratios, $\chi_{ij}=(q_{i}-1)/(q_{j}-1)$, obtained from theoretical models PYTHIA8 and kTpQCD\_v20.}
\centering
\begin{tabular}{cccc}
\toprule
\textbf{Hadron Ratio} & \textbf{PYTHIA8} & \textbf{kTpQCD\_v20} & \textbf{Colaescence} \\
\midrule
$\chi_{K\pi}$ & $ 1.09 \pm 0.01 $ & $ 1.10 \pm 0.01 $ & $1.0$ \\
$\chi_{Kp}$ & $ 0.95 \pm 0.01 $ & $ 1.06 \pm 0.01 $ & $1.5$ \\
$\chi_{\pi p}$& $ 0.87 \pm 0.01 $ & $ 0.94 \pm 0.01 $ & $1.5$ \\
\bottomrule
\end{tabular}
\label{tab:qij}
\end{table}
The~kTpQCD\_v20- and PYTHIA8-calculated $(q_{i}-1)/(q_{j}-1)$ points have the same order:
\begin{equation}
\chi_{K\pi}>\chi_{Kp}>\chi_{\pi p}\ \ \ .
\end{equation}
For $\chi_{K\pi}$ both models give the same value, $10\%$ larger than the improved coalescence expectation of 1. For $\chi_{meson,baryon}$ kTpQCD\_v20 has slightly higher values than PYTHIA8, but both are far below the expected value 3/2. Comparing the experimental fit {\em curves} and the theoretically calculated {\em points}, both theory meets the experimental values of $\chi_{K\pi}$. However, for $\chi_{meson,baryon}$ the kTpQCD\_v20 model agrees with the experimental values only at the highest $\sqrt{s}$ c.m. energies.
In summary the improved quark-coalescence model prediction might be reached only beyond the LHC energies, now they seem to support the smaller values. Allegedly, constituent-quark scaling is a~high-$\sqrt{s}$ property. Experimental data support the trends, the~very hadronizaton model needs further~investigation.
\subsection{Investigating the $T_{Slope}$ in the Quark-Coalescence Model}
\label{sec:coal-q}
In the quark-coalescence model, $T_{hadron}=T_{parton}=T$, thus $T_{meson}=T_{baryon}$ is also assumed, c.f.~Section~\ref{sec:coal}. Using the Tsallis distribution we consider the logarithmic slope of $E_i$ spectra:
\begin{equation}
T_{slope}= \left[ -\frac{ {\textrm d}}{ {\textrm d} E_i} \ln P_i \right]^{-1} = T + (q-1)E_i \ \ \ .
\label{eq:tslope}
\end{equation}
This may explain the mass ordering found in Equation (\ref{eq:T-order}). A possible way to read off this effect would be to determine the slope of $(m_{T,i}-m_i)$ spectra. Estimating $E_i$ by $\sqrt{s}-m_i$ one obtains results as seen in Figure~\ref{fig:tslope}.
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{all_tslope_mT2}
\caption{The~$T_{slope}(\sqrt{s})$ curves defined by Equation~(\ref{eq:tslope}), fitted on the theoretical PYTHIA8 and kTpQCD\_v20 data (empty points) and on the experimental values (solid points), for all investigated hadron species.}
\label{fig:tslope}
\end{figure}
\section{Summary and Discussion}
\label{sec:fifth}
In this study we analyzed identified hadron spectra measured in proton-proton collisions from RHIC to LHC energies in the range $62.4$ GeV $\leq \sqrt{s} \leq 7$ TeV. We showed that the Tsallis--Pareto distributions originated from non-extensive thermodynamics describe the spectra very well in wide $m_T$ regions, typically at $p_T\lesssim$ 10--20 GeV/$c$ using the distribution in the form of Equation~(\ref{eq:TS}).
We provided a~comprehensive and detailed analysis of the state-of-the-art experimental data which will be used also to make predictions about the forthcoming 13 TeV and 14 TeV spectra. The~$\sim\log(\sqrt{s})$-like evolution of the parameters $q_i$ and $T_i$ were tested on the identified hadron spectra data measured for charge averaged $\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $p$, and~$\bar{p}$. We observed that both the non-extensivity parameter $q_i$ and temperature-like $T_i$ parameters agree with the suggested QCD-inspired evolution pattern. However, the~temperature has almost a~constant value within the investigated center-of-mass energy regime. We found a~mass-ordered hierarchy in the evolution parameters of the experimental fits, i.e.,~lighter hadron spectra have the more non-extensive $q_{i}>1$ and heavier hadron spectra fit with larger $T_i$ values. However, we~note that the deduced $T_i$ values might be sensitive not just the hadron mass but also on the chosen distribution function. This might result different values extracted from similar data, e.g.,~in reference~\cite{artic:cleymansTS}.
We compared the experimental data fit results with theoretical predictions. The~c.m. energy evolution of the fit parameters were calculated by PYTHIA8~\cite{artic:pythia6,artic:pythia8} and kTpQCD\_v20~\cite{artic:ktpqcd}.
We~found theoretical fit parameters to be more compact in the $(T,q)$ space than the experimental ones: \mbox{$T_i \in [80,240]$ MeV,} while non-extensivity is wider than the measurement-based $q_i \in [1.08,1,23]$ region. The~most deviating points arise from limitations of the theoretical models, i.e.,~where statistics is low or the phase-space is limited. Energy evolution in the theoretical models were investigated as well. In~agreement with our expectations we conclude that
\begin{enumerate}[leftmargin=21pt,labelsep=7pt]
\item[(i)] for the $\sqrt{s}$ evolution, kTpQCD\_v20 agrees more with the power-law related non-extensivity parameter $q_i$;
\item[(ii)] PYTHIA8 results correspond well with the measured $T_i(\sqrt{s})$ evolution.
\end{enumerate}
The~study of these models reflected the lack of the proper handling of the hadron-mass, since all assumptions fail for more massive hadron species.
At the highest energies and momenta, in~the infinite momentum frame, constituent quark number scaling is assumed to get stronger. To test this idea in the framework of the non-extensive approach, we~applied and investigated an~improved quark-coalescence model, inserting Tsallis-like energy distribution kernels where constituent quark scaling appears explicitly. Experimental~data present a~slight monotonic-increase with c.m. energy, but the saturation ridge of the ratio \mbox{$(q_{meson}-1)/(q_{baryon}-1)$} is lower than predicted by the coalescence-theory (3/2) while the reference ratio $(q_{meson}-1)/(q_{meson'}-1)$ is only slightly apart from the expected value $1$.
The~fit parameters calculated by PYTHIA8 and kTpQCD\_v20 models both have almost no $\sqrt{s}$ evolution, but only the ratio values for light mesons are in agreement with the experimental data especially at the highest LHC energies.
In summary, our detailed analysis aimed to investigate how we can provide physical meaning for experimentally-fitted parameters, based on well-known theoretical models and phenomena. Our~results motivate us to improve the model of hadronization in high-energy collisions, using spectra with exponential shape at low-$p_T$, keeps the power-law tail at thigh $p_T$, and~takes care of the meson/baryon spectra ratios and/or the experimentally observed $(q_{meson}-1)/(q_{barion}-1)$.
\vspace{6pt}
\acknowledgments{This work has been supported by Hungarian OTKA grants K104260, NK106119, K120660, by~the MTA-UA bilateral mobility program NKM-81/2016, and~by the Hungarian-Chinese Collaboration NKFIH TET 12 CN-1-2012-0016. G.G.B. thanks the J\'anos Bolyai Research Scholarship of the Hungarian Academy of Sciences for support. G.B. thanks for the support of Wigner GPU Laboratory. Author \'A.T. is supported by the \'UNKP-16-2 New National Excellence Program of the Ministry of Human Capacities, Hungary.}
\authorcontributions{G.B. collected and elaborated the experimental data and wrote the manuscript. G.G.B. conceived the study, interpreted the data and wrote the first version of the paper. \'A.T. performed the theoretical model calculations and analyzed the simulated data. T.S.B. developed the theoretical background and improved the formulations of the manuscript. K.\"U. initiated the method used for interpretation of the data presented in this~work.}
\conflictsofinterest{The~authors declare no conflict of interest.}
\abbreviations{The~following abbreviations are used in this manuscript:\\
\noindent
\begin{tabular}{@{}ll}
ALICE & A Large Ion Colliding Experiment \\
BNL & Brookhave National Laboratory \\
CERN & Conseil Européen pour la Recherche Nucléaire \\
CM & Center-of-mass \\
DGLAP & Dokshitzer--Gribov--Lipatov--Altarelli--Parisi \\
FF & Fragmentation Function\\
\end{tabular}}
\noindent
\begin{tabular}{@{}ll}
LHC & Large Hadron Collider \\
NBD & Negative Binomial Distribution \\
NDF & Number of Degrees of Freedom \\
PHENIX & A Physics Experiment at RHIC \\
PDF & Parton Distribution Function\\
RHIC & Relativistic Heavy Ion Collider \\
(p)QCD & (perturbative) Quantum Chromo Dynamics \\
STAR & Solenoidal Tracker at RHIC \\
QGP & Quark Gluon Plasma
\end{tabular}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,727 |
\section{Introduction}
A \emph{semiring} is an algebra with two associative binary operations $+,\cdot$, in which $+$ is commutative and $\cdot$ distributes over $+$ from the left and right. Sometimes an additive and multiplicative $0$ element is required, though in the theme of many earlier investigations in the theory of semiring varieties, we do not ask this here. The special case of \emph{additively idempotent semirings} (henceforth, \emph{ai-semirings}), where $+$ is idempotent, has received particular attention in the study of varieties and equational properties; these are also often called \emph{semilattice ordered semigroups}. Several of the most famous semirings are ai-semirings: the Kleene semiring of regular languages (see Conway \cite{con} for example), the max-plus and min-plus semirings of tropical analysis (see Aceto, \'Esik and Ing\'olfsd\'ottir~\cite{AEI}, for example), the powerset semirings of semigroups (see Dolinka~\cite{dol1} for example), and semirings of binary relations under composition and union (Andr\'eka and Mikul\'as \cite{andmik} for example). The $+$ operation is interpreted as union in all these cases except for the max-plus and min-plus algebras where it is max and min, respectively. There is a further combinatorial appeal to ai-semirings, as distributivity of $\cdot$ over $+$ and the commutative and idempotent properties of $+$ easily reveal that ai-semiring terms correspond in natural ways to sets of (multiplicative) semigroup terms. This already suggests a strong affinity between semigroup varieties and semiring varieties, and so it is perhaps not surprising that the vast majority of investigations relating to the general theory of ai-semiring varieties have built on the very heavily developed theory of semigroup varieties. We have in mind in particular, results concerning the nonfinite axiomatisability properties of ai-semirings.
An algebra $A$, or variety $V$ is said to be \emph{finitely based} (FB) if the set of its valid equations is derivable from some finite subset; otherwise it is said to be \emph{nonfinitely based} (NFB). A series of papers by Igor Dolinka \cite{dol0,dol1,dol2,dol3} showed that early ideas of Peter Perkins~\cite{per} for finite semigroups and the very deep contributions of Mark Sapir \cite{sap1,sap2} for inherently nonfinitely based semigroups (precise definition later) could be translated, with some care, to some significant and natural examples of ai-semirings. An exception to this theme are the known results of flat extensions of finite groups, which are FB always as semigroups (they are simply Clifford semigroups), yet are FB as semirings only when the underlying group omits any nonabelian nilpotent subgroups, Jackson \cite[Theorem 7.3]{jac:flat}. In this instance it is instead the quasi-equational theory of the underlying multiplicative semigroup that translates to the equational theory of the semiring. All of the other example references \cite{AEI,andmik,con} in the opening paragraph of the article also consider NFB issues for these classically arising ai-semirings, though the focus in the present article will be on finite semirings.
In the meantime, a quite extensive investigation into syntactically defined classes of ai-semirings has been performed, predominantly relating to establishing finite axiomatisability and describing variety lattices. Early work by McKenzie and Romanowska \cite{mckrom} showed that when multiplication (as well as addition) is idempotent and commutative, there is always a finite equational basis. The broader class of semilattice-ordered bands is considered in \cite{GPZ,pas,paszha}, where it is ultimately shown that there are precisely 78 varieties of multiplicatively idempotent ai-semirings, all having the FB property. Additively idempotent semirings satisfying $x^n\approx x$ were studied by Ku\v{r}il and Pol\'ak \cite{kurpol}, and there have been significant advances, particularly in the commutative case. The second and third authors \cite{renzha} showed that there are 9 distinct varieties of ai-semirings satisfying $x^3\approx x, xy\approx yx$, and more recently with Shao \cite{RZS} have extended this to higher periods, showing that when $n-1$ is square free, the lattice of subvarieties of the variety of ai-semirings defined by $x^n\approx x, xy\approx yx$ has $2+2^{r+1}+3^r$ elements, where $r$ denotes the number of prime divisors of $n-1$. Dropping the commutative condition, the second and third authors with Wang \cite{RZW} identify exactly 179 ai-semiring varieties satisfying $x^3\approx x$. Very recently, the second and third authors with Crvenkovi\'c, Shao and Dapi\'c \cite{ZRCSD} have provided a detailed investigation of ai-semirings of order $3$, revealing that of the 61 possibilities (as well as 6 of order $2$), all but perhaps one has a finite basis for its identities. The remaining unresolved semiring is denoted by $S_7$ in the classification there and has the following table, and is the starting point for the contributions of the present article.
\begin{center}
\begin{tabular}{c|ccc}
$+$&1&$a$&0\\
\hline
$1$&1&$0$&0\\
$a$&0&$a$&0\\
$0$&0&$0$&0\\
\end{tabular}\qquad
\begin{tabular}{c|ccc}
$\cdot$&1&$a$&0\\
\hline
$1$&1&$a$&0\\
$a$&$a$&$0$&0\\
$0$&0&$0$&0\\
\end{tabular}
\end{center}
The multiplicative semigroup reduct of $S_7$ is a seemingly innocuous commutative monoid, with identity basis $x^2\approx x^3,xy\approx yx$, so that it is not initially an obvious candidate for challenging equational behaviour as a semiring. Despite this we find that not only does $S_7$ have the NFB property, it holds this property rather infectiously. We derive a quite general condition that implies the NFB property provided that $S_7$ is contained in the variety. This is held by many examples that, like $S_7$, are FB as multiplicative semigroups, as well as others. One consequence of particular interest is the natural semiring structure on the combinatorial Brandt monoid $B_2^1$, which we show is NFB. Multiplicatively, $B_2^1$ can be given by the following matrices under matrix multiplication:
\[
\begin{tabular}{cccccc}
$\left(\begin{matrix} 0&0\\0&0\end{matrix}\right)$
&
$\left(\begin{matrix} 1&0\\0&1\end{matrix}\right)$
&
$\left(\begin{matrix} 0&1\\0&0\end{matrix}\right)$
&
$\left(\begin{matrix} 0&0\\1&0\end{matrix}\right)$
&
$\left(\begin{matrix} 1&0\\0&0\end{matrix}\right)$
&
$\left(\begin{matrix} 0&0\\0&1\end{matrix}\right)$
\\
\rule{0cm}{.5cm}$0$&$1$&$a$&$b$&$ab$&$ba$
\end{tabular}
\]
There is a unique semilattice order on $B_2^1$ that makes it a semiring, and it is precisely the order that arises from the usual natural order as an inverse semigroup, albeit using $\geq$ rather than $\leq$; see Figure \ref{fig:B21}.
\begin{figure}
\begin{tikzpicture}
\node [element,fill=black] (0) at (2.5,2) [label=$0$] {};
\node [element,fill=black] (ab) at (2,1) [label=below left:$ab$] {};
\node [element,fill=black] (ba) at (3,1) [label=below right:$ba$] {};
\node [element,fill=black] (1) at (2.5,0) [label=right:$1$] {};
\node [element] (a) at (1,1) [label=left:$a$] {};
\node [element] (b) at (4,1) [label=right:$b$] {};
\draw (0) -- (ab) -- (1) -- (ba)--(0)--(a);
\draw (0) -- (b);
\end{tikzpicture}
\caption{The semigroup $B_2^1$ becomes a semiring if $+$ is the join operation given by this order.}\label{fig:B21}
\end{figure}
Ironically, our arguments for $S_7$ has its roots in combinatorial developments made specifically for $B_2^1$ as a semigroup in \cite{jac:SAT}, but our arguments for $B_2^1$ as a semiring depend on the $S_7$ adaptation, rather than the original semigroup-theoretic methods presented in \cite{jac:SAT}. A further consequence of our method is that membership problem for finite algebras in the variety of $S_7$ is \texttt{NP}-complete; this is arguably the simplest example of a finite algebra known with non-polynomial time variety membership (assuming $\texttt{P}\neq \texttt{NP}$). The result again is somewhat infectious, and extends to the semiring $B_2^1$, again paralleling a result in~\cite{jac:SAT} for $B_2^1$ as a semigroup, though again via $S_7$.
\begin{remark}
The NFB property for the natural semiring structure of~$B_2^1$ has also been independently and contemporaneously established by Mikhail Volkov, using an entirely unrelated approach to that given here: this time relating to the nonfinite axiomatisability of $B_2^1$ as an inverse semigroup \cite{vol21}.
\end{remark}
We then return to the group-theoretic approach of \cite{jac:flat}, which again does not require the underlying multiplicative semigroup be NFB. The method of \cite{jac:flat} applies only to flat extensions of finite groups, which are exactly the subdirectly irreducible semirings arising from (finite) Clifford semigroups whose natural order as an inverse semigroup produces a semilattice order \cite[\S7]{jacsto:PM}. We show that, much more generally, any finite ai-semiring containing a nonabelian nilpotent subgroup in its multiplicative reduct is without a finite identity basis.
The structure of the article is as follows. We begin in Section \ref{sec:SW} with the definition and development of some basic semirings that will be the basis of the main results; aside from $S_7$, we make particular use of semigroups built from finite sets $W$ of commutative words, denoted $S_c(W)$; there is also a monoid version, which we denote by $M_c(W)$.
In Section \ref{sec:hypergraphs} we present the key \emph{hypergraph semiring} construction that we require for our proofs. This construction is similar to, though not identical to, the construction $S_c(W)$, where $W$ is a set of words corresponding to hyperedges. Our first main results are contained in Section~\ref{sec:hypersemiringvariety}, where we find that membership of the hypergraph semirings in varieties of our basis semirings is tied to their colourability properties.
Theorem~\ref{thm:hard} shows that $S_7$, $S_c(abb)$ and $B_2^1$ (amongst others) all have \texttt{NP}-hard membership problem for their variety; the case of $S_7$ is the smallest possible size for such an example, and is only the second known example on three elements.
The property of \texttt{NP}-hard membership implies the nonfinite basis property under the assumption of $\texttt{P}\neq \texttt{NP}$, however we provide a more direct, and more general, nonfinite basis result in Theorem~\ref{thm:p3}. This theorem demonstrates the FB property for a very broad range of finite ai-semirings, including all but a small class of the form $S_c(W)$, and all of the form $M_c(W)$, for nonempty $W$. Another example consequence is that a flat semiring with identity is FB if and only if it is the flat extension of a finite group whose nilpotent subgroups are abelian (Corollary~\ref{cor:monoid}).
The specific case of $S_7$ completes the classification of the finite basis property for ai-semirings on at most three elements: 60 of the 61 ai-semirings on three elements are known to be finitely based~\cite{ZRCSD}, while $S_7$ is NFB (see Corollary~\ref{cor:3element}). Given the prominence of $S_7$ in some of these results, in Section~\ref{sec:S7} we provide some finer analysis of its equational properties, in particular giving a combinatorial description of the equational theory (which has co-\texttt{NP}-complete membership). In Section~\ref{sec:group} we return to the classification of the FB property for flat groups given by the first author in~\cite{jac:flat}.
Our Corollary~\ref{cor:monoid} already extends this to cover all flat monoid semirings, however in Section~\ref{sec:group} we provide a different extension, by demonstrating that the requirement of being a flat group can be dropped: any finite ai-semiring is NFB if its multiplicative reduct contains a nonabelian Sylow subgroup (Theorem~\ref{thm:pgroup}).
We later demonstrate this result continues to hold in the signature of ai-semirings with~$0$ (Theorem~\ref{thm:pgroup0}).
We conclude the article in Section~\ref{sec:problems} with an extensive list of problems which we feel will provide useful directions to this area.
\section{Semirings}\label{sec:SW}
The main technical construction of hypergraph semirings will given in Section~\ref{sec:hypergraphs}, but in this section we give some further important examples and constructions.
The constructions in this section will all be \emph{flat semirings}: the order is of height one, with the top element equal to a multiplicatively and additively absorbing zero element. Thus the addition is idempotent and has $x+y:=0$ whenever $x\neq y$. Notationally, we use $0$ for this absorbing element by default, as it is a multiplicative $0$. As the top element in the $+$-order however, the notation $\infty$ is a further sensible notation, and we will use this in the rare cases where we wish to discuss semirings with $0$ (where $0$ plays the role of an additive identity and a multiplicative zero). We also briefly consider semirings with identity, where the constant $1$ will be used to denote a distinguished multiplicative identity element.
Flat algebras have been frequently considered in the literature, including \cite{jac:flat,kunver,sze,wil} for example. As we prove, the following result follows quickly from \cite[Theorem 3.1(3)]{jac:flat}.
\begin{lem}\label{lem:flatvariety}
The flat semirings generate a variety whose subdirectly irreducible members are precisely the flat semirings. Within the variety of ai-semirings, this variety can be defined by the equations
\[
x_1ux_2+y_1uy_2+y_1vy_2\approx x_1vx_2+y_1uy_2+y_1vy_2\tag{$*$}\label{eq:flat}
\]
where any of $x_1,x_2,y_1,y_2$ may be empty.
\end{lem}
\begin{proof}
It is immediate consequence of Theorem 3.1(3) of \cite{jac:flat} that the variety generated by flat semirings can be axiomatised by the ai-semiring axioms along with
\[
t(u+v,x_1,\dots,x_n)+s(u,x_1,\dots,x_m)\approx t(u+v,x_1,\dots,x_n)+s(v,x_1,\dots,x_m)\tag{$*'$}\label{eq:flat2}
\]
where $s(x_0,\dots,x_n)$ and $t(x_0,\dots,x_m)$ are any terms, where $x_0$ appears explicitly in $t$. Up to change of letter names and applications of distributivity of $\cdot$ over $+$, Equation \eqref{eq:flat} in the lemma statement is an example of this, where $s(x_0,x_1,x_2)$ is $x_1x_0x_2$ and $t(y_0,y_1,y_2)$ is $y_1y_0y_2$ (and where we allow $x_1,x_2,y_1,y_2$ to be omitted; Equation \eqref{eq:flat} is really a family of equations). However any equation of the form of Equation \eqref{eq:flat2} easily follows by repeated applications of Equation \eqref{eq:flat} once distributivity of $\cdot$ over $+$ as been used to rewrite $s$ and $t$ as a sum of multiplicative terms.
\end{proof}
The following lemma explains when a semigroup with $0$ becomes a flat semiring under the flat semiring addition.
\begin{lem}\label{lem:cancellative}
A semigroup with $0$ becomes a flat semiring if and only if it satisfies the \emph{0-cancellative laws} $xy\approx xz\not\approx 0\rightarrow y\approx z$ and $xy\approx zy\not\approx 0\rightarrow x\approx z$.
\end{lem}
\begin{proof}
We need only check the distributive laws, as $\cdot$ and $+$ are associative, and $+$ is a semilattice operation. By symmetry we consider distributivity from the left only. We have $xy+xz\leq x(y+z)$ always, so need to show that $x(y+z)\leq xy+xz$. If $y=z$ there is nothing to prove, so assume $y\neq z$; then $x(y+z)=0$. Thus left distributivity is equivalent to the condition that $xy+xz=0$ whenever $y\neq z$. In the flat setting, this is equivalent to either $xy=0$ or $xz=0$ or $xy\neq xz$, which is precisely the contrapositive of the first 0-cancellative law.
\end{proof}
The $0$-direct join of two semigroups $S,T$ with zero element $0$ is the semigroup on the disjoint union $S\backslash\{0\}\mathbin{\dot\cup}T\backslash\{0\}\mathbin{\dot\cup}\{0\}$ where all products within $S$ and $T$ are as before, but products between elements of $S$ with elements of $T$ (in either order) are $0$. It is clear that if $S$ and $T$ are $0$-cancellative, then their $0$-direct join is $0$-cancellative, so that the following lemma is a corollary of Lemma \ref{lem:cancellative}
\begin{lem}\label{lem:0directjoin}
The $0$-direct join of two flat semirings is a flat semiring.
\end{lem}
Every group $G$ is cancellative, and becomes a semiring $\flat(G)$ (isomorphic to $G^0$ as a multiplicative semigroup) satisfying the condition of Lemma \ref{lem:cancellative} by adjoining a multiplicative zero element that forms the top element of the $+$-semilattice. These \emph{flat groups} have previously been studied by the first author in \cite[\S7.7,7.8]{jac:flat} and \cite{jac:eqncomp}.
\begin{pro}\label{pro:monoid}
A finite flat semiring $S$ with multiplicative identity $1$ either contains $S_7$ as a subsemiring, or is the flat extension of a finite group. This is true in any of the signatures $\{+,\cdot\}$, $\{+,\cdot,0\}$, $\{+,\cdot,1\}$ or $\{+,\cdot,0,1\}$.
\end{pro}
\begin{proof}
Assume that $S$ is a finite flat semiring with multiplicative identity $1\neq 0$. If $S$ consists of only $1$ and $0$ then it is the flat extension of the trivial group. Otherwise, assume $S$ contains $a\notin\{1,0\}$. The idempotent power $e$ of $a$ has $ee=e$ and then because $S$ is flat and $e=1e+ee=(1+e)e$ it follows that $e=1$ or $e=0$. If all elements $a\neq 0$ have idempotent power equal to $1$ then $S$ is the flat extension of a finite group. Otherwise, there is $a$ with $a\neq 0$ but $a^n=0$ for some $n>1$. Choose $k\geq 1$ maximum such that $a^k\neq0$. Then $\{1,a^k,0\}$ is a subsemiring isomorphic to~$S_7$ (in any of the listed signatures).
\end{proof}
We mention that the flat extension of the one-generated free monoid $\{a\}^*$ is a flat semiring with identity that does not contain $S_7$ as a subsemiring.
We now consider a further rich source of examples. First we note that whenever~$S$ is an ai-semiring with a proper nonempty subset $J$ that is both a multiplicative absorbing ideal and an order-theoretic filter (with respect to $+$), then we may form the \emph{ideal quotient} $S/J$ obtained by identifying all elements in $J$. The semiring $S_7$ arises as the ideal quotient of $\flat(\{a\}^*)$ for example, using $J=\{a^k\mid k>1\}$.
Now consider the free semigroup $X^+$ on some alphabet $X$.
Let $W$ be a set of words from~$X^+$ and let $W^\leq$ denote the closure in $X^+$ of $W$ under taking (nonempty) subwords.
We define a semigroup on $W^\leq\cup\{0\}$ by letting $0$ be a multiplicative zero element, and for $\mathbf{u},\mathbf{v}\in W^\leq$ letting $\mathbf{u}\cdot \mathbf{v}=\mathbf{u}\mathbf{v}$ if $\mathbf{u}\mathbf{v}\in W^\leq$ and $0$ otherwise.
This construction will be denoted $S(W)$ and is easily seen to be the ideal quotient of $\flat(X^+)$ with respect to $J:=\{\mathbf{w}\in X^+\mid \mathbf{w}\notin W^{\leq}\}$.
If we start with $X^*$ (where the empty word is now automaticaly in $W^\leq$), then the corresponding semigroup is a monoid and we use the notation~$M(W)$, though we note that in many previous articles, the notation $S(W)$ was variously used for both the monoid version and the semigroup version. Following standard notation, we also let $S(\mathbf{w})$ abbreviate $S(\{\mathbf{w}\})$ in the case where $\mathbf{w}$ is a single word, and similarly for~$M(\mathbf{w})$.
There is a commutative variant of this idea. We let $S_c(W)$ (and~$M_c(W)$) denote the same construction but formed on the free commutative semigroup (or monoid, respectively) generated by $X$. These again satisfy the conditional cancellativity condition of Lemma \ref{lem:cancellative} so that the following lemma is immediate.
\begin{lem}
For any set of words $W$, the semigroups $S(W)$, $M(W)$, $S_c(W)$ and $M_c(W)$ are flat semirings if the zero element is taken as the top element.
\end{lem}
The semiring $S_7$ arises in this way as $S_7=M(a)=M_c(a)$, where $a$ is a single letter. When $a_1,\dots,a_n$ are distinct letters and $n>2$, the semiring $S_c(a_1\dots a_n)$ will be a special example of one of the hypergraph semirings to be constructed in Section~\ref{sec:hypergraphs}, and in fact this particular family of semirings holds an important place in the lattice of semiring varieties generated by semirings of the form $S_c(W)$ and~$M_c(W)$.
\begin{pro}\label{pro:SinM}
For every $n$ and any set of words $W$ containing at least one nonempty word, the semirings $S_c(a_1\dots a_n)$ and $M_c(a_1\dots a_n)$ are contained in the variety generated by $M_c(W)$.
\end{pro}
\begin{proof}
It is clear that $M_c(a)=S_7$ lies in the variety of $M_c(W)$, so it suffices to prove the result in the case that $W=\{a\}$. We use the notation $1_i^a$ to denote the $n$-tuple consisting of $1$ everywhere except in coordinate $i$ where it equals $a$. Consider the subsemiring $S$ of $S_7^n$ generated by $\{1_i^a\mid 1\leq i\leq n\}$ and let $J$ denote the set of elements of $S$ containing a $0$ coordinate (to obtain $M_c(a_1\dots a_n)$, add the constant tuple $1$ at this step). The set $J$ is trivially a multiplicative semigroup ideal because~$0$ is a multiplicative zero. It is also a filter in the $+$-order because $0$ is the maximum element. Thus we have a well defined ideal quotient $S/J$. It is routine to see that $S/J$ is isomorphic to $S_c(a_1\dots a_n)$ as required.
\end{proof}
We may also prove the following analogue of Proposition~\ref{pro:SinM} for the $S_c(W)$ construction.
\begin{pro}\label{pro:SinS}
Let $W$ be a set of words containing a word $\mathbf{w}$ that is not a power of a single letter. Then the semiring $S_c(a_1\dots a_n)$ is contained in the variety generated by $S_c(W)$, for any $n\leq |\mathbf{w}|$.
\end{pro}
\begin{proof}
Suppose $\mathbf{w}$ is a word in $W$ that has length exactly $n$ and that is not a power of a single letter. It is clear that $S_c(\mathbf{w})$ is in the variety of $S_c(W)$, so it suffices to show that $S_c(a_1\dots a_n)$ is in the variety of $S_c(\mathbf{w})$.
By commutativity of $\cdot$, we may assume that $\mathbf{w}$ is written in the form $b_1^{i_i}b_2^{i_2}\dots b_k^{i_k}$ for some distinct letters $b_1,\dots,b_k$ and with $i_1+i_2+\dots+i_k=n$.
There is no loss of generality also in assuming that $i_1\geq i_2\geq \dots\geq i_k$. Let us denote the set of all $\binom{n}{i_1,\dots,i_k}$ distinct rearrangements of the word $\mathbf{w}$ as $R$ and for each $i\leq n$ let $\mathsf{a}_i$ denote the $R$-tuple from $\big(S_c(\mathbf{w})\big)^R$ whose position at $r\in R$ is given by the $i^{\rm th}$ position of the rearrangement $r$.
Let $S$ denote the subsemiring of $S_c(\mathbf{w})^n$ generated by $\{\mathsf{a}_i\mid i=1,\dots,n\}$.
The product $\mathsf{w}:=\mathsf{a}_1\dots \mathsf{a}_n$ in $S$ has all coordinates equal to $\mathbf{w}$ (noting that $S_c(\mathbf{w})$ is commutative).
We let $J$ denote the set of all elements of $S$ that do not divide $\mathsf{w}$.
This is obviously a multiplicative ideal, but it is also an order-theoretic filter because it contains all tuples with a $0$-coordinate, and the remaining elements of $S$ are incomparable in the $+$-order.
Thus we may form the semiring $S/J$, which we will show is isomorphic to $S_c(a_1\dots a_n)$, under the obvious map defined on generators by $\mathsf{a}_i\mapsto a_i$.
Trivially, each subset $I\subseteq \{1,\dots,n\}$ has $\prod_{i\in I}\mathsf{a}_{i}\notin J$. We can use the following claim to show that these are the only products that give rise to elements not in $J$, from which the isomorphism property easily follows.
{\bf Claim 1.} If $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_\ell}$ is a product of generators equal to $\mathsf{w}$, then $\ell=n$ and $\{j_1,\dots,j_\ell\}=\{1,\dots,n\}$.
\begin{proof}[Proof of Claim 1.]
It is trivial that $\ell=n$ is required, as each coordinate of $\mathsf{w}$ is $\mathbf{w}$, a word of length $n$. Assume for contradiction that a product $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}=\mathsf{w}$ but $\{j_1,\dots,j_n\}\neq\{1,\dots,n\}$. Thus there is at least one number in $\{1,\dots,n\}$ missing from $\{j_1,\dots,j_n\}$ and at least one number appears twice in the sequence $j_1,\dots,j_n$. Without loss of generality we may assume that $1\notin \{j_1,\dots,j_n\}$ and $2$ appears at least twice. By commutativity we may assume that $j_1=j_2=2$. For each $i\leq k$ let $R_i\subseteq R$ denote those rearrangements that begin with the letter $b_i$ and let $R_{i,j}$ denote the subset of $R_i$ that have second letter equal to $b_j$. For $j\leq n$, let $m_j$ denote the multiplicity of $j$ in the sequence $j_1,\dots,j_\ell$; so $m_1=0$ and $m_2\geq 2$.
If $i_k=1$ then we are done because on coordinates $r\in R_k$, the tuples $\mathsf{a}_2,\dots,\mathsf{a}_n$ have no occurrences of $b_k$, contradicting $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}(r)=\mathbf{w}$. Thus we may assume that $i_k\geq 2$. If $i_k<m_2$ then we have an immediate contradiction, because $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}$ has at least $m_2>i_k$ occurrences of $b_k$ on all coordinates in $R_{1,k}$.
If $i_k=m_2$ then we obtain a similar contradiction, because the assumption that $i_k\geq 2$ ensures that for every rearrangement $r$ in $R_{1,k}$ there is a $j>2$ such that the $j^{\rm th}$ letter is $b_k$.
Then if $m_j>0$ there would be more than $m_2+m_j>i_k$ occurrences of $b_k$ in $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}(r)$, contradicting $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}(r)=\mathbf{w}$. Avoiding this contradiction requires $m_j=0$ for all $j>2$, so that $m_2=n$. But then $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}=\mathsf{a}_2^n$ is not equal to $\mathbf{w}$ on any coordinate, also a contradiction.
Thus we may assume that $i_k> m_2$. On each $r\in R_{1,k}$ there are always $i_k-1$ values $j$ from $\{3,4,\dots,n\}$ for which $\mathsf{a}_j(r)=b_k$, however in $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_n}(r)$ we have $m_2$ occurrences arising from the repetition of $\mathsf{a}_2$, thus only $i_k-m_2$ that are provided by $\mathsf{a}_{j_{m_2}+1}(r),\dots,\mathsf{a}_{j_n}(r)$.
It follows from $m_2>1$ that there is a number $j\in\{3,\dots,n\}$ with $m_j=0$; without loss of generality we may assume $m_3=0$.
Because $m_2=n$ leads immediately contradiction we may further assume without loss of generality that $m_4>0$.
Now consider two rearrangements $r_1,r_2$ in $R_{1,k}$ for which the positions of $b_k$ differ only between positions $3$ and $4$; for example, as $i_1\geq i_2\geq i_k\geq 2$, the rearrangement $r_1$ in $R_{1,k}$ may have $b_1$ in position $3$ and $b_k$ in position $4$, while $r_2$ has $b_k$ in position $3$, and $b_1$ in position $4$, but is otherwise identical.
Then the number of occurrences of $b_k$ in $\mathsf{a}_2^{m_2}\mathsf{a}_{j_{m_2}+1}\dots\mathsf{a}_{j_n}(r_1)$ differs by $m_4$ from the number in $\mathsf{a}_{j_{m_2}+1}\dots\mathsf{a}_{j_n}(r_2)$, contradicting the fact that both equal $\mathbf{w}$.
\end{proof}
Now we may complete the main proof of the proposition. We need to show that a product $\mathsf{a}_{j_1}\dots \mathsf{a}_{j_\ell}$ of generators divides $\mathsf{w}$ if and only if $j_1,\dots,j_n$ contains no repeats. The ``if'' direction is trivial. The ``only if'' direction follows from Claim 1, because a concatenation of letters that includes a repeat, cannot extend to one that avoids repeats. It now follows that $S_c(a_1\dots a_n)$ is isomorphic to $S/J$, under the map determined on generators by $a_i\mapsto \mathsf{a}_i$.
\end{proof}
\begin{remark}
In Proposition \ref{pro:SinS}, the condition that a word needs to consist of more than a single letter is necessary, because
\[
S(a^k)\models x_1\dots x_k\approx x_1\dots x_k+ x_1^k,
\]
which obviously fails on $S(a_1\dots a_k)$ when $k>1$.
\end{remark}
The semirings $S_c(a_1\dots a_n)$ (for $n\geq 3$) will play an important role in the article. As already noted, the semiring $S_7$ coincides with $M_c(a)$, while we later find a parallel role for the semiring $S_c(abb)$.
\begin{lem}\label{lem:incomp}
For $k>2$, neither the semiring $S_7$ nor the semiring $S_c(ab^{k-1})$ lies in the variety generated by the other.
\end{lem}
\begin{proof}
We have $S_7\models x^2\approx x^3$, while $S_c(ab^{k-1})$ fails this when $x$ is assigned $b$. Conversely, $S_c(ab^{k-1})$ is $(k+1)$-nilpotent (satisfying $x_1\dots x_{k+1}\approx y_1\dots y_{k+1})$, which fails on $S_7$ when the $x_i$ are assigned $1$ while the $y_i$ are assigned $0$.
\end{proof}
In contrast, Propositions \ref{pro:SinM} and~\ref{pro:SinS} together immediately imply the following.
\begin{lem} \label{lem:a1tok}
For $k>1$ we have $\mathsf{V}(S_c(a_1\dots a_{k}))\subseteq \mathsf{V}(S_7)\wedge \mathsf{V}(S_c(ab^{k-1}))$.
\end{lem}
The (multiplicative) semigroups, both $S_c(W)$ and $M_c(W)$ are commutative, so are finitely based as semigroups, Perkins \cite{per}. Similarly, the underlying semigroup of $\flat(G)$ is the Clifford semigroup $G^0$, which is finitely based as a semigroup (provided $G$ is finite), due primarily due to the fact that every finite group has a finite basis for its equations, Oates and Powell \cite{oatpow}. The semigroups $S(W)$ are also finitely based as semigroups (assuming $W$ is finite) as they are nilpotent, and every term is equivalent to one of length at most one more than the longest word in $W$. The semigroups $M(W)$ are variously finitely based and nonfinitely based, and since the original work of Perkins \cite{per}, have been amongst the richest source of challenging behaviour for the finite basis property and related problems in finite semigroups; see the work of Olga Sapir \cite{osap}, Jackson and Sapir \cite{jacsap}, amongst many others. Following the notation from~\cite{jacvol}, we say that an algebra $A$ (or variety $V$) satisfying a property $P$ is \emph{inherently nonfinitely based} (INFB) \emph{relative to $P$} if every variety satisfying $P$ and containing $A$ (or $V$, respectively) is without a finite basis for its equations. When $P$ is ``generates a locally finite variety'', then we omit reference to $P$ and say that $A$ (or $V$) is \emph{inherently nonfinitely based} (INFB). It is a trivial consequence of the work of Mark Sapir \cite{sap2}, that no semigroup $M(W)$ (for finite~$W$) is INFB. \ The following folklore result appears to be well known to researchers worldwide but does not appear to have been stated in published form in English.
\begin{thm}\label{thm:INFB}
\begin{enumerate}
\item Let $S=\langle S;+,\cdot\rangle$ be an ai-semiring whose multiplicative reduct $\langle S;\cdot\rangle$ is locally finite. Then $S$ is locally finite.
\item Let $\Sigma$ be a finite set of semigroup identities defining a locally finite variety of semigroups. Then, in conjunction with the axioms for ai-semirings, the identities $\Sigma$ define a locally finite variety of ai-semirings.
\item The multiplicative reduct of an ai-semiring $S$ is INFB if $S$ is INFB.
\end{enumerate}
\end{thm}
\begin{proof}
For (1), let $A$ be a finite subset of $S$. The subsemigroup $S_{A,(\cdot)}$ of $\langle S;\cdot\rangle$ generated by $A$ is finite, but by distributivity and idempotence, every element of the subsemiring $S_{A,(+,\cdot)}$ of $S$ generated by $A$ can be written as a sum of elements of~$S_{A,(\cdot)}$, without repeats, showing that $|S_{A,(+,\cdot)}|\leq 2^{|S_{A,(\cdot)}|}$.
Item (2) follows from (1) given that all semirings satisfying $\Sigma$ have locally finite semigroup reducts, and hence are locally finite. Item (3) is an immediate corollary of (2).
\end{proof}
\begin{remark}
Theorem \ref{thm:INFB} admits many generalisations and variations. The proof used only the fact that arbitrary elements of $S_{A,(+,\cdot)}$ can be written as sums of elements of $S_{A,(\cdot)}$. This requires only left and right distributivity of $\cdot$ over $+$, and enough properties of $+$ to ensure that that sums over a finite alphabet are equivalent sums of bounded length. As an example, the theorem generalises to semirings satisfying only a periodic law for addition, as the variety of commutativity $+$-semigroups of any given index and period is locally finite. These ideas generalise to larger signatures, subject to the availability of suitable distributivity properties. Something very similar also occurs, for example, in the class of (one-sided) restriction semigroups, see \cite[Theorem 9.2]{jon}. There is also no significant generating power obtained by adding (finitely many) constants, so that semirings with $0$ also satisfy a version of the theorem.
\end{remark}
\section{Hypergraphs and hypergraph semirings}\label{sec:hypergraphs}
Our primary construction is a reimagining of some hypergraph methods developed for semigroups in~\cite{jac:SAT} and then subsequently in \cite{jaczha}, however the methods in the present article are a parallel rather than direct application. In \cite{jac:SAT,jaczha} the constructions depend in a fundamental way on the structure of the non-commutative semigroup~$B_2^1$: hyperedge encodings are kept separate in long products by way of a noncommuting barrier. In contrast, the commutativity of $\cdot$ in~$S_7$ and~$S_c(abc)$ appears to play a central role in the arguments of the present article, and the hyperedge encodings are instead gathered together by long sums.
We begin with some basic background on hypergraphs and related concepts; there is some overlap with the articles \cite{hamjac:hyper} and \cite{jaczha}, as well as the manuscript~\cite{jac:SAT}, however we are not able to present our semiring-specific developments without presenting them here again.
We adopt standard notation and terminology, as in \cite{hamjac:hyper}. A \emph{hypergraph} is a pair $H=(V,E)$, where $E$ is a family of nonempty subsets of $V$. If $|e|\leq k$ for each $e\in E$, we say that $H$ is a \emph{$k$-hypergraph}. If $|e|=k$ for all $e\in E$ then $H$ is a \emph{$k$-uniform hypergraph}. We also view $k$-hypergraphs as relational structures in the signature of a single $k$-ary relation consisting of the $k$-tuples $\{(v_1,\dots,v_k)\mid \{v_1,\dots,v_k\}\in E\}$; we refer to a $k$-hypergraph as a \emph{$k$-hypergraph structure} when in this context. So, for a hyperedge $\{u,v\}$ in a $3$-hypergraph will, in the corresponding $3$-hypergraph structure, give rise to six tuples $uuv,uvu,vuu,uvv,vuv,vuu$; here we adopt a standard convention of abbreviating tuples $(x,y,z)$ as strings $xyz$. Similarly, any relational structure in the signature of a single $k$-ary relation $R$ becomes a $k$-hypergraph, by taking the hyperedges $\{v_1,\dots,v_k\}$ for each tuple $(v_1,\dots,v_k)\in R$. The $k$-hypergraph structure formed from a general $k$-relational structure is a kind of ``set-closure'' of the relations: a tuple is present in this closure if it contains the same elements as some other tuple in the relation.
A \emph{path} in a hypergraph is a sequence $v_0,e_0,v_1,e_1,\dots,v_{n-1},e_{n-1}$ alternating between vertices and hyperedges, with no repeats, and such that $v_0\in e_0$ and $v_{i+1}\in e_i\cap e_{i+1}$ (with addition in the subscript modulo $n$). A path is a \emph{cycle} if we also have $v_0\in e_{n-1}\cap e_0$; we often write $v_0,e_0,v_1,e_1,\dots,v_{n-1},e_{n-1},v_0$.
The \emph{girth} of hypergraph is the length of the shortest cycle; hypergraphs without cycles are called \emph{hyperforests} and are conventionally set to have infinite girth.
We will also make use of the $2$-element structure $\mathbbold{2}=\langle \{0,1\};R\rangle$, where $R$ is the ternary relation $\{110,101,011\}$. This is not quite a $3$-hypergraph structure, but it is similar, and of the same signature. The homomorphism problem for $\mathbbold{2}$ is the positive 2-in-3SAT problem and is well known to be \texttt{NP}-complete. This computational problem is more frequently encountered as \emph{positive 1-in-3SAT}, as in~\cite{jac:SAT} for example, however we are switching the role of $0$ and $1$ in the present article, as this notation better matches the semiring $S_7$.
The positive 2-in-3SAT problem corresponds to a specific kind of 3-hypergraph colouring problem: a function from $V\to\{0,1\}$ in which each 3-hyperedge is sent to one of the tuples $\{110,101,011\}$. This extends further to $k$-hypergraphs: a \emph{$(k-1)$-in-$k$ satisfaction} of a $k$-hypergraph is a function $V\to\{0,1\}$ mapping each hyperedge to $\{\overbrace{1\dots11}^{k-1}0,1\dots 101,\dots,011\dots 1\}$: that is, all but one vertex in each hyperedge is given the value $1$. The more general and more familiar notion of $\ell$-colouring of a $k$-hypergraph $\mathbb{H}=(V,E)$ is a function $\gamma$ from $V$ to $(\{0,\dots,\ell-1\},F)$ where $F$ consists of all subsets of $\{0,\dots,\ell-1\}$ that have cardinality between $2$ and~$k$ (inclusive) and such that $\gamma(e)\in F$ for all $e\in E$. (This definition carries across equivalently to the setting of $k$-hypergraph structures; see \cite[\S2]{hamjac:hyper}.) The hypergraph $2$-colourability of $3$-hypergraphs structures can be seen to be just the well known \emph{positive NAE3SAT} problem. One final notion is required. A $k$-hypergraph is \emph{${\leq}i$-robustly $(k-1)$-in-$k$ satisfiable} if every valid partial $(k-1)$-in-$k$ satisfaction of any set of at most $i$ vertices extends to a full $(k-1)$-in-$k$-satisfaction (see \cite{AGK,ham,hamjac,jac:SAT}). We will only use the case of $i=2$ here, so give some further details on what the concept means in this particular case. Any mapping from vertices $x,y$ to one of the pairs $(1,1),(0,1),(1,0)$ is a valid partial satisfaction, while $(0,0)$ is valid only if $\{x,y\}$ is \emph{not} a subset of a hyperedge: if $\{x,y\}$ was a subset of hyperedge $e$ then it is trivial that the partial assignment $x\mapsto 0$ and $y\mapsto 0$ cannot extend to a $(k-1)$-in-$k$ satisfying assignment, as any such assignment would give exactly one vertex in $e$ the value $0$.
Our hardness and nonfinite axiomatisability results will hinge on two convenient facts relating to $k$-hypergraph structures. The first item is proved by Jackson \cite[Theorem 6.2]{jac:SAT}. The second item is by Erd\H{o}s and Hajnal \cite{erdhaj}; see Theorems 2.6 and 2.7 of \cite{hamjac:hyper} for further discussion.
\begin{thm}\label{thm:facts}
\begin{enumerate}
\item The following promise-problem is \texttt{NP}-hard for any fixed $k\geq 3$. Given a $k$-uniform hypergraph $H=(V,E)$ with girth at least $5$\up:
\begin{itemize}
\item[Yes:] $H$ is ${\leq}2$-robustly 2-in-3 satisfiable.
\item[No:] $H$ is not 2-in-3 satisfiable.
\end{itemize}
\item For every $n$ and every $\ell\geq 2$ there exists a $k$-uniform hypergraph $H_{n,\ell}$ that is not $\ell$-colourable, but every $n$-element substructure is a $k$-uniform hyperforest.
\end{enumerate}
\end{thm}
We note that unfortunately it is not possible to extend the \texttt{NP}-hardness of the promise problem in (1) to the more extreme possibilities of (2): it is known that there is a polynomial time algorithm to distinguish $2$-in-$3$-colourable hypergraphs (the YES instances of positive 2-in-3SAT) from those that are not $2$-colourable (the NO instances of NAE3SAT); see \cite[Example 7.15 and Theorem 7.19]{bragur} or \cite[\S8]{BKO}.
The next lemma will be a $k$-hypergraph generalisation of \cite[Lemma 6.3]{jac:SAT}; the arguments are also quite similar, though we give them for completeness.
Let us say that a set of vertices $\{v_1,\dots,v_\ell\}$ is a \emph{subhyperedge} for some $k$-hypergraph $\mathbb{H}$ if it is a subset of a hyperedge of $\mathbb{H}$. Two subhyperedges of size $k-1$ will be said to be \emph{linked}, if there is a single vertex $w$ that completes both of the subhyperedges to full hyperedges of $\mathbb{H}$. The \emph{link graph} of $\mathbb{H}$ is the graph whose vertices are the $(k-1)$-element sets of vertices, with two such sets adjacent if they are linked. We make only marginal use of the link graph, though it is useful conceptual tool in understanding the consequences of some of the defining properties of our semiring construction later.
\begin{lem}\label{lem:hyperprop2}
Let $k>2$. If $\mathbb{H}$ is a $k$-uniform hypergraph of girth at least $4$, then the following are true.
\begin{itemize}
\item[(I)] No two distinct hyperedges share more than one common vertex.
\item[(II)] If $\ell>2$ and every proper subset of $\{v_1,\dots,v_\ell\}$ is a subhyperedge, then $\{v_1,\dots,v_\ell\}$ is a subhyperedge; in particular, $\ell\leq k$.
\item[(III)] For all $\ell>1$, a set $\{v_1,\dots,v_\ell\}$ is a subhyperedge if and only if all $2$-element subsets of $\{v_1,\dots,v_\ell\}$ are subhyperedges.
\item[(IV)] The link graph of $\mathbb{H}$ consists of a disjoint union of cliques.
\end{itemize}
\end{lem}
\begin{proof}
Item (I) is equivalent to the absence of $2$-cycles in a hypergraph: if $\{u,v\}$ is a subset of two distinct hyperedges $e,f$ then $u,e,v,f,u$ is a $2$-cycle.
For item~(II), assume that every proper subset of $\{v_1,\dots,v_\ell\}$ is a subhyperedge. If $\ell>3$ then we can use (I) to deduce that $\{v_1,\dots,v_\ell\}$ is a subhyperedge as follows. For each $(\ell-1)$-element subset $S$---a subhyperedge by assumption---identify a hyperedge $e_S$ containing $S$. As distinct $(\ell-1)$-element subsets $S,T$ of $\{v_1,\dots,v_\ell\}$ share $\ell-2\geq 2$ elements, it follows from (I) that $e_S=e_T$ so that there was in fact a single hyperedge $e$ containing every $\ell-1$ element subset of $\{v_1,\dots,v_\ell\}$. Thus $\{v_1,\dots,v_\ell\}\subseteq e$. For the case that $\ell=3$ use the fact that if all $2$-element subsets of $\{v_1,v_2,v_3\}$ are subhyperedges, then we have hyperedges $e_1$ containing $\{v_1,v_2\}$, $e_2$ containing $\{v_2,v_3\}$ and $e_3$ containing $\{v_1,v_3\}$. As there are no $3$-cycles by assumption, the sequence $v_1,e_1,v_2,e_2,v_3,e_3,v_1$ contains a repeat $e_i=e_j$, so that there is a hyperedge containing $\{v_1,v_2,v_3\}$.
Now for (III). The forward implication is trivial, so let assume now that $\{v_1,\dots,v_\ell\}$ is not a subhyperedge. If there is a subset of size at least $2$ that is not a subhyperedge we may solve the problem for this and obtain the desired solution for $\{v_1,\dots,v_\ell\}$. Thus there is no loss of generality in assuming that either $\ell=2$ or every subset of $\{v_1,\dots,v_\ell\}$ is a subhyperedge. But then (II) implies that $\ell=2$ and we are done.
If we let $\sim$ denote the edge relation of the link graph, then property (IV) is equivalent to
\[
A\sim B\sim C\rightarrow (A\sim C\vee A=C)
\]
for link graph vertices $A,B,C$. Assume that $A=\{u_1,\dots,u_{k-1}\}$ is linked to $B=\{v_1,\dots,v_{k-1}\}$ by way of some vertex $v$ and that $B$ is linked to $C=\{w_1,\dots,w_{k-1}\}$ by way of some vertex $w$. But then $v=w$ as $\{v_1,\dots,v_{k-1},v\}$ and $\{v_1,\dots,v_{k-1},w\}$ overlap by more than $1$ vertex so that either
\[
C=\{w_1,\dots,w_{k-1},w\}=\{u_1,\dots,u_{k-1},w\}=A
\]
or that $A$ links to $C$, as required.
\end{proof}
The following lemma is essentially a minor modification of Lemmas 2.8 and~2.9 of \cite{hamjac:hyper}, and the proof is included for completeness only. We use the familiar (and easy to prove) property that every finite hyperforest either contains no hyperedges at all, or it contains a leaf: a hyperedge that is has just at most one vertex in common with any other hyperedge.
\begin{lem}\label{lem:forest}
A $k$-uniform hyperforest $\mathbb{F}$ is ${\leq}2$-robustly $(k-1)$-in-$k$ satisfiable.
\end{lem}
\begin{proof}
This is by induction on the number of hyperedges of $\mathbb{F}$. For a single hyperedge the statement is trivial. Now assume that it is true for $k$-uniform hyperforests consisting of $n$ hyperedges, and consider a $k$-uniform hyperforest $\mathbb{F}$ of $n+1$ hyperedges. Identify a leaf $e$ of $\mathbb{F}$. If $e$ is an isolated hyperedge then the statement follows immediately, so we assume that $e$ does share one vertex, $v$ with the remaining part of $\mathbb{F}$. Let $\mathbb{F}_e$ be the result of removing the hyperedge $e$ and all of its vertices except $v$. Every $(k-1)$-in-$k$ satisfaction of $\mathbb{F}_e$ extends to a $(k-1)$-in-$k$ satisfaction of $\mathbb{F}$: if $v$ is coloured $0$ then this colouring is unique, and if $v$ is coloured $1$ then there are $k-1$ choices of a vertex in $e\backslash\{v\}$ to colour $0$ and the rest are coloured $1$. It is now easy to use the assumed ${\leq}2$-robust $(k-1)$-in-$k$ satisfiability of $\mathbb{F}_e$ to prove the same for $\mathbb{F}$. For pairs of vertices $x,y$ that lie in $\mathbb{F}_e$ the required property holds by induction. If exactly one is in $e$ (say, $x$) and neither are $v$, then we may fix any colouring of $\mathbb{F}_e$ that achieves the desired colour for $x$ and then use the free extendability to $e\backslash\{v\}$ just observed to give either of $0$ or $1$ to $y$. If both are in $e$ then we can $(k-1)$-in-$k$ satisfy $e$ to achieve any colouring of $x,y$ except the prohibited $(x,y)\mapsto (0,0)$ and then extend this to a $(k-1)$-in-$k$ satisfaction of the rest of $\mathbb{F}$ by choosing any colouring of $\mathbb{F}_e$ that takes the already determined colour for $v$ (which can be done by induction).
\end{proof}
Let us consider, for some $k>2$, a $k$-uniform hypergraph $\mathbb{H}$ of girth $m\geq 4$, noting that each of the properties in Lemma \ref{lem:hyperprop2} will hold for $\mathbb{H}$; we further assume that $\mathbb{H}$ contains no isolated vertices, as such vertices play no role in the colourability properties (nor in $(k-1)$-in-$k$ satisfiability properties) of $\mathbb{H}$. We now construct an ai-semiring $S_\mathbb{H}$ from $\mathbb{H}$; collectively we refer to these as \emph{hypergaph semirings}, though note that these are entirely different from the hypergraph algebras and flat hypergraph algebras of \cite{kunver}, which are rarely semirings. In the case that $k=3$, the construction is in surprisingly close analogy to the semigroup $S_I$ of \cite[\S6]{jac:SAT}\footnote{The similarity is because both encode the hypergraph hyperedges by way of products, but the algebraic scaffolding that encodes the broader homomorphism and colouring properties of hypergraphs are perhaps less similar: one uses the structure of the main regular $\mathscr{D}$-class of $B_2^1$, while the other depends fundamentally on the flat semilattice structure of $S_7$ and related semirings.} and the present ideas loosely follow those in \cite{jac:SAT}, as much as this is possible to make sense, given the different settings. Only $k=3$ is needed to prove the nonfinite basis property for~$S_7$, but the move to arbitrary $k>2$ seems required for stronger properties.
The generators of $S_\mathbb{H}$ consist of the following elements:
\begin{enumerate}
\item[(i)] $0$, a multiplicative and additive zero element;
\item[(ii)] for each vertex $v$ of $\mathbb{H}$ an element $\mathbf{a}_v$.
\end{enumerate}
The multiplicative part of the semiring is subject to the following rules:
\begin{enumerate}
\item $\mathbf{a}_u\mathbf{a}_v=0$ if $u=v$ or $\{u,v\}$ is not a subhyperedge.
\item $\mathbf{a}_u\mathbf{a}_v=\mathbf{a}_v\mathbf{a}_u$.
\item $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_k}=:\mathbf{a}$ whenever $\{u_1,\dots,u_k\}$ is a hyperedge of $\mathbb{H}$.
\item $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_{k-1}}=\mathbf{a}_{v_1}\dots \mathbf{a}_{v_{k-1}}$ if $\{u_1,\dots,u_{k-1}\}$ and $\{v_1,\dots,v_{k-1}\}$ are linked $(k-1)$-tuples.
\end{enumerate}
When $\{u_1,\dots,u_{k-1}\}$ is a $(k-1)$-element subhyperedge, then we let $[u_1\dots u_{k-1}]$ denote the set of all $(k-1)$-sets $\{x_1,\dots,x_{k-1}\}$ such that $\{u_1,\dots,u_{k-1}\}$ is linked to $\{x_1,\dots,x_{k-1}\}$; so $[u_1\dots u_{k-1}]=[x_1\dots x_{k-1}]$ in this case. We write $\mathbf{a}_{[u_1\dots u_{k-1}]}$ to denote the value $\mathbf{a}_{u_1}\mathbf{a}_{u_2}\dots \mathbf{a}_{u_{k-1}}$. Rule (4) shows that when $\{u_1,\dots,u_{k-1}\}$ is linked to $\{x_1,\dots,x_{k-1}\}$ we have
\[
\mathbf{a}_{[u_1\dots u_{k-1}]}=\mathbf{a}_{u_1}\mathbf{a}_{u_2}\dots \mathbf{a}_{u_{k-1}}=\mathbf{a}_{x_1}\mathbf{a}_{x_2}\dots \mathbf{a}_{x_{k-1}}=\mathbf{a}_{[x_1\dots x_{k-1}]},
\]
which is consistent with the fact that $[u_1\dots u_{k-1}]=[x_1\dots x_{k-1}]$. We note that item~(IV) of Lemma \ref{lem:hyperprop2} shows that we do not expect significant flow-on consequences of Rule (4); this is established in the course of the proofs anyway, but noting it may be useful to help establish intuition about the equalities defining $S_\mathbb{H}$. We mention also that Rule (4) is essentially a special instance of the 0-cancellativity condition in Lemma \ref{lem:cancellative} and so is required for a construction to be a flat semiring.
Additively, we let $S_\mathbb{H}$ be flat ai-semiring: a height $1$ semilattice, with $0$ being an absorbing element for addition. We also mention that it is possible to adjoin a multiplicative identity element $1$ to this semiring (also set to be order-theoretically incomparable to all other elements except $0$), and we denote this by $M_\mathbb{H}$.
Rules (1)--(4) (and the flat semiring property) are a semiring presentation, so $S_\mathbb{H}$ is certainly a semiring. The following lemma shows that there are ``no surprises'' to the structure of $S_\mathbb{H}$: the obvious equalities in the rules defining $S_\mathbb{H}$ are the only ones. This lemma is not used elsewhere in the article, and is given only in order to aid intuition of the construction.
\begin{lem}\label{lem:clarification}
Let $\mathbb{H}$ be a $k$-uniform hypergraph of girth at least $4$.
Every non-zero element in $S_\mathbb{H}$ is of the form $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_\ell}$ for some subhyperedge $\{u_1,\dots,u_\ell\}$ of size $\ell\leq k$. Two such products $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_i}$ and $\mathbf{a}_{v_1}\dots \mathbf{a}_{v_j}$ are distinct if they are distinct subhyperedges, unless
\begin{itemize}
\item $i=j=k$, so that both $\{u_1,\dots ,u_k\}$ and $\{v_1,\dots ,v_k\}$ are hyperedges\up{:} in this case $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_k}=\mathbf{a}_{v_1}\dots \mathbf{a}_{v_k}$\up{;} or
\item $i=j=k-1$ and $\{u_1,\dots ,u_{k-1}\}$ and $\{v_1,\dots ,v_{k-1}\}$ are linked\up{:} in this case $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_{k-1}}=\mathbf{a}_{v_1}\dots \mathbf{a}_{v_{k-1}}$.
\end{itemize}
\end{lem}
\begin{proof}
The flat semilattice property ensures every nonzero element can be written as a product of generators $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_\ell}$. (If we were to consider $M_\mathbb{H}$ then we would also have trivial extensions of this product involving $1$). Lemma \ref{lem:hyperprop2}(III) and applications of Rules (1)--(2) then easily show that $\{u_1,\dots,u_\ell\}$ is a subhyperedge and that there are no repeats in $u_1,\dots,u_\ell$, so that $|\{u_1,\dots,u_\ell\}|=\ell\leq k$. It now remains to show that except in the two itemised cases there are no further identifications. We give sketch details only, reiterating that it is a consequence of later arguments (not depending on the present lemma). Each of the Rules (1)--(4) either return $0$ or do not change the length of a product. Thus two nonzero products $\mathbf{a}_{u_1}\dots \mathbf{a}_{u_i}$ and $\mathbf{a}_{v_1}\dots \mathbf{a}_{v_j}$ must have the same length $i=j=:\ell$. Clearly Rule (2) only enables rearrangements up to commutativity. If $\ell<k-1$ then this is the only rule that applies. If $\ell=k$ then applications of Rule (4) apply universally to show equality of all nonzero products of length $k$, so there is nothing to prove. When $\ell=k-1$, applications of Rule (3) does not yield further equivalences beyond those covered directly in the rule itself because of Lemma \ref{lem:hyperprop2}(IV).
\end{proof}
\section{Hypergraph semirings in varieties}\label{sec:hypersemiringvariety}
The hypergraph semirings $S_\mathbb{H}$ encode hypergraphs in a transparent way. In this section we find how membership of these hypergraph semirings $S_\mathbb{H}$ in varieties will relate to various colouring conditions of the hypergraph $\mathbb{H}$, provided that the varieties containing some of the critical objects we encountered in Section \ref{sec:SW}.
\begin{lem}\label{lem:HinS7}
If $\mathbb{H}$ is ${\leq 2}$-robustly $(k-1)$-in-$k$ satisfiable, then $S_\mathbb{H}\in \mathsf{V}(S_7)\wedge \mathsf{V}(S_c(ab^{k-1}))$.
\end{lem}
\begin{proof}
We let $S$ denote either of $S_7$ or $S_c(ab^{k-1})$. At many steps of the proof it will make no difference as to which is chosen, but when there is a difference, we give details for the $S_7$ case, with the difference required for $S_c(ab^{k-1})$ given in brackets. In the case of $S=S_7$, we also invite the reader to consider the situation where the monoid construction $M_\mathbb{H}$ is used in place of $S_\mathbb{H}$: the arguments below carry through with trivial amendment.
Let $T$ denote the set of all valid $(k-1)$-in-$k$ satisfactions of the vertices of~$\mathbb{H}$. For each vertex $u$, define a tuple $a_u$ in $S^T$ by
\[
a_u(\phi)=\begin{cases}
a &\text{ if $\phi(u)=0$}\\
1 &\text{ if $\phi(u)=1$.}
\end{cases}
\]
(When $S=S_c(ab^{k-1})$, the value $a_u(\phi)=1$ is to be replaced by $a_u(\phi)=b$.)
Let~$A$ denote the subsemiring of $S^T$ generated by the elements of the form $a_u$. We claim that $S_\mathbb{H}$ can be obtained from $A$ by factoring out the ideal $J$ generated by all elements of $A$ having a coordinate equal to $0$. This is trivially an upset of the join semilattice structure on $A$, and an ideal of the multiplicative structure of $A$, so is a well-defined quotient. We now wish to show that this quotient is isomorphic to $S_\mathbb{H}$, which will show that $S_\mathbb{H}\in\mathsf{V}(S)$. The approach is as follows. We first show that each of the defining equations (1)--(4) of $S_\mathbb{H}$ holds, with $a_u$ replacing $\mathbf{a}_u$, and that $A/J$ is a flat semilattice. This shows that $A/J$ is a quotient of $S_\mathbb{H}$. To show that it is isomorphic, we show that all products of the generators $a_u$ that are not obviously equivalent due to (1)--(4) are distinct in $A/J$.
We begin by checking the defining laws (1)--(4) of $S_\mathbb{H}$ hold, with $a_u$ replacing $\mathbf{a}_u$, and noting that we do have the required generators: a $0$ element (namely $J$) and an element $a_u$ for each vertex $u$.
If $u=v$ or $\{u,v\}$ does not extend to a hyperedge, then by ${\leq 2}$-robust $(k-1)$-in-$k$ satisfiability, there is a $(k-1)$-in-$k$ satisfaction that gives both $u$ and $v$ the value $0$. Then $a_ua_v(\phi)=aa=0$ so that $a_ua_v\in J$ showing that (1) holds. The commutativity property (2) holds trivially, based on the commutativity of $S$. For item (3), consider any hyperedge $\{u_1,\dots,u_k\}$ and note that for any $(k-1)$-in-$k$ satisfaction $\phi$ we have that $\phi$ assigns exactly one of $u_1,\dots,u_k$ the value $0$, and the remainder are given the value $1$.
Up to commutativity, this yields $a_{u_1}\dots a_{u_k}(\phi)=a\overbrace{11\dots 1}^{k-1}=a$ in the case of $S=S_7$ and $a\overbrace{bb\dots b}^{k-1}=ab^{k-1}$ in the case of $S=S_c(ab^{k-1})$. Thus $a_{u_1}\dots a_{u_k}$ is the constant tuple $\underline{a}$, showing that (3) holds.
(In the case of $S=S_c(ab^{k-1})$ it is the constant tuple $\underline{ab^{k-1}}$.)
Finally, consider when $\{u_1,\dots,u_{k-1}\}$ and $\{v_1,\dots,v_{k-1}\}$ are linked by way of some vertex~$w$.
For any $(k-1)$-in-$k$ satisfaction $\phi$, if $\phi(w)=0$ then $\phi(u_1)=\dots=\phi(u_{k-1})=\phi(v_1)=\dots=\phi(v_{k-1})=1$ so that $a_{u_1}\dots a_{u_{k-1}}(\phi)=a_{v_1}\dots a_{v_{k-1}}(\phi)=1$ (or $b^{k-1}$ in the case of $S=S_c(ab^{k-1})$).
Alternatively, if $\phi(w)=1$, then precisely one of $u_1,\dots,u_{k-1}$ and one of $v_1,\dots,v_{k-1}$ is $0$ and the other are $1$; and this case $a_{u_1}\dots a_{u_{k-1}}(\phi)=a_{v_1}\dots a_{v_{k-1}}(\phi)=a$ (or $ab^{k-2}$ in the case of $S_c(ab^{k-1})$).
Either way, we have $a_{u_1}\dots a_{u_{k-1}}(\phi)=a_{v_1}\dots a_{v_{k-1}}(\phi)$ for all $\phi$ so that equality (4) holds.
The last remaining property of $S_\mathbb{H}$ that needs checked for $A/J$ is that it is a flat semilattice. This is immediate, given that $S$ is a flat semiring and each pair of distinct elements differ in some coordinate. The sum of two such elements will then take the value $0$ on that coordinate and hence lie in $J$.
In order to complete the proof that $A/J\cong S_\mathbb{H}$ we must show that no further equalities hold in $A/J$ beyond those that are forced by the equalities (1)--(4). At this point we still do not know if there are some subtle equalities that are consequences of (1)--(4), but in the course of the proof we will discover that only the obvious ones hold in $A/J$, which will imply that $A/J\cong S_\mathbb{H}$.
The elements in $A/J$ can all be written as either the set $J$ (a multiplicative $0$) or as a product of the form $a_{u_1}\dots a_{u_\ell}$ for some $\ell$. Thus, by the ${\leq 2}$-robust $(k-1)$-in-$k$ satisfiability property, all elements of $A$ have a coordinate equal to either $a$ or $0$. We can deduce that if the product $a_{u_1}\dots a_{u_\ell}$ in $A$ is not an element of $J$, then it contains no repeats, as we noted commutativity and the property $a_ua_u\in J$ already. We may also assume that $\{u_1,\dots,u_\ell\}$ is a subhyperedge: by Lemma~\ref{lem:hyperprop2}(III) we know that otherwise there is a pair $u,v\in \{u_1,\dots,u_\ell\}$ such that $\{u,v\}$ is not a subhyperedge, and hence the known equalities (1) and (2) show that the product lies in $J$. This also implies that $\ell\leq k$, as no set of size more than $k$ can extend to a $k$-element hyperedge.
Now let $\{u_1,\dots,u_{i}\}$ and $\{v_1,\dots,v_{j}\}$ be distinct subhyperedges of size $i$ and~$j$ respectively.
Our goal is to show that if $a_{u_1}\dots a_{u_i}=a_{v_1}\dots a_{v_j}$ then either $i=j=k$ or $i=j=k-1$ and $\{u_1,\dots,u_{i}\}$ and $\{v_1,\dots,v_{j}\}$ are linked.
We prove the contrapositive.
First, if $i=k$ and $j<k$ then we know that $a_{u_1}\dots a_{u_i}$ is the constant tuple $\underline{a}$ (or the constant tuple $\underline{ab^{k-1}}$ when $S=S_c(ab^{k-1})$). But there is a hyperedge $\{v_1,\dots,v_k\}$ extending $\{v_1,\dots,v_{j}\}$ and we may find a $(k-1)$-in-$k$ satisfaction $\phi$ of $\mathbb{H}$ that gives $v_k$ the value $0$.
Then $a_{v_1}\dots a_{v_j}(\phi)=1$ so that $a_{u_1}\dots a_{u_i}(\phi)=a\neq 1=a_{v_1}\dots a_{v_j}(\phi)$.
(In the case of $S=S_c(ab^{k-1})$ we have $a_{v_1}\dots a_{v_j}(\phi)=b^j$, while $a_{v_1}\dots a_{v_j}(\phi)=ab^{j-1}$.)
So now assume that both $i,j\leq k-1$ and that $\{u_1,\dots,u_{i}\}$ is not linked to $\{v_1,\dots,v_{j}\}$.
We now find a $(k-1)$-in-$k$ satisfaction~$\phi$ such that one of $\{\phi(u_1),\dots,\phi(u_{i})\}$ and $\{\phi(v_1),\dots,\phi(v_{j})\}$ is $\{1\}$, and the other contains $0$, which will complete the proof as then one of $a_{u_1}\dots a_{u_i}(\phi)$ and $a_{v_1}\dots a_{v_j}(\phi)$ is $1$
(or a power of $b$ in the case of $S=S_c(ab^{k-1})$) and the other is~$a$ (or contains an $a$).
For convenience, let $u_{i+1},\dots,u_k$ be such that $\{u_1,\dots,u_k\}$ is a hyperedge and $v_{j+1},\dots,v_k$ be such that $\{v_1,\dots,v_k\}$ is a hyperedge. Without loss of generality, we may assume that $i\geq j$ and that $u_1\notin \{v_1,\dots,v_j\}$.
It's possible that $\{u_1,\dots,u_k\}=\{v_1,\dots,v_k\}$ but then we may use ${\leq 2}$-robustness and choose $\phi$ to colour $u_1$ the value $0$, which will force all remaining vertices in $\{u_1,\dots,u_k\}=\{v_1,\dots,v_k\}$ to be coloured $1$, which shows that $\{\phi(u_1),\dots,\phi(u_{i})\}=\{0,1\}\neq \{1\}=\{\phi(v_1),\dots,\phi(v_{j})\}$ as required.
So we assume that $\{u_1,\dots,u_k\}\neq \{v_1,\dots,v_k\}$, in which case $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ share at most one common vertex.
If they are disjoint, then at most one of $\{u_1,v_k\}$ and $\{v_1,u_k\}$ is a subyperedge: otherwise we get a $4$-cycle in $\mathbb{H}$.
There is no loss of generality in assume that $\{u_1,v_k\}$ is not a subhyperedge, in which case we can choose $\phi$ to colour both $u_1$ and $v_k$ the value~$0$.
Then $\{\phi(u_1),\dots,\phi(u_{i})\}=\{0,1\}\neq \{1\}=\{\phi(v_1),\dots,\phi(v_{j})\}$ as required. Now consider the case where $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ share (exactly) one vertex $u$.
If $u$ is not $u_k$ then $\{u_k,v_1\}$ is not a subhyperedge (as otherwise we have a $3$-cycle in $\mathbb{H}$), and so we can choose a $(k-1)$-in-$k$ satisfaction $\phi$ giving both $u_k$ and $v_1$ the colour $0$, so that $\{\phi(u_1),\dots,\phi(u_{i})\}=\{1\}\neq \{0,1\}$, while $0\in\{\phi(v_1),\dots,\phi(v_{j})\}$.
If $u=u_k$ and $v_k\neq u_k$ then $\{u_1,v_k\}$ is not a hyperedge and we can choose a $(k-1)$-in-$k$ satisfaction $\phi$ giving both $u_1$ and $v_k$ the colour $0$ so that $0\in\{\phi(u_1),\dots,\phi(u_{i})\}$ while $\{\phi(v_1),\dots,\phi(v_{j})\}=\{1\}$.
If $u=u_k=v_k$, then the assumption that $i\geq j$ and $\{u_1,\dots,u_{i}\}$ and $\{v_1,\dots,v_{j}\}$ are not linked $(k-1)$-sets ensures $j<k-1$. Then use the fact that $\{u_1,v_{k-1}\}$ is not a subhyperedge, to again find the required $(k-1)$-in-$k$ satisfaction.
\end{proof}
Recall that $S_c(a_1\dots a_k)=S_\mathbb{E}$, where $\mathbb{E}$ denotes the $k$-uniform hypergraph with a single hyperedge. A $k$-uniform hyperforest $\mathbb{F}$ that is nontrivial contains at least one hyperedge, so that $\mathsf{V}(S_\mathbb{F})$ contains $S_\mathbb{E}$. The next lemma shows the converse, from which it follows that all $k$-uniform hyperforest semirings generate the same variety.
\begin{lem}\label{lem:forest2}
If $k>2$ and $\mathbb{F}$ is a $k$-uniform hyperforest, then
$S_\mathbb{F}\in\mathsf{V}(S_c(a_1\dots a_k))$.
\end{lem}
\begin{proof}
It can be shown by induction \cite[Lemma 2.8]{hamjac:hyper} that $\mathbb{F}$ is an induced sub-hypergraph of a direct power $\mathbb{E}^T$ of $\mathbb{E}$, for some index set $T$ that is finite in the case that $\mathbb{F}$ is finite. Thus each vertex $u$ of $\mathbb{F}$ may be associated with an $T$-tuple $\bar{u}$ of vertices of $\mathbb{E}$, in such a way that $\{u_1,\dots,u_k\}$ is a hyperedge of $\mathbb{F}$ if and only if for all $i\in T$ the tuple $\{\bar{u}_1(i),\dots,\bar{u}_k(i)\}$ is a hyperedge of $\mathbb{E}$. (Of course, there is just one hyperedge in $\mathbb{E}$, so this means $\{\bar{u}_1(i),\dots,\bar{u}_k(i)\}=\{a_1,\dots,a_k\}$.) In fact we may assume slightly more than this, as the representation $\mathbb{F}\leq \mathbb{E}^T$ can be shown to be ${\leq}2$-robust, in the sense that if $u$ and $v$ are distinct vertices that do not lie in the same hyperedge, then we may find $i,j\in T$ such that $\bar{u}(i)=\bar{v}(i)$ and $\bar{u}(j)\neq \bar{v}(j)$. This is not directly shown in the proof of \cite[Lemma 2.8]{hamjac:hyper}, but it is easy to verify as we now sketch. For every such $u,v$, we wish to find maps $\phi_1,\phi_2$ from $\mathbb{F}$ to $\mathbb{E}$ with $\phi_1(u)=\phi_1(v)$ and $\phi_2(u)=\phi_2(v)$; then we may include $\phi_1,\phi_2$ as coordinates in $T$, with $\bar{w}(\phi_i)=\phi_i(w)$ for all vertices $w$, just as in Lemma \ref{lem:HinS7}.
The proof of \cite[Lemma 2.8]{hamjac:hyper} is inductive, using the fact that every hyperforest contains a leaf. Specifically we may find a series $\mathbb{E}\cong \mathbb{F}_0\leq \mathbb{F}_1\leq\dots\leq \mathbb{F}_n=\mathbb{F}$ of hyperforests, each obtained from its predecessor by adjoining a new leaf. The ${\leq}2$-robustness condition holds at the base case, and if it holds in $\mathbb{F}_i$, then it is very straightforward to show it holds in $\mathbb{F}_{i+1}$ as there is almost complete freedom for how we extend our desired maps $\phi_1$ and $\phi_2$ to the new vertices added in the introduction of the new leaf $e$. The only constraint is that our bijection from $e$ to $\{a_1,\dots,a_k\}$ has to agree on the (at most one) vertex common to $e$ and $\mathbb{F}_i$, if such a vertex exists. The details are almost identical to that in \cite[Lemma 2.8]{hamjac:hyper} and we omit them.
The remaining arguments are very similar to those in the proof of Lemma \ref{lem:HinS7} so we again give only sketch details. We let $A$ be the subsemiring of $\mathbb{E}^T$ generated by the elements $\bar{u}$ for vertices $u$ in $\mathbb{F}$, and let $J$ denote the set of elements in $A$ with a zero coordinate, which is an additive and multiplicative ideal, so that we have a well defined quotient $A/J$. We claim that $A/J\cong S_\mathbb{F}$. Properties (1)--(4) for $S_\mathbb{F}$ are verified of $A/J$ immediately due to the fact that $A/J$ is commutative, and the ${\leq}2$-robustness property of the representation of $\mathbb{F}$ as an induced substructure of $\mathbb{E}^T$: property (4), on linked subhyperedges, is because if $\{u_1,\dots,u_{k-1}\}$ and $\{v_1,\dots,v_{k-1}\}$ are linked via $w
$, and $\bar{w}(i)=a_\ell$, then $\bar{u}_1\dots\bar{u}_{k-1}(i)=\prod_{j\neq \ell}a_j=\bar{v}_1\dots\bar{v}_{k-1}(i)$. Thus $A/J$ is a homomorphic image of $S_\mathbb{F}$. The fact that it is isomorphic follows by an almost identical process to the final stages of the proof of Lemma \ref{lem:HinS7}, using the ${\leq}2$-robustness of the representation of $\mathbb{F}$ as an induced substructure of $\mathbb{E}^T$. We omit the details.
\end{proof}
\begin{remark}\label{rem:monoidforest}
Lemma \ref{lem:forest2} holds also if the monoids $M_\mathbb{F}$ and $M_c(a_1\dots a_k)$ are used in place of $S_\mathbb{F}$ and $S_c(a_1\dots a_k)$, with trivial amendment to the proof.
\end{remark}
In the following results, a \emph{cyclic element} will mean an element $g$ satisfying $g^n=g$ for some
$n$, which in the periodic case is equivalent to lying a subgroup of the multiplicative reduct of a semiring.
\begin{defn}\label{def:1in3}
We say that the \emph{$1$-in-$3$ property} holds in a semiring $S$ at a point $c\in S$ if $c$ is noncyclic and there is an element $d$ such that whenever $x,y,z\in S$ have all permutations of the product $xyz$ below $c$ in the $+$-semilattice order, then precisely one of $x,y,z$ equals $d$. The $2$-in-$3$ property is defined similarly but with precisely two of $x,y,z$ equal to $d$. We say that the $1$-in-$3$ (respectively, the $2$-in-$3$) property holds in $S$ if it holds at every noncyclic element of $S$.
\end{defn}
The 1-in-3 property holds in $S_7$ for the unique noncyclic element $c:=a$ via the element $d:=a$, while the $2$-in-$3$ property holds using $d=1$. The $2$-in-$3$ property holds in $S_c(abb)$ at the element $c:=abb$ via $d:=b$, while the $1$-in-$3$ property holds when $d:=a$. The $1$-in-$3$ and $2$-in-$3$ properties hold vacuously at all other choices of $c$ in $S_c(abb)$ as all noncyclic elements admit no $x,y,z$ with $xyz=c$. The $1$-in-$3$ property also holds in $B_2^1$: the noncyclic elements are $a$ and $b$, and up to symmetry we may consider $xyz,xzy,yxz,yzx,zxy,zyx\leq a$, from which it easily follows that $(x,y,z)\in\{(a,1,1),(1,a,1),(1,1,a)\}$. Moreover the same holds more generally in $B_2^1(G)$ for any group $G$: as a semigroup this is just the Brandt semigroup over the group $G$, with adjoined identity. As with $B_2$, the semigroup $B_2^1(G)$ becomes a flat semiring by way of the natural inverse semigroup order, and the added identity element in $B_2^1(G)$ will sit beneath the two nonzero idempotents of $B_2(G)$ as in Figure~\ref{fig:B21}. All other elements are pairwise incomparable and are covered by $0$, as $a$ and $b$ are in Figure~\ref{fig:B21}.
\begin{lem}\label{lem:H23}
Let $\mathbb{H}$ be a $3$-uniform hypergraph of girth at least $5$, and let $S$ be an ai-semiring of finite $+$-height, finite period and satisfying the $1$-in-$3$ or $2$-in-$3$ property. Assume further that the noncyclic elements form an order ideal in the $+$-order.
If $S_\mathbb{H}\in \mathsf{V}(S)$ then $\mathbb{H}$ is $2$-in-$3$-satisfiable.
\end{lem}
\begin{proof}
It suffices to assume that all vertices of $\mathbb{H}$ lie within a hyperedge, as we may trivially extend any $2$-in-$3$-satisfaction on the non-isolated vertices to any isolated vertices.
Assume $S_\mathbb{H}\in \mathsf{V}(S)$. So there is a substructure $A\leq S^P$ for some finite set $P$ and $A$ is a homomorphic to $S_\mathbb{H}$, under some homomorphism $\psi:A\to S_\mathbb{H}$. For every $x\in S_\mathbb{H}$, let $\bar{x}$ denote the largest element of $\psi^{-1}(x)$, which exists because $S$ has finite height (in the $+$-order). As $\mathbf{a}^2=0$ there is at least one $p\in P$ such that $\bar{\mathbf{a}}(p)$ is noncyclic in $S$; call this element $c$. We now invoke the 1-in-3 property, though note the argument for 2-in-3 as we go. Using the 1-in-3 property, there is an element $d$ such that whenever $x,y,z$ have all permutations of the product $xyz$ beneath $c$, then precisely one of $x,y,z$ is $d$. We colour the vertices of $\mathbb{H}$ as follows: if $\bar{\mathbf{a}}_v(p)=d$ then we colour $v$ by $0$, and otherwise we colour $v$ by $1$. (In the case of the $2$-in-$3$ property, the role of $0$ and $1$ is switched.) We claim this is a $2$-in-$3$ colouring. Now, all elements $\mathbf{a}_u$ lie within a hyperedge $\{u,v,w\}$ and $\bar{\mathbf{a}}_u\bar{\mathbf{a}}_v\bar{\mathbf{a}}_w(p)\leq \bar{\mathbf{a}}(p)=c$. So the $1$-in-$3$ property ensures that exactly one of $u,v,w$ is coloured $0$, as required. Thus $\mathbb{H}$ is $2$-in-$3$-colourable, as required.
\end{proof}
Again the reader will readily verify that this result and proof also holds if $S_\mathbb{H}$ is replaced by $M_\mathbb{H}$ so that the result holds in the setting of semirings with identity also.
Now we may prove one of our main nonfinite axiomatisability results.
\begin{thm}\label{thm:hard}
Let $S$ be a periodic ai-semiring of finite $+$-height with the following properties\up:
\begin{enumerate}
\item The noncyclic elements are an order ideal in the $+$-order\up;
\item The $1$-in-$3$ property holds on $S$ or the $2$-in-$3$ property holds on $S$\up;
\item Either $S_7$ or $S_c(abb)$ is contained in $\mathsf{V}(S)$.
\end{enumerate}
Then it is \texttt{NP}-hard to distinguish the finite algebras in the subvariety $\mathsf{V}(S_c(abb))\wedge \mathsf{V}(S_7)$ from those that are not in $\mathsf{V}(S)$.
\end{thm}
\begin{proof}
Theorem \ref{thm:facts} shows that it is \texttt{NP}-hard to distinguish those $3$-uniform hypergraphs of girth at least $5$ that are ${\leq 2}$-robustly 2-in-3 satisfiable from those that are not 2-in-3 satisfiable.
Lemma \ref{lem:HinS7} shows that when a $3$-uniform hypergraph $\mathbb{H}$ of girth at least $5$ is ${\leq 2}$-robustly 2-in-3 satisfiable then $S_\mathbb{H}\in \mathsf{V}(S_c(abb))\wedge \mathsf{V}(S_7)$, while Lemma \ref{lem:H23} shows that when $\mathbb{H}$ is not 2-in-3 satisfiable then $S_\mathbb{H}\notin \mathsf{V}(S)$.
\end{proof}
As noted, the semirings $S_c(abb)$, $S_7$ and ${B}_2^1(G)$ (for any group $G$ of finite exponent) satisfy the $1$-in-$3$ property, and all have finite $+$-height. The non-cyclic elements in each case form an order ideal, and either $S_7$ or $S_c(abb)$ are subsemirings. Thus we have the following corollary to Theorem~\ref{thm:hard}.
\begin{cor}\label{cor:hard}
Each of the ai-semirings $S_c(abb)$, $S_7$ and ${B}_2^1(G)$ \up(for any group $G$ of finite exponent\up) generates a variety with \texttt{NP}-hard variety membership.
\end{cor}
\begin{remark}
The computational problem of deciding membership in the variety generated by a finite flat algebra of finite signature and with absorbing $0$ is in \texttt{NP}. Thus the membership problem for finite algebras in $\mathsf{V}(S_c(abb))$ and in $\mathsf{V}(S_7)$ is \texttt{NP}-complete.
\end{remark}
\begin{proof}
\texttt{NP}-hardness follows immediately from Theorem \ref{thm:hard}. For membership in $\texttt{NP}$, let $A$ and $B$ be finite ai-semirings and with $B$ a flat semiring. A certificate for membership of $B$ in $\mathsf{V}(A)$ is as follows: for each pair $a\neq b$ in $B$ we give a congruence $\theta$ such that $B/\theta$ is a flat semiring, $a\notin b/\theta$ and separating homomorphisms witnessing membership of the partial algebra $(B/\theta)\backslash\{0\}$ in $\mathsf{SP}(A\backslash \{0\})$; see Willard \cite[Theorem 1.2]{wil}.
\end{proof}
Corollary \ref{cor:hard} shows that $S_7$, $S_c(abb)$ and ${B}_2^1$ (and many other semirings) do not have a finite basis for their equations: if they did, then the finite basis would provide a polynomial time algorithm to determine membership in their variety, which would contradict Corollary \ref{cor:hard}. Technically this assumes $\texttt{P}\neq \texttt{NP}$, though if we verify first order reductions throughout, we could observe that a finite basis would place the membership problem in $\texttt{AC}^0$ and use the known fact that $\texttt{AC}^0\subsetneq \texttt{NP}$. We are, however, in a position to give a direct proof that these semirings, and those of the form $S_c(a_1\dots a_k)$ for $k>2$ are NFB and in some cases INFB relative to a broad class of ai-semirings, including all that satisfy the conditions of Corollary~\ref{cor:hard}, and in fact all finite flat semirings. The \emph{index} of a semiring $S$, if it exists, will be the smallest $k$ such $S\models x^k\approx x^{k+p}$ for some $p$. When $S$ is finite then its index exists and is obviously at most $|S|$.
\begin{thm}\label{thm:p3}
Let $S$ be any finite ai-semiring whose noncyclic elements form an order ideal $I$, and let $k'$ be the index of $S$. If $S_c(a_1\dots a_{k})\in\mathsf{V}(S)$ for some $k\geq \max(k',3)$ then $S$ has no finite basis for its equational theory.
\end{thm}
\begin{proof}
Let $n$ be an arbitrary positive integer, and let $m$ denote the number $k\binom{kn}{2}$. Let $\ell$ denote the number of elements in $S$. By Theorem~\ref{thm:facts}(2), there exists a $k$-uniform hypergraph $\mathbb{H}_{n,\ell}=(V_{n,\ell},E_{n,\ell})$ of chromatic number greater than $\ell$ and girth greater than $m$. Let $T$ be a subsemiring of $S_{\mathbb{H}_{n,\ell}}$ generated by at most $n$ elements; let $T_{\rm gen}$ denote the chosen set of generators. For each $t\in T_{\rm gen}$ we may select $k$ generators $\mathbf{a}_{u_{t,1}},\dots, \mathbf{a}_{u_{t,k}}$ of $S_{\mathbb{H}_{n,\ell}}$ such that $t$ is equal to $\mathbf{a}_{u_{t,1}}\dots\mathbf{a}_{u_{t,i}}$ for some $i$ between $1$ and $k$ and $\{u_{t,1},\dots,u_{t,k}\}$ is a hyperedge. In the case that $1<i<k-1$ the choice of $u_{t,i+1},\dots,u_{t,k}$ is unique because Lemma \ref{lem:hyperprop2}(I) implies that $\{u_{t,1},\dots,u_{t,i}\}$ extends to a unique hyperedge of $\mathbb{H}_{n,\ell}$ in this case. In the case where $i=k-1$ the choice of $u_{t,k}$ is unique, but (by the defining property (4) for~$S_{\mathbb{H}_{n,\ell}}$) we may replace $\{u_{t,1},\dots,u_{t,k-1}\}$ by any other subhyperedge of size $k-1$ that is linked to $\{u_{t,1},\dots,u_{t,k-1}\}$. In the case where $i=k$ then we may choose any hyperedge, while when $i=1$ then we may choose any hyperedge extending $\{u_{t,1}\}$.
In all cases however, we fix a choice and can generate $t$ from $\mathbf{a}_{u_{t,1}},\dots, \mathbf{a}_{u_{t,k}}$. We consider now the subsemiring $T^+$ of $S_{\mathbb{H}_{n,\ell}}$ generated by $\{\mathbf{a}_{u_{t,i}}\mid t\in T_{\rm gen}, i\leq k\}$, and let $\mathbb{G}$ denote the subhypergraph of $\mathbb{H}_{n,\ell}$ induced by the vertex set $V_\mathbb{G}:=\{u_{t,i}\mid t\in T_{\rm gen}, i\leq k\}$.
Now $|V_\mathbb{G}|\leq nk<m=k\binom{nk}{2}$ so that the condition on the girth of $\mathbb{H}_{n,\ell}$ ensures that~$\mathbb{G}$ is a hyperforest.
By Lemma \ref{lem:forest2}, we have that $S_\mathbb{G}$ lies in $\mathsf{V}(S_c(a_1\dots a_k))\subseteq \mathsf{V}(S)$.
Unfortunately, this information is not enough for us to conclude the proof.
The semiring $T^+$ is \emph{almost} equal to $S_\mathbb{G}$ except that, due to of the nature of the defining rules for the constructions, the semiring $T^+$ may carry the ``shadows'' of hyperedges in $\mathbb{H}_{n,\ell}$ that are not themselves in $\mathbb{G}$. The issue arises in Rules~(1) and~(4) in the definition.
In the case of Rule~(1), there may be pairs of vertices $\{u,v\}$ in $\mathbb{G}$ that are subhyperedges in $\mathbb{H}_{n,\ell}$ but not in~$\mathbb{G}$, so that $\mathbf{a}_u\mathbf{a}_v=0$ in $S_\mathbb{G}$ but not in $T^+$.
In the case of Rule~(4) there may be pairs of $(k-1)$-element sets of vertices $\{u_1,\dots,u_{k-1}\}$ and $\{v_{1},\dots,v_{k-1}\}$ in~$\mathbb{G}$ that are linked in $\mathbb{H}_{n,\ell}$ but not in~$\mathbb{G}$: this happens when the unique $k^{\rm th}$ vertex $w$ that completes these as hyperedges is not in the vertex set of $\mathbb{G}$.
In this situation $T^+$ will have $\mathbf{a}_{u_1}\dots\mathbf{a}_{u_{k-1}}=\mathbf{a}_{v_1}\dots\mathbf{a}_{v_{k-1}}\neq 0$ while in $S_\mathbb{G}$ these two products will be~$0$.
To circumvent these issues we instead construct a larger subhyperforest $\mathbb{G}^+$ of $\mathbb{H}_{n,\ell}$ that contains all the corrections to these anomalies for~$\mathbb{G}$ in comparison to $\mathbb{H}_{n,\ell}$ so that $T^+$ embeds into $S_{\mathbb{G}^+}$.
We do not need to argue that $S_{\mathbb{G}^+}$ is a subsemiring of $S_{\mathbb{H}_{n,\ell}}$---it will not typically be so---only that $S_{\mathbb{G}^+}$, and therefore $T^+$, lies in the variety of $S_c(a_1\dots a_k)$.
But this last fact follows immediately from Lemma \ref{lem:forest2} and the fact that $\mathbb{G}^+$ is a hyperforest.
So it remains to actually construct $\mathbb{G}^+$. We define $\mathbb{G}^+$ as the induced subhypergraph of $\mathbb{H}_{n,\ell}$ on the following set $V_{\mathbb{G}^+}$ of vertices:
\[
\bigcup\{e\in E_{n,\ell}\mid 2\leq |e\cap V_\mathbb{G}|\}.
\]
Note that $V_\mathbb{G}\subseteq V_{\mathbb{G}^+}$ because we ensured that every $v\in V_\mathbb{G}$ is contained in some hyperedge $e$ of $\mathbb{G}$ and then $2\leq k=|e\cap V_\mathbb{G}|$ gives $v\in e\subseteq V_{\mathbb{G}^+}$. We now claim that $|V_{\mathbb{G}^+}|\leq m$. To see this, note that if $e,f\in E_{n,\ell}$ have $|e\cap V_\mathbb{G}|\geq 2$ and $|f\cap e|\geq 2$ then $e=f$ because no two distinct hyperedges share more than one vertex. Thus we have
\[
V_{\mathbb{G}^+}=\bigcup\{e\in E_{n,\ell}\mid \text{$e$ extends a 2-element subset of $V_\mathbb{G}$}\},
\]
so that $|V_{\mathbb{G}^+}|\leq k\binom{|V_\mathbb{G}|}{2}\leq k\binom{kn}{2}=m$. Again then, the condition on the girth of~$\mathbb{H}_{n,\ell}$ ensures that $\mathbb{G}^+$ is a hyperforest, so that $S_{\mathbb{G}^+}\in \mathsf{V}(S_c(a_1\dots a_k))$ by Lemma~\ref{lem:forest2}. This time however we have that~$T^+$ is a subsemiring of $S_{\mathbb{G}^+}$ because the nonzero elements of $T^+$ are products of generators that correspond to subhyperedges of~$\mathbb{H}_{n,\ell}$ but lying within~$V_\mathbb{G}$, and every subset of $V_\mathbb{G}$ that is a subhyperedge in $\mathbb{H}_{n,\ell}$ is also a subhyperedge in~$\mathbb{G}^+$. Of course, there may now be subsets of $V_{\mathbb{G}^+}$ that are subhyperedges in $\mathbb{H}_{n,\ell}$ but not in~$\mathbb{G}^+$, but we did not need to correct these: only those that corresponded to elements of $T^+$.
We have shown that $n$-generated subsemirings of $S_{\mathbb{H}_{n,\ell}}$ lie in $\mathsf{V}(S_c(a_1\dots a_k))\subseteq \mathsf{V}(S)$. Now we show that $S_{\mathbb{H}_{n,\ell}}\notin \mathsf{V}(S)$; we attempt to follow the proof of Lemma~\ref{lem:H23}. As in that proof, let $A\leq S^Q$ for some finite set $Q$ be such that $\phi:A\to S_{\mathbb{H}_{n,\ell}}$ is a surjective homomorphism, and let $\bar{a}$ be the maximum element of $\phi^{-1}(\mathbf{a})$. As $\mathbf{a}$ is not cyclic it follows that there is $q\in Q$ such that $\bar{a}(q)$ is not cyclic.
For each $u\in V_{n,\ell}$, fix an element $a_u\in \phi^{-1}(\mathbf{a}_u)$. Because $\mathbb{H}_{n,\ell}$ is not $\ell$-colourable and the map $u\mapsto a_u(q)$ maps into $S$ (of size $\ell$), there must be a hyperedge $\{u_1,\dots,u_k\}$ such that $a_{u_1}(q)=\dots=a_{u_k}(q)$. Then $(a_{u_1}\dots a_{u_k})(q)$ is cyclic, because $A\models x^k\approx x^{k+p}$. But $(a_{u_1}\dots a_{u_k})(q)\leq \bar{a}(q)$ and $\bar{a}(q)$ is not cyclic, which then contradicts the assumption that the noncyclic elements form an order ideal of~$S$. Thus $S_{\mathbb{H}_{n,\ell}}\notin \mathsf{V}(S)$ as required.
\end{proof}
\begin{remark}\label{rem:p3monoid}
Theorem \ref{thm:p3} also holds in the monoid signature, where $M_c(a_1\dots a_{k})$ replaces $S_c(a_1\dots a_{k})$.
\end{remark}
\begin{proof}
We have noted throughout that the relevant results and arguments to prove Theorem \ref{thm:p3} hold when $M_c(a_1\dots a_{k})$ replaces $S_c(a_1\dots a_{k})$ and the proof uses $M_{\mathbb{H}_{n,\ell}}$ in place of $S_{\mathbb{H}_{n,\ell}}$; see Remark \ref{rem:monoidforest} (in place of Lemma \ref{lem:forest}), and note that the final paragraph of the proof of Theorem \ref{thm:p3} holds because if $M_{\mathbb{H}_{n,\ell}}\in \mathsf{V}(S)$ in the signature $\{+,\cdot,1\}$, then $S_{\mathbb{H}_{n,\ell}}\in \mathsf{V}(S)$ in the signature $\{+,\cdot\}$, which we showed is not true.
\end{proof}
While $S_7$ is not INFB, Theorem \ref{thm:p3} shows it is relatively INFB for a broad class of semirings.
\begin{cor}\label{cor:rinfb}
The semiring $S_7$, is inherently nonfinitely based relatively to the property of being generated by a flat semiring.
\end{cor}
\begin{proof}
This follows immediately from Proposition \ref{pro:SinM} (with $S_7=M_c(a)$) and Theorem \ref{thm:p3}.
\end{proof}
Lemma \ref{lem:0directjoin} shows that we may also replace ``a flat semiring'' by ``a finite family of flat semirings'' in Corollary \ref{cor:rinfb}. We may also visit the remaining cases from Corollary \ref{cor:hard}; the result follows already from Corollary \ref{cor:hard}, assuming $\texttt{P}\neq\texttt{NP}$, but follows unconditionally from Theorem \ref{thm:p3}.
\begin{cor}\label{cor:NFBtake2}
The ai-semirings $S_c(abb)$ and ${B}_2^1(G)$ \up(for any group $G$ of finite exponent\up) are not finitely based.
\end{cor}
The next corollary provides a complete classification of the finite basis property in one natural case.
\begin{cor}\label{cor:monoid}
If a finite flat semiring has a multiplicative identity element, then it is finitely based if and only if it is the flat extension of a finite group whose nilpotent subgroups are abelian.
\end{cor}
\begin{proof}
This is an immediate consequence of Proposition \ref{pro:monoid}, Corollary \ref{cor:rinfb} and the fact that a flat extension of a finite group is finitely based if and only if all of its nilpotent subgroups are abelian.
\end{proof}
In Section \ref{sec:group} we will see that we can add the further equivalent condition of ``not being INFB relative to being generated by a finite flat semiring'' as well.
\section{The semiring $S_7$}\label{sec:S7}
In \cite{ZRCSD} it is shown that all ai-semirings of order at most $3$, with the possible exception of $S_7$, are finitely based.
Combining this with Corollary \ref{cor:rinfb} we can state the following result.
\begin{cor}\label{cor:3element}
A finite ai-semiring of order at most $3$ is finitely based if and only if it is not $S_7$.
\end{cor}
Three-elements is also the smallest possible size of a generator for non-polynomial time variety membership given that all $2$-element algebras (of finite signature) have a finite basis for their equations \cite{lyn}; the only other known example was given in \cite[\S8]{jac:SAT}.
In view of the special status that $S_7$ appears to have in terms of small ai-semirings, it seems pertinent to provide some further investigation to the equational properties of $S_7$.
First we observe that it is easily seen that $S_7$ as co-$\texttt{NP}$-complete equation checking problem; it is not the smallest with this property, as even the two element distributive lattice $D_2$ has this property \cite{BHR}.
Let $\mathscr{S}$ be a family of nonempty subsets of a set $\{x_1,\dots,x_n\}$. For each $s\in \mathscr{S}$, let $\mathbf{w}_s$ denote the product of the variables in $s$ (so, if $s=\{x,y,z\}$, then $\mathbf{w}_s=xyz$, noting that as $S_7$ is commutative, the order of appearance in the product is not important). Let $t_\mathscr{S}$ denote the term
\(
\sum_{s\in \mathscr{S}}\mathbf{w}_s.
\)
\begin{lem}
Let $\mathscr{S}$ be a family of nonempty subsets of a set $\{x_1,\dots,x_n\}$, with each $x_i$ appearing in at least one element of $\mathscr{S}$. Then $S_7\models t_\mathscr{S}\approx t_\mathscr{S}^2$ if and only if there is a subset $Y\subseteq \{x_1,\dots,x_n\}$ that intersects each $s\in\mathscr{S}$ exactly once.
\end{lem}
\begin{proof}
For an assignment $\nu:\{x_1,\dots,x_n\}\to S_7$, let $Y$ denote those variables assigned $a$. Then $\nu(t_\mathscr{S})$ is nonidempotent
if and only if $\nu(t_\mathscr{S})=a$, if and only if $\nu(x_i)=a$ for all $i$, if and only if $Y$ intersects each $s\in\mathscr{S}$ exactly once.
\end{proof}
When each $s\in\mathscr{S}$ has size exactly $3$, this is the well known \texttt{NP}-complete problem 1-in-3SAT. Thus we have the following consequence.
\begin{cor}
Equation checking is co-\texttt{NP}-complete for $S_7$.
\end{cor}
This idea generalises to a full combinatorial characterisation of the equational theory of $S_7$. By distributivity, all ai-semiring terms in variables $X=\{x_1,x_2,\dots\}$ are finite sums of words in $X^+$, thus for the purposes of characterisation we may consider an \emph{ai-semiring identity} (${\bf AI}$-identity for short) over $X$ as an
expression of the form $\mathbf{u}\approx \mathbf{v}$, where $\mathbf{u}, \mathbf{v}\in \wp_{\rm fin}f(X^+)$, the nonempty finite subsets of $X^+$.
Thus we consider
$\mathbf{u}_1+\cdots+\mathbf{u}_k\approx \mathbf{v}_1+\cdots+\mathbf{v}_\ell$ synonymously with the ${\bf AI}$-identity
$\{\mathbf{u}_i \mid 1\leq i\leq k\}\approx \{\mathbf{v}_j \mid 1\leq j\leq \ell\}$.
Let $\mathbf{w}\in X^+$ and $x\in X$. Then let
\begin{itemize}
\item $c(\mathbf{w})$ (the \emph{content} of $\omega$) denote the set of variables that occur in $\mathbf{w}$.
\item $\operatorname{occ}(x, \mathbf{w})$ denotes the number of occurrences of $x$ in $\mathbf{w}$.
\end{itemize}
These notations extend to sums of words in an obvious way though we need only
\[
c\Big(\sum_{1\leq i\leq n}\mathbf{w}_i\Big)=\bigcup_{1\leq i\leq n}c(\mathbf{w}_i).
\]
Now consider an ${\bf AI}$-term $\mathbf{w}:= \mathbf{w}_1 + \dots + \mathbf{w}_m$ , where each $\mathbf{w}_i\in X^+$. We let $\delta(\mathbf{w})$ denote the set of nonempty subsets $Z$ of $c(\mathbf{w})$ such that both
\begin{itemize}
\item
$Z \cap c(\mathbf{w}_i)$ is a singleton for every $\mathbf{w}_i$ and
\item $\operatorname{occ}(x, \mathbf{w}_i ) =1$ if $\{x\}=Z \cap c(\mathbf{w}_i)$.
\end{itemize}
Let $M_2$ denote $2$-element flat semiring on $\{1,0\}$ with $0$ a multiplicative and additive zero and $1\cdot 1=1$; in the $M(W)$ notation, we have $M_2=M(1)$ where $1$ is the empty word. The solution of the equational problem for $M_2$
can be found in~\cite{sharen}.
\begin{lem}\label{l1} \up(\cite[Lemma~1.1~(iii)]{sharen}.\up)
Let $\mathbf{u}\approx \mathbf{v}$ be an ${\bf AI}$-identity. Then
$\mathbf{u}\approx \mathbf{v}$ holds in $M_2$ if and only if $c(\mathbf{u})=c(\mathbf{v})$.
\end{lem}
\begin{pro}\label{npro1}
Let $\mathbf{u}\approx \mathbf{v}$ be an ${\bf AI}$-identity. Then
$\mathbf{u}\approx \mathbf{v}$ holds in $S_7$ if and only if $c(\mathbf{u})=c(\mathbf{v})$ and $\delta(\mathbf{u})=\delta(\mathbf{v})$.
\end{pro}
\begin{proof}
Suppose that $\mathbf{u}\approx \mathbf{v}$ holds in $S_7$. Since $\{0, 1\}$ forms a subsemiring
of $S_7$ and is isomorphic to $M_2$, it follows from Lemma \ref{l1} that $c(\mathbf{u})=c(\mathbf{v})$.
Let $Z$ be an arbitrary element of $\delta(\mathbf{u})$.
Then $Z\subseteq c(\mathbf{u})$. It follows immediately that $Z\subseteq c(\mathbf{v})$.
Now consider the substitution $\varphi_Z: X\rightarrow S_7$: $\varphi_Z(x)=a$
if $x\in Z$ and $\varphi_Z(x)=1$ otherwise. Then $\varphi_Z(\mathbf{u})=a$.
Since $\mathbf{u}\approx \mathbf{v}$ holds in $S_7$, it follows that
$\varphi_Z(\mathbf{v})=\varphi_Z(\mathbf{u})=a$. This implies that $\varphi_Z(\mathbf{v}_j)=a$
for every $\mathbf{v}_j$ in $v$. Furthermore, for each $\mathbf{v}_j$ in $\mathbf{v}$,
there exists $x_j$ in $X$ such that $Z \cap c(\mathbf{v}_j)=\{x_j\}$
and $\operatorname{occ}(x_j, \mathbf{v}_j) =1$.
Thus $Z\in \delta(\mathbf{v})$ and so $\delta(\mathbf{u})\subseteq\delta(\mathbf{v})$.
Similarly, $\delta(\mathbf{v})\subseteq\delta(\mathbf{u})$.
We now conclude that $\delta(\mathbf{u})=\delta(\mathbf{v})$.
Conversely, assume that $c(\mathbf{u})=c(\mathbf{v})$ and $\delta(\mathbf{u})=\delta(\mathbf{v})$.
Let $\psi: X\rightarrow S_7$ be an arbitrary substitution.
Consider the following two cases:
\begin{itemize}
\item $\{x\mid x\in c(\mathbf{u}), \psi(x)=0\}\neq \varnothing$.
Since $c(\mathbf{u})=c(\mathbf{v})$, it follows that $\psi(\mathbf{u})=\psi(\mathbf{v})=0$.
\item $\{x\mid x\in c(\mathbf{u}), \psi(x)=0\}=\varnothing$.
Then $\psi(x)=a$ or $1$ for every $x$ in $c(\mathbf{u})\cup c(\mathbf{v})$.
Let $Z$ denote the set $\{x\mid x\in c(\mathbf{u}), \psi(x)=a\}$.
If $Z=\varnothing$, then $\psi(\mathbf{u})=\psi(\mathbf{v})=1$. Otherwise,
we need to consider the following two subcases:
\begin{itemize}
\item[$\diamond$] $Z\in \delta(\mathbf{u})$. Then $\psi(\mathbf{u})=a$.
Since $\delta(\mathbf{u})=\delta(\mathbf{v})$, it follows that $Z\in \delta(\mathbf{v})$.
This implies that $\psi(\mathbf{v})=a=\psi(\mathbf{u})$.
\item[$\diamond$] $Z\notin \delta(\mathbf{u})$. Then $\psi(\mathbf{u})=0$.
Since $\delta(\mathbf{u})=\delta(\mathbf{v})$, it follows that $Z\notin \delta(\mathbf{v})$.
This implies that $\psi(\mathbf{v})=0=\psi(\mathbf{u})$.
\end{itemize}
\end{itemize}
We therefore have that $\mathbf{u}\approx \mathbf{v}$ holds in $S_7$.
\end{proof}
\section{Non-abelian $p$-groups imply nonfinitely based}\label{sec:group}
It follows from results in \cite[Theorem 7.3]{jac:flat} (stated slightly more directly in \cite[Remark~4.9]{jac:eqncomp}) that the flat extension of a finite group with a nonabelian Sylow subgroup is INFB relative to the property of being a variety generated by a finite flat semiring. Unlike $S_7$, these examples are not commutative, but they do have a multiplicative reduct that is FB, due primarily to the Oates-Powell Theorem \cite{oatpow}. We now observe that the same ideas yield a significantly stronger nonfinite basis result.
\begin{thm}\label{thm:pgroup}
Let $S$ be a finite ai-semiring. If the multiplicative reduct of $S$ contains a nonabelian nilpotent subgroup, then every finite ai-semiring whose variety contains $S$ is nonfinitely based.
\end{thm}
This theorem will be proved over the remainder of this section. We first establish some basic lemmas; the first is well known.
\begin{lem}\label{lem:order}
If $G$ is a periodic group and $\leq$ an order that is preserved by the multiplication of $G$, then $\leq$ is an antichain. In particular, a subgroup of the multiplicative reduct of a finite semiring is always an antichain relative to the $+$ order.
\end{lem}
\begin{proof}
Assume $g\leq h$. Then $gh^{-1}\leq e$. Let $k$ denote $gh^{-1}$; we have $k\leq e$ and $k^n=e$ for some $n$. Then $kk\leq ke=k$. Then $k^3\leq k^2\leq k\leq e$ and so on: $k^i\leq k^j$ whenever $i>j$. But then $e=k^n\leq k^{n-1}\leq\dots\leq k^2\leq k\leq e$, so that all of these inequalities are equalities. Thus $gh^{-1}=e$, or equivalently, $g=h$.
\end{proof}
In the following $\mathsf{HS}(S)$ will denote, as usual, those algebras obtained as a homomorphic images of subalgebras of $S$.
\begin{lem}\label{lem:flatin}
Let $G$ be a finite nontrivial subgroup of the multiplicative reduct of an ai-semiring $S$. Then $\flat(G)\in\mathsf{HS}(S)$.
\end{lem}
\begin{proof}
Let $S_G$ be the subsemiring generated by $G$, which consists of the elements of $G$ along with finite sums of elements of $G$ (this is clearly generated by $G$ and also a subsemiring). Let $G^+$ denote all elements that can be written as a proper sum; that is, a sum involving at least two distinct elements of $G$; equivalently, $G^+=S_G\backslash G$, which is nonempty as $G$ is nontrivial. We claim that $G^+$ is a multiplicative ideal of $\langle S_G;\cdot\rangle$ and also a filter of $\langle S_G;+\rangle$. For the multiplicative reduct claim, note that if $s=g_1+\dots +g_n$ is a sum of distinct elements of $G$, where $n>1$, and $t$ is also a sum of distinct elements $h_1+\dots +h_m$, then $st=\sum_{i\leq n, j\leq m}g_ih_j$ and $g_1h_1\neq g_2h_1$, so that $st\in G^+$; by symmetry, $ts\in G^+$ too. For the filter property, simply note that if $s\leq g$ for some $g\in G$ and $s=g_1+\dots +g_n$ is as before, then $g_1\leq g$ and $g_2\leq g$, which contradicts Lemma \ref{lem:order}. Thus $G^+$ is a multiplicative ideal and additive filter, so that we may take a Rees quotient (identifying all elements of $G^+$) and obtain an isomorphic copy of $\flat (G)$.
\end{proof}
The following lemma seems quite counterintuitive, as it shows that when a group arises within a quotient of (the multiplicative reduct of) a finite ai-semiring $S$, then it was already present as a subgroup of $S$.
\begin{lem}[The group quotient embedding lemma]\label{lem:reverse}
Let $S$ and $A$ be finite ai-semirings and let $G$ be a finite subgroup of the multiplicative reduct of~$S$. If there is a surjective semiring homomorphism $\phi:A\to S$, then there is an injective semigroup homomorphism from $G$ into $A$.
\end{lem}
\begin{proof}
First recall that the semigroup-theoretic $\mathscr{J}$-order is defined by $x\leq_\mathscr{J} y$ if there exists $u,v$ (possibly empty) such that $x=uyv$; see a test such as Howie \cite{how} for example.
Observe that $\phi^{-1}(G)$ is a subsemigroup $A_G$ of the multiplicative reduct of $A$. Because $A$ is finite, $A_G$ contains a minimal idempotent $e_A$ with respect to the $\mathscr{J}$-order; evidently we must have $\phi(e_A)=e$, as $e_Ae_A=e_A$. Moreover, $e_AA_Ge_A$ is a finite group $H$ with $\phi(H)=G$. The proof will then be complete if we can show that $\phi$ is injective on $H$. Let $N\leq H$ denote $\phi^{-1}(e)\cap H$; the kernel of $\phi$ restricted to $H$. Consider any $g_1,g_2\in N$, and observe that $g_1+g_2$ is also in $N$ because $\phi(g_1+g_2)=\phi(g_1)+\phi(g_2)=e+e=e$ and $e_A(g_1+g_2)e_A=g_1+g_2$ by distributivity. But then $g_1\leq g_1+g_2$ and $g_2\leq g_1+g_2$ implies $g_1=g_2=g_1+g_2$ by Lemma \ref{lem:order}, which shows that $|N|=1$ as required.
\end{proof}
\begin{lem}\label{lem:Gin}
Let $S$ and $T$ be finite ai-semirings with $S\in\mathsf{V}(T)$ and let $G$ be a subgroup of the multiplicative reduct of $S$. Then $G$ lies in the quasivariety of the multiplicative reduct of $T$.
\end{lem}
\begin{proof}
There is $k\in \mathbb{N}$ and $A\leq T^k$ such that $\phi:A\twoheadrightarrow S$. By Lemma \ref{lem:reverse} we have that $G$ embeds into $A$ as a multiplicative subgroup. Then $G$ is a subgroup of a power of the multiplicative reduct of $T$, or in other words is in the quasivariety generated by $T$ (as a multiplicative semigroup), as claimed.
\end{proof}
Now we may prove Theorem \ref{thm:pgroup}.
\begin{proof}[Proof of Theorem \ref{thm:pgroup}]
Let $S$ be a finite ai-semiring containing a non-abelian but nilpotent subgroup $G$. Let $T$ be a finite ai-semiring with $S\in\mathsf{V}(T)$. Ol$'$shanski\u{\i}~\cite{ols} proved that for each $n$ there is a finite group $G_n$ that does not lie in the quasivariety of $T$, but whose $n$-generated subgroups lie in the quasivariety of $G$. (Technically, this is proved by Ol$'$shanski\u{\i} when $T$ is a group, but the argument uses only the cardinality of $T$, so holds in the broader class of all semigroups.) Then the $n$-generated subsemirings of $\flat(G_n)$ lie in the variety of $\flat(G)$, which lies in $\mathsf{V}(S)\leq \mathsf{V}(T)$ by Lemma~\ref{lem:flatin}. We wish to show that $\flat(G_n)\notin \mathsf{V}(T)$. But this follows immediately from Lemma~\ref{lem:Gin}, because $G_n$ is not in the quasivariety of~$T$.
\end{proof}
Note that we also showed the following.
\begin{lem}\label{lem:localflat}
Let $G$ be a finite group and $S$ a finite semiring. The following are equivalent.
\begin{enumerate}
\item $\flat(G)\in \mathsf{V}(S)$,
\item $\flat(G)$ lies in the variety generated by the flat semirings in $\mathsf{HS}(S)$,
\item $G$ lies in the quasivariety of the subgroups of $S$.
\end{enumerate}
\end{lem}
\begin{proof}
Using Lemma \ref{lem:Gin} we have that (1) implies that $G$ is in the quasivariety of the multiplicative reduct of $S$, which implies that $G$ is in the quasivariety generated by the class of subgroups of the multiplicative reduct of~$S$; that is (3). In turn (3) implies that $\flat(G)$ is in the variety generated by the flat extensions of subgroups of~$S$ by \cite{jac:flat}; but these are amongst the flat semirings in $\mathsf{HS}(S)$ by Lemma \ref{lem:flatin}; so (2) holds. That (2) implies (1) is trivial.
\end{proof}
As noted in \cite[Remark~4.9]{jac:eqncomp}, the smallest nonabelian nilpotent groups have 8 elements, so the smallest nonfinitely based ai-semirings from this group-theoretic approach have $9$ elements. It also follows immediately from \cite{jac:flat} and \cite{ols} that there are infinitely many examples of \emph{limit varieties} of semirings that are generated by flat extensions of groups: nonfinitely based varieties whose subvarieties are all finitely based. Subsequent work in preparation will give greater detail of these and other kinds of limit variety of ai-semiring, though we observe that the properties of the 27-element group $G_3$ described in \cite[\S3.4]{jac:continuum} show that it yields a concrete example after applying the flat extension construction. (The group $G_3$ is just the $2$-generated free Burnside group $B(2,3)$ of exponent~$3$, as described in Burnside \cite{bur} or Hall \cite{hal}.)
The power semiring $\wp(G)$ of a finite group contains $G$ as a subgroup, so Theorem~\ref{thm:pgroup} will apply to show that $\wp(G)$ is not finitely based whenever $G$ contains a nonabelian Sylow subgroup.
\begin{example}\label{eg:q}
The power semiring of the quaternion group is nonfinitely based as a semiring, and is inherently nonfinitely based relative to the property of being generated by a finite ai-semiring, but not relative to generating a locally finite variety.
\end{example}
\begin{proof}
Theorem \ref{thm:pgroup} shows that $\wp(Q)$ is inherently nonfinitely based relative to the property of being generated by a finite semiring.
The idempotents in $\wp(G)$ are precisely the subgroups (or $\varnothing$) of $G$. When all subgroups of $G$ are normal, such as for $G=Q$, the idempotents of $\wp(Q)$ are central. This is not possible for a finite INFB semigroup: it is a trivial consequence of the first author's analysis \cite{jac:INFB} (based on Mark Sapir's celebrated classification of INFB finite semigroups \cite{sap1,sap2}), as no minimal finite INFB divisor has central idempotents. Indeed, it follows from the main classification there, and the fact that $\wp(G)$ is a block group (see \cite{pin} for example), that~$\wp(G)$ is INFB as a semigroup if and only if $B_2^1$ is a divisor. Thus $\wp(Q)$ is not INFB as a semigroup, hence not as a semiring either by Theorem~\ref{thm:INFB}(3).
\end{proof}
The arguments for Example \ref{eg:q} apply equally to any finite nonabelian group whose subgroups are all normal (that is, a \emph{Hamiltonian} group).
To conclude this section we note an extension of Corollary \ref{cor:monoid}, which gives a rare example of a variety for which the finite basis problem is moderately complex but for which we can give an effective characterisation.
\begin{thm}\label{thm:classification}
In the signature $\{+,\cdot,1\}$ of semirings with identity, the variety generated by flat semirings with identity is defined by the ai-semiring laws, along with the flat semiring laws \up(Lemma \ref{lem:flatvariety}\up) and $1x\approx x1\approx x$. The following are equivalent for a finite semiring with identity $S$ in this variety\up:
\begin{enumerate}
\item $S$ is FB\up;
\item $S$ is completely regular and all nilpotent subgroups are abelian\up;
\item all nilpotent subgroups of $S$ are abelian and the variety of $S$ avoids $S_7$.
\end{enumerate}
\end{thm}
\begin{proof}
The variety generated by $S$ is equal to the variety generated by the set $\mathscr{S}$ of subdirectly irreducible quotients of $S$, which by Lemma \ref{lem:flatvariety} is a finite set of flat monoids. But by Proposition \ref{pro:monoid}, a flat semiring with identity either contains $S_7$ as a subsemiring (even when $1$ is in the signature) or is a flat extension of a group. If $S_7$ is a subsemiring of a member of $\mathscr{S}$, then we may choose any $k$ greater than the index of $S$, and use the monoid version of Theorem \ref{thm:p3} (Remark \ref{rem:p3monoid}), using Proposition \ref{pro:SinM} to show that $S$ is not finitely based. If $S$ (or equivalently, a member of $\mathscr{S}$) contains a nonabelian nilpotent subgroup, then $S$ is NFB by Theorem \ref{thm:pgroup}. If $\mathscr{S}$ consists of flat extensions of groups whose nilpotent subgroups are abelian, then the variety of $S$ can be generated by the flat extension of the finite group $G$ obtained as the direct product of all subgroups of $S$. All nilpotent subgroups of this semiring are abelian, so that $S$ is FB by \cite[Theorem 7.3]{jac:flat}.
\end{proof}
\section{Open problems and future directions}\label{sec:problems}
In this article we have shown that finite ai-semirings can have no finite basis for their equations for reasons that are nothing to do with the corresponding property for their semigroup reducts. While a great many ai-semirings are covered by the results, there is a sense in which this is a start rather than a conclusion. The finite basis property is clearly very rich for ai-semirings and the present results arguably open up more new directions than they close off. An obvious overall goal is a complete classification of the finite basis property for ai-semirings, though this is probably too ambitious to provide good direction for future progress.
While the finite basis problem for general algebras is undecidable (McKenzie~\cite{mck}), we offer no speculations within the class of all ai-semirings despite the promising positive steps in Theorem \ref{thm:classification}. Instead we list a number of more achievable problems that we feel might shape the area.
We have shown that $M_c(W)$ and $M(W)$ is always nonfinitely based (for finite, nonempty sets of words $W$), and that $S_c(W)$ is very often nonfinitely based.
\begin{problem}\label{prob:1}
\begin{enumerate}
\item For which finite sets of words $W$ is $S(W)$ finitely based as an ai-semiring?
\item When $W$ consists of powers of letters and words of length at most $2$, when is $S_c(W)$ finitely based?
\item More generally than 1 and 2\up: when a finite nilpotent semigroup carries a flat semiring structure, under what conditions is it finitely based?
\item More generally still, which finite flat semirings are finitely based?
\end{enumerate}
\end{problem}
It is also of interest to investigate Problem \ref{prob:1} in the context of the computational complexity of the membership problem.
\begin{problem}\label{prob:2}
When is the power semiring of a finite semigroup finitely based as an ai-semiring?
\end{problem}
The examples in the present article show that, unlike for semigroups, commutativity is not a barrier for the nonfinite basis property in the world of finite ai-semirings. To some extent it even acts as a trigger for our arguments, as the hypergraph encoding makes inherent use of the commutativity of products formed from vertices. On the other hand, the situation for flat groups \cite{jac:flat} depended explicitly on non-commutativity (of Sylow subgroups). The following problem may be achievable.
\begin{problem}
Which finite multiplicatively-commutative ai-semirings are finitely based? The restriction of this problem to nilpotent semirings, to flat semirings and to power semirings are natural restrictions of interest.
\end{problem}
Most of our examples do not have an additive zero element, while the important earlier investigations of Igor Dolinka \cite{dol0,dol1,dol2,dol3} included $0$ in the signature and axiomatically required that it be an additive identity and multiplicative zero. This is of course not compatible with any flat ai-semiring structure though one can always extend a flat semiring to an ai-semiring by adjoining an additive identity (so that the $+$-semilattice reduct has height $2$). It is not entirely clear how our methods apply in this case, and Problems \ref{prob:1}, \ref{prob:2} remain of interest (adding the required $0$ in the case of Problem \ref{prob:1}, and letting $\varnothing$ be the~$0$ in Problem \ref{prob:2}). The finite basis property is even unclear in the case of adjoining a $0$ to $S_7$.
\begin{problem}\label{prob:3}
\begin{enumerate}
\item Is $S_7^0$ finitely based or not finitely based?
\item In the signature $\{+,\cdot\}$, what is the cardinality of the interval $[\mathsf{V}(S_7),\mathsf{V}(S_7^0)]$ in the lattice of semiring varieties?
\item When is the finite basis (nonfinite basis) property stable under adjoining an additive $0$?
\end{enumerate}
\end{problem}
In this context, we now show that Theorem \ref{thm:pgroup} continues to hold in the setting of semirings with $0$.
We first reprove a version of Lemma \ref{lem:flatin} in the semiring with~$0$ setting.
\begin{lem}\label{lem:flatin0}
Let $G$ be a finite nontrivial subgroup of the multiplicative reduct of an ai-semiring $S$ with $0$. Then $\flat(G)^0\in\mathsf{HS}(S)$.
\end{lem}
\begin{proof}
The proof is very similar to that of Lemma \ref{lem:flatin}, so we condense steps where possible.
First note that $0\notin G$, as $0$ is a multiplicative zero for $S$ and $G$ is nontrivial.
Let $S_G$ be the subsemiring with $0$ generated by $G\cup\{0\}$, and let $G^+$ denote all elements that can be written as a proper sum of elements of $G$ and note that again $0\notin G^+$, as $0\leq s$ for all $s\in S_G$. As before, $G^+$ is a multiplicative ideal of $\langle S_G\backslash \{0\};\cdot\rangle$ and also an additive filter of $\langle S_G;+\rangle$. Then the equivalence relation $\theta:=\{(x,x)\mid x\in G\cup \{0\}\}\cup (G^+\times G^+)$ is a congruence and $S_G/\theta\cong \flat(G)^0$.
\end{proof}
We can now prove Theorem \ref{thm:pgroup} in the signature $\{+,\cdot,0\}$.
\begin{thm}\label{thm:pgroup0}
Theorem \ref{thm:pgroup} holds in the signature $\{+,\cdot,0\}$ of semirings with $0$.
\end{thm}
\begin{proof}
Lemma \ref{lem:order} holds without change. In place of Lemma \ref{lem:flatin} we may use Lemma~\ref{lem:flatin0}. The Group Quotient Embedding Lemma~\ref{lem:reverse} holds immediately, because every homomorphism in the signature $\{+,\cdot,0\}$ is a homomorphism in the reduct signature $\{+,\cdot\}$ and the conclusion of the lemma concerns only the semigroup reduct. Similarly, Lemma \ref{lem:Gin} holds in the semiring with $0$ setting because if $S\in\mathsf{V}(T)$ there, then so also does $S\in\mathsf{V}(T)$ in the reduct signature $\{+,\cdot\}$, and the conclusion again concerns only the $\cdot$-reduct.
Thus we have all of the ingredients required for the proof of Theorem \ref{thm:pgroup}, now using $\flat(G)^0$ in place of $\flat(G)$.
\end{proof}
Every inverse semigroup carries a natural order given by $x\leq y\Leftrightarrow xx^{-1}y=x$ and a number of authors have investigated when this order has compatible greatest lower bounds $\wedge$; see Leech \cite{lee}, but also Garvac$'$ki\u{\i} \cite{gar}. This is also the basis of the recent approach by Volkov \cite{vol21}, and the starting point for the first author's investigations in \cite[\S7.8]{jac:flat} via \cite[\S7]{jacsto:PM}. The standard Wagner-Preston representation for inverse semigroups extends to a representation of $\wedge$ as intersection, which gives rise to an ai-semiring structure where the role of addition is played by $\wedge$. Garvac$'$ki\u{\i}'s results show that even without inverse, the law $xv+uv+uy\leq xy$ (within ai-semirings) characterises semigroups of injective partial maps with intersection, where $+$ is intersection of relations. The order on $B_2^1$ given in Figure \ref{fig:B21} is an example of these approaches. Subsemirings include $B_2$ (dropping the element $1$), as well as the non-inverse subsemirings $B_0$ (dropping $b$ from $B_2$) and $P$ (dropping $ba$ from $B_0$). The semigroups $B_2$, $B_0$, and $P$ have each played an important role in semigroup variety considerations: see \cite{tra}, \cite{edm,leeB0} and \cite{golsap} (respectively) for example. The semiring $P$ appears as $S_4$ in \cite{ZRCSD}, where a finite basis is given for its equations.
\begin{problem}\label{prob:4}
\begin{enumerate}
\item Resolve the finite or nonfinite basability of $B_0$ and $B_2$ as ai-semirings.
\item Is $B_2^1$ inherently nonfinitely based as an ai-semiring?
\item Which finite naturally semilattice-ordered inverse semigroups are finitely based, in either of the signatures $\{+,\cdot\}$ or $\{+,\cdot,0\}$?
\end{enumerate}
\end{problem}
Parts (1) and (2) of Problem \ref{prob:4} will be significant steps toward (3), especially given that the finitely based naturally semilattice-ordered Clifford semigroups are completely classified in \cite{jac:flat}. Problem \ref{prob:4}(1) is also a particular case of Problem~\ref{prob:1}(4), as $B_0$ and $B_2$ are flat semirings.
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,350 |
R 'n' B
Whack Job: 1980s BPAMI Formanta Solo Two
An Eighties-era gem, examined in this edition of "Whack Job."
Terry Carleton
During the cold War, anything to do with Western pop culture was strictly forbidden in the U.S.S.R. That included rock and roll music and the equipment required to make it. The only way for would-be Soviet rockers to get a well-made electric guitar was through the black market. Unfortunately, the cost of an American, European or even Japanese model was equal to about six months' rent, payable in cash. Then, once you owned the contraband, you ran the risk of having it discovered by the authorities.
As a result, aspiring musicians were left with the only other option: a much cheaper Soviet-made guitar, such as the Formanta Solo Two pictured here. While the offerings of the country's guitar makers improved over the years, the early builds were atrocious. The Formanta Solo Two was made in Belarus in the mid-Eighties by BPAMI, which stands for Belarusian Production Association of Musical Instruments. That rolls off the tongue, eh?
WEIRDO FACTOR
It's Russian! BPAMI was one of a couple dozen or so independent companies in the federation that manufactured electric guitars. The existence of these Soviet-made instruments is odd enough, considering that, in 1956, just as the electric guitar was coming to prominence, Soviet premier Nikita Khrushchev denounced capitalism, telling a group of visiting Western ambassadors, "We will bury you!" (Maybe he meant to say, "We will rock you"?)
What makes these instruments even more unusual is that they have a vibe unfamiliar to Westerners. The Formanta Solo Two's body shape is tame compared to its contemporaries (look up the Tonika or Roden), but the fancy headstock, the truss-rod cover, the three-panel red sparkle pickguards and the overbuilt steel pickups with matching red sparkle inlays belie the eccentricity at the heart of Soviet electric guitar design.
PLAYABILITY & SOUND
Like so many Soviet-era guitars, it plays just rotten. The back of the neck is squared off, so using it in conventional fashion is like trying to fret a lap-steel guitar. But, as with most things, you can probably get used to it — just check out Mike Dugan's YouTube channel, drowninginguitars. The pickups are voiced nicely, are fairly musical and can be selected using the trio of black buttons that look like organ rocker switches. Engaging them reveals the guitar has a pleasant rhythm tone, as well as a trebley Tele-like sound. The vibrato is Bigsby-like, and as you've probably noticed, the output is a five-pin DIN jack, necessitating an adaptor if you plan to amplify it.
But I'll bet you're asking, "How does it sound with distortion?" I'm glad you asked, because it has a built-in fuzz tone, which is what the upper set of knobs control. The sound it produces is reminiscent of the Big Muff Pi pedal — very cool, and perfect for garage rock.
There are a lot of these guitars available online, and most can be purchased for under $200. Unfortunately, due to their weird necks, DIN output jack and weight (this particular specimen is 13 pounds), the Formanta Solo Two is unlikely to become one of your favorite players, even at budget-friendly prices.
WHY IT RULES
So if it doesn't play well, why is it worth having? Because it's an interesting historical piece and will adorn your studio wall like no other guitar. As a collector, I was happy to have the Formanta Solo Two represent yet another country in my international Whack Job posse. To learn more about Russian guitars, visit sovietguitars.com.
Got a whack job? Get in touch! Contact rtcarleton@gmail.com
terry carleton1980s BPAMI Formanta Solo Two | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,249 |
\section{Introduction}
Inclusive hadron production in collisions of quasi-real photons
can be used to study the structure of photon interactions
complementing similar studies of jet production in $\gamma\gamma$
collisions~\cite{bib-opalgg}.
The photons are radiated by the beam electrons\footnote{Positrons
are also referred to as electrons} carrying only
small negative
squared four-momenta $Q^2$. They can therefore be considered to be
quasi-real ($Q^2 \approx 0$) if the
electrons are scattered at very small angles where they
are not detected. For the ``anti-tagged'' event sample, events
are rejected if one or both scattered electrons have been detected.
The interactions of the photons can be modelled by assuming that
each photon can either interact directly or appear resolved
through its fluctuations into hadronic states.
In leading order Quantum Chromodynamics (QCD) this
model leads to three different event classes for the $\gamma\gamma$
interactions: direct, single-resolved and
double-resolved. In resolved events partons
(quarks or gluons) from the hadronic fluctuation of the photon
take part in the hard interaction.
The probability to find a parton in the photon
carrying a certain momentum fraction of the photon is parametrised
by parton density functions.
We measure differential production cross-sections
as a function of the transverse momentum and the pseudorapidity
of charged hadrons and neutral $\mbox{K}^0_{\rm S}$ mesons.
Since the distributions are fully corrected for
losses due to event and track selection cuts, the acceptance and
the resolution of the detector, they are directly comparable
to leading order Monte Carlo models
and to next-to-leading order (NLO) perturbative QCD calculations by
Binnewies, Kniehl and Kramer~\cite{bib-binnewies}.
Until now, transverse momentum distributions of charged hadrons have only been
measured for single-tagged events by TASSO~\cite{bib-tasso} and
MARK~II~\cite{bib-mark2} at an average $\langle Q^2 \rangle$
of 0.35~GeV$^2$ and 0.5~GeV$^2$, respectively.
We present the first measurement
in anti-tagged collisions of quasi-real photons. Furthermore,
the transverse momentum distributions in $\gamma\gamma$ interactions are expected to
have a harder component than in photon-proton or meson-proton
interactions due to the direct photon interactions. This will be
demonstrated by comparing our data to the photo- and hadroproduction
data measured by WA69~\cite{bib-wa69}.
At large transverse momenta (after crossing the charm threshold)
the production of $\mbox{K}^0_{\rm S}$ mesons in photon-photon collisions
is sensitive to the direct production of primary charm quarks
in addition to the production of primary strange quarks, since
the photon couples to the quark charge.
$\mbox{K}^0_{\rm S}$ production in anti-tagged $\gamma\gamma$ collisions has
previously been measured by TOPAZ~\cite{bib-topaz} and
in single-tagged events by MARK~II~\cite{bib-mark2}.
In this paper, charged hadron and $\mbox{K}^0_{\rm S}$ production are studied
using the full data sample taken in 1996 at $\mbox{e}^+\mbox{e}^-$ centre-of-mass
energies of 161 and 172~GeV corresponding to an
integrated luminosity of about 20~pb$^{-1}$.
\section{The OPAL detector}
\label{sec-dec}
A detailed description of the OPAL detector
can be found in Ref.~\cite{opaltechnicalpaper}, and
therefore only a brief account of the main features relevant
to the present analysis will be given here.
The central tracking system is located inside
a solenoidal magnet which
provides a uniform axial magnetic field of 0.435~T along the beam
axis\footnote{In the OPAL coordinate system
the $z$ axis points in the direction of the e$^-$ beam. The
polar angle $\theta$, the azimuthal angle $\phi$
and the radius $r$ denote the usual spherical coordinates.}.
The detection efficiency for charged particles is close to 100~$\%$
within the polar angle range $|\cos\theta|<0.92$.
The magnet is surrounded in the barrel region ($|\cos\theta|<0.82$)
by a lead glass electromagnetic
calorimeter (ECAL) and a hadronic sampling calorimeter (HCAL).
Outside the HCAL, the detector is surrounded by muon
chambers. There are similar layers of detectors in the
endcaps ($0.81<|\cos\theta|<0.98$).
The small angle region from 47 to 140 mrad
around the beam pipe on both sides
of the interaction point is covered by the forward calorimeters (FD)
and the region from 25 to 59 mrad by the silicon tungsten luminometers (SW).
From 1996 onwards, including the data presented in this paper,
the lower boundary of the acceptance has been increased to 33 mrad
following the installation of a low angle shield to protect the
central detector against possible synchrotron radiation.
Starting with the innermost components, the
tracking system consists of a high precision silicon
microvertex detector, a vertex
drift chamber, a large volume jet chamber with 159 layers of axial
anode wires and a set of $z$ chambers measuring the track coordinates
along the beam direction.
The transverse momenta $p_{\rm T}$ of tracks are measured with a precision
parametrised by
$\sigma_{p_{\rm T}}/p_{\rm T}=\sqrt{0.02^2+(0.0015\cdot p_{\rm T})^2}$ ($p_{\rm T}$ in GeV/$c$)
in the central region. In this paper ``transverse''
is always defined with respect to the $z$ axis.
The jet chamber also provides
measurements of the energy loss, ${ \rm d} E/ {\rm d}x$,
which are used for particle identification~\cite{opaltechnicalpaper}.
The barrel and endcap sections of the ECAL are
both constructed from lead glass blocks with a depth of
$24.6$ radiation lengths in the barrel region and more than
$22$ radiation lengths in the endcaps.
The FD consist of cylindrical lead-scintillator calorimeters with a depth of
24 radiation lengths divided azimuthally into 16 segments.
The electromagnetic energy resolution is about
$18\%/\sqrt{E}$, where $E$ is in GeV.
The SW detectors~\cite{bib-siw} consist
of 19 layers of silicon detectors and 18
layers of tungsten, corresponding to a total of 22 radiation
lengths. Each silicon layer consists of 16 wedge
shaped silicon detectors. The electromagnetic energy resolution is about
$25\%/\sqrt{E}$ ($E$ in GeV).
\section{Kinematics and Monte Carlo simulation}
The properties of the two interacting photons ($i=1,2$)
are described by their negative four-momentum transfers $Q_{i}^2$.
Each $Q_i^2$ is related to the electron
scattering angle $\theta'_i$ relative to the beam direction by
\begin{equation}
Q_i^2=-(p_i-p'_i)^2\approx 2E_i E'_i(1-\cos\theta'_i),
\label{eq-q2}
\end{equation}
where $p_i$ and $p'_i$ are the four-momenta of the beam
electrons and the scattered electrons, respectively,
and $E_i$ and $E'_i$ are their energies.
Events with detected scattered electrons (single-tagged or
double-tagged events) are excluded from
the analysis. This anti-tagging condition
is met when the scattering angle $\theta'$ of
the electron is less than 33~mrad between the beam axis and
the inner edge of the SW detector.
It defines an effective upper
limit, $Q^2_{\rm max}$, on the values of $Q_{i}^2$ for both photons.
The hadronic final state is described by its invariant mass $W$.
The spectrum of photons with an energy fraction $y$ of
the electron beam may be obtained
by the Equivalent Photon Approximation (EPA)~\cite{bib-wwa}:
\begin{equation}
f_{\gamma/{\rm e}}(y)=
\frac{\alpha}{2\pi}\left(\frac{1+(1-y)^2}{y}
\log\frac{Q^2_{\rm max}}{Q^2_{\rm min}}
-2m^2_{\rm e} y\left( \frac{1}{Q^2_{\rm min}}-\frac{1}{Q^2_{\rm max}} \right)\right),
\end{equation}
with $\alpha$ being the electromagnetic coupling constant.
The minimum kinematically allowed negative squared four-momentum
transfer $Q^2_{\rm min}$ is determined by the electron mass $m_{\rm e}$:
\begin{equation}
Q^2_{\rm min}=\frac{m_{\rm e}^2y^2}{1-y}.
\end{equation}
The Monte Carlo generators
PYTHIA~\cite{bib-pythia} and PHOJET~\cite{bib-phojet} have been
used to simulate quasi-real photon-photon interactions.
More details about the event generation can be found in Ref.~\cite{bib-opalgg}.
All possible hard interactions relevant to photon-photon
interactions are included. The fragmentation is handled by
JETSET~\cite{bib-pythia}.
PYTHIA uses the SaS-1D parametrisation~\cite{bib-sas1d} for the
parton densities of the photon and PHOJET uses the GRV
parametrisation~\cite{bib-grv}. An approximation is used for
the processes with primary charm quarks, i.e. where
the charm quark is produced in the hard interaction.
These processes are simulated
using the matrix elements for light quarks.
Subsequently the charm quarks are put on the mass-shell.
\section{Event selection and background}
\label{sec-evsel}
The production of charged hadrons and $\mbox{K}^0_{\rm S}$ mesons was studied
using the data taken at $\mbox{e}^+\mbox{e}^-$ centre-of-mass energies, $\sqrt{s}_{\rm ee}$,
of 161 and 172~GeV with an integrated luminosity of about 9.9~pb$^{-1}$
and 10.0~pb$^{-1}$, respectively.
Photon-photon events are selected with the following set of cuts:
\begin{itemize}
\item
The sum of all energy deposits
in the ECAL and the HCAL has to be less than 45 GeV.
\item
The visible invariant hadronic mass, $W_{\rm ECAL}$, calculated
from the position and the energy of the clusters measured
in the ECAL, has to be greater than 3 GeV.
\item
The missing transverse energy of the event
measured in the ECAL and the forward
calorimeters has to be less than 5 GeV.
\item
At least 3 tracks must have been found in the tracking chambers.
A track is required to have a minimum transverse momentum
of 120 MeV/$c$,
at least 20 hits in the central jet chamber,
and the innermost hit of the track must be within a radius of 60 cm
with respect to the $z$ axis.
The distance of the point of closest approach to the origin
in the r$\phi$ plane must be
less than 30 cm in the $z$ direction and less than
2 cm in the $r\phi$~plane.
Tracks with a momentum error larger than the momentum itself
are rejected if they have fewer than 80 hits.
The number of measured
hits in the jet chamber must be more than half of the number of possible hits.
The number of possible hits
is calculated from the polar angle $\cos\theta$ of the track, assuming
that the track has no curvature.
\item
To remove events with scattered electrons in the FD or SW,
the total energy measured in the FD has to be less than
50 GeV and the total energy measured in the SW
has to be less than 35 GeV.
These cuts also reduce contamination from multihadronic events with
their thrust axis close to the beam direction.
\item
To reduce the background due to beam-gas and beam-wall interactions,
$|\langle z_{0}\rangle|$ must be smaller than 10 cm where
$\langle z_{0}\rangle$ is the error-weighted
average of the track's $z$ coordinates at the point of closest approach to
the origin in the $r\phi$~plane.
Beam-wall events with a vertex in the beam-pipe are rejected
by requiring the radial position of the primary vertex in the $r\phi$~plane
to be less than 3~cm.
\end{itemize}
After all cuts 56732 events remain.
All relevant background processes
apart from beam-gas and beam-wall events
were studied using Monte Carlo
generators. Multihadronic events ($\mbox{e}^+\mbox{e}^-\rightarrow \mbox{q}\overline{\mbox{q}}(\gamma)$) were
simulated with PYTHIA 5.722~\cite{bib-pythia}.
KORALZ 4.02~\cite{bib-koralz} was used
to generate the process $\mbox{e}^+\mbox{e}^-\rightarrow\tau^+\tau^-(\gamma)$ and
BHWIDE~\cite{bib-bhwide} to generate the Bhabha process
$\mbox{e}^+\mbox{e}^-\rightarrow\mbox{e}^+\mbox{e}^-(\gamma)$.
Processes with four fermions in the final state, including
W pair production, were simulated with grc4f~\cite{bib-grc4f},
EXCALIBUR~\cite{bib-excal},
VERMASEREN~\cite{bib-vermaseren} and FERMISV~\cite{bib-fermisv}.
All signal and background Monte Carlo samples were generated
with a full simulation of the OPAL detector~\cite{bib-gopal}.
They were analysed using the same reconstruction algorithms as for the data.
The main background processes are multi-hadronic $\mbox{e}^+\mbox{e}^-$ annihilation events and
$\mbox{e}^+\mbox{e}^-\rightarrow\mbox{e}^+\mbox{e}^-\tau^+\tau^-$ events.
Other background processes are found to be negligible.
The multihadronic background is mainly reduced by the cut on the sum of the
energy measured by the HCAL and the ECAL and by the cut on the
missing transverse energy.
The background from all these processes after the selection cuts
amounts to less than $1\%$.
The cut on the energy in SW and FD
rejects photon-photon events with electrons scattered
at angles $\theta'$ larger than 33 mrad and with an energy greater than 35~GeV
in the SW or greater than 50~GeV in the FD.
From the Monte Carlo, the rate of events
with $\theta'>33$~mrad and energies less than 50 GeV
is estimated to be negligible. The effective anti-tagging
condition is therefore $\theta'<33$~mrad.
\section{Analysis}
\subsection{Correction procedure}
The measured transverse momentum and pseudorapidity
distributions of the charged hadrons and the
$\mbox{K}^0_{\rm S}$ mesons have to be corrected for
losses due to the event and track selection cuts, for the acceptance and
for the resolution of the detector. This is done with
Monte Carlo events which were generated with PYTHIA 5.722 and
PHOJET 1.05c. The data are corrected
by multiplying the experimental distribution, e.g.~of
the transverse momentum $p_{\rm T}$, with correction
factors which are calculated as the bin-by-bin
ratio of the generated and the reconstructed Monte Carlo
distributions:
\begin{equation}
\left(\frac{{\rm d}\sigma}{{\rm d} p_{\rm T}}
\right)_{\rm corrected}=
\frac{\left(\frac{{\rm d}\sigma}{{\rm d} p_{\rm T}}
\right)^{\rm MC}_{\rm generated~~~~}}{
\left(\frac{{\rm d}\sigma}{{\rm d} p_{\rm T}}
\right)^{\rm MC}_{\rm reconstructed}}
\left(\frac{{\rm d}\sigma}{{\rm d} p_{\rm T}}
\right)_{\rm measured}.
\label{eq1}
\end{equation}
As a correction factor the mean value from PYTHIA and PHOJET
is used.
The distributions of the pseudorapidity $\eta=-\ln\tan(\theta/2)$ are
corrected in the same way.
This method only yields reliable results if the migration
between bins due to the finite resolution is
small. The bins of the $p_{\rm T}$ and $|\eta|$ distributions have therefore
been chosen to be significantly larger than the resolution expected from
the Monte Carlo simulation.
The average transverse
momentum, $\langle p_{\rm T} \rangle$, and the average pseudorapidity,
$\langle |\eta| \rangle$, in each bin
is calculated directly from the data, since detector corrections
are small compared to the statistical errors.
The visible invariant mass, $W_{\rm vis}$, is determined from all tracks
and calorimeter clusters, including the FD and
the SW detectors. An algorithm is applied
to avoid double-counting of particle momenta
in the central tracking system and the calorimeters~\cite{bib-opalgg}.
All distributions are shown for $10<W<125$~GeV where
$W$ is the hadronic invariant mass corrected for
detector effects.
To minimize migration effects when using Eq.~\ref{eq1}
for the detector correction, the bins in
$W$ must be larger than the experimental resolution
and the average reconstructed hadronic invariant mass,
$\langle W_{\rm rec}\rangle$, should be approximately equal to the
average generated hadronic invariant mass, $\langle W_{\rm gen}\rangle$.
The average $\langle W_{\rm vis}\rangle $ and the resolution
on $W_{\rm vis}$ as a function of the generated hadronic invariant mass
$W_{\rm gen}$ are therefore shown in Fig.~\ref{fig-wcorr}a, where
the vertical bars show the standard deviation (resolution) in each bin.
The average $\langle W_{\rm gen}\rangle $ as a function of
$W_{\rm vis}$ is plotted in
Fig.~\ref{fig-wcorr}b, where the vertical bars give the
error on the mean. This plot is used to determine
a correction function so that
$\langle W_{\rm gen}\rangle/W_{\rm rec} \approx 1$.
The value of $W_{\rm vis}$ measured in the detector is on average
significantly smaller than $W_{\rm gen}$.
The relation between $W_{\rm gen}$ and
$W_{\rm vis}$ shown in Fig.~\ref{fig-wcorr}b is almost
independent of the beam energy and the Monte Carlo generator used.
A single polynomial is
therefore used to calculate $W_{\rm rec}$ from $W_{\rm vis}$.
The polynomial is obtained from the fit shown
in Fig.~\ref{fig-wcorr}b. It is applied to the data and the Monte Carlo.
The efficiency to reconstruct photon-photon events in the detector,
estimated by the Monte Carlo,
is greater than $20\%$ for $W_{\rm gen}>10$~GeV and
greater than $60\%$ for $W_{\rm gen}>50$~GeV.
The trigger efficiency is defined as the ratio of the number
of selected and triggered events to the number of selected events.
It was studied using data samples which
were obtained using nearly independent sets of triggers.
On average the trigger efficiency for the lowest $W$ range, $10<W<30$~GeV,
is greater than $97\%$ and it approaches $100\%$ for larger values of $W$.
Only lower limits on the trigger efficiency can be determined with
this method and therefore no correction factor is applied.
\subsection{Charged hadron production}
For the charged hadron analysis
only particles with a proper lifetime $\tau>0.3$~ns
are used to define the primary charged hadronic multiplicity
in the Monte Carlo.
The primary charged hadrons originate either directly from the
primary interaction or from the decay of particles with a lifetime
$\tau<0.3$~ns including $\Lambda$ and $\mbox{K}^0_{\rm S}$ decay products.
The track selection criteria are defined as in Section~\ref{sec-evsel}.
In order to avoid regions where the detector has little or no
acceptance, all measurements of charged hadrons
were restricted to the range $|\eta|<1.5~(|\cos\theta|\sleq~0.9)$.
In this range, the resolution on $p_{\rm T}$ is given by
$\sigma_{p_{\rm T}}/p_{\rm T} \approx 0.02$ (see Section~\ref{sec-dec}) and
the resolution on $\eta$ by $\sigma_{\eta}\approx 0.02$.
For the $p_{\rm T}$ distribution in the range $10<W<125$~GeV
the correction factors as defined in Eq.~\ref{eq1} decrease from about
1.7 for $p_{\rm T}>120$~MeV/$c$ to about $1.1-1.4$ for $p_{\rm T}>2$~GeV/$c$.
The correction factor of about 1.6 for the $\eta$ distribution is nearly
constant for $|\eta|<1.5$.
The PHOJET and PYTHIA correction factors differ by about $3-10~\%$.
\subsection{K\boldmath $^0_{\rm S}$ production}
The $\mbox{K}^0_{\rm S}$ mesons are reconstructed using
the decay channel $\mbox{K}^0_{\rm S}\rightarrow \pi^+\pi^-$ which has a branching ratio
of about 69$\%$~\cite{bib-pdg}. The reconstruction procedure is similar to
the procedure described in Ref.~\cite{bib-z01995}. It has been
optimised to increase the efficiency for finding $\mbox{K}^0_{\rm S}$ mesons in photon-photon
events.
Tracks of opposite charge are paired together.
In addition to other quality cuts
the tracks must have a minimum transverse momentum of 120 MeV/$c$
and at least 20 jet chamber hits.
The intersection of the tracks in the $r\phi$~plane is
considered as a secondary
vertex candidate if it satisfies the following criteria:
\begin{itemize}
\item the radial distance between primary vertex and the intersection point
must be greater than 0.5 cm and less than 150 cm.
For events with at least 6 tracks the primary
vertex is fitted and for events with less than 6 tracks
the beam spot reconstructed from tracks collected from many consecutive
$\mbox{e}^+\mbox{e}^-$ events during a LEP fill is taken as primary
vertex~\cite{bib-prim}.
\item the difference between the radial coordinate of the secondary vertex and
the radial coordinate of the first jet-chamber hit associated with either of
the two tracks has to be less than 10 cm;
\item the radial coordinate of the tracks
at the point of closest approach to
the primary vertex has to be greater than 0.2~cm;
\item the angle between the direction of flight from primary to
secondary vertex and the combined momentum vector
of the two tracks at the intersection point
has to be less than~5$^{\circ}$.
\end{itemize}
In addition, a fit was performed for track pairs passing all
these cuts, constraining them to originate from a common vertex.
A correction procedure was used to compensate for the energy
loss of the pions in the inactive material of the detector.
All secondary vertices satisfying
$|M(\pi^+\pi^-)-0.4977\mbox{~GeV}/c^2|< 0.02$~GeV/$c^2$
are considered to be $\mbox{K}^0_{\rm S}$ decay vertices, where
the mass $M$ is calculated assuming that both tracks are pions.
Finally, the residual background is reduced by requiring
at least 20 ${\rm d}E/{\rm d}x$ hits.
The two tracks are identified as pions if the ${\mathrm d}E/{\mathrm d}x$
probability for the pion hypothesis, that is the probability that
the specific ionisation energy loss in the
jet chamber $({\rm d}E/{\rm d}x)$ is compatible with that
expected for a pion, exceeds $5\%$.
In Fig.~\ref{fig-mpp} the $\pi^+\pi^-$ invariant mass $M$
is shown for all identified secondary vertices in the selected
events before and after applying the ${ \rm d} E/ {\rm d}x$
cuts.
After all cuts the reconstruction efficiency for $\mbox{K}^0_{\rm S}\rightarrow\pi^+\pi^-$
decays is about $35.5\%$ and the purity is about $95.5\%$
for $p_{\rm T}(\mbox{K}^0_{\rm S})>1$~GeV/$c$, $|\eta(\mbox{K}^0_{\rm S})|<1.5$ and $10<W<125$~GeV.
\section{Systematic errors}
The following systematic errors, common to the charged
hadron and $\mbox{K}^0_{\rm S}$ measurements, are taken
into account:
\begin{itemize}
\item
The correction factors are obtained using PHOJET and PYTHIA,
separately.
The resulting distributions are averaged to get the final result.
The differences between the two distributions are used to
define the systematic error.
\item The lower limit on the
trigger efficiency is taken into account by an additional
systematic error of $3\%$ on the cross-section in the range $10<W<30$~GeV.
\item
Systematic errors due to the modelling of the detector resolution
for the measurement of tracks
were found to be negligibly small in comparison to the other errors.
The systematic error due to the uncertainty in the energy scale
of the electromagnetic calorimeter was estimated by varying
the reconstructed ECAL energy in the Monte Carlo by $\pm 5\%$.
\item The limited statistics of the Monte Carlo samples,
especially at large transverse momenta $p_{\rm T}$, is also included
in the systematic error.
\item The systematic error of the luminosity measurement is
negligible compared to the other systematic errors.
\end{itemize}
The systematic error of the Monte Carlo modelling and of
the ECAL energy scale and, for the low $W$ region, the
error from the trigger efficiency contribute about equally
to the total systematic error.
In the $\mbox{K}^0_{\rm S}$ reconstruction additional
systematic errors were studied by varying the parameters
of the secondary vertex finder and the ${ \rm d} E/ {\rm d}x$ cuts.
The full difference between the results is used to estimate
the contribution to the total systematic error
from the $\mbox{K}^0_{\rm S}$ reconstruction, the Monte Carlo model dependence and
the ECAL energy scale.
The systematic error affecting the $\mbox{K}^0_{\rm S}$ reconstruction
and the error from comparing the PHOJET and PYTHIA
correction factors are of similar magnitude.
The total systematic error was obtained by adding all
systematic errors in quadrature. The total systematic
errors are highly correlated from bin to bin.
\section{Results}
\label{sec-results}
The differential inclusive cross-section $\dspt$
for charged hadrons in the region $|\eta|<1.5$ is
shown in Fig.~\ref{fig-pt1} for different corrected $W$ ranges
together with the statistical and systematical errors.
The corrected cross-sections are given in
Tables~\ref{tab-pt1a} and \ref{tab-pt1b}.
The measured differential cross-sections are compared to NLO
calculations by Binnewies, Kniehl and Kramer~\cite{bib-binnewies}.
The cross-sections are calculated using the QCD partonic cross-sections
to NLO for direct, single- and double-resolved processes.
The hadronic cross-section is a convolution of the Weizs\"acker-Williams
effective photon distribution, the parton distribution functions and
the fragmentation functions of Ref.~\cite{bib-bkk} which are obtained
from a fit to $\mbox{e}^+\mbox{e}^-$ data from TPC and ALEPH.
The NLO GRV parametrisation of the parton densities of the
photon~\cite{bib-grv} is
used with $\Lambda^{(5)}_{\overline{\rm MS}}=131$ MeV
and $m_{\rm c}=1.5$~GeV$/c^2$.
The renormalization and factorization scales
in the calculation are set equal to $\xip_{\rm T}$ with $\xi=1$.
The change in slope around $p_{\rm T}=3$~GeV/$c$ in the
NLO calculation is due to the charm threshold, below which
the charm distribution in the resolved photon and the charm
fragmentation functions are set to zero.
The cross-section calculation was
repeated for the kinematic conditions of the data presented here
at an average $\mbox{e}^+\mbox{e}^-$ centre-of-mass energy
$\sqrt{s}_{\rm ee}=166.5$~GeV and for scattering angles $\theta'<33$~mrad.
For the differential cross-section $\dspt$
a minimum $p_{\rm T}$ of 1~GeV/$c$ is required to ensure
the validity of the perturbative QCD calculation.
For the same reason the differential cross-section $\dseta$
is restricted to the region $p_{\rm T}>1.5$~GeV/$c$.
The scale dependence of the NLO calculation was studied
by setting $\xi=0.5$ and 2. This leads to a variation of the cross-section
of about $30\%$ at $p_{\rm T}=1$~GeV/$c$ and of about $10\%$ for $p_{\rm T}>5$~GeV/$c$.
The NLO calculations lie significantly below the data
for $W<30$~GeV for $\dspt$ and $\dseta$.
The agreement with the data improves in the higher $W$ bins.
The NLO calculation is shown separately for double-resolved,
single-resolved and direct
interactions. At large $p_{\rm T}$ the direct interactions dominate.
It should be noted that these classifications are scale dependent in NLO.
The $p_{\rm T}$ distribution for $10<W<30$~GeV
is compared in Fig.~\ref{fig-wa69} to $p_{\rm T}$ distributions
in $\gamma$p and hp (h$=\pi,$K) interactions measured by the experiment
WA69~\cite{bib-wa69}. The hp data are
weighted by WA69 in such a way that they contain $60\%$ $\pi$p and
$40\%$ Kp data to match the expected mixture of non-strange
and strange quarks in the photon beam of the $\gamma$p data.
The WA69 data is normalised to the $\gamma\gamma$ data in the low
$p_{\rm T}$ region
at $p_{\rm T}\approx 200$~MeV/$c$ using the same factor for the hp and the
$\gamma$p data.
The $p_{\rm T}$ distribution of WA69 has been measured
in the Feynman-$x$ range $0.0<x_{\rm F}<1.0$. The hadronic invariant
mass of the hp data is $W=16$~GeV and the average $\langle W\rangle$
is of similar
size for the $\gamma$p data.
In the $\gamma\gamma$ Monte Carlo the average $\langle W \rangle$ is about
17~GeV in the range $10<W<30$~GeV,
i.e.~the average values of $W$ in the different data samples are
approximately the same. Whereas only a small increase is observed
in the $\gamma$p data compared to the $\pi$p and K$\pi$ data at large $p_{\rm T}$,
there is a significant increase of the relative rate in the range
$p_{\rm T}>2$~GeV/$c$ for $\gamma\gamma$ interactions due to the
direct process.
A clear deviation is seen at large $p_{\rm T}$ from the exponential
fall-off expected for purely hadronic interactions.
The differential cross-section $\dseta$ is compared to
the predictions of the Monte Carlo generators PHOJET 1.10
and PYTHIA 5.722 in Fig.~\ref{fig-eta1} taking into account
the anti-tagging condition $\theta'<33$~mrad.
In PHOJET the $Q^2$ suppression of the total $\gamma\gamma$ cross-section
is parametrised using Generalised Vector Meson Dominance (GVMD) and
a model for the change of soft hadron production and diffraction with
increasing photon virtuality $Q^2$ is also included.
The photon-photon mode of PYTHIA only simulates the interactions
of real photons with $Q^2=0$. The virtuality of the photons defined
by $Q^2$ enters only through the equivalent photon
approximation in the generation of the photon
energy spectrum, but the electrons are scattered at zero angle.
This model is not expected to be correct for larger values of $Q^2$.
We have therefore simulated events with $Q^2<1$~GeV$^2$ with the
photon-photon mode of PYTHIA and events with $Q^2>1$~GeV$^2$ and
$\theta'<33$~mrad with the electron-photon mode of PYTHIA.
The differential cross-section $\dseta$ shown in Fig.~\ref{fig-eta1}
is nearly independent of $|\eta|$ in the measured range.
The $|\eta|$ distribution is reasonably well described by
PYTHIA and PHOJET for $p_{\rm T}>120$~MeV$/c$,
apart from the high $W$ region where PHOJET appears to be below the data.
For transverse momenta $p_{\rm T}>1.5$~GeV$/c$ (Fig.~\ref{fig-eta2})
both Monte Carlo models underestimate the data significantly.
The same behaviour is observed for the NLO calculation at low $W$, but
the agreement of the NLO calculation with the data improves in the
high $W$ bins.
The corrected cross-sections are given in
Tables~\ref{tab-eta2} and \ref{tab-eta1}.
The differential inclusive cross-sections $\dspt$
and $\dseta$ have been measured for $\mbox{K}^0_{\rm S}$ mesons
with $p_{\rm T}(\mbox{K}^0_{\rm S})>1$~GeV$/c$ and $|\eta(\mbox{K}^0_{\rm S})|<1.5$.
The $p_{\rm T}$ and $\eta$ dependent cross-sections
are presented in the $W$ range $10~<~W~<~125$~GeV
(Fig.~\ref{fig-ks1} and Tables~\ref{tab-ks1}--\ref{tab-ks2}).
In addition the $p_{\rm T}$ distribution is shown for
two separate $W$ ranges (Fig.~\ref{fig-ks2} and Table~\ref{tab-ks3}).
The results are compared to PHOJET and PYTHIA.
Based on the PYTHIA simulation using SaS-1D, about half of the
$\mbox{K}^0_{\rm S}$ are expected to be produced from charm quarks
at large $p_{\rm T}$ where direct processes are dominant.
Both Monte Carlo models significantly underestimate the $\mbox{K}^0_{\rm S}$ production
cross-section in the low $p_{\rm T}$ region where most $\mbox{K}^0_{\rm S}$ are
expected to originate from primary strange quarks.
The distributions are reasonably well
described by the NLO calculations
which use the $\mbox{K}^0_{\rm S}$ fragmentation function fitted
to MARK II~\cite{bib-markk0} and ALEPH~\cite{bib-alephk0}
data in Ref.~\cite{bib-fk0}.
The change in slope between $p_{\rm T}=2$ and $p_{\rm T}=3$~GeV/$c$ in the
NLO calculation is again due to the charm threshold.
The variation of the calculated cross-section
as a function of $p_{\rm T}$ for different choices of scales, $\xi=0.5$ and $2$,
is largest around the charm threshold,
about $30-40\%$, and $10-20\%$ elsewhere.
\section{Conclusions}
We present measurements of differential cross-sections
as a function of transverse momentum and pseudorapidity
for charged hadrons and $\mbox{K}^0_{\rm S}$ mesons produced in photon-photon
collisions at LEP. The data were taken at $\mbox{e}^+\mbox{e}^-$ centre-of-mass
energies of 161 and 172~GeV.
The differential cross-section $\dspt$ for charged hadrons
is compared to NLO calculations.
In the range $10<W<30$~GeV more charged hadrons are found
at large $p_{\rm T}$ than predicted.
Good agreement between the NLO calculation and the data is
found in the highest $W$ range, $55<W<125$~GeV.
The Monte Carlo models PYTHIA and PHOJET both underestimate
the cross-section for tracks with $p_{\rm T}>1.5$~GeV and $|\eta|<1.5$
in all $W$ ranges.
The shape of the differential cross-section $\dseta$ is
well reproduced by the NLO calculations and the Monte Carlo models.
A comparison of the $p_{\rm T}$ distributions of the $\gamma\gamma$ data
to $p_{\rm T}$ distributions measured in $\gamma$p and ($\pi$,K)p processes
at similar invariant masses shows the relative increase of hard interactions
in $\gamma\gamma$ processes due to the direct component.
The transverse momentum and pseudorapidity distributions
of the $\mbox{K}^0_{\rm S}$ mesons are reasonably well reproduced by the NLO calculations,
but they are significantly underestimated by the Monte Carlo models
PHOJET and PYTHIA.
\medskip
\bigskip\bigskip\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,536 |
Q: Chrome won't redirect back to URL after Authentication handling For at least a couple of years, I've been using code similar to this in my MVC solutions...
[Authorize]
public class HomeController : Controller
{
[HttpGet]
public ActionResult Index()
{
..........
Then in my Authentication code
myAuthenticationProperties = new Microsoft.Owin.Security.AuthenticationProperties();
myAuthenticationProperties.AllowRefresh = true;
myAuthenticationProperties.ExpiresUtc = DateTime.UtcNow.AddMinutes(60);
myAuthenticationManager.SignIn(myAuthenticationProperties, myClaimsIdentity);
return RedirectToAction("Index", "Home");
And in my Startup..
public void Configuration(IAppBuilder app)
{
CookieAuthenticationOptions myAuthOptions = new CookieAuthenticationOptions();
myAuthOptions.AuthenticationType = "ApplicationCookie";
myAuthOptions.CookieHttpOnly = true;
myAuthOptions.SlidingExpiration = true;
myAuthOptions.LoginPath = new PathString("/Authentication/LogIn");
//This is what was added for the Owin cookie "fix"
myAuthOptions.CookieSameSite = SameSiteMode.Strict;
myAuthOptions.CookieSecure = CookieSecureOption.Always;
app.UseCookieAuthentication(myAuthOptions);
}
And life has been dandy... until now. I've been chasing my tail all over the place trying to figure out why when I attempt to log in, sometimes it works, and other times it just hangs. Using some Debug Messages, I found that my Authentication process finishes, but when the RedirectToAction occurs, nothing happens.. just hangs.
Then I had a break through, I tried using IE and Edge and it seems to work every time. Only Chrome hangs and it does it at least 75% of the Time if not more.
** UPDATE **
I have used both Fiddler and Chrome's Debugging (Console and Network tabs) and when the RedirectToAction occurs, as far as the website is concerned it is done. However nothing, and I mean nothing, comes back on the network to my client (according to Fiddler and Chrome's Networking).
Yet, if I manually change the url to go back home, Chrome is happy, I'm now authenticated and my [Authorize] now allows the controller to load.
I have looked into the new Chrome cookie thing, and although the "fix" seems to be as clear as mud, I was able to find someone who used code to force the SameSite cookie to report something other than LAX. I implemented that, actually having it set to "Strict" and still.... Chrome Hangs.
** The Band-Aid **
I don't know how much time this is going to buy me, but I have kludged the problem by using a Javascript timer that when the user clicks the submit button, the timer starts, waits 6 seconds, and then redirects back to the Home/Index.
If the issue isn't there (IE, Edge) the Client redirects automatically before the timer gets a chance to take hold. If they are using Chrome and it decides to hang, 6 seconds later it will behave as if their browser is just slow and will also take them to the correct place.
** Fixed (maybe) **
So even though no network traffic can be seen coming back to the client, I ended up (in addition to my band-aid above), implementing some additional changes so now both the Owin and Asp.net cookies are reporting Secure and sameSite = Strict. This seems to make a difference with my problem, and in cases where it still wants to hang, my Timed re-direct finishes off the problem.
For those that may experience this oddity as well, the gist of the Cookie fix is this...
*
*Update your Owin packages to make sure you are using version 4.1
*Adjust your CookieAuthenticationOptions in your Startup.cs to the items I added above to make the Owin cookie compliant.
*Update the following in your Web.config to make your Asp.net cookie compliant
<system.web>
<sessionState cookieSameSite="Strict" />
<httpCookies requireSSL="true" />
</system.web>
Doing those 3 things, (along with running your project under SSL) will result in Chrome reporting both cookies as Secure and Strict.
A: Google made a change to how Chrome handles cookies without the SameSite attribute. Before, Chrome treated not having the SameSite attribute set on a cookie as the same as having SameSite=None, which meant the browser would accept all cookies. Now, they're treating it as having SameSite=Lax, which will only accept cookies from the same domain. To get the same effect as the old method, the attribute must be set as SameSite=None; Secure.
I can't tell if this is what's impacting you, but if this is the case you'll see an error in the Chrome console.
The official release was February 4th IIRC, but they're doing a phased rollout to judge an issues being caused.
Some resources:
Microsoft ASP.NET blog about the upcoming changes / Archived copy
Chromium Blog / Archived Copy
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,337 |
Q: How to use REGEX to capitalise the first letter of each word in a sentence? I would like to use regular expressions (REGEX) to capitalise the first letter of each word in a sentence.
I have achieved the same result in programming languages, but it seems that using regular expressions would be more concise.
A: Example using sed command.
~$ echo "foo bar" | sed 's/^\(.\)/\U\1/'
Where:
*
*the ^ represents the start of a line.
*. matches any character.
*\U converts to uppercase.
*\( ... \) specifies a section to be referenced later (as \1 in this case).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,758 |
\section{Introduction}
For the last decades, there has been an increasing number of studies
in travelling wave fronts for delayed diffusion equations, and
several methods
to prove their existence have been developed.
In this paper, we are concerned with the existence and positiveness
of travelling waves connecting two equilibria, for a class of
$N$-dimensional systems of reaction-diffusion equations with
distributed delay in the reaction terms, of the form
\begin{equation}\label{e1.1}
{{\p u}\over {\p t}}(t,x)=\Delta u(t,x)+f(u_t(\cdot ,x)),\quad t\in
\mathbb{R},\ x\in\mathbb{R}^p.
\end{equation}
Here, $f:{\cal C}:=C([-\tau ,0];\mathbb{R}^N)\to \mathbb{R}^N$ is
continuous, ${\cal C}$ is equipped with the norm
$\|\var\|_\infty=\sup_{\th\in [-\tau,0]} |\var(\th)|$, for some
fixed norm $|\cdot|$ in $\mathbb{R}^N$, and $\tau >0$. As usual,
$u_t(\cdot ,x)$ denotes the restriction of a solution $u(t,x)$ to
the time interval $[t-\tau,t]$, i.e., $u_t(\th,x)=u(t+\th,x)$ for
$-\tau\le \th \le 0, x\in \mathbb{R}^p$. For simplicity, we consider
all the diffusion coefficients equal to 1 in (\ref{e1.1}), but all
our results apply to the more general case of the diffusion term
given by $D\Delta u(t,x)$, where $D=diag \, (d_1,\dots,d_N)$ with
$d_i>0$.
We are mostly interested in situations where (1.1) represents a
population dynamics model, or another biological model. Typically,
we want to obtain conditions for the existence of a travelling front
connecting two steady-states, zero and a positive equilibrium
$K\in\mathbb{R}^N$. Due to the biological interpretation of the
model, only non-negative solutions are meaningful, therefore we look
for {\it positive} travelling wave solutions, connecting 0 to $K$ as
$t$ goes from $-\infty$ to $\infty$.
With the method presented here, such positive travelling waves are
obtained for large wave speeds, as perturbations of a positive
heteroclinic solution for the corresponding functional differential
equation (FDE) without diffusion,
\begin{equation}\label{e1.2}
u'(t)=f(u_t),\quad t\in \mathbb{R}
\end{equation}
(where $u_t\in {\cal C}$ denotes the function $u_t(\th)=u(t+\th)$
for $\th \in [-\tau,0]$), whose existence we shall prove under some
requirements on $f$. This idea is not original, and has been
exploited in the literature (see e.g. \cite{FHW, FT}). When compared
with \cite{FHW}, in the present paper the major novelty is that we
give conditions for the travelling waves to be {\it positive}. We
also note that \cite{FHW} considers delayed reaction-diffusion
equations with a global space interaction, a situation not
considered here, for the sake of simplicity. Our results can however
be extended easily, to take into account non-local effects. On the
other hand, \cite{FT} deals with {\it scalar} reaction-diffusion
equations with one single {\it discrete} delay of the form
\begin{equation}\label{e1.3}
{{\p u}\over {\p t}}(t,x)=d{{\p^2 u}\over {\p x^2}}(t,x)+f(u(t,x),
u(t-\tau,x)),
\end{equation}
where $f(u(t,x), u(t-\tau,x)) =- u(t,x)+g(u(t-\tau,x))$ and
$g:[0,\infty)\to [0,\infty)$ is $C^2$-smooth, $g(0)=0, g(K)=K$ for
some $K>0$, and $g'(0)>1$. Assuming that the two equilibria $0$ and
$K$ are hyperbolic, under some further assumptions the existence
of positive and in general non-monotone travelling wavefronts
connecting 0 to $K$ was established in \cite{FT}.
Recently, several techniques have been developed to prove the
existence of travelling wave fronts for delayed diffusion
equations. They are often based on the application of a fixed point
theorem in an adequate Banach space, which requires a {\it
quasi-monotonicity} condition, either for the original equation
(\ref{e1.3}) \cite{M,WZ} or, more recently, for some auxiliary
equations \cite{MA}. These methods are usually combined with a
monotonic iteration scheme, associated with the construction of a
pair of upper and lower solutions. See \cite{M,MA,WZ} and references
therein. We emphasize that our method applies to systems
(\ref{e1.1}) with non-monotone nonlinearities, in the sense that we
do not impose on $f$ any type of quasi-monotonicity condition, as
defined in \cite{HS,WZ}.
Before introducing our hypotheses, we set some standard notation.
For $d=(d_1,\dots ,d_N)\in\mathbb{R}^N$, we say that $d>0$
(respectively $d\ge 0$) if $d_i>0$ (respectively $d_i\ge 0$) for
$i=1,\dots,N$. In ${\cal C}$, we consider the partial order
$\phi\ge \psi$ if and only if $\phi(\th)- \psi(\th)\ge 0$ for
$\th\in [-\tau,0]$; in a similar way, $\phi >\psi$ if $\phi(\th)-
\psi(\th)> 0$ for $\th\in [-\tau,0]$. As usual, ${\cal C}_+$ denotes
the positive cone $C([-\tau,0];[0,\infty)^N)$.
\med For $f$, the following hypotheses will be considered:
\begin{description}
\item [{(H1)}] $f(0)=f(K)=0$, where $K$ is some positive vector;
\item [{(H2)}] (i) $f$ takes bounded sets of ${\cal C}$ into bounded sets of $\mathbb{R}^N$ and is $C^2$-smooth; furthermore,
(ii) for all $M>0$ there is $\be >0$ such that $f_i(\var)+\be
\var_i(0)\ge 0,$ $ i=1,\dots,N$, for all $\var\in {\cal C}$ with
$0\le \var\le M$;
\item [{(H3)}] for Eq. (\ref{e1.2}), the equilibrium $u=K$
is locally asymptotically stable and
globally attractive in the set of solutions of (1.2) with initial conditions $\var\in {\cal C}_+,\var(0)> 0$;
\item [{(H4)}] for Eq. (\ref{e1.2}), its linearized equation about
the equilibrium 0 has a real characteristic root $\la_0>0$, which is
simple and dominant (i.e., $\Re\, z<\la_0$ for all other
characteristic roots $z$); moreover,
there is a characteristic eigenvector ${\bf v}>0$ associated with $\la_0$.
\end{description}
We summarize the main results in this paper as follows. In Section
2, we assume (H1)-(H4) and establish the existence of a positive
heteroclinic solution $u^*(t)$ to (\ref{e1.2}), with
$u^*(-\infty)=0, u^*(\infty)=K$, and asymptotic behaviour
$O(e^{\la_0 t})$ at $-\infty$. In Section 3, for large wave speeds
we prove the existence of travelling wave solutions for
(\ref{e1.1}), connecting 0 to $K$. The profiles of these waves are
obtained as perturbations of $u^*(t)$ via a contraction principle
argument. For this, we generalize the procedure in \cite{FHW}, and
give an abstract formulation of the wave profiles as solutions of an
operational equation, acting in suitable Banach spaces, which
incorporate a desirable exponential decay $O(e^{\mu t})$ at
$-\infty$, $0\le \mu<\la_0$. Some nice results of Hale and Lin
\cite{HL} on the index of some associated Fredholm operators are
used, and a Liapunov-Schmidt reduction effected, to set up the right
framework for the application of a contraction principle. As
mentioned above, an existence result of travelling waves connecting
two hyperbolic equilibria was already obtained in \cite{FHW}, for a
class of reaction-diffusion equations with global response, but for
such waves no exponential decay at $-\infty$ was derived in
\cite{FHW}, nor their positiveness. By a careful analysis of the
behaviour of the wave profiles at $-\infty$, in Section 4 we prove
that there are {\it positive} travelling waves if the wave speed is
large enough, and explicitly give their asymptotic decay at
$-\infty$. Section 5 is dedicated to applications, which include the
Fisher-KPP equation with delay and a 2-dimensional chemostat model.
An important theorem on the asymptotic behaviour of solutions of
perturbed linear autonomous ordinary FDEs is given in the Appendix.
This result generalizes to the case of FDEs with distributed delay
a result by Mallet-Paret \cite{MP}, for FDEs with discrete
time-delays (or time-shifts), and is often used in Sections 2 and 4.
\section{Positive heteroclinic solution for Eq. (\ref{e1.2})}
In this section, we prove the existence of a positive solution
$u^*(t)$ of the ordinary FDE (\ref{e1.2}) connecting the equilibrium
0 to the positive equilibrium $K$. We recall that a function $u(s)$
defined on a set $S$ and with values in $\mathbb{R}^N$ is said to be
positive if all its components $u_1(s),\dots ,u_N(s)$ are positive
functions on $S$.
\begin{thm}\label{T2.1} Assume (H1)-(H4). Then:\vskip 0cm
i) There exists a heteroclinic
solution $u^*(t),t\in \mathbb{R},$ of Eq. (\ref{e1.2}), with
$u^*(-\infty )=0, u^*(\infty)=K$;\vskip 0cm (ii) $u^*(t)$ is
positive, $t\in\R$;\vskip 0cm (iii) $u^*(t)=ce^{\la_0t} {\bf
v}+O(e^{(2\la_0-\vare)t})$ at $-\infty$, for some $c>0$ and each
fixed $\vare>0$. \end{thm}
\noindent {\it Proof}. (i) Consider the linearization of (1.2) about
$0$,
\begin{equation}\label{e2.1}
u'(t)=Lu_t,\quad {\rm where}\quad L=Df(0),
\end{equation}
and its characteristic equation
\begin{equation}\label{e2.2}
\det \Delta_0 (\la )=0,\quad {\rm where}\quad \Delta_0 (\la)=L(e^{\la
\cdot}I)-\la I.
\end{equation}
Recall that $\la $ is a solution of (\ref{e2.2}) if and only if
$\la\in \sigma (A)$, where $A$ is the infinitesimal generator
associated with the semi-flow of (\ref{e2.1}).
Let $\la_0>0$ be the leading (simple) eigenvalue of (\ref{e2.1})
given in (H4), and ${\bf v}\in\R^N, {\bf v}>0$, such that
$\Delta_0(\la_0){\bf v}=0$. Choose $\gamma>0$ with $\gamma
<\la_0<2\gamma$ and such that the strip $\gamma \le Re\, \la <\la_0$
does not contain any root of (\ref{e2.2}). Define
$\chi_0(\th)=e^{\la_0\th}{\bf v},\th \in [-\tau,0]$, and decompose
the phase ${\cal C}$ as ${\cal C}=P\oplus Q$, where $P=<\chi_0>$ and
$Q$ is the complementary space given by the formal adjoint theory of
Hale \cite{HVL}. Then there are neighbourhoods $N_0,N_1$ of 0 in
$P,Q$, respectively, and a $C^1$ map $w:N_0\to N_1$ with
$w(0)=0,Dw(0)=0$ such that the local $\gamma$-unstable manifold of 0
for Eq. (\ref{e1.2}) is given by
$$W(0)=\{ \phi +w(\phi): \phi\in N_0\}.$$
Note that $\var \in W(0)$ if and only if there is a full
trajectory $u_t=u_t(\var)\, ( t\in\mathbb{R})$ of (\ref{e1.2}) with
$u_0=\var$, $u_t\in N_0+N_1$ for $t\le 0$ and $u(t)e^{-\gamma t}\to
0$ as $t\to -\infty$. See Krisztin et al. \cite{KWW}, Hale and Lunel
\cite[Sec. 10.1-10.2]{HVL}, and Diekmann et al. \cite[Sec. 8.4]{DW}.
We now argue as in \cite{OW}. Write $w(t)=(w_1(t),\dots,w_N(t)),
{\bf v}=({\bf v}_1,\dots ,{\bf v}_N)$. Since $Dw(0)=0$, then
$\lim_{\|\phi\|\to 0, \phi\in N_0}{{\|w(\phi)\|}\over {\|\phi\|}}=0$
and we have $\lim_{c\to 0}{{\|w(c \chi_0)\|}\over {|c|}}=0$. Thus,
there is $c_0>0$ such that $|w_i(c\chi_0)|_\infty\le ce^{-\la_0
\tau}{\bf v}_i/2$ for $c\in (0,c_0], i=1,\dots,N$, which implies
that for $0<c\le c_0$ we have
\begin{equation}\label{e2.3}
\min_{-\tau\le \th \le 0} \Big (ce^{\la_0 \th}{\bf
v}_i+w_i(c\chi_0)(\th)\Big )\ge ce^{-\la_0\tau}{\bf v}_i/2>0,\quad
i=1,\dots,N,
\end{equation}
and therefore $c\chi_0+w(c\chi_0)\in W(0)\cap {\cal C}_+$ for all
$c\in (0,c_0]$. Fix e.g. $c=c_0$, denote
$\phi=c_0\chi_0+w(c_0\chi_0)$ and consider the full trajectory
$u_t^*=u_t(\phi),\ t\in \mathbb{R}$. We have $u_t^*\in W(0)$ for
$t\le 0$, hence $u_t^*$ has the form
$u_t^*=c(t)\chi_0+w(c(t)\chi_0)$. Since the map $t\mapsto u_t^*$ and
the canonical projection of ${\cal C}$ on $P$ are continuous, $c(t)$
is continuous as well, with $c(t)\to 0$ as $t\to -\infty$. This
implies that there is $T<0$ such that $c(t)\le c_0$ for $t< T$. On
the other hand, if $c(t_0)=0$ for some $t_0<T$, then $u_{t_0}^*=0$,
which is not possible. From (\ref{e2.3}) it follows that $u^*(t)>0$
for $t<T$. Now, from (H3) we have $u^*(t)\to K$ as $t\to\infty$.
This means that $u^*(t)$ is a heteroclinic solution of (\ref{e1.2})
connecting the two equilibria $0,K$, with $u^*(t)$ positive on some
interval $(-\infty,T)$.
(ii) Choose $M>0$ such that $u_i^*(t)\le M, \ t\in \mathbb{R},\
i=1,\dots,N$. For the sake of contradiction, suppose there is $t\ge
T$ and $i\in\{ 1,\dots,N\}:=I $ with $u_i^*(t)\le 0$. Define
$t^*=\min \{ t\ge T: u_j^*(t)= 0$ for some $j\in I\}$ and take $i\in
I$ such that $u_i^*(t^*)=0$. For $M$ as above, let $\be$ be as in
(H2), i.e., $f_j(\var)+\be \var_j(0)\ge 0,$ for $j\in I$ and $0\le
\var\le M$. Writing $u_i^*(t)$ in integral form,
\begin{equation*}
u_i^*(t)=\int_{-\infty}^t e^{-\be (t-s)} (f_i(u^*_s))+\be
u_i^*(s))\, ds,\quad t\in \mathbb{R},
\end{equation*}
we obtain
\begin{equation*}
0=\int_{-\infty}^{t^*} e^{-\be (t^*-s)} (f_i(u^*_s))+\be u_i^*(s))\,
ds,
\end{equation*}
where $f_i(u^*_s)+\be u_i^*(s)\ge 0$ for $s\le t^*$. Hence
$f_i(u^*_s)+\be u_i^*(s)= 0$ for $s\le t^*$, and in particular
$u_i^*$ satisfies the scalar ODE $y'=-\be y$ for $s\le t^*$. Thus
$u_i^*(s)\equiv 0$ for $s\le t^*$, which is not possible. \smal
(iii) We note that $u_t^*$ belongs to $W(0)$ for $t\le 0$, thus
$u_t^*=O(e^{\gamma t})$ at $-\infty$ and $u^*(t)$ satisfies
$u'(t)=Lu_t+h(t),$ with $h(t)=f(u_t^*)-Lu_t^*=O(e^{2\gamma t})$ at
$-\infty$. From Theorem \ref{TA.1} (see Appendix), for each $\vare
>0$ we deduce that $u^*(t)=z(t)+O(e^{(2\gamma -\vare) t})$ at
$-\infty$, where $z(t)=ce^{\la_0 t}{\bf v}$ for some (positive)
$c\in \mathbb{R}$. Thus, $u^*(t)=O(e^{\la_0 t})$ at $-\infty$. \hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\begin{rem}\label{rem2.1} In fact, one could use \cite[Lemma 4]{FT} and
its constructive proof to derive that there is a complete solution
$u^*(t)$ of (\ref{e1.2}), with $u^*(-\infty )=0, u^*(\infty)=K$, and
$u^*(t)> 0$ for $t\le 0$. This proves assertion (i) of Theorem
\ref{T2.1}. In order to prove that $u^*(t)=O(e^{\la_0 t})$ at
$-\infty$ it is however more convenient to explicitly construct
$u^*(t)$ as a perturbation of the eigenfunction $e^{\la_0 t}{\bf v}$
as above. This asymptotic result will be crucial to prove the
existence of {\it positive} travelling waves for (\ref{e1.1}), if
the wave speed is sufficiently high. On the other hand, if we
assume that the interior of the positive cone $ {\cal C}_+$ is
positively invariant for the flow of (\ref{e1.2}), as an alternative
to hypothesis (H2)(ii), then the positiveness of $u^*(t)$ on
$\mathbb{R}$ follows immediately from the fact that $u^*(t)$ is
positive in the vicinity of $-\infty$.
\end{rem}
\section{Existence of travelling waves and their asymptotic
decay at $-\infty$} Throughout this section, for simplicity we
assume that (H1)-(H4) are fulfilled, but in fact some of the
hypotheses can be weakened (cf. Remark \ref{R3.2}). We shall prove
the existence of travelling waves for (\ref{e1.1}) which will be
obtained as perturbations of the heteroclinic solution $u^*(t)$ of
(\ref{e1.2}). The asymptotic behaviour at $-\infty$ of $u^*(t)$
given in Theorem \ref{T2.1}(iii) will be important to study the
asymptotic decay of such waves at $-\infty$; however its
positiveness is irrelevant here, and will be only used for the analysis in Section 4.
For a unit vector $w\in \mathbb{R}^p$, we look for wave solutions
of (\ref{e1.1}) with direction $w$ and speed $c>0$, connecting the
equilibria 0 to $K$, i.e., solutions of the form $u(t,x)=\phi
(ct+w\cdot x)$ with $\phi(-\infty)=0,\phi(\infty)=K$.
The equation for the travelling wave profile $\phi$ is given by
\begin{equation}\label{e3.1}
\phi''(t)-c\phi'(t)+f_c(\phi_t)=0,\quad t\in \mathbb{R},
\end{equation}
where $f_c(\phi)=f(\phi (c\cdot))$, with $\phi$ subject to the
conditions
\begin{equation*}
\phi(-\infty)=0, \quad \phi(\infty)=K.
\end{equation*}
With $\vare=1/c$, (\ref{e3.1}) is equivalent to
\begin{equation}\label{e3.2}
\vare^2\phi''(t)-\phi'(t)+f(\phi_t)=0.
\end{equation}
We also consider Eq. (\ref{e3.2}) with $\vare =0$, in which case it
reduces to Eq. (\ref{e1.2}).
Let $C_b(\mathbb{R},\mathbb{R}^N)$ be the space of all continuous
and bounded functions from $\mathbb{R}$ to $\mathbb{R}^N$, with the
supremum norm $\|y\|_{\infty}=\sup _{s\in \mathbb{R}}|y(s)|$. As a
particular case of the framework in \cite{FHW}, we have the
following result:
\begin{thm}\label{thm3.1}\cite{FHW} Let $f$ have the form $f(\phi)=F(\phi
(0),g(N\phi)),\, \phi\in {\cal C},$ for some bounded linear
operator $N:{\cal C}\to \mathbb{R}^N$ and $g:\mathbb{R}^N\to
\mathbb{R}^N, \, F:\mathbb{R}^{2N}\to \mathbb{R}^N$ $C^2$-smooth
functions. Suppose also that:\vskip 0cm (i) $f(0)=f(K)=0$ for some
$K\in\mathbb{R}^N$,\vskip 0cm (ii) for Eq. (\ref{e1.2}), the
equilibrium $u=0$ is hyperbolic and unstable, and the equilibrium
$u=K$ is locally asymptotically stable. \vskip 0cm Then, if there is
a heteroclinic solution $u^*(t)$ for (\ref{e1.2}) connecting 0 to
$K$, for each unit $w\in \mathbb{R}^p$ there are a neighbourhood
${\cal V}$ of $u^*(t)$ in $C_b(\mathbb{R},\mathbb{R}^N)$ and a
constant $c^*>0$, such that for $c>c^*$ the set of travelling waves
for (\ref{e1.1}) in ${\cal V}$, with direction $w$ and wave speed
$c$, constitutes a $C^1$-manifold of dimension $m$, where $m$ is the
dimension of the unstable space for $\dot u(t)=Df(0)u_t$. \end{thm}
In this section, the idea is to retrace some arguments in \cite{FHW}
for the proof of Theorem \ref{thm3.1} adapted to the case of
(\ref{e1.1}), but in appropriate Banach spaces, which will allow us
to deduce not only the existence of travelling wave solutions for
(\ref{e1.1}), but also their asymptotic behaviour at $-\infty$.
This behaviour will be used in Section 4, to prove the existence of
{\it positive} travelling waves.
\med In addition to $C_b:=C_b(\mathbb{R},\mathbb{R}^N)$, we
introduce the following Banach spaces:
\smal
$C_b^1:=C_b^1(\mathbb{R},\mathbb{R}^N)=\{ y\in C_b:y'\in C_b\}$
with the norm $\|y\|_1=\|y\|_{\infty}+\|y'\|_\infty$;
$C_0=\{ y\in C_b:\lim _{s\to\pm \infty}y(s)=0\}$ is considered as a
subspace of $C_b$;
$C_0^1=\{ y\in C_b^1: y,y'\in C_0\}$ is considered as a subspace of
$C_b^1$;
$C_\mu=\{ y\in C_b:\sup_{s\le 0}e^{-\mu s}|y(s)|<\infty\}$ (for
$\mu>0$) with the norm
\begin{equation*}
\|y\|_\mu =\max \{\|y\|_\infty,\|y\|_{\mu}^- \}\quad {\rm where} \quad
\|y\|_\mu^-=\sup_{s\le 0}e^{-\mu s}|y(s)|;
\end{equation*}
$C_\mu ^1=\{ y\in C_b^1 : y,y'\in C_\mu\}$, with the norm $\|y\|_{1,\mu}=\|y\|_{\mu}+\|y'\|_\mu$;
$C_{\mu,0}=C_{\mu}\cap C_0$ is considered as a subspace of $C_\mu$.
By the change of variables $\phi(t)=w(t)+u^*(t)$, (\ref{e3.2})
becomes
\begin{equation}\label{e3.3}
\vare^2w''(t)-w'(t)-w(t)=-w(t)-Df(u_t^*)w_t-G(\vare,t,w),\quad t\in
\mathbb{R},
\end{equation}
where
\begin{equation}\label{e3.4}
G(\vare,t,w)=f(w_t+u_t^*)-f(u_t^*)-Df(u_t^*)w_t+\vare^2 {u^*}''(t),
\end{equation}
subject to the conditions $w(-\infty)=w(\infty)=0.$ The roots of
the characteristic equation associated with
$\vare^2w''(t)-w'(t)-w(t)=0$ are
\begin{equation*}
\al(\vare)={{1-\sqrt{1+4\vare^2}}\over {2\vare^2}},\quad
\be(\vare)={{1+\sqrt{1+4\vare^2}}\over {2\vare^2}},
\end{equation*}
and satisfy $\al(\vare)\to -1^+,\be(\vare)\to \infty$ as $\vare \to
0^+$. In the case of different diffusion coefficients
$d_i>0,i=1,\dots,N,$ instead of $\al(\vare),\be(\vare)$ one has to
consider $\al_i(\vare),\be_i(\vare)$, the solutions of the
characteristic equations $d_i\vare^2z^2-z-1=0,i=1,\dots,N,$ but the
arguments are similar (cf. \cite{FHW}).
A bounded function $w:\mathbb{R}\to\mathbb{R}^N$ is a solution of
(\ref{e3.3}) if and only if
\begin{equation}\label{e3.5}
Jw(t)=H(\vare,w)(t),\quad t\in\mathbb{R},
\end{equation}
where
\begin{equation*}
Jw(t)=w(t)-\int_{-\infty}^te^{-(t-s)} [w(s)+Df(u_s^*)w_s]\, ds
\end{equation*}
and
$$\displaylines{
H(\vare,w)(t)=\int_{-\infty}^t\left [{{e^{\al(\vare)(t-s)}}\over
{\sqrt{1+4\vare^2}}}-e^{-(t-s)}\right ](w(s)+Df(u_s^*)w_s)\, ds+\cr
{1\over {\sqrt{1+4\vare^2}}}\left [ \int_{-\infty}^t
e^{\al(\vare)(t-s)} G(\vare,s,w)\, ds+ \int^{+\infty}_t
e^{\be(\vare)(t-s)} [w(s)+Df(u_s^*)w_s+G(\vare,s,w)]\,
ds\right].\cr}$$
\smal
Our purpose is to apply a contraction principle argument in order to
obtain a solution of Eq. (\ref{e3.5}), for $\vare>0$ small and $w$
close to 0, in adequate spaces $C_\mu$. We first analyse the
linearity $J$, by introducing some auxiliary equations and
operators.
Define
\begin{equation*}
(Ty)(t)=y'(t)-Df(u_t^*)y_t,\quad y\in C_b^1,t\in \mathbb{R}.
\end{equation*}
We easily see that $J:C_0\to C_0,$ $T:C_b^1\to C_b$ are linear
bounded operators and $w\mapsto H( w,\vare)$ maps $C_0$ in $C_0$,
for $\vare>0$ (cf. \cite{FHW}). For $\mu>0$, we also define
\begin{equation*}
T_\mu:=T|_{C_\mu ^1}:C_\mu ^1\to C_\mu.
\end{equation*}
\begin{lem}\label{L3.1} Let $\mu>0$. Then $T_\mu$ and $J|_{C_{\mu
,0}}:{C_{\mu ,0}}\to {C_{\mu ,0}}$ are bounded operators. \end{lem}
{\it Proof}. Since the map $t\mapsto \|Df(u_t^*)\|$ is continuous
for $t\in \R$ and $\|Df(u_t^*)\|\to \|Df(0)\|$ as $t\to -\infty$,
$\|Df(u_t^*)\|\to \|Df(K)\|$ as $t\to \infty$, then
$M:=\sup_{t\in\R} \|Df(u_t^*)\| <\infty.$
It follows that
$|(Ty)(t)|\le \max (1,M)\|y\|_{1,\mu} $ for $t\ge 0$ and $e^{-\mu
t}|(Ty)(t)|\le \max (1,M)\|y\|_{1,\mu} $ for $t\le 0, y\in C_\mu ^1
$.
For $y\in C_\mu$, we now have $|(Jy)(t)|\le (2+M)\|y\|_\mu$ for
$t\ge 0$ and $e^{-\mu t}|(Jy)(t)| \le [1+(1+M)(\mu
+1)^{-1}]\|y\|_\mu$, hence $J(C_{\mu ,0})\subset C_{\mu ,0}$.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
Consider the linear variational equation around the heteroclinic
solution $u^*(t)$,
\begin{equation}\label{e3.6}
y'(t)=Df(u_t^*)y_t.
\end{equation}
Define the operators $L(t):=Df(u_t^*)$ in (\ref{e3.6}), with
$L(-\infty)=Df(0)$ and $L(\infty)=Df(K)$. Hence, Eq. (\ref{e3.6}) is
asymptotically autonomous, with limiting equations (\ref{e2.1}) and
$y'(t)=Df(K) y_t$, respectively at $-\infty$ and $\infty$.
\begin{lem}\label{L3.2} Consider $\mu\in (0,\la_0)$ such that there are no
characteristic roots $\la$ of (\ref{e2.1}) with $\Re\, \la =\mu$.
For $T_\mu:C_\mu ^1\to C_\mu$ defined as above,
\begin{equation*}
Im (T_\mu)=C_\mu,\quad \dim\, Ker\, (T_\mu)=r_\mu,
\end{equation*}
where $r_\mu=\# \{ \la\in \mathbb{C} :\det \Delta_0(\la)=0, \Re\,
\la
>\mu\}.$ In particular, $r_\mu=1$ for $\mu$ close to $\la_0$.
Moreover, $Ker\, (T_\mu)\subset C_{\mu,0}$.
\end{lem}
{\it Proof}.
Clearly, equation $y'(t)=Df(K) y_t$ is asymptotically stable,
and the autonomous equation (\ref{e2.1})
admits a ``shifted exponential dichotomy" in $\mathbb{R}$ with the
splitting made at $\mu$ and exponents $\mu-\delta, \mu+\delta$,
for $\delta>0$ small. See Hale and Lin \cite{HL} for definitions,
and note that $C_\mu=C^0(\mu, 0)$ in the notation in \cite{HL}. From
\cite[Lemma 4.3]{HL}, there is $T>0$ such that (\ref{e3.6}) has a
shifted exponential dichotomy on $(-\infty, -T]$ and $[T,\infty)$.
We now apply Lemma 4.6 of \cite{HL} to (\ref{e3.6}). It follows that
$T_\mu$ is a Fredholm operator, with index $Ind(T_\mu)$ given by
\begin{equation*}
Ind(T_\mu)=\dim Im(P_u^-(-t))-\dim Im (P_u^+(t)),\quad t\ge T,
\end{equation*}
where $P_u^-(-t), P_s^-(-t)$ and $P_u^+(t), P_s^+(t)\, (t\ge T)$ are
the projections associated with the (shifted) exponential
dichotomies for $y'(t)=Df(0) y_t$ and $y'(t)=Df(K) y_t$,
respectively. From \cite[Lemma 4.3]{HL}, we also have that
$P_u^-(-t)\to P_u^-, P_u^+(t)\to P^+_u$ as $t\to\infty$, where
$P_u^-$ is the canonical projection from ${\cal C}$ onto the
$\mu$-unstable space $E_\mu^-$ for $y'(t)=Df(0) y_t$, and $P_u^+$
is the canonical projection from ${\cal C}$ onto the unstable space
$E_u^+$ for for $y'(t)=Df(K) y_t$. We have $E_u^+=\{ 0\}$ and $\dim
E_\mu^-=r_\mu$, where $r_\mu$ is the number of characteristic values
for (\ref{e2.1}) (counting multiplicities) with real parts greater
than $\mu$. Hence $Ind (T_\mu)=r_\mu$. On the other hand, the index
of $T_\mu$ is defined by $Ind (T_\mu)=\dim Ker(T_\mu)-{\rm codim}\,
Im(T_\mu)$. Again by \cite[Lemma 4.6]{HL} we have $\dim
Ker(T_\mu)=\dim E_\mu^-=r_\mu$, yielding that $Im (T_\mu)=C_\mu$.
For $y\in Ker\, (T_\mu)$, from the definition of shifted
exponential dichotomy we have $\lim_{t\to\infty} y(t)=0.$ Thus,
$Ker\, (T_\mu)\subset C_{\mu,0} $. \hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
Similarly to what was done for $T$, we now restrict the domain and range of the operator $J$.
With $D:=d/dt +id$, consider the commutative diagram
\begin{equation*}
\begin{array}[c]{ccccc}
C^1_\mu &&\stackrel{T_\mu}\longrightarrow&&C_\mu\\
& {_{J}}{\searrow}&&{\nearrow}_D\\
&&C^1_\mu
\end{array}
\end{equation*}
It is easy to check that this diagram is well defined, and that $D$
is one-to-one and surjective. Since $T_\mu = D\circ J$ is
surjective, we may conclude that $J$ is also surjective. Moreover,
\begin{lem}\label{L3.3} Consider $\mu\in (0,\la_0)$ such that there are no
characteristic roots $\la$ of (\ref{e2.1}) with $\Re\, \la =\mu$.
Then, for the operator $J|_{C_{\mu ,0}}:{C_{\mu ,0}}\to {C_{\mu
,0}}$ we have $Ker\, (J|_{C_{\mu ,0}})=Ker\, (T_\mu)$ and $Im\,
(J|_{C_{\mu ,0}})=C_{\mu ,0}$.
\end{lem}
{\it Proof}. Recall that $Ker\, (T_\mu)\subset C_{\mu,0}$.
Clearly, for $w\in C_{\mu ,0}$ we have $Jw=0$ if and only if $w'(t)=Df(u_t^*)w_t$, and then
$w'\in C_\mu$. We therefore deduce that $(Ker\, J)\cap C_\mu =(Ker\, J)\cap C_\mu^1$, and $Ker\, (J|_{C_{\mu ,0}})=Ker\, (T_\mu)$.
We now prove that $Im\, (J|_{C_{\mu ,0}})=C_{\mu ,0}$. Indeed, for
$y \in C_{\mu,0}$ we have that $\xi:= y-Jy \in C^1_{\mu}$ and $D\xi
(t)=y(t)+Df(u_t^*)y_t$, hence $D\xi \in C_{\mu,0}$. Equation $Jw =
y$ is equivalent to $ J(w-y) = \xi ,$ and therefore it possesses a
solution $\chi= w-y \in C^1_\mu$. After applying $D$ to both sides
of the latter equation, we get $ T_\mu \chi = D\xi \in C_{\mu,0}$.
Since the $\omega$-limit operator $T_\mu(\infty)$ is hyperbolic, we
may invoke Lemma 3.3 from \cite{FHW} to conclude that
$\chi(\infty)=0$. Thus $w(\infty)= 0$, and $J: C_{\mu,0} \to
C_{\mu,0}$ is surjective. \hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
We now focus our attention on the non-linearity $H$ of Eq. (\ref{e3.5}). Proceeding as in \cite{FHW},
one sees that $H(w,\vare)\in C_0$ for each $\vare > 0$ and $w\in
C_0$. We want however to consider the maps $H(\cdot ,\vare)$
restricted to some neighbourhood of zero in $C_{\mu,0}$, for
$\vare>0$ and $\mu >0$. We start with an auxiliary lemma:
\begin{lem}\label{L3.4} Let $X, Y$ be normed spaces and $C\subset O \subset X$.
Suppose that the set $C$ is compact, $O$ is open and $F:O \to Y$ is
a continuous map. Then for every $\sigma >0$ there exists $\delta
>0$ such that
\begin{equation*}
|F(x+z)-F(x)| \leq \sigma, \quad x \in C, \ |z| \leq \delta.
\end{equation*}
\end{lem}
{\it Proof}. By the continuity of $F$, for each $x\in C$ there is
$\delta(x)>0$ such that if $|z| \leq 2\delta(x)$ then $x+z \in O$
and $ |F(x+z)-F(x)| \leq \sigma/2$. Since $C \subset \cup_{x \in
K}B_{\delta(x)}(x)$ is compact, there is a finite subcover
$\{B_{\delta(x_j)}(x_j)\}_{j=1}^m$ of $C$. For each $x \in
B_{\delta(x_j)}(x_j)\cap C$ and $|z|\leq \delta := \min
\{\delta(x_j)\}$, we have $ |F(x+z)-F(x)| \leq |F(x+z)-F(x_j)|+
|F(x)-F(x_j)| \leq \sigma,$ which proves the lemma. \hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\begin{lem}\label{L3.5} Assume (H1)--(H4) and consider $\mu\in (0,\la_0)$.
For any $\delta>0$, there are $\vare ^*>0$ (independent of $\mu$) and $\sigma >0$ such that $H(w,\vare)\in C_{\mu ,0}$
for any $\vare >0$ and $w\in C_{\mu ,0}\cap B_\sigma^\mu(0)$, and
\begin{eqnarray}\label{e3.7}
\nonumber \fl \|H(w,\vare)\|_\mu \le \delta (\|w\|_\mu +1), && \\
\fl \|H(w,\vare)-H(v,\vare)\|_\mu \le \delta \|w-v\|_\mu, &&\quad {\rm for}\quad w,v\in C_{\mu,0} \cap
B_\sigma^\mu(0), \vare\in (0,\vare^*)
\end{eqnarray}
where $B_\sigma^\mu(0)$ is the $\sigma$-neighbourhood of 0 in $C_\mu$.
\end{lem}
{\it Proof}.
We write $H=H_1+H_2+H_3$, where
$$\eqalign{
H_1(\vare,w)(t)&=\int_{-\infty}^t\left [{{e^{\al(\vare)(t-s)}}\over
{\sqrt{1+4\vare^2}}}-e^{-(t-s)}\right ](w(s)+Df(u_s^*)w_s)\, ds,\cr
H_2(\vare,w)(t)&={1\over {\sqrt{1+4\vare^2}}} \int_{-\infty}^t
e^{\al(\vare)(t-s)} G(\vare,s,w)\, ds,\cr H_3(\vare,w)(t)&={1\over
{\sqrt{1+4\vare^2}}}\int^{+\infty}_t e^{\be(\vare)(t-s)}
[w(s)+Df(u_s^*)w_s+G(\vare,s,w)]\, ds,\cr }$$ and $G$ is given by
(\ref{e3.4}). Let $M=\sup_{t\in\R}\|Df(u_t^*)\|$ as before. For
$t\in\mathbb{R},\ \vare> 0$ and $\mu \ge 0$, we have
$$\eqalign{
&\left |\int_{-\infty}^t\left [{{e^{\al(\vare)(t-s)}}\over
{\sqrt{1+4\vare^2}}}-e^{-(t-s)}\right ]e^{\mu s}\, ds\right |\cr
&\le {1\over {\sqrt{1+4\vare^2}}}\int_{-\infty}^t \left |
e^{\al(\vare)(t-s)} (1-\sqrt{1+4\vare^2})+\sqrt{1+4\vare^2}\Big
(e^{\al(\vare)(t-s)}-e^{-(t-s)}\Big )\right | e^{\mu s} \, ds \cr &=
{1\over {\sqrt{1+4\vare^2}}}\left [
(\sqrt{1+4\vare^2})-1)\int_{-\infty}^t e^{\al(\vare)(t-s)} e^{\mu
s} \, ds + \sqrt{1+4\vare^2} \int_{-\infty}^t
(e^{\al(\vare)(t-s)}-e^{-(t-s)}\Big ) e^{\mu s} \, ds \right ] \cr
&= {1\over {\sqrt{1+4\vare^2}}}\left [
{{2\sqrt{1+4\vare^2}-1}\over {\mu -\al(\vare)}}- { {\sqrt{1+4\vare^2}}\over {\mu +1}}\right] e^{\mu
t}\cr}
$$
\begin{equation}\label{e3.8}
\fl = {1\over {\sqrt{1+4\vare^2}}} \left [ {{\sqrt{1+4\vare^2}-1}\over
{\mu -\al(\vare)}}+ {{(1+\al(\vare))\sqrt{1+4\vare^2}}\over
{(\mu-\al(\vare))(\mu+1)}} \right ] e^{\mu t}
\le C_1(\vare)e^{\mu t},
\end{equation}
where
\begin{equation*}
C_1(\vare)= -{1\over {\al(\vare)}} \left ( 1- {1\over
{\sqrt{1+4\vare^2}}}+1+\al(\vare)\right ) \to 0\quad {\rm as}\quad \vare
\to 0^+.
\end{equation*}
From (3.8), we obtain
\begin{equation}\label{e3.9}
\fl \|H_1(\vare,w)-H_1(\vare, v)\|_\mu\le C_1(\vare)
(1+M)\|w-v\|_\mu,\quad w,v\in C_\mu,\vare> 0.
\end{equation}
Since $H_1(\vare, 0)=0$, in particular $H_1(\vare,w)\in C_\mu$ for
$w\in C_\mu$ and $\vare >0$.
For $0\le \mu<\be (\vare)$ and $t\in\mathbb{R}$, we now have
\begin{equation}\label{e3.10}\fl
\int_{-\infty}^t e^{\al(\vare)(t-s)}e^{\mu s}\, ds={{e^{\mu t}}\over
{\mu-\al(\vare)}},\ \int^{+\infty}_t e^{\be(\vare)(t-s)} e^{\mu s}
\, ds={{e^{\mu t}}\over {\be (\vare)-\mu}}.
\end{equation}
Consider e.g. $\mathbb{R}^N$ equipped with the maximum norm. For
$t\in\mathbb{R},$ $\vare > 0,$ $ w,v\in C_{\mu,0},$ $ i=1,\dots,N$,
we have
\begin{eqnarray}\label{e3.11}
\nonumber \fl & |G_i(\vare, t,w)|&\le
\vare^2|{u_i^*}''(t)|+|f_i(w_t+u_t^*)-f_i(u_t^*)-Df_i(u_t^*)w_t| \\
\fl & & \le
\vare^2|{u_i^*}''(t)|+\|Df_i(u^*_t+\xi_{i,t}w_t)-Df_i(u_t^*)\|\|w_t\|_\infty
\end{eqnarray}
and
\begin{equation}\label{e3.12}\fl
|G_i(\vare, t,w)-G_i(\vare,t,v)|\le
\|Df_i(v_t+u_t^*+\th_{i,t}(w_t-v_t))-Df_i(u_t^*)\|\|w_t-v_t\|_\infty,
\end{equation}
for some $\xi_{i,t},\th_{i,t} \in (0,1)$ for $t\in\mathbb{R}$.
Note that ${u^*}''\in C_\mu$ for $0<\mu \le \la_0$. In fact,
$u^*\in C_\mu$ from Theorem \ref{T2.1}, hence Eq. (\ref{e1.2}) and
the smoothness of $f$ lead to $|{u^*}'(t)|\le M_0\|u^*_t\|_\infty$,
from which we derive $\|{u^*}'\|_\mu \le M_0 \|u^*\|_\mu,$ for some
$M_0>0$. By differentiating, we obtain ${u^*}''(t)=Df(u_t^*)
({u^*}')_t$, thus $\|{u^*}''\|_\mu \le M \|{u^*}'\|_\mu.$
In order to simplify the notation, for each $\mu ,\sigma >0$ write
$C_{\mu,0}\cap B_{\sigma}(0)$ to denote the $\sigma$-neighbourhood
of 0 in $C_{\mu ,0}$. Since $u^*$ is uniformly bounded on
$\mathbb{R}$ and $f$ transforms bounded sets of ${\cal C}$ into
bounded set of $\mathbb{R}^N$, then ${u^*} '$ is uniformly bounded
on $\mathbb{R}$ and $u^*$ uniformly continuous on $\mathbb{R}$.
Thus, ${\cal K}= \overline{\{{u_t^*}, \ t \in \mathbb{R}\}} \subset
{\cal C}$ is compact. The continuity of $Df_i: {\cal C} \to
\cal{L}(C,C)$ and Lemma \ref{L3.4} imply that, for each $\delta>0$
fixed, there is $\sigma =\sigma(\de,\mu)>0$ such that
$\|Df_i(v_t+u_t^*+\th_{i,t}(w_t-v_t))-Df_i(u_t^*)\|<\delta$ for
$w,v\in C_{\mu,0}\cap B_\sigma(0), \ t \in \R$. From (\ref{e3.11}),
(\ref{e3.12}), we get
$$\displaylines{
|G(\vare, t,w)|\le \vare^2|{u^*}''(t)|+\delta \|w_t\|_\infty, \quad
w\in C_{\mu,0}\cap B_{\sigma}(0),\cr |G(\vare, t,w)-G(\vare,t,v)|\le
\delta \|w_t-v_t\|_\infty,\quad w,v\in C_{\mu,0}\cap B_\sigma(0).\cr}
$$
From these estimates and (\ref{e3.10}), we conclude that
$H_2(\vare,w), H_3(\vare,w)\in C_\mu$ for all $w\in C_\mu\cap
B_{\sigma}(0)$ and $\vare >0,\mu \in (0,\la_0)$, with
\begin{eqnarray}\label{e3.13}
\nonumber \fl & & \|H_2(\vare,w)\|_\mu \le {1\over
{(\mu-\al(\vare))\sqrt{1+4\vare^2}}}(\vare^2\|{u^*}''\|_\mu+\delta
\|w\|_\mu), \\
\fl & & \|H_3(\vare,w)\|_\mu \le {1\over
{(\be(\vare)-\mu)\sqrt{1+4\vare^2}}} \Big
[\vare^2\|{u^*}''\|_\mu+(1+M+\delta )\|w\|_\mu \Big].
\end{eqnarray}
Furthermore, for $\|w-v\|_\mu, w,v\in C_{\mu,0}\cap
B_\sigma(0),\vare>0,$ we get
\begin{eqnarray}\label{e3.14}
\nonumber \fl & & \|H_2(\vare,w)-H_2(\vare, v)\|_\mu\le{\delta \over
{(\mu-\al(\vare))\sqrt{1+4\vare^2}}} \|w-v\|_\mu, \\
\fl & & \|H_3(\vare,w)-H_3(\vare, v)\|_\mu\le {{1+M+\delta }\over
{(\be(\vare)-\mu)\sqrt{1+4\vare^2}}}.
\end{eqnarray}
On the other hand,
$${1\over {\sqrt{1+4\vare^2}}}\left ({1\over {\be(\vare)-\mu}}+{1\over {\mu -\al(\vare)}} \right ) ={1\over {1+\mu-\vare^2\mu^2}}< 1$$
if $\vare^2\mu <1$. From (\ref{e3.9}), (\ref{e3.13}) and
(\ref{e3.14}), for $\vare>0$ small enough and $\mu\in (0,\la_0)$ we
obtain
\begin{equation}\label{e3.15}
\|H(w,\vare)\|_\mu\le C(\vare)\|w\|_\mu +D(\vare),\quad w\in
C_{\mu,0}\cap B_{\sigma}(0)
\end{equation}
and
\begin{equation}\label{e3.16}
\|H(w,\vare)-H(v,\vare)\|_\mu\le C(\vare) \|w-v\|_\mu,\quad w,v\in
C_{\mu,0} \cap B_\sigma(0),
\end{equation}
where $C(\vare), D(\vare)$ do not depend on $\mu$ and are given by
$$C(\vare)=C_1(\vare)(1+M)+\delta+
{{1+M }\over {(\be(\vare)-\la_0)\sqrt{1+4\vare^2}}},\quad D(\vare)=
\vare^2\|{u^*}''\|_{\la_0} .$$ Since $C_1(\vare)\to 0,
\be(\vare)\to\infty$ as $\vare\to 0^+$, by replacing $\delta$ by
$\delta /2$ in (\ref{e3.15}), (\ref{e3.16}), we obtain (\ref{e3.7})
for $\vare>0$ sufficiently small. \hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
We now return to Eq. (\ref{e3.5}). Let $0< \mu <\la_0$. For $\vare
>0$ small, we look for a solution $w\in C_{\mu ,0}$ of (\ref{e3.5}). For
the case $\mu=0$, where the space $C_{0,0}$ denotes $C_0$, this
question was addressed in \cite{FHW}. Our purpose is to solve this
problem for $\mu\in (0,\la_0)$.
We first apply a Liapunov-Schmidt reduction. From Lemmas \ref{L3.2}
and \ref{L3.3}, $X_\mu:=Ker\, (J|_{C_{\mu ,0}})$ is finite
dimensional, hence there is a complementary subspace $Y_\mu$ in
$C_{\mu ,0}$,
\begin{equation*}
C_{\mu ,0}=X_\mu \oplus Y_\mu.
\end{equation*}
For $w\in C_{\mu ,0}$, write $w=\xi+\phi$ with $\xi\in X_\mu,\phi\in
Y_\mu$. Define $S_\mu:=J|_{Y_\mu}$. Since $S_\mu:Y_\mu\to C_{\mu,0}$
is bounded and bijective, then $S_\mu^{-1}$ is bounded. In the space
$C_{\mu,0}$, Eq (\ref{e3.5}) is equivalent to $\phi
=S_\mu^{-1}H(\vare, \xi+\phi)$, therefore we look for fixed points
$\phi\in Y_\mu$ of the map
\begin{equation}\label{e3.17}
{\cal F}_\mu(\vare,\xi,\phi)=S_\mu^{-1}H(\vare, \xi+\phi).
\end{equation}
For simplicity, in what follows we write $S, {\cal F}, B_\sigma(0)$
instead of $S_\mu,{\cal F}_\mu,B_\sigma^\mu (0)$, respectively, when
there is no risk of misunderstanding.
\begin{rem}\label{R3.1}
For $0<\mu_1<\mu_2 <\la_0$ with $\mu_1,\mu_2\notin \Re\, \sigma
(A)$, where $ \sigma (A)$ is the set of solutions of (\ref{e2.2}),
it is clear that $C_{\mu_2}\subset C_{\mu_1}$ with $\|y\|_{\mu_1}\le
\|y\|_{\mu_2}$, and $X_{\mu_2}\subset X_{\mu_1}$. Together with
Lemmas \ref{L3.2} and \ref{L3.3}, this implies that for each
interval $I:=[\mu_1,\mu_2] \subset (0,\la_0)\setminus \Re\, \sigma
(A)$, we have $X_{\mu_2}= X_{\mu_1}$. We now show that the
complementary subspaces $Y_\mu$ can be chosen so that
$Y_{\mu_2}\subset Y_{\mu}\subset Y_{\mu_1}$ for $\mu\in I$. In fact,
let $X_\mu=span\, \{ y_1,\dots, y_r\}$, where $y_1,\dots, y_r\in
C_{\mu,0}$ and $r=r_\mu$ for $\mu\in I$. From the Hahn-Banach
theorem, let $h_i\in (C_{\mu_1,0})'$ be such that $h_i(y_i)=1,
h_i(y_j)=0$ for $j\ne i, i,j=1,\dots, r$. Define the natural
injections $i(\mu,\mu_1): C_{\mu,0}\to C_{\mu_1,0}$, which are
continuous, and the subspaces $Y_\mu = \{ y\in C_{\mu,0}: h_i\circ
i(\mu,\mu_1)(y)=0, i=1,\dots, r\}$. Hence $Y_\mu$ is a closed
subspace of $C_{\mu,0}$, and for $y\in C_{\mu,0}$ we have
$\sum_{i=1}^r h_i(y)y_i\in X_\mu, y-\sum_{i=1}^r h_i(y)y_i\in
Y_\mu$, from which the decompositions $C_{\mu ,0}=X_\mu \oplus
Y_\mu$ follow, with $Y_{\mu_2}\subset Y_{\mu}\subset Y_{\mu_1}$ for
$\mu\in I$.
\end{rem}
\begin{thm} \label{Thm3.2} Assume (H1)-(H4),
and denote by $\sigma(A)$ the set of characteristic values for
(\ref{e2.1}). Fix an interval $I:=[\mu_1\mu_2]\subset (0,\la_0)
\setminus \Re\, \sigma (A)$, and denote $r=r_\mu$ for all $\mu\in
I$. Then, there exist $\vare^*>0$ and $\sigma >0$, such that for
$0<\vare\le \vare^*$, the following holds: for each unit vector
$w\in \mathbb{R}^p$ and all $\mu\in I$, in a neighbourhood
$B_\sigma^\mu(0)$ of $u^*(t)$ in $C_\mu$, the set of all travelling
wave solutions $u(t,x)=\psi (ct+w\cdot x)$ of (\ref{e1.1}) with
speed $c=1/\vare$ and connecting 0 to $K$ forms a $r$-dimensional
manifold (which does not depend on $\mu$), with the profile
$\psi\in {\cal M}_{I,\vare}$, where
$${\cal M}_{I,\vare}=\{ \psi: \psi (t)=u^*(t)+\xi +\phi (\vare, \xi),\ {\rm for}\ \xi\in X_\mu\cap B_\sigma ^\mu(0)\} ,$$
where $\phi (\vare, \xi)=\phi_\mu (\vare, \xi)$ is the fixed point
of ${\cal F}_\mu(\vare,\xi,\cdot)$ in $Y_\mu\cap B_\sigma^\mu (0)$,
and is continuous on $(\vare,\xi)$.
\end{thm}
{\it Proof}. In the sequel, we shall use the simplified notation
$S, {\cal F}, B_\sigma(0)$, for $S_\mu, {\cal F}_\mu,B_\sigma^\mu
(0)$, respectively. Fix $\mu \in I$ and $k\in (0,1)$. From Lemma
\ref{L3.5} (cf. (\ref{e3.15}) and (\ref{e3.16})), for $\delta>0$
small there are $\sigma=\sigma(\de,\mu)>0$ and
$\vare^*=\vare^*(\de)>0$ such that for $0<\vare<\vare^*$, $\xi \in
X_\mu\cap \overline{B_\sigma (0)}$ and $\phi_1,\phi_2\in Y_\mu\cap
\overline{B_\sigma (0)}$ we have
\begin{equation}\label{e3.18} \fl
\|S^{-1}H(\xi+\phi_1,\vare)\|\le \|S^{-1}\| \big (C(\vare)\|\xi+\phi_1\|_\mu +D(\vare)\big )\le \delta \|S^{-1}\| (\|\xi+\phi_1\|_\mu+1)
\end{equation}
and
\begin{equation}\label{e3.19} \fl
\|S^{-1}(H(\xi+\phi_1,\vare)-H(\xi+\phi_2,\vare))\|_\mu \le \|S^{-1}\| C(\vare) \|\phi_1-\phi_2\|_\mu \le \delta\|S^{-1}\| \|\phi_1-\phi_2\|_\mu
\end{equation}
with $\delta (1+2\sigma)\|S^{-1}\|\le \sigma$ and $\delta \|S^{-1}\|
\le k$. From (\ref{e3.18}) and (\ref{e3.19}), it follows that
${\cal F}:(0,\vare^*)\times (X_\mu\cap \overline{B_\sigma
(0)})\times (Y_\mu\cap \overline{B_\sigma (0)})\to Y_\mu\cap
\overline{B_\sigma (0)}$ is a uniform contraction map of $\phi \in
Y_\mu\cap \overline{B_\sigma (0)}$, hence for $(\vare, \xi)\in
(0,\vare^*)\times (X_\mu\cap \overline{B_\sigma (0)})$ there is a
unique solution $\phi (\vare, \xi)=\phi_\mu (\vare, \xi)\in Y_\mu$
of (\ref{e3.17}), with $\phi (\vare, \xi)$ continuous. Define the
$r$-dimensional manifold ${\cal M}_{\mu,\vare}=\{ \psi: \psi
(t)=u^*(t)+\xi +\phi (\vare, \xi),\ {\rm for}\ \xi\in X_\mu\cap
B_\sigma (0)\} $. Choose $\sigma=\sigma(\de,\mu_2)$, independent of
$\mu\in I$. From the uniqueness of the fixed point and Remark
\ref{R3.1}, it follows that $\phi_\mu (\vare, \xi)=\phi_{\mu_2}
(\vare, \xi)$ does not depend on $\mu\in I$, as well as ${\cal
M}_{\mu, \vare}:={\cal M}_{I, \vare}$.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
We observe that if 0 is a hyperbolic equilibrium of (\ref{e1.2}) and
$f$ has the particular form $f(\phi)=F(\phi(0), g(N\phi))$, then
Theorem \ref{thm3.1} asserts that the result in Theorem \ref{Thm3.2}
is valid for $\mu=0$.
\begin{cor}\label{Cor3.1} Under the assumptions of Theorem \ref{Thm3.2} and
with the same notation, for $0<\mu<\la_0 $ such that the strip $\{
\la\in \mathbb{C}: \Re\, \la \in (\mu, \la_0)\}$ does not intersect
$\sigma (A)$, the manifold ${\cal M}_{\mu, \vare}$ is 1-dimensional.
\end{cor}
\begin{cor}\label{Cor3.2} Under the assumptions of Theorem
\ref{Thm3.2} and with the same notation, for an interval
$I:=[\mu_1\mu_2]\subset (0,\la_0) \setminus \Re\, \sigma (A)$, there
are $\vare^*>0,\sigma
>0$ and $C>0$ such that the travelling profiles $\psi (\vare,\xi)$
satisfy
\begin{equation*}
\|\psi (\vare,\xi)\|_\mu \le C, \quad \|\psi '(\vare,\xi)\|_\mu \le C\quad
{\rm for}\quad 0<\vare<\vare^*, \xi \in X_\mu\cap \overline{B_\sigma
(0)},
\end{equation*}
where $C$ does not depend on $\mu\in I$. In particular $|\psi
(\vare, \xi)(t)|\le Ce^{\mu t}$ for $t\le 0, 0<\vare<\vare^*,$ $ \xi
\in X_\mu\cap \overline{B_\sigma (0)}.$ \end{cor}
{\it Proof}. For all $\mu\in I$, the profiles are given by $\psi
(\vare,\xi)=u^*+\xi+ \phi (\vare,\xi)$, where $\phi
(\vare,\xi)=\phi_{\mu_2} (\vare,\xi)$ is the fixed point of $ {\cal
F}_{\mu_2}(\vare,\xi,\cdot).$ Since $\|y\|_\mu \le \|y\|_{\mu_2}$
for $y\in C_{\mu_2,0}$, we only need to prove the result for
$\mu=\mu_2$. In what follows, we write $S_{\mu_2}^{-1}=S^{-1}$.
Fix $k\in (0,1)$, and consider $\vare_1>0$ such that $\|S^{-1}\|
C(\vare) \le k$ for $0<\vare<\vare_1$ where $C(\vare)$ is as in
(\ref{e3.16}). From (\ref{e3.19}), for $w_1,w_2\in C_{\mu_2 ,0}\cap
B_\sigma (0)$ we have
$$
\fl \|S^{-1}(H(w_1,\vare)-H(w_2,\vare))\|_{\mu_2} \le k
\|w_1-w_2\|_{\mu_2},
$$
and the contraction principle yields
\begin{equation*}
\fl \|\phi (\vare,\xi)\|_{\mu_2} \le {1\over {1-k}} \| {\cal
F}_{\mu_2}(\vare,\xi,0)\|_{\mu_2} \le {1\over {1-k}} \|S^{-1}\|
\|H(\vare,\xi)\|_{\mu_2},\quad \xi \in X_{\mu}\cap \overline{B_\sigma
(0)}.
\end{equation*}
For $\vare,\sigma >0$ small enough, from (\ref{e3.15}) we get
$$\| H(\vare, \xi)(t)\|_{\mu_2} \le C(\vare)\|\xi\|_{\mu_2} +\vare^2 \| {u^*}''\|_{\mu_2},\quad
\xi \in X_{\mu}\cap \overline{B_\sigma (0)}.$$
We thus obtain $\| \psi (\vare,\xi)\|_{\mu_2} \le
\|u^*\|_{\mu_2}+(1+ C(\vare))\sigma +\vare^2 \| {u^*}''\|_{\mu_2}\le
C_1$ for $\vare$ small, where $C_1$ does not depend on $\vare,\xi$.
Now we want to prove a similar estimate for the derivates
$d\psi(\vare,\xi)/dt$. For simplicity, we only prove the result for
$\xi=0$.
Since $\psi (t):=\psi (\vare,0)(t)$ is a solution of (\ref{e3.2}),
then $\psi (t)$ is given by the integral formula
\begin{equation*}
\fl \psi (t)={1\over {\sqrt{1+4\vare^2}}}\left ( \int_{-\infty}^t
e^{\al(\vare)(t-s)} [\psi (s)+f(\psi_s)]\, ds+ \int^{+\infty}_t
e^{\be(\vare)(t-s)} [\psi (s)+f(\psi_s)]\, ds\right ),
\end{equation*}
from which we derive
\begin{equation*}\fl \psi'(t)={1\over {\sqrt{1+4\vare^2}}}\Big
(\al(\vare)\hspace{-2mm} \int\limits_{-\infty}^t
e^{\al(\vare)(t-s)}[\psi (s)+f(\psi_s)]ds
-\be(\vare)\hspace{-2mm}\int\limits^{+\infty}_t e^{\be(\vare)(t-s)}[\psi (s)+f(\psi_s)]
ds\Big ).
\end{equation*}
Since $f$ is bounded on bounded sets of ${\cal
C}=C([-\tau,0];\mathbb{R}^N)$, there is $\ell$ such that
$|f(\phi)|\le \ell $ for $\phi\in {\cal C}$ with $\|\phi\|_\infty
\le C_1$. Thus, $|f(\psi_s)|\le \ell $ for $s\in\mathbb{R}$, where
$\ell$ does not depend on $\mu,\vare$, and $\|\psi'\|_\infty\le
2(C_1+\ell)/\sqrt{1+4\vare^2}.$ From (\ref{e3.10}), the
$C^1$-smoothness of $f$ and $f(0)=0$, we easily deduce that there is
$C_2>0$ such that $\|\psi'\|_\mu \le C_2$. This completes the
proof.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
In fact a stronger result can be proven:
\begin{cor} \label{Cor3.3} Assume (H1)-(H4), take $\mu \in I:=[\mu_1\mu_2]\subset (0,\la_0) \setminus \Re\, \sigma (A)$,
and consider the travelling wave profiles $\psi (\vare,\xi)=u^*+\xi+\phi (\vare, \xi)$
for $\vare\in (0,\vare^*),\xi\in X_\mu\cap B_\sigma (0)$, given in
Theorem \ref{Thm3.2}. For $\xi=0$ and $\mu\in I$, the profile $\psi
(\vare,0)$ satisfies
\begin{equation}\label{e3.20}
\psi (\vare,0)\to u^*\ {\rm in} \ C_\mu\quad {\rm as} \quad \vare\to 0^+.
\end{equation}
\end{cor}
{\it Proof.} Let $\vare^*,\sigma>0$ be as in the statement of
Theorem \ref{Thm3.2}, and recall that $\psi (\vare,0)=
\psi_\mu(\vare,0)$ only depends on $\vare$. Next, we deduce some
estimates as in Lemma \ref{L3.5}, so details are omitted.
For $\vare=0$, define
$$H(0,w)(t)=\int_{-\infty}^t e^{-(t-s)} [f(w_s+u_s^*)-f(u_s^*)-Df(u_s^*)w_s]\, ds.$$
We write $H(0,w)(t)=\int_{-\infty}^t e^{-(t-s)} G(0,t,w)\, ds:=H_2(0,w)$, where $G(0,t,w)$ is given by (\ref{e3.4}).
After some computations, we observe that the function $H$ restricted to $[0,\vare^*)\times (C_{\mu,0}\cap B_\sigma^\mu (0))$ satisfies
$$\|H(\vare,w)-H(0,w)\|_\mu\le C_0(\vare)\|w\|_\mu+D_0(\vare),
$$
with $C_0(\vare),D_0(\vare)$ independent of $\mu$, $C_0(\vare),D_0(\vare)\to 0$ as $\vare\to 0^+$.
This means that the function $(\vare,w)\mapsto H(\vare,w)$ converges, uniformly on $w\in C_{\mu,0}\cap B_\sigma^\mu (0)$, to $H(0, \cdot )$ in $C_{\mu,0}$ as $\vare \to 0^+$.
Moreover, for $\vare =0$ and $\xi=0$ the fixed point of (3.17) is
$\phi (0,0)=0$. Therefore, the application of the contraction
principle as in the proof of Theorem \ref{Thm3.2} leads to
(\ref{e3.20}).\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\begin{rem}\label{R3.2} As seen in Section 2, the existence of a positive
eigenvector ${\bf v}\in\mathbb{R}^N$ associated with the
characteristic root $\la_0$ of (\ref{e2.2}) was crucial to prove the
existence of a {\it positive} heteroclinic solution $u^*(t)$ of
(\ref{e1.2}), connecting the equilibria 0 to $K$. For all the
results in this section the positiveness of such heteroclinic
solution is irrelevant, and therefore it is not necessary to impose
the above requirement in (H4) that ${\bf v}$ is positive. For the
same reason, in Section 3 assumption (H2)(ii) is not needed as well.
\end{rem}
\section{Positiveness of travelling waves}
Consider the characteristic equation for the linearization of
(\ref{e3.2}) at 0,
\begin{equation}\label{e4.1}
\det \Delta_\vare (z)=0,\quad {\rm where}\quad \Delta_\vare (z):=\vare^2
z^2I-zI+L(e^{z\cdot}I),
\end{equation}
where $L=Df(0)$. Recall that for $\vare=0$ the characteristic
matrix-valued function $\Delta _0(z)$ was defined in (\ref{e2.2}).
Since $\la_0>0$ is a simple root of the characteristic equation
$\det \Delta_0 (z)=0$, from the implicit function theorem, for
$\vare >0$ small there is a simple real root $\la(\vare)$ of
(\ref{e4.1}), with $\la(\vare)\to \la_0$ as $\vare\to 0^+$.
\begin{lem}\label{L4.1} For $\de >0$ sufficiently small and $\de_1
>0$, there exists $\vare_0>0$ such that, for $0<\vare<\vare_0$,
$\la(\vare)$ is the only root of the characteristic equation
(\ref{e4.1}) on the vertical strip $\la_0-\de \le Re\, z\le \la_0
+\delta_1$.
\end{lem}
{\it Proof}. Let $\delta >0$ be such that $\la_0$ is the only root
of $\det \Delta_0 (z)=0$ on the strip $S=\{ z:\la_0-\delta \le
\Re\, z\le \la_0+\delta_1\}$. If $z(\vare)\in S$ is a root of $\det
\Delta_\vare (z)=0$, then there is a unit vector
$w=w(\vare)\in\mathbb{R}^N$ such that
$(\vare^2z(\vare)^2-z(\vare))w=L(e^{z(\vare)\cdot }w)$, hence
\begin{equation*}\fl
\|L\| \ge |\vare^2z(\vare)^2-z(\vare)|\ge |\Im\,
(\vare^2z(\vare)^2-z(\vare))|=
|\Im\, z(\vare)| |2\vare^2 \Re\, z(\vare)-1|.
\end{equation*}
Choose $\vare_0>0$ such that $ |2\vare^2 \Re\, z-1|>1/2$ for all
$z\in S, |\vare|<\vare_0$. For $|\vare|<\vare_0$, we have
\begin{equation*}
|\Im\, z(\vare)|<2\|L\|.
\end{equation*}
Thus, for $|\vare|<\vare_0$ the solutions $z(\vare)\in S$ of $\det
\Delta_\vare (z)=0$ are necessarily inside the rectangle
$\Gamma=[\la_0-\de, \la_0 +\de_1 ]\times \big [-2\|L\|,2\|L\|\big
]$.
Now, let $F(z,\vare):=\det \Delta_\vare (z), z\in\mathbb{C},
\vare\in \mathbb{R}$. Clearly, $F(z,\vare)\to F(z,0)$ as $\vare\to
0$, for all $z\in\mathbb{C}$. Moreover, since $\Delta_\vare
(z)=\vare^2z^2I+\Delta_0(z)$, one easily deduces that the function
$F(\cdot ,\vare)$ converges uniformly to $F(\cdot , 0)$ on bounded
sets of $\mathbb{C}$, as $\vare\to 0$.
We now apply Rouch\'e's Theorem on the boundary $\partial\Gamma$ of
$\Gamma$. Set $m=\min _{z\in \partial\Gamma}
|F(z,0)|>0$. For $|\vare|$ small, we have $|F(z,\vare)-F(z,0)|<m ,$
$ z\in\partial\Gamma,$ hence $F(z,\vare)$ and $F(z,0)$ have the same
number of zeros inside $\Gamma$. Thus, for $\vare>0$ sufficiently
small $\la(\vare)$ is the only solution of (\ref{e4.1}) in the strip
$S$.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\smal
For (\ref{e3.2}) written as a system in $\R^{2N}$, its linearized
equation at zero is
\begin{equation}\label{e4.2}
x'(t)={\cal L}_\vare (x_t),
\end{equation}
where
\begin{equation*}
{\cal L}_\vare \pmatrix{\phi_1\cr \phi_2\cr}=\pmatrix {\phi_2(0)\cr
-{1\over {\vare^2}}L(\phi_1)+{1\over {\vare^2}} \phi_2(0)\cr},\quad
\phi_1,\phi_2\in {\cal C}=C([-\tau ,0];\mathbb{R}^N).
\end{equation*}
For the linear system (\ref{e4.2}), the characteristic equation is
given by
\begin{equation}\label{e4.3}
\det D_\vare (s)=0, \quad {\rm where} \quad D_\vare (s) =\pmatrix
{sI&-I\cr {1\over {\vare^2}}L(e^{s\cdot }I)& (s-{1\over
{\vare^2}})I\cr},
\end{equation}
\smal and $I$ is the $N\times N$ identity matrix. Clearly, $\det
\Delta_\vare(s)=\vare^{2N} \det D_\vare(s)$, hence for $\vare >0$
(\ref{e4.3}) is equivalent to (\ref{e4.1}).
\begin{lem} \label{L4.2} Consider $b\in\mathbb{R}$ and $\vare_0\in (0,1)$
such that $\det \Delta_\vare(s)\ne 0$ for all $\vare \in [0,\vare_0]$ and $s$ on the vertical line
$\Sigma =\{ s=b+iy:y\in\mathbb{R}\}$.
Then there is $\vare_1 \in (0,\vare_0]$ such that
$$\sup \Big \{ |s|\, \|\Delta_\vare(s)^{-1}\|: 0\le \vare\le \vare_1, s\in \Sigma \Big \}<\infty,$$
and
$$\sup \Big \{ |s|\, \|D_\vare(s)^{-1}\|: 0< \vare\le \vare_1, s\in \Sigma \Big \}<\infty.$$
\end{lem}
{\it Proof.} For $\vare \in (0,\vare_0], s\in \Sigma$, we have
\begin{equation}\label{e4.4}\fl
G(\vare,s):=D_\vare(s)^{-1}=\pmatrix {\Delta_\vare(s)^{-1}&0\cr 0&\Delta_\vare(s)^{-1}\cr}
\pmatrix {(\vare^2 s-1)I&\vare^2I\cr -L(e^{s\cdot}I)&\vare^2sI\cr}.
\end{equation}
Clearly,
\begin{equation*}
\fl G(\vare,s)\to \pmatrix {\Delta_0(s)^{-1}&0\cr
0&\Delta_0(s)^{-1}\cr}
\pmatrix {-I&0\cr -L(e^{s\cdot}I)&0\cr}:=G(0,s)\quad {\rm as}\quad \vare\to 0^+,
\end{equation*}
with $G(\vare,s)$ continuous on $[0,\vare_0]\times \Sigma.$ For
$s\in \Sigma$, we have $\|L (e^{s\cdot}I)\|\le \max (1,
e^{-b\tau})\|L\|$. It follows that
$$
\|G(\vare,s)\|\le c (\vare^2|s|+1) \|\Delta_\vare(s)^{-1}\|,\quad 0<\vare\le \vare_0, s\in \Sigma,$$
for some $c>0$. Since $\Delta_\vare(s)=(\vare^2 s^2-s)I+L(e^{s\cdot}I)$, then
$$ \|\Delta_\vare(s)^{-1}\| \le {1\over {|s||\vare^2s-1|-\|L(e^{s\cdot}I)\|}}$$
if $|s||\vare^2 s-1|-\|L(e^{s\cdot}I)\|>0$. Choose $\vare_1\in (0,\vare_0]$ such that $1-\vare_1^2b\ge 1/2$. Then for $s\in\Sigma$ and $0< \vare\le \vare_1$, if $|s|\ge 4\max (1,e^{-b\tau})\|L\|:=c_1$, it follows that $|s||\vare^2 s-1|-\|L(e^{s\cdot}I)\|\ge |s|/4>0$,
thus $|s|\, \|\Delta_\vare(s)^{-1}\| \le 4$ and
$$
|s|\,\|G(\vare,s)\|\le {{c |s|(\vare^2|s|+1)}\over {|s|
|\vare^2s-1|- \|L(e^{s\cdot}I)\|}} \le {{c\vare^2|s|^2}\over {|s|
|\vare^2s-1|- \|L(e^{s\cdot}I)\|}}+4c
$$
for $\vare \in (0,\vare_1]$ and $s\in \Sigma, |s|\ge c_1.$ Now, $|s|
|\vare^2s-1|\ge |s| \sqrt {\vare^4|s|^2+1/2}\ge \vare^2 |s|^2$, and
$${{\vare^2|s|^2}\over {|s| |\vare^2s-1|- \|L(e^{s\cdot}I)\|}}\le 2$$
if $\vare^2|s|^2\ge 2 \|L(e^{s\cdot}I)\|$; and if $\vare^2|s|^2\le 2 \|L(e^{s\cdot}I)\|$, then
$${{\vare^2|s|^2}\over {|s| |\vare^2s-1|- \|L(e^{s\cdot}I)\|}}\le
{{ 2 \|L(e^{s\cdot}I)\|}\over {|s|/4}}\le 2;
$$
hence, $|s|\,\|G(\vare,s)\|\le 6c$ if $|s|\ge c_1$.
On the other hand, on the compact set $\{(\vare, s)\in
[0,\vare_1]\times \Sigma: |s|\le c_1\}$ the continuous functions
$|s|\, \|\Delta_\vare(s)^{-1}\|$ and $|s|\, \|G(\vare,s)\|$ attain
their suprema, and the conclusion follows.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
We are finally in a position to prove the main result of this
section, on the existence of positive travelling wave solutions of
Eq. (\ref{e1.1}) for large wave speeds.
\begin{thm}\label{Thm4.1} Assume (H1)-(H4). Then, there is $c^*>0$,
such that for $c>c^*$ Eq. (\ref{e1.1}) has a positive travelling
wave solution of the form $u(t,x)=\psi (ct+w\cdot x)$ for each unit
vector $w\in \mathbb{R}^p$, with $\psi(-\infty)=0,\psi(\infty)=K$.
Moreover, the components of the profile $\psi$ are increasing in the
vicinity of $-\infty$ and it satisfies $\psi(t)=O(e^{\la (\vare)t}),
\psi'(t)=O(e^{\la (\vare)t})$ at $-\infty$, where $\vare=1/c$ and
$\la (\vare)$ is the real solution of (\ref{e4.1}) with
$\la(\vare)\to \la_0$ as $\vare\to 0^+$. \end{thm}
{\it Proof}.
Consider Eq. (\ref{e3.2}), where $\vare=1/c$. Let $\mu \in (0,
\la_0)$ be as in the statement of Corollary \ref{Cor3.1} and satisfy
$\la_0< \mu+2\delta$ for some fixed $\delta \in (0,\mu/4)$. Suppose
also that $\la(\vare)$ is the unique solution of $\det \Delta_\vare
(z)=0$ on the strip $\mu-\delta \le \Re\, z < 2\mu$ for all $\vare
\in [0,\vare^*]$. Fix the profiles
\begin{equation}\label{e4.5}
\psi_\vare=\psi(\vare,0), \quad \psi_0(t):= u^*(t)=e^{\la_0 t}{\bf v}
+O(e^{2\mu t}), \quad \vare \in [0, \vare^*],
\end{equation}
as in Corollary \ref{Cor3.3} and Theorem \ref{T2.1}. Recall that
${\bf v}$ is a positive eigenvector associated with $\la_0$. The
proof is now divided in several steps.
\smal
{\it Claim 1}. There is $\vare_0>0$ such that $(\psi_\vare
(t),\psi_\vare'(t))=(e^{\la (\vare)t}{\bf v}^1(\vare
),\la(\vare)e^{\la (\vare)t} {\bf v}^1(\vare ))+w_\vare (t),$ $\vare
\in [0,\vare_0)$, with continuous ${\bf v}^1(\vare ) >0, {\bf
v}^1(0) ={\bf v},$ and $w_\vare (t)=O(e^{(\la_0+\delta)t})$ at
$-\infty$. \smal
To prove the above claim, note that $x_\vare(t):=(\psi_\vare(t),
\psi_\vare '(t))$ is a solution of the system
\begin{eqnarray}\label{e4.6}
\nonumber & & x_1'(t)=x_2(t)\\
& & \vare^2 x_2'(t)= x_2(t)-L(x_{1,t})-h_{\vare}(t),
\end{eqnarray}
where $L=Df(0)$ and
$h_{\vare}(t)=f((\psi_\vare)_t)-L((\psi_\vare)_t)$. For $\vare
>0$, equivalently we write (\ref{e4.6}) as
\begin{equation*}
x'(t)={\cal L}_\vare (x_t)-{1\over {\vare^2}}\pmatrix{0\cr
h_\vare(t)\cr},
\end{equation*}
where ${\cal L}_\vare$ is as in (\ref{e4.2}). Since $f$ is a $C^2$
function, using the Taylor formula for $f$ (cf. e.g. \cite[p.
23]{CH}), we have the estimate
$$|h_{\vare}(t)|\le \int_0^1 (1-s) \|D^2f(s (\psi_\vare)_t)\|
\|(\psi_\vare)_t\|_\infty^2\, ds, \quad t\in \mathbb{R}, \vare \in
[0,\vare^*].$$ Since $\|\psi_\vare-u^*\|_\mu \to 0$ as $\vare\to
0^+$ and $u^*(t)\to 0$ as $t\to -\infty$, the continuity of $D^2f$
at 0 implies that $\| D^2f(s (\psi_\vare)_t)\|$ is uniformly bounded
on $\vare \in [0,\vare_1] \subset [0,\vare^*), \ t \leq 0, \ s \in
[0,1]$. Together with Corollary \ref{Cor3.2}, this leads to
\begin{equation}\label{e4.7}
|x_\vare(t)|\le Ce^{\mu t},\quad |h_{\vare}(t)|\le D e^{2\mu t}\quad {\rm for}\quad t\le 0,
\end{equation}
for some constants $C,D$ independent of $\vare \in [0,\vare_1]$.
We now apply Theorem \ref{TA.1} to (\ref{e4.6}) at $-\infty$ (see
Appendix), and derive that for $\vare >0$
\begin{equation}\label{e4.8}
x_\vare(t)=(\psi_\vare (t), \psi_\vare '(t))=z_\vare(t)+w_\vare (t)
\end{equation}
where $z_\vare(t)$ is an eigenfunction for the linear system
$x'(t)={\cal L}_\vare (x_t)$ corresponding to the set $\Lambda
_\vare=\{ z\in \mathbb{C}: \det \Delta_\vare (z)=0, \mu \le \Re\, z
<2\mu\}$ and $w_\vare(t)=O(e^{(\la_0+\delta))t})$ at $-\infty$.
From Lemma \ref{L4.1}, let $\vare$ be on some interval
$(0,\vare_0)\subset (0,\vare_1)$ such that that $\Lambda _\vare=\{
\la(\vare)\}$. Then, $z_\vare(t)$ is an eigenfunction for
$x'(t)={\cal L}_\vare (x_t)$ associated with the root $\la (\vare)$
of (\ref{e4.3}), hence $z_\vare(t)=e^{\la(\vare)t}{\bf v}(\vare)$
with ${\bf v}(\vare)=({\bf v}^1(\vare),{\bf v}^2(\vare))\in
\mathbb{R}^{2N}$ satisfying $D_\vare (\la(\vare)){\bf v}(\vare)=0$
for $D_\vare$ as in (\ref{e4.3}). From this we obtain ${\bf
v}^2(\vare)=\la(\vare){\bf v}^1(\vare)$ and $\Delta_\vare
(\la(\vare)){\bf v}^1(\vare)=0.$ Furthermore, from Theorem
\ref{TA.1} (with $a=\mu, b=\la_0+2\delta, \epsilon=\delta$) and
formulae (\ref{eA.5}) and (\ref{eA.7}) (adapted to the situation
$-\infty$) we get
\begin{equation*}\fl
z_\vare(t)=Res\, (e^{t\cdot}\widetilde {\bf
x}_\vare,\la(\vare))=e^{\la(\vare)t}\lim_{s\to \la(\vare)}
(s-\la(\vare))\widetilde {\bf x}_\vare(s)=e^{\la(\vare)t}({\bf
v}^1(\vare),\la(\vare){\bf v}^1(\vare))
\end{equation*}
and
\begin{equation}\label{e4.9}
w_\vare(t)={1\over {2\pi i}} \int_{b+{\delta\over
2}-i\infty}^{b+{\delta\over 2}+i\infty} e^{st}\widetilde {\bf
x}_\vare(s)\, ds,
\end{equation}
with
\begin{equation}\label{e4.10}
\widetilde {\bf x}_\vare (s):=\widetilde{ x_\vare (-\cdot
)}(-s)=G(\vare,s)\left (r_\vare(s)-\pmatrix {0\cr \widetilde {\bf
h}_\vare (s)\cr}\right ),
\end{equation}
for $G(\vare,s)=D_\vare (s)^{-1}$ and
\begin{eqnarray}\label{e4.11}
\nonumber & & r_\vare(s)=x_\vare(0)+ {\cal L}_\vare \left (
e^{s\cdot} \int _{\cdot}^0 e^{-su} x_\vare(u)\, du \right ),\\
& & \widetilde {\bf h}_\vare (s):=\widetilde {h_\vare (-\cdot
)}(-s)=\int_{-\infty}^0e^{-su}h_\vare(u)\, du,\quad \Re\,
s<\la_0+2\delta.
\end{eqnarray}
Here, $\widetilde{ x_\vare (-\cdot )},\widetilde{ h_\vare (-\cdot
)}$ denote the Laplace transforms of the functions $t\mapsto
x_\vare(-t) ,t\mapsto h_\vare(-t)$, respectively. Note that
$\widetilde {\bf x}_\vare (s)$ is meromorphic for $\Re\, s<2\mu$
with a unique singularity at $s = \la (\vare)$ which is a simple
pole of $\Delta_\vare (s)^{-1}$ (cf. Appendix).
From (\ref{e4.8}), we get
\begin{equation}\label{e4.12}
\psi_\vare (t)=e^{\la (\vare)t} {\bf v}^1(\vare)+w^1_\vare (t),\
\psi_\vare'(t)=\la(\vare)e^{\la (\vare)t} {\bf v}^1(\vare)+w^2_\vare
(t),
\end{equation}
with $ w^1_\vare (t)=O(e^{(\la_0+\delta) )t})$ at $-\infty$ and
$w^2_\vare (t)=(w^1_\vare (t))'$. Next, the definition of
$r_\vare(s)$ in (\ref{e4.11}) yields
$$r_\vare(s)=x_\vare(0)+\pmatrix {0\cr -{1\over\vare^2} L\Big (e^{s\cdot } \int _{\cdot}^0 e^{-st} \psi_\vare(t)\, dt\Big )\cr},$$
hence from (\ref{e4.4}) and (\ref{e4.10}) we obtain
\begin{equation}\label{e4.13}\fl
\widetilde {\bf x}_\vare (s)= G(\vare,s)x_\vare(0)- \pmatrix
{\Delta_\vare (s)^{-1} L\Big (e^{s\cdot } \int _{\cdot}^0 e^{-st}
\psi_\vare(t)\, dt\Big )\cr s\Delta_\vare (s)^{-1} L\Big (e^{s\cdot
} \int _{\cdot}^0 e^{-st} \psi_\vare(t)\, dt\Big )\cr} -\pmatrix {
\Delta_\vare (s)^{-1} \widetilde {\bf h}_\vare(s)\cr s\Delta_\vare
(s)^{-1} \widetilde {\bf h}_\vare(s)\cr}.
\end{equation}
We now extend naturally this situation for $\vare=0$. In
(\ref{e4.5}), denote $\la(0)=\la_0, {\bf v}^1(0)={\bf v} $. Write
$\widetilde {\bf x}_\vare= (\widetilde {\bf x}_\vare^1,\widetilde
{\bf x}_\vare^2)$ for $\vare \in (0,\vare_0)$, and let $\widetilde
{\bf x}_0^1(s)$ be defined by (\ref{e4.13}) for $\vare=0$. Note
that formula (\ref{e4.10}) can still be used to obtain $\widetilde
{\bf x}_0^1(s)$ (cf. Appendix for more details),
$$
\widetilde {\bf x}_0^1(s)=\Delta_0(s)^{-1}[r_0^1(s)+\tilde {\bf
h}_0(s)],$$ where $r_0^1(s)=u^*(0)+L\big (e^{s\cdot } \int
_{\cdot}^0 e^{-st} u^*(t)\, dt\big )=\lim_{\vare\to
0^+}r_\vare^1(s)$.
For each $\vare\in [0,\vare_0)$, $s=\la(\vare)$ is a pole of order
one of $G(\vare,s)$, and from (\ref{e4.10}) we deduce that for
$\vare \in [0,\vare_0)$ and $\mu\le \Re\, s<\la_0+2\de$ the
function $A(\vare,s)$ defined by
$A(\vare,s)=(s-\la(\vare))\widetilde {\bf x}_\vare^1(s)$ for $s\ne
\la(\vare)$, $A(\vare,\la(\vare))={\bf v}^1(\vare)$ is analytic on
$s$ and continuous on $(\vare,s)$. In particular, $\lim_{\vare\to
0^+}A(\vare,\la(\vare))= A(0,\la_0)={\bf v}>0$, hence ${\bf
v}^1(\vare)\to {\bf v}$ as $\vare\to 0^+$. Moreover, ${\bf
v}^1(\vare)>0$ for $\vare>0$ sufficiently small. This proves Claim
1.
\med
{\it Claim 2}. For $\vare_0^*>0$ sufficiently small, there exists a
constant $D_0>0$ such that
\begin{equation}\label{e4.14}
|w^1_\vare (t)|\le D_0e^{(\la_0+\delta)t}\quad {\rm for\ all }\quad t\le
0,0<\vare <\vare_0^*.
\end{equation}
\med
To prove Claim 2, once more we shall use some formulae and estimates
in the proof of Theorem \ref{TA.1} in the Appendix, changed
accordingly to account for the asymptotic behaviour at $-\infty$,
rather than $\infty$.
Define $v_\vare
(t)=(v_\vare^1(t),v_\vare^2(t))=e^{-(\la_0+\delta)t}w_\vare (t)$ and
$u_\vare
(t)=(u_\vare^1(t),u_\vare^2(t))=e^{-(\la_0+3\delta/2)t}w_\vare (t)$.
Note that $v_\vare^1(0)=w_\vare^1(0)$ an
$v_\vare^1(t)=w_\vare^1(0)-\int_t^0 (v_\vare^1)'(s)\, ds$, with
$w_\vare^1(0)=\psi_\vare(0)-{\bf v}^1(\vare)$. Then
$|w_\vare^1(0)|\le C+|{\bf v}^1(\vare)|$, where $C>0$ is as in
(\ref{e4.7}). We need to prove that $v_\vare^1(t)$ is uniformly
bounded for $t\le 0$ and $\vare>0$ small enough.
In order to achieve this, we shall show that
there are constants $C_0,D_0>0$ and $\vare_0^*>0$, such that
\begin{equation}\label{e4.15}
\|v_\vare^1\|_{L^1(-\infty,0]}\le C_0,\quad \vare\in (0,\vare_0^*)
\end{equation}
and
\begin{equation}\label{e4.16}
\|(v_\vare^1)'\|_{L^1(-\infty,0]}\le D_0/2,\quad \vare\in
(0,\vare_0^*),
\end{equation}
so that (\ref{e4.14}) follows immediately from (\ref{e4.16}) and
$|w_\vare^1(0)|\le D_0/2$ for $\vare\in (0,\vare_0^*)$. These
uniform estimates require a careful analysis of the explicit
formulae for $w_\vare$ given in (\ref{e4.9}) and (\ref{e4.10}). We
shall prove (\ref{e4.15}) beforehand, and then use (\ref{e4.15}) to
prove (\ref{e4.16}).
First, observe that $x_\vare(t)=z_\vare(t)+w_\vare(t)$ is a solution
of (\ref{e4.6}), with $z_\vare(t)$ being an eigenfunction for the
linear system $x'(t)={\cal L}_\vare (x_t)$, hence $w_\vare(t)$ is a
solution of system (\ref{e4.6}) as well. The definition of
$v_\vare(t)$ yields now
\begin{equation}\label{e4.17}
(v_\vare^1)'(t)=-(\la_0+\delta) v_\vare^1(t)+v_\vare^2(t)
\end{equation}
and
\begin{equation}\label{e4.18}
\vare^2(v_\vare^1)''(t)-\al \, (v_\vare^1)'(t)+P_\vare(t)=0,
\end{equation}
where $\al=1-2\vare^2(\la_0+\delta)$ and
\begin{equation}\label{e4.19}\fl
P_\vare(t)=[\vare^2(\la_0+\delta)^2-(\la_0+\delta)]
v_\vare^1(t)+L\Big (e^{(\la_0+\delta)\cdot} (v_\vare^1)_t\Big )+
e^{-(\la_0+\delta)t}h_\vare(t).
\end{equation}
Next, similarly to (\ref{eA.9}) and Remark \ref{RA1}, using
(\ref{e4.9}) and the Plancherel theorem, we obtain
$$\|v_\vare^1\|_{L^1(-\infty, 0]}=\|w^1_\vare(t)e^{-(\la_0+\delta)t}\|_{L^1(-\infty,
0]}= \| {{e^{\delta t/2}}\over {2\pi}} \int_{-\infty}^{+\infty}
e^{iut}\widetilde {\bf x}^1_\vare(\la_0 + 3\delta/2+iu)\,
du\|_{L^1(-\infty, 0]}
$$
\begin{equation}\label{e4.20}
\le C_1(\delta) \| \widetilde {\bf
x}_\vare^1(\la_0+3\delta/2-i\cdot)\|_{L^2(\R)}\quad {\rm for}\quad \
\vare\in (0,\vare_0).
\end{equation}
From (\ref{e4.7}), $ |x_\vare(0)|$ is uniformly bounded for $\vare
\in (0,\vare_0)$; and since $2\mu>\la_0+2\delta$, (\ref{e4.7}) also
implies that $\| L\Big (e^{s\cdot } \int _{\cdot}^0 e^{-st}
\psi_\vare(t)\, dt\Big )\|$ and $|\widetilde {\bf h}_\vare(s)|$ are
uniformly bounded for $\vare \in (0,\vare_0)$ and $\Re\,
s=\la_0+3\delta/2$. Using now Lemma \ref{L4.2}, from (\ref{e4.13})
we conclude that there is $K_1>0$ such that
\begin{equation*}
|\widetilde {\bf x}_\vare^1(s)|\le K_1/|s|,\quad {\rm for }\
s=b+{{3\delta}\over 2}+iy,\ y\in\mathbb{R},\ {\rm and}\
0<\vare<\vare_0,
\end{equation*}
from which we derive
$$\| \widetilde {\bf x}_\vare^1(\la_0+3\delta/2-i\cdot)\|_{L^2(\R)}^2\le
\int_{\R} {K_1\over { (b+{{3\delta}\over 2})^2+y^2}}\, dy<\infty.$$
Together with (\ref{e4.20}), this leads to the estimate
(\ref{e4.15}) with $\vare_0^*= \vare_0$.
In order to prove (\ref{e4.16}), we now use (\ref{e4.18}) and some
ideas from Aguerrea et al. \cite[Lemma 4.1]{ATV}. After partial
integration of Eq. (\ref{e4.18}), we obtain
\begin{equation}\label{e4.21}
(v_\vare^1)'(t)=e^{(\al/\vare^2)t} (v_\vare^1)'(0)+
{{e^{(\al/\vare^2)t}}\over \vare^2} \int_t^0 e^{-(\al/\vare^2)s}
P_\vare (s)\, ds.
\end{equation}
Let $\vare>0$ be small, so that $\al>0$. From (\ref{e4.17}) and
Claim 1,
$$ (v_\vare^1)'(0)=-(\la_0+\de) w_\vare^1(0)+w_\vare^2(0)=
-(\la_0+\de)\psi_\vare(0)+(\la_0+\de -\la(\vare)){\bf
v}^1(\vare)+\psi_\vare'(0),$$ with ${\bf v}^1(\vare)\to {\bf v}$ as
$\vare\to 0^+$. Corollary \ref{Cor3.2} implies that
$|(v_\vare^1)'(0)|$ is uniformly bounded on $\vare >0$ sufficiently
small, hence
\begin{equation}\label{e4.22}
|(v_\vare^1)'(0)|\int_{-\infty}^0 e^{(\al/\vare^2)t}\, dt\le K_2,
\end{equation}
for some $K_2>0$ and $\vare >0$ sufficiently small. On the other
hand, interchanging the order of integration leads to
$$
{1\over {\vare^2}}\int_\sigma^0 e^{(\al/\vare^2)t} \left ( \int_t^0
e^{-(\al/\vare^2)s}P_\vare (s)\, ds\right ) dt={1\over {\vare^2}}
\int_\sigma^0 e^{-(\al/\vare^2)s}P_\vare (s)\left ( \int_\sigma^s
e^{(\al/\vare^2)t}\, dt\right ) ds, $$ hence
$$ {1\over {\vare^2}}\int_{-\infty}^0 e^{(\al/\vare^2)t} \left ( \int_t^0 e^{-(\al/\vare^2)s}|P_\vare (s)|\, ds\right ) dt\le {1\over \al} \int _{-\infty}^0 |P_\vare (s)|\, ds.$$
From (\ref{e4.7}), (\ref{e4.15}) and the definition of $P_\vare
(s)$ in (\ref{e4.19}) , we easily see that
$$\|P_\vare\|_{L^1(-\infty,0]}\le K_3,\quad 0<\vare <\vare_0^*$$
for some $\vare_0^*>0$. Together with (\ref{e4.21}) and
(\ref{e4.22}), this yields the estimate (\ref{e4.16}), and therefore
Claim 2 is proven.
\med
We finally prove:
\med
{\it Claim 3}. There is $\vare_1^*>0$ such that $\psi_\vare (t)>0$
for $t\in\mathbb{R}$ and $\vare \in (0,\vare_1^*)$.
\med
From Claims 1 and 2, for $\vare >0$ small enough
$$\psi_\vare (t)\ge e^{\la(\vare)t}{\bf v}^1(\vare)-D_0e^{(\la_0+\delta)t} \vec{\bf 1}
\ge e^{\la(\vare)t}[{\bf v}^1(\vare)-D_0 e^{(\delta /2)t}\vec {\bf
1}],\quad t\le 0,$$ where $\vec {\bf 1}=(1,\dots, 1)$ and
$|\la(\vare)-\la_0|<\delta/2$. Choose $T^*\le 0$ and $\vare_1^*>0$
such that
$${\bf v}^1(\vare)> {\bf v}/2 \quad {\rm if}\quad 0<\vare<\vare_1^*, \quad {\rm and}\quad D_0 e^{(\delta /2)T^*}\vec {\bf 1}\le {\bf v}/4.$$
Then
$$\psi_\vare (t)\ge e^{\la(\vare)t} {\bf v}/4>0, \quad 0<\vare<\vare_1^*,\ t\le T^*.$$
On the other hand, since $\| \psi_\vare -u^*\|_\infty\to 0$ as
$\vare\to 0^+$, we define $\eta:=\inf_{t\ge T^*} u^*(t)>0$, and
suppose that $\vare_1^*$ was chosen so that
$\| \psi_\vare -u^*\|_\infty<\eta$ for $0<\vare<\vare_1^*$.
It follows that $\psi_\vare (t)>0$ for all $t\in\mathbb{R}$ and $\vare\in (0,\vare_1^*)$. The proof of
the theorem is complete.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\med
The above proof shows that the requirement that there is a positive
eigenvector ${\bf v}\in \mathbb{R}^N$ for the dominant
characteristic value $\la_0$ of (\ref{e2.1}) is crucial to deduce
the {\it positiveness} of the travelling wave fronts, for large wave
speeds. Nevertheless, the {\it existence} of such waves and their
asymptotic behaviour at $-\infty$ can be deduced from our Theorem
\ref{Thm3.2}, as well as the auxiliary results in Section 3, and
the proof of the above Claim 1. We summarize these remarks in the
following theorem:
\begin{thm} \label{Thm4.2} Assume (H1), (H2)(i), and\vskip 0cm
(i) for Eq. (\ref{e1.2}) the equilibrium $u=K$ is localy asymptotically stable;\vskip 0cm
(ii) for Eq. (\ref{e1.2}), the linearized equation about 0 has a real characteristic root $\la_0>0$, which is simple and dominant;\vskip 0cm
(iii) Eq. (\ref{e1.2}) has a heteroclinic solution $u^*(t), t\in\mathbb{R}$,
with $u^*(-\infty)=0, u(\infty)=K$ and $u^*(t)= O(e^{\la_0 t})$ at
$-\infty$.\vskip 0cm Then, there is $c^*>0$, such that for $c>c^*$,
Eq. (\ref{e1.1}) has a travelling wave solution of the form
$u(t,x)=\psi (ct+w\cdot x)$ for each unit vector $w\in
\mathbb{R}^p$, with
$\psi(t)=O(e^{\la (\vare)t}), \psi'(t)=O(e^{\la (\vare)t})$ at $-\infty$, where $\vare=1/c$
and $\la (\vare)$ is the real solution of (\ref{e4.1}) with $\la(\vare)\to \la_0$ as $\vare\to 0^+$.
\end{thm}
\section{Applications}
\subsection{A diffusive generalized logistic equation with
distributed delay}
As a first application, we consider a scalar reaction-diffusion equation with distributed delays in the reaction-terms, which includes
the Fisher-KPP equation with delay as a particular case.
Let ${\cal C}=C([-\tau,0];\mathbb{R}), \tau >0,$ and consider
\begin{equation}\label{e5.1}
{{\p u}\over {\p t}}(t,x)={{\p u^2}\over {\p x^2}} (t,x)+bu(t,x)[1-L(u_t(\cdot ,x))],\quad t\in \mathbb{R},\ x\in \mathbb{R},
\end{equation}
where $b>0$ and $L:{\cal C}\to \mathbb{R}$ is a nonzero positive
linear operator, i.e., $L\ne 0$ is linear and $L(\var)\ge 0$
whenever $\var \ge 0$. In particular, $L$ is bounded and
$\|L\|=L(1)>0$.
The corresponding delayed ODE model,
\begin{equation}\label{e5.2}
u'(t)=bu(t)[1-Lu_t],\quad t\in\mathbb{R},
\end{equation}
has two equilibria, $u=0$ and $u=L(1)^{-1}:=K$. Here $f$ in (1.2)
reads as $f(\var)=b\var (0)[1-L(\var)], \var\in {\cal C}$, for which
it is easy to verify that conditions (H1) and (H2) are satisfied.
The linearized equation about zero is the ODE $u'(t)=bu(t)$, with
characteristic value $b>0$.
Now, $u'(t)=-bKLu_t$ is the linearized equation about $K$, with
characteristic equation $P(\la ):=\la +bK L(e^{\la \cdot})=0$. If
$\la=i\omega, \omega>0,$ is a solution of $P(\la )=0$, then $L(\cos
(\omega \cdot))=0$ and $0= \omega +bK L(\sin (\omega \cdot))\ge
\omega-b$. If $\omega\tau <\pi /2$, we deduce that there is $\de >0$
such that $\cos (\omega \th)\ge \de$ for $\th \in [-\tau,0]$, hence
$L(\cos (\omega \cdot))\ge \de L(1)>0$, which is a contradiction. On
the other hand, if $b\tau\le 3/2$, from \cite{F} we conclude that
the positive equilibrium $K$ is globally attractive in the set of
all positive solutions of (\ref{e5.2}) with initial conditions
$\var\in{\cal C}_+,\var (0)>0$. We thus conclude that (H3) holds if
$b\tau\le 3/2$. Therefore, the following result is an immediate
consequence of Theorem \ref{Thm4.1}
\begin{thm}\label{T5.1} If $b\tau\le 3/2$, there exists $c^*>0$ such that
for $c>c^*$ equation (\ref{e5.1}) has a positive travelling wave
solution $u(t,x)=\psi (x+ct)$ with $\psi (-\infty )=0,\psi
(\infty)=K$. Moreover, $\psi$ is increasing in a vicinity of
$-\infty$ and it has the asymptotic decay $\psi (t)=O(e^{b(c)t}),
\psi '(t)=O(e^{b(c)t})$ at $-\infty$, where
$b(c)=2bc/(c+\sqrt{c^2-4b})$. \end{thm}
We note that (\ref{e5.1}) includes as a particular case the
Fisher-KPP equation with a single delay,
\begin{equation}\label{e5.3}
{{\p u}\over {\p t}}(t,x)={{\p u^2}\over {\p x^2}}
(t,x)+bu(t,x)[1-u(t-\tau ,x)/K],\quad t\in \mathbb{R},\ x\in
\mathbb{R}.
\end{equation}
By using a pair of upper-lower solutions and a monotone iterative
method, Wu and Zou \cite{WZ} proved that if $c>2\sqrt b$, then there
exists $\tau^*(c)>0$ such that, for a delay $\tau\le\tau ^*(c)$,
(\ref{e5.3}) has a non-decreasing travelling wave front connecting 0
to $K$ with wave speed $c$. Our approach does not allow us to
determine the minimal wave speed $c^*$, but, on the contrary, we
explicitly exhibit the maximal delay $\tau^*=3/(2b)$, under which we
can assure the existence of such positive (but not necessarily
monotone) travelling solutions.
Travelling waves for the Fisher-KPP equation (\ref{e5.3}) with $b=1$
were also considered in \cite[Corollary 6.6]{FHW}, where it was
shown that, for $\tau \le e^{-1}$, there exists $c^*>0$ such that
there is a travelling wave front with wave speed $c>c^*$. We remark
that Theorem \ref{T5.1} applied to (\ref{e5.3}) with $b=1$ clearly
improves this result, since it guarantees the existence of
travelling wave solutions for $\tau\le 3/2$, and, most relevant in
biological terms, it does assert that such travelling waves are {\it
positive}.
For some other recent results and references on the Fisher-KPP
equation, see \cite{BNPR, GT}.
\subsection{A chemostat model with delayed growth response }
Consider the following model for the growth of bacteria in a
well-stirred chemostat supplied by a single essential nutrient (cf.
Ellermeyer \cite{E1} and Ellermeyer et al. \cite{E2}):
\begin{eqnarray}\label{e5.4}
\nonumber & & S'(t)=D(S^0-S(t))-f(S(t))u(t),\\
& & u'(t)=e^{-D\tau} f(S(t-\tau))u(t-\tau)-Du(t).
\end{eqnarray}
Here $S(t)$ and $u(t)$ are the concentration of nutrient in the
growth vessel and the biomass concentration of bacteria at time $t$,
respectively, $D>0$ is the dilution rate of the chemostat, $S^0>0$
is the input concentration of nutrient, and $\tau\ge 0$ is the delay
in the growth response, to account for the lag in the nutrient
conversion into biomass due to cellular absorption; $f$ is the
specific functional response for the bacteria, and typically the
Michaelis-Menten response is chosen, $ f(s)=ms/(a+s),\ s\ge 0, $
with $m,a>0$. More generally, one can consider a continuously
differentiable and bounded function $f:[0,\infty)\to [0,\infty)$
with
\begin{equation}\label{e5.5}
f(0)=0,\quad f'(s)>0\ {\rm for}\ s\in\mathbb{R}.
\end{equation}
For an unstirred chemostat, the nutrient is added to the vessel but
not mixed, so one has to introduce diffusion terms. The diffusion
rates $d_1,d_2>0$ for the nutrient and the organisms may be
different, and model (\ref{e5.4}) becomes
\begin{eqnarray}\label{e5.6}
\nonumber & & {{\p S}\over {\p t}}(t,x)=d_1\Delta
S(t,x)+D(S^0-S(t,x))-f(S(t,x))u(t,x)\\
& & {{\p u}\over {\p t}}(t,x)=d_2\Delta u(t,x)+e^{-D\tau}
f(S(t-\tau,x))u(t-\tau,x)-Du(t,x),
\end{eqnarray}
for $t\in \mathbb{R}$ and $x\in (0,L)\ (L>0)$ (or more generally
$x\in\Omega$, where $\Omega \subset \mathbb{R}^3$ is an open
domain). There is an extensive literature on chemostat models with
``delayed growth response", with and without diffusion. We refer to
\cite{BR,E1,E2,GK,SW,WXR}, for results, other related chemostat
models, biological explanations, and further references.
For both (\ref{e5.4}) and (\ref{e5.6}), there is always the
equilibrium $(S^0,0)$, corresponding to the ``washout state". If
\begin{equation}\label{e5.7}
f(S^0)>De^{D\tau},
\end{equation}
there is another nonnegative equilibrium $(\bar S,\bar u)$, called
the ``survival state", given by $(\bar S,\bar
u)=(f^{-1}(De^{D\tau}), e^{-D\tau}(S^0-\bar S))$. Condition
(\ref{e5.7}) imposes a restriction on the size of the time-delay
$\tau$, which should satisfy $\tau<D^{-1}\log(f(S^0)D^{-1})$ for
$f$ as in (\ref{e5.5}). Moreover, (\ref{e5.7}) implies that the
equilibrium $(S^0,0)$ of (\ref{e5.4}) is unstable and $(\bar S,\bar
u)$ is asymptotically stable and a global attractor of all solutions
with initial conditions $(S_0,u_0)=(\phi_1,\phi_2)\in {\cal C}_+$,
$\phi_2(0)>0$ \cite{E1,E2}.
In order to apply our results, we first observe that both the
positive cone ${\cal C}_+$ and the set $\{ (\phi_1,\phi_2)\in {\cal
C}_+: \phi_2(0)> 0, \phi_1(0)< S^0\}$ are positively invariant for
(\ref{e5.4}). Translating the washout state to the origin, by
setting $s(t)=S^0-S(t)$, we rewrite (\ref{e5.4}) as
\begin{eqnarray}\label{e5.8}
\nonumber & & s'(t)=-Ds(t)+f(S^0-s(t))u(t)\\
& & u'(t)=e^{-D\tau} f(S^0-s(t))u(t-\tau)-Du(t).
\end{eqnarray}
If $f$ is $C^2$-smooth and (\ref{e5.7}) holds, then Eq. (\ref{e5.8})
satisfies (H1) and (H2). The set $A=\{ (\phi_1,\phi_2)\in {\cal
C}_+: \phi_2(0)> 0, 0<\phi_1(0)\le S^0\}$ is positively invariant,
and (\ref{e5.8}) has equilibria $E_0=(0,0)$ and $K:= (\bar s,\bar
u)>0$, with $\bar s=S^0-\bar S, \bar u=e^{-D\tau} \bar s$, the first
one being unstable and the second one being locally stable and a
global attractor of all solutions with initial conditions in $A$
(cf. \cite{E1}). It remains to verify that (H4) holds.
The characteristic equation for the linearization of (\ref{e5.8}) at
$(0,0)$ is
\begin{equation}\label{e5.9}
(\la+D)(\la+D-e^{-D\tau} f(S^0)e^{-\la \tau})=0.
\end{equation}
Define $h(x)=(x+D)e^{(x+D)\tau}, x\in \mathbb{R}$. Under
(\ref{e5.7}), there are two real roots of (\ref{e5.9}), $-D$ and
$\la_0$, where $\la_0>0$ is the unique solution of $h(x)=f(S^0)$.
For $\la\notin \mathbb{R}$ a solution of (\ref{e5.9}), we have
$h(\Re\, \la)<f(S^0)$. Since $h'(x)>0$ for $x>0$, if follows that
$\Re\, \la <\la_0$, and we conclude that $\la_0$ is a dominant
eigenvalue for the linearization of (\ref{e5.8}) about $(0,0)$, with
${\bf v}=(f(S^0)(\la_0+D)^{-1}, 1)>0$ as an associated eigenvector.
From Theorems \ref{T2.1} and \ref{Thm4.1}, we therefore obtain the
following result:
\begin{thm}\label{Thm5.2} Consider Eq. (\ref{e5.4}), where $S^0,D,\tau >0$,
the function $f:[0,\infty)\to [0,\infty)$ is bounded, $C^2$-smooth
and satisfies (\ref{e5.5}) and (\ref{e5.7}). Then, there exists a
heteroclinic solution $(S^*(t), u^*(t))$ connecting the washout
state $(S^0,0)$ to the survival state $(\bar S,\bar u)$ of
(\ref{e5.4}), with $0<S^*(t)<S^0, u^*(t)>0$ for $t\in\mathbb{R}$ and
$(S^*(t), u^*(t))=(S^0,0)+O(e^{\la_0 t})$ at $-\infty$, where
$\la_0>0$ satisfies $\la_0+D=f(S^0)e^{-(\la_0+D)\tau}$. For the
diffusion model (\ref{e5.6}) with $d_1,d_2>0$, there is $c^*>0$ such
that for $c>c^*$ (\ref{e5.8}) has a positive travelling wave
solution of the form $(S(t,x),u(t,x))=(\psi_1(ct+x),\psi_2(ct+x))$,
with $\psi_1(-\infty)=S^0,\psi_2(-\infty)=0$ and
$\psi_1(\infty)=\bar S,\psi_2(\infty)=\bar u$; moreover,
$\psi_1'(t)<0,\psi_2'(t)>0$ in the vicinity of $-\infty$, and
$(\psi_1(t),\psi_2(t))=(S^0,0)+O(e^{z (c) t})$ at $-\infty$, where
$z(c)$ is the real solution of
\begin{equation}\label{e5.10}\fl
z^2-cz -c(D-e^{-(D+\la)\tau}f(S^0))=0.
\end{equation}
\end{thm}
{\it Proof.} Let $\vare=1/c$, for $c>0$ large. With the notation in
(\ref{e4.3}), we have that
\begin{equation}\label{e5.11}\fl
\det D_\vare (\la)={1\over {\vare^2 d_1d_2}}( \vare d_1\la^2-\la -D)
\Big (\vare \la^2-\la -D+ e^{-(D+\la)\tau}f(S^0)\Big ).
\end{equation}
Define $\la(\vare)$ as the real solution of (\ref{e5.11}) such that
$\la(\vare)\to \la_0$ as $\vare\to 0^+$. Then $z(c)=\la(\vare)$
satisfies (\ref{e5.10}).\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip \smal
Theorem \ref{Thm5.2} asserts the existence of {\it positive}
travelling wavefronts for (\ref{e5.6}) with $S(t,x)<S^0$ for all
$t\in\R,x\in\R$. Here, due to the change of variables $s=S-s^0$, the
positivity of the component $s(t)$ (or $s(t,x)$) translates as the
nutrient concentration being smaller than $S^0$. We emphasize that
biologically significant solutions of (\ref{e5.4}) and (\ref{e5.6})
must be positive and have a nutrient concentration $S$ not larger
than the input concentration $S^0$.
\begin{rem}\label{R5.1} The existence of a positive eigenvector ${\bf v}$
associated with the dominant eigenvalue $\la_0$, as prescribed in
(H4), may seem a quite restrictive requirement, since it is not
satisfied by many populations dynamics systems, namely Kolmogorov
type models with $N>1$. We however observe that if the
characteristic matrix for (\ref{e2.1}) at $\la_0$,
$\Delta_0(\la_0)$, is an irreducible matrix with non-negative
off-diagonal entries, then there is a positive eigenvector for
$\Delta_0(\la_0)$ associated with $\la_0$ (see e.g. \cite[p.
258]{SW}). This property will be exploited in a forthcoming paper,
where Theorems \ref{Thm3.2} and \ref{Thm4.1} will be applied to
several population models.
\end{rem}
\section{Appendix}
In this appendix, we extend Proposition 7.1 of Mallet-Paret
\cite{MP} to systems with distributed delays.
Consider the FDE
\begin{equation}\label{eA.1}
x'(t)=L_0x_t+h(t),\quad t\in\mathbb{R}
\end{equation}
and the homogeneous system
\begin{equation}\label{eA.2}
x'(t)=L_0x_t,
\end{equation}
where $L_0:C([-\tau, 0];\mathbb{R}^N)\to \mathbb{R}^N$ is a bounded
linear operator and $h:\mathbb{R}\to\mathbb{R}^N$ is continuous. For
(\ref{eA.2}), write the characteristic equation
\begin{equation*}
\det \Delta_0(s)=0,\quad {\rm where}\quad \Delta_0(s)=sI-L_0(e^{s\cdot
}I).
\end{equation*}
It is well known that the solutions of the characteristic equation
are exactly the eigenvalues for the homogeneous system (\ref{eA.2}),
i.e., the eigenvalues for the infinitesimal generator $A$ associated
with the semiflow of (\ref{eA.2}). Furthermore, the spectrum $\sigma
(A)$ of $A$ is only composed of the point spectrum.
\begin{lem}\label{LA.1} If $f$ is a holomorphic function on a disc $\{
s: |s-\la|<\epsilon\} $, where $\la$ is an eigenvalue of
(\ref{eA.2}) and $\epsilon>0$ is small, then
\begin{equation*}
x(t)=Res\, (e^{t\cdot} \Delta_0^{-1}f,\la)={1\over {2\pi
i}}\int_{|s-\la|=\epsilon} e^{st} \Delta_0(s)^{-1}f(s)\, ds
\end{equation*}
is an eigenfunction of (\ref{eA.2}) corresponding to $\la$.
\end{lem}
{\it Proof}. The proof follows the arguments of Mallet-Paret
\cite[Section 7]{MP}, so we omit it.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\begin{thm}\label{TA.1} Let $x(t)$ be a solution of (\ref{eA.1}) on
$[T,\infty)$ for some $T\in\mathbb{R}$. Assume there are
$a,b\in\mathbb{R}, a<b$, such that
\begin{equation*}
x(t)=O(e^{-at}),\quad h(t)=O(e^{-bt})\quad {\rm as}\quad t\to\infty.
\end{equation*}
Then, for every $\epsilon >0$, we have
\begin{equation*}x(t)=z(t)+O(e^{-(b-\epsilon)t})\quad {\rm as}\quad t\to\infty,
\end{equation*}
where $z(t)$ is an eigenfunction of (\ref{eA.2}) associated with the
set of eigenvalues $\Lambda=\{ \la \in \sigma (A):-b<\Re\, \la \le
-a\}.$ Analogously, if $x(t)$ is a solution of (\ref{eA.1}) on
$(-\infty, T]$ for some $T\in\mathbb{R}$ and
\begin{equation*}
x(t)=O(e^{at}),\quad h(t)=O(e^{bt})\quad {\rm as}\quad t\to -\infty,
\end{equation*}
with $a<b$, then for every $\epsilon >0$ we have
\begin{equation*}
x(t)=z(t)+O(e^{(b-\epsilon)t})\quad {\rm as}\quad t\to -\infty,
\end{equation*}
where $z(t)$ is an eigenfunction of (\ref{eA.2}) associated with the
set of eigenvalues $\Lambda=\{ \la \in \sigma (A):a\le \Re\, \la
<b\}.$
\end{thm}
{\it Proof}. We only prove the result for $+\infty$; for $-\infty$
it is analogous. Without loss of generality, take $T=0$. In what
follows, for $f:[0,\infty)\to \mathbb{C}, f(t)=O(e^{-at})$ at
$+\infty$, we denote the Laplace transform of $f$ by
$$({\cal L}f)(s)=\tilde f(s)=\int_0^\infty e^{-st} f(t)\, dt,\quad {\rm for}\quad \Re\, s>-a.$$
Write $L_0(\var)=\int_{-\tau}^0 d\eta (\th) \var (\th)$, where $\eta
(\th)$ is an $N\times N$ matrix-valued function of bounded
variation. Applying the Laplace transform to (\ref{eA.1}), we
obtain
\begin{equation}\label{eA.3}
-x(0)+s\tilde x(s)={\cal L} (L_0 x_t)(s)+\tilde h(s), \quad \Re\, s>-a,
\end{equation}
where \begin{eqnarray}\label{eA.4}
\nonumber\fl & & {\cal L} (L_0 x_t)(s)=\int _0^\infty e^{-st} \left
(\int _{-\tau}^0 d\eta (\th) x(t+\th )\right ) dt=\int _{-\tau}^0
d\eta (\th)
\left ( \int _0^\infty e^{-s(t+\th)} x(t+\th)\, dt\right )\\
\fl & & = \int _{-\tau}^0 d\eta (\th) \left ( \int_\th ^0 e^{-su}
x(u)\, du +\tilde x(s)\right )=L_0\left ( e^{s\cdot} \int _{\cdot}^0
e^{-su} x(u)\, du \right )+L_0(e^{s\cdot} I)\tilde x(s).
\end{eqnarray}
From $(\ref{eA.3}),(\ref{eA.4})$, we obtain
$$
\Delta_0(s)\tilde x(s)=r(s)+\tilde h(s),\quad {\rm where}\quad
r(s)=x(0)+L_0\left ( e^{s\cdot} \int _{\cdot}^0 e^{-su} x(u)\, du
\right).
$$
We observe that $r(s)$ is an entire function, $\tilde h(s)$ is
defined and holomorphic for $\Re\, s>-b$, and
\begin{equation}\label{eA.5}
\tilde x(s)=\Delta_0(s)^{-1} [r(s)+\tilde h(s)]
\end{equation}
is holomorphic for $\Re\, s>-b$ with the exception of finitely many
poles.
Take $\epsilon >0$ and $k>-a$. On any strip of the form
$-b+\epsilon\le \Re\, \la \le k$, the functions $r(s)$ and $\tilde
h(s)$ are uniformly bounded. Since $k$ is greater than the real
part of all singularities of $\tilde x(s)$, we can use the inverse
formula for the Laplace transform,
\begin{equation}\label{eA.6}
x(t)={1\over {2\pi i}} \int_{k-i\infty}^{k+i\infty} e^{st} \tilde
x(s)\, ds.
\end{equation}
Choose $\epsilon>0$ small such that $-b+\epsilon/2<k$ and $\sigma
(A)\cap \{ s:-b<\Re\, s\le -b+\epsilon/2\} =\emptyset.$ Note that in
the strip $-b\le \Re\, s\le -a$, the only possible poles of $\tilde
x$ lie on $-b+\epsilon/2<\Re\, s\le -a$.
In the strip $-b+\epsilon/2\le \Re\, s\le k$, the functions
$|r(s)|,| \tilde h(s)|$ are bounded, and $\|L_0(e^{s\cdot }I)\|\le
\max (1, e^{-k\tau})\|L_0\|$, hence the operator norm for the
inverse of $\Delta_0(s)=sI-L_0(e^{s\cdot }I)$ satisfies
\begin{equation*}
\| \Delta_0(s)^{-1}\| \le {1\over {|s|-\max (1,
e^{-k\tau})\|L_0\|}}\quad {\rm for}\quad |s|>\max (1, e^{-k\tau})\|L_0\|.
\end{equation*}
From this estimate and (\ref{eA.5}), we conclude that $| e^{st} \tilde x(s)| \to 0$ as $|\Im\, s|\to \infty$, uniformly
in the strip $-b+\epsilon/2\le \Re\, s\le k$, and that $\tilde x(s)$ is $L^2$-integrable in any straight line
$s=x_0+iy,y\in\mathbb{R}$, for any fixed $x_0\in [ -b+\epsilon/2,
k]$.
We may shift the path of integration in (\ref{eA.6}) to the left, and obtain
\begin{equation*}
x(t)=z(t)+w(t),\quad {\rm where}
\end{equation*}
\begin{equation}\label{eA.7}
z(t)=\sum_{\la \in\Lambda} Res\, (e^{t\cdot} \tilde x,\la),\quad
w(t)={1\over {2\pi i}} \int_{-b+{\epsilon\over 2}-i\infty}^{-b+{\epsilon\over 2}+i\infty} e^{st} \tilde x(s)\, ds.
\end{equation}
From the previous lemma, $z(t)$ is an eigenfunction of (\ref{eA.2})
associated with the set of eigenvalues in $\Lambda$. It remains to
prove that
\begin{equation}\label{eA.8}
w(t)=O(e^{-(b-\epsilon)t})\quad {\rm at}\quad +\infty.
\end{equation}
Define $u(t)=e^{(b-\epsilon/2)t}w(t), v(t)=e^{(b-\epsilon)t}w(t)$.
We first prove that $v\in L^2[0,\infty)$. Here, $L^p[0,\infty)=L^p([0,\infty);\mathbb{C}^N),p=1,2$.
We have
\begin{equation*}
\fl u(t)={1\over {2\pi i}} \int_{-b+{\epsilon\over
2}-i\infty}^{-b+{\epsilon\over 2}
+i\infty} e^{(s+b-\epsilon/2)t} \tilde x(s)\, ds={1\over {2\pi}}\int _{\R} e^{-ist} \tilde x(-b+\epsilon/2-is)\, ds.
\end{equation*}
By Plancherel Theorem, $u(t)\in L^2[0,\infty)$, so that
$v(t)=e^{-\epsilon t/2} u(t)$ implies
\begin{equation}\label{eA.9}\fl
\|v\|_{L^1[0,\infty)}
\le \|u\|_{L^2[0,\infty)}
\|e^{-\epsilon t/2}\|_{L^2[0,\infty)}
\le C\| \tilde x(-b+\epsilon/2+i\cdot )\|_{L^2(\mathbb{R})} <\infty,
\end{equation}
for some $C>0$. Hence, $v\in L^1[0,\infty).$ Define now
\begin{equation*}
V(t)=L_0(e^{-(b-\epsilon)\cdot }v_t)=\int_{-\tau}^0d\eta(\th)
e^{-(b-\epsilon)\th}v(t+\th).
\end{equation*}
Then,
\begin{equation}\label{eA.10}
\int_0^{+\infty} |V(t)|\, dt\le \max (1, e^{(b-\epsilon)\tau})\|L_0\|\|v\|_{L^1[-\tau,\infty)}.
\end{equation}
In particular, $V\in L^1[0,\infty)$.
We now observe that $x(t)=z(t)+w(t)$, with $x(t)$ a solution of
(\ref{eA.1}) and $z(t)$ a solution of (\ref{eA.2}). Hence $w(t)$
satisfies (\ref{eA.1}), $w'(t)=L_0(w_t)+h(t)$, and we obtain
\begin{equation*}
v'(t)=(b-\epsilon)v(t)+L_0(e^{-(b-\epsilon)\cdot}v_t)+e^{(b-\epsilon)t}h(t),
\end{equation*}
with $e^{(b-\epsilon)t}h(t)=O(e^{-\epsilon t})$. We conclude that
$v'\in L^1[0,\infty)$. Since $|v(t)|\le |v(0)|+\int_0^t |v'(s)|\,
ds$ for $ t\ge 0$, then $v$ is bounded on $[0,\infty)$, and
(\ref{eA.8}) holds.\hfill \vrule width 5 pt height 7 pt depth - 2 pt\smallskip
\begin{rem}\label{RA1} For the situation $x(t)=O(e^{-at}), h(t)=O(e^{-bt})\ (a<b)$ as
$t\to\infty$, denote $v(t)=(v_1(t),\dots,v_N(t))$ as in the above
proof. Clearly, one can obtain componentwise estimates similar to
(\ref{eA.9}) or (\ref{eA.10}). In fact, one concludes that for
$\epsilon>0$ small such that $\sigma (A)\cap \{ s:-b<\Re\, s\le
-b+\epsilon/2\} =\emptyset$ and $t\ge 0, j=1,\dots,N,$
\begin{equation*}
\|v_j\|_{L^1[0,\infty)}\le (2\pi \sqrt{\epsilon})^{-1}\| \tilde
x_j(-b+\epsilon/2+i\cdot )\|_{L^2(\mathbb{R})}
\end{equation*}
and $|v_j(t)|\le |v_j(0)|+\| v_j'\|_{L^1[0,\infty)}$ with
\begin{equation*}
\| v_j'\|_{L^1[0,\infty)}\le C \| \tilde
x_j(-b+\epsilon/2+i\cdot)\|_{L^2(\mathbb{R})}+\|e^{(b-\epsilon)\cdot}h_j\|_{L^1[0,\infty)},
\end{equation*}
where $C={{(b-\epsilon)+e^{(b-\epsilon/) \tau}\|L_0\|}\over {2\pi
\sqrt {\epsilon}}} $. Similar estimates hold for the case
$x(t)=O(e^{at}), h(t)=O(e^{bt})$ at $ -\infty.$ \end{rem}
\ack This research was supported by FCT (Portugal), Financiamento
Base 2008-ISFL-1-209 (Teresa Faria) and by FONDECYT (Chile),
projects 7080045 (Teresa Faria) and 1071053 (Sergei Trofimchuk). S.
Trofimchuk was also partially supported by CONICYT (Chile) through
PBCT program ACT-56 and by the University of Talca, program
``Reticulados y Ecuaciones".
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,937 |
Developers dug into beta firmware and found new features for the mobile shooter.
Smartphones may have effectively killed off dedicated point-and-shoot cameras, but Apple is looking to them for inspiration with iOS 11. Developers have dug through beta firmware for the HomePod, and tucked inside the code for Apple's smart speaker, there are hints that the next version of its mobile OS will feature something called "SmartCam."
If you've ever used a point-and-shoot camera, the feature should sound pretty familiar: different scene modes and photo settings depending on what you're shooting. So, one each for fireworks, foliage, pets, skies, snow, sports and others, as SlashGear notes. There's even one for documents.
The "smart" in its name suggests that maybe machine learning will play a role here as well, potentially analyzing the scene for you and picking the best settings. This might not use machine learning to improve photography a la what Google does with the Pixel, but it could make Apple's woefully basic camera app a little more full featured.
Whether this will be exclusive to Apple's next round of mobile hardware -- whenever it's announced -- or if it'll apply to legacy handsets too is hard to tell. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,863 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/68551\/find-sets-with-given-union","text":"# Find sets with given union\n\nWe are given positive integers $n$, $m$ and a set $S$ of subsets of the set $\\{1,2,...,n\\}$ with $|S| = m$.\n\nFor example:\n\n$n = 5,\\,\\, m = 4,\\,\\, S = \\{\\, \\{1,2\\}, \\{1,3\\}, \\{4,5\\}, \\{2,3\\} \\,\\}.$\n\nWhat is the fastest way to find the minimal subset $S'$ of $S$ such that the union of sets in $S'$ is equal to $\\{1,2,...,n\\}$.\n\nIn our example, there are 3 possible answers:\n\n$S' = \\{\\,\\{1,2\\}, \\{1,3\\}, \\{4,5\\}\\,\\} \\\\ S' = \\{\\,\\{1,2\\}, \\{2,3\\}, \\{4,5\\}\\,\\} \\\\ S' = \\{\\,\\{1,3\\}, \\{2,3\\}, \\{4,5\\}\\,\\}$\n\n\u2022 Take a look at Set cover problem. \u2013\u00a0Eugene Jan 11 '17 at 11:05\n\u2022 Your problem is a variant of the set cover problem, and is probably NP-complete. So it probably has no polynomial time solution. \u2013\u00a0Yuval Filmus Jan 11 '17 at 11:54\n\nGiven $$U=\\{1,\\ldots , n\\}$$ and $$S\\subseteq 2^U$$, integer $$k$$\nDecide whether $$C\\subseteq S$$ exists s.t. $$\\cup_{X \\in C} X = U$$ and $$|C| \\leq k$$","date":"2021-08-06 00:29:09","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\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.872200608253479, \"perplexity\": 167.888291135548}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046152085.13\/warc\/CC-MAIN-20210805224801-20210806014801-00129.warc.gz\"}"} | null | null |
Minador do Negrão is a municipality located in the Brazilian state of Alagoas. Its population is 5,322 (2020) and its area is 167 km².
References
Municipalities in Alagoas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,333 |
Patients Are The Top Priority At Dentistry In The Village!
One of the cornerstones of our practice is that we believe every patient is important to our practice. But without exceptional treatment and superior services, it is unlikely that we would have satisfied patients. So how do we know if we are succeeding at providing these important elements? One key way is by looking at what our patients in the Mississauga, ON area and beyond are saying about our practice. Take a look and see for yourself why so many patients are pleased with the care they have received at both of our offices, Dentistry in the Village and Mavis Dental Centre. The testimonials below are just a small sample of the quality of care you will receive from our amazing staff.
"First off, I never review offices/restaurants because I don't feel like I need to. I posted this review on yelp but it won't show for some reason. I decided to write this review because the last appointment I had reminded me why I love this office. Office? Beautiful. Staff? Friendly. The dentists are thorough and the hygienist is probably the nicest hygienist you will ever meet. The Doctors are great. You can always depend on them to be honest with you. Options are always given. Sometimes appointments start a little late but when they do take you in you are out just as quickly. I have only met the lady doctor once before and it was pleasant but Dr.Michael is usually the one I see when I get a check up and he is quick and efficient. The hygienist is very honest, patient and thorough. She wants to know how she can make your experience better throughout the appointment. She always welcomes you with a smile and leaves you with the receptionist with a smile. The experience is completely different from dental offices 15-20 years ago. My whole family, including my kids, love her. You can see the difference when you sit with Karen because she cares. My first appointment with the office several years ago started with her. I used to hate dental offices and Karen completely reassured that dental offices are not something to fear. She actually said to my son "Am I scary?" which she is not, she knew how to diffuse a "scary" moment. She is gentle when cleaning and thorough. Overall atmosphere is great. The Receptionist is friendly and experience is well worth it. I am hoping that google will post this review so I can praise my favourite Dental Office."
"As someone who requires regular dental maintenance, I would not hesitate to recommend this dental office to anyone. The owners and staff are amazing. They always make me feel relaxed, comfortable, and in great hands. They have two offices - one at Mavis and Eglinton and one in Port Credit. If you're looking for a new dentist, give them a call." | {
"redpajama_set_name": "RedPajamaC4"
} | 4,264 |
Examining the relationship of vaping to smoking initiation among US youth and young adults: a reality check
David T Levy Cancer Prevention and Control, Lombardi Comprehensive Cancer Center, Georgetown University, Washington, District of Columbia, USA PubMed articlesGoogle scholar articles
Kenneth E Warner Department of Health Management and Policy, School of Public Health, University of Michigan, Ann Arbor, Michigan, USA PubMed articlesGoogle scholar articles
K Michael Cummings Department of Psychiatry and Behavioral Sciences, Medical University of South Carolina, Charleston, South Carolina, USA PubMed articlesGoogle scholar articles
David Hammond School of Public Health and Health Systems, University of Waterloo, Waterloo, Ontario, Canada PubMed articlesGoogle scholar articles
Charlene Kuo Cancer Prevention and Control, Lombardi Comprehensive Cancer Center, Georgetown University, Washington, District of Columbia, USA PubMed articlesGoogle scholar articles
Geoffrey T Fong Department of Psychology, School of Public Health and Health Systems, Ontario Institute for Cancer Research, Toronto, Ontario, Canada PubMed articlesGoogle scholar articles
James F Thrasher Health Promotion, Education, and Behavior, School of Public Health, University of South Carolina, Columbia, South Carolina, USA PubMed articlesGoogle scholar articles
Maciej Lukasz Goniewicz Department of Health Behavior, Roswell Park Comprehensive Cancer Center, Buffalo, New York, USA PubMed articlesGoogle scholar articles
Ron Borland Nigel Gray Distinguished Fellow in Cancer Prevention, Cancer Council Victoria, Carlton, Victoria, Australia PubMed articlesGoogle scholar articles
Correspondence to Dr. David T Levy, Cancer Prevention and Control, Lombardi Comprehensive Cancer Center, Georgetown University, Washington DC 20007, USA; dl777{at}georgetown.edu
Levy DT, Warner KE, Cummings KM, et al
Tobacco Control 2019;28:629-635.
Received April 16, 2018
Revised September 19, 2018
First published November 20, 2018.
Previous version (20 November 2018).
© Author(s) (or their employer(s)) 2019. Re-use permitted under CC BY-NC. No commercial re-use. See rights and permissions. Published by BMJ. This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0/. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,878 |
package jjwu.xdeveloper.spring.aop;
import javax.annotation.Resource;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.springframework.test.context.ContextConfiguration;
import org.springframework.test.context.junit4.SpringJUnit4ClassRunner;
@RunWith(SpringJUnit4ClassRunner.class)
@ContextConfiguration(value = { "classpath*:/conf/applicationContext.xml" })
public class AppTest {
@Resource
private AppDao appDao;
@Test
public void testApp() {
System.out.println(appDao.getUserName());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
\section{Introduction}
\label{sec:intro}
Bidirectional transformations (BXs) are a ``mechanism for maintaining the consistency of two (or more) related sources of information''~\cite{GRACE:09}. The challenge of writing BXs has been long known in the database community since the seminal studies on view-update translation by Bancilhon and Spyratos~\cite{Bancilhon:81} and Dayal and Bernstein~\cite{Dayal:82}.
More recently, the pioneering work of Foster et al. on combinatorial languages for BXs~\cite{Foster:07} has recast a lot of attention towards this challenge, and given the impulse to the birth of a new research field on BXs, uniting researchers from diverse computer science communities including programming languages, model-driven engineering and databases.
In the last ten years, this burgeoning interest in BXs has led to the proposal of a vast number of approaches~\cite{GRACE:09,Schurr:2008}, inspired by different visions of the problem and motivated by different contexts where the need for bidirectionality arises.
In face of the multitude and diversity of existing approaches, each tool has been tailored to answer the challenges of its particular bidirectional scenarios, and has evolved to support different formal properties and specification styles that best suit its needs. Therefore, many of the fundamental problems of the field are not yet well established, mostly due to the non-existence of a universal classifying system and to the lack of common theoretical grounds between many of the approaches.
This makes it hard to compare the various solutions, to understand precisely their advantages and limitations and provide effective criteria for assessing progress in the field.
To move forward, this pressing unification need has been gaining voice in the BX subcommunities, and such a maturation effort has been slowly undergoing through a series of seminars and workshops, displayed in publications such as~\cite{GRACE:09,HSST11,BX12,Terwilliger:2012}.
Unfortunately, despite such significant effort, the community is still far from reaching a consensus on the terminology and properties that are desirable and satisfiable by BX tools.
For example, some BX programming languages satisfy specific properties that are hard to correlate with the properties satisfied by other languages or whose practical implications on the expressiveness and behavior of the corresponding BXs are hard to understand. Moreover, although some properties are already well understood in the databases or programming languages communities, their meaning and applicability in the MDE community remains unclear. On the other side, the specification style and deployment level of most BX programming languages are still not adequate for tackling real-world scenarios currently supported by model-driven BX approaches.
In this paper, we propose a generic BX scheme in which concrete interfaces can be instantiated by choosing the desired representation of the update and traceability information. On top of that scheme, we propose a set of generic semantic properties that embody the desirable bidirectional behavior of the transformations.
In addition, we walk through a number of
BX frameworks and instantiate them in our scheme by incrementally exploring its design space, and the expected properties naturally emerge from such exercise.
As a first attempt to validate our generic BX scheme, we present a comparative survey of up to 40 existing BX approaches emerging from diverse BX subcommunities, classified according to their interface and semantic properties. Besides providing an insightful high-level picture of the state-of-art of the field of BXs, it raises interesting questions about the key design features for classifying BXs.
Section~\ref{sec:framework} presents our generic scheme and explores the instantiations of some popular frameworks (namely mappings, lenses, maintainers, trigonal systems, edit lenses and symmetric delta-lenses). Section~\ref{sec:properties} presents the generic bidirectional properties which are then instantiated for the frameworks enumerated above, while Section~\ref{sec:survey-existing-bx} presents a first effort to survey existing BX techniques under the proposed axes.
Sections~\ref{section:relwork} and~\ref{section:conclusion} discuss related work on other classification efforts
and set forth the path towards a more complete survey on the design space of BXs.
\section{Scheme}
\label{sec:framework}
We begin by clarifying what we mean by BX. The goal of a BX between \ensuremath{\Conid{A}} and \ensuremath{\Conid{B}} is to enforce
consistency between values of types \ensuremath{\Conid{A}} and \ensuremath{\Conid{B}}. The term ``type''
should be understood as a broad placeholder for type, schema or
metamodel, with ``value'' denoting value, instance, or model,
accordingly. Consistency recovery is achieved by means of two
transformations \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}} whose purpose is, respectively,
to propagate \ensuremath{\Conid{A}} updates into consistent \ensuremath{\Conid{B}} updates and vice-versa.
Some BX frameworks derive the two transformations from an explicitly declared \emph{consistency
relation} \ensuremath{\mathsf{R}\subseteq\Conid{A}\;\!\!\times\!\!\;\Conid{B}} between both types. Often, conversely, the consistency relation is actually
expressed in terms of the underlying transformations. In these cases, usually one of the transformations
must be specified by the user, the opposite one being derived from
it. In some rare cases the consistency relation is an implicit notion of the system.
Table~\ref{tab:consistency} summarizes these options.
\begin{table}[!t]
\begin{center}
\begin{tabular}{|l|c|p{8cm}|}
\hline
\textbf{Explicit} & E & There is an explicitly declared consistency relation. \\
\hline
\textbf{Transformation} & T & The consistency relation is one of the transformations.\\
\hline
\textbf{Implicit} & I & The consistency relation is implicit.\\
\hline
\end{tabular}
\end{center}
\caption{Consistency Relation.}
\label{tab:consistency}
\vspace{-.2in}
\end{table}
This definition precludes some frameworks sometimes said to also be
BX. That is the case of frameworks with general synchronization
procedures that recover consistency between values that were updated
concurrently.
Notice that our goal is not to give a definitive definition of
what is BX (thus rejecting such frameworks as not being BX), but just to
clarify and limit the scope of this~paper.
We depict the two application scenarios of a BX between \ensuremath{\Conid{A}} and \ensuremath{\Conid{B}} as follows:
\begin{displaymath}
\xymatrixrowsep{1pc}
\xymatrix{
A \ar[dd] & B \ar@{..>}[l] \ar[dd]\\
\ar@{=>}[r]^{\ensuremath{\mathsf{to}}} & \\
A \ar@{..>}[r] & B
}
\qquad
\xymatrix{
A \ar@{..>}[r] \ar[dd] & B \ar[dd]\\
& \ar@{=>}[l]_{\ensuremath{\mathsf{from}}} \\
A & \ar@{..>}[l] B
}
\end{displaymath}
In the left scenario, we first have \ensuremath{\Varid{a}\mathbin{:}\Conid{A}} and \ensuremath{\Varid{b}\mathbin{:}\Conid{B}} that are
somehow consistent. The objective of \ensuremath{\mathsf{to}} is to transform an update on
\ensuremath{\Varid{a}} into an update on \ensuremath{\Varid{b}} such that consistency is restored. Updates
are depicted using solid arrows (although they are not necessarily functions, as clarified in Section~\ref{sec:update}).
The transformation \ensuremath{\mathsf{to}} may also
receive some extra information concerning the system state prior to
the update on \ensuremath{\Varid{a}}. Namely, it might have access to some trace
information testifying how \ensuremath{\Varid{a}} and \ensuremath{\Varid{b}} were consistent. Such
\emph{traceability} is depicted using a dotted arrow, and when not
trivially derived from the updated values must also be returned by
\ensuremath{\mathsf{to}}. The right scenario is dual.
In general, transformations \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}} can be typed as follows:
\begin{hscode}\SaveRestoreHook
\column{B}{@{}>{\hspre}l<{\hspost}@{}}%
\column{7}{@{}>{\hspre}l<{\hspost}@{}}%
\column{E}{@{}>{\hspre}l<{\hspost}@{}}%
\>[B]{}\mathsf{to}{}\<[7]%
\>[7]{}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})\;\!\!\times\!\!\;\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \overleftarrow{\mathsf{U}}(\Conid{B})\;\!\!\times\!\!\;\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B}){}\<[E]%
\\
\>[B]{}\mathsf{from}{}\<[7]%
\>[7]{}\mathbin{:}\overleftarrow{\mathsf{U}}(\Conid{B})\;\!\!\times\!\!\;\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \overrightarrow{\mathsf{U}}(\Conid{A})\;\!\!\times\!\!\;\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B}){}\<[E]%
\ColumnHook
\end{hscode}\resethooks
Here, \ensuremath{\overrightarrow{\mathsf{U}}} is a parameterized type constructor that denotes the type
of \ensuremath{\Conid{A}} updates that \ensuremath{\mathsf{to}} propagates, and \ensuremath{\overrightarrow{\mathsf{T}}} a type constructor that
denotes the type of the traceability \ensuremath{\mathsf{to}} should produce (to be
received by \ensuremath{\mathsf{from}}). Dually, we have \ensuremath{\overleftarrow{\mathsf{U}}} and \ensuremath{\overleftarrow{\mathsf{T}}} for \ensuremath{\mathsf{from}}. As we
will see in the next sections, the interface of existing (and
potential) BX frameworks can be obtained by giving concrete
definitions for these type constructors.
We should also clarify at this point that by transformation we mean a
\emph{partial function}. Frameworks differ on the degree of
totality of both transformations, which will be characterized by a specific
property (Section
\ref{sec:properties}).
This formal characterization encompasses both \emph{symmetric} and
\emph{asymmetric} frameworks (see Table~\ref{tab:symmetry}).
Asymmetric frameworks are often
biased towards transformation scenarios where one of the types is
``larger'' and contains more information than the other, whereas
symmetric frameworks are more balanced and tend to consider that both
types contain roughly the same information or that each may contain
information not present in the other.
\begin{table}[!t]
\begin{center}
\begin{tabular}{|l|c|p{9cm}|}
\hline
\textbf{Symmetric} & S & The update propagation nature is the same in both directions. We have equal definitions for \ensuremath{\overrightarrow{\mathsf{U}}} and \ensuremath{\overleftarrow{\mathsf{U}}}, for \ensuremath{\overrightarrow{\mathsf{T}}} and \ensuremath{\overleftarrow{\mathsf{T}}}, and similar laws for both \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}}. \\
\hline
\textbf{Asymmetric} & A & The update propagation nature is different in both directions. This may lead to different definitions for \ensuremath{\overrightarrow{\mathsf{U}}} and \ensuremath{\overleftarrow{\mathsf{U}}}, for \ensuremath{\overrightarrow{\mathsf{T}}} and \ensuremath{\overleftarrow{\mathsf{T}}}, and different laws for \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}}. \\
\hline
\end{tabular}
\end{center}
\caption{Symmetry.}
\label{tab:symmetry}
\vspace{-.2in}
\end{table}
For each instance of the arrows in the above diagrams it may be useful to
reason about its source and target values.
We will use (overloaded)
operators \ensuremath{ \delta } and \ensuremath{ \rho } to denote them.
Namely, for each update constructor \ensuremath{\overrightarrow{\mathsf{U}}}
(and dually for \ensuremath{\overleftarrow{\mathsf{U}}}), we have
\ensuremath{ \delta \mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})\to \Conid{A}} and \ensuremath{ \rho \mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})\to \Conid{A}} that, given an update, denote the value in
its pre- and post-state, respectively.
Similarly for traceabilities,
\ensuremath{ \delta \mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \Conid{A}} and \ensuremath{ \rho \mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \Conid{B}} denote the source and
target values related by an instance of \ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})}, respectively.
\ensuremath{\overleftarrow{\mathsf{T}}} traceabilities
are seen in the other direction, so \ensuremath{ \delta \mathbin{:}\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \Conid{B}} and
\ensuremath{ \rho \mathbin{:}\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \Conid{A}}.
We will denote updates and traceabilities by bold
characters (\ensuremath{{\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}},{\textbf {\textit {\textrm s}}},\mathbin{...}}) in contrast to non-bold characters
for the values (\ensuremath{\Varid{a},\Varid{b},\mathbin{...}}) returned by these operators.
Although it is assumed that every arrow has a pre- and post-state value, that does not mean that they are retrieved directly from the update representation (that information may not even be present in the constructor).
When a transformation \ensuremath{\mathsf{to}\;({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})\mathrel{=}({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})} occurs, updates and
traceabilities must agree on the respective sources and targets.
This can be captured by the following properties, coined
\emph{incidence conditions} in~\cite{Diskin:2011}.
\begin{align}
\label{eq:incidence}
\ensuremath{ \delta {\textbf {\textit {\textrm a}}}\mathrel{=}\rho {\textbf {\textit {\textrm s}}}\quad\quad \delta {\textbf {\textit {\textrm b}}}\mathrel{=} \delta {\textbf {\textit {\textrm s}}}\quad\quad\rho {\textbf {\textit {\textrm a}}}\mathrel{=} \delta {\textbf {\textit {\textrm r}}}\quad\quad\rho {\textbf {\textit {\textrm b}}}\mathrel{=}\rho {\textbf {\textit {\textrm r}}}}
\end{align}
The laws for \ensuremath{\mathsf{from}} are dual. These properties enable the retrieval of states even if not explicitly present in the constructor. For instance, an update \ensuremath{{\textbf {\textit {\textrm a}}}} may not have information about its pre-state, but it could for instance be retrieved from the traceability \ensuremath{{\textbf {\textit {\textrm s}}}}, since \ensuremath{ \delta {\textbf {\textit {\textrm a}}}\mathrel{=}\rho {\textbf {\textit {\textrm s}}}}.
We also assume that the input traceability truly testifies the consistency relation,
i.e., \ensuremath{(\rho {\textbf {\textit {\textrm s}}}, \delta {\textbf {\textit {\textrm s}}})\;\!\in\! \;\mathsf{R}}, which will be denoted by \ensuremath{{\textbf {\textit {\textrm s}}}\;\!\in\! \;\mathsf{R}}. Due
to the incidence conditions, \ensuremath{\rho {\textbf {\textit {\textrm s}}}} and \ensuremath{ \delta {\textbf {\textit {\textrm s}}}} can instead be accessed by \ensuremath{ \delta {\textbf {\textit {\textrm a}}}} and \ensuremath{ \delta {\textbf {\textit {\textrm b}}}},
respectively, whenever they are not present in traceability.
\subsection{Update representation}
\label{sec:update}
One of the main axes distinguishing existing frameworks is
\emph{update representation}, i.e., what are the concrete definitions
of \ensuremath{\overrightarrow{\mathsf{U}}} and \ensuremath{\overleftarrow{\mathsf{U}}} in the above generic scheme. Some possible
definitions are presented in Table~\ref{tab:update}.
\begin{table}[!t]
\begin{center}
\begin{tabular}{|l|c|p{7.5cm}|}
\hline
\parbox[t]{4cm}{\textbf{Post-state} \\\ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\Conid{A}}} & S & An update is represented only by the post-state.\\
\hline
\parbox[t]{4cm}{\textbf{Both states} \\ \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\Conid{A}\;\!\!\times\!\!\;\Conid{A}}} & \ensuremath{\mathbb{S}} &The update is represented by the pre- and the post-state. \\
\hline
\parbox[t]{4cm}{\textbf{Delta} \\ \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\Conid{A}\;\!\!\times\!\!\;\Conid{A}\;\!\!\times\!\!\;\mathsf{D}(\Conid{A},\Conid{A})}} & D & Beside the pre- and the post-state, an update representation also comes with a \emph{sameness relation} stating which components of both are conceptually the same. \\
\hline
\parbox[t]{4cm}{\textbf{Edit} \\ \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\mathsf{O}(\Conid{A})^\star}} & E & An update is represented by the sequence of edit operations that was performed. \\
\hline
\parbox[t]{4cm}{\textbf{State + Edit} \\ \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\Conid{A}\;\!\!\times\!\!\;\mathsf{O}(\Conid{A})^\star}} & \ensuremath{\mathbb{E}} & An update is represented by the pre-state and the sequence of edit operations that was performed. \\
\hline
\parbox[t]{4cm}{\textbf{Function} \\ \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})\mathrel{=}\Conid{A}\to \Conid{A}}} & F & An update is represented by a semantic value that models it as an endo-function. \\
\hline
\end{tabular}
\end{center}
\caption{Update.}
\label{tab:update}
\vspace{-.2in}
\end{table}
Frameworks that fall within the first two categories are usually known
as \emph{state-based}, since only the value in the post- (and
sometimes also the pre-) state of an update is considered. When some
knowledge of the exact changes that were (or have to be) performed in
an update is represented, we have an \emph{operation-based}
framework. A possible way to represent such changes is via a
\emph{sameness relation} tracing back components of the post-state to
corresponding components in the pre-state. In Table~\ref{tab:update}
we encapsulate the concrete definition of such a sameness relation in
the parameterized type constructor \ensuremath{\mathsf{D}}, since it depends on the
domains involved in the transformation\footnote{\ensuremath{\mathsf{D}(\Conid{A},\Conid{B})} should not
be confused with the set of all binary relations between \ensuremath{\Conid{A}} and
\ensuremath{\Conid{B}}, which is denoted by \ensuremath{\mathcal{P}(\Conid{A}\;\!\!\times\!\!\;\Conid{B})}.}. Notice that a sameness
relation carries more information than the conjunction of pre- and
post-state alone: for example, if a component is deleted and a new one
inserted with the same content, these components would be
unconnected in the sameness relation, but indistinguishable
otherwise. A sequence of edit operations is another possible
representation for updates. Again, since the set of edit operations
supported by each type varies, we abstract away its concrete
definition in the parameterized type constructor \ensuremath{\mathsf{O}}. Another alternative
is to store the pre-state along with the edit-sequence, in which case
the post-state could also be calculated by applying the edit-sequence to
the pre-state. When updates
are performed programmatically, one might only have access to the
executable that performed them instead of a syntactic
representation. This information might still be exploited by
frameworks implemented on top of semantic BX techniques.
Although the extra knowledge in operation-based frameworks can lead to
BXs that satisfy more precise properties (see discussions
in~\cite{Diskin:2011,Hu:04journal}), they usually
demand a tight coupling with applications, so that they can track the
changes that characterize an update. Transforming updates onto updates
also makes BXs more natural with
incrementality~\cite{Giese:2006} (rather than recomputing a
new model when the correlated model changes, only a small ``delta'' is
propagated). State-based frameworks, on the other hand, are more
flexible and support more usage scenarios, like integration with
off-the-shelf applications that have not been designed with
bidirectionality in mind, and are moreover less sensitive to ``noise''
in the updates. The distinction between state- and operation-based
approaches is not always obvious. Hybrid approaches may build a
state-based system with a richer operation-based core.
As they discard all update information, some
model differencing procedure is required to infer new hypothetical
update operations. Similarly, an incremental system can have a simple
state-based core, but keep track of operations merely as an
optimization, to exploit the locality of updates.
\subsection{Traceability representation}
Likewise to update representation, Table~\ref{tab:traceability}
presents possible definitions for \emph{traceability representation},
i.e., the concrete definitions of \ensuremath{\overrightarrow{\mathsf{T}}} and \ensuremath{\overleftarrow{\mathsf{T}}} in the above generic
scheme. In the first category we have frameworks without any trace
information. This is a very limiting scenario, since only the to be translated
update itself is known when attempting to recover consistency. Not
even information about the current state of the opposite domain is
known: in a state-based framework this means that only ``fresh''
values must be produced, ruling out any sort of incremental
updating. In the second category traceability amounts precisely to the
value of the opposite domain. This is the case, for example, of
frameworks that tackle the view-update problem, and that require (specifically, \ensuremath{\mathsf{from}} requires) access
to the (previous) value of the source (\ensuremath{\Conid{A}}) to fetch information not
recoverable from the view (\ensuremath{\Conid{B}}).
Traceability can also be represented by
means of a \emph{complement}: \ensuremath{\mathsf{C}(\Conid{A},\Conid{B})} is a parameterized type
constructor that encapsulates the complement of a type \ensuremath{\Conid{A}} with
respect to another type \ensuremath{\Conid{B}}. Essentially, elements of \ensuremath{\mathsf{C}} will contain
some of the components of one (or both) value(s) not present in the other. Finally, we can
also use a sameness relation to trace the execution of a previous
update translation, pinpointing the exact pairs of components in the
source and target that testify the consistency between them.
Sometimes the returned traceability is redundant and can be computed
from the remaining available information. A possible instantiation where this occurs is when \ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\text{S}} and \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\text{S}}. Since \ensuremath{\mathsf{to}\;({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})\mathrel{=}({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})}, the \ensuremath{{\textbf {\textit {\textrm r}}}\mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})} will actually be
equivalent to \ensuremath{ \delta {\textbf {\textit {\textrm r}}}\mathbin{:}\Varid{a}}, which from~\eqref{eq:incidence} is
equivalent to \ensuremath{\rho {\textbf {\textit {\textrm a}}}\mathbin{:}\Conid{A}}, the post-state of the source update that
is precisely one of the inputs of \ensuremath{\mathsf{to}}.
In such cases, the
transformations will be typed just as follows:
\begin{equation}
\label{eq:short:scheme}
\ensuremath{\mathsf{to}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})\;\!\!\times\!\!\;\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \overleftarrow{\mathsf{U}}(\Conid{B})\quad\quad\mathsf{from}\mathbin{:}\overleftarrow{\mathsf{U}}(\Conid{B})\;\!\!\times\!\!\;\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\to \overrightarrow{\mathsf{U}}(\Conid{A})\quad}
\end{equation}
\begin{table}[!t]
\begin{center}
\begin{tabular}{|l|c|p{7cm}|}
\hline
\parbox[t]{4.5cm}{\textbf{None} \\\ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\mathrel{=}\mathrm{1}}} & N & No trace information is represented. \\
\hline
\parbox[t]{4.5cm}{\textbf{State} \\ \ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\mathrel{=}\Conid{A}}} & S & Only the source state is represented. \\
\hline
\parbox[t]{4.5cm}{\textbf{Complement} \\ \ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\mathrel{=}\mathsf{C}(\Conid{A},\Conid{B})}} & C & Some complement of the source and/or target values is represented. \\
\hline
\parbox[t]{4.5cm}{\textbf{Delta} \\ \ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})\mathrel{=}\Conid{A}\;\!\!\times\!\!\;\Conid{B}\;\!\!\times\!\!\;\mathsf{D}(\Conid{A},\Conid{B})}} & D & Beside both consistent states, a sameness relation between them is represented. \\
\hline
\end{tabular}
\end{center}
\caption{Traceability.}
\label{tab:traceability}
\vspace{-.2in}
\end{table}
\subsection{Exploring the design space}
By instantiating these two axes we get different flavors of BX
frameworks. As we will present next, some instantiations correspond to
well-known existing frameworks, but others have not been instantiated
yet, and an interesting question is whether they might be useful or
not.
Not surprisingly, one of the most popular schemes is that of
bidirectional \emph{mappings}, where no traceability information is
represented and updates are typically represented by only the
post-state. Formally, we have \ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\text{S}} and \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{N}}, leading
to the scheme \ensuremath{\mathsf{to}\mathbin{:}\Conid{A}\to \Conid{B}} and \ensuremath{\mathsf{from}\mathbin{:}\Conid{B}\to \Conid{A}}. In these frameworks,
the consistency relation typically corresponds to one of the
transformations and thus is omitted. A symmetric instantiation of this category is that of
bijective languages,
whose transformations establish a bijection between subsets of \ensuremath{\Conid{A}} and
\ensuremath{\Conid{B}}. These subsets contain essentially the same information but just
present it differently. Such languages promote the interoperability
between different formats and are easy to reason about because
bijectivity is preserved by composition and inversion. Less
restrictive asymmetric mapping frameworks encompass transformations
that are only reversible in a particular direction, for example when
\ensuremath{\Conid{B}} refines \ensuremath{\Conid{A}}.
The most popular asymmetric scheme, \emph{lenses}~\cite{Foster:07},
was proposed as a solution to the classical view-update problem from database theory~\cite{Bancilhon:81}.
Updates are still represented using just the post-state, but \ensuremath{\mathsf{from}}
requires traceability information to deal with missing information,
namely the original source value of type \ensuremath{\Conid{A}}.
More precisely, we have \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\text{S}}
and \ensuremath{\overleftarrow{\mathsf{T}}\mathrel{=}\text{N}}. Since the returned \ensuremath{\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})} is equal to the input of \ensuremath{\mathsf{to}}, we
end up with the simplified scheme~\eqref{eq:short:scheme}, resulting in
the interface \ensuremath{\mathsf{to}\mathbin{:}\Conid{A}\to \Conid{B}} and \ensuremath{\mathsf{from}\mathbin{:}\Conid{B}\;\!\!\times\!\!\;\Conid{A}\to \Conid{A}} (known, respectively, as \ensuremath{\Varid{get}}
and \ensuremath{\Varid{put}} in the lens framework).
The consistency relation is assumed to be \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}} iff \ensuremath{\mathsf{to}\;\Varid{a}\mathrel{=}\Varid{b}}.
In order to guarantee stronger properties or support particular
transformation scenarios, some
lens-like
approaches are operation-based
, or use an additional sameness relation, providing a traceability
between views and sources.
The asymmetric treatment of lenses
only works well for (essentially) surjective (information decreasing)
transformations, since \ensuremath{\mathsf{to}} cannot access \ensuremath{\Conid{B}} details without
counterpart in \ensuremath{\Conid{A}}. For more general transformations without a
dominant flow of information, where each of the source and target
models may contain information not present in the other, we end up
with symmetric schemes. Among those, \emph{maintainers}~\cite{Meertens:98} are among the
most popular, where \ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\text{S}}
and \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{S}}, leading
to the scheme \ensuremath{\mathsf{to}\mathbin{:}\Conid{A}\;\!\!\times\!\!\;\Conid{B}\to \Conid{B}} and \ensuremath{\mathsf{from}\mathbin{:}\Conid{B}\;\!\!\times\!\!\;\Conid{A}\to \Conid{A}}, where
updates are propagated given knowledge of the pre-state of the
respective opposite transformations. Likewise to
lenses, since the output traceability can trivially be derived from input
updates, it is not returned by the transformations. Also, unlike the
previous frameworks, there is now an explicitly declared consistency relation
\ensuremath{\mathsf{R}}, from which \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}} are somehow inferred.
\emph{Trigonal systems} were proposed in~\cite{Diskin:08} to avoid the
recalculation of the whole state values when updates are
incremental. As a generalization of maintainers, besides having
knowledge of the target pre-state, information about the source
pre-state is also present. Concretely, we now have \ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\mathbb{S}} and
\ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{S}}. The existence of an explicit
consistency relation \ensuremath{\mathsf{R}} is also assumed. Like the previous schemes, the content
captured by \ensuremath{\overrightarrow{\mathsf{T}}} is trivially derived from the input information, so
it is not returned by \ensuremath{\mathsf{to}}. However, since the value of the pre-state
of \ensuremath{\overleftarrow{\mathsf{U}}} can also be directly retrieved from \ensuremath{\overleftarrow{\mathsf{T}}}, it is also omitted
from the output. Putting it all together, we have the scheme \ensuremath{\mathsf{to}\mathbin{:}(\Conid{A}\;\!\!\times\!\!\;\Conid{A})\;\!\!\times\!\!\;\Conid{B}\to \Conid{B}} and \ensuremath{\mathsf{from}\mathbin{:}(\Conid{B}\;\!\!\times\!\!\;\Conid{B})\;\!\!\times\!\!\;\Conid{A}\to \Conid{A}}.
\emph{Symmetric lenses}~\cite{Hofmann:2011} assume \ensuremath{\overleftarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{S}} and represent the traceability as a complement (\ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{C}}). Thus, we have \ensuremath{\mathsf{to}\mathbin{:}\Conid{A}\;\!\!\times\!\!\;\mathsf{C}(\Conid{A},\Conid{B})\to \Conid{B}\;\!\!\times\!\!\;\mathsf{C}(\Conid{A},\Conid{B})} (and vice-versa), and the complement \ensuremath{\mathsf{C}(\text{S},\Conid{T})} stores both the information of \ensuremath{\Conid{A}} not present in \ensuremath{\Conid{B}} (passed as input to \ensuremath{\mathsf{from}}) and vice-versa.
\emph{Edit lenses}~\cite{Hofmann:12} are an operation-based formulation of symmetric lenses, with edit-sequences as updates (\ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\text{E}}).
We have \ensuremath{\mathsf{to}\mathbin{:}\mathsf{O}(\Conid{A})^\star\;\!\!\times\!\!\;\mathsf{C}(\Conid{A},\Conid{B})\to \mathsf{O}(\Conid{B})^\star\;\!\!\times\!\!\;\mathsf{C}(\Conid{A},\Conid{B})} and the opposite for \ensuremath{\mathsf{from}}.
This time, the transformations do not process states and the complement \ensuremath{\mathsf{C}(\Conid{A},\Conid{B})} stores only some extra information about \ensuremath{\Conid{A}} and \ensuremath{\Conid{B}} that is used to disambiguate updates, but not sufficient to restore the original states.
Among symmetric schemes, \emph{symmetric
delta-lenses}~\cite{Diskin:11a} are one of the most general. Here
\ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\text{D}} and \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{D}}: besides the pre-state values of \ensuremath{\Conid{A}}
and \ensuremath{\Conid{B}}, we have a sameness relation between them in the traceability,
and likewise for the update itself. We have \ensuremath{\mathsf{to}\mathbin{:}(\Conid{A}\;\!\!\times\!\!\;\Conid{A}\;\!\!\times\!\!\;\mathsf{D}(\Conid{A},\Conid{A}))\;\!\!\times\!\!\;(\Conid{A}\;\!\!\times\!\!\;\Conid{B}\;\!\!\times\!\!\;\mathsf{D}(\Conid{A},\Conid{B}))\to (\Conid{B}\;\!\!\times\!\!\;\Conid{B}\;\!\!\times\!\!\;\mathsf{D}(\Conid{B},\Conid{B}))\;\!\!\times\!\!\;(\Conid{A}\;\!\!\times\!\!\;\Conid{B}\;\!\!\times\!\!\;\mathsf{D}(\Conid{A},\Conid{B}))} and the opposite for \ensuremath{\mathsf{from}}. An
explicit consistency relation \ensuremath{\mathsf{R}} between updates is also present.
\section{Properties}
\label{sec:properties}
The interface of a framework gives some hints about its expressivity
but says little about the actual behavior of the transformations. Such
behavior is usually specified by high-level algebraic properties that
enforce some predictability on the system (namely concerning
bidirectionality). Table~\ref{tab:properties} identifies several generic
properties that, independently of the framework, might be desirable
from an end-user perspective. We only present the properties from the
perspective of \ensuremath{\mathsf{from}} (i.e., propagating updates from the \ensuremath{\Conid{B}} side to
the \ensuremath{\Conid{A}} side). The dual properties can also be specified for \ensuremath{\mathsf{to}}. All
free variables are implicitly universally quantified. This
formalization is to some extent
a textual version of the graphical
\emph{tile algebra}~\cite{DiskinGTTSE:11}, previously used to
formalize some of the laws presented here. Since the transformation
can be partial, the properties are only required to hold when they yield
a result. Given a transformation $f$, $f\ x \downarrow$ holds when $f$
is defined on $x$, and $f\ x \sqsubseteq y$ holds if $f\ x
\downarrow\ \Rightarrow f\ x = y$.
\begin{table}[!t]
\begin{center}
\begin{align*}
& \frac{}{\ensuremath{\mathsf{from}\;({\mathsf{id}_\Conid{B}},{\textbf {\textit {\textrm r}}})\sqsubseteq({\mathsf{id}_\Conid{A}},{{{\textbf {\textit {\textrm r}}}}^\circ})}} \tag{\ensuremath{\mathsf{from}}-Stability}
\\
& \frac{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})}}{\ensuremath{\mathsf{to}\;({\textbf {\textit {\textrm a}}},{{{\textbf {\textit {\textrm r}}}}^\circ})\sqsubseteq({\textbf {\textit {\textrm b}}},{{{\textbf {\textit {\textrm s}}}}^\circ})}}\tag{\ensuremath{\mathsf{from}}-Invertibility}
\\
& \frac{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})}}{\ensuremath{\mathsf{from}\;({{{\textbf {\textit {\textrm b}}}}^\circ},{{{\textbf {\textit {\textrm s}}}}^\circ})\sqsubseteq({{{\textbf {\textit {\textrm a}}}}^\circ},{{{\textbf {\textit {\textrm r}}}}^\circ})}}\tag{\ensuremath{\mathsf{from}}-Undoability}
\\
& \frac{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm s}}}_{1})} \quad \ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{2},{{{\textbf {\textit {\textrm s}}}_{1}}^\circ})\mathrel{=}({\textbf {\textit {\textrm a}}}_{2},{\textbf {\textit {\textrm s}}}_{2})}}{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{2}\mathbin{\circ}{\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm r}}})\sqsubseteq({\textbf {\textit {\textrm a}}}_{2}\mathbin{\circ}{\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm s}}}_{2})}}\tag{\ensuremath{\mathsf{from}}-History-ignorance}
\\
& \frac{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})}}{\ensuremath{{\textbf {\textit {\textrm s}}}\;\!\in\! \;\mathsf{R}}}\tag{\ensuremath{\mathsf{from}}-Correctness}
\\
& \frac{\ensuremath{{\textbf {\textit {\textrm b}}}\mathbin{\circ}{\textbf {\textit {\textrm r}}}\;\!\in\! \;\mathsf{R}}}{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\sqsubseteq({\mathsf{id}_\Conid{A}},{{({\textbf {\textit {\textrm b}}}\mathbin{\circ}{\textbf {\textit {\textrm r}}})}^\circ})}}\tag{\ensuremath{\mathsf{from}}-Hippocraticness}
\\
& \frac{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})} \quad \ensuremath{{{{\textbf {\textit {\textrm s}}}}^\circ}\mathbin{\circ}{\textbf {\textit {\textrm a}}}\mathrel{=}{{{\textbf {\textit {\textrm s}}}_{1}}^\circ}\mathbin{\circ}{\textbf {\textit {\textrm a}}}_{1}} \quad \ensuremath{{\textbf {\textit {\textrm s}}}_{1}\;\!\in\! \;\mathsf{R}}}{\ensuremath{{\textbf {\textit {\textrm a}}} \leq {\textbf {\textit {\textrm a}}}_{1}}}\tag{\ensuremath{\mathsf{from}}-Least-update}
\\
& \frac{}{\ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})} \downarrow}\tag{\ensuremath{\mathsf{from}}-Totality}
\end{align*}
\end{center}
\caption{Properties.}
\label{tab:properties}
\vspace{-.2in}
\end{table}
\emph{Stability} imposes that null updates must be translated to
null updates, in the sense that if a \ensuremath{\Conid{B}} is not modified, then no
change shall be performed on the consistent \ensuremath{\Conid{A}}. We represent a null
update on \ensuremath{\Conid{A}} by a constant \ensuremath{{\mathsf{id}_\Conid{A}}}, where \ensuremath{ \delta {\mathsf{id}_\Conid{A}}\mathrel{=}\rho {\mathsf{id}_\Conid{A}}}.
\emph{Invertibility} states that it shall be possible to
revert the application of a transformation by applying the opposite
transformation. This law
implies that updates
on the \ensuremath{\Conid{B}} side are translated faithfully to \ensuremath{\Conid{A}}, otherwise they could
not be inverted. In asymmetric frameworks \ensuremath{\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})} might be different
from \ensuremath{\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})}. In those cases, we assume that traceability \ensuremath{{\textbf {\textit {\textrm r}}}\mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})} can be reversed as \ensuremath{{{{\textbf {\textit {\textrm r}}}}^\circ}\mathbin{:}\overleftarrow{\mathsf{T}}(\Conid{A},\Conid{B})}, where \ensuremath{ \delta {{{\textbf {\textit {\textrm s}}}}^\circ}\mathrel{=}\rho {\textbf {\textit {\textrm s}}}} and \ensuremath{\rho {{{\textbf {\textit {\textrm s}}}}^\circ}\mathrel{=} \delta {\textbf {\textit {\textrm s}}}}. \emph{Undoability} ensures that an update
translation can be undone by re-applying the same transformation with
an inverse update. Likewise to traceability, given an update \ensuremath{{\textbf {\textit {\textrm a}}}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})} its inverse will be denoted by \ensuremath{{{{\textbf {\textit {\textrm a}}}}^\circ}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})}, where \ensuremath{ \delta {{{\textbf {\textit {\textrm a}}}}^\circ}\mathrel{=}\rho {\textbf {\textit {\textrm a}}}} and \ensuremath{\rho {{{\textbf {\textit {\textrm a}}}}^\circ}\mathrel{=} \delta {\textbf {\textit {\textrm a}}}}. Also, two
updates \ensuremath{{\textbf {\textit {\textrm a}}}_{1}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})} and \ensuremath{{\textbf {\textit {\textrm a}}}_{2}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})} can be sequentially composed
as \ensuremath{{\textbf {\textit {\textrm a}}}_{2}\mathbin{\circ}{\textbf {\textit {\textrm a}}}_{1}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{A})}.
\emph{History-ignorance} states that update
translation does not depend on the past history. In practice, this
means that two consecutive update translations can be performed at
once on the composed update.
The following three properties involve the
consistency relation. \emph{Correctness} simply states that a
transformation restores consistency. \emph{Hippocraticness} is a
stronger version of stability (although not always
desirable~\cite{Diskin:08}), stating that an update that does not
break the consistency should be ignored. In the generic formulation we
assume that a traceability \ensuremath{{\textbf {\textit {\textrm r}}}\mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})} can be composed with an
update \ensuremath{{\textbf {\textit {\textrm b}}}\mathbin{:}\overrightarrow{\mathsf{U}}(\Conid{B})} to yield a traceability \ensuremath{{\textbf {\textit {\textrm b}}}\mathbin{\circ}{\textbf {\textit {\textrm r}}}\mathbin{:}\overrightarrow{\mathsf{T}}(\Conid{A},\Conid{B})}
relating the original source \ensuremath{\Conid{A}} with the updated \ensuremath{\Conid{B}}. Due to the
incidence conditions, checking the consistency of \ensuremath{{\textbf {\textit {\textrm b}}}\mathbin{\circ}{\textbf {\textit {\textrm r}}}} is
equivalent to checking the consistency of these values.
In schemes where the consistency relation is one of the transformations,
correctness and hippocraticness degenerate into invertibility and stability, respectively.
The \emph{least-update} property can be seen as an additional \emph{quality property} entailing that the returned update must be
the smallest among all \ensuremath{\mathsf{R}}-consistent ones that could have been returned.
To compare updates, we assume the existence of a total
preorder \ensuremath{ \leq } on \ensuremath{\overrightarrow{\mathsf{U}}(\Conid{A})}. When \ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}},{\textbf {\textit {\textrm r}}})\mathrel{=}({\textbf {\textit {\textrm a}}},{\textbf {\textit {\textrm s}}})} then
\ensuremath{{\textbf {\textit {\textrm a}}}} must be smaller than every update \ensuremath{{\textbf {\textit {\textrm a}}}_{1}} that could lead to a
value consistent with the post-state of \ensuremath{{\textbf {\textit {\textrm b}}}} (testified by a
consistent traceability \ensuremath{{\textbf {\textit {\textrm s}}}_{1}}). Assuming that, for an already consistent state, the null target update
is the unique minimal update, least-update subsumes hippocraticness.
So far, we presented the properties modulo undefinedness of the
unidirectional transformations. This is because totality requirements
are by themselves an important feature in the design of a BX
framework. In practice, it is often convenient to acknowledge that
the type system might not be expressive enough to capture all
constraints induced by the transformations, and to allow the source
and target types to be larger than the actual domains of the
transformations, leading to partially defined transformations.
While this might be a ``show stopper'' for batch applications that are
expected to always produce results, it is usually acceptable for
interactive applications: an editor does not need to handle every
update and can signal an error to the user disallowing a specific
modification. Partiality is not adequate for security
applications~\cite{Foster:2009b} though, since users might extract
information about the hidden data from the cases for which the
transformations fail. As such, to allow a finer-grain comparison of
frameworks, we choose to factor out \emph{totality} as an orthogonal
property: it holds for a transformation if it is defined for every
possible combination of update and traceability.
Sometimes, weaker versions of the above laws may be satisfied instead. For
example, we can have weaker versions of \emph{invertibility} where the
final state is equal to the original one modulo another update
translation. This particular \emph{weak invertibility} is a kind of
\emph{convergence} law (or \emph{bi-idempotence}
in~\cite{Hu:04journal}), since it entails that update translation
eventually converges into stable states. Other \emph{weak} variants
of the laws occur when value comparison ignores details
that are inessential for an application scenario, like ordering, whitespaces or structure sharing.
For operation-based frameworks, these
may also mean that round-tripping does not preserve the full update,
but only its post-state.
An interesting weaker
version of totality is \emph{safety}~\cite{Pacheco:12} (also known as \emph{domain correctness}
in~\cite{Diskin:08}), which entails that a transformation is defined
at least for the range of the opposite one, independently of its
pre-state. Taking into account the consistency relation, safety can also be stated as follows: a transformation must be defined for every source value that has at least one consistent target.
Weaker versions of \emph{correctness} include allowing the creation of inconsistent states if no consistent ones exists.
\subsection{Revisiting the design space}
To instantiate the generic properties for a concrete framework, one
must first devise how to express null, inversion and composition of
updates and traceability. In some frameworks some of these might not
be expressible, meaning that some properties may not be applicable.
For example, \emph{mappings} have no traceability and updates are
represented just by the post-state, hence there is no way to reason
about pre-states. This implies that neither null updates, nor update
inversion can be defined, and \ensuremath{\mathsf{from}}-Stability, \ensuremath{\mathsf{from}}-Undoability,
and \ensuremath{\mathsf{from}}-Hippocractiness are not expressible. \ensuremath{\mathsf{from}}-Invertibility
is just \ensuremath{\mathsf{to}\;(\mathsf{from}\;\Varid{b})\sqsubseteq\Varid{b}}, ensuring \ensuremath{\mathsf{from}} to be left-invertible
(injective). Without any knowledge of the pre-state,
\ensuremath{\mathsf{from}}-History-ignorance holds trivially. Assuming the consistency relation \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\equiv (\Varid{b}\mathrel{=}\mathsf{to}\;\Varid{a})}, \ensuremath{\mathsf{from}}-Correctness
degenerates to \ensuremath{\mathsf{from}}-Invertibility, and \ensuremath{\mathsf{from}}-Least-update amounts
to checking that if \ensuremath{\mathsf{from}\;\Varid{b}\mathrel{=}\Varid{a}}, then \ensuremath{\Varid{a}} is smaller than every \ensuremath{\Varid{a}_{1}}
leading to the same \ensuremath{\Varid{b}}, that is \ensuremath{\mathsf{from}\;\Varid{b}\mathrel{=}\Varid{a}\;\wedge\;\mathsf{to}\;\Varid{a}\mathrel{=}\mathsf{to}\;\Varid{a}_{1}\mathrel{=}\Varid{b}\Rightarrow \Varid{a} \leq \Varid{a}_{1}}.
Unlike mappings, in \emph{lenses} there is some traceability
information when applying \ensuremath{\mathsf{from}} that allows us to reason about
pre-states. In particular, when \ensuremath{\mathsf{from}} is applied to traceability \ensuremath{\Varid{a}}
(the original source value), due to the consistency relation
\ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\equiv (\Varid{b}\mathrel{=}\mathsf{to}\;\Varid{a})}, we know that the pre-state of the input
update is \ensuremath{\mathsf{to}\;\Varid{a}}. Hence, a null update has the same
post-state, and \ensuremath{\mathsf{from}}-Stability can be instantiated as \ensuremath{\mathsf{from}\;(\mathsf{to}\;\Varid{a},\Varid{a})\sqsubseteq\Varid{a}} (known in this framework as \textsc{GetPut}).
Instantiation of \ensuremath{\mathsf{from}}-Invertibility is more
straightforward --- just ignore unused traceability --- yielding \ensuremath{\mathsf{from}\;(\Varid{b}_{1},\kern0.06em \vbox{\hrule\@width.5em} )\mathrel{=}\Varid{a}_{1}\Rightarrow \mathsf{to}\;\Varid{a}_{1}\sqsubseteq\Varid{b}_{1}} (known as \textsc{PutGet}).
To instantiate \ensuremath{\mathsf{from}}-Undoability, we follow the same approach as in \ensuremath{\mathsf{from}}-Stability, resulting in \ensuremath{\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\mathrel{=}\Varid{a}_{1}\Rightarrow \mathsf{from}\;(\mathsf{to}\;\Varid{a},\Varid{a}_{1})\sqsubseteq\Varid{a}}.
Since the composition of two state-based updates \ensuremath{\Varid{b}\mathbin{\circ}\Varid{a}} is just \ensuremath{\Varid{b}},
instantiation of \ensuremath{\mathsf{from}}-History-Ignorance is \ensuremath{\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\mathrel{=}\Varid{a}_{1}\mathrel{\wedge}\mathsf{from}\;(\Varid{b}_{2},\Varid{a}_{1})\mathrel{=}\Varid{a}_{2}\Rightarrow \mathsf{from}\;(\Varid{b}_{2},\Varid{a})\sqsubseteq\Varid{a}_{2}} (known as
\textsc{PutPut}). Due to the consistency
relation being \ensuremath{\mathsf{to}}, \ensuremath{\mathsf{from}}-Correctness and \ensuremath{\mathsf{from}}-Hippocraticness degenerate
into \ensuremath{\mathsf{from}}-Invertibility and \ensuremath{\mathsf{from}}-Stability, respectively. Likewise
to mappings, \ensuremath{\mathsf{from}}-Least-update is formulated as \ensuremath{\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\mathrel{=}\Varid{a}_{1}\;\wedge\;\mathsf{to}\;\Varid{a}_{2}\mathrel{=}\Varid{b}_{1}\Rightarrow \Varid{a}_{1} \leq \Varid{a}_{2}}.
Like in the previous frameworks, the value of the pre-state of the
input update is not explicitly represented in \emph{maintainers}.
However, unlike lenses, the declared consistency relation of a maintainer may
not be functional (deterministic), and a value of \ensuremath{\Conid{A}} is not uniquely related to another value
of \ensuremath{\Conid{B}}. This means that it is impossible to identify a null update given
only the original source value present in the traceability, and
\ensuremath{\mathsf{from}}-Stability cannot be formulated.
In the case of \ensuremath{\mathsf{from}}-Invertibility, to revert a transformation \ensuremath{\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\mathrel{=}\Varid{a}_{1}} we need to invert the traceability \ensuremath{\Varid{a}}, and recover the
original consistent \ensuremath{\Varid{b}}. As discussed above, in general this is not
possible, but we can generalize this property assuming that the
transformation can be reverted for any consistent \ensuremath{\Varid{b}}, that is \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\;\wedge\;\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\mathrel{=}\Varid{a}_{1}\Rightarrow \mathsf{to}\;(\Varid{a}_{1},\Varid{b})\sqsubseteq\Varid{b}_{1}}.
Following a similar approach, \ensuremath{\mathsf{from}}-Undoability can be formulated as \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\;\wedge\;\mathsf{from}\;(\Varid{b}_{1},\Varid{a})\sqsubseteq\Varid{a}_{1}\Rightarrow \mathsf{from}\;(\Varid{b},\Varid{a}_{1})\sqsubseteq\Varid{a}}.
Since the \ensuremath{\mathsf{from}} interface is similar to lenses, \ensuremath{\mathsf{from}}-History-ignorance is exactly the same.
Due to the explicit consistency relation,
\ensuremath{\mathsf{from}}-Correctness is directly formulated as \ensuremath{\mathsf{from}\;(\Varid{b},\kern0.06em \vbox{\hrule\@width.5em} )\mathrel{=}\Varid{a}\Rightarrow (\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}} and \ensuremath{\mathsf{from}}-Hippocraticness as \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\Rightarrow \mathsf{from}\;(\Varid{b},\Varid{a})\sqsubseteq\Varid{a}}. Lastly, \ensuremath{\mathsf{from}}-Least-update is again similar to that of lenses but with an explicit consistency relation, i.e., \ensuremath{\mathsf{from}\;(\Varid{b},\Varid{a})\mathrel{=}\Varid{a}_{1}\;\wedge\;(\Varid{a}_{2},\Varid{b})\;\!\in\! \;\mathsf{R}\Rightarrow \Varid{a}_{1} \leq \Varid{a}_{2}}.
In \emph{trigonal systems}, updates are represented by both pre- and
post-state value. As such, null updates and composition and inversion
of updates can be directly defined as \ensuremath{{\mathsf{id}_\Conid{B}}\mathrel{=}(\Varid{b},\Varid{b})},
\ensuremath{(\Varid{b}_{1},\Varid{b}_{2})\mathbin{\circ}(\Varid{b}_{2},\Varid{b}_{3})\mathrel{=}(\Varid{b}_{1},\Varid{b}_{3})} and \ensuremath{{{(\Varid{b}_{1},\Varid{b}_{2})}^\circ}\mathrel{=}(\Varid{b}_{2},\Varid{b}_{1})}, and the
instantiation of properties becomes rather straightforward. Namely,
\ensuremath{\mathsf{from}}-stability is \ensuremath{\mathsf{from}\;((\Varid{b},\Varid{b}),\Varid{a})\sqsubseteq\Varid{a}}, \ensuremath{\mathsf{from}}-Invertibility is
\ensuremath{\mathsf{from}\;((\Varid{b},\Varid{b}_{1}),\Varid{a})\mathrel{=}\Varid{a}_{1}\Rightarrow \mathsf{to}\;((\Varid{a},\Varid{a}_{1}),\Varid{b})\sqsubseteq\Varid{b}_{1}}, \ensuremath{\mathsf{from}}-Undoability
is \ensuremath{\mathsf{from}\;((\Varid{b},\Varid{b}_{1}),\Varid{a})\mathrel{=}\Varid{a}_{1}\Rightarrow \mathsf{from}\;((\Varid{b}_{1},\Varid{b}),\Varid{a}_{1})\sqsubseteq\Varid{a}}, and
\ensuremath{\mathsf{from}}-History-ignorance is \ensuremath{\mathsf{from}\;((\Varid{b},\Varid{b}_{1}),\Varid{a})\mathrel{=}\Varid{a}_{1}\;\wedge\;\mathsf{from}\;((\Varid{b}_{1},\Varid{b}_{2}),\Varid{a}_{1})\mathrel{=}\Varid{a}_{2}\Rightarrow \mathsf{from}\;((\Varid{b},\Varid{b}_{2}),\Varid{a})\sqsubseteq\Varid{a}_{2}}. Likewise to
maintainers, due to the explicit consistency relation, the remaining
instantiations are also immediate: \ensuremath{\mathsf{from}}-Correctness is \ensuremath{\mathsf{from}\;((\kern0.06em \vbox{\hrule\@width.5em} ,\Varid{b}_{1}),\kern0.06em \vbox{\hrule\@width.5em} )\mathrel{=}\Varid{a}_{1}\Rightarrow (\Varid{a}_{1},\Varid{b}_{1})\;\!\in\! \;\mathsf{R}}, \ensuremath{\mathsf{from}}-Hippocracticness is \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\Rightarrow \mathsf{from}\;((\kern0.06em \vbox{\hrule\@width.5em} ,\Varid{b}_{1}),\Varid{a})\sqsubseteq\Varid{a}} and \ensuremath{\mathsf{from}}-Least-update is \ensuremath{\mathsf{from}\;((\kern0.06em \vbox{\hrule\@width.5em} ,\Varid{b}_{1}),\Varid{a})\mathrel{=}\Varid{a}_{1}\;\wedge\;(\Varid{a}_{2},\Varid{b}_{1})\;\!\in\! \;\mathsf{R}\Rightarrow (\Varid{a},\Varid{a}_{1}) \leq (\Varid{a},\Varid{a}_{2})}. Likewise, the instantiation
of the properties in the framework of \emph{symmetric delta-lenses} is
straightforward, since both the pre- and post-state values of
updates are represented and have an explicit consistency relation.
In \emph{edit lenses}, updates are represented by sequences of edit
operations: the null update is the empty sequence \ensuremath{[\mskip1.5mu \mskip1.5mu]}, composing
updates \ensuremath{{\textbf {\textit {\textrm a}}}} and \ensuremath{{\textbf {\textit {\textrm b}}}} amounts to concatenation \ensuremath{{\textbf {\textit {\textrm a}}}\mathbin{+\!\!\!+} {\textbf {\textit {\textrm b}}}}, and assuming each
edit operation to be undoable, an update \ensuremath{{\textbf {\textit {\textrm a}}}\mathrel{=}[\mskip1.5mu \Varid{a}_{1},\mathinner{\ldotp\ldotp},\Varid{a}_{\Varid{n}}\mskip1.5mu]} could be
inverted, for example, as \ensuremath{{{{\textbf {\textit {\textrm a}}}}^\circ}\mathrel{=}[\mskip1.5mu {{\Varid{a}_{\Varid{n}}}^\circ},\mathinner{\ldotp\ldotp},{{\Varid{a}_{1}}^\circ}\mskip1.5mu]}.
Traceability information (the complement) is ``symmetric'' in the sense that \ensuremath{{{{\textbf {\textit {\textrm c}}}}^\circ}\mathrel{=}{\textbf {\textit {\textrm c}}}}.
Equipped with these definitions, some properties can be
directly instantiated: \ensuremath{\mathsf{from}}-Stability is \ensuremath{\mathsf{from}\;([\mskip1.5mu \mskip1.5mu],{\textbf {\textit {\textrm c}}})\sqsubseteq([\mskip1.5mu \mskip1.5mu],{\textbf {\textit {\textrm c}}})},
\ensuremath{\mathsf{from}}-Invertibility is \ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}}_{1})\Rightarrow \mathsf{to}\;({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}})\sqsubseteq({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}}_{1})},
\ensuremath{\mathsf{from}}-Undoability is \ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}}_{1})\Rightarrow \mathsf{from}\;({{{\textbf {\textit {\textrm b}}}_{1}}^\circ},{\textbf {\textit {\textrm c}}}_{1})\sqsubseteq({{{\textbf {\textit {\textrm a}}}_{1}}^\circ},{\textbf {\textit {\textrm c}}})}, and \ensuremath{\mathsf{from}}-History-ignorance is \ensuremath{\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}}_{1})\;\wedge\;\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{2},{\textbf {\textit {\textrm c}}}_{1})\mathrel{=}({\textbf {\textit {\textrm a}}}_{2},{\textbf {\textit {\textrm c}}}_{2})\Rightarrow \mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1}\mathbin{+\!\!\!+} {\textbf {\textit {\textrm b}}}_{2},{\textbf {\textit {\textrm c}}})\sqsubseteq({\textbf {\textit {\textrm a}}}_{1}\mathbin{+\!\!\!+} {\textbf {\textit {\textrm a}}}_{2},{\textbf {\textit {\textrm c}}}_{2})}.
In \ensuremath{\mathsf{from}}-Correctness the existence of the initial source-target pair \ensuremath{(\Varid{a},\Varid{b})} that gave origin to complement \ensuremath{{\textbf {\textit {\textrm c}}}} is assumed, over which the resulting edit-sequences are applied, resulting in \ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\mathrel{\wedge}\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}}_{1})\Rightarrow ({\textbf {\textit {\textrm a}}}_{1}\;\Varid{a},{\textbf {\textit {\textrm b}}}_{1}\;\Varid{b})\;\!\in\! \;\mathsf{R}}, where \ensuremath{({\textbf {\textit {\textrm a}}}\;\Varid{a})} denotes the application of the edit-sequence \ensuremath{{\textbf {\textit {\textrm a}}}} to the value \ensuremath{\Varid{a}}.
The information about the system state is external to the transformations, as updates are only represented by edit-sequences.
\ensuremath{\mathsf{from}}-Hippocracticness applies to transformations over the complement of states that were already consistent, represented as
\ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\Rightarrow \mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\sqsubseteq([\mskip1.5mu \mskip1.5mu],{\textbf {\textit {\textrm c}}})}. Lastly, \ensuremath{\mathsf{from}}-Least-update is formulated as
\ensuremath{(\Varid{a},\Varid{b})\;\!\in\! \;\mathsf{R}\mathrel{\wedge}\mathsf{from}\;({\textbf {\textit {\textrm b}}}_{1},{\textbf {\textit {\textrm c}}})\mathrel{=}({\textbf {\textit {\textrm a}}}_{1},{\textbf {\textit {\textrm c}}}_{1})\mathrel{\wedge}({\textbf {\textit {\textrm a}}}_{2}\;\Varid{a},{\textbf {\textit {\textrm b}}}_{1}\;\Varid{b})\;\!\in\! \;\mathsf{R}\Rightarrow {\textbf {\textit {\textrm a}}}_{1} \leq {\textbf {\textit {\textrm a}}}_{2}}. Note that the pre-order compares only edit-sequences.
Each framework has a notion of what is a \emph{well-behaved}
transformation, i.e., the minimum properties it must satisfy to be
considered reasonable. Typically, a transformation is
\emph{well-behaved} if it is at least stable and correct, i.e.,
preserves null updates and recovers consistency. In mappings
\ensuremath{\mathsf{to}}-Correctness degenerates into \ensuremath{\mathsf{to}}-Invertibility, meaning that
\ensuremath{\mathsf{to}} is injective and \ensuremath{\Conid{B}} is a \emph{refinement} of \ensuremath{\Conid{A}} (it
contains more information). In a \emph{well-behaved} symmetric
mapping, both \ensuremath{\mathsf{to}}-Invertibility and \ensuremath{\mathsf{from}}-Invertibility hold and
the BX is an \emph{isomorphism}, such that \ensuremath{\mathsf{to}} and \ensuremath{\mathsf{from}}
are bijections (when restricted to the respective domains).
In asymmetric lenses, \ensuremath{\mathsf{from}}-Invertibility together with
\ensuremath{\mathsf{from}}-Stability are the typical laws required for a lens to be
well-behaved.
The law \ensuremath{\mathsf{from}}-Invertibility implies that \ensuremath{\mathsf{to}} is surjective (again when restricted to the respective domain) and \ensuremath{\Conid{B}} is an
\emph{abstraction} of \ensuremath{\Conid{A}} (also called a view), meaning that the target contains less information than the source. If a framework also satisfies
history ignorance, then it is usually considered \emph{very well-behaved}. Since stability and history ignorance entail undoability,
very well-behaved frameworks are also undoable.
Some approaches do not enforce any totality requirements
. However, this can easily be abused: a (partial) BX can be
trivially well-behaved if both transformations are always
undefined.
For asymmetric frameworks, one transformation generally dominates the data flow and has stronger totality requirements than the other, and thus approaches usually assume \ensuremath{\mathsf{to}} to be total and \ensuremath{\mathsf{from}} either partial
or safe.
For symmetric frameworks, there is not generally a dominant data flow (for example, not every Java feature can be represented with a relational database schema, and vice-versa), and both transformations may be plausibly partial.
For \emph{total} BXs,
the types capture the exact domains over which the transformation is defined and guaranteed to behave well, ensuring that update translation cannot fail at run time.
\section{A survey of existing BX frameworks}
\label{sec:survey-existing-bx}
\begin{table}[!thb]
\begin{center}
\makebox[\textwidth]{%
\begin{tabular}{|c!{\vrule width 1.5pt}c|cc|cc|c||c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{
\begin{picture}(130,75)(0,0)
\put(85,55){\bf Feature}
\put(-6,75){\line(5,-3){140}}
\put(10,-10){\bf Approach}
\end{picture}}
& \multicolumn{6}{c||}{\hyperref[sec:framework]{Scheme}} & \multicolumn{9}{c|}{\hyperref[sec:properties]{Properties}} \\
\cline{2-16}
&\raiserot{\hyperref[tab:symmetry]{Symmetry}} & \raiserot{\hyperref[tab:update]{\ensuremath{\overrightarrow{\mathsf{U}}}}} & \raiserot{\hyperref[tab:update]{\ensuremath{\overleftarrow{\mathsf{U}}}}} & \raiserot{\hyperref[tab:traceability]{\ensuremath{\overleftarrow{\mathsf{T}}}}} & \raiserot{\hyperref[tab:traceability]{\ensuremath{\overrightarrow{\mathsf{T}}}}} & \raiserot{\hyperref[tab:consistency]{\ensuremath{\mathsf{R}}}}
&\raiserot{\hyperref[tab:properties]{Stable}} & \raiserot{\hyperref[tab:properties]{Invertible}} & \raiserot{\hyperref[tab:properties]{Convergent}} & \raiserot{\hyperref[tab:properties]{Undoable}}
&\raiserot{\hyperref[tab:properties]{History Ignorant}} & \raiserot{\hyperref[tab:properties]{Correct}} & \raiserot{\hyperref[tab:properties]{Hippocratic}} & \raiserot{\hyperref[tab:properties]{Least-update}} & \raiserot{\hyperref[tab:properties]{Total}} \\
\hline\hline
Brabrand et al. (2008)~\cite{Brabrand:2008}& S& \mcb{S}& \mcb{N}& E& & $\dasharrows$& & & & $\arrows$& & & $\arrows$ \\
Kawanaka and Hosoya (2006)~\cite{Kawanaka:2006} & S& \mcb{S}& \mcb{N}& E& & & & & & $\arrows$& & & $\arrows$ \\
Ehrig et al. (2007)~\cite{Ehrig:2007}& S& \mcb{S}& \mcb{D}& E& & $\arrows$& & & & $\arrows$& & & $\arrows$ \\
Wadler (1987)~\cite{Wadler:1987} & S& \mcb{S}& \mcb{N}& T& & $\arrows$& & & & & & & \\
Atanassow and Jeuring (2007)~\cite{Atanassow:2007}& S& \mcb{S}& \mcb{N}& I& & $\arrows$& & & & & & & \\
Kennedy (2004)~\cite{Kennedy:2004} & A& \mcb{S}& \mcb{N}& T& & $\ldasharrows$& & & & & & & \\
Terwilliger et al. (2007)~\cite{Terwilliger:2007} & A& S& E& \mcb{N}& T& & $\longrightarrow$& & & & & & & $\ldasharrows$ \\
Cunha et al. (2012)~\cite{Cunha:12} & A& \mcb{E}& \mcb{N}& E& $\arrows$& $\longrightarrow$& & & $\arrows$& $\arrows$& $\longleftarrow$& & $\arrows$ \\
Mu et al. (2004)~\cite{Mu:04} & A& S& E& \mcb{N}& T& & $\longrightarrow$& $\dashleftarrow$& & & & & & \\
Berdaguer et al. (2007)~\cite{Berdaguer:07} & A& \mcb{S}& \mcb{N}& T& & $\longrightarrow$& & & & & & & $\ldasharrows$ \\
Wang et al. (2010)~\cite{Wang:2010} & A& \mcb{S}& \mcb{N}& T& & $\longleftarrow$& & & & & & & $\arrows$ \\
Foster et al. (2007)~\cite{Foster:07} & A& \mcb{S}& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Bohannon et al. (2006)~\cite{Bohannon:06} & A& \mcb{S}& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Bohannon et al. (2008)~\cite{Bohannon:08} & A& \mcb{S}& N& S& T& $\ldasharrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Foster et al. (2008)~\cite{Foster:2008} & A& \mcb{S}& N& S& T& $\dasharrows$& $\dashleftarrow$& & & & & & & $\arrows$ \\
Barbosa et al. (2010)~\cite{Barbosa:2010} & A& S& D& N& D& T& $\arrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Hu et al. (2008)~\cite{Hu:04journal} & A& S& E& N& S& T& & & $\dasharrows$& & & & & & $\longrightarrow$ \\
Liu et al. (2007)~\cite{Liu:07} & A& S& E& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\longrightarrow$ \\
Hidaka et al. (2010)~\cite{Hidaka:10} & A& S& E& N& S& T& $\longleftarrow$& $\dashleftarrow$& & & & & & & $\longrightarrow$ \\
Takeichi (2009)~\cite{Takeichi:09} & A& \mcb{S}& N& S& I& $\longleftarrow$& & & & & & & & $\longrightarrow$ \\
Matsuda et al. (2007)~\cite{Matsuda:07} & A& \mcb{S}& N& C& T& $\arrows$& $\longleftarrow$& & & $\longleftarrow$& & & & $\longrightarrow$ \\
Voigtl\"{a}nder (2009)~\cite{Voigtlander:09} & A& \mcb{S}& N& S& T& $\arrows$& $\longleftarrow$& & & $\longleftarrow$& & & & $\longrightarrow$ \\
Voigtl\"{a}nder et al. (2010)~\cite{Voigtlander:10}& A& \mcb{S}& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\longrightarrow$ \\
Fegaras (2010)~\cite{Fegaras:10}& A& S& E& N& D& T& $\longleftarrow$& $\dashleftarrow$& & & & & & & $\longrightarrow$ \\
Melnik et al. (2007)~\cite{Melnik:07} & A& \mcb{S}& N& S& E& $\arrows$& $\longleftarrow$& & & & & & & $\rdasharrows$ \\
Diskin et al. (2011)~\cite{Diskin:11a}* & A& \mcb{D}& N& S& T& $\arrows$& $\longleftarrow$& & & $\arrows$& & & & $\ldasharrows$ \\
Wang et al. (2011)~\cite{Wang:2011}* & A& S& F& N& S& T& $\arrows$& $\longleftarrow$& & $\longleftarrow$& & & & & $\longrightarrow$ \\
Pacheco and Cunha (2010)~\cite{PachecoCunha:10} & A& \mcb{S}& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Pacheco and Cunha (2012)~\cite{PachecoCunha:12} & A& \mcb{D}& N& S& T& $\arrows$& $\longleftarrow$& & & & & & & $\arrows$ \\
Meertens (1998)~\cite{Meertens:98} & S& \mcb{S}& \mcb{S}& E& & & & & & $\arrows$& $\arrows$& & $\arrows$ \\
Meertens (1998)~\cite{Meertens:98}& S& \mcb{E}& \mcb{S}& E& & & & & & $\arrows$& & $\arrows$& $\arrows$ \\
Stevens (2007)~\cite{Stevens:07}* & S& \mcb{S}& \mcb{S}& E& & & & $\arrows$& & $\arrows$& $\arrows$& & $\arrows$ \\
Macedo and Cunha (2013)~\cite{MacedoCunha:13} & S& \mcb{S}& \mcb{S}& E& & & & & & $\arrows$& & $\arrows$& $\dasharrows$ \\
Hofmann et al. (2011)~\cite{Hofmann:2011} & S& \mcb{S}& \mcb{C}& I& $\arrows$& & & & & & & & $\arrows$ \\
Hofmann et al. (2012)~\cite{Hofmann:12}& S& \mcb{E}& \mcb{C}& I& & & & & & $\arrows$& & & $\dasharrows$ \\
Diskin et al. (2011)~\cite{Diskin:2011}* & S& \mcb{D}& \mcb{D}& T& & & $\arrows$& $\dasharrows$& & $\arrows$& & & \\
Hermann et al. (2011)~\cite{Hermann:2011} & S& \mcb{D}& \mcb{D}& E& & & $\arrows$& & & $\arrows$& & & $\dasharrows$ \\
Cicchetti et al. (2011)~\cite{Cicchetti:11} & S& \mcb{S}& \mcb{S}& E& $\arrows$& & & & &$\dasharrows$ & & & $\arrows$ \\
Ennals and Gay (2007)~\cite{Ennals:2007} & S& \mcb{S}& \mcb{S}& I& $\dasharrows$& & & & & $\arrows$& & & $\arrows$ \\
\hline
\end{tabular}}
\end{center}
\caption{Comparison of existing BX approaches.}
\label{table:stateofart}
\end{table}
Based on the proposed generic scheme and properties, we attempted a
comprehensive survey of existing BX tools and
frameworks. Table~\ref{table:stateofart} presents the results of this
first effort\footnote{A more detailed classification of BX approaches related to this paper can be found in~\cite[Chapter 3]{Pacheco:12}. Although the mentioned scheme regards only 3 specific frameworks (mappings, lenses and maintainers), this complementary work also proposes a taxonomy for the particular deployment features of BX frameworks and a textual justification for each of the entries in Table~\ref{table:stateofart}.}.
Regarding the scheme, the different axes were classified
according to Tables~\ref{tab:consistency}, \ref{tab:symmetry},
\ref{tab:update}, and \ref{tab:traceability}. Regarding the semantic
properties, for every property of Table~\ref{tab:properties}, we use
right arrows to denote that the property holds for \ensuremath{\mathsf{to}} and a left arrow
for \ensuremath{\mathsf{from}}. A normal arrow denotes that a property is satisfied by all well-behaved BXs in a given approach, while
a dashed arrow signals that well-behaved BXs only satisfy a weaker version. The absence
of an arrow means that well-behaved BXs do not satisfy a particular property, or that such property is not explicitly stated by the authors or implied by other
properties. This also means that laws implied by others are not depicted. For instance, as we have seen, correctness and hippocraticness in lenses degenerates into invertibility and stability, due to the consistency relation \ensuremath{\Varid{b}\;\mathsf{R}\;\Varid{a}\equiv \Varid{b}\mathrel{=}\mathsf{to}\;\Varid{a}}.
For totality, the absence of an arrow means that the
transformation is partial, and a dashed arrows means that the
transformation is safe.
Some particular entries in Table~\ref{table:stateofart} do not represent concrete BX tools but proposals of BX frameworks.
Such entries are signaled with an asterisk \texttt{*}: marked properties indicate the criteria for well-behaved BXs in such frameworks. The duplicate entry of~\cite{Meertens:98} is due to the fact that two different approaches are presented in it.
Table~\ref{table:stateofart} is not intended to be complete, but rather to provide a high-level picture of existing BX tools.
It must be read with some caution, though,
since it does not capture specific intricacies of particular
frameworks that are not representable in our generic scheme, and since
some of them are omissive or ambiguous regarding particular features,
which leads to some subjectivity in the classification. To alleviate
this we intend to publish the current survey online, and engage the
authors in the classification of their own tools and frameworks. With
completeness in mind, we also appreciate any suggestion of additional
classification axes and BX frameworks to include in the survey, in
order to reach a detailed global picture of the state of the art of
the field.
\section{Related work}
\label{section:relwork}
Acknowledging the heterogeneity of the field,~\cite{GRACE:09} surveys the related literature on BXs, grouping existing work by subcommunities, identifying some of the grand challenges of the field and providing a modest discussion on the terminology, key concepts and semantic properties used across the represented communities.
A more focused picture on bidirectional model transformations is given in~\cite{Stevens:2008b}, with emphasis on tool support and inherent open challenges.
Acknowledging the growing effort in the BX community towards unification, \cite{Terwilliger:2012} compares five particular BX tools in terms of their strengths and weaknesses for particular scenarios. For such comparison, it proposes a simple taxonomy
to analyze the behavior of BX tools from a model-driven perspective, but that does not consider particular BX properties.
A detailed feature model for the classification of model transformation approaches is proposed in \cite{Czarnecki:06}, together with a survey of a vast number of existing approaches.
Nevertheless, this classification is focused on design features rather than semantic properties, and does not pay particular detail to bidirectionality.
A detailed scheme for classifying a wide spectrum of bidirectional model synchronization axiomatizations and features is provided in \cite{Antkiewicz:2008}, illustrating particular instantiations with examples of existing systems. Still, only design properties are considered, and not the semantic properties of each instance.
In contrast, \cite{Diskin:08} proposes a classifying system in which some state-based BX frameworks can be compared and analyzed in terms of their semantic laws. Despite most laws proposed there for lenses, maintainers and trigonal systems matching our own instantiations (see Section~\ref{sec:properties}), invertibility and undoability do not. In fact, the laws for maintainers assume some surjectivity constraints on the transformations, while those for trigonal systems do not. This is an indication that no consistent method for the instantiation of the laws was followed.
A general framework for building delta-based model synchronization frameworks is proposed in~\cite{DiskinGTTSE:11}, which
takes the shape \ensuremath{\overrightarrow{\mathsf{U}}\mathrel{=}\overleftarrow{\mathsf{U}}\mathrel{=}\text{D}} and \ensuremath{\overrightarrow{\mathsf{T}}\mathrel{=}\overleftarrow{\mathsf{T}}\mathrel{=}\text{D}} and considers generic laws similar to those from Table~\ref{tab:properties}, except for invertibility and least-update. However, it is not explored how existing frameworks can be instantiated under this general framework, and how the choices made on the scheme affect the properties. We follow a bottom-up characterization of the laws from state-based mappings to symmetric delta lenses, and also provide an extensive comparison of existing BX tools.
\section{Conclusions and future work}
\label{section:conclusion}
In this paper, we have presented a generic scheme and several generic
properties of BX frameworks. The generic scheme can be instantiated
along two main axes (update and traceability representation), and the
presented properties cover the most for bidirectional laws proposed in
the literature. We have also shown how this generic presentation can
be instantiated to obtain and compare most of the existing concrete BX
frameworks, such as lenses or maintainers. We have applied
this comparative study not only to such broad framework categories,
but to a large set of concrete BX tools and techniques proposed in the
literature. In the future we intend to
extend this survey effort with more entries and classification axes (covering, for
example, deployment features, such as data domain or
bidirectionalization technique) to achieve a detailed global picture
of the state of the art of the field.
One of the key roles of properties is to ensure some degree of
predictability concerning the behavior of BXs. As seen in our survey, most frameworks only guarantee
stability, correctness and invertibility, but unfortunately these
properties still leave a lot of room for unpredictable and sometimes
unreasonable behavior, defeating their goal as a means for comparing
the effectiveness of two BX frameworks. In the long term,
this problem should be solved with the study of new properties
that better characterize more refined behavior, like minimization of update translation. Meanwhile, in the
continuation of~\cite{GRACE:09}, we intend to address the problem by
developing a suite of paradigmatic examples for each application
domain, so that the user can compare the behavior of different
frameworks within that domain.
Moreover, citing~\cite{Terwilliger:2012}:
\begin{quote}
A more ambitious goal would be a truly unified theoretical foundation to BX. [...]
Such a unification would not be trivial to accomplish, since it will require a huge collaborative effort, involving researchers from distinct communities, countries and scientific cultures. However, such a unification would be desirable, both for better addressing existing bidirectional scenarios and for tackling largely unexplored, yet important scenarios.
\end{quote}
We believe our work presents a solid base towards such unified theoretical foundation. In its current form, we think it might help developers when designing new frameworks and end-users when comparing existing ones. Finally, we intend to improve the
accuracy and completeness of our survey by enrolling the
BX community: the taxonomy and current results
will be published online, and authors will be invited to discuss and
extend it by classifying their own frameworks.
\section*{Acknowledgements}
This work is funded by ERDF - European Regional Development Fund through the COMPETE Programme (operational programme for competitiveness) and by national funds through the FCT - Funda{\c c}{\~ a}o para a Ci{\^ e}ncia e a Tecnologia (Portuguese Foundation for Science and Technology) within project FCOMP-01-0124-FEDER-020532. The first author of the paper is also sponsored by FCT grant SFRH/BD/69585/2010.
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,875 |
Cameron declares himself a Zionist
UK Conservative leader talks tough on Hamas, blasts boycott campaigns.
By JONNY PAUL Jerusalem Post correspondent
jp.services2
"I am a Zionist," Conservative Party leader David Cameron told an audience of party supporters of Israel in London on Tuesday. "If what you mean by Zionist, is someone who believes that the Jews have a right to a homeland in Israel and a right to their country then, yes, I am a Zionist and I'm proud of the fact that Conservative politicians down the ages have played a huge role in helping to bring this about," Cameron declared. The Conservative leader was guest of honor at the Conservative Friends of Israel annual business lunch, which was attended by some 500 people - including half the parliamentary party, 30 Conservative parliamentary candidates, former leaders, lords and Israel's ambassador.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,607 |
Q: How can I get row colors to change in standard grid based on data criteria I have a situation where I want a grid row to be a specific color, based on what one of the fields in that row is set to.
I've looked at this case:
How can we set color for particular column filed value conditionally in Acumatica
Also here:
http://asiablog.acumatica.com/2016/12/using-colors-in-acumatica.html
And it works - to a point. Unfortunately, it only changes the color when you click on one of the rows, not when the screen loads or when the field is updated and the save is clicked. Does the event "grid_RowDataBound" not fire when the page is refreshed or a save even happens?
Bottom line: How can I get this to set the row color when the screen is refreshed or when the field is updated / screen saved?
If a user has to click on one of the rows for this functionality to occur, it's not of much use.
Thanks...
A: A better place to put the javascript method is under the PXDataSource aspx control:
<px:PXDataSource>
<ClientEvents Initialize="HighlightLines" CommandPerformed="HighlightLines"/>
</px:PXDataSource>
<script type="text/javascript">
function HighlightLines ()
{
if(px_all && px_all["ctl00_phG_grid <=the html ID of the grid"] && px_all["ctl00_phG_grid"].rows)
{
let lines = px_all["ctl00_phG_grid"].rows.items;
for(let i=0;i<lines.length;i++)
{
let currentLine=lines[i];
if(currentLine.getCell("DAC__Field").getValue() == 'SomeValue')
{
currentLine.style = 'background-color: green';
currentLine.repaint();
}
else if (currentLine.getCell("DAC__Field").getValue() == 'SomeOtherValue')
{
currentLine.style = 'background-color: yellow';
currentLine.repaint();
}
else if (currentLine.getCell("NGSampleTest__Test").getValue() == 'PGTAND')
{
currentLine.style = 'background-color: red';
currentLine.repaint();
}
}
}
}
</script>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,219 |
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\section*{Acknowledgments}
The authors are grateful to FAPESP grants \#2013/07375-0, \#2014/12236-1, \#2017/25908-6, \#2019/07825-1, and \#2019/07665-4, as well as CNPq grants \#307066/2017-7, and \#427968/2018-6. This study was financed in part by the Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior – Brasil (CAPES) – Finance Code 001.
\bibliographystyle{splncs04}
\subsection{Deep Belief Networks}
\label{ss.dbn}
Restricted Boltzmann Machines can also be employed to compose more complex models. They are commonly used as building blocks to generate the so-called Deep Belief Networks~\cite{Hinton:06}, which are composed of a visible and a set of $L$ hidden layers. In this model, each layer is connected to the next through a weight matrix $\textbf{W}^{(l)}$, $l \in [1,L]$. In short, DBNs consider each set of two subsequent layers as an RBM trained in a greedy fashion, where the hidden layer of the bottommost RBM feeds the next RBM's visible layer. For classification purposes, a Softmax layer is appended to the model. Afterwards, the model is fine-tuned using the backpropagation algorithm, as depicted in Figure~\ref{f.dbn}. Notice that $\textbf{h}^{(l)}$ stand for the $l$-th hidden layer.
\begin{figure}[!ht]
\centerline{\begin{tabular}{c}
\includegraphics[width=3.7cm]{./figs/dbn.eps} \\
\end{tabular}}
\caption{DBN architecture with two hidden layers for classification purposes.}
\label{f.dbn}
\end{figure}
\section{Theoretical Background}
\label{s.theoretical}
In this section, we provide a brief description of the main concepts regarding RBM and DBN formulations, as well as their discriminative variant to deal with classification problems.
\input{./sections/rbm.tex}
\input{./sections/dbn.tex}
\section{Conclusions and Future Works}
\label{s.conclusion}
This paper dealt with the problem of human intestinal parasites classification through RBM and DBN approaches. Experiments conducted over three distinct scenarios composed of Larvae, Eggs, and Protozoa, which are also partially surrounded by fecal impurities, confirmed the robustness of the models for classification purposes. Additionally, the performance of RBMs was also compared against Autoencoders for data augmentation since the datasets are highly unbalanced. Regarding future works, we intend to analyze the behavior of the models over a broader spectrum using colored images, as well as employing other RBM-based models, such as the Infinite RBMs (iRBMs) and the DBMs, to the task of human intestinal parasites classification.
\section{Methodology}
\label{s.methodology}
In this section, we introduce the dataset employed in this work, as well as the technical details concerning the experimental setup.
\subsection{Dataset}
\label{ss.datasets}
The experiments consider datasets from human intestinal parasites divided into three groups: (i) \textbf{Helminth eggs} (i.e., Eggs) with $12,691$ images, (ii) \textbf{Helminth larvae} (i.e., Larvae) with $1,598$ images, and (iii) \textbf{Protozoan cysts} (i.e., Protozoa) with $37,372$ images. Notice that all datasets contain fecal impurities, which is a diverse class that looks alike to some parasites. Each dataset comprises the following categories and their respective label in parenthesis:
\begin{sloppypar}
\begin{itemize}
\item \textbf{Helminth eggs}: \emph{H.nana} (1), \emph{H.diminuta} (2), \emph{Ancilostomideo} (3), \emph{E.vermicularis} (4), \emph{A.lumbricoides} (5), \emph{T.trichiura} (6), \emph{S.mansoni} (7), \emph{Taenia} (8), and impurities (9).
\item \textbf{Helminth larvae}: larvae (1) and impurities (2); and
\item \textbf{Protozoan cysts}: \emph{E.coli} (1), \emph{E.histolytica} (2), \emph{E.nana} (3), \emph{Giardia} (4), \emph{I.butschlii} (5), \emph{B.hominis} (6), and impurities (7).
\end{itemize}
\end{sloppypar}
These are the most common species of human intestinal parasites in Brazil, and they are also responsible for public health problems in most tropical countries~\cite{Suzuki:2013a}. Notice that all datasets are unbalanced with considerably more impurity samples. The objects of interest were first segmented from the background, converted to grayscale, and further resized to $50 \times 50$ pixels. Table~\ref{t.dist}(a) presents the distribution of samples per class.
\subsection{Data augmentation}
\label{ss.dataAugmentation}
In this paper, we proposed two different synthetic data generation approaches to overcome the class imbalance problem: (i) an Autoencoder (AE) and (ii) an additional RBM for image reconstruction purposes. In all cases, the models were trained with examples of the class to be oversampled only. Further, to allow a fair comparison, both the RBM and the AE contain similar architectures. Table~\ref{t.generator_hyperparameters} presents the hyperparameters employed while training the models for data augmentation.
\begin{table}[ht]
\renewcommand{\arraystretch}{1}
\centering
\begin{tabular}{llcc}
\hline
\textbf{Model} &
\textbf{Hyper-parameter} &
\textbf{Search interval} &
\textbf{Best value} \\
\hline
\multirow{4}{*}{AE}
& $\eta$ & $[10^{-5}, 10^{-2}]$ & $10^{-3}$ \\
& $p_{drop}$ & $[0, 0.4]$ & $0.2$ \\
& Hidden dim & $\{250, 500, 2000\}$ & $500$ \\
& Batch size & $\{16, 32, 128\}$ & $32$ \\
\hline
\multirow{3}{*}{RBM}
& $\eta$ & $[10^{-5}, 10^{-2}]$ & $10^{-4}$ \\
& Hidden dim & $\{500, 2000\}$ & $500$ \\
& Batch size & $\{4, 8, 16\}$ & $8$ \\
\hline
\end{tabular}
\\~\\
\caption{Hyper-parameter setting up.}
\label{t.generator_hyperparameters}
\end{table}
Regarding the synthetic data generation, our policy is to oversample the minority classes in which the sum of total samples generated, for all classes, does not overpass approximately $50\%$ of the majority class (impurities). Table~\ref{t.dist}(b) presents the augmentation results.
\begin{table}[!ht]
\centering
\resizebox{\textwidth}{!}{%
\subfigure[Original]{
\begin{tabular}{crrr}
\hline
\multirow{2}{*}{\textbf{Class}} &
\multicolumn{3}{c}{\textbf{\# samples}} \\
&
\multicolumn{1}{c}{Eggs} &
\multicolumn{1}{c}{Larvae} &
\multicolumn{1}{c}{Protozoa} \\
\hline
1 & 500 & 246 & 868 \\
2 & 83 & 1,352 & 659 \\
3 & 286 & -- & 1,783 \\
4 & 103 & -- & 1,931 \\
5 & 835 & -- & 3,297 \\
6 & 435 & -- & 309 \\
7 & 254 & -- & 28,525\\
8 & 379 & -- & -- \\
9 & 9,816 & -- & -- \\
\hline
\textbf{Total} &
\textbf{12,691} &
\textbf{1,598} &
\textbf{37,372} \\
\hline
\end{tabular}}
\centering
\subfigure[Augmented]{
\begin{tabular}{crrr}
\hline
\multirow{2}{*}{\textbf{Class}} &
\multicolumn{3}{c}{\textbf{\# samples}} \\
&
\multicolumn{1}{c}{Eggs} &
\multicolumn{1}{c}{Larvae} &
\multicolumn{1}{c}{Protozoa} \\
\hline
1 & 1,000 (500) & 738 (492) & 868 \\
2 & 415 (332) & 1,352 & 1,977 (1,318) \\
3 & 572 (286) & -- & 1,783 \\
4 & 412 (309) & -- & 1,931 \\
5 & 835 & -- & 3,297 \\
6 & 870 (435) & -- & 1,236 (927) \\
7 & 2,508 (2,254) & -- & 28,525 \\
8 & 379 & -- & -- \\
9 & 9,816 & -- & -- \\
\hline
\textbf{Total} &
\textbf{14,807 (2,116)} &
\textbf{2,090 (492)} &
\textbf{39,619 (2,245)} \\
\hline
\end{tabular}}}
\caption{Class frequency regarding the (a) original and (b) augmented datasets. The values in parenthesis stand for the number of samples generated artificially.}
\label{t.dist}
\end{table}
\subsection{Experimental Setup}
\label{ss.experimental_setup}
Three different models were considered in this paper: one RBM with $500$ hidden neurons and two DBNs, i.e., the first with two hidden layers (DBN-2) containing $500$ neurons each, and the other comprising three hidden layers (DBN-3) with $2,000$ neurons in the first two levels and $500$ neurons in the uppermost layer\footnote{In case of acceptance, we shall provide the link to the source-code.}. All models were trained for $100$ epochs considering each RBM stack with a learning rate $\eta=10^{-5}$ and mini-batches of $64$ samples. Further, the networks were fine-tuned for an additional $100$ epochs with mini-batches of size $128$.
\subsection{Evaluation procedure}
\label{ss.evaluation}
Since we have unbalanced datasets, the standard accuracy (ACC) may not be suitable to evaluate the proposed models since it favors classifiers biased towards the most common classes. To address such an issue, we considered the Balanced Accuracy score (BAC)~\cite{Brodersen:2010} implemented in \texttt{sklearn}\footnote{Available at \url{https://scikit-learn.org}.}. Additionally, the Cohen's kappa coefficient~\cite{fleiss1973equivalence} is employed to assess the degree of agreement between the classifier and the ground truth labels. Such a value lies in the interval $[-1, 1]$, where the lower and upper boundaries represent a complete disagreement and an agreement, respectively. Finally, we employed the Wilcoxon signed-rank test~\cite{Wilcoxon:45} with significance of $5\%$ to evaluate the statistical similarity among the best results.
\color{black}
\section{Experimental Results}
\label{s.results}
In this section, we present the experimental results concerning automatic human parasites classification.
\subsection{Classification results}
\label{ss.ClassificationResults}
Table~\ref{t.larvae_summary} presents the mean results, concerning the standard accuracy, the balanced accuracy, and the Kappa value with respect to the Larvae dataset. Results are presented over the RBM, DBN-2, and DBN-3 techniques using three distinct configurations, i.e., the original dataset and its augmented versions using RBM (Aug-RBM) and AE (Aug-AE). Moreover, the best ones regarding Wilcoxon test are in bold.
The results confirm the robustness of the proposed approaches since all models with RBM Augmentor achieved more than $94\%$ of BAC. One can highlight the DBN-2 results using the Aug-RBM with $95\%$ and $0.901$ of mean accuracy and Kappa values, respectively. Such results provide good shreds of evidence towards the relevance of data augmentation with respect to the baseline, once Aug-RBM supported an improvement of around $5.6\%$ concerning the standard accuracy, $17.3\%$ regarding BAC, and $38\%$ considering the Kappa value. Although Aug-AE provided some improvements, RBM figures as the most accurate approach for such a task.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & 94.03$\pm$0.30 & 77.03$\pm$1.85 & 90.14$\pm$0.14 & \textbf{95.05$\pm$0.34} & 90.66$\pm$0.87 & 90.53$\pm$0.23 & 94.85$\pm$0.33 & 92.15$\pm$0.65 & 89.61$\pm$1.26 \\
\multicolumn{1}{c}{\textbf{BAC}} & 94.07$\pm$0.28 & 69.71$\pm$2.95 & 80.19$\pm$0.38 & \textbf{95.09$\pm$0.33} & 90.63$\pm$0.75 & 81.24$\pm$0.41 & \textbf{94.87$\pm$0.34} & 91.40$\pm$0.79 & 80.99$\pm$2.29 \\
\multicolumn{1}{c}{\textbf{Kappa}} & 0.880$\pm$0.005 & 0.445$\pm$0.053 & 0.637$\pm$0.006 & \textbf{0.901$\pm$0.007} & 0.804$\pm$0.018 & 0.653$\pm$0.007 & \textbf{0.897$\pm$0.007} & 0.832$\pm$0.014 & 0.630$\pm$0.041 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Larvae dataset using the proposed approaches.}
\label{t.larvae_summary}
\end{table*}
Table~\ref{t.eggs_summary} presents the results regarding the Eggs dataset. In this scenario, DBN-3 obtained the best results concerning the ACC and Kappa values, while the standard RBM performed better over the BAC measure. This behavior is surprising since both Kappa and BAC were proposed to cope with unbalanced data evaluation, thus expecting to behave similarly to the other models.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & 93.54$\pm$0.37 & 84.25$\pm$1.13 & 90.30$\pm$0.052 & 94.03$\pm$0.19 & 92.13$\pm$0.99 & 91.91$\pm$0.45 & \textbf{94.41$\pm$0.32} & 94.01$\pm$0.19 & 93.08$\pm$0.31 \\
\multicolumn{1}{c}{\textbf{BAC}} & \textbf{92.09$\pm$0.68} & 67.15$\pm$2.54 & 79.94$\pm$0.55 & 90.98$\pm$0.77 & 88.36$\pm$1.77 & 78.34$\pm$1.33 & 91.06$\pm$0.62 & 90.39$\pm$0.30 & 78.67$\pm$1.75 \\
\multicolumn{1}{c}{\textbf{Kappa}} & 0.884$\pm$0.006 & 0.685$\pm$0.025 & 0.769$\pm$0.009 & 0.891$\pm$0.004 & 0.857$\pm$0.015 & 0.794$\pm$0.009 & \textbf{0.897$\pm$0.006} & 0.890$\pm$0.003 & 0.820$\pm$0.009 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Eggs dataset using the proposed approaches.}
\label{t.eggs_summary}
\end{table*}
The behavior observed in the Protozoa dataset, presented in Table~\ref{t.proto_summary}, highlights an interesting scenario. One of the best ACC ($87.51\%$) and Kappa ($0.736$) results were achieved with the simplest model, i.e., an RBM using Aug-RBM. Such behavior points out that, for such a dataset, we can compress the input data into a small latent space, thus extracting useful and representative features with only $500$ units, while the performance is still remarkable even with unbalanced classes. Moreover, concerning BAC values, one can observe that DBN-2 and DBN-3 with data augmentation by Restricted Boltzmann Machines, as well as DBN-3 using AE for synthetic data generation, obtained similar results.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & \textbf{87.51$\pm$0.14} & 75.85$\pm$0.13 & 86.21$\pm$0.30 & 86.97$\pm$0.31 & 87.01$\pm$0.22 & 85.97$\pm$0.50 & 85.97$\pm$0.59 & \textbf{87.29$\pm$0.37} & 84.73$\pm$0.94 \\
\multicolumn{1}{c}{\textbf{BAC}} & 77.84$\pm$0.82 & 43.85$\pm$0.84 & 63.77$\pm$1.15 & \textbf{78.84$\pm$1.22} & 73.83$\pm$0.74 & 62.97$\pm$2.88 & \textbf{77.66$\pm$1.88} & \textbf{77.87$\pm$1.58} & 60.55$\pm$2.85 \\
\multicolumn{1}{c}{\textbf{Kappa}} & \textbf{0.736$\pm$0.004} & 0.368$\pm$0.009 & 0.662$\pm$0.006 & \textbf{0.731$\pm$0.007} & 0.710$\pm$0.005 & 0.659$\pm$0.012 & 0.711$\pm$0.010 & 0.724$\pm$0.009 & 0.615$\pm$0.023 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Protozoa dataset using the proposed approaches.}
\label{t.proto_summary}
\end{table*}
\subsection{Training Analysis}
\label{ss.trainingAnalysis}
Regarding the training analysis, we considered the datasets aumented with RBMs only since these models outperformed the ones using Autoencoders. Figure~\ref{f.trainingProgress} depicts the evolution of the Kappa values over the testing set during training. One can notice that: (i) data augmentation provided a considerable improvement in the results, (ii) training with data augmentation led to more stable results (Figures~\ref{f.trainingProgress}a and~\ref{f.trainingProgress}b), and (iii) differently from the other two datasets, techniques over Protozoa kept learning up to $80$ epochs (Figure~\ref{f.trainingProgress}c). Such behavior is somehow expected since Protozoa dataset poses a more challenging scenario. The stable results provided by data augmentation may allow us to apply some criteria for convergence analysis during training, such as early stop.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/larvae_kappa.eps}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/eggs_kappa.eps}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/protozoan_kappa.eps}}
\caption{Average Kappa values over the testing set concerning (a)~Larvae, (b)~Eggs, and (c)~Protozoa datasets.}
\label{f.trainingProgress}
\end{figure}
\subsection{Data Augmentation Analysis}
\label{ss.imageReconstruction}
Figure~\ref{f.augmentation} shows some synthetic data generated by RBMs using $500$ hidden neurons. One can observe that RBMs were able to generate useful samples, which corroborates the aforementioned results, i.e., such a process improved the parasites classification. Besides, the less accurate results concern the ones related to the Larvae dataset since we have a small subset of samples and their shape change considerably among the parasites.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cccccc}
\includegraphics[width=1.9cm,height=1.9cm]{./figs/larvae_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/larvae_aug.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/eggs_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/eggs_aug.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/proto_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/proto_aug.png} \\
(a) & (b) & (c) & (d) & (e) & (f)
\end{tabular}}
\caption{Data augmentation analysis: (a)~real and (b)~synthetic Larvae samples, (c)~real and (d)~synthetic Eggs samples, and (e)~real and (f)~synthetic Protozoa samples.}
\label{f.augmentation}
\end{figure}
\subsection{Restricted Boltzmann Machines}
\label{ss.rbm}
Restricted Boltzmann Machines stand for energy-based neural networks that can learn the probability distribution over a set of input vectors. Such models are named after the Boltzmann distribution, a measurement that uses the system's energy to obtain the probability of a given state. Energy-based models are inspired by physics since they assign a scalar energy value for each variable configuration, thus learning by adjusting their parameters to minimize the energy of the system. Moreover, they are modeled as a bipartite graph, i.e., there are no connections between units from the same layer. Such a technique assumes binary-valued nodes, although there are extensions to real- and even complex-valued inputs~\cite{srivastava2012,nakashika2017}.
Given an initial configuration $(\bm{v},\bm{h})$, the energy of the system can be computed as follows:
\begin{equation}
\label{e.energy}
E(\bm{v},\bm{h}) = -\sum_{i=1}^mb_iv_i-\sum_{j=1}^nc_jh_j-\sum_{i=1}^m\sum_{j=1}^nW_{ij}v_ih_j,
\end{equation}
where $\bm{v}\in\Re^m$ and $\bm{h}\in\Re^n$ stand for the visible and hidden layers, respectively, and $\bm{b}\in\Re^m$ and $\bm{c}\in\Re^n$ denote their bias vectors. Additionally, $\bm{W}_{m\times n}$ corresponds to the weight matrix concerning the connections between layers $\bm{v}$ and $\bm{h}$.
The learning procedure aims at finding $\bm{W}$, $\bm{a}$, and $\bm{b}$ in such a way Equation~\ref{e.energy} is minimized. However, calculating the joint probability of the model is intractable since it requires computing every possible initial configuration. Moreover, one can estimate the conditional probabilities using alternated iterations over a Monte Carlo Markov Chain (MCMC) approach, where the probabilities of both input and hidden units can be computed as follows:
\begin{equation}
\label{e.ph}
p(h_j=1|\bm{v}) = \sigma\left(c_j + \sum_{i=1}^mW_{ij}\bm{v}_i\right),
\end{equation}
and
\begin{equation}
\label{e.pv}
p(v_i=1|\bm{h}) = \sigma\left(b_i + \sum_{j=1}^nW_{ij}\bm{h}_j\right),
\end{equation}
where $\sigma$ stands for the logistic-sigmoid function. Since the visible and hidden units are conditionally independent, one can train the network using the MCMC algorithm with Gibbs sampling through Contrastive Divergence (CD)~\cite{Hinton:02}.
\section{Introduction}
\label{s.introduction}
Estimates reveal that around $4$ billion people in the world are infected with some intestinal parasite~\cite{paho:2007}. The human intestinal parasitism is a public health problem, especially in tropical countries~\cite{who:2010}, in which such infections can lead children and immunodeficient adults to death. The detection and diagnosis of human intestinal parasitosis depend on the visual analysis of optical microscopy images obtained from fecal samples mostly. However, the manual analysis of those images is time-consuming and error-prone. In order to circumvent this problem, Suzuki et al.~\cite{Suzuki:2013a} proposed a fully automated enteroparasitosis diagnosis system via image analysis, which addressed the $15$ most common species of protozoa and helminths in Brazil. The proposed approach is composed of three main steps: (i) image segmentation, (ii) object delineation, and its further (iii) classification.
Previous works have also investigated protozoa and helminth parasites classification. Suzuki et al.~\cite{Suzuki:2013b}, for instance, introduced the Optimum Path Forest~\cite{PapaIJIST:09,PapaPR:12} classifier for such a task, with results that outperformed Support Vector Machines and Artificial Neural Networks. Later on, Peixinho et al.~\cite{Peixinho:2015} explored Convolutional Neural Networks (CNNs) in this context. Further, Peixinho et al.~\cite{peixinho2018delaunay} proposed generating synthetic samples to increase the number of images for under-represented classes by adding points onto a 2D projection space. Furthermore, Benato et al.~\cite{benato2018semi} investigated an approach to cope with the lack of supervised data by interactively propagating labels to reduce the user effort in data annotation. Finally, Castelo et al.~\cite{Castelo:2019} used bag of visual words to extract key points from superpixel-segmented images and further build a visual dictionary to automatic classify intestinal parasites.
Apart from the techniques mentioned earlier, Restricted Boltzmann Machines (RBMs)~\cite{Smolensky:86} obtained notorious attention due to their promising results in a wide variety of tasks such as data reconstruction~\cite{passosIJCNN:19}, exudate identification in retinal images~\cite{khojasteh2019exudate}, and collaborative filtering~\cite{salakhutdinov2007restricted}, to cite a few. Moreover, RBMs can be used as the building block for more complex and deep models such as Deep Belief Networks (DBNs)~\cite{Hinton:06} and Deep Boltzmann Machines (DBMs)~\cite{salakhutdinov2009deep}.
However, as far as we are concerned, no work has employed RBM-based models in the task of intestinal parasite classification to date. Therefore, the main contributions of this work are threefold: (i)~to propose an effective method for parasite classification using RBMs and DBNs; (ii)~to evaluate the ability of Restricted Boltzmann Machines ability for data augmentation; and (iii)~to foster the scientific literature concerning both RBM-based applications and intestinal parasites identification.
The remainder of this paper is organized as follows: Section~\ref{s.theoretical} introduces the theoretical background concerning RBMs and DBNs, while Sections~\ref{s.methodology} and~\ref{s.results} present the methodology and the experimental results, respectively. Finally, Section~\ref{s.conclusion} states conclusions and future works.
\section*{Acknowledgments}
The authors are grateful to FAPESP grants \#2013/07375-0, \#2014/12236-1, \#2017/25908-6, \#2019/07825-1, and \#2019/07665-4, as well as CNPq grants \#307066/2017-7, and \#427968/2018-6. This study was financed in part by the Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior – Brasil (CAPES) – Finance Code 001.
\bibliographystyle{splncs04}
\subsection{Deep Belief Networks}
\label{ss.dbn}
Restricted Boltzmann Machines can also be employed to compose more complex models. They are commonly used as building blocks to generate the so-called Deep Belief Networks~\cite{Hinton:06}, which are composed of a visible and a set of $L$ hidden layers. In this model, each layer is connected to the next through a weight matrix $\textbf{W}^{(l)}$, $l \in [1,L]$. In short, DBNs consider each set of two subsequent layers as an RBM trained in a greedy fashion, where the hidden layer of the bottommost RBM feeds the next RBM's visible layer. For classification purposes, a Softmax layer is appended to the model. Afterwards, the model is fine-tuned using the backpropagation algorithm, as depicted in Figure~\ref{f.dbn}. Notice that $\textbf{h}^{(l)}$ stand for the $l$-th hidden layer.
\begin{figure}[!ht]
\centerline{\begin{tabular}{c}
\includegraphics[width=3.7cm]{./figs/dbn.eps} \\
\end{tabular}}
\caption{DBN architecture with two hidden layers for classification purposes.}
\label{f.dbn}
\end{figure}
\section{Theoretical Background}
\label{s.theoretical}
In this section, we provide a brief description of the main concepts regarding RBM and DBN formulations, as well as their discriminative variant to deal with classification problems.
\input{./sections/rbm.tex}
\input{./sections/dbn.tex}
\section{Conclusions and Future Works}
\label{s.conclusion}
This paper dealt with the problem of human intestinal parasites classification through RBM and DBN approaches. Experiments conducted over three distinct scenarios composed of Larvae, Eggs, and Protozoa, which are also partially surrounded by fecal impurities, confirmed the robustness of the models for classification purposes. Additionally, the performance of RBMs was also compared against Autoencoders for data augmentation since the datasets are highly unbalanced. Regarding future works, we intend to analyze the behavior of the models over a broader spectrum using colored images, as well as employing other RBM-based models, such as the Infinite RBMs (iRBMs) and the DBMs, to the task of human intestinal parasites classification.
\section{Methodology}
\label{s.methodology}
In this section, we introduce the dataset employed in this work, as well as the technical details concerning the experimental setup.
\subsection{Dataset}
\label{ss.datasets}
The experiments consider datasets from human intestinal parasites divided into three groups: (i) \textbf{Helminth eggs} (i.e., Eggs) with $12,691$ images, (ii) \textbf{Helminth larvae} (i.e., Larvae) with $1,598$ images, and (iii) \textbf{Protozoan cysts} (i.e., Protozoa) with $37,372$ images. Notice that all datasets contain fecal impurities, which is a diverse class that looks alike to some parasites. Each dataset comprises the following categories and their respective label in parenthesis:
\begin{sloppypar}
\begin{itemize}
\item \textbf{Helminth eggs}: \emph{H.nana} (1), \emph{H.diminuta} (2), \emph{Ancilostomideo} (3), \emph{E.vermicularis} (4), \emph{A.lumbricoides} (5), \emph{T.trichiura} (6), \emph{S.mansoni} (7), \emph{Taenia} (8), and impurities (9).
\item \textbf{Helminth larvae}: larvae (1) and impurities (2); and
\item \textbf{Protozoan cysts}: \emph{E.coli} (1), \emph{E.histolytica} (2), \emph{E.nana} (3), \emph{Giardia} (4), \emph{I.butschlii} (5), \emph{B.hominis} (6), and impurities (7).
\end{itemize}
\end{sloppypar}
These are the most common species of human intestinal parasites in Brazil, and they are also responsible for public health problems in most tropical countries~\cite{Suzuki:2013a}. Notice that all datasets are unbalanced with considerably more impurity samples. The objects of interest were first segmented from the background, converted to grayscale, and further resized to $50 \times 50$ pixels. Table~\ref{t.dist}(a) presents the distribution of samples per class.
\subsection{Data augmentation}
\label{ss.dataAugmentation}
In this paper, we proposed two different synthetic data generation approaches to overcome the class imbalance problem: (i) an Autoencoder (AE) and (ii) an additional RBM for image reconstruction purposes. In all cases, the models were trained with examples of the class to be oversampled only. Further, to allow a fair comparison, both the RBM and the AE contain similar architectures. Table~\ref{t.generator_hyperparameters} presents the hyperparameters employed while training the models for data augmentation.
\begin{table}[ht]
\renewcommand{\arraystretch}{1}
\centering
\begin{tabular}{llcc}
\hline
\textbf{Model} &
\textbf{Hyper-parameter} &
\textbf{Search interval} &
\textbf{Best value} \\
\hline
\multirow{4}{*}{AE}
& $\eta$ & $[10^{-5}, 10^{-2}]$ & $10^{-3}$ \\
& $p_{drop}$ & $[0, 0.4]$ & $0.2$ \\
& Hidden dim & $\{250, 500, 2000\}$ & $500$ \\
& Batch size & $\{16, 32, 128\}$ & $32$ \\
\hline
\multirow{3}{*}{RBM}
& $\eta$ & $[10^{-5}, 10^{-2}]$ & $10^{-4}$ \\
& Hidden dim & $\{500, 2000\}$ & $500$ \\
& Batch size & $\{4, 8, 16\}$ & $8$ \\
\hline
\end{tabular}
\\~\\
\caption{Hyper-parameter setting up.}
\label{t.generator_hyperparameters}
\end{table}
Regarding the synthetic data generation, our policy is to oversample the minority classes in which the sum of total samples generated, for all classes, does not overpass approximately $50\%$ of the majority class (impurities). Table~\ref{t.dist}(b) presents the augmentation results.
\begin{table}[!ht]
\centering
\resizebox{\textwidth}{!}{%
\subfigure[Original]{
\begin{tabular}{crrr}
\hline
\multirow{2}{*}{\textbf{Class}} &
\multicolumn{3}{c}{\textbf{\# samples}} \\
&
\multicolumn{1}{c}{Eggs} &
\multicolumn{1}{c}{Larvae} &
\multicolumn{1}{c}{Protozoa} \\
\hline
1 & 500 & 246 & 868 \\
2 & 83 & 1,352 & 659 \\
3 & 286 & -- & 1,783 \\
4 & 103 & -- & 1,931 \\
5 & 835 & -- & 3,297 \\
6 & 435 & -- & 309 \\
7 & 254 & -- & 28,525\\
8 & 379 & -- & -- \\
9 & 9,816 & -- & -- \\
\hline
\textbf{Total} &
\textbf{12,691} &
\textbf{1,598} &
\textbf{37,372} \\
\hline
\end{tabular}}
\centering
\subfigure[Augmented]{
\begin{tabular}{crrr}
\hline
\multirow{2}{*}{\textbf{Class}} &
\multicolumn{3}{c}{\textbf{\# samples}} \\
&
\multicolumn{1}{c}{Eggs} &
\multicolumn{1}{c}{Larvae} &
\multicolumn{1}{c}{Protozoa} \\
\hline
1 & 1,000 (500) & 738 (492) & 868 \\
2 & 415 (332) & 1,352 & 1,977 (1,318) \\
3 & 572 (286) & -- & 1,783 \\
4 & 412 (309) & -- & 1,931 \\
5 & 835 & -- & 3,297 \\
6 & 870 (435) & -- & 1,236 (927) \\
7 & 2,508 (2,254) & -- & 28,525 \\
8 & 379 & -- & -- \\
9 & 9,816 & -- & -- \\
\hline
\textbf{Total} &
\textbf{14,807 (2,116)} &
\textbf{2,090 (492)} &
\textbf{39,619 (2,245)} \\
\hline
\end{tabular}}}
\caption{Class frequency regarding the (a) original and (b) augmented datasets. The values in parenthesis stand for the number of samples generated artificially.}
\label{t.dist}
\end{table}
\subsection{Experimental Setup}
\label{ss.experimental_setup}
Three different models were considered in this paper: one RBM with $500$ hidden neurons and two DBNs, i.e., the first with two hidden layers (DBN-2) containing $500$ neurons each, and the other comprising three hidden layers (DBN-3) with $2,000$ neurons in the first two levels and $500$ neurons in the uppermost layer\footnote{In case of acceptance, we shall provide the link to the source-code.}. All models were trained for $100$ epochs considering each RBM stack with a learning rate $\eta=10^{-5}$ and mini-batches of $64$ samples. Further, the networks were fine-tuned for an additional $100$ epochs with mini-batches of size $128$.
\subsection{Evaluation procedure}
\label{ss.evaluation}
Since we have unbalanced datasets, the standard accuracy (ACC) may not be suitable to evaluate the proposed models since it favors classifiers biased towards the most common classes. To address such an issue, we considered the Balanced Accuracy score (BAC)~\cite{Brodersen:2010} implemented in \texttt{sklearn}\footnote{Available at \url{https://scikit-learn.org}.}. Additionally, the Cohen's kappa coefficient~\cite{fleiss1973equivalence} is employed to assess the degree of agreement between the classifier and the ground truth labels. Such a value lies in the interval $[-1, 1]$, where the lower and upper boundaries represent a complete disagreement and an agreement, respectively. Finally, we employed the Wilcoxon signed-rank test~\cite{Wilcoxon:45} with significance of $5\%$ to evaluate the statistical similarity among the best results.
\color{black}
\section{Experimental Results}
\label{s.results}
In this section, we present the experimental results concerning automatic human parasites classification.
\subsection{Classification results}
\label{ss.ClassificationResults}
Table~\ref{t.larvae_summary} presents the mean results, concerning the standard accuracy, the balanced accuracy, and the Kappa value with respect to the Larvae dataset. Results are presented over the RBM, DBN-2, and DBN-3 techniques using three distinct configurations, i.e., the original dataset and its augmented versions using RBM (Aug-RBM) and AE (Aug-AE). Moreover, the best ones regarding Wilcoxon test are in bold.
The results confirm the robustness of the proposed approaches since all models with RBM Augmentor achieved more than $94\%$ of BAC. One can highlight the DBN-2 results using the Aug-RBM with $95\%$ and $0.901$ of mean accuracy and Kappa values, respectively. Such results provide good shreds of evidence towards the relevance of data augmentation with respect to the baseline, once Aug-RBM supported an improvement of around $5.6\%$ concerning the standard accuracy, $17.3\%$ regarding BAC, and $38\%$ considering the Kappa value. Although Aug-AE provided some improvements, RBM figures as the most accurate approach for such a task.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & 94.03$\pm$0.30 & 77.03$\pm$1.85 & 90.14$\pm$0.14 & \textbf{95.05$\pm$0.34} & 90.66$\pm$0.87 & 90.53$\pm$0.23 & 94.85$\pm$0.33 & 92.15$\pm$0.65 & 89.61$\pm$1.26 \\
\multicolumn{1}{c}{\textbf{BAC}} & 94.07$\pm$0.28 & 69.71$\pm$2.95 & 80.19$\pm$0.38 & \textbf{95.09$\pm$0.33} & 90.63$\pm$0.75 & 81.24$\pm$0.41 & \textbf{94.87$\pm$0.34} & 91.40$\pm$0.79 & 80.99$\pm$2.29 \\
\multicolumn{1}{c}{\textbf{Kappa}} & 0.880$\pm$0.005 & 0.445$\pm$0.053 & 0.637$\pm$0.006 & \textbf{0.901$\pm$0.007} & 0.804$\pm$0.018 & 0.653$\pm$0.007 & \textbf{0.897$\pm$0.007} & 0.832$\pm$0.014 & 0.630$\pm$0.041 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Larvae dataset using the proposed approaches.}
\label{t.larvae_summary}
\end{table*}
Table~\ref{t.eggs_summary} presents the results regarding the Eggs dataset. In this scenario, DBN-3 obtained the best results concerning the ACC and Kappa values, while the standard RBM performed better over the BAC measure. This behavior is surprising since both Kappa and BAC were proposed to cope with unbalanced data evaluation, thus expecting to behave similarly to the other models.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & 93.54$\pm$0.37 & 84.25$\pm$1.13 & 90.30$\pm$0.052 & 94.03$\pm$0.19 & 92.13$\pm$0.99 & 91.91$\pm$0.45 & \textbf{94.41$\pm$0.32} & 94.01$\pm$0.19 & 93.08$\pm$0.31 \\
\multicolumn{1}{c}{\textbf{BAC}} & \textbf{92.09$\pm$0.68} & 67.15$\pm$2.54 & 79.94$\pm$0.55 & 90.98$\pm$0.77 & 88.36$\pm$1.77 & 78.34$\pm$1.33 & 91.06$\pm$0.62 & 90.39$\pm$0.30 & 78.67$\pm$1.75 \\
\multicolumn{1}{c}{\textbf{Kappa}} & 0.884$\pm$0.006 & 0.685$\pm$0.025 & 0.769$\pm$0.009 & 0.891$\pm$0.004 & 0.857$\pm$0.015 & 0.794$\pm$0.009 & \textbf{0.897$\pm$0.006} & 0.890$\pm$0.003 & 0.820$\pm$0.009 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Eggs dataset using the proposed approaches.}
\label{t.eggs_summary}
\end{table*}
The behavior observed in the Protozoa dataset, presented in Table~\ref{t.proto_summary}, highlights an interesting scenario. One of the best ACC ($87.51\%$) and Kappa ($0.736$) results were achieved with the simplest model, i.e., an RBM using Aug-RBM. Such behavior points out that, for such a dataset, we can compress the input data into a small latent space, thus extracting useful and representative features with only $500$ units, while the performance is still remarkable even with unbalanced classes. Moreover, concerning BAC values, one can observe that DBN-2 and DBN-3 with data augmentation by Restricted Boltzmann Machines, as well as DBN-3 using AE for synthetic data generation, obtained similar results.
\begin{table*}[!htb]
\renewcommand{\arraystretch}{1.85}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllll}
\hline
& \multicolumn{3}{c}{\textbf{RBM}} & \multicolumn{3}{c}{\textbf{DBN-2}} & \multicolumn{3}{c}{\textbf{DBN-3}} \\ \hline
& Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline & Aug-RBM & Aug-AE & Baseline \\ \hline
\multicolumn{1}{c}{\textbf{ACC}} & \textbf{87.51$\pm$0.14} & 75.85$\pm$0.13 & 86.21$\pm$0.30 & 86.97$\pm$0.31 & 87.01$\pm$0.22 & 85.97$\pm$0.50 & 85.97$\pm$0.59 & \textbf{87.29$\pm$0.37} & 84.73$\pm$0.94 \\
\multicolumn{1}{c}{\textbf{BAC}} & 77.84$\pm$0.82 & 43.85$\pm$0.84 & 63.77$\pm$1.15 & \textbf{78.84$\pm$1.22} & 73.83$\pm$0.74 & 62.97$\pm$2.88 & \textbf{77.66$\pm$1.88} & \textbf{77.87$\pm$1.58} & 60.55$\pm$2.85 \\
\multicolumn{1}{c}{\textbf{Kappa}} & \textbf{0.736$\pm$0.004} & 0.368$\pm$0.009 & 0.662$\pm$0.006 & \textbf{0.731$\pm$0.007} & 0.710$\pm$0.005 & 0.659$\pm$0.012 & 0.711$\pm$0.010 & 0.724$\pm$0.009 & 0.615$\pm$0.023 \\ \hline
\end{tabular}}
\\~\\
\caption{Effectiveness over Protozoa dataset using the proposed approaches.}
\label{t.proto_summary}
\end{table*}
\subsection{Training Analysis}
\label{ss.trainingAnalysis}
Regarding the training analysis, we considered the datasets aumented with RBMs only since these models outperformed the ones using Autoencoders. Figure~\ref{f.trainingProgress} depicts the evolution of the Kappa values over the testing set during training. One can notice that: (i) data augmentation provided a considerable improvement in the results, (ii) training with data augmentation led to more stable results (Figures~\ref{f.trainingProgress}a and~\ref{f.trainingProgress}b), and (iii) differently from the other two datasets, techniques over Protozoa kept learning up to $80$ epochs (Figure~\ref{f.trainingProgress}c). Such behavior is somehow expected since Protozoa dataset poses a more challenging scenario. The stable results provided by data augmentation may allow us to apply some criteria for convergence analysis during training, such as early stop.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/larvae_kappa.eps}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/eggs_kappa.eps}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{./figs/protozoan_kappa.eps}}
\caption{Average Kappa values over the testing set concerning (a)~Larvae, (b)~Eggs, and (c)~Protozoa datasets.}
\label{f.trainingProgress}
\end{figure}
\subsection{Data Augmentation Analysis}
\label{ss.imageReconstruction}
Figure~\ref{f.augmentation} shows some synthetic data generated by RBMs using $500$ hidden neurons. One can observe that RBMs were able to generate useful samples, which corroborates the aforementioned results, i.e., such a process improved the parasites classification. Besides, the less accurate results concern the ones related to the Larvae dataset since we have a small subset of samples and their shape change considerably among the parasites.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cccccc}
\includegraphics[width=1.9cm,height=1.9cm]{./figs/larvae_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/larvae_aug.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/eggs_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/eggs_aug.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/proto_orig.png} &
\includegraphics[width=1.9cm,height=1.9cm]{./figs/proto_aug.png} \\
(a) & (b) & (c) & (d) & (e) & (f)
\end{tabular}}
\caption{Data augmentation analysis: (a)~real and (b)~synthetic Larvae samples, (c)~real and (d)~synthetic Eggs samples, and (e)~real and (f)~synthetic Protozoa samples.}
\label{f.augmentation}
\end{figure}
\subsection{Restricted Boltzmann Machines}
\label{ss.rbm}
Restricted Boltzmann Machines stand for energy-based neural networks that can learn the probability distribution over a set of input vectors. Such models are named after the Boltzmann distribution, a measurement that uses the system's energy to obtain the probability of a given state. Energy-based models are inspired by physics since they assign a scalar energy value for each variable configuration, thus learning by adjusting their parameters to minimize the energy of the system. Moreover, they are modeled as a bipartite graph, i.e., there are no connections between units from the same layer. Such a technique assumes binary-valued nodes, although there are extensions to real- and even complex-valued inputs~\cite{srivastava2012,nakashika2017}.
Given an initial configuration $(\bm{v},\bm{h})$, the energy of the system can be computed as follows:
\begin{equation}
\label{e.energy}
E(\bm{v},\bm{h}) = -\sum_{i=1}^mb_iv_i-\sum_{j=1}^nc_jh_j-\sum_{i=1}^m\sum_{j=1}^nW_{ij}v_ih_j,
\end{equation}
where $\bm{v}\in\Re^m$ and $\bm{h}\in\Re^n$ stand for the visible and hidden layers, respectively, and $\bm{b}\in\Re^m$ and $\bm{c}\in\Re^n$ denote their bias vectors. Additionally, $\bm{W}_{m\times n}$ corresponds to the weight matrix concerning the connections between layers $\bm{v}$ and $\bm{h}$.
The learning procedure aims at finding $\bm{W}$, $\bm{a}$, and $\bm{b}$ in such a way Equation~\ref{e.energy} is minimized. However, calculating the joint probability of the model is intractable since it requires computing every possible initial configuration. Moreover, one can estimate the conditional probabilities using alternated iterations over a Monte Carlo Markov Chain (MCMC) approach, where the probabilities of both input and hidden units can be computed as follows:
\begin{equation}
\label{e.ph}
p(h_j=1|\bm{v}) = \sigma\left(c_j + \sum_{i=1}^mW_{ij}\bm{v}_i\right),
\end{equation}
and
\begin{equation}
\label{e.pv}
p(v_i=1|\bm{h}) = \sigma\left(b_i + \sum_{j=1}^nW_{ij}\bm{h}_j\right),
\end{equation}
where $\sigma$ stands for the logistic-sigmoid function. Since the visible and hidden units are conditionally independent, one can train the network using the MCMC algorithm with Gibbs sampling through Contrastive Divergence (CD)~\cite{Hinton:02}.
\section{Introduction}
\label{s.introduction}
Estimates reveal that around $4$ billion people in the world are infected with some intestinal parasite~\cite{paho:2007}. The human intestinal parasitism is a public health problem, especially in tropical countries~\cite{who:2010}, in which such infections can lead children and immunodeficient adults to death. The detection and diagnosis of human intestinal parasitosis depend on the visual analysis of optical microscopy images obtained from fecal samples mostly. However, the manual analysis of those images is time-consuming and error-prone. In order to circumvent this problem, Suzuki et al.~\cite{Suzuki:2013a} proposed a fully automated enteroparasitosis diagnosis system via image analysis, which addressed the $15$ most common species of protozoa and helminths in Brazil. The proposed approach is composed of three main steps: (i) image segmentation, (ii) object delineation, and its further (iii) classification.
Previous works have also investigated protozoa and helminth parasites classification. Suzuki et al.~\cite{Suzuki:2013b}, for instance, introduced the Optimum Path Forest~\cite{PapaIJIST:09,PapaPR:12} classifier for such a task, with results that outperformed Support Vector Machines and Artificial Neural Networks. Later on, Peixinho et al.~\cite{Peixinho:2015} explored Convolutional Neural Networks (CNNs) in this context. Further, Peixinho et al.~\cite{peixinho2018delaunay} proposed generating synthetic samples to increase the number of images for under-represented classes by adding points onto a 2D projection space. Furthermore, Benato et al.~\cite{benato2018semi} investigated an approach to cope with the lack of supervised data by interactively propagating labels to reduce the user effort in data annotation. Finally, Castelo et al.~\cite{Castelo:2019} used bag of visual words to extract key points from superpixel-segmented images and further build a visual dictionary to automatic classify intestinal parasites.
Apart from the techniques mentioned earlier, Restricted Boltzmann Machines (RBMs)~\cite{Smolensky:86} obtained notorious attention due to their promising results in a wide variety of tasks such as data reconstruction~\cite{passosIJCNN:19}, exudate identification in retinal images~\cite{khojasteh2019exudate}, and collaborative filtering~\cite{salakhutdinov2007restricted}, to cite a few. Moreover, RBMs can be used as the building block for more complex and deep models such as Deep Belief Networks (DBNs)~\cite{Hinton:06} and Deep Boltzmann Machines (DBMs)~\cite{salakhutdinov2009deep}.
However, as far as we are concerned, no work has employed RBM-based models in the task of intestinal parasite classification to date. Therefore, the main contributions of this work are threefold: (i)~to propose an effective method for parasite classification using RBMs and DBNs; (ii)~to evaluate the ability of Restricted Boltzmann Machines ability for data augmentation; and (iii)~to foster the scientific literature concerning both RBM-based applications and intestinal parasites identification.
The remainder of this paper is organized as follows: Section~\ref{s.theoretical} introduces the theoretical background concerning RBMs and DBNs, while Sections~\ref{s.methodology} and~\ref{s.results} present the methodology and the experimental results, respectively. Finally, Section~\ref{s.conclusion} states conclusions and future works.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,638 |
Girl Child Education Fund™ (GCEF)
The Florence Nightingale International Foundation (FNIF), the charitable arm of ICN, launched the Girl Child Education Fund (GCEF) in 2005 with the mission of supporting access to education for the orphaned daughters of nurses in developing countries. Currently, GCEF is present in four countries from the sub-Saharan Africa region: Eswatini (Swaziland), Kenya, Uganda and Zambia.
Since the initiation of the programme over 400 girls have been enrolled in the GCEF and has enable enabled 285 girls to graduate from secondary school. Currently, 77 girls are being supported through this initiative.
An average of 1,400 USD ensures one year of school fees, uniform, shoes and books to an orphaned girl. Your generous donation will support the Girl Child Education Fund and ensure that these orphaned daughters of nurses complete their school education.
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Former Girl Child Education Fund student wins second award for fighting HIV stigma
The ICN Girl Child Education Fund: Diana Mumbula's story
Diana Mumbula, who has been supported on the scheme for as long as she can remember, has recently followed in her mother's footsteps and qualified as a nurse. Her story illustrates her determination to succeed in life, and the powerful difference that individuals' donations to the GCEF can make to girls who might otherwise miss out on the vital education that they need to lead full, healthy lives.
International Day of the Girl Child – ICN supports the empowering of young women through its Girl Child Education Fund | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,712 |
package systemd
import (
"errors"
"math/big"
"strconv"
"strings"
)
// RangeToBits converts a text representation of a CPU mask (as written to
// or read from cgroups' cpuset.* files, e.g. "1,3-5") to a slice of bytes
// with the corresponding bits set (as consumed by systemd over dbus as
// AllowedCPUs/AllowedMemoryNodes unit property value).
func RangeToBits(str string) ([]byte, error) {
bits := new(big.Int)
for _, r := range strings.Split(str, ",") {
// allow extra spaces around
r = strings.TrimSpace(r)
// allow empty elements (extra commas)
if r == "" {
continue
}
ranges := strings.SplitN(r, "-", 2)
if len(ranges) > 1 {
start, err := strconv.ParseUint(ranges[0], 10, 32)
if err != nil {
return nil, err
}
end, err := strconv.ParseUint(ranges[1], 10, 32)
if err != nil {
return nil, err
}
if start > end {
return nil, errors.New("invalid range: " + r)
}
for i := start; i <= end; i++ {
bits.SetBit(bits, int(i), 1)
}
} else {
val, err := strconv.ParseUint(ranges[0], 10, 32)
if err != nil {
return nil, err
}
bits.SetBit(bits, int(val), 1)
}
}
ret := bits.Bytes()
if len(ret) == 0 {
// do not allow empty values
return nil, errors.New("empty value")
}
return ret, nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,090 |
\section{Constants}
\section{The Kauffman skein algebra}
\label{sect:SkeinAlgebra}
Let $S$ be an oriented surface (without boundary) with finite topological type. Namely, $S$ is obtained by removing finitely many points from a compact oriented surface $\bar S$. We consider \emph{framed links} in the thickened surface $S\times [0,1]$, namely unoriented 1--dimensional submanifolds $K\subset S \times [0,1]$ endowed with a continuous choice of a vector transverse to $K$ at each point of $K$. A \emph{framed knot} is a connected framed link.
The \emph{framed link algebra} $\mathcal K(S)$ is the vector space (over $\mathbb C$, say) freely generated by the isotopy classes of all framed links $K \subset S \times [0,1]$.
The vector space $\mathcal K(S)$ can be endowed with a multiplication, where the product of $K_1$ and $K_2$ is defined by the framed link $K_1 \cdot K_2 \subset S\times[0,1]$ that is the union of $K_1$ rescaled in $S\times [0, \frac12]$ and $K_2$ rescaled in $S\times [\frac12, 1]$. In other words, the product $K_1\cdot K_2$ is defined by superposition of the framed links $K_1$ and $K_2$.
This \emph{superposition operation} is compatible with isotopies, and therefore provides a well-defined algebra structure on $\mathcal K(S)$.
Three links $K_1$, $K_0$ and $K_\infty$ in $S\times[0,1]$ form a \emph{Kauffman triple} if the only place where they differ is above a small disk in $S$, where they are as represented in Figure~\ref{fig:SkeinRelation} (as seen from above) and where the framing is vertical and pointing upwards (namely the framing is parallel to the $[0,1]$ factor and points towards $1$).
For $A\in \mathbb C-\{0\}$, the \emph{Kauffman skein algebra} $\mathcal S^A(S)$ is the quotient of the framed link algebra $\mathcal K(K)$ by the two-sided ideal generated by all elements $K_1 - A^{-1}K_0 - A K_\infty$ as $(K_1, K_0, K_\infty)$ ranges over all Kauffman triples. The superposition operation descends to a multiplication in $\mathcal S^A(S)$, endowing $\mathcal S^A(S)$ with the structure of an algebra. The class $[\varnothing]$ of the empty link is an identity element in $\mathcal S^A(S)$.
An element $[K]\in \mathcal S^A(S)$, represented by a framed link $K \subset S \times [0,1]$, is a \emph{skein} in $S$. The construction is defined to ensure that the \emph{skein relation}
$$
[K_1] = A^{-1}[K_0] + A [K_\infty]
$$
holds in $\mathcal S^A(S)$ for every Kauffman triple $(K_1, K_0, K_\infty)$.
\section{The Chekhov-Fock algebra and the quantum trace homomorphism}
\subsection{The Chekhov-Fock algebra}
\label{sect:CheFock}
The Chekhov-Fock algebra (defined in \cite{BonLiu} and inspired by \cite{CheFoc1, CheFoc2, Foc}) is the avatar of the quantum Teichm\"uller space associated to an ideal triangulation of the surface $S$. If $S$ is obtained from a compact surface $\bar S$ by removing finitely many points $v_1$, $v_2$, \dots, $v_s$, an
\emph{ideal triangulation} of $S$ is a triangulation $\lambda$ of $\bar S$ whose vertex set is exactly
$\{ v_1,v_2,
\dots, v_s \}$. The surface admits an ideal triangulation if and only if it admits at least one puncture, and its Euler characteristic is negative; we will consequently assume these properties satisfied throughout the article. If the surface has genus $g$ and $s$ punctures, an ideal triangulation then has $n=6g+3s-6$ edges and $4g+2s-4$ faces.
Let $\lambda_1$, $\lambda_2$, \dots, $\lambda_n$ denote the edges of $\lambda$. Let $a_i \in \{0,1, 2\}$ be the number of times an end of the edge $\lambda_j$ immediately succeeds an end of $\lambda_i$ when going counterclockwise around a puncture of $S$, and set $\sigma_{ij}=a_{ij}-a_{ji}\in \{-2, -1, 0, 1, 2\}$. The \emph{Chekhov-Fock algebra} $\mathcal T^\omega(\lambda)$ of $\lambda$ is the algebra defined by generators $Z_1^{\pm1}$, $Z_2^{\pm1}$, \dots, $Z_n^{\pm1}$ associated to the edges $\lambda_1$, $\lambda_2$, \dots, $\lambda_n$ of $\lambda$, and by the relations
$$
Z_iZ_j = \omega^{2\sigma_{ij}} Z_jZ_i.
$$
(The actual Chekhov-Fock algebra $\mathcal T^q(\lambda)$ that is at the basis of the quantum Teichm\"uller space uses the constant $q=\omega^4$ instead of $\omega$. The generators $Z_i$ of $\mathcal T^\omega(\lambda)$ appearing here are designed to model square roots of the original generators of $\mathcal T^q(\lambda)$.)
An element of the Chekhov-Fock algebra $\mathcal T^\omega(\lambda)$ is a linear combination of monomials $Z_{i_1}^{n_1}Z_{i_2}^{n_2} \dots Z_{i_l}^{n_l}$ in the generators $Z_i$, with $n_1$, $n_2$, \dots, $n_l\in \mathbb Z$. Because of the skew-commutativity relation $Z_iZ_j = \omega^{2\sigma_{ij}} Z_jZ_i$, the order of the variables in such a monomial does matter. It is convenient to use the following symmetrization trick.
The \emph{Weyl quantum ordering} for $Z_{i_1}^{n_1}Z_{i_2}^{n_2} \dots Z_{i_l}^{n_l}$ is the monomial
$$
[Z_{i_1}^{n_1}Z_{i_2}^{n_2} \dots Z_{i_l}^{n_l}] = \omega^{-\sum_{u<v} n_un_v\sigma_{i_ui_v}} Z_{i_1}^{n_1}Z_{i_2}^{n_2} \dots Z_{i_l}^{n_l}.
$$
The formula is specially designed that $[Z_{i_1}^{n_1}Z_{i_2}^{n_2} \dots Z_{i_l}^{n_l}] $ is invariant under any permutation of the $Z_{i_u}^{n_u}$.
\subsection{The quantum trace homomorphism}
\begin{thm}[\cite{BonWon1}]
\label{thm:QTrace}
For $A=\omega^{-2}$, there exists an injective algebra homomorphism
$$\mathrm{Tr}_\lambda^\omega \kern -3pt : \mathcal S^A_{\mathrm s}(S) \to \mathcal T^\omega(\lambda).$$
\end{thm}
The specific homomorphism $\mathrm{Tr}_\lambda^\omega$ constructed in \cite{BonWon1} is the \emph{quantum trace homomorphism}. It is uniquely determined by certain properties stated in that article, but we will only need to know that it exists and that it satisfies the properties given in \S \ref{sect:ChebFrob} below.
\subsection{The Chebyshev and Frobenius homomorphisms}
\label{sect:ChebFrob}
We now assume that $A^2$ is a primitive $N$--root of unity with $N$ odd. Set $\epsilon = A^{N} =\pm1$. Recall that $T_N$ denotes the $N$--th normalized Chebyshev polynomial, defined by the property that $\cos N\theta = \frac12 T_N(2\cos \theta)$ for every $\theta$.
\begin{thm}[\cite{BonWon3}]
\label{thm:ChebSkeinRelation}
When $A^2$ is a primitive $N$--root of unity with $N$ odd and $\epsilon = A^{N} =\pm1$, there is a unique algebra homomorphism $\mathbf T^A \kern -3pt : \mathcal S^{\epsilon}(S) \to \mathcal S^A(S)$ such that
$$
\mathbf T^A \bigl( [K] \bigr) = T_N \bigl( [K] \bigr)
$$
for every framed knot $K\subset S \times [0,1]$ whose projection to $S$ has no crossing and whose framing is vertical. In addition, the image of $\mathbf T^A$ is central in $\mathcal S^A(S)$. \qed
\end{thm}
The homomorphism $\mathbf T^A $ provided by Proposition~\ref{thm:ChebSkeinRelation} is the \emph{Chebyshev homomorphism}. It is a key ingredient in the definition of the invariants of Theorem~\ref{thm:InvariantsExist}.
There is an analogous and much simpler homomorphism at the level of the Chekhov-Fock algebra, namely the following \emph{Frobenius homomorphism}.
\begin{prop}
\label{prop:Frobenius}
If $\iota = \omega^{N^2}$, there is an algebra homomorphism
$$
\mathbf F^\omega \kern -3pt : \mathcal T^{\iota}(\lambda) \to \mathcal T^\omega(\lambda)$$
which maps each generator $Z_i\in \mathcal T^{\iota}(\lambda)$ to $Z_i^N\in \mathcal T^\omega(\lambda)$, where in the first instance $Z_i\in \mathcal T^{\iota}(\lambda)$ denotes the generator associated to the $i$--th edge $\lambda_i$ of $\lambda$, whereas the second time $Z_i\in \mathcal T^\omega(\lambda)$ denotes the generator of $\mathcal T^\omega(\lambda)$ associated to the same edge~$\lambda_i$. \qed
\end{prop}
The key to our construction is the following compatibility result, which connects the Chebyshev homomorphism to the Frobenius homomorphism through appropriate the quantum trace homomorphism.
\begin{thm}[\cite{BonWon3}]
\label{thm:ChebyQTracesFrob}
The diagram
$$
\xymatrix{
\mathcal S^A(S)
\ar[r]^{\mathrm{Tr}_{\lambda}^\omega}
& \mathcal T^\omega(\lambda)\\
\mathcal S^{\epsilon}(S)
\ar[r]^{\mathrm{Tr}_{\lambda}^\iota}
\ar[u]^{\mathbf T^A}
& \mathcal T^{\iota}(\lambda)
\ar[u]_{\mathbf F^\omega}}
$$
is commutative. Namely, for every skein $[K]\in \mathcal S^{\epsilon}(S)$, the quantum trace $\mathrm{Tr}_\lambda^\omega\bigl([K^{T_N}]\bigr)$ of $[K^{T_N}] = \mathbf T^A\bigl([K]\bigr)$ is obtained from the classical trace polynomial $\mathrm{Tr}_\lambda^\iota \bigl([K]\bigr)$ by replacing each generator $Z_i \in \mathcal T^{\iota}(\lambda)$ by $Z_i^N\in \mathcal T^\omega(\lambda)$.
\end{thm}
\section{The balanced Chekhov-Fock algebra}
\subsection{The balanced Chekhov-Fock algebra}
The quantum trace map $\mathrm{Tr}^\omega_\lambda$ of Theorem~\ref{thm:QTrace} (and \cite{BonWon1}) is far from being surjective. Indeed, for a skein $[K]\in \mathcal S^A(S)$, the exponents of the monomials $Z_1^{k_1}Z_2^{k_2} \dots Z_n^{k_n}$ appearing in the expression of $\mathrm{Tr}^\omega_\lambda\bigl([K]\bigr)$ are \emph{balanced}, in the sense that they satisfy the following {parity condition}: for every triangle $T_j$ of the ideal triangulation $\lambda$, the sum $k_{i_1}+ k_{i_2}+ k_{i_3}$ of the exponents of the generators $Z_{i_1}$, $Z_{i_2}$, $Z_{i_3}$ associated to the sides of $T_j$ is even.
Let $\mathcal Z^\omega(\lambda)$ denote the sub-algebra of $\mathcal T^\omega(\lambda)$ generated by all monomials satisfying this parity condition. By definition, $\mathcal Z^\omega(\lambda)$ is the \emph{balanced Chekhov-Fock algebra} of the ideal triangulation $\lambda$.
\begin{figure}[htbp]
\SetLabels
( .3* .25) $T_j $ \\
( .5*.57 ) $\tau_\lambda $ \\
\endSetLabels
\centerline{\AffixLabels{ \includegraphics{TriangleTrainTrack.eps}}}
\caption{}
\label{fig:TriangleTrainTrack}
\end{figure}
To keep track of the parity condition, it is convenient to consider a train track $\tau_\lambda$ which, on each triangle $T_j$ of the ideal triangulation $\lambda$, looks as in Figure~\ref{fig:TriangleTrainTrack}. In particular, $\tau_\lambda$ has one switch for each edge of $\lambda$, and three edges for each triangle of $\lambda$. Let $\mathcal W(\tau_\lambda; \mathbb Z)$ be the set of integer edge weight systems $\alpha$ for $\tau_\lambda$, assigning a number $\alpha(e) \in \mathbb Z$ to each edge $e$ of $\tau_\lambda$ in such a way that, at each switch, the weights of the edges incoming on one side add up to the sum of the weights of the edges outgoing on the other side. There is a natural map $\mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb Z^n$ which, given an edge weight system, associates to each of the $n$ switches of $\tau_\lambda$ the sum of the weights of the edges incoming on one side of the switch. Then, an element $(k_1, k_2, \dots, k_n) \in \mathbb Z^n$ satisfies the above parity condition if and only if it is in the image of this map. Also, the map $\mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb Z^n$ is easily seen to be injective. Since the image of this map has finite index, it follows that $\mathcal W(\tau_\lambda; \mathbb Z)$ is isomorphic to $\mathbb Z^n$ as an abelian group.
This enables us to give a different description of $\mathcal Z^\omega(\lambda)$. For a weight system $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$, assigning a weight $\alpha_i\in \mathbb Z$ to the $i$--th edge $\lambda_i$ of $\lambda$ (= the $i$--th switch of $\tau_\lambda$), define
$$
Z_\alpha = [ Z_1^{\alpha_1}Z_2^{\alpha_2} \dots Z_n^{\alpha_n}] \in \mathcal Z^\omega(\lambda)
$$
where the bracket $[\enspace]$ denotes the Weyl quantum ordering defined in \S \ref{sect:CheFock}.
The above discussion proves the following fact.
\begin{lem}
\label{lem:TrainTrackSqRootCheFock}
As $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$ ranges over all weight systems for the train track $\tau_\lambda$, the associated $Z_\alpha$ form a basis for the vector space $\mathcal Z^\omega(\lambda)$. \qed
\end{lem}
\subsection{The algebraic structure of the balanced Chekhov-Fock algebra}
\label{subsect:AlgStructCheFock}
We first describe the multiplicative structure of the balanced Chekhov-Fock algebra $\mathcal Z^\omega(\lambda)$ in the context of Lemma~\ref{lem:TrainTrackSqRootCheFock}.
The space $ \mathcal W(\tau_\lambda; \mathbb Z)$ carries a very natural antisymmetric bilinear form
$$
\Theta \kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \times \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb Z,
$$
the \emph{Thurston intersection form} defined by the property that, for $\alpha$, $\beta\in \mathcal W(\tau_\lambda; \mathbb Z)$,
$$
\Theta(\alpha, \beta) = \sum_{e \text{ right of }e'} {\textstyle\frac12} \bigl( \alpha(e)\beta(e') - \alpha(e')\beta(e) \bigr)
$$
where the sum is over all pairs $(e,e')$ of edges of $\tau_\lambda$ such that $e$ comes out to the right of $e'$ at some switch of $\tau_\lambda$. See Lemma~\ref{lem:ThurstonHalfIntersection} in the Appendix for a more conceptual interpretation of $\Theta$, and for a proof that $\Theta(\alpha, \beta)$ is really an integer.
\begin{lem}
\label{lem:SqRootCheFockThurston}
For every $\alpha$, $\beta\in \mathcal W(\tau_\lambda; \mathbb Z)$,
$$
Z_\alpha Z_\beta = \omega^{2\Theta(\alpha, \beta)} Z_{\alpha+\beta}.
$$
In particular, $Z_\alpha Z_\beta = \omega^{4\Theta(\alpha, \beta)} Z_\beta Z_\alpha$.
\end{lem}
\begin{proof}
The second statement, that $Z_\alpha Z_\beta = \omega^{4\Theta(\alpha, \beta)} Z_\beta Z_\alpha$, is a simple computation. The first statement, that $Z_\alpha Z_\beta = \omega^{2\Theta(\alpha, \beta)} Z_{\alpha+\beta}$, then follows by definition of the Weyl quantum ordering.
\end{proof}
This is particularly simple if we replace $\omega$ by $\iota=\omega^{N^2}$, with the assumption that $A^{2N}=1$ so that $\iota^{4} = \omega^{4N^2} =A^{-2N^2}=1$.
\begin{cor}
\label{cor:SquareRootChefockCommutativeIota}
If $\iota^4=1$, the algebra $\mathcal Z^\iota(\lambda)$ is commutative. \qed
\end{cor}
In general, the key to understanding the algebraic structure of $\mathcal Z^\omega(\lambda)$ is Lemma~\ref{lem:StructureThurston} below.
For $k=1$, \dots, $s$, the $k$--th puncture of $S$ specifies an element $\eta_k \in \mathcal W(\tau_\lambda; \mathbb Z)$, defined by the property that $\eta_k(e)\in \{0,1,2\}$ is the number of sides of the edge $e$ that are adjacent to the same component of $S-\tau_\lambda$ as this puncture.
Recall that the surface $S$ has genus $g$ and $s$ punctures.
\begin{lem}
\label{lem:StructureThurston}
The lattice $ \mathcal W(\tau_\lambda; \mathbb Z) \cong \mathbb Z^n$ admits a basis in which the matrix of the Thurston intersection form $\Theta$ is block diagonal with $g$ blocks
$\begin{pmatrix}
0&1\\-1&0
\end{pmatrix}$,
$2g+s-3$ blocks
$\begin{pmatrix}
0&2\\-2&0
\end{pmatrix}$
and $s$ blocks
$\begin{pmatrix}
0
\end{pmatrix}$. In addition, the kernel of $\Theta$ is freely generated by the elements $\eta_1$, $\eta_2$, \dots, $\eta_s \in \mathcal W(\tau_\lambda; \mathbb Z) $ associated to the punctures of $S$.
\end{lem}
\begin{proof}
This is a consequence of a more general result given by Theorem~\ref{thm:StructureThurston} in the Appendix, which computes the algebraic structure of the Thurston intersection form for a general train track $\tau$. When applying this result to the train track $\tau_\lambda$, the numbers $h$, $n_{\mathrm{even}}$ and $n_{\mathrm{odd}}$ of Theorem~\ref{thm:StructureThurston} are respectively equal to the genus $g$ of $S$, to the number $s$ of punctures of $S$, and to the number $4g+2s-4$ of triangles of the ideal triangulation $\lambda$.
\end{proof}
The combination of Lemmas~\ref{lem:TrainTrackSqRootCheFock}, \ref{lem:SqRootCheFockThurston} and \ref{lem:StructureThurston} now provides the complete algebraic structure of the balanced Chekhov-Fock algebra $\mathcal Z^\omega(\lambda)$. Let $\mathcal W^q$ denote the algebra, known as the quantum 2--torus, defined by generators $X^{\pm1}$, $Y^{\pm1}$ and by the relation $XY = q YX$.
\begin{cor}
\label{cor:StructureSquareRootCheFock}
For $q=\omega^4$, the balanced Chekhov-Fock algebra $\mathcal Z^\omega(\lambda)$ is isomorphic to
$$
\mathcal W^q_1 \otimes \mathcal W^q_2 \otimes \dots \otimes \mathcal W^q_{g} \otimes
\mathcal W^{q^2}_{g+1} \otimes \mathcal W^{q^2}_{g+2} \otimes \dots \otimes \mathcal W^{q^2}_{3g+s-3} \otimes \mathbb C[H_1] \otimes \mathbb C[H_2] \otimes \dots \otimes \mathbb C[H_s]
$$
where each $\mathcal W^q_i$ is a copy of the quantum $2$--torus $\mathcal W^q$, each $\mathcal W^{q^2}_j$ is a copy of $\mathcal W^{q^2}$, and each $ \mathbb C[H_k] $ is a polynomial algebra in the variable $H_k$. \qed
\end{cor}
In addition, the $s$ generators $H_k=Z_{\eta_k}$ are associated to the punctures of $S$ as in Lemma~\ref{lem:StructureThurston}.
\subsection{Representations of the balanced Chekhov-Fock algebra}
\label{sect:RepBalancedCF}
The algebraic structure of the balanced Chekhov-Fock algebra $\mathcal Z^\omega(\lambda)$ determined in Corollary~\ref{cor:StructureSquareRootCheFock} is relatively simple. This makes it easy to classify its irreducible representations.
As usual, we assume that $A^2=\omega^{-4}$ is a primitive $N$--root of unity. For every $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$, Lemma~\ref{lem:SqRootCheFockThurston} shows that the element $Z_\alpha^N$ is central in $\mathcal Z^\omega(\lambda)$. In particular, if $\rho\kern -3pt : \mathcal Z^\omega(\lambda) \to \mathrm{End}(V)$ is an irreducible representation of $\mathcal Z^\omega(\lambda)$, there is a number $\zeta_\rho(\alpha)\in \mathbb C^*$ such that
$
\rho(Z_\alpha^N) = \zeta_\rho(\alpha) \,\mathrm{Id}_V
$. This proves:
\begin{lem}
If $\omega^4$ is a primitive $N$--root of unity, an irreducible representation $\rho\kern -3pt : \mathcal Z^\omega(\lambda) \to \mathrm{End}(V)$ determines a group homomorphism $\zeta_\rho\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$, from $\mathcal W(\tau_\lambda; \mathbb Z)$ to the multiplicative group $\mathbb C^*= \mathbb C-\{0\}$, by the property that
$$
\rho(Z_\alpha^N) = \zeta_\rho(\alpha) \,\mathrm{Id}_V
$$
for every $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$. \qed
\end{lem}
Note that the data of a group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ is just a fancy way of expressing the fact that we are given a non-zero complex number for each of the $n$ generators of $ \mathcal W(\tau_\lambda; \mathbb Z) \cong \mathbb Z^n$.
The balanced Chekhov-Fock algebra $\mathcal Z^\omega(\lambda)$ admits $s$ other central elements $H_k=Z_{\eta_k}$, coming from the elements $\eta_k\in \mathcal W(\tau_\lambda; \mathbb Z)$ associated to the punctures of $S$ as in Lemma~\ref{lem:StructureThurston} and Corollary~\ref{cor:StructureSquareRootCheFock}. For each puncture of the surface $S$, this determines a number $h_k\in \mathbb C^*$ such that
$$
\rho(H_k) = h_k\,\mathrm{Id}_V.
$$
These numbers $h_k$ are constrained by the group homomorphism $\zeta_\rho\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$, because $h_k^N = \zeta_\rho(\eta_k)$.
\begin{prop}
\label{prop:RepsSquareRootCheFock}
Assuming that $\omega^4$ is a primitive $N$--root of unity with $N$ odd, suppose that we are given a group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ and, for each of the punctures of $S$, a number $h_k\in \mathbb C^*$ such that $h_k^N = \zeta_\rho(\eta_k)$. Then, up to isomorphism, there exists a unique irreducible representation $\rho\kern -3pt : \mathcal Z^\omega(\lambda) \to \mathrm{End}(V)$ such that
\begin{enumerate}
\item $\zeta_\rho = \zeta$, namely $\rho(Z_\alpha^N) = \zeta(\alpha) \,\mathrm{Id}_V$ for every $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$;
\item $\rho(H_k) = h_k\,\mathrm{Id}_V$ for $k=1$, \dots, $s$.
\end{enumerate}
In addition, for such a representation, the vector space $V$ has dimension $N^{3g+s-3}$.
\end{prop}
\begin{proof}
Using elementary linear algebra, this is an immediate consequence of Corollary~\ref{cor:StructureSquareRootCheFock}.
More precisely, consider the isomorphism
$$
\mathcal Z^\omega(\lambda) \cong\mathcal W^q_1 \otimes \dots \otimes \mathcal W^q_{g} \otimes
\mathcal W^{q^2}_{g+1} \otimes \dots \otimes \mathcal W^{q^2}_{3g+s-3} \otimes \mathbb C[H_1] \otimes \dots \otimes \mathbb C[H_s]
$$
provided by Corollary~\ref{cor:StructureSquareRootCheFock}.
For $1\leq i\leq 3g+s-3$, let $X_i^{\pm1}$ and $Y_i^{\pm1}$ denote the generators of $\mathcal W^q_i$ or $\mathcal W^{q^2}_i$ (satisfying the relation $X_iY_i = qY_iX_i$ if $1\leq i\leq g$ and $X_iY_i = q^2Y_iX_i$ if $g< i\leq 3g+s-3$). In particular, these generators are of the form $X_i=Z_{\alpha_i}$ and $Y_i = Z_{\beta_i}$ for some $\alpha_i$, $\beta_i \in \mathcal W(\tau_\lambda; \mathbb Z)$ such that $\Theta(\alpha_i, \beta_j)=0$ if $i \not=j$, $\Theta(\alpha_i, \beta_i)=1$ if $1\leq i\leq g$, and $\Theta(\alpha_i, \beta_i)=2$ if $g< i\leq 3g+s-3$.
Because $N$ is odd, $q=\omega^4$ and $q^2$ are both primitive $N$--root of unity. Arbitrarily pick $N$--roots $\zeta(\alpha_i)^{\frac1N}$ and $\zeta(\beta_i)^{\frac1N}$, and define $\rho_i\kern -3pt : \mathcal W^q_i \to \mathrm{End}(V_i)$ by the property that, if $v_1$, $v_2$, \dots, $v_N$ form a basis for $V_i \cong \mathbb C^N$,
\begin{itemize}
\item[] $\rho_i(X_i) (v_j) = \zeta(\alpha_i)^{\frac1N} q^j v_j$ and $\rho_i(Y_i) (v_j) = \zeta(\beta_i)^{\frac1N} v_{j+1}$
if $1\leq i \leq g$, and
\item[]
$\rho_i(X_i) (v_j) = \zeta(\alpha_i)^{\frac1N} q^{2j} v_j$ and $ \rho_i(Y_i) (v_j) = \zeta(\beta_i)^{\frac1N} v_{j+1}$
if $g< i\leq 3g+s-3$.
\end{itemize}
Then, for $V = V_1 \otimes V_2\otimes \dots \otimes V_{3g+s-3}$, define $\rho \kern -3pt : \mathcal Z^\omega(\lambda) \to \mathrm{End}(V)$ by the property that $\rho$ coincides with $\rho_1 \otimes \rho_2\otimes \dots \otimes \rho_{3g+s-3}$ on $ \mathcal W^q_1 \otimes \mathcal W^q_2 \otimes \dots \otimes \mathcal W^q_{3g+s-3} $, and $\rho(H_k) = h_k\,\mathrm{Id}_V$ for every $k=1$, \dots, $s$.
It is immediate that $\rho$ satisfies the required properties. The fact that $\rho$ is irreducible, and that every irreducible representation is isomorphic to $\rho$, is easily proved by elementary linear algebra; see for instance \cite[\S 4]{BonLiu} for details.
\end{proof}
\begin{rem}
When $q=\omega^4$ is a primitive $N$--root of unity with $N$ even, the irreducible representations of $\mathcal Z^\omega(\lambda)$ can be classified by similar arguments, but the result is somewhat more complicated to state. Compare \cite[\S 4]{BonLiu}.
\end{rem}
\section{Character varieties and trace maps}
\label{sect:CharVar}
We begin with the character variety
$$
\mathcal R_{\mathrm{SL}_2(\C)}(S) = \{ \text{group homomorphisms }r\kern -3pt : \pi_1(S) \to \mathrm{SL}_2(\mathbb C) \}/\kern -4pt/ \mathrm{SL}_2(\mathbb C)
$$
where $\mathrm{SL}_2(\mathbb C)$ acts on homomorphisms by conjugation, and where the quotient is to be taken in the sense of geometric invariant theory \cite{Mum}.
For a group homomorphism $r\kern -3pt : \pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ and a closed curve $K\subset S \times [0,1]$, the trace $\mathrm{Tr}\,r(K)\in \mathbb C$ depends only on the class of $r$ in the character variety $\mathcal R_{\mathrm{SL}_2(\C)}(S)$. An observation of Doug Bullock, Charlie Frohman, Jozef Przytycki and Adam Sikora \cite{Bull1, Bull2, BFK1, BFK2, PrzS} then shows that this defines an algebra homomorphism
$$
\mathrm{Tr}_r \kern -3pt : \mathcal S^{-1}(S) \to \mathbb C
$$
by the property that
$$
\mathrm{Tr}_r \bigl( [K] \bigr) = - \mathrm{Tr}\, r(K)
$$
for every (connected) framed knot $K\subset S\times [0,1]$. This $\mathrm{Tr}_r \bigl([K] \bigr)$ is independent of the framing of $K$.
The twisted version of the character varieties involves the space $\mathrm{Spin}(S)$ of isotopy classes of spin structures on $S$. Any two spin structures differ by an obstruction in $H^1(S;\mathbb Z_2)$, which defines an action of $H^1(S;\mathbb Z_2)$ on $\mathrm{Spin}(S)$. The cohomology group $H^1(S;\mathbb Z_2)$ also acts on the character variety $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ by the property that, if $\widehat r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$ and $\alpha \in H^1(S;\mathbb Z_2)$, then $\alpha \widehat r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$ is defined by
$$
\alpha \widehat r(\gamma) = (-1)^{\alpha(\gamma)} \widehat r(\gamma) \in \mathrm{SL}_2(\mathbb C)
$$
for every $\gamma \in \pi_1(S)$.
The \emph{twisted character variety} $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ is then defined as the quotient
$$
\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S) =\left( \mathcal R_{\mathrm{SL}_2(\C)}(S) \times \mathrm{Spin}(S) \right) / H^1(S;\mathbb Z_2) .
$$
An element $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ represented by $(\widehat r, \sigma)\in \mathcal R_{\mathrm{SL}_2(\C)}(S)\times \mathrm{Spin}(S)$ then defines an algebra homomorphism
$$
\mathrm{Tr}_{r} \kern -3pt : \mathcal S^{+1}(S) \to \mathbb C,
$$
which to a connected skein $[K]\in \mathcal S^{+1}(S)$ associates $$\mathrm{Tr}_r \bigl( [K] \bigr) = (-1)^{\sigma(K)} \mathrm{Tr}\, \widehat r(K) $$
where $\sigma(K)\in \mathbb Z_2$ denotes the monodromy of the framing of $K$ with respect to the spin structure $\sigma$. Note that the right-hand side of the above formula is invariant under the action of $H^1(S;\mathbb Z_2)$ on the product $\mathcal R_{\mathrm{SL}_2(\C)}(S) \times \mathrm{Spin}(S)$, so that it depends only on the image $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ of $(\widehat r, \sigma)$. See \cite[\S2]{BonWon1} or \cite[\S 5.1]{BonWon2} for details.
By analogy with the case of $\mathcal R_{\mathrm{SL}_2(\C)}(S)$, we will also write
$$
\mathrm{Tr} \, r(K) = - \mathrm{Tr}_r \bigl( [K] \bigr)
$$
for every $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ and every framed knot $K\subset S \times [0,1]$.
\section{Shear-bend coordinates and their square roots}
\label{sect:ShearCoord}
Let us see what happens if in Proposition~\ref{prop:RepsSquareRootCheFock} we replace $\omega$ by $\iota=\omega^{N^2}$. Since $\iota^4=1$, this amounts to considering the special case where $N=1$; in particular, the puncture invariants become irrelevant. Then Proposition~\ref{prop:RepsSquareRootCheFock} associates to any group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ a representation $\rho_\zeta\kern -3pt : \mathcal Z^\iota(\lambda) \to \mathrm{End}(\mathbb C)$. By composition with the trace homomorphism $\mathrm{Tr}_\lambda^\iota \kern -3pt : \mathcal S^{\epsilon}(S) \to \mathcal Z^\iota(\lambda) $ of Theorem~\ref{thm:QTrace}, we now have a homomorphism
$$
\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota \kern -3pt : \mathcal S^{\epsilon}(S) \to \mathrm{End}(\mathbb C) = \mathbb C
$$
for $\epsilon = \iota^2 =\pm1$. This representation of $\mathcal S^{\epsilon}(S)$ is clearly irreducible since its dimension is 1. We can therefore apply the case $N=1$ of Theorem~\ref{thm:InvariantsExist}, which provides a point $r_\zeta$ of the character variety $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$, according to whether $\epsilon=-1$ or $+1$, such that
$$
\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota \bigl([K]\bigr) = - \mathrm{Tr}\, r_\zeta(K)
$$
for every framed knot $K \subset S\times[0,1]$ whose projection to $S$ has no crossing and whose framing is vertical.
It is natural to ask which elements of $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ are obtained in this way. The answer involves the following geometric definition.
Let $\widetilde S$ be the universal cover of $S$, and let $\widetilde\lambda$ be the ideal triangulation of $\widetilde S$ obtained by lifting the edges and faces of $\lambda$. Identify $\mathrm{PSL}_2(\mathbb C)$ to the isometry group of the hyperbolic 3--space $\mathbb H^3$. A \emph{pleated surface} with \emph{pleating locus} $\lambda$ is the data $(\widetilde f, \bar r)$ of a map $\widetilde f \kern -3pt : \widetilde S \to \mathbb H^3$ and a group homomorphism $\bar r\kern -3pt : \pi_1(S) \to \mathrm{PSL}_2(\mathbb C)$ such that:
\begin{enumerate}
\item $\widetilde f$ homeomorphically sends each edge of $\widetilde \lambda$ to a complete geodesic of the hyperbolic space $\mathbb H^3$, and every face of $\widetilde \lambda$ to a totally geodesic ideal triangle of $\mathbb H^3$, with vertices on the sphere at infinity;
\item $\widetilde f$ is $\bar r$--equivariant, in the sense that $\widetilde f( \gamma \widetilde x) = \bar r(\gamma) \bigl( \widetilde f(\widetilde x) \bigr)$ for every $\gamma \in \pi_1(S)$ and every $\widetilde x \in \widetilde S$.
\end{enumerate}
A pleated surface $(\widetilde f, \bar r)$ determines, for each edge $\widetilde \lambda_i$ of the ideal triangulation $\widetilde \lambda$ of $\widetilde S$, a complex weight $\widetilde x_i\in \mathbb C^*$ defined as follows: If $\widetilde Q_i \subset \widetilde S$ is the quadrilateral formed by the two triangles of $\widetilde \lambda$ meeting along $\widetilde\lambda_i$, then $\widetilde x_i$ is the cross-ratio of the 4 vertices of $\widetilde f(\widetilde Q_i)$ in the sphere at infinity $\mathbb C \cup \{ \infty \}$ of $\mathbb H^3$. There edge weights $\widetilde x_i$ are equivariant under the action of $\pi_1(S)$, and therefore descend to a system of weights $x_i$ for the edges $\lambda_i$ of $\lambda$. The edge weights $x_i \in \mathbb C^*$ are the \emph{shear-bend coordinates} of the pleated surface $(\widetilde f, \bar r)$.
Conversely, the pleated surface can be completely reconstructed from the data of these edge weights, up to post-composition by an isometry of $\mathbb H^3$ and pre-composition by an isotopy of $\widetilde S$ respecting $\widetilde S$. See for instance \cite{Thu, Bon97}. In particular, these shear-bend coordinates uniquely determine an element $\bar r \in \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$. We will see in the proof of Proposition~\ref{prop:RealizeIdealTriang} that the issue of lifting $\bar r \in \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ to an element $r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ is strongly connected to choosing appropriate square roots $\sqrt{x_i}$ for the shear-bend coordinates.
Following the terminology introduced in \cite{Thu}, we say that the group homomorphism $\bar r\kern -3pt : \pi_1(S) \to \mathrm{PSL}_2(\mathbb C)$ \emph{realizes} the ideal triangulation $\lambda$ if there exists a pleated surface $(\widetilde f, \bar r)$ with pleating locus $\lambda$. By extension, a point in the character variety $\mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ \emph{realizes} $\lambda$ if it can be represented by a homomorphism $\bar r\kern -3pt : \pi_1(S) \to \mathrm{PSL}_2(\mathbb C)$ realizing $\lambda$. Finally, a point of one of the character varieties $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ \emph{realizes} $\lambda$ if, for the canonical projections $\mathcal R_{\mathrm{SL}_2(\C)}(S)\to \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ and $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)\to \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$, it projects to a point of $\mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ realizing $\lambda$.
We are now ready to state the result promised. At the beginning of this section, we associated to each group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ a point $r_\zeta$ of the twisted character variety $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ if $\epsilon=+1$, and a point of $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ if $\epsilon=-1$.
\begin{prop}
\label{prop:RealizeIdealTriang}
A point of the character varieties $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ or $\mathcal R_{\mathrm{SL}_2(\C)}(S)$, according to whether $\epsilon =+1$ or $-1$, is associated to a group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ as above if and only it realizes the ideal triangulation $\lambda$.
\end{prop}
\begin{proof} To avoid having to keep distinguishing cases, let us focus on the case where $\epsilon =+1$. The case where $\epsilon =-1$ will be identical.
Consider a point $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ associated to a group homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$, namely such that the trace homomorphism $\mathrm{Tr}_r \kern -3pt : \mathcal S^{+1}(S) \to \mathbb C$ is equal to $\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota$ for the algebra homomorphism $\rho_\zeta\kern -3pt : \mathcal Z^\iota(\lambda) \to \mathbb C$ determined by~$\zeta$.
Recall that an element of $ \mathcal W(\tau_\lambda; \mathbb Z) $ can also be interpreted as the assignment of an integer weight to each edge of $\lambda$, satisfying a certain parity condition on each face of $\lambda$. Let $\alpha_i\in \mathcal W(\tau_\lambda; \mathbb Z) $ correspond to the weight system assigning weight $2$ to the $i$--th edge $\lambda_i$ and weight $0$ to every other edge. Set $x_i=\zeta(\alpha_i) \in \mathbb C^*$.
The edge weights $x_i \in \mathbb C^*$ determine a pleated surface $(\widetilde f, \bar r)$ whose shear-bend coordinates are these $x_i$. For the monodromy $\bar r \in \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ of this pleated surface and for a closed curve $K$ in $S\times [0,1]$, the trace $\mathrm{Tr}\, \bar r(K)$ can be expressed up to sign as an explicit Laurent polynomials in square roots $z_i = \sqrt{x_i}$; see \cite[\S1.3]{BonWon1}. Note that this trace $\mathrm{Tr}\,\bar r(K)$ is only defined up to sign because $\bar r$ is valued in $\mathrm{PSL}_2(\mathbb C)$. This is consistent with the ambiguities in the choice of square roots $z_i = \sqrt{x_i}$; indeed, in the explicit formula of \cite[\S1.3]{BonWon1}, choosing different square roots multiply all terms of this Laurent polynomial by the same power of $-1$.
The homomorphism $\mathrm{Tr}_\lambda^\iota \kern -3pt : \mathcal S^{+1}(S) \to \mathcal Z^\iota(\lambda)$ was specially designed in \cite{BonWon1} to mimic this formula. In particular, $
\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota \bigl( [K] \bigr) = \pm \mathrm{Tr}\, \bar r(K)
$
for every framed knot $K$ in $S\times [0,1]$. Since $\mathrm{Tr}_r= \rho_\zeta \circ \mathrm{Tr}_\lambda^\iota$, we conclude that the monodromy $\bar r$ of the pleated surface $(\widetilde f, \bar r)$ is equal to the projection of $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ to $\mathcal R_{PSL)\mathbb C)}(S)$. By definition, this means that $r$ realizes the ideal triangulation $\lambda$, which is what we wanted to prove.
Conversely, let $r$ be a point in $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ that realizes $\lambda$. This means that there exists a pleated surface $(\widetilde f, \bar r)$ whose monodromy $\bar r\in \mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ is the projection of $r$. Let $x_1$, $x_2$, \dots, $x_n\in \mathbb C^*$ be the shear-bend coordinates of this pleated surface.
Pick square roots $z_i=\sqrt {x_i}$ for each of these shear-bend coordinates. These square roots define a homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ by the property that, if $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z) $ associates a weight $\alpha(s_i)\in \mathbb Z$ to the switch $s_i$ of $\tau_\lambda$ corresponding to the $i$--th edge $\lambda_i$ of $\lambda$, then $\zeta(\alpha) = z_1^{\alpha(s_1)} z_2^{\alpha(s_2)}\dots z_n^{\alpha(s_n)}$. This group homomorphism defines an algebra homomorphism $\rho_\zeta \kern -3pt : \mathcal Z^\iota(\lambda) \to \mathbb C$, which in turn determines an element $r_\zeta \in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ such that the two homomorphisms
$\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota $, $\mathrm{Tr}_{r_\rho} \kern -3pt : \mathcal S^{+1}(S) \to \mathbb C$ are equal.
Again, by design of the homomorphism $\mathrm{Tr}_\lambda^\iota$,
$
\rho_\zeta \circ \mathrm{Tr}_\lambda^\iota \bigl( [K] \bigr) = \pm \mathrm{Tr}\, \bar r(K)
$
for every framed knot $K$ in $S\times [0,1]$. As a consequence, the projection of $r_\zeta \in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ to $\mathcal R_{\mathrm{PSL}_2(\mathbb C)}(S)$ is also equal to $\bar r$, and the two elements $r$, $r_\zeta\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ only differ by an obstruction $o\in H^1(S; \mathbb Z_2)$. More precisely, if we use the same spin structure $\sigma\in \mathrm{Spin}(S)$ to represent $r$ and $r_\zeta$, respectively, by $(\widehat r, \sigma)$ and $(\widehat r_\zeta, \sigma)$ with $\widehat r$ and $\widehat r_\zeta\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$, then
$
\widehat r(\gamma ) = (-1) ^{o(\gamma)} \widehat r_\zeta (\gamma)
$
for every $\gamma \in \pi_1(S)$.
We now define a new homomorphism $\zeta'\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ by the property that, for every $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$, $\zeta'(\alpha) = (-1)^{o([\alpha])} \zeta(\alpha)$, where $[\alpha]\in H_1(S;\mathbb Z_2)$ is the homology class represented by $\tau_\lambda$ endowed with the edge multiplicities defined by $\alpha$. Now $\widehat r=\widehat r_{\zeta'}$, which proves that $r$ is associated to the homomorphism $\zeta'\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$.
This concludes the proof of Proposition~\ref{prop:RealizeIdealTriang} when $\epsilon = +1$. The case where $\epsilon = -1$ is essentially identical.
\end{proof}
\section{Representations of the skein algebra}
We are now ready to prove Theorem~\ref{thm:RealizeInvariantsIntro}.
We begin with an elementary lemma about the Chebyshev polynomials $T_n$. Remember that the polynomial $T_n$ is defined by the property that
$\mathrm{Tr}\, M^n = T_n (\mathrm{Tr}\,M)$ for every $M\in \mathrm{SL}_2(\mathbb C)$.
\begin{lem}
\label{lem:ChebElementaryProp}
$ $
\begin{enumerate}
\item If $x=a+a^{-1}$, then $T_n(x) = a^n + a^{-n}$;
\item If $y=b+b^{-1}$, the set of solutions to the equation $T_n(x)=y$ consists of the numbers $x=a+a^{-1}$ as $a$ ranges over all $n$--roots of $b$.
\end{enumerate}
\end{lem}
\begin{proof}
For a matrix $M\in \mathrm{SL}_2(\mathbb C)$, the data of its trace $x$ is equivalent to the data of its spectrum $\{a, a^{-1}\}$. The first property is then a straightfoward consequence of the fact that $\mathrm{Tr}\, M^n = T_n (\mathrm{Tr}\,M)$. The second property immediately follows.
\end{proof}
Recall from \S \ref{sect:CharVar} that, for a point $r$ in the character variety $\mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ and for a framed knot $K \subset S\times [0,1]$, there is a well-defined trace $\mathrm{Tr}\,r(K)\in \mathbb C$. When $r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$, this trace is the usual trace, and is independent of the framing of $K$. However, when $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$, the definition $\mathrm{Tr}\,r(K)$ uses both the framing of $K$ and the spin information of $r$.
In particular, $r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$ or $\mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ associates a number $\mathrm{Tr}\,r(P_k)$ to the $k$--th puncture of $S$, where $P_k$ is a small loop going around the puncture and endowed with the vertical framing.
\begin{thm}
\label{thm:RealizeInvariants}
Assume that the surface $S$ has at least one puncture, that $A^2$ is a primitive $N$--root of unity with $N$ odd, and that we are given:
\begin{enumerate}
\item a point $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ if $A^N=+1$, or a point $r\in \mathcal R_{\mathrm{SL}_2(\C)}(S)$ if $A^N=-1$, such that $r$ realizes some ideal triangulation of $S$;
\item a number $p_k\in \mathbb C$ such that $T_N (p_k) =- \mathrm{Tr}\,r(P_k)$ for each of the punctures of~$S$.
\end{enumerate}
Then, there exists an irreducible representation $\rho \kern -3pt : \mathcal S^A(S) \to \mathrm{End}(V)$ whose classical shadow is equal to $r$ and whose puncture invariants are the $p_k$.
\end{thm}
\begin{proof}Again, let us first focus on the case where $A^N=+1$.
Given $r\in \mathcal R_{\mathrm{PSL}_2(\C)}^{\mathrm{Spin}}(S)$ realizing some ideal triangulation $\lambda$, Proposition~\ref{prop:RealizeIdealTriang} provides a homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$, linearly extending to a representation $\rho_\zeta \kern -3pt :\mathcal Z^\iota(\lambda) \to \mathrm{End}(\mathbb C)=\mathbb C$, such that the composition $\rho_\zeta \circ \mathrm{Tr}^\iota_\lambda \kern -3pt : \mathcal S^{+1}(S) \to \mathbb C$ coincides with the trace homomorphism $\mathrm{Tr}_{r}\kern -3pt : \mathcal S^{+1}(S) \to \mathbb C$ associated to $r$.
In Theorem~\ref{thm:InvariantsExist} and in \S \ref{sect:RepBalancedCF}, we associated to the $k$--th puncture of $S$ a skein $[P_k]\in \mathcal S^A(S)$ and an element $H_k \in \mathcal Z^\omega(\lambda)$ of the balanced Chekhov-Fock algebra. We will use the same letters for the similarly defined elements $[P_k] \in \mathcal S^{+1}(S)$ and $H_k\in \mathcal Z^\iota(\lambda) $.
The image of $[P_k]\in \mathcal S^A(S)$ under the quantum trace homomorphism $\mathrm{Tr}^\omega_\lambda \kern -3pt : \mathcal S^A(S) \to \mathcal Z^\omega(\lambda)$ is easily computed by using Theorem~\ref{thm:QTrace}, and $\mathrm{Tr}_\lambda^\omega\bigl([P_k]\bigr)=H_k + H_k^{-1}$ in $\mathcal Z^\omega(\lambda)$. Similarly, $\mathrm{Tr}_\lambda^\iota\bigl([P_k\bigr])=H_k + H_k^{-1}$ in $\mathcal Z^\iota(\lambda)$. (Beware that we are using the same symbols to denote different objects in different spaces.)
Then, for $[P_k] \in \mathcal S^{+1}(S)$,
$$
\mathrm{Tr}_{r}\bigl([P_k]\bigr) = \rho_\zeta \circ \mathrm{Tr}^\iota_\lambda([P_k]) = \rho_\zeta(H_k + H_k^{-1}) = g_k +g_k^{-1}
$$
if $g_k = \rho_\zeta(H_k) \in \mathrm{End}(\mathbb C)=\mathbb C$. If we are given a number $p_k\in \mathbb C$ such that $T_N(p_k) = \mathrm{Tr}_{r}\bigl([P_k]\bigr) $, Lemma~\ref{lem:ChebElementaryProp} then implies that there exists an $N$--root $h_k$ of $g_k = \rho_\zeta(H_k)$ such that $p_k = h_k + h_k^{-1}$.
Proposition~\ref{prop:RepsSquareRootCheFock} now associates to the homomorphism $\zeta\kern -3pt : \mathcal W(\tau_\lambda; \mathbb Z) \to \mathbb C^*$ and to the $N$--root $h_k$ of $ \rho_\zeta(H_k)$ an irreducible representation $\mu\kern -3pt : \mathcal Z^\omega(\lambda) \to \mathrm{End}(V)$ such that
\begin{enumerate}
\item $\mu(Z_\alpha^N) = \zeta(\alpha) \,\mathrm{Id}_V$ for every $\alpha \in \mathcal W(\tau_\lambda; \mathbb Z)$;
\item $\mu(H_k) = h_k\,\mathrm{Id}_V$ for every $k=1$, \dots, $s$.
\end{enumerate}
The first property can also expressed in terms of the Frobenius homomorphism $\mathbf F^\omega \kern -3pt : \mathcal T^{\iota}(\lambda) \to \mathcal T^\omega(\lambda)$ of \S \ref{sect:ChebFrob} as follows: $\mu \circ \mathbf F^\omega(Z) = \rho_\zeta(Z)\, \mathrm{Id}_V$ for every $Z\in \mathcal Z^\iota(\lambda)$.
Composing with the quantum trace map $\mathrm{Tr}^\omega_\lambda \kern -3pt : \mathcal S^A(S) \to \mathcal Z^\omega(\lambda)$, we now obtain a representation
$$
\rho = \mu \circ \mathrm{Tr}^\omega_\lambda \kern -3pt : \mathcal S^A(S) \to \mathrm{End}(V).
$$
If we already knew that $\rho$ was irreducible, its non-quantum shadow would be computed by composing $\rho$ with the Chebyshev homomorphism $\mathbf T^A \kern -3pt : \mathcal S^{\epsilon}(S) \to \mathcal S^A(S)$.
For a skein $[K]\in \mathcal S^A(S)$, Theorem~\ref{thm:ChebyQTracesFrob} shows that
\begin{align*}
\rho \circ \mathbf T^A \bigl([K]\bigr)
&= \mu \circ \mathrm{Tr}^\omega_\lambda \circ \mathbf T^A \bigl([K]\bigr)
= \mu \circ \mathbf F^\omega \circ \mathrm{Tr}^\iota_\lambda \bigl([K]\bigr)\\
&= \rho_\zeta \circ \mathrm{Tr}^\iota_\lambda \bigl([K]\bigr)\, \mathrm{Id}_V
= \mathrm{Tr}_{r} \bigl ([K] \bigr)\, \mathrm{Id}_V .
\end{align*}
In particular, if $K$ is a knot whose projection to $S$ has no crossing and whose framing is vertical,
$$
T_N \bigl( \rho([K])\bigr) = \rho \bigl( T_N([K])\bigr) =
\rho \circ \mathbf T^A \bigl([K]\bigr)
= \mathrm{Tr}_{r} \bigl ([K] \bigr)\, \mathrm{Id}_V
=-\mathrm{Tr}\,r(K) \, \mathrm{Id}_V
$$
where the first equality comes from the fact that $\rho$ is an algebra homomorphism. By definition of the classical shadow in Theorem~\ref{thm:InvariantsExist}, this is exactly what was needed to show that the classical shadow of $\rho$ is equal to $r$.
Similarly,
$$
\rho([P_k]) = \mu \circ \mathrm{Tr}^\omega_\lambda([P_k]) = \mu(H_k + H_k^{-1}) = (h_k + h_k^{-1})\, \mathrm{Id}_V = p_k\, \mathrm{Id}_V.
$$
Therefore, if $\rho$ is irreducible, it has the classical shadow and cusp invariants required.
When $\rho$ is not irreducible, it suffices to consider an irreducible component $\rho' \kern -3pt : \mathcal S^A(S) \to \mathrm{End}(W)$ with $W\subset V$. Restricting the above computations to $W$ show that $\rho'$ has classical shadow $r$ and puncture invariants the $p_k$.
This concludes the proof when $\epsilon = A^{N^2}$ is equal to $+1$. The case when $\epsilon=-1$ is identical.
\end{proof}
\begin{rem}
We conjecture that, when $r$ is sufficiently generic, the representation $\rho = \mu \circ \mathrm{Tr}^\omega_\lambda $ used in the proof of Theorem~\ref{thm:RealizeInvariants} is already irreducible, and that there is no need to restrict to an irreducible factor.
\end{rem}
\begin{rem}
In the very non-generic case there $r(P_k)$ is the identity for some punctures, the representation $\rho = \mu \circ \mathrm{Tr}^\omega_\lambda $ is definitely reducible. This is at the basis of the ``filling in the punctures'' process of \cite{BonWon4}.
\end{rem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,410 |
{"url":"https:\/\/socratic.org\/questions\/what-is-the-derivative-of-f-t-t-sint-cost","text":"What is the derivative of f(t) = (t-sint , cost ) ?\n\nDec 30, 2015\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = \\sin \\frac{t}{\\cos t - 1}$\n\nExplanation:\n\n$x ' \\left(t\\right) = 1 - \\cos t$\n$y ' \\left(t\\right) = - \\sin t$\n\nThe derivative of the parametric function is\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = \\frac{y ' \\left(t\\right)}{x ' \\left(t\\right)} = \\frac{- \\sin t}{1 - \\cos t} = \\sin \\frac{t}{\\cos t - 1}$","date":"2022-07-05 07:47:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 4, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8320289850234985, \"perplexity\": 2604.967944358066}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104514861.81\/warc\/CC-MAIN-20220705053147-20220705083147-00300.warc.gz\"}"} | null | null |
Jun. 1, 2016 9:05 am
Here's a directory of 2,400+ bots on Slack, Facebook Messenger and more
It's the work of Center City web services firm Stuzo. Here's why the company is betting on bots.
By Roberto Torres / staff
About Audacy Find Out More
Stuzo's offices in Center City.
On the list of buzzwords for 2016, bots and artificial intelligence (AI) are surely near the top. But recent hype is not the reason why Center City-based web services company Stuzo is putting its money on bots.
"Bots are better from an end-user experience perspective, and more seamless from a brand experience perspective," said CEO and cofounder Gunter Pfau on the potential bots have as customer service solutions and more.
But to reach a broader number of users as service providers, bots must jump over a first hurdle: they must be easier to find.
That's what led the company to create Bot Finder, a directory for bots that houses 2,462 artificial intelligence creations for platforms like Slack, Telegram, Skype and Facebook's 900-million-user-strong Messenger.
For the latter platform, Stuzo — which has three satellite offices and a staff of 45 — created an in-house bot that allows users to contact the company and find out more about their what they do.
"We also built a prototype for a customer service bot using Comcast as an example," said Pfau. "Customers can receive billing information, perform speed tests and reset their routers through the platform."
What an Xfinity Messenger bot might look like. (Image courtesy of Stuzo)
One advantage for companies taking this approach to customer service: bots could better reach millennials, who are said to prefer text-based communications over phone conversations.
So, where's the money in all of this?
Pfau says that right now the goal isn't making money from a product release. "It's more about showcasing a new platform to be built on top of," he said. "We see Messenger as a platform that will provide a great deal of flexibility and can handle several functions within its space. It's not just a simple lightweight thing: experiences can be built around this."
Stuzo's CEO goes even further, and bets the future of artificial intelligence for customer service lies not only in one bot helping users get stuff done, but several bots talking to each other and feeding off each other's intelligence.
"Today, really good customer service is 20 percent led by bots and AI, with 70 percent of processes still done in person. That ratio will be reversed over the next five years," he said. "Human interaction will be a lot less, if anything at all."
Companies: Facebook, Stuzo
People: Gunter Pfau
Software Development | philly
To set yourself up for web design success, follow this dev's 3 rules
Company Culture | philly
Tempest helps destination orgs stake their 'little flags on the internet'
Free bootcamp LaunchCode is bringing 2 new web dev courses to Philly this year
Startups | philly
15-year-old web dev firm P'unk Ave was just acquired by Yes& | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,434 |
Denver Property Flip Need To Sell Your House Fast? We Buy Houses Fast, Any Condition!
Have questions? Great. We're glad you do!
What's It Cost To Work With Us?
We buy houses in any condition or shape. You'd be amazed at some of the houses we've bought before. Why do we buy even rundown houses? Simple, most ugly houses just need a little TLC, and then they're pretty houses that someone would love to live in once again. We pay for all renovations and repairs after we buy the house — so you don't have to worry about any of that.
We're located in the Denver area. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,459 |
package org.uberfire.provisioning.docker.runtime.provider;
import javax.enterprise.context.ApplicationScoped;
import org.uberfire.provisioning.runtime.providers.base.BaseProviderType;
@ApplicationScoped
@Docker
public class DockerProviderType extends BaseProviderType {
public DockerProviderType() {
super( "docker", "1.9.1", DockerProvider.class, DockerProviderService.class, DockerRuntimeService.class );
}
public DockerProviderType( String providerName, String version, Class provider, Class providerService, Class runtimeService ) {
super( providerName, version, provider, providerService, runtimeService );
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,657 |
Q: Redirect in PHP - Show txt file content I'm just learning to code and I don't know my way around yet.
I already know how to redirect to another page with php.
for example you go to http://example.com/test.php/ then you are redirected to http://example.com/test.txt with the following code:
"header('Location: http://example.com/test.txt);"
But now I'm not sure how to show the content of different files on your domain if the requested url has
?data=1 or ?seid=2 after the php. For example:
"http://example.com/test.php?data=request1" shows the text from
"http://example.com/test1.txt"
"http://example.com/test.php?data=request2" shows the text from
"http://example.com/test2.txt"
Have informed me so far that I have found something with $DataArray but don't know exactly how to use it? I tried something with it:
<?php
$DataArray = array(
"request1" => "test1",
"request2" => "test2"
if(isset($_GET['data'])) {
$data = str_replace(" ", "+", $_GET["data"]);
if(array_key_exists($data, $DataArray))
echo trim(json_decode(file_get_contents('data/'. $DataArray[$data] .'.txt'),JSON_UNESCAPED_SLASHES), '"');
else
echo "badrequest";
}
?>
Sadly that doesn't work for me so I don't know how to manage that.
edit: was able to fix it thanks for your help.
result:
<?php
$targets = array("1" => "http://redirect-new.com/", "2" => "http://redirect-old.com/", /* ... */);
if (isset($_GET["data"]) && array_key_exists($_GET["data"], $targets)) {
header("Location: {$targets[$_GET["data"]]}");
exit;
}
?>
A: Pass the file name in the link without using the extension and try this code:
if(isset($_GET['data']))
{
$data = $_GET['data'];
$filename = $data.'.txt'; // here we will get the name of the file with the extension (.Text)
$FilePath = '/www/path/to/file'; // your directory files here
$Link = $FilePath.'/'.$filename;
/*
* Here we will find out whether the file exists or not :
*/
// if use include :
if(file_exists($Link)){
include $Link;
}else{
print 'No File Exist !';
}
// if use header location
if(file_exists($Link)){
header('Location: http://example.com/'.$Link);
}else{
print 'No File Exist !';
}
}
This will work with you, but you should read more about the HTTP protocol It will benefit you more in the future .
A: You need to learn a few things, let me try to help you complete a 'reading list' that will help you to understand what you're trying to do and how to do it.
First, you want to learn your way around HTTP protocol. It's a basic knowledge that will get you far - it's great to be aware of this protocol - it will help you identify what you need in different scenarios.
https://developer.mozilla.org/en-US/docs/Web/HTTP/Overview
pay attention to http messages part.
Then, see how PHP abstracts different parts of HTTP protocol, here's the GET parameters:
https://www.php.net/manual/en/reserved.variables.get.php - something you will have to make use of.
Also you'll have to read files: https://www.php.net/manual/en/function.file-get-contents.php
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Finding aids (10688)
Images (1319)
Archival materials (10688)
Collection descriptions (10688)
Manuscripts (4647)
Pages (3283)
Black-and-white photographs (1399)
Clippings (1305)
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Color photographs (797)
Color transparencies (495)
Electronic records (488)
Audiotapes (400)
Floor plans (383)
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National Portrait Gallery (4)
Query: manuscripts
MS 3261 Manuscripts relating to Winnebago music
Densmore, Frances, 1867-1957
NAA.MS3261
Material includes manuscript "Winnebago Music," 362 typed pages, 50 illustrations (filed separately in original prints file, Bureau of American Ethnology File, Number 3261 part), transcriptions of 205 Winnebago songs, and 2 flute melodies (ca. 116 pages) marked "ready for publication" and submitted November 28, 1939; and original copies of 205 Winnebago songs received from the Densmore estate, ca. 1962. This …
in NAA.MS3261 for manuscripts
Editorial Manuscript Files
2 cu. ft. (2 record storage boxes)
This accession consists of referee files documenting the review process for accepted manuscripts for the publication "Isis" primarily during the tenure of editor H. Floris Cohen, 2014-2019. Materials include correspondence, manuscripts, peer reviews, and related materials.
in Accession 21-103 for manuscripts
This accession consists of referee files documenting the review process for accepted manuscripts for the publication "Isis" during the tenure of editor Bernard V. Lightman, 2004-2014. Materials include correspondence, manuscripts, peer reviews, and related materials.
Smithsonian Institution. Badianus Manuscript Project
0.25 cu. ft. (1 half document box)
Record Unit 7269
Materials consist of approximately 115 gouache copies by Marie-Therese Missonnier-Vuillemin of illustrations in the "Badianus Manuscript"; and a descriptive pamphlet by Emily Walcott Emmart concerning the "Badianus Manuscript," an Aztec Herbal, "Codex Barberini, Latin 24 (Vatican Library), 1935. Additional documentation concerning the project exists in Record Unit 46 and at Johns Hopkins University.
in Record Unit 7269 for manuscripts
George Brown Goode Papers, Unpublished Manuscript on the History of American Science
Goode, G. Brown (George Brown), 1851-1896
0.5 cu. ft. (1 document box)
Accession T89001
This accession consists of manuscript drafts for chapters on the unpublished volume concerning the history of American science. These appear to be earlier drafts of manuscripts found in Record Unit 7050, collection division 6 (Box 24).
in Accession T89001 for manuscripts
MS 2630 Tobacco, pipe, corn, etc. among the stocks of Mexico and Central America
McGuire, Dr.
850 Items (ca. 850 pages)
1,775 Items (cards )
Subject: 1. "Tobacco," "pipe," and occasionally "maize" in a variety of Indian languages, by language family. Approximately 600 pages. Originally from Manuscript Number 2630. 2. "History of Tobacco." Draft. 74 pages. Footnotes, 18 pages. Originally from Manuscript Number 1927. 3. "Tobacco and its Mixtures." Draft. 50 pages. Originally from Manuscript Number 2630. 4. "Sacrificial Offerings." Draft. 14 pages. Originally from Manuscript 2630. 5. "Kinnikinnick …
John Wesley Manuscript
Wesley, John
circa 1920s-1930s
0.5 Linear feet ((1 box))
ACMA.06-120
Anacostia Community Museum Archives
This collection contains a handwritten manuscript of approximately 200 pages by John Wesley. The manuscript gives a history of several generations of the Wesley family in Maryland.
in ACMA.06-120 for manuscripts
Sven M. Gronberger Manuscript
Gronberger, Sven M., 1866-1916
This manuscript was apparently Gronberger's doctoral dissertation on European and Asiatic species in Greenland from the eighteenth century until the date of writing.
MS 3941 Materials assembled by Hewitt for preparation of articles in Bureau of American Ethnology Bulletin 30 and for replies to inquires from the public
Hewitt, J. N. B. (John Napoleon Brinton), 1859-1937
Bogaskie, F.
MacKinley, W. E. W., Captain
Skinner, Alanson, 1886-1925
Contents: Adirondack tribe (St Lawrence River) Old Manuscript Number 3553. Adoption Old Manuscript Number 4007. Refers to Algonquian method of counting -only; see Haas note 2/18/72; Old Manuscript Number 3864. "Alligewi"; Animism Old Manuscript Number 3867 and 2842-c, box 6. Blood Indians, origin of name; Brant, Joseph Old Manuscript Number 3874. Chippewa, origin of name Old Manuscript …
Editorial Files
Society for Marine Mammalogy
15 cu. ft. (15 record storage boxes)
This accession consists of records relating to the Society for Marine Mammology's journal, Marine Mammal Science. Materials include manuscripts, editorial correspondence and reviews for manuscripts that were published under the third journal editor, W. F. Perrin. Perrin stepped down as editor in 1999 and was succeeded by W. Don Bowen. These …
10688 records — Page 1 of 1069 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,562 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2084704\/what-boundary-condition-is-imposed-when-fourier-transform-is-used-for-solving-di","text":"What boundary condition is imposed when Fourier transform is used for solving differential equation on infinite domain?\n\nIt's a question that has puzzled me for a long time.\n\nEvery PDE textbook I've ever seen tells me that, Fourier transform can be used to solve linear constant-coefficient differential equations on an infinite domain, but none of them includes an explanation about what boundary condition is actually used when Fourier transform \"kills\" the derivative. Some materials, for example this seems to suggest that, the boundary condition is $0$ at $\\pm\\infty$, but it's not true. A counter example is\n\n$$y'(x)+y(x)=\\sin (x)$$\n\nThe general solution of this equation is\n\n(* Here's the corresponding Mathematica code *)\nDSolve[y'[x] + y[x] == Sin[x], y[x], x]\n\n\n$$y(x)= c_1 e^{-x}+\\frac{1}{2} (\\sin (x)-\\cos (x))$$\n\nwhile the solution given by Fourier transform and inverse Fourier transform is\n\n(* Here's the corresponding Mathematica code *)\nfou = FourierTransform[#, x, w] &;\nfou[y'[x]] + fou@y[x] == fou@Sin[x] \/. HoldPattern@FourierTransform[__] :> Y[w]\nSolve[%, Y[w]][[1, 1, -1]]\nInverseFourierTransform[%, w, x]\n\n\n$$y(x)=\\frac{1}{2} (\\sin (x)-\\cos (x))$$\n\nClearly the boundary condition isn't $y(\\pm\\infty)=0$ or $y'(\\pm\\infty)=0$.\n\nWhat boundary condition \/ restriction is imposed when Fourier transform is used for solving differential equations?\n\nWhen a given PDE can be solved using the Fourier transform in $L^2(\\mathbb{R})$, then the cited material is morally right: If a function in $L^2(\\mathbb{R})$ has a limit at $\\pm \\infty$, then this limit equals $0$. However, as noted by user TrialAndError below, there are functions in $L^2(\\mathbb{R})$ which do not converge to $0$ at infinity. Such examples do almost converge to $0$ though, e.g. we have for any $\\epsilon>0$ that $|\\{x\\in \\mathbb{R}\\setminus B_r(0): f(x)>\\epsilon\\}|\\to 0$ as $r\\to\\infty$.\nAs you noted, the solution of your example is not at all converging to zero. This is because arriving at this solution requires the calculus of Fourier transforms of tempered distributions! For example, $\\sin(x)$ is not in $L^2(\\mathbb{R})$ and its distributional Fourier transform involves Dirac deltas, which are also not in $L^2(\\mathbb{R})$. While tempered distributions are a larger space than $L^2(\\mathbb{R})$, they still exclude functions that diverge exponentially, which is why Mathematica gives you the particular solution you mention.\n\u2022 I don't know if it is worth the effor, at the end the message is easy: the usual Fourier transform of the 19th century is defined for $L^2$ functions and these trivially vanish at infinity. Tempered distributions are objects that in particular include functions that do not vanish (like he constant 1 function) for which a Fourier transform can be defined with some more effort as well. If it turns out that you cannot solve a PDE with the old Fourier transform, for example because the right hand side is not in $L^2$, then you always know that the solution is not necessarily decaying at infty \u2013\u00a0Bananach Jan 5 '17 at 16:22\n\u2022 @Bananach : There are $L^2(\\mathbb{R})$ functions that don't vanish at $\\infty$. You can construct an example with unit height step functions of decreasing widths placed at the integers. \u2013\u00a0Disintegrating By Parts Jan 8 '17 at 2:07","date":"2020-12-04 00:24:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.84909987449646, \"perplexity\": 126.4844441864458}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141732835.81\/warc\/CC-MAIN-20201203220448-20201204010448-00538.warc.gz\"}"} | null | null |
<?php
require_once('../views/header.php');
require_once ('../views/nav_login.php');
?>
<!-- Full Width Image Header with Logo -->
<!-- Image backgrounds are set within the full-width-pics.css file. -->
<header class="image-bg-fluid-height">
<img class="img-responsive img-center" src="" alt="">
</header>
<!-- Content Section -->
<section>
<div class="container">
<div class="row">
<div class="col-xs-12">
<!-- Form um sich zu registrieren-->
<form class="form-horizontal" action="../passiv/eintragen.php" method="post">
<h3 class="text-center">Créer un profil :</h3>
<p class="lead section-lead"></p>
<div class="form-group">
<label class="col-xs-4 control-label" for="vname">Prénom</label>
<div class="col-xs-4">
<input name="vname" type="text" placeholder="à compléter"
class="form-control" required="">
</div>
</div>
<div class="form-group">
<label class="col-xs-4 control-label" for="nname">Nom</label>
<div class="col-xs-4">
<input name="nname" type="text" placeholder="compléter si cela doit apparaître dans le profil"
class="form-control" >
</div>
</div>
<div class="form-group">
<label class="col-xs-4 control-label" for="uname">Nom d'utilisateur</label>
<div class="col-xs-4">
<input name="uname" type="text" placeholder="à compléter"
class="form-control" required="">
</div>
</div>
<div class="form-group">
<label class="col-xs-4 control-label" for="upw">Mot de passe</label>
<div class="col-xs-4">
<input name="upw" type="password" placeholder="à compléter"
class="form-control" required="">
</div>
</div>
<div class="form-group">
<label class="col-xs-4 control-label" for="email">E-mail</label>
<div class="col-xs-4">
<input name="email" type="text" placeholder="à compléter, cela n'apparaîtra pas dans le profil"
class="form-control" required="">
</div>
</div>
<div class="form-group">
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<?php
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?> | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,760 |
One of GYPO's golden rules is to always shop your closet first. If you have a solid base, then it's easy to mix and match what you have to create endless possibilities. Another golden rule at GYPO is that there are certain pieces totally worth investing in. These are pieces that are classic and timeless, they will be worn year after year and it's worth it to have something that will stand up to the challenge of being worn that much. But, as we all know, it takes time to build up a go-to closet and even longer to obtain all the investment pieces we long for. If you're going to invest in one piece this spring, may we make a few suggestions? Below are some wonderful investment options to add to your closet and if you're still not ready to make the plunge, I've included a budget-friendly option as well.
White denim goes with everything and can be worn through multiple seasons. When selecting the perfect pair, opt for ones that are simple in design and don't include a lot of trendy elements that may date them. We know the challenges of finding the right pair – too thick and you're sweating all spring/summer long, too thin and they become see through and no one wants that! But I promise, don't stop until you find the perfect pair, you'll be happy you did and then you'll be able to enjoy them with all the other staples in your closet!
Whether you choose booties, sandals, wedges or flats, having a fabulous pair of taupe shoes is an essential closet staple. They go with everything and can easily elevate any outfit. Just make sure they satisfy your level of comfort or they'll never make it out of your closet!
This is one of my favorite closet staples because it goes with EVERYTHING from your favorite dress, color bottoms and yes, even your jeans. Just be sure you are wearing a wash different from your jacket and it will work, I promise you! Plus, a great denim jacket can be worn at least two, maybe even three seasons out of the year depending on where you live. When choosing a denim jacket, pick one that has some stretch to it, that way it will move with you and you won't feel so constricted.
The best part about all these closet staples is that they all pair up perfectly together! Just add your favorite graphic tee, striped shirt or even a colorful tank!
Your turn! What is your favorite spring closet staple? What investment piece do you hope to add? | {
"redpajama_set_name": "RedPajamaC4"
} | 2,973 |
\section{ Introduction}
The new data of TOTEM collaboration\cite{TOTEMRHO1,TOTEMRHO2,TOTEMRHO3,
TOTEMRHO4}
drew attention to the state with negative signature and with an
intercept which is close to unity
(see Refs.\cite{KMRO,BJRS,TT,MN,BLM,SS,KMRO1,KMRO2,GLP,CNPSS}).
This state is known as the Odderon, and it appears naturally in
perturbative QCD (see Ref.\cite{KOLEB} for the review)
with the intercept $\alpha_{\rm Odd}\Lb t=0\right)\,\,=\,\,1$\cite{BLV,KS}.
In a number of papers it was shown that such a state could be
helpful for describing the experimental
data\cite{KMRO,BJRS,TT,MN,BLM,SS,KMRO1,KMRO2,GLP,CNPSS}.
However, in perturbative QCD the dependence of the Odderon on energy
is crucially affected by the shadowing corrections, which lead to a
substantial decrease of the Odderon contribution
with increasing energy\cite{KS,KOLEB,CLMS}.
In this paper we wish to study the shadowing corrections to the
Odderon contribution using the model that we proposed in
Ref.\cite{GLPPM}. The model is based (i) on
Pomeron calculus in 1+1 space-time, suggested in Ref. \cite{KLL},
and
(ii) on
simple assumptions of hadron structure, related to the impact
parameter
dependence of the scattering amplitude. This parton model stems from QCD,
assuming that the unknown non-perturbative corrections lead to
determining
the
size of the interacting dipoles. The advantage of this approach is that
it
satisfies both the $t$-channel and $s$-channel unitarity, and can be
used
for summing all diagrams of the Pomeron interaction, including Pomeron
loops. In other words, we can use this approach for all possible
reactions: dilute-dilute (hadron-hadron), dilute-dense (hadron-nucleus)
and dense-dense (nucleus-nucleus), parton systems scattering.
The model gives a fairly good description of
three
experimental
observables: $\sigma_{\rm tot}$,$\sigma_{\rm el}$ and $B_{\rm el}$ for proton-proton scattering,
in the eikonal model for
the structure of hadrons at high energy.
The goal of this paper is study the influence of the shadowing
corrections on the Odderon contribution in our model.
\section{ The model (brief review)}
Our model includes three essential ingredients: (i) the new parton
model for the dipole-dipole scattering amplitude that has been discussed
above; (ii) the simplified one channel model that enables us to
take
into account diffractive production in the low mass region, and (iii)
the assumptions for impact parameter dependence of the initial
conditions.
\subsection{New parton model.}
The model that we employ \cite{GLPPM,KLL} is based on
three
ingredients:
1. The Colour Glass Condensate
(GCC) approach (see Ref.\cite{KOLEB} for a review), which can be
re-written in an equivalent form as the interaction of BFKL
Pomerons\cite{AKLL} in a limited range of rapidities
( $Y \leq Y_{\rm max}$):
\begin{equation} \label{RAPRA}
Y \,\leq\,\frac{2}{\Delta_{\mbox{\tiny BFKL}}}\,\ln\Lb
\frac{1}{\Delta^2_{\mbox
{\tiny BFKL}}}\right)
\end{equation}
$\Delta_{\mbox{\tiny BFKL}}$ denotes the intercept of the BFKL
Pomeron\cite{BFKL}. In our model $ \Delta_{\mbox{\tiny BFKL}}\,
\approx\,0.2 - 0.25$ leading to $Y_{max} = 20 - 30$, which covers
all collider energies.
2. The following Hamiltonian:
\begin{equation}\label{HNPM}
{\cal H}_{\rm NPM}=-\frac{1}{\gamma}\bar PP\end{equation}
where NPM stands for ``new parton model''. $P$ and $\bar P$ denote
the BFKL
Pomeron fields.
The fact that it is self dual is evident. This Hamiltonian in the limit
of small $\bar P$ reproduces the Balitsky-Kovchegov Hamiltonian
${\cal H}_{\rm BK}$
( see Ref.\cite{KLL} for details). This condition is
important for determining the form of
${\cal H}_{\rm NPM}$. $\gamma$ in \eq{HNPM} denotes the dipole-dipole
scattering amplitude, which in QCD is proportional to $\bar{\alpha}_S^2$.
3. The new commutation relations:
\begin{equation}\label{CRCOR}
\Big(1\,\,-\,\,P\Big)\Big(1\,\,-\,\,\bar P \Big)\,\,=\,\,(1-\gamma)\Big(1\,\,-\,\,\bar P\Big) \Big(1\,\,-\,\,P\Big)
\end{equation}
For small $\gamma$ and in the regime where $P$ and $\bar P$ are also
small, we obtain
\begin{equation}
[P,\bar P]=-\gamma +...
\end{equation}
consistent with the standard BFKL Pomeron calculus (see Ref.\cite{KLL}
for details) .
In Ref.\cite{KLL}, it was shown that the scattering matrix
for the model
is
given by
\begin{eqnarray}\label{classs}
S^{\rm NPM}_{m\bar n}(Y)&=&e^{\frac{1}{\gamma} \int_0^Yd\eta\left[
\ln(1-p)\frac{\partial}{\partial \eta}\ln (1-\bar p)
+\bar pp\right]}[1-p(Y)]^m[1-\bar p(0)]^{\bar n}|_{p(0)=1-e^{-\gamma
\bar n};\ \bar p(Y)=1-e^{-\gamma m}}\nonumber\\
&=&[1-p(Y)]^m\,e^{\frac{1}{\gamma}\int_0^Yd\eta \left[\ln(1-\bar p)+\bar
p\right]p}
\end{eqnarray}
where $p(\eta)$ and $\bar p(\eta)$ are solutions of the classical equations
of motion and have the form:
\begin{equation} \label{H03}
P (\eta)\,=\,\frac{ \alpha +\beta e^{ (1 - \alpha) \eta} }{1 + \beta e^{ (1
- \alpha) \eta}}; \ \ \ \ \bar P(\eta)= \frac{ \alpha (1+\beta e^{
(1 - \alpha) \eta}) }{\alpha + \beta e^{ (1 - \alpha) \eta}};
\end{equation}
where the parameters $\beta$ and $\alpha$ should be determined from
the
boundary conditions:
\begin{equation} \label{H0BC}
P (\eta= 0)\,=\,p_0;\,\,\,\,\,\,\,\, \bar P (\eta= Y)\,=\,\frac{\alpha}{P
(\eta= Y)}\,=\,\bar p_0
\end{equation}
\subsection{ Eikonal approximation }
In the eikonal approximation we neglect the contribution of the
diffractive production and assume that the hadron wave function
diagonalize the matrix of interaction. In this model
the unitarity constraints take the form
\begin{equation} \label{UNIT}
2\,\mbox{Im}\,A\left(s,b\right)=|A\left(s,b\right)|^2
+G^{in}(s,b),
\end{equation}
where $G^{in}$ denotes the contribution of all inelastic processes.
In \eq{UNIT}
$\sqrt{s}=W$ denotes the energy of the colliding hadrons and $b$ denotes
the
impact parameter. In our approach we used the solution to \eq{UNIT}
given by \eq{classs} and
\begin{equation} \label{EIK}
A \,=\,1 - S^{\rm NPM}(Y, b)\,\,\,\equiv\,\,\,1\,\,-\,\,\exp\Big(- \Omega\Lb Y,b\right)\Big)
\end{equation}
\subsection{ The general formulae.}
{\it Initial conditions:}
Following Ref.\cite{GLPPM} we chose the initial conditions in the form:
\begin{equation} \label{IC}
p(b') = p_{0 } \,S(b',m)~~~\mbox{with}~~S(b,m)= m b K_1(m\, b );~~~~~
\bar{p}(\vec{b} - \vec{b}) = p_{0} S( \vec{b} - \vec{b}',m) ~~~~~~~z_m = e^{\Delta(1 - p_{0})Y}
\end{equation}
Both $p_{0} $ and mass $m$, as well as the Pomeron intercept
$\Delta$, are parameters of the model, which are determined by
fitting to the relevant data. Note, that
$S\Lb b, m\right) \xrightarrow{m_i\,b \gg 1}\,\exp\Lb - m\,b\right)$
in accord with the Froissart theorem\cite{FROI}.
From \eq{IC} we find that
\begin{eqnarray}
a(b,b')\,\equiv\, a\Lb p, \bar{p}, z_m\right) &=&\,\frac{1}{2}\Lb p + \bar{p}\right) \,+\,\frac{1}{2\,z_m}\Lb (1-p)(1-\bar{p} ) \,-\, D\right);\label{ALEQ}\\
b(b,b') \,\equiv\,b\Lb p, \bar{p}, z_m\right) \,\,&=&\, \frac{1}{2} \frac{p- \bar{p}}{1 - p} -\frac{1}{2 z_m (1 - p)}\Lb (1- p) (1 - \bar{p}) - D\right);\label{BEEQ}\\
~~
D &=& \sqrt{4 p (1 - p) (1- \bar{p}) z_m - \Lb (1 - p) (1- \bar{p}) - (p - \bar{p}) z_m\right)^2};\label{D}
\end{eqnarray}
These equation are the explicit solutions to \eq{H03} and \eq{H0BC}.
\par{\it Amplitudes:}
In the following equations $ p \equiv p ( b')$ and $\bar{p}
\equiv \bar{p}(\vec{b} - \vec{b} ')$.
~
$$z = e^{\Delta\,(1- p_{0})\,y}$$
~
$S (a,b,z) \equiv S(a(b,b'),b(b,b'),z_m)$, \,\, $ X( a, b,z) \equiv X(a(b,b'),b(b,b'),z_m)$
\begin{equation}
X(a,b,z)) = \frac{a + b z}{1 + b z}
\end{equation}
\begin{eqnarray}
&&SS(a,b,z)=\\
&&-(a-1) \text{Li}_2(-b z)+a
\text{Li}_2\left(-\frac{b
z}{a}\right)+(a -1)
\text{Li}_2\left(\frac{a+b
z}{a -1}\right)+\frac{1}{2} a \log
^2((1-a) b z)\nonumber\\
&& -(a -1) \log (b z+1)
\log ((1-a) b z)
-\left(a \log
(z)-(a-1) \log \left(-\frac{b
z+1}{a -1}\right)\right) \log (a + b
z)\nonumber\\
&& +a \log (z) \log \left(\frac{b
z}{a+1}\right)\nonumber
\end{eqnarray}
\begin{equation} \label{FIN}
S(a,b,z) \,\,=\,\,SS(a, b, z) \,-\,SS(a , b,z=1) \end{equation}
~
The amplitude is given by
\begin{eqnarray} \label{AIK}
\hspace{-1cm}&&A(s, b)\,\,\,=\,\,\,1\,\,\,-\,\,\,e^{ - \Omega\Lb W, b\right) }\,\,=\\
\hspace{-1cm}&&\,1 - \exp\Bigg( \frac{1}{p_{0}}\int \frac{m^2 d^2 b'}{4 \pi} \Big( S(a,b,z_m) \,\,+\,\, a(b,b') \Delta (1 - p_0) Y\Big) - \int \frac{m^2 d^2 b'}{4 \pi} \bar{p}( \vec{b} - \vec{b}',m)\,X(a, b,z_m) \Bigg)\nonumber
\end{eqnarray}
\section{The Odderon contribution}
\subsection{ Odderon exchange}
As has been mentioned, we view the Odderon as a reggeon with negative
signature and with the intercept
$\alpha_{\rm Odd}(t=0)$=1. Generally speaking its contribution to the
scattering amplitude has the following form:
\begin{equation} \label{ODD1}
O_{i k}(s,b)\,\,=\,\,\eta_{-}(t) \,g^i_{\rm Odd}(b)\,g^k_{\rm Odd}(b)\,\,s^{\alpha_{\rm Odd}(t)\,\,-\,\,1}
\end{equation}
where $\eta_{-}$ is a signature factor $\eta\,\,=\,\,\tan\Lb \frac{1}{2} \pi\,
\alpha_{\rm Odd}(t)\right)\,\,-\,\,i$
, $g^i_{\rm Odd}$ is the vertex for the interaction of the Odderon with
state $i$, and $\alpha_{\rm Odd}$ denotes the trajectory. The Odderon
appears naturally in perturbative QCD. As one can see from
\fig{odqcd} the QCD Odderon describes the exchange of three
gluons and all the interactions between them. The QCD Odderon has
the trajectory with the intercept equal to 1 and which does
not depend on $t$\cite{BLV,KS}. Hence, the Odderon only contributes
to the real part of the scattering amplitude. For an
estimate we will use the following form of the Odderon contribution:
\begin{equation} \label{ODD2}
O_{i k}(s,b)\,\,=\,\,\pm \, \sigma_0 e^{ - \frac{b^2}{4 \,B}}
\end{equation}
where sign plus corresponds to proton-antiproton scattering, while
sign minus describes the proton-proton collisions.
The value of $\sigma_0$ was evaluated in Ref.\cite{RYODD} (see also
Ref.\cite{LERY90}) in the framework of perturbative QCD. It turns
out that $\sigma_0\,\,=\,\,20.6\,\bar{\alpha}_S^3\,mb$. In perturbative QCD,
we expect that $B$ is smaller than for the elastic scattering. We
choose $ B=5.6-6 \,GeV^{-2}$ for our estimates\cite{GLPPM,KMRO}.
In \eq{ODD2} we assume that $g^i_{\rm Odd}(b)$
in \eq{ODD1} does not depend on $i$.
\begin{figure}
\centering
\includegraphics[width=14cm]{OddQCD.pdf}
\caption{QCD Odderon for two dipoles scattering: the wavy
lines describe gluons and the solid lines correspond to quarks. }
\label{odqcd}
\end{figure}
\subsection{Shadowing corrections}
In the eikonal model the elastic amplitude is equal to
(see \fig{sc}-a)
\begin{eqnarray} \label{SC1}
A_{\rm el}\Lb s, b\right) \,\,\,&=&\,\,\,\,1\,\,- \,\,\exp\Big( -\,\,\Omega\Lb s, b\right)\Big)
\end{eqnarray}
\eq{SC1} is the series whose general term is proportional to
$\Omega^n/n!$. In the case of Odderon exchange we
need to replace one of $\Omega$ by $O(s,b)$. Hence
$\Omega^n/n!$
should be replaced by $ O(s,b) \,n\, \Omega^{n - 1}/n!
\,\,=\,\,O(s,b) \Omega^{n - 1}/(n - 1)! $. Finally, we
have (see also Ref.\cite{GLP})
\begin{equation} \label{SC2}
O^{\rm SC}\Lb s, b\right) \,\,\,=\,\,\,\, \,O(s,b)\,e^{- \Omega\Lb s, b\right)} \,\,=\,\,
\,\,O(s,b)\,\Bigg( 1\,\,-\,\,A_{\rm el}\Lb s, b\right)\Bigg)
\end{equation}
\begin{figure}
\centering
\includegraphics[width=12cm]{OddEik.pdf}
\caption{Shadowing corrections to the Odderon exchange:
\fig{sc}-a: elastic amplitude in the two channel model.
\fig{sc}-b: the shadowing corrections in our model.
The wavy lines describe the Pomeron exchanges while
the zigzag line corresponds to the exchange of the Odderon. }
\label{sc}
\end{figure}
\subsection{Numerical estimates}
In this section we make estimates using our model
for $\Omega$, with parameters that are given by Table I.
\begin{figure}
\centering
\includegraphics[width=12cm]{Bdep.pdf}
\caption{$O\Lb W, b\right)$ versus $b$ for different energies.
The red line corresponds to the contribution of \eq{ODD2}. }
\label{b}
\end{figure}
In \fig{b} we plot the $b$ dependence of the Odderon contribution.
One can see that the shadowing corrections lead to a considerable
suppression of the Odderon contribution at small $b$ in comparison
with \eq{ODD2} (see red line in \fig{b}). This suppression
is much smaller than in our approach, based on CGC\cite{GLP}. The reason
for this is that in our model the value of $A_{\rm el}\Lb s,
b\right) $
turns out to be smaller than 1 even at very high energies. Due to this
$O\Lb W, b=0\right)\,\,\neq\,\,0$ even at $W \approx 20 TeV$.
In \fig{rho} we plot the contribution of the Odderon to the ratio of
$\rho\,\,=\,\,{\rm Re}/{\rm Im}$ parts of the scattering amplitude as
function of energy. One sees the influence of the shadowing
corrections, which induce the energy dependence of this ratio
on energy. \eq{ODD2} shows that the Odderon does not depend
on energy without these corrections. This induced energy dependence
turns out to be rather large causing a decrease of $\rho$
in the energy range: W = 0.5 $\div$ 20 TeV. However, this effect is
much smaller than in our previous estimates \cite{GLP} and the value
of $\rho$ does not contradict the experimental data
\cite{TOTEMRHO1,TOTEMRHO2,TOTEMRHO3, TOTEMRHO4}.
The shadowing correction has a remarkable
effect on the $t$-dependence
of the scattering amplitude (see \fig{q} ). We see that the shadowing
corrections lead to a narrower distribution over $t$, than the
input
given by \eq{ODD2}, which is shown in \fig{q} by the red line.
\begin{figure}
\centering
\includegraphics[width=12cm]{RhovsW1.pdf}
\caption{$\rho = {\rm Re}/{\rm Im}$ due to the Odderon
contribution versus W in our model. The solid line presents the estimates,
using \eq{SC2}, while the dashed line describes the evaluation in two
channel model of Ref.\cite{GLPPM2}. }
\label{rho}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|l|l|l|}
\hline
\hline
$\Delta_{\rm dressed}$ & $p_{0}$ & $m(GeV)$ & $\chi^2$/d.o.f\\
\hline
0.331 $\pm$ 0.030 & 0.483 $\pm$ 0.030 &0.867 $\pm$ 0.005& 1.3\\
\hline
\hline
\end{tabular}
\caption{Fitted parameters.$\Delta_{\rm dressed} = \Delta\Lb 1 - p_{0}\right)$.}
\label{t2}
\end{table}
\begin{figure}
\centering
\includegraphics[width=8cm]{Qdep.pdf}
\caption{$O\Lb W,q=\sqrt{|t|}\right)$ versus $q =
\sqrt{|t|}$ for different energies. The red line corresponds
to the contribution of \eq{ODD2}. }
\label{q}
\end{figure}
\begin{boldmath}
\section{Dependence of the elastic cross sections on $t$ and the Odderon}
\end{boldmath}
In this section we will look at another facet of the Odderon
contribution: it could contribute to the real part of the
scattering amplitude at the value of $t=t_{min}$, where $d
\sigma_{\rm el}/dt$ has a minimum.
We attempt to describe the elastic cross section for $|t|=0 \div
1\,GeV^2$ . Our model predicts the existence of a
minimum
in the elastic cross sections, however its position occurs at
$|t |\,\approx\,0.3\,GeV^2$, which is much smaller than was
observed experimentally by TOTEM collaboration\cite{TOTEMLT}.
Assuming that this discrepancy is due to the simplified
form of
$b$ dependence of our amplitude which is given by \eq{IC}, we
changed the initial conditions of \eq{IC} to the following equations
\begin{equation} \label{IC1}
p(b') = p_{0 i} \,S(b',m,\mu,\kappa_i)~~~\mbox{with}
~~S(b,m,\mu,\kappa_i)= \Lb 1 - \kappa\right) \Lb m\,b\right)^{\nu_1} K_{\nu_1}(m \,b )\,\,+\,\,\kappa \frac{\Lb m \,b\right)^{\nu_2} K_{\nu_2}(\mu \,b )}{2^{\nu_2\,-\,1} \,\Gamma\Lb \nu_2\right)}
\end{equation}
In Table II we present the parameters that we found
for the fit. \fig{dsdt} shows the comparison with TOTEM data
of Ref.\cite{TOTEMLT}. One can see that we obtain good agreement
with the experimental data for $|t|\,<\,|t|_{min}$ and for
$|t|\,>\, |t|_{min}$. However, for $ |t| \approx\,|t|_{min}$ the real
part
of the scattering amplitude turns out to be small, and we obtain
a value of the $d \sigma_{el}/d \,t$ approximately an order of
magnitude smaller than the experimental one. It should be stressed that
we do not use any of the simplified approaches to estimate the
real part of
the amplitude, but using our general expression of \eq{AIK} for $A_{i
k}\Lb s,t\right)$, we consider the sum $A_{ik}\Lb s,+ i \epsilon t\right) +
A_{ik}\Lb u-i \epsilon,t\right)$, which corresponds to positive signature,
and calculated the real part of this sum.
In \fig{dsdt} we estimate the contribution of the $\omega$ -reggeon ,
using the
description taken from the paper of Ref.\cite{DOLA}(note the difference
between green dashed line and the blue solid curve). This
contribution is small, and can be neglected.
\begin{figure}[ht]
\centering
\leavevmode
\includegraphics[width=11cm]{DSDT.pdf}
\caption{$d \sigma_{el}/dt$ versus $t$. The black green line
describes the result of our fit. The dashed line corresponds to the
contribution of the imaginary part of the scattering amplitude to the
elastic cross section. The dotted line relates to the real part of our
amplitude.The red solid line takes into account the contribution of the
odderon to the real part of the $p p$ amplitude, as is shown in
\eq{ODD}. The data , shown in grey, include systematic errors. They are
taken from Ref.\cite{TOTEMLT}. } \label{dsdt}
\end{figure}
To evaluate the real part of the amplitude we use the relation:
\begin{equation} \label{DSDT1}
{\rm Re}A_{11}(s,t)\,\,=\,\,\frac{1}{2} \,\pi\,\frac{\partial}{\partial\,\ln\Lb
s/s_0\right)}\, {\rm Im}A_{11}\Lb s,t\right)|_{\eq{AIK}}
\end{equation}
\eq{DSDT1} correctly describes the real part of the amplitude only for
small $\rho={\rm Re}A/{\rm Im} A$. In \fig{dsdt1} we plot
the $d \sigma/dt$ with these estimates for the real part.
The real part from \eq{DSDT1} turns
out to be almost
twice larger than the experimental data in the vicinity of $t_{min}$.
Therefore, at the minimum, where
${\rm Im}\, A \,\ll\,{\rm Re}A$, \eq{DSDT1} cannot be used
for the real part. However, replacing \eq{DSDT1} by
\begin{equation} \label{DSDT2}
{\rm Re}A_{11}(s,t)\,\,=\,\,\tan\Lb\rho\right)\, {\rm Im}A_{11}\Lb s,t\right)|_{ \eq{AIK}}
\end{equation}
we obtain the same result, that the real part of the amplitude turns
out to be too large. Actually,\eq{DSDT2} assumes that the scattering
amplitude depends on energy as a power $A\Lb s,
t\right)\,\propto\,s^{2\,\rho/\pi}$. Our amplitude
is a rather complex function of energy, and depends
on $\ln(s)$.
\begin{figure}[ht]
\centering
\leavevmode
\includegraphics[width=10cm]{DSDT1.pdf}
\caption{$d \sigma_{el}/dt$ versus $t$. The solid line
describes the result of our fit. The dotted line corresponds
to the contribution of the real part of the scattering amplitude
to the elastic cross section, which is calculated using \eq{DSDT1},
with added contribution of the
exchange of the $\omega$ - reggeon, which is taken from
Ref.\cite{DOLA}. We do not show the contribution of the real part
without the $\omega$-reggeon as it coincides with the dotted
line. The dashed line is the contribution of the imaginary part
of the amplitude.
The data are taken from Ref.\cite{TOTEMLT}) }
\label{dsdt1}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Variant of the fit &$\Delta_{\rm dressed}$ & $p_{0}$ & $m$
(GeV) &$\mu$(GeV)& $\nu_1$& $\nu_2$&$\kappa$\\
\hline
one channel model &0.48 $\pm$ 0.01&0.8 $\pm$ 0.05&0.860 &7.6344&0.9&0.1 &0.48\\\hline
\hline
\end{tabular}
\caption{Fitted parameters for $d \sigma_{el}/d t$
dependence.$\Delta_{\rm dressed} = \Delta\Lb 1 - p_{01}\right)$.}
\label{t3}
\end{table}
Concluding, we see that to describe the TOTEM experimental data in
the framework of our model, the contribution to
the real part of the amplitude from the exchange of the odderon\cite{ODD}
is needed.
Hence, our
estimates confirm the conclusions of Ref.\cite{ODDSC}. In
\fig{dsdt} we plot the description of the elastic cross section in
which we have added the odderon contribution to the amplitude of
\eq{AIK}
(red solid curve in \fig{dsdt}):
\begin{equation} \label{ODD}
f\Lb s,t\right)\,\,=\,\,f\Lb s,t; \eq{AIK}\right)\,\,\pm\,\, O(s,t) \end{equation}
where we consider a QCD odderon\cite{ODD}: the state with odd signature and
with the intercept $\alpha_{\rm odd}(t=0)=1$, which contributes only
to the real part of the scattering amplitude. $O(s,t)$ is given by
\eq{ODD2}. Our odderon parameters are in
accord with the estimates in Ref.\cite{KMR}. The amplitude $f(s,t)$ is
related to $a_{el}\Lb s,b\right)$ as
\begin{equation} \label{OBSEL}
\frac{d \,\sigma_{el}}{d t}\,\,=\,\,\pi\,|f(s,t)|^2;\,\,\,\,\,a_{el}(s,b)\,\,=\,\,\frac{1}{2 \,\pi}\int d^2 q\, e^{- i \vec{q}\cdot\vec{b}}\,f\Lb s,t\right) \mbox{where}\,\, t\,=\,- q^2
\end{equation}
In \fig{dsdt2} we show the prediction for proton-antiproton scattering.
One can conclude that in our model the measurements of the elastic cross
sections for $p\,p$ and $\bar{p} p$ scattering can provide the estimates
for
the odderon contribution. It should be stressed that the contribution
of the $\omega$-reggeon leads to negligible contribution at
$W= 7\,TeV$ (see \fig{dsdt1}).
\begin{figure}[ht]
\centering
\leavevmode
\includegraphics[width=10cm]{DSDT2.pdf}
\caption{$d \sigma_{el}/dt$ versus $t$.
The solid line describes the elastic cross sections
for $p p$-scattering with the odderon contribution
(see \eq{ODD}), while the dashed line shows the
elastic cross section for $\bar{p} p$-scattering
using \eq{ODD}. The data are taken from Ref.\cite{TOTEMLT})
}
\label{dsdt2}
\end{figure}
~
~
\section{Conclusions}
In this paper we discussed the Odderon contribution in
our model\cite{GLPPM} that provides a fairly good description of
$\sigma_{\rm tot}$,$\sigma_{\rm el}$
and $B_{\rm el}$, especially as related to the energy dependence
of these observables. We showed that the shadowing
corrections are large and induce considerable dependence
on energy for the Odderon contribution, which
in perturbative QCD is energy independent . However, this energy
dependence does not contradict the experimental data for
$\rho = {\rm Re}/{\rm Im}$, if we assume that the Odderon
gives a contribution of about $1 \div 4$ mb at W=7 TeV (see \fig{rhoexp}).
\begin{figure}
\centering
\includegraphics[width=8cm]{RhoExp.pdf}
\caption{ $\rho$ = Re/Im for proton-proton scattering
versus $W =\sqrt{s}$. Data are taken from PDG \cite{PDG} and
from the TOTEM papers \cite{TOTEMRHO1,TOTEMRHO2,TOTEMRHO3,TOTEMRHO4}. The
solid line shows the
predictions of our model, while the dashed line presents the estimates
for the
value of $\rho$, adding the odderon contribution $4 mb$ at W=13 TeV,
to our model. }
\label{rhoexp}
\end{figure}
This fact is
in striking contrast to our estimates for the CGC based
model\cite{GLP}.
The reason for this difference is that the elastic scattering amplitude
in our eikonall model does not reach the unitarity limit ($A_{el}\Lb
W, b=0\right)=1$, even at very high energies. The contrast turns out
to be more pronounced in the two channel model\cite{GLPPM2}, which is shown
by dashed lines in \fig{rho}. However, we need to point out here
that
the
comparison with the experimental data in the two channel model is worse
than in the eikonal one, especially for $\sigma_{el}$.
Our attempt to describe the $t$-dependence of the elastic
cross section shows that we can reproduce the main features of
the $t$-dependence that are measured experimentally: the slope
of the elastic cross section at small $t$, the existence of the
minima in $t$-dependence which is located at $|t|_{min} = 0.52\,GeV^2$
at W= 7 TeV; and the behaviour of the cross section at $|t|\,>\,|t|_{min}$.
It should be stressed that
we do not use any of the simplified approaches to estimate the
real part of
the amplitude which we show ( in our model ), that they do not
reproduce
correctly the real part of the amplitude at large $t$.
In our model the real
part turns out to be much smaller
than the experimental one. Consequently, to achieve a description of the
data, it is necessary to add an
odderon
contribution. Hence, our model
corroborates the conclusion of Ref.\cite{ODDSC}.
Summarizing, in this paper we have presented estimates
resulting from a simple eikonal model,
which provides a fair description of the data on total and elastic cross
sections.
We are aware, that this is a simplified approach, which could be a good
first approximation, but we need to go further. We plan to
develop a model which will also describe diffraction production.
We have made the first effort \cite{GLPPM2}, but we consider it as
not very successful, and we need to continue our search. We also
plan to re-visit our model based on CGC approach \cite{GLP}, with
the goal to improve it so that , we can also introduce
the Odderon contribution. We believe that we can base these attempts
on results for QCD Odderon \cite{KOLEB,BLV,KS,LRRW,YHH,CLMS}.
{\it Acknowledgements.} \\
We thank our colleagues at Tel Aviv University and UTFSM for
encouraging discussions. Our special thanks go to
Tamas Cs\"org\H o and Jan Kasper for discussion of the odderon contribution
and elastic scattering during the Low x'2019 WS.
This research was supported by
CONICYT PIA/BASAL FB0821(Chile) and Fondecyt (Chile) grants
1170319 and 1180118 .
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,660 |
Q: iOS App Preview videos - are all 3 iPhone resolutions required I've just completed an App Preview video for my forthcoming universal iOS app. I did it by capturing an iPhone 6 using QuickTime then editing in iMovie.
If I only submit my iPhone 6 App Preview video (750x1334), will iPhone 5/5s and 6 Plus users be able to see the video on the App Store?
Or do I have to create and upload separate iPhone 5/5s & iPhone 6 Plus videos?
Thanks.
A: In the end I only uploaded 3 videos - iPhone 5 resolution, iPhone 6 resolution and iPad resolution. When I visited the app on the App Store on an iPhone 6 PLUS - the iPhone 6 video was shown.
So it looks like larger iPhones will fallback to smaller videos if the larger preview video is not present. Presumably the iPhone 6 will fallback to the iPhone 5 video as well.
A: An App Preview is an optional short video demonstrating your app. Your app may have one App Preview per device. The specifications for App Previews are given in App Preview Properties.
Source: developer.apple.com
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,175 |
'P2' another stalker 'romance'
Tim Miller
I'd like to say "P2" should be called "P.U.," but it really isn't that bad. It just isn't very good.
The premise is simple: Thomas (Wes Bentley), a psycho overnight-parking attendant, traps Angela (Rachel Nichols), an attractive businesswoman, in the underground garage of an office building.
It's Christmas Eve, you see, and Thomas thinks it's the perfect time to declare his love for Angela during a nice, quiet dinner in his small office. So when she's the last to leave the office, can't start her car (hmmm ...) and comes to him for help, he gives her the big invite. She looks at him like he's out of his skull, so he acts like he was just kidding. But soon he drugs her, and when she wakes up she finds she's down to her undergarments and chained to a table, with Thomas' trusty dog growling and snapping at her. Thomas, meanwhile, is sitting across from her — and dressed like Santa Claus.
Ah yes, romance.
From there you pretty much get what you expect: She eventually escapes, but is still trapped in the parking lot, with Thomas and his pup chasing her around. It's intense, it's a bit scary, it's one-note.
The performances are better than a story like this deserves.
Nichols doesn't play Angela as a sweet innocent turned victim, as you might expect, but as a rather cold young woman who would look right through a person in Thomas' position. Tears stream down her face, but they're more out of fear and anger than self-pity, and that somehow makes Angela seem different from the typical movie victim in these circumstances.
Bentley, who you'd think after "American Beauty" would have better material to work with, is a little more predictable as the crazed suitor, and there are times when his nuttiness seems a bit forced — as when he does an impersonation of Elvis singing "Blue Christmas" over the garage loudspeaker. Overall, though, he's convincing as a guy who thinks he's Mr. Right and is so, so wrong.
Which, if you want to bother to think about it, is at the heart of what this story is really about. Often in movies about unrequited love, we sympathize with the person who feels the passion, not the person who inspires it. In "The Graduate" (to reach back a bit), most moviegoers identify with, or at least root for, Benjamin, not Elaine. But is passion for another person enough? The romantic side of us might want to say yes, but then we'd be rooting for Thomas here — a chilling thought. "P2," like "The Collector" or "Play Misty for Me" or "Misery" before it, reveals the dark side of longing, the side in which desire is purely selfish, and turns to bullying, with no regard for what's best for the supposed loved one.
I have no idea whether the makers of "P2" — first-time writer-director Franck Khalfoun and co-writers Alexandre Aja and Gregory Levasseur, who both worked on the remake of "The Hills Have Eyes" (yuck) — thought about any of this, or whether they just thought, "Hey, let's try to scare people."
Whatever their intentions, they've come up with a horror flick that's almost instantly forgettable — until, at least, the next time you walk into an empty parking garage.
Tim Miller is the Times' entertainment editor.
What: "P2"
Star rating: HH (out of four)
Starring: Rachel Nichols and Wes Bentley
Directed by: Franck Khalfoun
Written by: Khalfoun, Alexandre Aja and Gregory Levasseur
Rating: R (for strong violence/gore, terror and language)
Where: Entertainment Cinemas in South Dennis, and Regal Cinemas in Harwich, Hyannis (Cape Cod Mall) and Mashpee (Mashpee Commons) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,418 |
February 4, 1969: Music Box, Omaha
CROWD JOINS ACT OF GRATEFUL DEAD
In tones sometimes strident and often satiric, and nearly always driven with fantastic force, The Grateful Dead entertained an estimated eight hundred persons Tuesday night at the Music Box.
The "acid rock" performance was sponsored by KOWH-FM which hopes to bring in other groups, said Tom Rambler, program and music director.
The Tuesday night performance by the seven-member Grateful Dead from San Francisco was "theater" rather than "concert" - with the audience giving as much as it took, adding to the evening's drama.
Some youths sat on the ballroom floor.
Others ringed the balcony, their feet - sometimes bare but more often wrapped in boots - dangling over the edge.
Men in the audience wore everything from Edwardian jackets to Army surplus field jackets. Many had hats - mostly black and frequently western. The women came in everything from capes to lounging pajamas to the simplest of skirts and blouses with thigh-high boots.
A haze of smoke from cigarettes and sticks of incense hung over the performers and audience.
About the only lighting came from a few dim bulbs around the floor's edge and brightly lighted, multi-colored panels behind the stage.
The Grateful Dead was preceded on stage by a group from St. Louis called the Unknown.
The Unknown completed its session with a long farewell number begun by having the audience chant "peace, love and freedom."
Then came the highly amplified Grateful Dead: Jerry Garcia, lead guitar; Phil Lesh, bass guitar; Bob Weir, rhythm guitar; "Pigpen" McKernan, vocals; Micky Hart and Bill Kreutzman, drums; and Tom Constanten, organ.
After one or two numbers broken by brief pauses and appeals for water or soft drinks to quench their thirst, The Grateful Dead lunged forward into an "unending" series of complex renditions that went from the blues to even a brief flirtation with Latin rhythms and lasted at least 30 minutes without pause.
(by Gerald Wade, from the Omaha World-Herald, 5 February 1969)
https://archive.org/details/gd69-02-04.sbd.barbella.7294.sbeok.shnf
After a year of being out of the picture, the rock group that started it all, the Grateful Dead, is back on concert tour. Three weeks ago I had the privilege of hearing them in Omaha in an evening concert which was unlike anything ever witnessed in Lincoln.
The concert was held at the Music Box, the one time posh dance hall of Omaha's elite. The audience spaced out on the floor in an atmosphere of subdued lights and the fragrance of incense...and they waited, and waited.
Five musicians from St. Louis, the Unknown, started the music at 8:15 and layed a variety of old and new hard rock. They brought the house down with a song called "Don't trust your woman with your grass," a highly country and western-flavored number.
After a brief intermission, the nine musicians came on stage and began sorting out the mass of equipment, each choosing his favorite instrument, and began to tune. To the beat of two complete sets of drums as well as an organ, three guitars and assorted percussion effects, they began their first epic piece of music.
Approximately three songs and some 30 minutes later they broke into a rendition of "Turn on your love light," and invited everyone in the group to "get up on your feet." It didn't take long for a large circle of dancers to form and start expanding through the crowd, absorbing new members as it grew. This wild group then began to snake through itself in a backbreaking routine that kept everyone jumping, all to the strains of music.
Nothing more welcome than an intermission which allowed everyone to relax before the band played on well into the night.
Since that night the air has been filled with rumors about the next concert, when and where. The where is easy, it will be at the Music Box again. When? Next month...and who? The official word now has it that it will be the Vanilla Fudge, or most likely, The Rotary Connection.
The Rotary Connection broke it up at Christmas with an album of Christmas carols which told it like it was, rock and all. Their second album, Aladdin, has been billed as the first space operetta ever recorded. Their record sales are jumping and, accordingly, they are in great demand.
Watch this column for the exact time and place, and how much bread you should set aside in preparation and anticipation of a big night of rock in Omaha.
(by J.L. Schmidt, from the Daily Nebraskan, Lincoln, 27 February 1969)
The World-Herald also ran a short review of Live-Dead at the end of the year:
Many performers seem to freeze when they get near a recording microphone and aren't able to create the quality music for which they are known.
The records of the Grateful Dead, one of the best known of San Francisco's rock groups, are a case in point. Seldom do the group's albums convey force and purpose.
Most of the combo's best recorded moments have come in live performances, so the Dead's new, two-record album "Live-Dead" (Warner Bros. - Seven Arts 1830) is among its best. However, the music still lacks the immediacy one comes to expect.
Light Into Ashes October 5, 2017 at 6:15 PM
This was reviewed by an older mainstream reporter, as evidenced by his close attention to the clothes of the audience and the shady atmosphere. Clearly "acid rock" wasn't his main beat; he's mainly struck by how loud, strident and forceful the Dead are, and says it was more theater than concert. (He doesn't quite describe how the audience "joined the act" and "added to the evening's drama," unless it was just by their appearance.)
He does note that after the first couple numbers (Schoolgirl & Dew), the Dead "lunged forward into an unending series of complex renditions," the Dark Star suite, from which he could make out some blues (Death Don't) and some Latin rhythms (probably the Eleven). When the "unending" music "lasted at least 30 minutes without pause," I'd guess that it wore him out and he left early to write his review (unless it was cut short for the paper), since he doesn't mention that they played yet another half-hour suite after that!
He's a careful reviewer, though - though it's evident he didn't much like them, he actually doesn't say anything negative - he's reporting, rather than judging.
Deadlists says the opening group was the Liberation Blues Band (a Nebraska band), but this review says otherwise. The Unknown are, well, unknown to me; but my guess is the LBB actually opened for the Dead on their return to the Music Box on 4/15/69.
The Live/Dead review isn't really notable (or enthusiastic), just saying that the Dead are better live than on their records, so the live album is an improvement but still "lacks immediacy."
Light Into Ashes October 19, 2017 at 3:48 PM
I found another review to add! From the Daily Nebraskan, the University of Nebraska student paper.
It's a massive difference - this time from a young reviewer who enjoyed the atmosphere at the Music Box, appreciated the "spaced out" audience, subdued lights & incense, and couldn't wait for the next show there.
He calls the Dead "the rock group that started it all" - I'm a little puzzled by his saying there were nine musicians (newspaper error?), or that they were "back on tour" after being "out of the picture" for a year, but those are small points.
What's really exciting is that he shows there's a massive cut in the tape - he says after three numbers (about 30 minutes), they played a Lovelight that got the crowd dancing, then took an intermission before a long second set.
Our tape cuts off with Death Don't Have No Mercy, picking up with the Other one suite. Lovelight must have followed Death, and the Other One presumably started the second set.
He misjudges the time - Lovelight would have come over an hour into the set, after a lot more than three numbers - but there's no way he could mistake the song, or its effect on the audience. This is a nice description of how the Dead got the crowd on their feet - and it explains why Lovelight kept getting longer through the year. (Circle dances can't be appreciated on tape.)
The same writer also did a review of the Dead's 4/15/69 return to the Music Box, which I'll post next.
July 8, 1970: Mississippi River Festival, Edwardsv...
April 17, 1969: Quadrangle, Washington University,...
April 11, 1969: University of Arizona, Tucson
July 4, 1969: Kinetic Playground, Chicago
July 11, 1969: New York State Pavilion, Flushing M...
April 15, 1969: Music Box, Omaha
July 9, 1970: Fillmore East
June 19, 1970: Mid-South Coliseum, Memphis
July 1970: Warner Brothers Promotion
January 23-24, 1970: Civic Auditorium, Honolulu
February 23, 1970: Municipal Auditorium, Austin
February 22, 1970: Coliseum, Houston
July 8, 1969: Rock Pile, Toronto, Ontario
February 11-12, 1969: Fillmore East
August 5, 1967: O'Keefe Center, Toronto, Ontario | {
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Oxylamia chopardi är en skalbaggsart som först beskrevs av Jean François Villiers 1942. Oxylamia chopardi ingår i släktet Oxylamia och familjen långhorningar. Inga underarter finns listade i Catalogue of Life.
Källor
Långhorningar
chopardi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,059 |
Le Curé de Cucugnan est un sermon recueilli par Auguste Blanchot de Brenas en 1858, qu'Alphonse Daudet a rendu populaire sous la forme d'une nouvelle publiée dans L'Événement du , puis dans les Lettres de mon moulin en 1869. Daudet y traduit Lou curat de Cucugnan de Roumanille.
Résumé
À Cucugnan, dans l'Aude, la foi n'est plus présente. Le curé raconte dans un sermon qu'il a rêvé qu'il allait au Paradis puis au Purgatoire et n'y trouvait pas les habitants décédés de Cucugnan ; il les a trouvés en Enfer. Il fait alors le projet de confesser tout le village et de redonner la foi à tous les habitants.
Auteurs
En 1858 un jeune voyageur, Auguste Blanchot de Brenas, entend le sermon dans un village des Corbières. Il relate son voyage sous la forme d'un feuilleton publié sous le titre Avec mon ami Félix dans l'hebdomadaire La France littéraire, artistique et scientifique. Le sermon du curé de Cucugnan apparaît dans le numéro du . Blanchot y affirme que la scène se situe dans un hameau où la ferveur était en décroissance et qu'il appelle Cucugnan. Il précise en note que « l'anecdote n'a pas eu lieu à Cucugnan : ce nom a été pris au hasard pour ne froisser aucune susceptibilité ».
En 1866, le félibre Joseph Roumanille rédige une version provençale du texte de Blanchot de Brenas qu'il publie dans l'Armana prouvençau sous le titre Lou curat de Cucugnan.
Alphonse Daudet traduit le texte de Roumanille qu'il publie la même année, accompagné de l'incipit suivant : « Tous les ans, à la Chandeleur, les poètes provençaux publient en Avignon un joyeux petit livre rempli jusqu'aux bords de beaux vers et de jolis contes. Celui de cette année m'arrive à l'instant, et j'y trouve un adorable fabliau que je vais essayer de vous traduire en l'abrégeant un peu… ». Il ajoute dans l'explicit : « Et voilà l'histoire du curé de Cucugnan, telle que m'a ordonné de vous la dire ce grand gueusard de Roumanille, qui la tenait lui-même d'un autre bon compagnon ». Daudet a raccourci le texte de Roumanille, omettant un passage où est décrit le stratagème utilisé par le curé pour que tout le village vienne écouter son sermon (la découverte d'un trésor).
La version de Daudet devient immédiatement célèbre. Aussi Blanchot de Brenas réclame à Daudet et Roumanille la paternité du texte. Sans réponse de leur part il menace Roumanille d'un procès pour plagiat. Celui-ci arrive à faire traîner les choses et échappe au procès grâce à la mort de Blanchot en 1877.
Depuis, de nombreuses versions ont vu le jour, presque toutes inspirées du texte de Daudet, notamment celle de l'audois Achille Mir, dans le tome 3 de ses œuvres complètes, Countes en proso e en vèrs, sous le titre Lou sermou dal Curat de Cucugna en 1884 et celle de Frédéric Estre, sous le titre Lou curat de Cucugnan en prouvençau en 1878. Quant à Blanchot de Brenas, il reste oublié et sa paternité du texte souvent contestée à tort. Victime d'un véritable plagiat que Roumanille a reconnu, Blanchot de Brenas n'a cependant pas inventé le sermon puisqu'il l'a recueilli auprès d'un habitant des Corbières. Mais il l'a rédigé à sa manière et l'a doté d'un titre amusant qui a contribué au succès de l'histoire.
Car ce sermon est un récit exemplaire que les curés des Corbières racontaient sous diverses variantes. Charles Pélissier affirme que l'abbé Ruffié, curé de Cucugnan au milieu du , aurait prononcé en chaire un sermon de la même veine. On connaît par ailleurs une variante moins connue du sermon, intitulée le Sermon du père Bourras de Ginestas, recueillie dans les années 1850 par le narbonnais Hercule Birat, qui l'a adaptée et publiée en 1860. Dans le premier volume de ses Poésies narbonnaises, dans le « Cinquième entretien », l'auteur annonce à l'Aristarque, à propos de la commune de Ginestas : « Je vais travailler à un sermon que je ferai prononcer au père Bourras » et il invoque la « tradition patoise » qui exposait ainsi l'arrivée du curé aux portes du Paradis, puis du Purgatoire :
Ce thème se retrouve chez Daudet : « — Pan, pan ! — Qui frappe me fait une voix rauque et dolente. — Le curé de Cucugnan ».
Dans le deuxième volume des Poésies narbonnaises, dans le « Sixième entretien », l'auteur dit « « à l'ami lecteur » : « Tu ne prendras que ce que tu voudras de notre bavardage ; mais ne vas pas te dispenser au moins de jeter les yeux sur le sermon si pathétique et si orthodoxe du révérend père Bourras qui en fait partie car il contient des choses très-profitables ; les survivants de ces vieilles ouailles de Ginestas qui, par la négligence, sans doute, de leurs précédents pasteurs, abion toutos saoutat lou parré, s'étaient toutes échappées du bercail et y furent ramenées, sans qu'il en manquât une, par ses salutaires exhortations, peuvent en porter témoignage. » Suit quelques pages plus loin, Le Sermon du père Bourras, en octosyllabes (et en français).
Adaptations
Après l'adaptation cinématographique de trois Lettres de mon moulin en 1954 par Marcel Pagnol, Le Curé de Cucugnan fait l'objet d'une nouvelle adaptation de l'écrivain et cinéaste provençal, en 1968, sous forme de téléfilm de moyen métrage dans lequel le rôle du curé est tenu par Fernand Sardou.
Fernandel a enregistré le Curé de Cucugnan.
Articles connexes
Liste de prêtres catholiques de fiction
Bibliographie
.
Notes et références
Liens externes
Présentation du sermon et sources numérisées sur la bibliothèque Occitania.
Nouvelle française parue en 1866
Prêtre catholique de fiction
Nouvelle d'Alphonse Daudet
Enfer
Purgatoire | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,818 |
Praise for _Outsmarting Overeating_
" _Outsmarting Overeating_ does more than offer a solid, step-by-step approach to building a healthy relationship with food and eating. It offers a thoughtful, compassionate, effective path for healing your life, emotions, and relationships. Highly recommended!"
— Donald Altman, MA, LPC, author of _One-Minute Mindfulness_ and _Art of the Inner Meal_
"If you find that your relationship to food and eating is a problem and you want to find a way to change it without dieting, _Outsmarting Overeating_ is sure to add some wonderful tools to your toolbox. Karen R. Koenig's newest book elevates the self-help genre to a whole new level of writing that will benefit every reader!"
— Megrette Fletcher, MEd, RD, CDE, cofounder of The Center for Mindful Eating and coauthor of _Eat What You Love, Love What You Eat with Diabetes_
"This is a nurturing, realistic system for developing healthy life skills and then applying them to eating behaviors."
— Anna Jedrziewski, _Retailing Insight_
Praise for Karen R. Koenig's Previous Books
"What [ _The Rules of 'Normal' Eating_ ] offers is a perspective on working through some of the many misbeliefs and setbacks that prevent many of us from eating a 'normal' diet. Karen outlines what each rule would look like in real life, and gives practical advice for how to start to change lifelong habits into healthier ones."
— Rebecca Bitzer, The Nutrition Experts, EmpoweredEatingBlog.com
"Using humor, plain talk, examples from her clinical experience, reflection exercises, case studies, and homework, Koenig lets troubled eaters know that their yo-yo patterns of eating and self-care are due to conflicts. She shies away from easy answers and, instead, provides hope and concrete actions to developing a _permanent, positive relationship with food."_
_— Midwest Book Review_
"Since women, at least those of us in the Western World, are socialized to be pleasers, Karen Koenig has written a wonderful book to help us save ourselves from ourselves. . . . [ _Nice Girls Finish Fat_ ] is deceptively simple and chock-full of stories to help readers see themselves in the lessons she teaches. She is a master clinician."
— Dr. Beth Erickson, author of _Marriage Isn't for Sissies_
" _What Every Therapist Needs to Know about Treating Eating and Weight Issues_ is a wonderful tool for therapists to gain more insight on the occasional eating and weight problems in clients."
_— International Journal of Psychotherapy_
Outsmarting
Overeating
Also by Karen R. Koenig
_The Food and Feelings Workbook:_
_A Full Course Meal on Emotional Health_
_Nice Girls Finish Fat:_
_Put Yourself First and Change Your Eating Forever_
_The Rules of "Normal" Eating:_
_A Commonsense Approach for Dieters, Overeaters, Undereaters_ ,
_Emotional Eaters, and Everyone in Between!_
_Starting Monday:_
_Seven Keys to a Permanent, Positive Relationship with Food_
_What Every Therapist Needs to Know
about Treating Eating and Weight Issues_
| New World Library
14 Pamaron Way
Novato, California 94949
---|---
Copyright © 2015 by Karen R. Koenig
All rights reserved. This book may not be reproduced in whole or in part, stored in a retrieval system, or transmitted in any form or by any means — electronic, mechanical, or other — without written permission from the publisher, except by a reviewer, who may quote brief passages in a review.
The material in this book is intended for educational purposes only. No expressed or implied guarantee of the effects of the use of the recommendations can be given nor liability taken.
Text design by Tona Pearce Myers
Library of Congress Cataloging-in-Publication Data
Koenig, Karen R., date.
Outsmarting overeating : boost your life skills, end your food problems / Karen R. Koenig, LCSW, MEd.
pages cm
Includes bibliographical references and index.
ISBN 978-1-60868-316-1 (paperback)
1. Eating disorders—Psychological aspects. 2. Food—Psychological aspects. I. Title.
RC552.E18K634 2014
616.85'26—dc23 2014034245
First printing, January 2015
ISBN 978-1-60868-316-1
Printed in Canada on 100% postconsumer-waste recycled paper
| |
New World Library is proud to be a Gold Certified Environmentally Responsible Publisher. Publisher certification awarded by Green Press Initiative. www.greenpressinitiative.org
---|---|---
10 9 8 7 6 5 4 3 2 1
_For my mother_
## Contents
Introduction
Life Skills Preassessmen
Chapter 1. The Definition and Purpose of Life Skills
_Is Excelling at Cleaning My Plate a Life Skill?_
Chapter 2. Wellness and Physical Self-Care
_You Mean My Body's Not Like a Self-Cleaning Oven?_
Chapter 3. Handling Emotions
_I Thought That's What a Spoon and Fork Were For!_
Chapter 4. Living Consciously
_I'm Conscious Only of Wanting to Go Unconscious!_
Chapter 5. Building and Maintaining Relationships
_I Already Have a Great Relationship. . .with My Refrigerator!_
Chapter 6. Self-Regulation
_There's Something Besides an On-Off Switch?_
Chapter 7. Problem Solving and Critical Thinking
_Is Critical Thinking Different from Thinking Critically about Myself?_
Chapter 8. Setting and Reaching Goals
_What If I Can't Get There from Here?_
Chapter 9. Balancing Work and Play
_All Work and No Play Makes Jack. . .Crave a Snack!_
Life Skills Postassessment
Chapter 10. Integrating Life Skills into Eating "Normally"
_I Get It — Gain the Life Skills, Lose the Food Problem!_
Acknowledgments
Notes
Index
About the Author
## Introduction
I bet you picked up this book with a groan of disbelief, thinking, here you are, _again_ , reading _yet another_ book to improve your eating habits. Perhaps you have been struggling for decades to make peace with the refrigerator and the scale. You forge a bit of progress, only to see it vanish into thin air as a crisis strikes or as you get fed up with working so darned hard to eat "normally." You may be sick to death of thinking about food — all the "shoulds" and "shouldn'ts," all the instructions telling you to "eat this" and "don't eat that" — that you're ready to throw up your hands in despair and accept your relationship with food, no matter how crummy it is.
Almost. The fact that you're reading these words means you still feel a spark of optimism — that maybe this book will turn your eating habits around once and for all, that it will tell you why you keep snacking when you're not hungry, eating past the point of being full, obsessing about food and weight, and choosing foods that endanger your health and happiness. So here you are, hoping against hope that this book will tell you how to put your eating problems behind you and leave them there.
It will! — because _Outsmarting Overeating_ isn't a diet book. It's not even primarily about eating. It's about the rest of your life, the part that has been mucking up your relationship with food. Read on to discover the _real_ reasons you have an eating problem and to learn a tried-and-true approach to putting food in its rightful place while simultaneously creating a better life for yourself.
During my three decades of working with troubled eaters, it's become glaringly obvious to me that a person's misguided relationship with food is a symptom of deeper, darker problems. The truth is, biology and heredity aside (and both can play a huge role in determining eating patterns and weight parameters), if you engage in unwanted, unhealthy eating with such frequency and ferocity that it ruins your self-esteem and damages your health, you're likely lacking a set of essential skills needed to better manage your life. Sure, you can benefit from nutritional information and behavioral techniques that will help you eat more healthfully. But you'll never succeed if the skills you use to cope with life's ups and downs are so weak and ineffective that eating has become your major strategy for facing your daily challenges, and if food is your greatest source of pleasure, most loved reward, most pursued passion, day's highlight, and bosom buddy.
Troubled eaters like you engage in mindless eating because that's the best way — the _only_ way — you know how to get through the day, the week, the weekend, your life!
Though desperate to improve your relationship with food, you won't succeed merely by using the tired, old, well-meaning advice to count calories or fat grams, shop only the supermarket's perimeter, cut food portions way back, and weigh in daily. That's like expecting alcoholics to know how to deal with life without taking a drink, when they've always dealt with life by tossing down a few. Where, pray tell, would such know-how come from if alcohol has been keeping them afloat for years or decades? Where would it come from if food is the crutch you've been using to limp through life?
Attempts to eat better won't amount to a hill of beans unless you _also_ possess essential skills to maneuver through life effectively and successfully — to beat the blahs and the blues, manage stress, achieve goals, establish and maintain functional relationships, take care of yourself physically and emotionally, think rationally, and create a passionate, meaningful life that makes you want to get out there and live rather than rent space in your refrigerator. Without new, effective strategies for being good to yourself and interacting with the world, you're condemned to old, destructive eating patterns no matter how grand your intentions or how strong your motivation. Without acquiring essential _life skills_ that you lack, you're doomed to continue struggling with food and the scale and to remain unhappy, unhealthy, and unfulfilled.
What are life skills? According to the World Health Organization, they're "abilities for adaptive and positive behaviour that enable individuals to deal effectively with the demands and challenges of everyday life." Based on the list of skills that WHO developed, I've created a select skill set targeted specifically to troubled eaters. The life skills taught in this book are the same ones that I teach in my therapy and coaching sessions, my lectures and workshops, and I have touched on them in previous books. They are attitudes and actions that we're fortunate to learn in childhood. But many of us didn't learn them then because our parents taught us from their own, often dysfunctional, histories, distorted perspectives, limited knowledge base, and imperfect abilities.
Life skills are the missing piece of the eating-disorder puzzle and can be learned by anyone at any age. Bulletin: It's not just troubled eaters who need these skills; we _all_ need them.
This book will teach them to you. Read on and you'll learn how to relax and let loose; soothe your ruffled emotions; sustain your motivation and achieve your goals; make effective decisions without driving yourself crazy before, while, and after you make them; create and maintain healthy relationships; find purpose and passion that suit your unique talents, interests, and abilities; and take such exquisite care of your body and mind that you wouldn't think of using food or inadequate self-care to harm them.
This book covers eight essential life skills, explaining the purpose of each one and giving examples to prove their necessity. It also illustrates how lacking each skill propels you toward unnecessary eating, and how, in turn, unwanted eating prevents skill development. For example, with poor problem-solving skills, you might raid the pantry at the first whisper that your company may start downsizing your department. Thrown into a panic, you may feel so overwhelmed and paralyzed that your brain stops working, while your desire for sweets and treats shifts into overdrive. And while you're chowing down, you're missing an opportunity to learn how to truly solve the problems posed by the possibility of losing your job.
Here are a few more examples. If you find your mind drifting back toward the past or zooming off into the future on a regular basis without your explicit permission, you may feel so full of regret or anxiety that you miss each marvelous moment of today. And, if you're not present when you're eating (or grocery shopping or preparing food), it's unlikely that you're going to attend to the task of feeding your body exactly what it wants in a quantity that suits it. By learning the skill of anchoring your mind to the present, you'll be able to consider the past or future consciously for specific reasons, then quickly recenter yourself in the here and now when you're eating or doing other activities.
If you've grown into an adult who believes she can't depend on people and tries to handle everything alone, you're at a distinct disadvantage when life gets rough-and-tumble, and your likelihood of turning to food for comfort and stabilization is good to excellent. When you're more skilled in trusting and depending on people — the right people, the ones who'll be there for you gladly and with wisdom — you'll find real comfort and security and won't even think about turning to food when you're in distress.
Here's a bit of information about the format of this book. Before you begin reading chapters, you'll find the "Life Skills Preassessment" questionnaire — a set of sixty questions to help you evaluate your life-skill strengths and weaknesses. At the end of the book, you'll find its companion, the "Life Skills Postassessment" questionnaire, which will show you the progress you've made and your remaining proficiency challenges.
Chapters include "Get Smart!" questions, which provide information about how you're doing with a particular life skill, help you identify barriers to gaining expertise, and teach you how to speed up skill acquisition. You'll also find "Skill Booster" activities at the end of every chapter, which will keep you practicing life skills between chapters.
After practicing these life skills for a while (okay, a _long_ while), they'll become more automatic. You'll be less stressed, calmer, happier, centered, and well-balanced, as well as more competent to face life head-on. Once these skills are in place and have become part of your repertoire, you'll find them so powerful and effective for living your best life that food abuse will gradually become a distant memory. As author Maya Angelou succinctly puts it, "You did the best that you knew how. Now that you know better, you'll do better."
## Life Skills Preassessment
_Life skills are abilities for adaptive and positive behaviour that enable individuals to deal effectively with the demands and challenges of everyday life_.
— World Health Organization
This questionnaire is designed to help you recognize your life-skill proficiencies and deficiencies — that is, to gauge strengths and weaknesses in managing life effectively and successfully. At the end of this book, you will find a Life Skills Postassessment that will help you determine what you've learned and which specific, challenging skill sets need more attention.
Although there are no right or wrong responses to the statements in this questionnaire, please give each one some thought before answering, and be honest. No judging yourself, puh-lease, if you're not as competent as you'd like to be. Instead of being hard on yourself, stay curious and maintain a mind-set that lets you know you'll feel a good deal better about your skills after doing the postassessment than you do right now. Remember, you're reading this book for self-knowledge that will help you permanently repair your relationship with food. And it will, I promise!
**Instructions:** Circle the number that best describes your response to each statement, with the number 1 representing _least true_ and 10 representing _most true_.
1. Overall, I have effective life skills.
1 2 3 4 5 6 7 8 9 10
2. I surround myself with people who have effective life skills.
1 2 3 4 5 6 7 8 9 10
3. Improved life skills would help me eat "normally" and attain and maintain a healthy weight.
1 2 3 4 5 6 7 8 9 10
4. I take excellent care of my health.
1 2 3 4 5 6 7 8 9 10
5. I have routine medical tests, follow doctors' orders, and take care of emergency medical concerns right away.
1 2 3 4 5 6 7 8 9 10
6. I get sufficient sleep most nights.
1 2 3 4 5 6 7 8 9 10
7. I take vitamins and supplements or medication consistently.
1 2 3 4 5 6 7 8 9 10
8. I get exercise (formal or informal) on a regular basis.
1 2 3 4 5 6 7 8 9 10
9. I am generally in touch with and can identify my feelings.
1 2 3 4 5 6 7 8 9 10
10. I value and am willing to experience all my feelings.
1 2 3 4 5 6 7 8 9 10
11. Rather than judging them, I am curious about my feelings.
1 2 3 4 5 6 7 8 9 10
12. I can tolerate intense and uncomfortable or conflicting feelings.
1 2 3 4 5 6 7 8 9 10
13. I express emotions appropriately and effectively.
1 2 3 4 5 6 7 8 9 10
14. I comfort and calm myself effectively.
1 2 3 4 5 6 7 8 9 10
15. For the most part, I live consciously and in the present.
1 2 3 4 5 6 7 8 9 10
16. I don't spend time unnecessarily worrying about the past.
1 2 3 4 5 6 7 8 9 10
17. I don't spend time unnecessarily worrying about the future.
1 2 3 4 5 6 7 8 9 10
18. I know that whatever befalls me in life, I will manage.
1 2 3 4 5 6 7 8 9 10
19. I plan for the future, then let it take care of itself.
1 2 3 4 5 6 7 8 9 10
20. I am comfortable in most social situations.
1 2 3 4 5 6 7 8 9 10
21. I am generally honest and share my feelings with intimates.
1 2 3 4 5 6 7 8 9 10
22. Intimates care about me as much as I care about them.
1 2 3 4 5 6 7 8 9 10
23. I set boundaries with intimates, and they respect them.
1 2 3 4 5 6 7 8 9 10
24. I take care of myself as well as I take care of others.
1 2 3 4 5 6 7 8 9 10
25. I'm good at knowing who to trust and who not to trust.
1 2 3 4 5 6 7 8 9 10
26. I know how to make and keep wonderful friends.
1 2 3 4 5 6 7 8 9 10
27. I'm pretty even emotionally and avoid emotional extremes.
1 2 3 4 5 6 7 8 9 10
28. I generally know when enough is enough.
1 2 3 4 5 6 7 8 9 10
29. I don't tend to frequently overdo or underdo.
1 2 3 4 5 6 7 8 9 10
30. I don't depend on others to regulate me emotionally.
1 2 3 4 5 6 7 8 9 10
31. When I'm with difficult people, I stay on an even keel.
1 2 3 4 5 6 7 8 9 10
32. I don't think in all-or-nothing, black-or-white terms.
1 2 3 4 5 6 7 8 9 10
33. I value both structure and freedom.
1 2 3 4 5 6 7 8 9 10
34. I'm skilled at troubleshooting and problem solving.
1 2 3 4 5 6 7 8 9 10
35. I'm neither too cautious nor impulsive in decision making.
1 2 3 4 5 6 7 8 9 10
36. I don't second-guess myself after making a decision.
1 2 3 4 5 6 7 8 9 10
37. I don't put off decisions because they're hard to make.
1 2 3 4 5 6 7 8 9 10
38. I have confidence in my critical-thinking skills.
1 2 3 4 5 6 7 8 9 10
39. I generally reach my goals.
1 2 3 4 5 6 7 8 9 10
40. I'm good at creating concrete, realistic goals for myself.
1 2 3 4 5 6 7 8 9 10
41. I know how to divide big goals into smaller ones.
1 2 3 4 5 6 7 8 9 10
42. I'm skilled at sustaining the effort needed to reach my goals.
1 2 3 4 5 6 7 8 9 10
43. I'm good at maintaining my hard-won achievements.
1 2 3 4 5 6 7 8 9 10
44. I don't sabotage my progress or achievements.
1 2 3 4 5 6 7 8 9 10
45. I recognize that all progress consists of baby steps.
1 2 3 4 5 6 7 8 9 10
46. My goals are composed of my wants, not "shoulds."
1 2 3 4 5 6 7 8 9 10
47. I can easily ask for or accept help in reaching my goals.
1 2 3 4 5 6 7 8 9 10
48. I always speak kindly to myself about myself.
1 2 3 4 5 6 7 8 9 10
49. I don't let other people be unkind or hurtful to me.
1 2 3 4 5 6 7 8 9 10
50. I don't have to be perfect.
1 2 3 4 5 6 7 8 9 10
51. I value myself and expect others to value me too.
1 2 3 4 5 6 7 8 9 10
52. I both love myself unconditionally and strive to do better.
1 2 3 4 5 6 7 8 9 10
53. I'm generally more curious about, than critical of, my mistakes.
1 2 3 4 5 6 7 8 9 10
54. I take failure in stride and learn from it.
1 2 3 4 5 6 7 8 9 10
55. I have a stable sense of myself.
1 2 3 4 5 6 7 8 9 10
56. I live a life balanced between obligations and play.
1 2 3 4 5 6 7 8 9 10
57. I can relax and let go fairly easily in healthy ways.
1 2 3 4 5 6 7 8 9 10
58. I know when I'm ready to work or play or relax.
1 2 3 4 5 6 7 8 9 10
59. I rarely procrastinate or rebel against commitments.
1 2 3 4 5 6 7 8 9 10
60. I live purposely and know the meaning of my life.
1 2 3 4 5 6 7 8 9 10
## CHAPTER 1
## The Definition and Purpose of Life Skills
## Is Excelling at Cleaning My Plate a Life Skill?
You may think food is your main problem, but it isn't. Rather, food is the misguided _solution_ to your real difficulties, and over the years misguided eating has morphed into a whopper unto itself. The truth is, your problem is that you never learned the skills and strategies everyone needs in order to live effectively and successfully. Think about it: you need skills for employment, relationships, parenting, and living in a community, for play and recreation, driving a car, and balancing your checkbook. No activity I can think of precludes having _some_ degree of competence — not even tying your shoes!
So, agreed: everyone needs life skills? Of course, the bad news is that simply wishing for them won't make them magically appear. The good news is that these skills are learnable by anyone at any age at any time. We're _all_ learning them to one degree or another as we muddle along, so join the crowd.
What are these must-have life skills? They're a set of universal competencies we all need to learn and practice to get the best out of life, rather than letting life get the best of us. Call them strategies or methods; tactics, tools, or techniques; competencies or abilities. What they boil down to are the essential maneuvers human beings must employ to engage with life successfully.
Specifically, here are the five basic life skills that span every culture, as identified by the World Health Organization's Department of Mental Health: (1) decision making and problem solving, (2) creative thinking and critical thinking, (3) communication and interpersonal skills, (4) self-awareness and empathy, and (5) coping with emotions and coping with stress.
I've taken the liberty of devising skill sets targeted to an audience of troubled eaters and focusing on the expertise they often lack regarding (1) wellness and physical self-care, (2) handling emotions, (3) living consciously, (4) building and maintaining relationships, (5) self-regulation, (6) problem solving and critical thinking, (7) setting and reaching goals, and (8) balancing work and play.
* * *
Get Smart!
Do you still think you have only eating problems, or are you starting to recognize that your difficulties are due to not managing life all that well? Does that make you more, or less, optimistic about developing a positive, healthy relationship with food? Come up with a sentence that will put you in a positive frame of mind for reading this book and learning life skills.
* * *
### What Are Essential Life Skills, and Where Do They Come From?
According to the 2003 _World Book Dictionary_ , a skill is defined as an "ability gained by practice or knowledge; expertness." The most significant aspect of this definition is that a skill is something _gained by practice_. Like money, skills don't grow on trees, ready for you to pluck them off. Knowledge or expertise comes at the _end_ of a process, not at the beginning, and life skills do not miraculously appear through wishin' and hopin' and thinkin' and prayin'. Unfortunately, many troubled eaters expect to excel at a task after one attempt or a few brief learning forays. Others believe that no matter how motivated they are and how much they practice, they'll never learn the skills others possess, because there's something inherently defective and unfixable about them.
The question of how life skills originate is complex. Certain talents and proclivities — art, music, math, language, dance, sports, writing, and so on — come to us through genetics. For example, I have two artist friends, one of whom is the son of a prolific portrait painter. The child of these two friends could, at eight years of age, outdraw and outpaint the adult me by a mile. We all know, or know of, families whose members shine in a particularly gifted way, and we generally accept that a certain amount of talent is innate. Alternatively, perfectly ordinary parents at times produce the most extraordinarily endowed children, and we wonder how _that_ happened.
Although it's clear that heredity plays some role in our abilities, it's impossible to pinpoint how much is due to nature (genetics) and how much is due to nurture (socialization) — and I'm not here to debate or resolve the question. A useful way to think of the process is nature _via_ nurture — that is, our environment helps us express the genetic tendencies with which we're born. In the end, we all have to make the best of what we bring into the world and how it's shaped by our personal history — by our parents, extended family, schooling, geography, social status, finances, race, ethnicity, gender, fortune and misfortune, culture, and other factors.
* * *
Get Smart!
Take a look at my life-skills list and consider how your parents' skills stacked up. Can you see how you came to lack certain skills because of your upbringing? The idea is not to blame your parents (or yourself) but to understand how your deficits came to exist. It's a simple process of cause and effect. Remember, you didn't choose to miss out on learning essential life skills!
* * *
It would be naive to believe that life skills are learned on an even playing field. Obviously, if our parents were replete with these skills, especially those that improve a person's parenting abilities, we will be far better off than if they bumbled and stumbled through child rearing. Although I'm no sociologist, I can say from practicing psychology for more than thirty years, and from living on the planet for more than twice that long, that no particular race, gender, social class, ethnicity, or locale produces people with more effective life skills than any other.
Here's a case in point. When I was doing my first internship in social work, I worked with a lovely couple, both of whom had schizophrenia. In spite of their illness and living a substantial distance apart, these twenty-somethings were in love and seriously committed to each other. They both had low-level jobs, took their psychiatric medications consistently, attended therapy appointments regularly and on time (in spite of having to travel by public transportation in Boston), worked their tails off in sessions, and, in my presence at least, treated each other with respect and kindness. On the whole, although limited in some ways by mental illness, they had some pretty darned good life skills.
Compare this couple to psychiatrists who have sexual relations with their patients, to parents who charge onto the field where their child is playing a sport to berate a coach or an umpire, or to legislators shouting abuses at each other in the halls of Congress. Get my point? These examples show how people we believe _should_ have better life skills very often don't. Moreover, I've met folks who had it fairly easy coming up who grew into adults with weak life skills, and survivors of the most egregious childhood abuses who manage their lives impressively by anyone's standards.
As far as I can ascertain, nothing inherent in anyone's upbringing is a surefire indicator that they're going to have stellar life skills. If I had to cite just one factor, however, it would be the life-skill level of a child's parents. What a boy or girl sees modeled at home goes a long way toward teaching him or her how to engage effectively with the world. Moreover, a child whose parents manage their own lives well will undoubtedly receive better treatment and socialization than a child whose parents' skill sets are spotty and ineffectual.
However, even that is not the whole story. Let's say a child grows up in a household in which her parents move from low-paying job to low-paying job, manage money poorly, drink or do illicit drugs regularly, scream at each other and their children, and mindlessly careen through life, bouncing from one crisis to another. Does this automatically mean that their child is doomed to miss out on learning how to negotiate life? Not necessarily, if this child has a neighbor, teacher, or close relative who cares enough and has sufficient life skills to help shape the child's attitudes and behaviors. Sometimes, all a child needs is one caring, competent person, called a mentor, to make the difference between growing up with or without effective skills. Of course, the longer that children blunder through life without observing or being taught appropriate ways of managing it, the harder it will be to reverse their habits. Hard, mind you, but never impossible.
* * *
Get Smart!
Aside from your parents, where else did you learn life skills as a child: from your favorite TV shows, books, movies, your best friend's parents, your parents' best friends, your teachers, relatives, a coach, or a religious leader?
* * *
Sometimes a child doesn't even need a real person to be guided in the right direction. A book, movie, TV series, or other story might spark an interest in or model acting appropriately. Here's a case in point. As a preteen, I angered fairly easily. Not that I had a wicked temper, but my low-grade irritation often leaked out in inappropriate ways. Then, I read _Little Women_ by Louisa May Alcott, and one of her characters, Jo, changed my life. Jo, too, was snippy and snappish and determined to get a handle on her temper. Inspired by her hard work and ultimate success, I followed in her fictional footsteps. Of course I didn't realize it at the time, but Alcott via Jo was an excellent life-skills teacher!
The truth is that most people have no clue what life skills are or whether they possess them. It's not as if folks go to bed at night and ask themselves, "Gee, how'd I do with my life skills today?" Instead, they muddle through their days dodging difficulties, winging it, hoping for change and success, and fervently wishing that enlightenment and competence would suddenly descend on them. Moreover, too many folks display their skill deficiencies by tsk-tsking about the deficits of others rather than assessing and improving their own qualities.
As far as I can tell, the key factors in developing life skills are recognizing which ones you lack, being highly motivated to acquire them, and practice, practice, practice. (More on the importance of practice in a bit.) Of course, the earlier you jump on the life-skills bandwagon, the easier learning them will be. Our brains are most malleable in childhood, when they're being shaped or "pruned" by what we (often unconsciously) learn from other people and through the experiences we encounter. But the neural circuitry of our brains is far from fixed, and we can learn new tricks at any point in our lives.
### How Does Motivation Affect My Ability to Learn Life Skills?
This is probably a good time to stop and consider how motivated you are to acquire the skills you're missing. Are you psyched, ambivalent, begrudgingly willing, or mildly enthusiastic; or do you feel as if you're dragging yourself through this book kicking and screaming? How will your drive level affect your ability to learn? What could you do to ratchet up your motivation? What do you suppose will happen if your enthusiasm remains low — or wanes?
Here are three steps to increase and sustain motivation:
**Step 1.** Recognize why you're not champing at the bit, ready to add new competencies to your repertoire, especially ones that not only will help you have a more positive relationship with food but also will undoubtedly enhance many areas of your life. Maybe you're scared that you won't succeed in learning them, that since you've already failed at reaching life goals so many times, you don't want to even bother trying. If so, let me assure you that as long as you have at least midlevel intelligence, there's nothing stopping you _but_ a fear of failure — that is, there's no earthly reason why you won't be able to learn these skills over time. We're not talking quantum physics here but the everyday actions that you see being taken by intimates and strangers. So if you fear you won't succeed, lay that misconception to rest. Most important, push that fear out of your mind, because the one reason — the sole reason — you might fail is the belief that you will.
**Step 2.** Focus on what learning life skills will get you. Break down the rewards, rather than saying, "I'll eat better" or "I'll be happier." What do those words really mean in concrete terms? Write down ten specific changes that will happen when you have improved your life skills, such as: "I will choose more appropriate friends; I won't turn to food so much when I'm stressed; I'll be better able to handle distressing emotions; I'll make wiser decisions; I'll treat myself better; I won't feel so bored and dissatisfied with life." See what I mean? Get down to specifics so you can keep in mind the rewards you'll reap by sticking with the learning process.
**Step** **3**. Stop thinking you have to learn all your skills perfectly — and right this minute. Instead, plan on letting the process inch along slowly but steadily. Helpful mantras include: "Baby steps, nothing but baby steps" and (my favorite) "I'm doing the best I can and that's all I can do." Perfection and impatience are the enemies of progress. They're the attitudes that will most likely make you think you can't learn and, therefore, cause you to stop trying. Expect learning to be frustrating, slow, and incremental, and you won't be disappointed. Cultivate realistic optimism that says your competence and expertise will come in good time — not in a short time.
Think of yourself as entering college or a training program. In your first semester, everything will be a bit new and mind-boggling and you'll feel at sea, thinking you'll never catch on to what you're supposed to learn. That's what the first semester is all about — learning how to learn — in terms of setting expectations and knowing how to pace yourself. You wouldn't expect to know, as a freshman, all that seniors know, would you? Okay, then, acknowledge that you're at the _beginning_ of learning, not at the _end_ , and don't fault yourself for not getting things right away. You're not supposed to, nor is anyone else. Learning is a process, not an event!
* * *
Get Smart!
Do you believe you must know everything right away, and that if you don't you're a failure? Do you fear that everyone else will "get it" but you won't? How do these faulty beliefs affect your ability to learn and sustain your motivation? What beliefs could you develop to ensure that your thinking about learning is helpful and won't impede your progress?
* * *
### How Do I Know What Life Skills I Need to Learn?
There are two answers to this question. The first, broad answer is that you have to learn _all_ the life skills necessary to achieve your goals. The narrower answer is that you have to learn all the ones you lack. If you're like most folks, you probably do better in some areas than others. That said, life skills aren't optional, nor can you pick and choose among them as if they're items on a menu. It makes sense that the more skills you possess and the more proficient you are at using them, the better your life — and (not incidentally) your eating — will be. I've never met anyone with a solid set of life skills who continued to hang on to their eating difficulties. Having effective life skills makes _all_ the difference.
The second and unique answer to the question of what you need to learn comes from your responses on the preassessment questionnaire. Based on the assumption that everyone needs all the itemized skills, you can identify the work you have ahead by noting the areas in which you could use improvement. Remember, there's no shame in admitting that you don't excel in every area — or in any area. If you did, you'd be doing something other than reading this book right now, wouldn't you? No one gets straight As when it comes to life skills, me included, I assure you. There's always something for us to learn and improve on. So, join the club!
### How Long Will I Need to Practice before I Feel Adept at Life Skills?
This question assumes that all adults begin their learning on a level playing field. Nothing could be further from the truth. As I said previously, some fortunate folks have an excellent life-skills education growing up (acquired through family, relatives, school, community, and mentors) and some unfortunate folks are raised in environments that model and provide abysmal skills. So the answer to how long you need to practice must take your starting point into account.
In addition to assessing your current skill level by means of the preassessment questionnaire to determine how your progress might go, you'll want to answer these questions:
• Is my motivation strong enough to power me through the tough times of gradual skill learning?
• Do I already have abilities that can transfer from one life-skill area to another? (For example, if you're good at problem solving, acquiring other competencies may happen faster.)
• Am I able to ask for support and feedback as I learn these skills, or do I believe I must bumble along alone?
• If I'm not learning fast enough to meet my unrealistic expectations, will I call myself a failure and give up, or decide to plug along until I get it?
• Will my friends, family, or coworkers be supportive of my skill-learning process, or will they intentionally or unintentionally stand in the way?
Whatever your answers, I can assure you that life-skill learning will take longer than you hope it will. What learning doesn't? But so what? Since the time will fly by anyway, you might as well be doing something constructive to improve your relationship with food and then some. Even if you master only some of the skills, you'll be ahead of the game. Even if you remain at an intermediate skill level, you'll still be better off than you were as a novice. Don't talk yourself out of being able to succeed; instead, talk yourself into it!
If you really want to know how long it takes to gain proficiency in a subject, listen to what Malcolm Gladwell has to say in his enlightening book _Outliers: The Story of Success_. In it, he cites a study by psychologist Anders Ericsson, who carefully reviewed the histories of successful violinists at the Berlin Academy of Music and concluded that those who performed the best (according to judges) spent the most time practicing. Gladwell expands on this theory by asserting that ten thousand hours is the average number of hours the violinists spent practicing their craft in order to learn it. Ten thousand hours — equivalent to practicing straight through for almost 417 days, just shy of a year and two months' worth of solid practice time! He found this to be true, across the board, among other professionals learning their craft — athletes, composers, writers, artists, and even criminals.
Ericsson's and Gladwell's point is that _the one attribute that can catapult you to success in any discipline is the amount of practice time you are willing to put into it_. That practice makes progress is a highly hopeful assertion, because it helps cancel out the unevenness of the playing field we started on. In fact, practice is good news of the tallest order when it comes to learning life skills, proving that if you're hell-bent on learning how to build and maintain relationships, and refuse to give up before you've become an ace self-regulator or critical thinker, you will succeed. Taking this concept one step further, we might say that you can't help but succeed if you continue to practice, because the behaviors we do repeatedly — good, bad, or indifferent — are the ones that become habits. And the more we do 'em, the better we do 'em.
* * *
Get Smart!
Assess your ability to remain motivated. When you slack off instead of practicing new learning, do you let yourself off the hook or, conversely, beat up on yourself? What excuses do you use when you don't want to practice? What rejoinder could you come up with to gently counter your excuses? (Remember, no "shoulds," puh-lease.)
* * *
Although there's been some research — based on the importance of certain genetic traits in learning and success — that challenges the ten-thousand-hour proficiency theory, I'd keep it in mind when you get tired of working hard to acquire a life skill, become frustrated that it's not coming easily, and want to throw in the towel and do things the old, more familiar way. If you wish to acquire new skills, whether eating or otherwise, you must keep practicing them. No matter what your history of dysfunction or trauma, if you are willing to keep on keeping on, mastery will be yours. You may be lucky — maybe proficiency will take you only nine thousand hours!
### How Does Possessing or Lacking Life Skills Affect Eating, and Vice Versa?
How does it _not_? If you take a look at the life skills covered in this book, you'll see what I mean.
#### 1. Skills for Wellness and Physical Self-Care
That's an easy one. If you don't have the concern and motivation necessary to take care of your body and keep it in excellent working order, eating unhealthily on a steady basis, or regularly overeating, isn't going to change. If you can't drag yourself off to bed when you're tired, or early enough to get a solid seven to nine hours' sleep most nights, you're more likely to have the munchies the next day, scientific studies say. If you tell yourself that you're hungry but too busy to eat breakfast, or if you usually skip lunch, you'll wind up so famished by dinner that you won't be able to prevent yourself from making poor food choices.
On the other hand, if you go to the doctor when you're sick or injured, you might head off physical problems that prevent you from being active (and you know what you'll end up doing if you're stuck in the house all day or night with nothing to do!). If you pay attention to physical messages from your body, you won't be as likely to stuff yourself with food when your body really wants rest and relaxation or to be charged up by activity. If you pride yourself in the terrific care you take of your body and your health, you won't think of engaging in abusing that well-valued body with food.
#### 2. Skills for Handling Emotions
Need I say a great deal about the necessity of having skills to handle your emotions effectively? You know well enough what happens when you feel stressed or distressed, run from emotional discomfort, are sucked back into recalling old wounds, or are keyed up about your future — and feel confused and know no road to take but the one that passes by the cupboard with the Ring Dings. Having top-notch life skills for self-soothing — containing your feelings when appropriate, distracting yourself when necessary, pacing activities and demands so you don't get overly taxed, and comfortably asking for and accepting emotional support from others — goes a long way toward helping you become and remain a "normal" eater.
When you're emotionally challenged, life can devolve into a depressing quagmire, a minefield of crises that barely let you recover from one catastrophe before you're faced with the next. When you're emotionally skilled, you learn to avoid creating crises and know you can manage and bounce back from the few that come your way.
#### 3. Skills for Living Consciously
Living on autopilot, not only do you make unwise choices, but you also generally dig yourself into the kind of deep holes from which it is difficult to climb out. You eat mindlessly, freak out, then starve yourself until you're almost too weak to pry off the top of a peanut butter jar. You say yes to requests you mean to decline, break plans at the last minute owing to exhaustion, and pinball from one day to the next without finding meaning in life or getting much enjoyment out of it. When you fail to take charge of your life, you end up feeling like a victim and treating yourself like one.
However, when you live out each moment with intention and attention, you're more engaged with the world. You don't miss danger signals warning that your relationships are not what you deserve, red flags that others can spot a mile away. Nor do you sleepwalk through life waiting and longing for some future event to rock your world. When you live consciously, you don't blame others for your poor choices, but take responsibility for growing and learning how to live up to your potential and strive to make each day better than the previous one.
As for food, when you live consciously, you eat only when you're hungry, and you recognize how to meet other needs effectively. You choose foods by considering their gustatory appeal and nutritional value. When you eat, you are mindful of every bite: you recognize how food tastes and feels in your body, realize when you are no longer hungry, and sense when you reach satisfaction. Mindful eating is an automatic outgrowth of conscious living.
#### 4. Skills for Building and Maintaining Relationships
If we weren't meant to be interdependent, there wouldn't be so darned many of us on the planet. One of the key skill sets required for improving your relationship with food is improving your relationship with people. When you have the wherewithal to discern which folks are good relationship material for friends, dates, or mates; when you possess the know-how to both listen actively and share deeply and intimately; when you can hold others accountable yet not take everything personally; when you can tell others what you expect from them and live up to what they reasonably expect of you, then food will seem like a poor substitute for heart-to-heart human bonding.
Without these skills, you can't help but be disappointed, lonely, emotionally isolated, and more likely to make poor decisions, succumb to depression, and miss out on a great deal of fun. What you're more likely to do when you're sick at heart is to turn to food and perpetuate a vicious cycle that you know all too well. When you have only tenuous relational skills, food takes center stage while real life — the good life, your best life — is taking place somewhere off in the wings.
#### 5. Skills for Self-Regulation
Whether you know it or not, if you are a compulsive and emotional binge eater, undereater, overeater, or chronic dieter, you lack adequate self-regulation skills. This means you have difficulty making small, incremental shifts in one direction or another to stay centered, to remain in balance, and to pace yourself well. I've never met a troubled eater who didn't have self-regulation difficulties regarding food and otherwise. A foolproof sign that you're not up to snuff with self-regulation is thinking and acting in all-or-nothing, extreme ways, or yo-yoing between the two, whether the subject is work, self-care, exercise, money, chores, fun, risk taking, commitment, drugs or alcohol, romance, or — you name it.
Self-regulation is a must-have skill for troubled eaters to develop. When you can sense what your mind or body is telling you and tune into "enoughness" (sufficiency), you'll be in better balance. You won't overdo, then underdo, and continue this yo-yo cycle. You won't fear excess or scarcity, too much freedom or too much structure. You'll feel more at peace and in sync with yourself and won't get exhausted ping-ponging back and forth between extremes. Plus, when you improve your skill at regulating yourself in nonfood areas, your proficiency can't help but rub off on your eating habits.
#### 6. Skills for Problem Solving and Critical Thinking
You understand what the term _problem solving_ means, but it's highly likely that you aren't sure what would be involved in acts of critical thinking. According to Dr. Ronald J. Massey, "These are sophisticated methods of assessing beliefs, opinions, and assertions using science, logic, and reliable information. Instead of simply accepting arguments and conclusions, one questions and evaluates in an organized manner."
Critical thinking teaches you how to separate fact from fiction, how to distinguish opinions from evidence-based truth, and how to put events and actions in context. A rigorous way of processing information that tells you the proof is in the pudding, critical thinking is pretty much the opposite of intuition and gut feeling. Too many people make important — more than that, critical — decisions based solely on how they _feel_. They screen out information that makes them uneasy using what's called confirmation bias — taking in only information that supports what they already believe or think — so that they need not be uncomfortable. They vote, pick jobs or partners, change jobs or partners, have children, parent, and plan entire lives without using rationality. Needless to say, mindless eating, or eating because you simply _feel_ like it, is the antithesis of critical thinking.
When you start developing and utilizing critical-thinking skills, you'll begin making better decisions in all aspects of life. You'll weigh pros and cons, inform yourself of all your options and all their consequences, seek out evidence even if it makes you uncomfortable or shows you are wrong, recognize what's best for you rather than what feels good in the moment, and be willing to change your mind when facts point you in a different or better direction. Can you imagine living so skillfully and rationally? More important, can you imagine eating so skillfully and rationally?
#### 7. Skills for Setting and Reaching Goals
The skills for setting and reaching goals encompass more than simply saying you want to do something, then charging off to do it. How many times have you gotten yourself up to the starting gate of "normal" eating and begun moving forward, only to falter before you've made it halfway through your first course? How often have you vowed to eat this and not eat that, and had the idea fly out of your head at the initial taste of this or that?
Without effective goal-setting skills and a clear understanding of the science and art of sustaining motivation and making progress, you're doomed to fail. And, if you're reading this book, you've had enough failures with food and weight goals to last ten lifetimes. Learning how to establish doable goals, divide big ones into smaller ones, reenergize yourself when your spirits flag, pace yourself, and recover from disappointment will go a long way toward helping you establish a positive relationship with food.
#### 8. Skills for Balancing Work and Play
You might not have thought so, but balancing work and play demands skill, and a lack of competence in this area often triggers unwanted eating. Too much work and skyrocketing stress can send you straight to the cookie jar. Too much play and you may lose your ability to mobilize your inner resources and forget that life (and eating) have consequences. Moreover, one significant problem of dysregulated eaters is that eating is their major sport and passion — that is, it's how they break loose and let it all hang out. In fact, most troubled eaters are pretty much at square one when it comes to letting their hair down in effective, appropriate ways.
When you _intentionally_ make a choice to work or play, you don't end up playing when you really wish to be working, or procrastinating so much that when it's time to play you need to hunker down and get work done pronto. As you can see, balancing work and play uses self-regulation skills and skills from living consciously. The more skilled you become at sensing what you need in terms of goal-directed activity (what we call work) and mindless fun (what we call play), the less you will turn to food to provide you with whichever you're lacking.
* * *
Get Smart!
Using my list of life skills, rank them in order of your proficiency with each, starting with the category you do best with and ending with the category that needs your attention the most.
* * *
I hope I've convinced you that by learning and perfecting eight essential skills, you'll forever change your eating habits and your life. Every time you turn to people rather than food for comfort; give your body the exquisite care it deserves; make a well-informed, rational decision; create balance in your life; plug away at reaching a worthwhile goal; and keep yourself centered and grounded no matter what craziness is going on around you, you grow wiser. And each hard-earned bit of wisdom will make skill-building that much easier and bring you closer to outsmarting overeating.
* * *
Skill Boosters
1. Read over the eight categories of life skills every day and work toward identifying when you are doing one or another. You might think, "Ah, critical-thinking skills are what I need right now," "Gee, I see some red flags in this relationship," or "I'm doing pretty well at self-regulating today."
2. Pick out one or two (three at most) skills to work on — say, problem solving and physical self-care, or building relationships and balancing work and play — and focus on developing them for one week. Then pick other skill sets for another week.
3. Notice the life skills of other people and consider whether they're better or worse than you'd expect. Pay special attention to people with excellent life skills and emulate them.
4. Make a list of the life skills you need or wish to improve in order to eat more "normally." If you engage in unwanted eating, recognize which skills you're not employing effectively enough — say, self-soothing when you're upset, playing enough, or asking others for help with your problems. Then put special attention on using these skills to improve your eating.
5. Make a list of skills you excel at — say, sociability, working effectively, creativity, parenting, athletic prowess — and read them over every day. Be detailed and specific when citing your skills — not "I'm good with people" but "I'm an attentive listener; not "I'm a pretty fair tennis player" but "I have a wicked backhand." These examples will help you feel less inadequate as you learn the skills in this book, by reminding you of how many things you already do well.
6. Monitor your motivation, and remember that it takes some ten thousand hours to become highly proficient at a skill. Notice when your motivation revs up and when it wanes, and reflect on what triggers the shift. Do more of what keeps you motivated and less of what doesn't. When you get impatient, focus on the present and what you've already achieved. Make sure to note all progress, tiny as it may seem, rather than focusing exclusively, as most dysregulated eaters do, on what you have yet to accomplish.
* * *
In chapter 2, you'll learn how to take care of your body from head to toe to improve your quality of life and longevity.
## CHAPTER 2
## Wellness and Physical Self-Care
## You Mean My Body's Not Like a Self-Cleaning Oven?
Taking care of your body is the life skill closest to eating healthfully. You might not even consider this a skill and instead may just take it for granted that, like an oven, your body will take care of itself. It may seem to, if you're in the bloom of youth — say, in your late teens or early twenties, when humans are in peak condition, their physical prime. But to get beyond that point and maintain some semblance of wellness for the rest of your days takes a substantial amount of consistent care.
That care starts in the womb, where, if you were fortunate, you didn't have toxic substances like alcohol, harmful drugs, and unhealthy foods dumped into your fetal bloodstream. And you weren't subjected to high amounts of cortisol from Mother's frequent and intense bouts of anxiety, fear, or anger. Instead, you were lovingly and safely ensconced inside her until you were ready to make your grand entrance into the world.
If your stars continued to align, you grew from an infant into a youth whose physical needs were taken seriously. If you had a boo-boo and it warranted a bandage (sometimes even if it didn't), you got one. Your parents and other relatives put great attention on preventing accidents from happening to you and took pains to grow you strong and healthy. They lathered you in sunscreen so that you wouldn't burn and harmful UV rays wouldn't damage your sensitive skin. They bundled you up from head to toe before you went out in the cold and snow and cozied you under their umbrella when it rained.
They took you to the doctor or dentist for regular checkups and to the hospital in emergencies. You received appropriate vaccinations, were compelled to take your medicine no matter how much you screamed and yelled (or hid from it), and were taught how to more or less stay out of harm's way. For example, your parents may have allowed you to climb trees but go only so far up, to play on your swing set but soar only so high, and to ride a bike or roller-skate but nowhere near heavily trafficked roads. As I tick off this list, you may take these things for granted, whereas others would start to notice that their parents, in fact, didn't make many of these necessary efforts.
There's another kind of physical care that speaks volumes about how much your caretakers valued you, their child: making sure that you sat in a car seat until you no longer _legally_ needed one, seeing that you buckled up your seat belt every time you got into the car, and ensuring that you wore a helmet when you were bike riding, roller-skating, or skiing. Actions like these speak volumes about how your parents valued you. They say, "You are so precious to me, so vital to my happiness and well-being, that I will do whatever is in my power to keep you safe." The message that would have come across to you, the young child, is this: I am loved, precious, of great significance and value to someone else; therefore, I must be lovable, precious, and valuable.
If parents are smokers it's unhealthy for them to smoke around their child, because the message this conveys is: "My pleasure and comfort are more important than your health." If parents must smoke, they should do it outdoors or at least not in rooms their children generally occupy. Having a parent step outside to have a cigarette says to a child, "Right now I can't help mistreating _my_ body, but I'm sure as heck not going to mistreat _yours_." (We'll get to confusing double messages regarding parental care versus child care in just a moment.)
One thing parents should never do is to hit or strike a child in any way. This reaction causes all kinds of problems, the obvious ones being that, if your parents struck you, they were not in control of themselves and were not taking the time to communicate with you effectively. Another obvious problem with hitting children is that it hurts and can damage them, and if done repeatedly it can dysregulate a child's nervous system. A slap on the head, a jab in the chest, a punch in the stomach — any of these actions can do real, though unintended, harm. Moreover, your parents crossed a line if they mistreated your body, because it gave you a double message. Maybe they slapped you because you frightened them when you ran into traffic, but the slap is confusing because it says, "I'm hurting you so you don't hurt yourself." Huh? A perplexing message at best and one that children internalize as: "It's okay to hurt my body if it's done in the service of taking care of it." It doesn't take a genius to see how that kind of mentality can lead to eating problems down the road — for example, to thinking that starving yourself is okay because it leads to a low weight, which is supposed to be healthy.
One more instance of how your parents may have unintentionally taught you that your body has little worth was by being neglectful. Maybe they were busy working two or three jobs or caring for your siblings or their elderly parents. Maybe they had their own physical or mental disabilities to deal with and they did what they could, but it was the bare minimum. Neglectful parents may have forgotten your doctor or dental appointments, pooh-poohed taking you to the emergency room even when you were badly injured (especially if it would have inconvenienced them), didn't make you take your medicine, or didn't refill your prescriptions on time. I'm not saying that parents intentionally fail to take physical care of their children; but whatever the reason, the impact is often the same. As a child, what you intuited from their actions (or, in this case, their lack of action) was that your body and well-being didn't matter awfully much so you must not be worth very much to them.
You may think I'm making too big a fuss over the specifics of how your parents took care of you physically and kept you well. What I mean to convey is less the significance of each specific act and more the aggregate impact: what is communicated by these actions during your first two decades of life. If you weren't made to wear a helmet while roller-skating, it doesn't mean your parents didn't love you or didn't care if you injured your body. Ditto if they didn't always reslather you with sunscreen after you finished swimming and toweled off. However, I cannot stress strongly enough that, aside from its importance to your health, _how your parents cared for your body went a long way toward teaching you how to value it_. If they neglected or mistreated your body, there is a good chance that you will too! Enough said on that subject.
* * *
Get Smart!
How well did your parents take care of your body, not just your health; how well did they protect you from physical harm? Were they attentive or neglectful? What lessons did you learn from how they treated your minor or major health concerns? How do those lessons play out in your care of your health and body today?
* * *
### What If Parents Take Care of Their Children's Bodies but Don't Take Care of Their Own?
If your parents took care of your body, but not theirs, they gave you a perplexing message: "Do as I say, not as I do." Children not only pick up information from how they're treated, but they also infer what to think and do from what they observe. For example, I knew a woman who would always carefully strap her toddler into her car with a seat belt but would never wear one herself. One day, I was driving her child somewhere alone when she blurted out, "How come Mommy makes me wear a seat belt but she never does?" How come indeed? Think about how confusing this is for a young child to sort out: the most important person in her life (her parent), whose behavior the child unconsciously watches like a hawk, fails to take care of her own body but makes sure to take the utmost care of her child's body.
How would a child interpret this contradictory behavior? She'd wonder why her body is so important but her mother's is less so. Does this mean that children are more important and should be better treated than grown-ups? Does it mean that when she becomes an adult she has to take care of her child's body, but she can slack off when it comes to her own? Should she do as her beloved parents do, in order to be like them, or should she do what is done to her by them? My head is spinning just writing about such puzzling dilemmas.
The message is also wildly confusing when parents lavish care on their own bodies but not on those of their children. They rush off to the doctor or take medication for every minor ache and pain, but ignore their children's fevers and even broken bones. What a terrible message that gives to a child: I, your parent, am worthy and valuable, and you, my child, are not.
A child needs to receive congruent messages from her parents: they take good physical care of her and of themselves. If she is to value her body, however, the message must be positive. Too often parents neglect their own bodies and health and those of their children. This is a heartbreaker of a message, but at least it's clear: bodies have little value, so why bother to care for them?
* * *
Get Smart!
Were there any double messages about wellness and physical self-care when you were growing up? What messages did you receive? How do the messages you received affect your physical self-care today? What impact have they had on your eating or weight?
* * *
### What If I Don't Like Going to the Doctor or the Dentist?
Please don't be offended, but this kind of question comes from not using critical-thinking skills. In fact, it's a great example of how people don't employ these skills to take care of their health. Over the decades I've heard this question countless times, more than I can recall, in fact. The question suggests that there are people who actually go wild over being poked and prodded with stethoscopes and needles by a relative stranger while sitting stiffly and devoid of clothing on a paper-covered sterile table in an office the temperature of an igloo. It also suggests that some strange folks positively swoon over flu shots and stress tests, never mind biopsies and colonoscopies.
Okay, you get my drift. Whether you like going to the doctor or the dentist is so beside the point. The point is that you go because you care about your physical well-being and are determined to stay as healthy as possible for as long as possible. Truly, I'm the first person to say that feelings have their place in life. But using them to make health care decisions is not one of them. You can't live your best life when you're guided by fear and discomfort. Getting medical or dental attention has nothing to do with how you feel about the _activity_ , and everything to do with how you feel about _yourself_ !
### What If My Health and Body Are Such a Mess That I'm Clueless about How to Start Taking Care of Myself?
I run into many clients who have let their health go so far that the thought of taking stock of their myriad ailments and learning how to properly deal with them makes them just want to cry. I sympathize, but that's not what these folks need. Nor do you. The best I can offer is hope and encouragement that, by taking small steps, you will feel a lot better both physically and emotionally. One problem is that you may not know where to begin — which doctor to call first, how to find a competent dentist who doesn't charge an arm and a leg, what annual tests to have for your age and gender, and how much insurance will pay for and what comes out of pocket. And another problem is that you may feel you have to make every appointment tomorrow or the next day, and you're afraid that if you don't you're going to put things off and these medical visits will never happen.
I recommend a few starting strategies. First, find a primary care doctor you like — really like — not just someone your neighbor raves about or the first name in the phone book or on your insurance plan. You have the right to doctor shop for high-quality medical care. Doctor or hospital shopping can be time-consuming, of course, but it's not impossible. Although you can't meet every physician and dentist available, you can go online and check out whatever information you can find on them, especially of the lawsuit and patient satisfaction variety. I know that sounds hypervigilant, but it's a place to start. Also, some communities — like mine — put out booklets that rate health practitioners. Try asking everyone you know for a referral that includes why they specifically like the health care professional they're recommending, as well as anything they don't care for about him or her.
Think about what's important to you: the doctor's expertise or specialization, gender, good listening skills, bedside manner, office hours, and proximity to your home; the ease of getting an appointment; the length of a visit; cost; after-hours availability or back-up coverage; how well the office is managed; and so on. If possible, be picky. I went to two primary care doctors after I moved from Boston to Sarasota, before finding a third one, with whom I was happy. Unless you live in a rural area with few health care providers, the more you know about what you want from one, the better you'll do at finding the right person. By the way, I suggest that, unless you're in an emergency situation, you start with identifying a primary care doctor before any others. Usually he or she can recommend specialists.
If you haven't been to the doctor in a long time, you might be fretting about what he or she will say about such a lapse, and this may incline you toward putting off making an appointment. If this is the case, you're proving my point about a need for improved life skills. For example, if you were better at self-soothing, you wouldn't be so anxious about a relatively innocuous, routine medical visit. If you were using effective problem-solving or critical-thinking skills, you wouldn't be struggling while deciding whether to avoid emotional discomfort by not going to the doctor or to ignore your anxiety and get your butt there posthaste. If you were living in the moment and not in the past or future, you wouldn't be ruminating about all the things the last doctor said to do that you didn't get done, or agitating about what your new doctor will think of your weight or poor health.
Your best bet is to make one or two appointments at a time — no more — so you don't get overwhelmed. Start with generalists. Take notes or, as soon as you leave the office, make a summary of what's been said. Better yet, see if you can get someone to go with you to jot down important points. Heads up: schedule your next appointment before you leave the office. This is where many folks fall down on the job. They forget their appointment book or digital device and say they'll call when they get home and don't. Repeat: do not leave the medical office without having scheduled your next visit. Better to reschedule it at a later date if it proves inconvenient than not to have an appointment at all.
If you don't know what kinds of tests you need, given your age and gender, ask the doctor. Or go online and see what's recommended by reputable sites. Try to schedule required tests as soon as possible (so you don't build up anxiety waiting for them). Write the test dates on a calendar, not on a scrap of paper that you might mistake for an old receipt and toss out. And if you don't schedule your next appointment ahead of time, then on your calendar, on the month before the appointment needs to take place, make a note to remind yourself to schedule it. For example, if you have your teeth cleaned every three months, and you last went in April, jot a note to yourself on the June page of your calendar about scheduling your next cleaning for July.
This is not rocket science here. It's using management and planning skills to take care of something you value — your health and body. You might think I'm crazy for being so picayune about all this health care minutiae, but many dysregulated eaters are so used to not taking care of their medical concerns that they lack the organizational skills to do so.
I suspect that you may have concerns that relate to your size, shape, and weight that keep you from getting medical attention. However, there are plenty of "normal" weight folks who take inadequate care of their bodies, and under- and overweight people who practice excellent physical self-care. Moreover, devaluing your body most likely preceded your eating problems and is the real issue: that is, thinking too little of yourself to carefully monitor and manage your health.
And while you're doctor shopping, it's important to think about obtaining health insurance coverage. In a practical sense, you will likely need it in the long run. Moreover, it says that you value yourself enough to spend money on making sure you can pay to have your medical concerns taken care of. And it speaks volumes about your ability to problem solve and use critical-thinking skills. The time to think of unexpected occurrences is not while they're happening but before they happen.
### What about Everyday Stuff That Shows I Care about My Physical Self?
I wish I had a dollar for every client I've heard complain about how much trouble it is to take care of himself or herself physically. Is that how you feel? Here are activities that appear to be the most burdensome: doing laundry, food shopping, cooking, keeping living quarters clean and neat, getting enough sleep, following a skin care regimen, practicing dental hygiene, taking vitamins, and buying and wearing appropriate size, comfortable clothes and shoes.
Wearing well-fitting clothes and shoes is a must, because it keeps your focus off what you're not wild about regarding your body — its size. Dressing suitably and attractively is part of daily physical self-care as well. I don't mean that you have to get all dolled or suited up to run across the street to grab a quart of milk in the 7-Eleven. I do mean making sure your clothes are clean and they fit well. Your shoes too. I understand that large people often have difficulty finding comfy footgear, but these days you can shop online for them and, at some sites, even get free shipping. Shoes are important: they ferry your body around all day long.
Clean, pressed clothes speak volumes about who you think you are. There's a huge difference between someone who's dressed attractively and well-kept and one who isn't — at any weight. Next time you go out, pay attention to well-appointed people and notice how you feel about them at first sight. Forget their size or shape; just notice their general appearance and the message it sends out. Telling yourself you're too fat or thin to dress nicely fools no one but yourself and is called pretzel logic. C'mon, does that really compute? If you're unhappy with your body, why compound your perceived problem by not looking presentable?
#### Getting Your Zzs
Most of us need between seven and nine hours' sleep most nights to function well. If you get less, your hormones will go awry, specifically the ones that influence your appetite. Sleep deficiency causes increased production of ghrelin, which generates sensations of hunger, and decreased production of leptin, which promotes satiation. If that isn't exactly the opposite of what you, as an overeater, want to happen, I don't know what is. I hate to push, but if there's one positive self-care action to take for your body, it's getting enough sleep.
#### Caring for Your Teeth, Skin, and Hair
When you're in a rush in the morning or exhausted at night, it can feel like a drag to pay attention to your teeth and gums, which means any or all of the following: brushing, flossing, rinsing, irrigating. If you don't continue to bemoan how long and boring the process is, you could probably do the whole shebang in about five minutes, give or take. That's ten minutes every day, and all of us can carve out that itty-bitty amount of time for oral hygiene. I bet you spend more time than that reading emails, posting on Facebook, and texting your friends. Need I say more?
I'm going to throw skin and hair care into the it's-too-much-trouble-so-I-don't-bother-with-it category. Again, we're talking about minutes at a time. Washing your face in the morning and evening (preferably with a clean washcloth, because dermatologists tell us that wet washcloths carry gross bacteria), using sunscreen before you go out and reapplying it during the day, and finding a moisturizer that will keep your skin feeling and looking fresh are all important. You don't need to spend hours fussing and primping, nor your entire paycheck on high-end products. Just the basics will take you a long way.
As to hair care, ask a stylist how often you need to wash your particular type of hair. Lose the all-or-nothing mentality, as in: No one's looking at my hair because they're staring at my ugly body, so why should I bother? A better perspective is to feel good about your haircut or hairstyle and not obsess so much about your body. Work on directing people's attention to what _you_ want them to see about your appearance. Here's a case in point: I was at a local blues club not long ago when a singer stepped onto the stage. She probably weighed well over three hundred pounds, and she had long, well-styled, shiny black hair and a modest amount of makeup, which emphasized her startling green eyes. The sparkly red top she was wearing was a knockout. And as soon as she belted out her first number, I can tell you, the audience was all ears.
One more point about appearance and I'll move on. I'm not saying that females must wear makeup. That's a personal choice and, for some, even a feminist issue. No woman has to wear eyeliner, blush, or lipstick to be attractive, and those who overdo may be expressing fear that they're not good-looking enough to go out in public without a little of this and a lot of that. Do whatever makes you feel attractive.
#### Creating a Peaceful Living Space
What, you might be wondering, does maintaining a peaceful living space have to do with physical self-care? I know if you think a minute, you'll come up with the answer. If your surroundings are pleasant to look at, you have a better shot at wanting to make the person who's living in them pleasant to look at as well. Sometimes when I meet someone who's poorly groomed, I come to find out that his or her house or apartment (or dorm room) is, quite frankly, a mess. Again, it's all about what you value. On the other hand, I've often been surprised to find that someone who seems as if he or she has not looked in a mirror in a long, long while lives in an environment that is lovely, attractive, neat, and creative. That tells me something too: this person loves beauty but doesn't feel that his or her body deserves it. Sadly, this person has an eye for attractiveness and good taste that gets turned outward, toward decor, and not inward, toward his or her body.
#### Buying and Making Food
I left grocery shopping, meal planning, and food preparation to the end of this section intentionally, because many dysregulated eaters feel a crushing fatigue and go mentally offline at the mere mention of what they call these "chores." What is it about the natural task of feeding your body that seems so grueling? Why do you have such antipathy toward activities that are so routine for most people? That is, many folks may not adore shopping and cooking, but they do it because they enjoy eating and know they're responsible for feeding themselves.
The problem is that some people feel they don't deserve to eat well, and so they deny their bodies pleasurable nourishment. Others wish someone would swoop in and make their meals for them so they'd feel well taken care of. Still others get so overwhelmed thinking about what's "right" to buy or cook that they give up before they're out the door and end up noshing on whatever's in the kitchen.
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Get Smart!
How would you rate your daily physical self-care? What activities are you proud of doing well and consistently? Which ones could you put more effort into? What gets in the way of taking excellent physical care of your body each and every day? What excuses do you use? How will you change your thinking to make these activities more appealing and integral to your life? Do you tell yourself the problem is your eating or weight or time or money, when it's really about how little you value yourself?
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### How Do I Develop Better Skills for Wellness and Physical Self-Care?
The first thing to do is to remind yourself that you've learned many skills throughout your lifetime, which means there are many things you do well, even exceptionally well. Very often the activities we're good at come easily to us, but sometimes not. I heard an interview with the late Lester Young, a saxophone great, on NPR not long ago in which he was complaining about how hard it was for him to learn to read music. It came naturally to his siblings, but not to him, though he could play by ear. However, reading music was a requirement if he wanted to play in the family band, so he painstakingly taught himself how to do it. Even at the time of the interview, he didn't much like doing it, but he understood the value of being a proficient music reader.
Learning to take better care of yourself physically cannot possibly be any harder than it was for Lester Young to become adept at reading music. The initial step comes from deciding how important this skill is to you. I will say that if you want to become a "normal" eater and be comfortable in and with your body, then becoming more skilled at physical self-care will be a gentle lead-in to it. If your dental and skin care are first-class, if you take pains with how you look because doing so is a reflection of how you feel about yourself, if you get a solid night's sleep most nights, and if you make sure that your environment is attractive, it will be a lot easier to ease into enhancing your feeding capabilities.
My point is that sometimes you have to set the stage for major change and let it come to you, rather than attack it head-on as if you're out to conquer the badlands. Start with making improvements in your physical arena (that means your body and the space you occupy a good part of the time), and it will feel more natural to care diligently for yourself. If you're stumped about how to dress up or spruce up your living quarters, ask a friend who has a flair for design and decor. Creative people generally love helping friends spiff up anything, anywhere. Don't be shy. You'll be doing some artistic person a favor; moreover, the more time you spend with this person, the more you'll see things through his or her eyes. You'll begin to notice what works and what doesn't, a skill in itself. Remember, people go to school to learn how to do this (although, to be fair, some folks seem to be born with the knack).
There are other practical ways to develop physical self-care skills. Make a folder for each of your health practitioners in which to file your appointment notes and test results; keep a calendar just for medical and dental visits or for regular appointments for haircuts and the like. Go through your closet at least twice a year to get rid of old clothes and bundle them up for the consignment store or to donate to charity. Have a clothes swap with friends or work colleagues at least annually. I once gave an adorable pair of loafers to a friend because they looked so wrong on my feet and so right on hers. I got a kick out of seeing her wear them.
Notice what other people are wearing, and make a mental note if you think you'd look good in that style or color. You don't have to spend a fortune on clothes, believe me. I confess that I'm a clotheshorse and buy over 50 percent of what's in my crammed closet at either consignment stores or Goodwill. I rarely pay full price for clothes.
So that you don't "forget" daily (or weekly) living activities, make a chart of everyday tasks and check them off: dental care, check; skin care, check; vitamins, check; laundry, check; dust, vacuum, mop, check. I have a client who has numerous health problems, but boy does she have a great system for taking her medication. She keeps all her pills in a box in the refrigerator, affixed to which is her pill-popping schedule and a pencil on a string. It can't get any simpler, or more doable, than that.
When it comes to food shopping and preparation, you might find yourself being squeamish or uneasy about stepping up to the plate (pun intended). That's because, as I mentioned earlier, feeding and nourishing yourself may be fraught with underlying conflict. Still, there are steps you can take. First and foremost, for goodness' sake, quit telling yourself how you hate to shop and cook. A former client spent every session telling me all about her abhorrence of anything related to grocery shopping and food preparation, then couldn't imagine why she avoided these opportunities for self-nurturing. Every word she uttered reinforced her negativity. Note: Tell yourself how you want to be tomorrow, not how you were yesterday or are today.
So, lighten up and brighten up. Find something pleasant about each task. In the supermarket, notice that you're getting exercise, enjoying a chance to people-watch, or just plain getting out of the house. As you're placing items in your cart, smile and tell yourself that what you're doing is part of taking care of yourself and those you love. Remind yourself that you're proud of going the extra mile to feed your body well. Keep smiling and staying positive — it's hard to feel resentful or angry when you're smiling. Vow never to tell yourself how much you can't stand these tasks. They're not odious; they're expressions of self-love.
As to cooking, I'm not a "foodie" and never will be. I confess that I keep it simple except when I have dinner guests, but I try to make mealtime or snacks tasty and nourishing. If you're unskilled in the feed-yourself department, get a 1-2-3 easy cookbook, take a class, or ask a friend who can cook to teach you how. Start out eating what you enjoy that's easy to make. Honestly, most of us eat the same things over and over anyway. Some people find this mostly satisfying (me, for instance), while others easily get bored. The way I see it, that's why there are restaurants (and doggy bags).
Someone asked me the other day what I think about when a large serving of food is placed in front of me in a restaurant. What else, but how many meals it might save me from making! Collect restaurant coupons and treat yourself once in a while, or occasionally get reasonably priced takeout; and if your appetite can manage it, make two meals out of one. Cook a bunch of stuff over the weekend and freeze it for the week ahead. Ask family members to take turns cooking or grocery shopping to give you a break. Potluck with neighbors. Whatever you do in the food department, remind yourself that it's like sending a care package from you to you, one based on pure love and an uncrushable desire to be well and healthy.
### So That's All There Is to It?
Well, no, not exactly. All the earlier suggestions will make it easier for you to become skillful at physical self-care, but they won't work if you're ambivalent about your self-worth. Sorry, but that's the way the cookie crumbles. If you harbor mixed feelings about your value as a human being and what you deserve, you may struggle with treating yourself well within, and outside of, the food arena. You may fail to act on any of the suggestions I've given here, or (more likely) you may do some of them for a while, then give up and go back to not attending to your physical needs. Then at some point, something will spur you to resume these routines, and, at another point, something else will cause you to drop them like a hot potato. You know how it is: you're in the groove until, poof, suddenly you're not. In no time flat, you go from self-care to "I don't care."
If you're curious about how you can get off this seesaw, that's the ticket. You're becoming skilled at problem solving, not merely sitting around lamenting what a sorry excuse for a human being you are. To end yo-yo behavior, you have to quit yo-yo thinking and, once and for all, come to a decision: Is you is or is you ain't, as the saying goes — worth it, that is. If you harbor doubts, you'll have to understand why, explore them, and toss them out with the garbage. Read self-help books, get into therapy, or join a support group. _You can't continue to be ambivalent about your value as a human being and expect to move forward and stay there_.
It's perfectly possible to go from conflicted to unconflicted. For years, I had self-worth conflicts, and now have none. I feel 100 percent worthy of the best in life 100 percent of the time. This change didn't happen overnight, but through an organic process. I went through it, and you can too. I assure you that when you're more single-minded about your value, self-care will be oodles easier. For more detailed instruction on how to use the organic process and resolve internal conflicts, read my book _Starting Monday: Seven Keys to a Permanent, Positive Relationship with Food_.
Beware: You can't sit around waiting to feel unconflicted, but must start acting this minute as if you know what you're doing is best for you. This means pretending or imagining and soon believing that you're just as deserving of health and happiness as anyone else on the planet. I suspect you'll say, "But how do I _know_ it's true?" Clients ask me this question all the time. They don't know, and neither do you. But you don't know that you're a worthless sack of sand, either. You fake it 'til you make it, and if you're not willing to do that, I guarantee you won't make it. You live in a way that's called "as if" — as if you were everything you want to be or think you should be, as if you're the greatest thing since sliced bread, as if you're the king or queen of the universe. Then gradually that's what you become.
* * *
Get Smart!
What's your current motivation to learn skills for wellness and physical self-care? What do you tell yourself that moves you forward? What do you say that prevents you from moving forward? Which physical skills will be easy to pick up? Which ones will be a challenge? Who can you recruit to help you?
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* * *
Skill Boosters
1. Reflect on the physical and health care you received as a newborn and as a child, and see if there's a connection between your treatment back then and how you treat your body now. If you weren't well taken care of physically, that means you never learned the skills you need, and you must stop blaming yourself for being a failure.
2. Consider the double messages you might have received about physical self-care, and begin to sort them out. These messages can be crazy-making if you try to straighten them out on your own, so, by all means, talk them out with someone you trust and let them help you. Then figure out how to retain only the message that is most constructive and beneficial for you.
3. Are there any health care appointments (medical, dental, tests) you need to make? Write them down, find the phone numbers you need, and put the list by the phone. Could you make one call right now? Rather than thinking of how anxious you are about making an appointment, think about how proud you'll feel after you do.
4. Pick one area of daily-living skills to take on — food shopping and preparation, keeping your living quarters clean and neat, getting enough sleep, caring for your skin and teeth, or dressing appropriately — and visualize how you will improve. Focus on the pride you'll feel after you've done your weekly grocery shopping or your laundry or cleaned the house.
5. If you're not sure you think well enough of yourself to treat yourself like the awesome person you are, figure out why before attempting to acquire new skills. It's fine to spend time doing this before jumping on the skills bandwagon. In fact, it's more than fine; it's necessary. Resolve your mixed feelings, and learning new skills will come far more easily. Therapy helps.
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In chapter 3, you'll learn what it means to have effective emotional management by building inner resources and getting support.
## CHAPTER 3
## Handling Emotions
## I Thought That's What a Spoon and Fork Were For!
You may never have thought that handling emotions effectively involves skill, and you may assume that some people are simply born better at it than you are. Maybe you've been amazed at how they seem to keep it together in the most depressing or hair-raising situations, or at how appropriately restrained they appear to be under one set of circumstances and assertive and dogged in another. Perhaps you envy the passionate, fully engaged lives that friends, family members, colleagues, or acquaintances enjoy without their becoming emotionally drained or overwhelmed. Maybe you're in awe of how people seemingly magically rebound from the most tragic events. I bet you never thought of their successes as skill based.
Yes, handling emotions effectively is a skill, or rather a set of skills that are learnable by anyone at any age. If you've experienced difficulty regulating emotions, I recommend that you read _The Food and Feelings Workbook: A Full Course Meal on Emotional Health_ , which I wrote specifically for emotional eaters. I guarantee that it will vastly improve the way you manage stress and distress — and eating! But when I tell you there's a correlation between food and feelings, don't simply take my word for it. Here's what Robert E. Thayer, PhD, author of _Calm Energy: How People Regulate Mood with Food and Exercise_ , has to say: "In my view, if you forget everything else — including the kinds of food to eat, their relative proportions and exact amounts — but you master your moods, you will go a long way toward controlling overeating."
What are these things we call emotions? Here's a scientific perspective. In her enlightening and delightful memoir, _My Stroke of Insight: A Brain Scientist's Personal Journey_ , neuroanatomist Jill Bolte Taylor, PhD, writes about her amazing journey back from a massive stroke and talks a great deal about emotions along the way. The book is both deeply personal and unflinchingly scientific. Taylor maintains that it takes "less than 90 seconds" for an emotion to get triggered, surge chemically through the bloodstream, then get flushed out. "What, emotions are chemicals?" you're thinking. "And the chemicals get discharged in under a minute and a half? Who'd have thunk it!" She goes on to assert that within this brief period of time, the automatic emotional response generates and arrives at completion, so that _whatever we feel after that is of our choosing_. Now that's something to think about, isn't it?
Let's look at an example of this process, taken from something that happened to one of my clients, who gave me a blow-by-blow description of a workplace problem he'd encountered decades before. He had worked for a company in which serious ethical violations had been going on for some time. Unbeknownst to management, he and his coworkers had contacted the company's board of directors to advise them of the violations and asked that they root out the rotten apples involved. At one point in a staff meeting, the head honcho (the rottenest apple of all!) looked around the table, making sure to stop and glare at each and every employee while breathing fire — or so my client swore! The boss announced that he'd heard rumors that a few employees were unhappy, and that he wanted them to identify themselves then and there and speak their minds.
Here's how my client described feeling. _Boom_ , his heart started pounding like a drum and his face felt as if it were on fire. He couldn't catch his breath, and his boss's voice sounded oddly muffled and far away. The feeling didn't last long; my client estimated that it went on for — you guessed it — about ninety seconds. After that, his heart rate began to slow down, his face cooled, and his breathing slowly returned to normal. Of course, he was still upset and feared being found out — and fired on the spot. Nothing of the kind happened. He reported no one noticing that he felt he might expire in his chair and never live to see the results of his office mutiny (which, by the way, was an empowering success).
For those of us who've held on to grief, grudges, ingratitude, rejection, insults, abandonment, bitterness, and the like for years or decades, Dr. Taylor's take on emotions may sound astonishing or even unbelievable. Even to someone like me who's spent decades in the field of psychology, the idea that an emotion from start to finish lasts for only about a minute and a half (far less time than it takes to do a respectable job of brushing my teeth with my electric toothbrush!) is astounding news with mind-blowing implications. It means that emotions function on a physical level that can be measured and monitored and are not all "in our heads." They're nothing but chemical surges that course through us like electrical currents. In fact, when you think about it, that's often how an emotion feels: a flash of searing pain, followed by rapidly escalating distress that floods through us and then, suddenly, recedes, similar to the reaction you get from an injection at the doctor's office. You wince an "ouch!" when the needle pierces your skin, and the pain surges, peaks, then dissipates. In no time flat, the ordeal is over.
### What's the Best Way to React to Emotions?
The first thing to do when you have a feeling is to notice it — really notice it — and see if you can give it a name. Noticing means _acknowledging_ that something is going on inside you. Don't pretend that nothing's wrong. Just let it happen. Remember, the chemical reaction of the emotion will be over within ninety seconds. Don't try to push away the feeling (a.k.a. avoidance), pretend it's no big deal (a.k.a. minimizing), explain it away (a.k.a. rationalizing), or switch to thinking mode rather than feeling mode (a.k.a. intellectualizing). Simply experience whatever is occurring within you, without having a gazillion judgments about it, especially about whether you should, or want to, be having the emotion or not. Think of it this way: If someone stomps on your foot, you feel pain, pure and simple. You don't spend time considering if you're better off not feeling the pain or what kind of ninny you are to have such a sensitive foot.
Dr. Taylor encourages us to be scientists like her, curious and totally nonjudgmental about what we're feeling. After all, emotions, like our senses, are meant only to guide us in the world. They're similar to musical notes and colors: neither good nor bad, they simply are, and some we like better than others. I bet you never thought of emotions from such a neutral perspective. Of course, there are feelings you go out of your way to experience, such as satisfaction and joy, and others you might not be so crazy about, such as disappointment and fear. But because of how emotions make you feel, you've unconsciously made the erroneous decision to welcome only the ones that cause you to feel "good," and you cold-shoulder the ones that cause you to feel "bad." You've confused the function of feelings with their effect on you, but in truth they all have the same raison d'être: to give you information about how to negotiate this world in order to have the best possible life.
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Get Smart!
How do you feel about experiencing intense emotions? What's your typical reaction to emotions that cause you to feel uncomfortable? How will you remind yourself to view them neutrally, exclusively as information transmitters?
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### What Can I Do with My Feelings?
This is a question I'm asked regularly, and it speaks to a uniquely Western notion that taking action is the best approach to whatever ails us, the operative word here being _do_. Fact is, sometimes there is something to do with a feeling, and sometimes there isn't. If your father died two weeks ago and you're feeling extremely sad and want to bawl like a baby, that's natural and normal and there is nothing to _do_ except whatever comes up for you. If your father died twenty years ago, and the anniversary of his death still throws you into a downward spiral that sends you to bed for a week, it's time to make adjustments so you'll experience less pain and remain more functional. My point is that how frequent, intense, and lasting the feeling or residue of feeling (recalling emotions via memory) is dictates whether you need to do anything with it.
Humans are creatures of habit and, therefore, we tend to settle into one way of acting or reacting even when it doesn't work particularly well for us. That's unfortunate, because flexibility is absolutely key to effective emotional management. Too many folks, troubled and nontroubled eaters alike, shut off their feelings automatically. They hurt, then deny or act in ways to numb their feelings and move on. That's the way they learned to cope with intense affect when they were children and didn't know any better. Alternatively, perhaps they have an inkling that shutting down isn't really beneficial but are terrified of experiencing emotions and determined to keep their distance.
However, avoiding feelings is not a great idea, because feelings have a function. Consider someone who mistreats you repeatedly. In order to let them know that either they must change or you will need to cut them out of your life, you first must acknowledge that they have hurt you. Without the recognition of hurt, you'd be the equivalent of emotionally deaf or blind. Emotions act as guides, but they're only effective if you pay attention to them the way you pay attention to your senses. When you poke your tongue at a tooth that's throbbing, the purpose of that pain is to tell you something is wrong. Conversely, when you take a whiff of roses and are enveloped in a lovely, heady scent, you're using your senses to tell you that something is bringing you pleasure.
One of the reasons we don't care for intense emotion is that it can easily and excessively dysregulate us. It's difficult to be zapped by a strong feeling and not have it shake you up internally. We call this effect emotional dysregulation — it knocks us off course, throws us for a loop, pulls the rug out from under us. However, it's the way we feel about and react to dysregulation that makes or breaks us, emotionally speaking. Rather than accept that we're momentarily (okay, sometimes a lot longer than momentarily) off-kilter, and that this is natural and normal, we try to reregulate ourselves immediately. If we just give ourselves a little time, our bodies and minds often move toward reregulation on their own. Instead, though, we rush off for a snack so it can perform its magic act. A better approach is to remind ourselves that it's okay to not always feel perfectly centered or calm, and that we'll regain our equilibrium in good time with the right practices. You'll read more about dysregulation and reregulation in chapter 6.
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Get Smart!
How do you know when you're emotionally dysregulated? What specific feelings cause this to happen? What experiences cause these feelings? How does food reregulate you? What else, other than eating, helps reestablish a comfortable stasis?
* * *
Dr. Taylor maintains that we don't _have_ to extend an emotional response beyond ninety seconds. What she's saying is that initial affective pain gets triggered automatically on a physical level owing to our nervous system. That's because the pain has a purpose, which I'll get to in a minute. The radical implication here is that experiencing pain beyond the initial stab of ouch is, in a sense, our choice. I say radical because, when we're in the throes of distress, few among us recognize that we are actually going out of our way to prolong our agony.
But heartache feels so real and so involuntary, you may insist — you wound me, I bleed. It does feel that way. However, once the initial zap has come and gone, we're responding not to authentic distress but to a _memory_ of the distress we've just been experiencing! We're responding to a recollection of the pain we experienced a moment ago, not the actual wounding, which is already history (though very recent), come and gone, over and done with. Once the initial sting of hurt lets up, instead of distancing ourselves from emotional distress we may circle back around and poke at and prod it until it starts in again. We tell ourselves: "It's not fair that I didn't get invited to Susie's dinner party! I invited her to mine." "How could he embarrass me in front of the whole department just for being a few minutes late? Like he's always on time." "Mom/Dad/Sister/Brother can be so cruel. I'm so hurt by what she [or he] said. Forget about my going to visit on Sunday."
Let's use my client's mutinous office situation to further illustrate this point. As he sat in my office talking about that awful moment of sheer terror decades before, when he had feared being exposed and branded a whistle-blower in his company, it was another gorgeous day in Sarasota, and he was gazing out my office window at the tangle of overgrowth that lets loose in a subtropical climate after rainy season. He was calm and related the story with moderate emotion.
Although he said he could _recall_ feeling as if his goose were about to be publicly cooked some thirty years earlier in that staff meeting, he certainly wasn't having the intense emotional response he'd had back then. That's because his memory had stored the data about that occurrence separately from the affect that it generated. Of course, maybe two days or two weeks after the interaction, he could still painfully recall being nearly found out, but even then he couldn't possibly feel the intense waves of affect he had experienced _at the time they occurred_.
Why? Because our automatic reactions have a very precise and singular reason for being. They grab our attention with a wallop of sensation to warn us of perceived immediate or imminent danger so that we can avoid, prevent, or minimize it. Now hear this: _The purpose of emotions — their sole raison d'être — is to steer us away from pain and keep us safe_. In evolutionary terms, that means they zap us to help us live — and breed — another day. Put another way, they generate pain to try to help us in the long run. Pain is acute to signal a warning to never forget a situation that has the potential to threaten our lives. The importance of grasping this concept cannot be overstated.
Like everything else in life, pain has a function — as does pleasure — and that function is first and foremost to preserve and propagate the species. On the pleasure front, for example, the neurotransmitter dopamine gets triggered in our brains when we eat high-fat, high-sugar foods, because many millennia ago they provided us with the greatest sustenance in the ancient, nutrient-scarce world in which we lived. We also experience a blast of dopamine during orgasm, a nifty incentive to keep on procreating. Our biochemistry developed over time — through evolution — the ability to fine-tune our reactions to pleasure and pain in order to keep us keeping on as a species.
Pain serves this purpose rather well. When something hurts, we tend to avoid doing whatever we think will cause more of it. We associate pain with perceived harm to self, and that's why we often have to talk ourselves into going to the dentist or the doctor, who might "hurt" us in the short run to heal us in the long run. When we experience pain, the primitive part of our brain registers the occurrence sharply and deeply so that we'll recall it automatically in case we find ourselves in a similar situation in the future.
### Why Do I Sometimes Still Feel Bad Long after I've Been Emotionally Wounded?
You're not alone in that. We all have this tendency. The question can be answered by taking the idea of "emotional pain as an alarm signaling a perceived threat" one step further. Then you'll understand what I mean when I say that emotional pain _after an event_ is not a response to reality but to recalling the occurrence that caused it. Let's backtrack a minute, to be sure you understand this important point. Remember that emotional pain — betrayal, rejection, fear, loneliness — and physical pain are intended as automatic warnings to _do_ something to ward off danger and protect yourself. Remember as well that, once the threat has passed, this warning no longer has a function. It loses its purpose when the threat is over, much as a signalman on a train track has nothing more to do after the caboose has passed him by. No need to wave those flags to alert the engineer about a steep incline, because the danger — and the train — are long gone.
Here's another example. If a lion rushed at you, you'd feel fear instinctively and immediately, which would mobilize your internal resources, enabling you to do something to prevent a mauling — duck, fight, run, hide. Comedian Richard Pryor had this brilliant routine about being frightened that says it all: "Feet, don't fail me now!" Isn't that more or less what you'd think if a lion charged at you? However, if the lion heard a sudden noise, stopped dead in its tracks, then hightailed it in the opposite direction from where you were standing and disappeared into the woods, your psychic alarm still might be clanging as the cortisol rush to energize you subsides. The cortisol rush subsides because, no longer having a function, alarm chemicals slowly stop being produced in your brain and circulated in your body as soon as they have served their purpose and you're out of danger.
Pain is a way of etching risk and threat into our brains. Remember, our biology is geared to thwart repeat performances of anything that puts us at perceived emotional or physical harm. So, down the road, having been charged by a lion (and scared out of your wits), it's likely that if you're at the zoo and you stroll past a lion in its cage rushing over to say hello, your heart may begin to race and fear may course through your body. That's because the memory of a lion almost mauling you years before comes instantly to the fore before you can realize that you're quite safe because you're outside the cage, not inside it with Leo the lion. There's no longer a need for psychic pain — in this instance imprinted as fright — to warn you of harm, but it happens anyway because the amygdala, where intense affect is stored, cannot distinguish between the initial unsafe situation, which happened years ago, and the current, strikingly similar situation in which you aren't in danger.
Returning once again to my whistle-blower-in-the-office anecdote in which my client feared being outed: it made sense for him to be scared at the time, but not for him to reexperience intense fear as he sat in my cozy office looking out at palm trees, Spanish moss, and giant scheffleras. There was no need for strong emotion to warn him of danger, because he was in no danger in the present. The danger had passed. So, he could choose to recall the visual and tactile impressions of the memory of the staff meeting (who was there, where he was sitting, what the room looked like, the time of day) without strong affective impression.
Of course, there are times when you want to review a previously threatening situation to make better sense of it or assess what you could have done differently, times when you want to access didactic or factual memory. You may wish to understand why your spouse left you, why you were bullied in school, or why your mother let your father beat her up. However, you can access the details of these memories without reexperiencing the emotional pain attached to them, by reminding yourself that the pain no longer has purpose because the event is over. Think of it this way: if you are safe now and feel fear, that fear goes with the memory that caused it in the first place.
Because pain is a signal to do something to make yourself safe, it's overkill to have it clanging an alarm when you're no longer in harm's way. You can stop this unnecessary alarm by using your cognitive abilities, the higher-functioning part of your brain that can think logically, to remind yourself that you are no longer at risk and are quite safe. For sure, you want your automatic pain-response system to do its job when you're in actual emotional danger, but _only_ when you are. If pain is coming from memory, not reality, that's a false alarm and your warning system needs a mental rewire.
* * *
Get Smart!
Put into your own words the function of emotional pain (explaining will help you remember it). Can you think of an experience in which you were perfectly safe but continued to feel distressed by emotional pain that was socked away in your memory? Can you see how the later experience was only a false alarm? What can you now say to yourself when you have this kind of faux reaction?
* * *
The reason I'm harping on this pain model is because much of the emotional discomfort we feel in the present isn't about the here and now at all. Our discomfort is a reaction to a memory triggered by a situation similar to one in which we had previously been in danger. Unfortunately, our amygdala, where intense memories are stored, can't filter itself or tell the difference between past and present circumstances. That's not in its job description. Rather, it senses threat and immediately begins releasing chemicals to snag our attention and alert us that we may be in danger. Its motto may as well be: "Watch out. Here we go again." That's when we need our higher-order brain structures to tease out whether there truly is danger in the present. Most of the time when we're reacting strongly to fear or upset, it's out of recall, not reality.
Here are some examples to illustrate my point. The fact that your wife is ignoring you and hanging out with her friends at a party may unconsciously trigger the neglect you felt as a child when your mother would have her friends over to visit and leave you all alone and insist you not bother her. Your boss chewing you out in front of a client may trigger how ashamed you felt when your father would attend your soccer games and yell insults at you in front of the coach and other players. Your neighbor dropping over at all hours of the day or night without being invited may trigger your memory of Grandma walking into your room, without knocking, whenever she felt like it, your privacy be damned.
Get the picture? Much of the time when we're upset in the here and now it's due to intense upset from the there and then. If you are to become skilled in managing your emotions, you can't ignore this cause-and-effect reaction. Instead, every time you feel a strong emotion, you'll want to stop and consider if _what you're experiencing is in proportion to the emotional danger you perceive_. Sometimes it is and sometimes it isn't, and it's essential to discern the difference so that you'll know how to respond.
### What Can I Do When I Feel All Discombobulated Inside?
The answer to this question has several parts. First, you want to acknowledge your discombobulation and _identify_ what you're feeling. Don't lump all your emotions together under the umbrella of hurt or upset. That would be like telling a friend you had a phone call from "someone" about "something." Of course, your friend would ask, "Well, who called and what did he say?" Be specific: you're feeling anxious, betrayed, inadequate, frustrated, rejected, ashamed, lonely, guilty, disappointed, helpless, enraged, or confused. Get the gist of it? The more specific you are, the more information you'll extract from the emotion, because — make no mistake — information is precisely what you're after.
Here's another dose of wisdom from Dr. Taylor, who advises us to be open to feelings from the here and now whenever they come, at whatever intensity we experience them. If we short-circuit them, we won't receive the full benefit of the message they're delivering. On the other hand, if we continue to stoke their fires, we'll end up holding on to discomfort unnecessarily and ruin the quality of our lives.
Sometimes, a feeling blasts you when you least expect it, but you know just what it is. Maybe you catch the guy you thought was your boyfriend while he's smooching your best friend and feel a stab of betrayal. Perhaps you hear that your brother is being sent overseas with a combat unit next week, leaving no time to say good-bye, and you feel a surge of fear and sadness in your gut. Alternatively, at times you may sense a vague emotion erupting and want to work to identify it. For example, say your colleagues all ducked out for a drink at 4:20 and you're left to meet a 5 PM deadline alone. You know that what they did doesn't sit right with you, but you're not exactly sure what you're feeling, until suddenly a wave of abandonment washes over you. You feel like Little Miss Muffet stranded on her tuffet.
### What about Feelings about My Feelings?
Clients often share with me how they feel about their painful feelings, which can be summed up as: not so good. I then ask them why they need to feel anything about their emotions. Why can't they simply experience a primary feeling, then not have it anymore, game over? Usually secondary emotions (feelings about feelings) are evaluative and judgmental about whether the initial, or primary, feeling is good or bad, tolerable or intolerable, acceptable or unacceptable. I'm not talking here about having two feelings at the same time — say, joy that your best friend got a new job and sadness that it's two states away from you. The secondary feelings that crop up are more than likely shame or guilt. You may feel shame that, for example, you're angry at a coworker who's gone out of his way to be nice to you but who recently ticked you off, or you may feel guilt for wanting to stay home when it's snowing outside and your kids want to go sledding.
Try your best to experience a feeling fully when it first comes around. Just showing up for it is all you have to do. Shoo away secondary feelings of guilt or shame and stick with the initial emotion until it subsides. Examples of primary emotions are as follows: shock at not getting a job for which you thought you were a shoo-in; sadness that your cat died; confusion over whether you should stay married after you find out your spouse has cheated on you; grief at being told you cannot bear children; loneliness after moving halfway around the world to where you don't know anyone; or guilt because, after having a fight with your neighbor, you came home and yelled at your children.
Stay with the feeling without trying to extend or curtail it, and by all means, ignore judgments about it. When the most intense part of the experience is over, congratulate yourself on a job well done, one that at times feels a bit like charging around on a bucking bronco. If you stayed in the saddle, give yourself a round of applause.
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Get Smart!
Do you often have secondary feelings about your primary feelings? Which emotions tend to generate a judgment from you? Is that judgment necessary or simply habit? What can you say when you judge your feelings that will allow you to remain curious, open, and neutral about them?
* * *
#### What Skills Are Needed for Effective Emotional Management?
There are several ways to slice this pie. We could say that what's needed are two major emotional skill sets to keep yourself emotionally regulated: the ability to contain feelings _and_ the ability to expand feelings as necessary. This involves trusting people enough to express and share emotional upset, take in honest feedback, and feel supported while also possessing various internal methods of responding to emotions when you're alone or it's not beneficial to express them. This is a tall order, to be sure, but doable. When handling emotions, many folks pick one approach only, and maybe you do, too. You may fear burdening others or not trust them sufficiently to be vulnerable in their presence, so you hold in emotions until you're ready to burst — or binge. Or perhaps you easily become overwhelmed with intense affect and call every friend in your iPhone desperately hoping that _someone_ will say _something_ to make you feel better.
Essentially, the goal is to comfortably ask for help and to also do all right without it. The ability to solicit and enjoy support comes from a mind-set in which you believe that many, most, or at least some people will understand what you're feeling and will empathize with you and validate what's going on within you. Being able to seek help also means you don't believe the ridiculous notion that you _should_ be able to manage intense emotions on your own. Likewise, knowing you have substantial emotional competence makes you feel strong and capable on your own and leads you to solicit help only when it's necessary. No surprise, huh, that you're looking for a balance of resources, within and without?
Another way to view what Harvard author Daniel Goleman, PhD, calls "emotional intelligence" is to assess your skill at engaging and disengaging with emotion when necessary. If you're in the middle of a job interview and a disagreement you had with your daughter that morning pops into your head, you want to be able to shelve it for the moment, rather than spend time right then figuring out who was at fault. If you continue to ruminate days later about your quarrel after you and your daughter have made up, you want to be skilled at a technique called mindfulness in order to merely observe, rather than engage with this particular memory that will kick up a whole lot of affective dander that is best left to settle and dissipate.
At the opposite end of the spectrum, it's also vital that you fully engage with and experience emotions when they plop themselves on your doorstep. For example, say you've planned a night out with a friend, and he calls at the last minute to say he's too tired or busy, and this is something he's done umpteen times before. This is a perfect moment to connect with your emotions, which are likely to include feeling disappointed, neglected, hurt, and misused. What vital information you're getting about the quality of your relationship with this so-called friend! You want his thoughtlessness to register deeply so that you can avoid getting hurt in the future. Perhaps you'll want to talk with him about his unacceptable behavior or even end the friendship.
* * *
Get Smart!
Do you commonly either contain your feelings or expand them? That is, do you feel you must handle them alone, or do you incline toward almost totally relying on others to help you reregulate yourself? Which skill would you like to improve, and how will you do this? Can you feel emotions deeply in the moment and absorb their wisdom? Do you fear and avoid your feelings? Do you dwell on and get lost in them beyond what's necessary in order to sort out and overcome distress?
* * *
### Will I Ever Improve at Soothing Myself without Food?
It's true that emotional eaters generally turn to food when their feathers are ruffled, and they don't practice more effective ways to smooth them out. There are several reasons for this. Studies on temperament reveal that some people may actually be more emotionally sensitive than others, owing to genetic tendencies involving neurotransmitter imbalances, among other things. We all recognize that some folks are simply born sunnier and more upbeat than others. That's hard wiring. Moreover, how our childhood role models, primarily our caretakers, soothed themselves is a strong predictor of how we'll soothe ourselves. Our parents can't teach us what they didn't know how to do.
If Dad stormed off in a huff to his basement workshop and Mom made a beeline for the refrigerator every time they had a tiff, we see two approaches to handling emotions — neither of which shows great competence or effectiveness. I will say that, culturally, women seem to reach out and men seem to withdraw, and this is part of our evolutionary heritage. In the days of the cave dwellers, out on a hunt for food with the guys was not the time or place to be complaining about the kids misbehaving and driving you crazy. How much easier it was for women sitting around the fire tending to those very children to start gabbing about them. Sadly, since cave dweller times, the strong, silent male and the overly emotional female have remained cultural stereotypes, and we, as both children and adults, absorb their characteristics and frailties without realizing it.
Moreover, when you're upset, it's so easy these days to turn to food that people often don't even try to develop effective skills to cope with feelings and comfort themselves. To do so means unlearning "I-feel-therefore-I-eat" behaviors and acquiring skills that involve a good deal of practice. The truth is, although you might wish to have these self-calming skills, the issue is not about desire but about how hard you're willing to work to become proficient at them. They absolutely will not become part of your repertoire simply because you lay yourself down to sleep and pray for them. Rather they will become more integral to your coping strategies the more you put attention on and repeat them.
The problem with acquiring these skills instead of grabbing a sweet or other treat is that at first you may not find them particularly helpful. The frustration of learning new methods of self-care can generate hopelessness and helplessness. You might even believe that you'll never learn how to comfort yourself or relax without food even if you were to live three lifetimes. You may practice these skills a few times and give up, or you may use them at times and turn to food at other times, so that you're not making true progress. Remember, mastery comes from doing the new behavior more times than the old one — many, many more times, I might add. The truth is, if you do any behavior often enough, you will lay down new neural pathways in your brain and it will become habit.
Here are some basic ways to calm down and soothe yourself without food. This list is not meant to be exhaustive; there are scores of books (including my _Food and Feelings Workbook_ ) that can give you detailed advice on the subject. There's no shortage of helpful information out there to be gained through psychotherapy (individual or group), support groups, workshops, podcasts, message boards, and even apps such as my free Facebook app, APPetite.
That said, to unwind or comfort yourself you can learn and practice the following to replace unwanted eating:
• In the physical arena: To relax your body, drink a cup of tea; take a bath or shower; close your eyes and take a nap; do neck rolls; take a walk, swim, or ride a bike; get up and dance or do kickboxing; do catlike stretches; sit quietly and visualize a happy scene from your life; walk the dog or play with the cat; practice yoga; do a relaxation progression from toe to head; massage your temples, scalp, or any tense spots on your body; get regular massages; garden or do light chores; rock in a chair or a hammock; or (my favorite) do deep breathing.
• In the sensual arena: To awaken your senses other than taste, light a scented candle and inhale the aroma; go outside and smell flowers or freshly mowed grass; play music you love; rub cream all over your body; immerse yourself in a book of photography or watch a video of exotic places; look through magazines that bring you visual pleasure; stroke a bunch of various soft fabrics; or pet the dog or the cat.
• In the mental arena: To relax by enlivening your thoughts and rejuvenating your energy, browse through a joke book; check out humorous videos; watch a funny sitcom; engage in a hobby or passion; read a book that's a page-turner; surf the web; check your email; turn on the history, science, or nature channel; or teach yourself something new.
• In the emotional arena: For self-soothing, say compassionate, kind, caring things to yourself; reread a loving letter from someone; develop and repeat a mantra about your lovability; give yourself a hug; curl up in a fetal ball; cry; call someone who will help you feel better; tell yourself that your mood or the crisis will pass; if you made a mistake, remind yourself that you're allowed to make them; acknowledge that you're feeling bad, but remind yourself that this doesn't mean there's anything wrong with you; picture a strong, bright candle within you that burns with self-love; or write a love letter to yourself.
Want an instant fix for stress or distress? Don't reinforce what you're feeling. Instead, shift your self-talk to how you _wish_ to feel, not whatever emotion you're experiencing. Some examples:
• If you're starting to feel overwhelmed, the last thing you want to say is, "Man, I am so overwhelmed. I feel awful. I can't stand this feeling." Instead, tell yourself, "I can handle this. It isn't so bad. I'm competent and will manage."
• If you're feeling frustrated, steer clear of messages like "This is too hard. I can't do it. I hate when I can't do things." Instead, say, "Oh, this isn't so bad. I'm getting the hang of it. Soon I'll succeed."
• If you're standing in front of the mirror hating your body, avoid insisting, "I look terrible. I'm so fat, I can't bear looking at myself." Instead, shift to: "My eyes are great, so blue and sparkly. I love that my body is getting healthier." (Then, step away from the mirror!)
• If you're falling into loneliness, mind that you don't make it worse by telling yourself, "I'll never have any friends. I'm just not good at relating to people. Who'd want to be my friend anyway, because I'm such a loser." Instead, show some self-compassion and offer yourself these kind words: "I have had friends, and I have some now. I can learn to feel more comfortable with people and have a circle of friends that feels just right to me."
• If your anxiety is skyrocketing, never, ever let these words pass your lips: "I'm feeling so panicky. I may be having a panic attack, and I don't know what I'll do. I can't bear feeling so anxious and out of control." Instead, soothe yourself by saying, "I'm fine and my anxiety is receding. I'm feeling calmer, cooler, and more collected." (For a really quick fix, do this while practicing deep breathing.)
You likely will have to try out many of these techniques to discover what works for you. Some options will bomb, and others will make you feel better only after you've practiced and are used to doing them. Find one or two strategies in each arena. Remind yourself that food is not the answer, and that you'd need to develop emotional management skills even if you didn't have an eating problem. Everyone needs them! Don't come down hard on yourself for not having these competencies. Merely remind yourself that all adults are learning life skills and that's how the game is played.
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Get Smart!
If you'd never heard of food, what would you do to soothe yourself or relax? Which of the alternatives to eating for de-stressing or comforting yourself that I've just described have you tried? If they didn't work, why not? What will you do to develop effective emotional management strategies? What stands in your way? What will help you succeed?
* * *
* * *
Skill Boosters
1. Notice whether an upset is mild, moderate, or intense. Rather than judge it, be curious about it and acknowledge, identify, and experience it with calmness and the certainty that it will yield important information about your life. Consider it a text message from you to you.
2. If you're afraid of experiencing emotions intensely, sink more deeply into them by assigning a number to an emotion you're feeling, to identify how deeply you're feeling it. Use a scale of 1 to 5, with 1 representing superficial engagement, 3 representing substantial engagement, and 5 indicating complete engagement. If you're at 1, move yourself to 2, then 3. If you're at 3, move yourself to 4, then 5.
3. Practice being curious about and compassionate with your emotions. If you slip into judgment, use mindfulness to notice this has happened and return to curiosity and compassion.
4. If you tend to always run to others for comfort when you're in emotional distress, rather than building effective coping resources within, see how far you can get while comforting yourself on your own.
5. If you tend to clam up, stay private, or lick your wounds alone when you're in emotional turmoil, reach out to a trusted someone. It doesn't matter if you're uncomfortable. That only means you're doing something unfamiliar.
6. Recognize feelings associated with stress — frustration, inadequacy, a sense of being overwhelmed — and talk yourself down from them. Practice extreme relaxation.
7. If you tend to ruminate and go over the same miseries again and again, remind yourself that your strong reaction is not from the present, that you're caught in a memory, and that replaying it will change nothing. Ground yourself in the present by reconnecting to your body in the current time and place and breathing deeply.
8. Every day, practice effective life skills from the arenas listed earlier. Don't wait until you're in a crisis. You must do these new activities _every single day_ for new neural pathways to form. In fact, building a preventive routine is your best bet for keeping yourself emotionally regulated.
9. Stop during the day to notice whether you're emotionally regulated or dysregulated, and be aware of which people, interactions, and events generate each state. Before engaging in an activity that would typically dysregulate you, or before being around a person who may do so, prepare yourself by relaxing and recognizing your concern. Work on reregulating yourself as soon as you've finished the activity or are away from dysregulating people.
10. If you have the urge to eat when you're not hungry, know that your desire is not for food. Ask yourself what you're feeling, and stay with it until you find the right approach to address whatever you're feeling (for example, sit with it, notice but don't engage with the feeling, de-stress, and so on).
* * *
In chapter 4, you'll learn how to avoid living in the past or future and stay anchored to the present.
## CHAPTER 4
## Living Consciously
## I'm Conscious Only of Wanting to Go Unconscious!
Humans have a funny orientation toward life. Although their heads remain securely attached to their bodies, their minds often drift off into the past or leap ahead into the future. You know what I mean. You're at your son's birthday party serving cake to all the four-year-olds, and instead of enjoying the moment, you're wondering what their parents will say about your party after they've gone. Will they think the games were fun and that your home was attractive, or will they be appalled that the cake was store bought and not homemade, and wonder why you couldn't have been more creative with the decorations?
Or, in an equally possible scenario, you might be doling out cake slices to those four-year-olds while agonizing about the fight you had on the phone that morning with your mother, who wanted to come to the party but didn't want to drive in the rain. She was angry that you wouldn't pick her up and take her home, and you're feeling guilty for saying no because you had so much to do. Although you're trying to have a good time on your son's behalf, you're so lost in your own thoughts of past and future that you can barely figure out who has cake and who doesn't.
This is not what we call living consciously — that is, with intent and awareness and a focus on the here and now. That only happens when all your energy — mental and physical — is directed toward the present moment. Of course, no one does this conscious-living thing perfectly, so the idea isn't to always be grounded in the present, but to be there as often as possible, certainly more often than not; know when you're slipping out of it; and be able to bring yourself back as quickly as possible.
The goal is to stay intentionally conscious except when you purposely want to shut off your mind and attention and give yourself a large or small mental vacation. Dysregulated eaters are often confused about when to intentionally focus on the reality at hand and when to zone out — what Geneen Roth, an eating-disorders author and educator, calls "going unconscious." Because focusing on and tuning out the present are mutually exclusive mental states, we can't do both at once. Consequently, not being clear on which choice to make can lead to a paradoxical state of becoming hyperconscious of food and craving it in order to surrender consciousness through eating. This contradiction is complicated, and I'll talk more about it in a moment. For now, suffice it to say that dysregulated eaters need to learn the skill of staying mentally in the present both with food and in other areas of life unless they have a valid purpose for intentionally thinking about the past or future.
* * *
Get Smart!
Do you live consciously in the here and now, or are you on autopilot most of the time? Have you tried to live more consciously? What more could you do to make this happen? What's the difference in your eating habits when you're aware or unaware? What's the difference in your life when you are mindful or functioning on automatic?
* * *
### Is It a Weird Kind of Magic Trick That We're Physically in the Present, Yet Manage to Mentally Disappear?
Yes, there is a kind of magic to thinking you're in the present while not being there mentally, or operating so automatically that you're unaware of being wherever you are. There are two ways we miss out on the here and now. Both are perfectly natural, human tendencies and cannot be entirely driven out of us like evil spirits. But with skill and practice, they can be limited in order to enhance our eating and our lives.
#### Living on Autopilot
When we live on autopilot, we're not necessarily dwelling on the past or catapulting our thoughts into the future. In fact, our minds are in a kind of netherworld, as if our brains have shifted, unbeknownst to us, into pause mode. We're not consciously thinking about other things, nor are we concentrating on what we're doing. If someone asks what we're thinking about, we might give her a startled look and laugh sheepishly. "I'm not sure," "Nothing really," or "I have no idea," we might say. Although we've disconnected our thinking from the present, we haven't really connected it to anything else either.
This experience often happens when you're doing routine tasks like mowing the lawn, ironing, or chopping vegetables. You're not cogitating on something else; in fact, you're not cogitating at all. Your body is definitely engaged in activity, but your mind isn't. This is not necessarily a negative occurrence. There's a distinct pleasure in doing repetitive tasks that demand little attention. They give your busy, weary mind a nice little break, perhaps the equivalent of taking an intentional catnap when you're tired in the middle of a hectic afternoon. When you purposely choose to shut off your mind and let your body do its thing, you may benefit greatly from the experience.
However, when you live on autopilot, whether you've intentionally chosen to or not, that's quite another thing. And this is often what dysregulated eaters do. They feel such pressure to get so many things done so perfectly that they whiz through the day without putting attention on any of them. If they're not zipping the kids to school, they're rushing off to work and racing through the day to get back to pick up the kids and throw together dinner. They do what they think they ought to do and miss out mentally on the actual experience of doing it. That's why at the dinner table when you ask your son how his day went, he says, "Mo-om, I told you all about it in the car on the way home. Weren't you listening?" Well, no, Mom, actually you weren't. In fact, you have no idea what was in your mind, except you seemed to have paid enough attention to driving to have ferried the kids home safely.
In _Care of the Soul: A Guide for Cultivating Depth and Sacredness in Everyday Life_ , a book that sounds religious but is quite secular, author Thomas Moore shines a light on how little attention we pay to everyday life and how greatly we suffer for it. He teaches us to see each moment as sacred, as precious, as a building block of life. As Moore sees it, we want to be as fully engaged in washing dishes as in life's grander moments, as present to suffering as to joy, as eager to partake in the mundane, routine aspects of life as we are to experience the extraordinary, unique, mind-blowing ones.
This makes sense, since there are many more humdrum moments than astounding or exceptional ones. If we pay attention only to the "big events," then we miss most of what happens to us. However, if we treat every moment as worthwhile, with full engagement, we live more richly and fully. Dysregulated eaters often are bored with their lives and are constantly seeking stimulation because they're cut off from the now. Moore says that stimulation is right there in front of your face, in everything you do — brushing your teeth, paying bills, changing a lightbulb, getting your hair cut, mowing the lawn, lifting weights, or filling your car with gas — and I heartily agree with him.
By fading in and out of life, you miss so much and often feel unfulfilled, not because your life isn't interesting or full, but because you exhibit very little interest in or connection to it. And this lack of attentiveness to each and every aspect of life leaves you feeling empty and apathetic, creating a kind of ennui that too often makes you look to food for engagement and excitement. When you remain emotionally connected to everything you do, you continuously fill up in small ways, and you avoid the emptiness that might drive you to mindless eating.
#### Living in the Past or the Future
Obviously, you recognize that I'm not referring to time travel — which, as far as I know, is not possible. When I talk about living in the past or future, I expect you understand that I mean shifting mental awareness away from the present and recalling what's come before or anticipating what lies ahead. This is a mental process, not a physical act, and there's nothing wrong with thinking about the past or the future if we do it intentionally and purposefully.
There are lots of excellent reasons to deliberately call up memories. We enjoy reminiscing about earlier times because it brings us pleasure and is often a bonding experience if we do it with others we knew way back when. Even on a practical level, thinking about the past is important for, say, recalling how we did something at work that turned out well, so that we can repeat our success. Moreover, as the adage says, how can we know where we're going if we don't know where we've been?
What I'm talking about here is the act of purposely moving your attention from the present to what came before. On a biological level, that means using the parts of our brains that store both the data and the affect from our history, what we call memory. For instance, while one part of your brain has stored a great recipe for the rice pudding you used to make as a teenager for Thanksgiving, another has stowed away the loving feelings you had toward your now deceased brother, who used to love making rice pudding with you. When we intentionally seek out memories, we generally recognize why we're doing so. You want to pull up that rice pudding recipe because you're having the family over for Thanksgiving and everyone loves it. You want to linger over memories of your brother because that makes you feel closer to him and miss him less.
But what about those times you suddenly experience a memory, and it snatches you up and plunks you on its merry-go-round and won't let you off? Like the time you totally forgot what you were talking about while you were chairing a business meeting, and you couldn't seem to find the point you wanted to make anywhere in your notes. How embarrassing was that! It's awful to remember your shame, yet somehow you can't stop recycling the memory. Or perhaps you keep going over the decision you made to give notice at your job at the end of the month — you find yourself hurtling back and forth between being sure you are doing the right thing in leaving work you hate and being terrified that you'll never find another job and will be living on the streets after you get evicted from your apartment.
Being hijacked by memory is an unpleasant experience. There's a sense of helplessness, panic, and frustration, which feels a great deal like being trapped in a compulsion. This discomfort is often caused by the overwhelming drive to travel back in time and undo what you did. To use the example of your mind going blank while you're chairing a meeting, all you want to do is return to standing in front of your subordinates and colleagues and make every point in your notes clearly and fluidly, so that everyone at the table is roused to applause, so impressed are they with your knowledge and leadership skills. It's as if your memory isn't playing fair with you, leading you on by bringing up the experience as if it's actually happening, but not letting you intervene to make the story come out with a happier ending.
Regularly engaging in this kind of mental-do-over mind game can consume so much energy that you have little left over for the present. Moreover, every time we get seduced into believing that we can change what came before by ruminating on it, we are disappointed. What an awful way to live! More than likely, these frustrating feelings will keep your mood spiraling further and further down until you're so miserable that you don't care if food isn't a great choice. It feels better than being trapped in what seems like all the wrongness of your life.
Notice how different ruminating about your business situation is from intentionally trying to recall a piece of information or a pleasurable memory. In the former case, you feel powerless to make your life better, and in the latter you use your power to actually make it better. What is the singular difference between these forays into memory? In one, you're inadvertently slipping into a memory fragment with the unrecognized desire to engage in an impossible task: altering the past. In the other, you're expressly replaying a memory that's perfectly acceptable as is.
As unpleasant as it is to be trapped in the time warp of memory, it's equally agonizing to try to catapult yourself into the future, another mission impossible. You not only can't rewind and relive your life, but you can't fast-forward it either. Our efforts to do this spring from wanting to make sure life will be okay (actually, that we will be okay) when we get to wherever we think we're going. Here are some examples. You obsess about being a specific weight or clothes size three months from the present, the date of the cruise you've booked with your high school friends. Or maybe you start planning out every minute of the visit your parents are making to your new house next weekend, so you can be sure they'll have a pleasant time. Or perhaps you can't stop thinking about your upcoming gastric bypass surgery.
Do you know what all these situations have in common? In each of them, you want to rush into the future and fix it, so that when you get there you'll be fine. Think about my explanation for a minute and see if it makes sense. Isn't that exactly what you do when you worry about the future? Your motivation is to think about it a lot _now_ so it will be fine _when it happens_. It's as if your thoughts could somehow travel through space to a time that has yet to happen, solely to ensure that things will go swimmingly when you get there.
Please don't feel bad if you engage in this kind of mental gymnastics on a regular basis. It's another one of those quirks of being human. But it doesn't make your life work better, does it? And for sure, putting pressure on yourself to make life come out right doesn't help you eat "normally." Immersing yourself in expending that kind of mental energy fruitlessly on a regular basis depletes you and is a recipe for abusing food.
* * *
Get Smart!
Do you ruminate about the past or worry incessantly about the future? Do you understand why you engage in these behaviors? How do they affect your eating?
* * *
### Am I Doomed to Wander through Time Forever, or Can I Learn to Stay Put in the Here and Now?
Of course you can learn the skill of living consciously, as long as you're willing to make some alterations in your life and not expect to be perfectly present all the time. One of the best books on the subject of staying conscious is _The Power of Now: A Guide to Spiritual Enlightenment_ by Eckhart Tolle. When I first heard of his book, it was already a huge bestseller, but I couldn't imagine reading an entire book whose premise was the need to pay attention to the present. The whole subject seemed so simplistic, and I was sure I'd be bored silly reading the book. But one day I found it at a yard sale and said what the heck.
How wrong I was! It still seems almost impossible to believe that Tolle could say something interesting and enlightening on the subject of the power of now for nearly two hundred pages — and he's written a bestselling sequel! What he does is shift the way we think about the world, our place in it, our abilities, and how we want to live. So the question is: do you wish to live consciously? If so, then it is certainly within your grasp. If you don't, well, then, that's okay too; but by giving up this quest, you may not be able to shed your eating problems.
Assuming you do yearn for a more conscious existence, please recognize that the now is always waiting for you. You don't have to search for it. And when you're in it, you know it. We've all experienced that "awesome" sensation of being fully engaged. I felt it, powerfully, twice last night — while mesmerized by a gorgeous Florida sunset and then while sitting and watching TV and petting my cat. I was totally awestruck and wanted those moments to last forever.
These moments can happen to you anytime — while you're tinkering with the car, feeding your goldfish, getting ready for a tennis serve, cooking dinner, or cooing at your infant son. Whenever you lose track of time because you're absorbed in an activity or creative process, you're in the now. In these moments, something happens in our brains that brings us to attention and floods us with enormous pleasure. People have many names for this experience, and the ones I use are _peace_ and _awe_. If you enjoy these feelings, you will certainly feel them more often when you spend more time in the here and now. No, not every moment will be filled with splendor and contentment, but you'll learn when you've left reality and how to get back there as you deepen your connection to life and to yourself.
* * *
Get Smart!
Describe what it feels like to be wholly conscious. Name three reasons you would like to live more consciously.
* * *
### How Do I Stay Conscious When I'm Pulled in So Many Directions and Don't Even Know When I Exit Reality?
One easy way to know that you've spaced out from reality is when you find yourself ruminating about the past or worrying about the future. Remember, you can't be in two places at once! To stay put, get in the habit of asking yourself routinely during the day: _Where am I?_ Asking this question breaks the spell cast over you by wherever you've been hanging out. If you've been caught up in some memory fruitlessly trying to plot a better ending for it, you'll be automatically bounced back to the present by simply noticing where you are. If you've been attempting to stealthily creep up on the future to make it turn out all right, asking yourself where you are will yank you back to where you belong.
Here's an example. As I've been writing, I've been pulled back into the memory of a bit of tension in a group I belong to. A member took one of my comments personally and blamed me for hurting her feelings. Although I recognized that she was only trying to make herself feel better, her snarky comment kept snagging my attention away from writing this book. I'd get a few sentences down, then find myself lost in thought about how this interaction might affect the group as a whole, or how I'll handle the situation if she continues blaming me for her insecurities. This jumping back and forth from past to present to future has not been helping me get this chapter written and, moreover, has all but ruined the deep pleasure I find in writing, which usually completely captures my attention and energy. That's because writing is a very now experience!
So, every time I found myself drifting backward or propelling myself forward, I noticed the fact and returned myself to the written word. The more I did this, the less often I mentally wandered, until I stopped it completely when I was halfway through this chapter. Of course, I could have stepped away from my computer and taken a break while I considered what happened in the group and how I might respond were it to happen again, but that would have given too much power to the situation. There are times, though, that this would be the appropriate choice when something is bothering me.
The point is to break your unconscious connection to any point in time except the current moment. Consciousness awakens when you cut the cord to anywhere but now, and you cut the cord by seeking consciousness. After practicing this process until it feels natural, your mind will respond by not drifting off so often; and even if it does drift, you're far more likely to notice that you've shifted your attention and you'll readjust your mental sights more quickly.
Another way to stay with or return to the present is to use your body and your senses. Try stamping your feet or singing a song when you're in the middle of dredging up some awful memory you're trying to refashion or you're scouting the future for some disaster you're struggling to avert. What happens? Physical activity brings you right back to the present. This is why meditation is so powerful in the way it uses the breath. Paradoxically, breathing is a process that we are usually not conscious of — and that's all well and good. However, attending to the breath works wonders for bringing ourselves back whenever our minds have busted open the barn door and galloped off.
One of the best ways to stay out of memory, except when you have a reason to go there, is to understand the attraction it has for you. This process reminds me of how overeaters, when bored, will troll their refrigerators and kitchen cabinets for food. You know what I mean: how you suddenly stop what you're doing, fling open the refrigerator door, and poke around for something to eat, until you finally realize there's nothing you really want and return to whatever you were doing. That's what happens with memory. Sometimes we simply find ourselves there for no good reason. One minute we're in reality and the next, poof, we're being left at the altar, asked to resign from our job, standing at the crematorium during Mom's funeral, or lying in that lumpy hospital bed with both our legs in casts.
Why, we need to ask ourselves, are we missing out on the precious present and creating misery for ourselves? Of course, there is no logical answer to this question. We don't do it for any beneficial reason, but we might do it because we think we ought to. Maybe we're still processing the shame of being left at the altar, the panic at losing our job, the guilt of not being around when Mom died, or the helplessness of being stuck in bed unable to walk. In that case, we may think that returning to the memory will help us work through what happened. It might, but only if we are consciously choosing to think about the past in order to learn from it. It won't help a whit if we've slipped into a memory trance and have no idea why we're wandering around in a self-imposed nightmare.
Think about learning how to swim. There's a huge difference between falling or being pushed into a lake and intentionally jumping in because you want to take the plunge. The first situation leaves you feeling frightened and helpless, while the second might well lead to pleasure. If you think of memory in this light, you'll want to spend time in it only if it has important or relevant information to cough up.
The antidote to straying from consciousness into thoughts of the future is to understand what you're seeking in this process. As I've said, when you worry, you go over dozens of scenarios in your mind to try to nail down every detail, to put everything in order to ensure that when you arrive at some not-now time, all will be well. The important question is why you have this desperate drive. The answer is that you believe you'll be fine only if all is well, and that you won't be fine if it isn't.
Let this concept sink in for a minute. Isn't it true you fear that if all is not well in the future, then the applecart that will be upset will have your name on it? What are you really afraid will happen if you don't get into the college of your choice, find the right job or the right partner (or any partner at all!), finish all your errands, clean your house, have a child that turns out the way you want, make your flight, lose a certain amount of weight, spend more time with your kids, or live out all your dreams?
You're worried that you'll feel distress and unhappiness. Put another way, you're worried that you won't feel fine. So, let's follow this line of thinking. You're in the present and are expending an inordinate amount of time and energy making sure that you'll be fine at some distant date. Does that not seem a bit pointless? Does that not seem nigh impossible? How can we ever ensure, no matter how much we agonize and overplan, that life will be as we wish it to be? We might be able to if we were the only ones living on the planet, but maybe not even then. After all, there are always Mother Nature and Father Time with which to contend.
Please understand that doing something now can never, ever ensure that we'll be fine in the future. That is an impossibility, so you might as well cross that wish off your list. Am I saying that this means we can never be okay with what happens? That's a horrid thought and definitely not the point I'm trying to make.
There is a way to be fine in the future, and it's grounded in living consciously today. The only way we can guarantee we'll be fine in the future is to make sure we're fine today and every day thereafter. Try this on for size: If you're embedded only in the present and do your utmost to make yourself feel fine every moment of every day, you can't help but be fine in the future, which is the now that hasn't happened yet. After all, why wouldn't you do your best to be fine all the time? It's far more pleasant than not feeling fine, isn't it? I mean, if you had a choice to make yourself feel fine or not fine, which would you pick? You'd pick _fine_ , of course.
And that, dear reader, is my point — you know you'll be fine in the future because your way of living is to always consciously work on feeling fine in the present. Feeling fine doesn't just happen — although some people are born with a tendency to be more upbeat, optimistic, and resilient than others, and still other folks have had awful things happen to them that make feeling fine more difficult. We make ourselves feel fine no matter what happens because, well, why wouldn't we do that?
As it happens, the way to be fine now and forever is to be conscious of how you feel in the moment. Consciousness of the present not only anchors you to it but also keeps you from a futile endeavor: worrying about the future. Some helpful advice: Make conscious decisions in the now that enable you to prepare for the future as much as anyone can prepare, and know that you can always choose to make yourself feel okay.
When I explain this process to clients or students in workshops, at this point I get a most useful question: Okay, so how _do_ I make myself feel fine? If you were thinking this question, slap a gold star on your forehead, because it means you understand that you can make choices when you're conscious. For exactly how to make yourself feel better when things don't work out, when you're disappointed or hurt, I refer you to chapter 3, on emotions.
Just remember that the way to take care of your emotions is to remain conscious — that is, to be present to reality, moment by moment. In fact, in the present is the only place you can tend to your emotions. You can't change the ones that you've already felt, nor can you alter the ones you have yet to feel, no matter how much you brace yourself for them. All you can do with emotions is work with whatever you feel at the moment.
Here's an effective way to stay conscious through mindfulness. Think of thoughts as trains, and think of your mind as a train station. If you never travel by train, recall a time when you took trains or buses regularly. Did you have a certain train that took you to work or back home? When other trains stopped at the station, did you get on or let them pass? Of course, you simply watched them go by and waited for the train that would take you to your destination.
Imagine thoughts as being like that. Why engage with a thought that isn't going to take you where you want to go? If it's taking you back to some misery in your life — a failed exam, a stupid comment that hurt someone's feelings, a time you gravely disappointed yourself or others — let it go by. If it's headed for some anxiety-provoking place in your future — knee surgery, a family visit that's bound to be tense, a long overdue appointment with your doctor, or a meeting with your child's principal — let it go by, too. Hop only onto trains that are going where you want to go, to sunny, delightful places where you really want to spend emotional time!
* * *
Get Smart!
What are the signs that you've left the present and are ruminating about the past? What are the signs that you've left the present and are worrying about the future? What is your specific plan for staying conscious? How will you feel fine?
* * *
### If I'm Always Paying Attention in and to the Moment, When Do I Give My Brain Time Off for Good Behavior?
An excellent question. The way you give your brain a rest is to make a conscious choice to take time out. You decide that you've been paying a great deal of attention to minding the kids, taking care of your sick mother, tending to your overworked partner, or busting your butt at your job, and you give yourself explicit permission to chill out. You don't aim to do everything perfectly, expect to be all things to all people, push yourself beyond what's healthy, or tune out your feelings of stress or exhaustion. You observe that you're starting to feel tired and cranky and would benefit from a break.
Living consciously doesn't mean never relaxing or letting loose. Au contraire! It's the precise path to making the right choice at the right time. In my experience, dysregulated eaters do the opposite. They tend to go overboard and never, or rarely, give themselves permission to wind down, and therefore they seek the release they yearn for in mindless eating. Some of them sleepwalk through the rest of their lives as well, while others are so hypervigilant that they can't ever let loose.
When you live consciously, you choose pleasures and pastimes that are beneficial for you mentally and physically. You don't just fall into doing things because someone else is doing them, or because you've always done them. That would be the antithesis of living consciously. Rather, you notice the activities that put your brain on pause in a healthy way. What you're looking for — and there will be more about this subject in chapter 9 — are ways to zone out that leave you proud of yourself, not ashamed. You could come home from work and get drunk every night, which would definitely numb your brain, but you wouldn't feel too swell physically or mentally once the booze wore off. Food or drugs would have a similar outcome.
The truth is that dysregulated eaters don't practice turning off their brains in healthy ways. In my experience, many of them have neurotransmitter imbalances that make it difficult for them to easily chill out or perk up. The chemicals that normally have these effects aren't working very well for them, so they grab whatever seems to work, which is often food — and when I say _food_ , I mean carbohydrates, because carbohydrates have inhibitory, anxiolytic, or excitatory, dopamine-generating properties. Unfortunately, people with biochemical imbalances must work extra hard to find healthy activities to change their moods.
Another way of talking about living consciously is to say we live mindfully. I find it sadly ironic that in order to be more mind _ful_ of food, dysregulated eaters often need to become more mind _less_ in other areas of life. Sometimes they're so vigilant about doing things right (or, worse, perfectly) everywhere else that the only time they feel they can let loose is while eating. So many dysregulated eaters keep themselves on a tight leash, never getting a chance to let it all hang out. By a _tight leash_ , I don't mean they consciously make well-informed choices for themselves at every moment. I'm talking about worrying so much about others' approval, and about doing things just so, that anything that doesn't contribute to those outcomes falls by the wayside.
Living consciously means trying to be present to every minute, and not being obsessively attached to future outcomes — to success or to avoiding someone else's displeasure down the line. It means not ruining your current enjoyment by feeling guilt, shame, or remorse. These emotions are the antithesis of enjoying the here and now. Be most connected to what's at your feet, not to what's down the road those feet are traveling on. Of course, you always want to keep in mind where you're heading, and there's nothing wrong with planning for the future. But you don't want to keep your eyes glued to it so that you miss the reality at hand.
Dysregulated eaters use food in a paradoxical way in terms of consciousness. On the one hand, they can be more or less oblivious (depending on the person, circumstance, and level of awareness) to what they're eating. I recall that, at times, in my world-class binge-eating days, I ate boxes and bags of foods I barely tasted. Yet at other times, I prolonged and savored every excessive mouthful as if it were the last bite of food I'd ever take. Although much nonhunger eating is totally mindless, some of it throws dysregulated eaters into a general hyperawareness of food — that is, their desire for it is intense, which drives thoughts of anything else out of their minds, while the actual eating experience is an unconscious or only vaguely conscious one. It's as if, when they finally arrive at the actual eating experience, their brains can finally shut off — which is what they wanted to do in the first place. My take: it's often the idea of food that dysregulated eaters yearn for, not the food itself.
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Get Smart!
What do you do consciously to turn off your mind? Are these behaviors effective and healthy? Is it difficult to give yourself permission to let go and tune out? What emotions get in the way? Make a list of healthy ways to "go unconscious" that don't involve eating or thinking about food or weight.
* * *
In sum, in order to gain skills to live consciously, you have to know when you slip out of being conscious and how to reground yourself in the present. It's also helpful to know why you're tempted to not meet the moment as is. Maybe it's unpleasant to think about being single, and you'd rather daydream endlessly about the coworker you have a crush on, who's shown no discernible romantic interest in you. Maybe you, as a parent, spend so much time doubting the decisions you've made about your children that you have no energy left to enjoy your kids in the present. Finally, perhaps you're so hell-bent on making the future right that you forget to experience the now.
By slowing down and focusing all your senses on the present moment, you can live more consciously. At any time of the day, ask yourself what you see, hear, smell, and feel through touch. When you find yourself racing through the day, intentionally slow down or stop for a moment and ask, What's the rush? Rather than transitioning from one task directly to another, pause between tasks and take a few delicious deep breaths. Ah, there, now you're living consciously.
* * *
Skill Boosters
1. When you awaken, take three long, deep breaths and set your intention to live consciously all day.
2. Before you go to sleep, assess how you did in your effort to stay conscious that day. Be curious, not judgmental. Note when you were thrilled to notice you were conscious and engaged in the moment, and when you observed yourself slipping into memory or galloping off into the future.
3. Make a list of your major worries and prioritize them. Does worrying improve life? If you gave up worrying, what would happen for better or for worse? Write down three ways to be less anxious.
4. Identify memories you return to repeatedly as if you could have a do-over, and make a point of avoiding them by using the train metaphor. Remember, board only the trains that will take you to a destination you want.
5. Note when you're most conscious during the day — morning, afternoon, or evening — and in what circumstances. What enhances or hinders consciousness?
6. Take a mindfulness-meditation class.
7. Identify and practice one to three activities that keep you conscious and in the moment, and engage in them as often as you can. (Scientific studies tell us that being in nature is an especially soothing and centering experience.)
8. The next time you have an urge to eat when you're not hungry, observe your desire to be mindless, and find something to do to fill that need. Or, if you're bingeing, pull yourself out of unconsciousness and focus your full attention on what you're doing, without judging it. Tell yourself, "I love you."
9. Read books on mindful eating, and put signs up around your dining area to remind yourself to stay conscious of food — "How hungry am I? Am I enjoying my food? Slow down and chew! Am I still hungry, or am I done?" Ask intimates to gently point out when you are not living consciously. Keep a journal of conscious living.
10. Use a timer, and see if you can stay conscious and alert while thinking of nothing, without your mind drifting to other thoughts. Practice increasing the amount of time you can do this.
11. Learn and practice ways to self-soothe and use positive self-talk, so that you're always moving toward feeling fine.
* * *
In chapter 5, you'll learn how to enjoy interdependent and intimate relationships.
## CHAPTER 5
## Building and Maintaining Relationships
## I Already Have a Great Relationship. . .with My Refrigerator!
Undoubtedly, at one time or another, you have thought of food as your best friend. It's there for the taking — or it's not more than a brief walk or car ride away — in all its glory, just waiting for you to pick it up for a hot date. Unlike certain people, food gives its all to you, and you perceive it as devoting itself completely to making you feel better. It has no needs of its own and offers no rebuffs or judgments. It lets you use it for pretty much whatever you please, and it never complains.
But if food were a true friend, you wouldn't be reading this book. I don't know where the idea of "food as friend" began, but it really is silly, when you think of what the term _friend_ means. Friends have our back, protect us from self-delusion, offer their wisdom, and want the best for us. Food may be a comfort, but it's never a friend.
Yet it's understandable that we may be drawn to it when true friends or intimates are not readily available — or when we lack a nurturing self to take care of us. Problems arise when you come to believe that food is better than people at helping you cope with life, you dream and fantasize about eating rather than enjoy real relationships, and you give up being with people in order to be with food. Sadly, you probably have had the experience of hanging out with friends or family, or even a date, and not having a bad time, when seemingly out of the blue, food cravings erupt even though you're not hungry. Maybe last night's leftovers are in the fridge, calling out to you, or maybe you have a vision of swinging by a fast-food joint and grabbing some takeout. Suddenly, the people around you seem to fall out of focus, and every fiber of your being is screaming to get _out of where you are_ and to cozy up to a sweet or other treat. That's how it goes when cravings overpower our food-warped minds.
Even the idea of having a _relationship_ with food is pretty bizarre, because we usually save the term for interactions in which there's an exchange, a give-and-take. Truth is, you're hardly having a "relationship" with food. You're using it (ineffectively, I might add) to meet your needs. If you want to become skilled at building and maintaining relationships, you'll need to give up the idea that you have one with things like Pop-Tarts, Doritos, and pizza. Remember, Ben, Jerry, and Mrs. Fields are someone's friends, but not yours. People who are skilled at enjoying relationships might enjoy Ben and Jerry or Mrs. Fields, but they don't mistake them for their best buds.
* * *
Get Smart!
Do you think of food as a friend? List the ways in which it isn't a friend to you.
* * *
Genuine relationships give you so much more than food can or ever will. When you turn to food, you're seeking comfort, validation, quality companionship, empathy, understanding, fun, laughs, stimulation, excitement, and nurturance; you want to feel loved and valued. But while you may not have received these things from many folks in your life, this doesn't mean that you, as a human renting space on the planet, don't need and deserve them — or that you can't get them from people. With practice, you will.
### What If I'm Not Great at Building and Maintaining Relationships, but I'm Super at Using Food to Make Myself Feel Better?
Learning a skill begins by admitting you know very little about a subject. So, you're in the right place, at the starting line. And you'll progress from here. Let's first figure out why you haven't been terrific at developing or holding on to relationships. After all, there must be a reason. The following questions will help.
1. What were your relationships with your parents like?
2. If you had siblings, what kind of relationships did you have with them?
3. How did your parents relate to each other?
4. How did family members relate to each other?
5. Did you feel loved and lovable as a child, that your boundaries were honored, and that you were respected, precious, and special?
6. Were your parents skilled at forming relationships outside the family? Did they have close, loving friendships and good relationships with their families?
7. While growing up, did you have solid friendships with classmates, kids in your neighborhood, teammates, or cousins?
8. Did you grow up in a stable community, or did you move from one place to another so that you had to make — and leave — friends over and over, and it was difficult to maintain ongoing relationships?
9. Were you shy or self-conscious around, or afraid of, other people, or were you bullied or teased a great deal as a child or adolescent?
10. Were you prevented or discouraged by your parents from making close friendships outside the home?
I hope these questions get you thinking about what modeling and instruction you received while growing up, which we might as well call Relationships 101. If you have trouble with closeness, you probably had a number of things going on in childhood. The first is that you may have had poor role models — parents with weak interpersonal skills who therefore couldn't teach you what you needed to know. If they didn't take much interest in you, or took too much interest and didn't give you sufficient space, you may not know how to regulate closeness and distance. If you grew up thinking that screaming and yelling is how people get their needs met, or that withdrawing and becoming isolated from others is how disagreements get settled, you were not taught a useful way of settling differences. If your childhood was replete with lies, deceit, rigid dictums, manipulation, neglect, mixed messages, intentional hurt, retaliation, betrayal, or physical or emotional abandonment, you likely came away from it with the wrong idea that people can't be trusted.
If fur was always flying in your home, it's easy to see why you might have technical difficulties with intimacy now. But there's another kind of early family dysfunction that occurs just as frequently and causes as much relational distress down the road. That's when parents deny their feelings or don't talk openly about them. When you grow up in a family that fights all the time, you at least know that expressing upset feelings is okay. But when you're raised in a household in which everything _seems_ fine (except for the vague, subliminal tensions you just can't quite put your finger on), and you're constantly assured that everything is "just fine," it's harder to see what's missing. I bet you can guess what was lacking: the expression of emotions. Many families act as if it's legitimate to put everything on the table except what's going on inside them.
Perhaps you'd come home from school and find Mom crying in her bedroom, but when you'd ask what was wrong, she'd sniffle and say, "Nothing," or smile and insist, "I'm okay, really." Maybe the chill in the air from Mom and Dad giving each other the silent treatment for days on end was enough to make your home feel like an igloo, and it was nerve-racking trying to figure out exactly how bad things were between them. At least in a family in which emotions run high and are openly expressed, you know that people have and are allowed to show feelings. When you're raised in a home in which emotions seem not to exist, you start to question your own feelings — and everyone else's.
I hope you're starting to understand the myriad ways that your parents' lack of relationship skills may have been passed down to you. If you learned that getting attached to people would only make life worse, no wonder you'd choose cheesecake over a crony any day of the week. If you view relationships as iffy at best, and downright dangerous and scary at worst, it makes sense that you'd steer clear of them.
* * *
Get Smart!
How did the way your parents related to you, each other, and other family members affect your ability to be in a healthy relationship now? How did their expression or nonexpression of emotions affect your relational skill set?
* * *
### What Skills Do I Need to Learn in Order to Have Great Relationships?
Before I answer this question, take a moment to consider someone or several people you know who seem successful in developing and holding on to quality relationships. Maybe this role model is a distant relative, neighbor, coworker, or former teacher or coach. Think specifically about what makes you believe this individual excels at relationships. If you know several "stars" in this department, use them all to help you develop a mental checklist of qualities that help people relate well to others. This checklist will help you see that it's not essential to be outgoing, charming, charismatic, or smart in order to have positive, lasting relationships. In truth, relationship skills are not determined by whether your still waters run deep or you enjoy being the belle of the ball, by whether you have a GED or a PhD, or by whether you're well traveled or have never left your hometown. I guarantee that you have what it takes to develop and keep high-quality relationships. All you need to do is recognize what to look for in people that tells you they have reasonably good relational skills and some practice.
#### 1. Look for People You Can Trust Emotionally
Trust is the basis of every relationship. If I get on an airplane, I trust that the pilot will know how to fly and land the plane. If I go to a dentist, I trust that he or she is competent to fill my cavities and adjust my bite correctly. I know that's not the kind of trust we think of when we talk about intimate relationships, but it has a strong bearing on the subject. What I'm saying is that we need to be able to trust people to represent themselves accurately. When folks say they're kind and caring, it's vital for them to back it up with behavior that supports this statement. To trust people, we must know that who they think and say they are is reinforced by their actions.
Sometimes people tell us right off the bat that they're not good at long-term relationships, are scared of intimacy, aren't the marrying kind, or can't seem to get the hang of monogamy, and so on. We can usually trust that they're being honest, and we should feel grateful that they've given us fair warning about their relational deficits. But it's harder to spot people who aren't honest when they portray themselves in a loving, caring light. It's natural to want to trust that we won't have the wool pulled over our eyes, and that someone will continue to be our Prince or Princess Charming.
Learning how to trust is a skill. There are scores of ways you could have learned not to trust people in childhood, and I'm not going to describe them all here. If you don't trust people, you usually know it, to a greater or lesser extent, so let's start from the premise that you don't, and that you want to become skilled at trusting.
Start by noticing your gut impression when you initially meet someone. Simply observe the feeling that comes up, without judging or liking it. It can be especially difficult to make romantic connections when you're immediately attracted to someone and forget there's more to a relationship than physicality. So, along with noticing that someone is hot, observe his or her appearance and what message it's giving. Listen to how this person speaks — both the words and the tone — and pay attention to how he or she interacts with you and with others. Is this person kind and empathetic, or does he or she ridicule people with sarcasm or barely respond when others are speaking? Does this person love to tease people even when they don't seem to enjoy it? Although you're not purposely asking yourself if you can trust him or her — whether we're talking about a neighbor, coworker, or date — you're registering whether there's anything that would indicate this individual is not emotionally trustworthy.
Many people go wrong by trusting folks that everybody else finds untrustworthy. If your boss has a reputation for flirting with supervisees, even if he's been a perfect gentleman around you, you don't want to ignore the possibility that he will come on to you at some point. If the neighbors on your right and left tell you that the swell guy across the street has borrowed money from them and not returned it, tuck that nugget of information away, because it's an indication of his character. If your coworker gossips about everyone, it's not smart to trust her with your darkest secrets. Well, you get the idea.
You'll also want to be on top of your own issues, because we often take things personally and jump to (wrong) conclusions. For example, if you ask a new friend for help and she explains why she can't give it to you, don't immediately assume she's selfish and can't be counted on. If you have every reason to believe she's friendship material, you would be wise to give her another try. And this brings us to the most important way we gain information about people: by noticing their patterns of behavior. When I first moved to Sarasota, I met lots of people and put effort into making friends — going out of my way to speak with those who appeared to have potential and suggesting we make plans to get together. But I was careful to go slowly and wait long enough to see acquaintances over time before I made up my mind about how they fit into my life. Patterns take a while to develop, from weeks to months. People don't have to be perfect to make it onto my friend list, but they need to show a pattern of honesty, caring, and reciprocity before I would even consider counting them as intimates.
* * *
Get Smart!
How trusting are you about personal matters? If you'd like to be more trusting, what holds you back? Fear of being hurt? Do you have a pattern of trusting too quickly, then getting disappointed?
* * *
#### 2. Look for People Who Know How and When to Share about Their Personal Lives and Are Eager to Hear about Yours
I was once sitting in the lobby of an office building when the woman next to me struck up a conversation, giving me way more information than I could possibly want about her dysfunctional life. We all know people like that, who are only too happy to spill their guts indiscriminately. Alternatively, I've had acquaintances for years who barely said a word about their personal lives. They're always "fine," and when others discuss their own problems, these people go silent. Sometimes they ask a million questions about others, but never quite get around to sharing what's going on in their own little world. They may be great friendship material, but not for me. If I'm going to open up, it'll have to be even steven.
Once again, this is where patterns are important. I've met people who take a while to warm up, and it may take several interactions for them to slowly let me into their personal world. That's fine with me. If I see they're moving in that direction — if each time we meet they seem a bit more forthcoming — I'm game for hanging in there. I've had a few friends over the years who grew up in families where no one ever talked about their feelings, so they go really slowly when moving from acquaintanceship to intimacy, and I don't blame them. Just telling me that about their early upbringing is a trust-generating disclosure of vulnerability.
Many dysregulated eaters with dysfunctional childhoods have difficulty sharing. They fear they'll be rebuffed or ridiculed, or that their feelings will be a burden to others. When you feel this way, it's time to move out of memory and into the present and objectively gauge the other person in reality. Do they complain about you dumping your problems on them? Do they shut you and others off every time something personal is said? If someone cuts you off every time you share something intimate, that's different from telling you they can't always talk when you phone to ask for their advice on a personal matter.
The biggest relational difficulty I've observed in dysregulated eaters is that they often get into one-sided relationships in which they're the helpers and others are the helped. Of course, you don't want to be making mental tick marks when someone takes care of you or you take care of them. But you do want the feeling of mutuality and reciprocity. One of the worst positions you can put yourself in is being everyone's confidant. When you are, you have not only your problems but also theirs to contend with, and nowhere to vent. And once you're in that situation, how long do you think it will take before you're speeding toward a carbohydrate fix?
* * *
Get Smart!
How well balanced are your relationships in terms of sharing? Are you more often the helper than the helped because you feel less vulnerable and more powerful in that role? If you don't share when you need to, how does that affect your eating habits and your emotional stability?
* * *
#### 3. Look for People Who Feel and Express Empathy
Clients often ask about the difference between sympathy and empathy. I explain it this way: with sympathy, someone feels bad _for_ you; and with empathy, he or she feels bad _with_ you. When something goes wrong, you want someone to validate what you are experiencing, which tells you that someone else has felt what you've felt or pretty darn close to it. For example, your friend may not have lost a child as you have, but he may have lost intimates that he loved deeply; and he can use those feelings to empathize with you.
Sympathy is fine as far as it goes, but when it comes to friendships, you really want someone who's able to slip on their hip boots and slog around in the muck with you. Another trait that is not empathy is what's called in clinical terms "parallel play." That's when you say that you just found out you may lose your job, and someone immediately shares their experience with having been laid off or fired. Usually they're trying to be helpful, and may eventually circle back to asking more about your situation and offering emotional validation and support. But sometimes they just continue on with their job-loss experience, and what you end up with is not a deep, emotional, satisfying connection but a recitation of your woes and a parallel recitation of theirs with little emotional exchange occurring.
This kind of dynamic is common in relationships and can be highly unsatisfying. You feel as if someone is in the ballpark with you — that is, on the same subject — but is not quite connecting with you. You may even sense that you've triggered something that makes it difficult for her to give you what you need — empathy. In these cases, the deficit is often not yours but hers. If someone isn't empathic and a good listener, she will never be the kind of friend you deserve.
* * *
Get Smart!
Do people you're close to empathize with you, or do you wind up with sympathy or parallel play? How do you feel when someone empathizes with you? Why would you stay in a close relationship with someone who doesn't?
* * *
#### 4. Look for People Who Have Good Boundaries
We first learn about boundaries through physical space. Your sister insists that you "stay out of my room!" Or your parents remind you not to leave your stuff in the common areas of the house. When you're raking leaves or shoveling snow, you learn pretty quickly where your neighbor's yard ends and yours begins. You come to recognize boundaries in school when you're told what you can and can't do on school property. Boundaries tell us: "this is my space" and "this is your space."
Of course, boundaries can be either rigid or flexible. If you're playing in a tennis tournament, you sure don't want to step over the line as you serve the ball. And there can be plenty of jurisdictional issues about where a crime has been committed, even if it's only a few yards over a state line. On the other hand, if you're sharing a dorm room, your things might accidentally end up on your roommate's side or his on yours, and you simply sort out what belongs to whom as you go along.
Boundaries regarding people are a bit more complex. Essentially, we're talking about folks knowing where their needs end and yours begin, and vice versa. Here are some examples of people who don't have good boundaries: A mother who tells her daughter, as I heard one say, "I'm cold. Put on a sweater." A father who blows up when you tell him that you decided not to attend his alma mater but wish to instead go to another college that is a better fit for you. A friend who steals your boyfriend. A coworker or boss who claims credit for a job well done — by you.
Some people simply think "what's yours is mine." They may also think "what's mine is yours," and maybe you can live with this and maybe you can't. But some people with poor boundaries really believe "what's yours is mine and what's mine is mine." As you notice, this leaves them with everything and you with nothing. This situation occurs when someone talks all about himself and doesn't let you get a word in edgewise. It arises when your roommate borrows your favorite purse but makes it clear you're to stay away from her stuff.
Boundary issues also surface when someone talks over you, eats your food without asking first if it's okay with you, copies your school or work ideas, and takes over a physical space that is yours. Needless to say, if someone touches you inappropriately, that's a mega boundary violation. Someone who wants to make all your decisions for you, or who always has to have things his way, is also exhibiting poor boundaries.
Two kinds of boundaries that portend relationship difficulties are those that are way too tight and those that are way too loose. Some people are inflexible; and when you're with them, things always have to be a certain way — their way. They need to drive and to pick the restaurant, won't share about a particular issue unless you've given your opinion first, and are generally pretty tightly wrapped in social settings. Other folks are loosey-goosey and have no clue that you're mortified when they share their sexual fantasies with your friends they've just met or flirt outrageously with your new boyfriend and then tell you later to stop being so uptight.
When assessing someone's boundaries, you'll pretty much want them to match yours. Sure, opposites attract, and someone with tight boundaries might get a kick out of someone who exhibits more flexibility, but these two personalities might not make it as best buds. Equally, if you're loud, and people at the other end of the bar can generally hear your every whisper, you might have difficulty being close friends with someone inclined to keep things to himself.
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Get Smart!
What kind of people are you drawn to, those with tight or loose boundaries? How have boundaries been a problem for you in relationships? How will you spot and avoid these problems in the future?
* * *
Although the skills I've just described will help you judge whether other people are relationship material, they also apply to you: are you trustworthy, empathic, willing to share and be vulnerable, and do you have reasonably good boundaries? More to the point for dysregulated eaters: do you set boundaries between yourself and others and hold people to them, or do you let others trample all over your borders and desires?
In my clinical experience, dysregulated eaters often don't have good boundaries or engage in relationships with people who do. It's more common for them to feel used and get stuck in a victim mentality, which leads to tremendous internal distress and an increased likelihood that they'll find solace in food.
You can't expect to connect with people who have good boundaries if you don't exhibit them yourself. Instead, you will likely find folks who have relationships _only_ with those who have poor boundaries. Who else would put up with them?
### Can't Expecting Too Much or Too Little of Relationships Drive Me to a Food Fix?
The answer, as you know too well, is that misplaced expectations certainly can trigger mindless eating. In a healthy partnership, the expectations of both parties are reasonable. Not all the time, of course, but most of the time. Work, romantic, and friendly relationships don't always start off with realistic expectations, but if both individuals are willing to talk about and work on improving their expectations, they can become a nonissue.
Say, for example, you are a teacher, and most of the teachers at your school work late and generally leave the building with a load of work to do at home. Naturally, the school principal is used to this and expects you, as the new teacher, to do the same. But if you became a teacher because you thought you'd get to leave school early and have time to play, well, I see storm clouds on the horizon. Or perhaps, because you're a terrific gourmet cook, your new housemates expect you to do most of the food prep, while you're thrilled to be living with three other people precisely because you're looking forward to the cooking being shared by all. Maybe you think that, because you've been dating the same woman for six months, the two of you are monogamous. That's the kind of subject you'd want to have a conversation about, not only because you don't want to get your heart broken but also because you don't want a sexually transmitted disease.
We all have conscious and unconscious expectations, and sometimes we don't even know we're expecting too little of a relationship. You may think the worst thing you can do is expect too much and be disappointed, but how about when you expect too little — and that's what you get? Either way, you're going to be disappointed! So often I hear clients describe not getting what they want from friends, lovers, spouses, bosses, parents, children, and coworkers, and it's because they've settled for so much less than they deserve. It's also crucial that you do not pick people you know can't give you what you want. Don't go into a relationship with the expectation that you can change a person who everyone, but you, can see has certain characteristics — ones that are clearly wrong for you (and maybe for anybody!).
One of the easiest skills to learn, in regard to expectations, is how to share them. Put your desires on the table and encourage the other person to do the same, whether that person is your supervisor, a budding friend, or your neighbor next door who seems to have more than a neighborly interest in you. It's fine to try to sense what someone wants, but there's nothing like having a heart-to-heart to make things perfectly clear.
* * *
Get Smart!
Do you go into relationships with your eyes wide open or tightly shut? Do you have realistic expectations in most relationships, or are you generally wearing rose-colored glasses? How comfortable are you when talking openly about what you expect and when asking other people about their expectations? Are you willing to speak openly in order to have better relationships?
* * *
### What Other Relationship Skills Do I Need?
Which skills will keep your relationships in good working order, and keep _you_ from eating your heart out? People who do well in relationships usually understand something about relational dynamics. For example, when people are stressed, in your experience, do they become upset and angry, or mellow? We all know exactly what happens, because we've experienced stress ourselves and recognize that the pressure can build up until we feel ready to explode. So try to understand the notion that, when your mother is preparing for a dinner party, she might not have time to check out your hot new shoes. Likewise, if someone lost his job last week, he might not want to hear that your boss is sending you to Paris next month to close a deal.
Also useful is your understanding that people often say things in the heat of the moment that they deeply regret, and usually feel awful about it, when they calm down. Don't get hung up on what someone said, but focus on her reaction when you explain how hurt you felt. Watch for patterns. If someone is defensive and often blows up at you, stay away — even if she apologizes profusely every time. To lean heavily on a few clichés, apologies are a dime a dozen, but someone who changes his behavior at your request is putting his money where his mouth is and is worth his weight in gold.
Remember, too, that what comes out of someone's mouth, even if it has your name attached to it, is always about the person speaking. It comes from _their_ brain, _their_ perception, and _their_ viewpoint. This means it's not necessarily true. As one person said on my Food and Feelings message board, "Just because someone says you're a car, do you sleep in the garage?" People can say all sorts of things about you, and every word may be untrue. The better you know yourself, and the more honestly you can identify your strengths and weaknesses, the better you'll recognize when what people say about you is spot on, or when they're just blowing smoke through their hats to make themselves feel better. I assure you, there's a great deal of that going around.
Here's a psychologically oriented skill that will help you weather relationships: recognizing that people often say rotten things about you that are true of themselves. This process is called _projection_ and is triggered by a person's discomfort about a trait they have — say, they can be unnecessarily sarcastic. They may accuse you, the least derisive person in the universe, of mocking others. Don't buy into it if it's not who you know yourself to be, no matter how much they insist. Understand that they're simply uncomfortable with who they are and will use this defense mechanism often to feel more comfortable about themselves.
Also remember that there are many Very Difficult People (VDPs) in this world. They are everywhere and, over time, are easily identifiable if you know how to spot them. In most cases, they're difficult for everyone, but not always. Sometimes their positive traits are so overwhelmingly lovable that you might put up with their negative ones. Say you know someone who can be counted on to make you laugh, someone who can't help looking on the bright side of life, but who always needs to be right. A person who can't stand being wrong is generally difficult to be around, but you may choose to tolerate this particular friend because of her upbeat, fun-loving nature.
Another kind of VDP may be your boss who is nitpicky and critical: no one wants to work for him when you're on vacation. The pleasing part of him is that if someone says anything bad about you, he'll go to bat for you every time. He's as loyal as your dog, Rover, and you can't say that about everyone. You know how difficult he can be, but you tolerate working for him because you also know he's got your back.
Then again, how about your next-door neighbor who's standoffish? The rest of the block thinks he's a cold fish, but he was the only one on the block to shovel your driveway when you were bedridden with pneumonia. You can barely get a wave out of him in the morning when you both head off to work, but there he was, knee deep in snow, digging out your car just in case you were feeling better and needed it, when all your so-called friendly neighbors were drinking hot cocoa inside their toasty kitchens.
My point is that people are complicated and so are relationships. Just because you have excellent skills in building and maintaining relationships doesn't mean they'll run smoothly or that things won't go awry. One thing I can promise you is that, even in the best of relationships, you will at times feel hurt, angry, taken aback, frustrated, and disappointed. In a healthy relationship, you won't feel these emotions often, and when you do you'll be able to talk through your feelings with the other person and usually come away feeling better about yourself and even the relationship. The fact is that a thorough airing of grievances can go a long way toward strengthening a relationship and making two people feel closer. All relationships have their ups and downs; that's the nature of the beast.
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Get Smart!
Do you take what others say too personally or gloss over hurts that occur repeatedly? Are you attracted to VDPs in the hopes of changing them? How astute are you at spotting and handling VDPs?
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* * *
Skill Boosters
1. Make a list of the reasons that food is not your friend. Make a list of the reasons that friendship is better than nonhunger eating.
2. List the qualities you would like in a friend, boss, coworker, date, and mate.
3. What did your parents tell you about trust in relationships growing up?
4. What behaviors or attitudes did your parents model with each other that gave you a positive or negative impression of relationships?
5. Reflect on what your own experiences with relationships have taught you, especially the part you played in creating negative experiences.
6. Notice the qualities of people with whom you're in relationships, and identify which qualities improve connection and which ones detract from it.
7. Share your expectations with a boss, date, friend, neighbor, or coworker and ask them to share their expectations of the relationship with you.
8. Identify the VDPs in your life, what makes them so difficult, and how you will handle them, which may include letting them go from your life.
9. Make a point of noting when people are projecting, which means blaming you for a quality you may or may not have but they certainly do.
10. How could you improve yourself and become a better partner in any relationship?
* * *
In chapter 6, you'll learn how to avoid going to extremes, to keep yourself pleasantly centered and balanced in terms of emotions, activities, and boundaries with others.
## CHAPTER 6
## Self-Regulation
## There's Something Besides an On-Off Switch?
Self-regulation is not a term we hear often in everyday life, but it's exactly what's not happening when people have difficulty managing eating and other behaviors. If you're a dysregulated eater you may find that you have trouble knowing when enough is enough in other areas of life as well. You may overwork or underexercise, overdo taking care of friends and neglect your self-care, or spend an inordinate amount of time piddling around and too little getting major tasks done.
Of course, biology and genetics play a huge role in our relationship to food and perhaps, as well, in our ability to regulate other activities. I suspect there's a biological component to many of our proclivities, one that we'll uncover one day, just as science has discovered that the trait of sensitivity (as in, "Oh, you're just too sensitive") may have an underlying biochemical basis for some individuals. But, biology aside, being able to self-regulate is an essential skill that, when done effectively, will make your whole life far more manageable and enjoyable.
At the heart of the self-regulation conflict is a yearning for, and yo-yoing back and forth between, structure and freedom. You know how it is when you have a wild feeding frenzy and the next day vow to go on a diet, or you lounge around the house all day as you shamefully, guiltily, give in to inertia, then get with the program and work far into the night to finish your chores. Think of it this way: When you become sufficiently uncomfortable with having no rules for eating, you think that imposing many rules is the answer and you begin your new diet. When you avoid your chores for long enough, you berate yourself for being a lazy head and, in a panic, barrel through everything on your to-do list until you're ready to drop. Alternatively, you get into a routine like going to the gym or cleaning up after yourself daily until you're sick of it, then you drop it completely. The problem is that you enjoy or value routine for only a short while, until it starts to feel like pressure — or prison — and then you long to break away and be footloose and fancy-free once more.
I hope you recognize that self-regulation is, in part, about acknowledging and respecting your need — or, rather, the very human need — for both freedom _and_ structure. One is not good and the other bad; one is not better than the other. Both, in fact, are value neutral. You require some of each to lead a balanced and satisfying life.
This may be a brand-new, shocking — and, perhaps even disturbing — concept to you. Rather than hurl yourself from structure to freedom and back again, knowing you'll never stop pinging and ponging if you keep it up, consider how you might move from freedom to a teensy bit more structure, or from structure to a tad more loosening up. No need to go whole hog in either direction. Better to avoid both extremes of dysregulation and simply pay attention to sensing your need for more structure or freedom. In this case, having a little of each goes a long way toward keeping you in balance.
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Get Smart!
How do you feel when your routines become too rigid or excessive? How do you feel when you don't have or don't follow routines at all? Specifically, what could you do to remain more in balance?
* * *
Now, back to the term _self-regulation_. If you're not exactly sure what I mean by it, try these examples on for size:
• You hardly ever exercise, but you sporadically go on intense weekend boot camps, which, the next day, make your body feel as if it's been run over by a truck.
• When you join a gym, you insist on going either every day or for two hours at a clip, even though neither is feasible given your schedule, so you eventually stop going altogether.
• You rarely buy clothes for yourself, but when you do it's a mega shopping spree and you come home with oodles of items that you look at and think, "When will I ever wear this?"
• You have a hard time leaving work on time almost every night, even though pretty much everyone else zips out the door at five sharp.
• Your home is either a mess or spotless.
• You usually let other people make decisions for you, which builds up a head of resentment in you until, one day, you put your foot down and freak out everyone when you insist, irrationally, that some petty thing must be done your way.
• You spend so much time with your social network in cyberspace that you miss out on real time with friends, so you unsubscribe to all your favorite sites, until you're in such deep withdrawal that you sign up again on them all.
• You either stuff your feelings or can't stop crying, screaming, or moping around.
• You pledge to go to sleep at a reasonable hour and do it self-righteously for a week, then lapse back into hitting the sack long after your body and mind have quit for the day.
• If you can't do something perfectly, or at least exceedingly well, you either don't do it at all or you're miserable the whole time you're doing it, because you feel like a failure.
These may seem like normal, everyday actions to you, but they're examples of dysregulation. It's as if you were born with only an on-off switch and no calibration in between. If you have dimmer switches on the lights in your home, you know what I mean by what's "in between": there's a whole range between blindingly bright light and dim light in which you can barely see your hand. Just as the volume control on your iPhone or TV is a continuum, you, as a human instrument, also possess a finely tuned control mechanism, but one that's been stuck in the on-off position. The skill you want to develop is to identify, value, and use its calibrations.
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Get Smart!
Do any of the examples here ring true? As you consider activities other than eating, do you think you might have more general problems with effectively regulating yourself, other than around food?
* * *
How wonderful would it be for you to feel nuances of hunger, to not simply feel famished or stuffed but to feel all the stops along the way — not quite hungry, mildly hungry, almost full, and so on? Can you imagine how satisfying it would be to know that you'll recognize and accept when a job is finished, even if it's not done perfectly? Can you imagine listening to your body so that you'll give it just the right amount of sleep, exercise, rest, and stimulation most of the time?
By acquiring both physical and emotional self-regulation skills, you'll learn to do all that and more. In order for this to happen, however, you'll need to deeply connect to your body. I don't mean connect in the way too many dysregulated eaters connect to their bodies by judging and obsessing about their size or appearance. I mean being aware of your internal shifts in physical sensation and noticing what you feel when your body is in different states.
Here are examples of dysregulated and regulated responses. When your dysregulated self is hiking with friends, you notice that you're sadly out of shape and not keeping up, that you're huffing and puffing your way along. Every time you fall behind, you admonish yourself to catch up. You pretend you don't feel tired, and you override your muscles' screams for you to slow down or stop and rest. Rather than speak up and ask your friends to sit down for a bit and enjoy the scenery, you stay mum until your calves are burning and you're so winded you can barely speak. When you've recovered enough breath, you angrily wheeze out your belief that your friends don't care about you or they would have walked more slowly, and you announce that you're giving up and going home.
A version of yourself more skilled at self-regulation might approach this outing as follows. First off, rather than impulsively agree to the hike, you'd think long and hard beforehand about whether you'll be able to keep up with your friends, based on how fit you are compared to them. You wouldn't let shame prevent you from telling your friends ahead of time that you just might need to go a little slower than everyone else, and you would ask whether that would be okay with the group. Then you'd try to do a bit more walking or other exercise every day before the hike in order to build up some stamina, and during the hike you would go at a pace comfortable for you. You'd pay serious attention to what your body was telling you every inch of the way and even mention to your friends how you were doing so they could take your pace into account.
Can you see the difference in these examples — how you're monitoring yourself appropriately or not? I use this situation to illustrate that regulation involves more than an on-off switch, and how keenly tuned into our bodies we must be to pace ourselves effectively in any kind of activity. Moreover, there's an emotional component going on here as well. In the first instance, you're ashamed and, therefore, silent until you abruptly blow up and give up. In the second, you nonjudgmentally stay in touch with what you're feeling along the way and are comfortable sharing with others. The hike becomes not an all-or-nothing challenge but an enjoyable, satisfying physical and social activity.
* * *
Get Smart!
Do you tend to make all-or-nothing choices and see only perfection or failure? Do you often override what your body and emotions are telling you and habitually either underdo or overdo?
* * *
### How Can I Learn to Better Self-Regulate?
Try this exercise. Get up and stand near something you can hold on to, like a table or the back of a chair. Now lift one foot slightly off the ground and let go of whatever you're holding on to. Observe how your body shifts a teensy bit in one direction, then the other, to stay balanced. Okay, end of exercise. Notice how naturally you shifted your body. You didn't let yourself fall down, then get up. Your mind and body made quick, automatic observations and adjustments to maintain balance.
The skill of self-regulation requires this kind of monitoring and correction. In fact, in many ways, you already do it every day. When you're cooking and the pan gets too hot, you adjust the temperature. You don't switch it off completely, do you? Ditto when the shower water threatens to either scald or freeze you. Similarly, you shift your body around on a crowded bus or subway so that you have enough space for yourself, but you also make room for others. You may not think of these behaviors as self-regulation, but they are.
The key is to stay connected to what you're thinking and feeling physically and emotionally at all times, except when you wish to intentionally go unconscious in a healthy way (for more about living consciously, see chapter 4). To become skilled at self-regulation, you also want to frequently ask yourself if you are getting or doing too much or too little. Eventually you will sense enoughness; but to learn how this sense of sufficiency is experienced in your mind and body, you will first have to pay exquisite attention.
In fact, right now, ask yourself these questions: Am I tired of reading this chapter or eager to continue learning more? Am I tired of this book or eager to read on? Am I hungry, and if so, am I hungry enough to eat? How is my body feeling — does it want to stay in the position it's resting in or move around a bit?
What you're doing is scanning yourself to identify what's going on inside you. For now, it's important to stop whatever you're doing when you're scanning, but when you get better at it, you'll be able to do it semi-consciously — although you'll still probably want to pause at times to get a better read on exactly what you're feeling. For example, as I'm typing away, I'm noticing that I'm slightly hungry as it approaches the noon hour, when I usually break for lunch. I don't have to do anything with this information right now except to note it. The next time I notice, my hunger will likely be stronger, and at some point it will tell me to get up and eat! This skill kicked in last night as well, when I was trying my darndest to finish a chapter and was getting more and more fatigued. Finally, my body gave me the signal to go to bed and I did, knowing that in the morning I'd pick up writing where I left off. That's how self-regulation works.
A warning about the one thing you don't want to do to enhance your sense of regulation, and which, unfortunately, I suspect is a method for determining enoughness that you've been using far too long: being guided by what others say is enough for you. This dynamic generally begins in childhood when you're habitually discouraged from sensing what's enough for yourself. When you're told by others what you feel or should feel or how much or how little you should have or do, you don't develop the ability to decide for yourself. Sadly, when parents refuse to validate the idea that only you know what's enough for yourself (age appropriately, of course) and you end up following their guidelines rather than your own, you start to lose your ability to sense sufficiency.
For example, let's say that as a child you used to love drawing and could do it for hours. Sure, you would have to stop once in a while for food, sleep, school, socializing, exercise, homework, and spending time with the family. If your parents were both jocks, they might have told you that you were spending too much time drawing and that this was bad for you. Or they even might have taken away your drawing materials because you wouldn't give them up voluntarily. Their actions could certainly have given you the impression that you didn't have a good sense of what was enough or too much for you. Of course, your parents may have simply wanted you to be a well-rounded child, but that doesn't mean this is the message you received. What you may have internalized is that, left to your own devices, you didn't know when enough was enough. When parents occasionally guide us toward enoughness, it doesn't become an issue. When they do it too often, we stop trying to sense adequacy within ourselves.
Alternatively, maybe you had parents who thought you had talent at the piano (perhaps you even did) and made you practice for hours on end when you hated it. Let's say that every time you tried to get out of practicing, you were told you weren't doing enough to be successful — when you didn't care a fig if you ever played "Chopsticks" again. You probably didn't understand that your parents wanted you to use your talent, and instead you internalized the message that you didn't know what was enough for yourself. Again, if on occasion they forced you to do certain activities against your will, that's okay. But if they frequently imposed their will on you and overrode your natural inclinations, you might have come to believe that others know better than you do when enough is enough.
In both of these examples, I'm not parent bashing. I'm trying to show you how subtle messages about enoughness creep into us in childhood. What if everyone is doing more of something and you want to do less? What if everyone is doing less and you want to do more? When you were growing up, were you, for the most part, encouraged to make your own decisions regardless of others' preferences, or was only homogeneity and conformity acceptable in your family? Could you eat as much or as little as you wanted most of the time, or was there a set portion you were given, which may have left you hungry or stuffed, that you had to eat — and never mind trying to change your parents' minds?
On the other hand, maybe your parents showed little interest in helping you figure out what was the right amount of anything for you, so that you were always struggling with what was too much or too little. For instance, maybe you tried so hard to do things that were way above your academic level that you ended up feeling like a failure. A little explanation from your parents about overreaching would have come in handy right there. Or perhaps you needed a little push to try harder to make friends when you moved to a new school, and you never received it from your parents so you gave up and simply spent all your time alone in your room. Had you been encouraged a bit, that might have been just enough of a nudge to help you become more sociable in an unfamiliar environment.
* * *
Get Smart!
How did your parents influence your ability to self-regulate when you were a child? Were you regularly coerced into doing or taking too much or too little? Were you allowed, at the appropriate age, to make decisions about what was enough for yourself? What kind of role models were your parents in the self-regulation department? What effect does their past modeling in your childhood have on your self-regulation today?
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### If Nobody Taught Me to Figure Out What's Enough, How Will I Ever Know?
I've written before about what I believe many dysregulated eaters suffer from, which I call an "enough" disorder. Good thing it's treatable. So, let's get started. First off, make a list of the areas in which you don't know when enough is enough — that is, areas in which you're not sure what amount is too little, just right, or too much. Here are some possibilities: food, sleep, rest, self-care, weight, work, play, child care, exercise, medical attention, taking medication, helping friends, and carving out time for yourself. To start practicing identifying nuance, note whether you have a lot of difficulty in an area or just a little. After all, I'm sure you're better at sensing sufficiency in some areas than in others.
Now, take one particular area in which you already do fairly well, and see if you can tweak it to improve. Let's say you usually get enough sleep, but not on vacation, when you just want to stay up and have a good time — something that doesn't work so well when you awaken in the morning bleary-eyed with a full day of activity planned. How could you arrange for a better balance of sleep and wakefulness when you're on vacation? Do you need to reduce your fear of missing out on fun, or pace yourself better during the day? Maybe you could take a brief nap during the afternoon to be able to stay up later. Or maybe you could restructure your vacation so that you wouldn't be trying to cram so many activities into each day.
When you're done with that, take a more difficult area — say, making sure that you take enough time for yourself and don't lavish it all on family members. Consider how you feel about caring for yourself, especially if it causes you to feel selfish and guilty. Explore where that wrongheaded conviction comes from, and counter it with a belief like: "It's not selfish to take care of myself" or "I need to take care of myself so I can take care of others." Then make a plan to start doing less for your family and more for yourself. Maybe the kids will get to make dinner one night a week, and maybe you'll buy takeout another night so that you have time to relax after a hard day's work. Perhaps they can ride their bikes rather than expect you to shuttle them in the car to all their activities, or they can carpool with other kids. Think hard: how can I carve out important time for myself, even if I only sit in my room and stare at my navel?
I hope you're getting the idea of how to develop the skill of knowing what's enough, which is another way of saying: what is exactly right for you. Remember Goldilocks and the porridge that was "not too hot and not too cold," and the bed that was "not too little and not too big but just right"? That's what you're trying to determine — what feels just right to you. Sufficiency is a felt sense, a sensation that hits you in your gut when you know you have reached it. Until then, a little voice in your head whispers, "Okay, you can do [or have] a little more." And when you get to "just right," that voice gets louder and insists, "There, stop there, that's precisely right." Consider when you've had that feeling — with food, time at the beach, completing a work project, decorating your house, playing tennis, hanging with friends, or working in the garden. If you listen closely, you will almost always know when enough is enough. Conversely, you can be sure that if you don't pay attention to it, you will soon hear another voice insisting, "You did it again — too much, too much!"
The good news is that as you get better at tuning into sufficiency in one area, this proficiency will spread throughout other areas. Some examples:
• When you regularly stop eating at the point when you reach satiation, you may start to realize you've had it with friends who go on and on about their problems and don't leave any time for you to talk about your life. Then you might nicely let them know that you have some things you'd like to share before a call ends, or you may even cut them off and tell them you have to go because you have things to do.
• You might be doing so many activities to benefit your community that you have little time to spend with your own family. In that case, you might try cutting back your hours as a volunteer at the women's shelter gift shop. If you still don't have enough family time, maybe you'll let your term as president of the PTA run out and not seek that office again. Soon you'll recognize that doing too much of anything doesn't feel right, and you'll become better attuned to being in balance.
• If you've been a jock, you might notice that you're no longer seriously interested in competitive skiing or tennis but are really looking to enjoy yourself socially, get a good workout, and engage in these sports at a less taxing level. As you pressure yourself less in athletic prowess, you might realize how hard you've pushed yourself in most areas and decide to take it a little easier in life, period.
* * *
Get Smart!
What beliefs would you like to change in order to encourage yourself to sense when you're "just fine"? Pay particular attention to unhealthy beliefs about not deserving much, about needing to give more to others than to yourself, and about self-care equaling selfishness. How do you feel about learning to identify enoughness — excited, a little nervous, a mix of both?
* * *
### So Is That It on the Self-Regulation Front?
Well, not exactly. Not only do you want to move away from all-or-nothing thinking, as well as from underdoing and overdoing certain behaviors, but you also want to develop skills that enable you to reregulate yourself fairly easily when you become emotionally dysregulated (as we all do at times).
Emotional dysregulation is something you're all too familiar with, although you may never have called it that. To comprehend the term, you first have to understand what it means to be emotionally regulated. In that state, you might say you're on an even keel, in stasis. Either nothing in particular is distressing you, or you might notice that you're a tad miffed at your spouse or anxious about an upcoming evaluation by your boss. Either way, the feeling isn't strong enough to upset or threaten your equilibrium and inner peace. Emotion isn't spilling over and making your heart beat like a drum, or making your thoughts flit around inside your head like bats in a cave.
Dysregulation happens when your nervous system responds as if you're under attack, when you suddenly feel as if you're losing centeredness or control of your emotions. You're intensely aware of a powerful affect, and you sense a strong need for action — that is, a need to do or say something to feel better. Dysregulation can happen when you recall something that upsets you (remember our discussion of memory triggers in chapter 3) — you think back to finding pot in your daughter's makeup case this morning when you were searching for the lipstick she borrowed from you; or you are nagged by the recollection of that horrible argument you had with your college roommate when she came to visit for the weekend, and how she stormed out the door without saying good-bye; or you remember that it's the one-year anniversary of your father's death and feel grief-stricken.
Because your memory is chock-full of events that could upset (or dysregulate) you, all you need to do is light upon one of them and, wham, you're flooded with unwanted affect. Your life at the moment may actually be fine and dandy; but according to the miscue you received internally, you believe it isn't, and you're off and running. For us to become dysregulated, we don't need "bad" things to happen externally. We can cause dysregulation all by our little old selves!
But, life being what it is (unpredictable, not arranged for our comfort or convenience), events from the environment do influence us and can easily cause dysregulation: Your son's principal may call to say little Freddy got into a fight and has a bloody nose. Your boss might toss a proposal on your desk at 4:40 that needs to be proofread posthaste, not caring (or even knowing) that you have to pick up your mother from the hairdresser's at 5:00 sharp. Or it's your birthday, and your best friend may cancel dinner plans at the last minute because she has the stomach flu and is heading home for the night, leaving you with an empty evening ahead.
Both recall-triggered and externally generated events can powerfully dysregulate us emotionally. There we are, tooling along thinking all is right with the world, and suddenly we're plunging down the rabbit hole like Alice. It may literally feel as if one moment we're fine and the next we're not. Suddenly you recall something upsetting and can't imagine how you'd forgotten it for even a minute, or an external event slams into you and knocks the wind out of you. Either way, if you don't have the skills to reregulate yourself quickly, you're going to be in a pickle.
### Is There Truly a Better Reregulator Than Food?
There had better be, hadn't there? Or you've wasted your money on this book. Of course there are better ways to reregulate yourself than eating. One question I ask clients is: What would you do to feel better if there were no such thing as food in the world? That is, what if it had never existed, and you'd never heard of such a thing? Please don't answer that you wouldn't do _anything_ if food weren't around. There _is_ something called self-preservation, which would propel you to find a way back to equilibrium. Maybe it wouldn't be a healthy activity; you might choose alcohol or drugs. Or perhaps you would do something only marginally healthy, like throw yourself into work even though you're exhausted, or call all your friends even though it's the middle of the night. My point is that you would find _something_ to help you regroup, rather than suffer ongoing dysregulation.
I am intentionally not listing all the ways that you can bring emotions back into stasis. Some techniques can be found in chapter 3. Others can be found in my _Food and Feelings Workbook_. You may already know ways to comfort yourself, and you may either practice them sporadically or wish you did even that much. There is no magic way to reestablish stasis, and people who are skilled in doing so recognize that fact. Sometimes you just have to wait for an intense feeling to pass like a storm, knowing that you will feel peace once again when it's spent. Other times you'll want to take quick action so that the intensity of the emotion doesn't escalate.
One of the worst things you can tell yourself is how awful you feel or how terrible the event you're reacting to is. The goal is to minimize the threat of whatever is upsetting you — by talking to yourself or distracting yourself with an activity that will draw your attention away from your inner turmoil. Think self-soothe, self-soothe, self-soothe. And you'll notice that, eventually, the cortisol level in your body will drop, along with your heart rate, and you'll feel calmer.
Sometimes you can will yourself to calm down by acknowledging that you're dysregulated. I actually remind myself that this is what's going on in me and instruct myself to reregulate myself. This statement points my mind and body in the right direction and helps move me toward tranquility. It also puts some distance between my perception of what's going on and what really is happening. Recognizing dysregulation is key to knowing what to do with it. You're not crazy, the world isn't going to end, and life as we know it isn't going to be extinguished. You may feel crazy inside, but when you give that sensation a name — dysregulation — then you have a chance to think clearly about your options, which are based on avoiding doing or saying anything to increase your distress, and doing and saying everything healthy you can think of to decrease it.
* * *
Get Smart!
How do you feel about the term _dysregulation_? What are your own physical and mental signs of dysregulation? What are the most effective ways, not involving food, that you've found to reregulate yourself? Which ones might you try that you don't do now?
* * *
### Can I Prevent Dysregulation?
You can prevent or avoid some, but not all, dysregulation. For example, what if you know that every time you visit Grandma, who barely knows you now, and whose dementia has turned her into a confused recluse, you leave distressed and drive through the Dairy Queen to console yourself? You now understand that visiting her dysregulates your nervous system, and your upset sometimes leaches into the rest of the day. You could certainly decide to stop visiting her, but that might make you feel bad and wouldn't be fair to Grandma.
What can you do to stop yourself from feeling so upset while you're with her? You can bring a magazine to leaf through while Grandma is resting her eyes, read to her, accept that her confusion and inability to recognize you is natural, take a break during the visit to call a friend, play some music your grandmother and you can enjoy together, do deep breathing exercises, or plan an enjoyable activity that you will do right after the visit. You can remind yourself that you'll sort out all your feelings when you get home — without food as a companion. These are some, certainly not all, of the things you can do to prevent yourself from careening over an emotional cliff.
And if you're able to contain your distress fairly well, you just may be able to get it back down to zero more quickly. Let's face it, we don't think rationally when we're upset; when we're in distress, our only drive is to reduce emotional discomfort posthaste. If you end your visit to Grandma in a clearer, more tranquil state of mind — in other words, if your distress level isn't rocketing off the charts — it's more likely that you'll make the wiser choice to circumvent the Dairy Queen.
You can also prevent dysregulation by avoiding unhappy memories and not triggering upset feelings. If thinking about the money you're losing on the cruise you had to cancel owing to acute appendicitis keeps making your blood boil — need I say it? — don't continue to retrieve and replay that recollection. Remind yourself that there's nothing you can do, and that you're only ruining the pleasant present by revisiting that memory. Would you keep rewatching a movie that upset you? Of course not. Well, treat memories the same way.
People who easily become dysregulated tend to assume that all instances of distress are of equal value, when they certainly are not. One comforting strategy is to remind yourself that some things are worth getting into a lather over, and some things simply are not. Temporarily losing your phone service, for example, is likely to affect you more than having your cable TV service go on the fritz. It's not a necessity to watch TV, but you might be waiting for an important business or personal call.
Moreover, people skilled in self-regulation generally try to avoid Very Difficult People (remember the VDPs from chapter 5?), who can be exceedingly dysregulating to be around. If you have the option to do so, choose not to hang around VDPs, who often lack appropriate interpersonal skills and boundaries. In fact, many VDPs are highly dysregulated themselves or cause others to be. Skilled self-regulators go out of their way to avoid VDPs as friends, and in social or work situations they try to minimize contact with them. If the chair of your condo board is a pain in the butt, you might have to put up with him during your monthly meetings, but you surely don't have to go out for a drink with him or have him and his wife over for dinner. If your doctor's nurse is an argumentative person with a lousy disposition, recognize this fact, don't take it personally, and make sure to avoid getting into a discussion with her about anything except your specific medical matters.
Now, I bet you're thinking, "Well, what about family? What's she going to do, tell me to never see my family again?" In some cases, absolutely, but not in most cases. How wonderful that we live in a huge nation, and in a world with so many countries and continents. The fact is, in most instances, we don't need to live near family. We may wrongly believe that it's our duty, and may feel guilty as hell if we don't choose to reside near them, but it is almost always a choice.
Frankly, some family members are just too abusive to be around, and you have every right to choose not to spend time with them. But if you're thinking of estranging yourself from family, it's a good idea to first talk about it with someone you trust, and perhaps even with a professional. This is a big step for most people who undertake it. But when family members dysregulate you so much that it's difficult for you to live the functional life you want and deserve, you may need to keep your distance — at least for now. This also means no phone calls, emails, and texts.
Many people don't need to escape their families but do need to proceed with caution around them. Knowing that your family has the ability to dysregulate you is helpful, and so are all the techniques for minimizing upset feelings and reregulating yourself as quickly and deftly as possible. There's a skill you can practice called maintaining "caring distance," which is exactly that: giving family members some of what they need and only what you can give in a detached way. This is an excellent strategy for avoiding doing either too much or too little with them. The goal is to try to do right by them and by yourself without much emotional intensity.
For example, when nurses' aides change the underwear of incontinent people, they do it without a great deal of emotional attachment to their charges; they see the task as part of the job. You can think of caring for people you don't much like in this same way. You can do your duty without having intimate feelings for them. People who are skilled at self-regulation use this approach in order to stay regulated while doing what they feel is necessary to keep their own self-respect and integrity — and sanity.
One more word about how people become dysregulated: it happens when they don't take care of business and, therefore, end up creating a crisis for themselves. If you wait until the last minute to pay your taxes, you're going to feel frazzled on April 14. If you keep putting off a tooth extraction in order to avoid the pain of surgery, then the more pain your tooth gives you, the more emotionally distressed you're going to feel. If you grew up in a household in which there were frequent crises — Mom's tantrums, Dad's unemployment, evictions, or parental separations — crises may be what you're familiar with, which is why you continue to generate them to this day.
If so, talking to a professional will help you understand the roots of this pattern, and you can learn how to bring it to an end. Just because you grew up on a roller coaster doesn't mean you want to continue riding it. You'll be amazed at how reducing the number of crises in your life helps you stay regulated.
* * *
Get Smart!
Is there anyone you need to maintain a caring distance from? How will doing so help you prevent emotional dysregulation? What can you do to avoid spending time locked into memories of disturbing events that distress you? Make a list of VDPs you want to steer clear of, and another list of those you want to minimize contact with whenever possible.
* * *
Trust me, you'll feel so much better when you're more adept at self-regulation. I don't mean you'll simply be happier with how you handle food; you'll also find yourself more relaxed. You'll feel more competent to face the world and confident in your ability to handle whatever crisis comes along. You don't have to be perfect at maintaining your equilibrium, but you do want to feel skilled at it. It will take time and practice — and numerous failures that you'll learn from — so don't expect success right away. Learn from each mistake, and make note of what keeps you regulated and returns you rapidly back to stasis when you become dysregulated.
* * *
Skill Boosters
1. Make a list of the behaviors you wish to regulate — for example, your behaviors around food, at work, while parenting, while doing more for yourself and less for others, and so on.
2. Notice things that exist on a continuum: volume controls, traffic that slows down and speeds up slowly, music that fades or grows louder, waning sunsets, and so on. This will help you recognize that most of life is measured in increments.
3. Ask yourself how you're feeling at the moment, to see if you're slowly starting to dysregulate, and then do something to nip the process in the bud.
4. Rather than hold in your feelings or opinions until you're like a balloon ready to burst, let feelings out little by little until you're fairly well deflated.
5. Examine your beliefs about success, perfection, and failure and change the ones that drive you to extremes.
6. Make a point of leaving some tasks unfinished, or done imperfectly, every once in a while.
7. At least a few times a day, scan your feelings and focus on whether you're approaching satisfaction or adequacy.
8. Practice stopping eating as soon as you're no longer hungry (you'll be surprised at how little food you need to actually quell hunger).
9. Find some humor in having an enoughness disorder, instead of being ashamed of or frustrated with it.
10. If getting to bed on time is a problem, check in with yourself every fifteen minutes and, without judging or fighting it, rate your tiredness level.
11. Stop looking to others to tell you what's enough for you, and look inward instead.
12. Make a list of the physical and mental signals that indicate you're becoming emotionally dysregulated.
13. Notice when you get bored with, or tired while doing, an activity and how much longer you continue with it beyond what is satisfying or sufficient.
14. Observe when people become dysregulated around you, and the physical and emotional changes that occur in them.
15. Make a list of VDPs in your family, circle of friends and acquaintances, workplace, and community, and decide how you're going to stay regulated around them.
* * *
In chapter 7, you'll learn how to think rationally, lucidly, and to make evidence-based decisions that are in your best interest.
## CHAPTER 7
## Problem Solving and Critical Thinking
## Is Critical Thinking Different from Thinking Critically about Myself?
Much of our view of life is shaped by how we look at what we call problems. Many people see problems as pesky annoyances that keep getting in the way of permanent happiness and success — like debris on the garden path. They believe that if they could only solve all their problems, life would be grand and they could simply kick back and enjoy themselves to the end of their days. Ah, what a world that would be. But, alas, it is not the one in which we live. Instead, our world is filled with large and small problems galore, so we might as well give up the dream that they'll disappear, and learn how to live with them.
You know what I mean — if it's not the washing machine breaking down, it's baby Alice getting an ear infection, or the new boss turning out to be a tyrant, or rabbits demolishing your first vegetable garden. I could go on, but I'm sure you are entirely familiar with what I mean. Problems are like weeds: as soon as one is vanquished, another pops up — at least that's the way it seems.
And this sense of "how it _seems_ " is what holds the key to unlocking one problem with problems. How often do you notice what goes wrong, but not what goes right? Many dysregulated eaters tend to look only at the negatives and disregard the positives, so much so that I often start my therapy or coaching sessions by saying, "So, tell me what's going on in your life that's positive, and a change for the better, since we last talked." For some clients that's really a showstopper!
Here's why my question about change is necessary, and how a conversation would go if I didn't instruct clients to share the bright side of their lives. I'd inevitably hear, "Well, I had the flu all week, and since I was home a lot with nothing to do, and was feeling sorry for myself, I noshed my way through the day when I really didn't even feel like eating, since I was so sick." My client would go on to tell me in excruciating detail every "bad" food she ate or overate and how frustrated she is with herself, complaining the entire session about her food failures, if I let her. Then, at the tail end of our conversation, she'd add, "Oh, and my husband and I started couples counseling, which went great. He was so into talking about improving our marriage that I cried right there in the therapist's office, and things have been really good between us since then."
The fact that this client had taken this gigantic step in a courageous, life-changing direction would have nearly slipped through the cracks. In her mind, how could it possibly take precedence over the mindless eating she'd been doing? This is often the way a dysregulated eater thinks, and I wouldn't be surprised if you do, too. I understand that when you speak with a therapist, you automatically cough up the things going wrong in your life — after all, isn't that the purpose of a therapist, to fix your problems? — but I don't believe this is all that's going on. In my experience, dysregulated eaters tend to evaluate their days, weeks, and entire lives by how well or poorly they've eaten, and they let that subject override pretty much everything else. In all areas of life, in fact, they pay much more attention to their "failures" than to their "successes."
Of note, science assures us that there is some biological basis for this common worldview: simplistically speaking, some people are blessed with happy genes and some aren't. Moreover, some people have had an upbringing in which life goes pretty smoothly, and because their parents modeled effective problem solving, they learned to manage whatever challenges come their way. Others are born with a tendency to see storm clouds on every horizon. And if they are born to parents who are downbeat or depressed, critical, anxious, or poor problem solvers, they don't get off to a very positive start in life.
We don't choose our genes; we have no choice in who our parents are, and scant power to affect our upbringing, so we come to adulthood as a composite of the heredity and socialization we received. If you were taught that problems are obstacles to pleasure, or that most of them were put there to ruin your happiness, you're certainly not going to view them as a normal part of life or as a challenge. If you were schooled to believe that your problems are worse than anyone else's, you're going to envy people who seem to have fewer of them. Moreover, if you were raised to believe that problems overshadow all the positive things in life, you're going to miss out on enjoying those positive things and insist that you're living under a permanent dark cloud.
I recall a client who really believed a nasty dark cloud chased her around. She had difficulty getting past the fact that she couldn't have children (yet she never raised the subject of possibly adopting them) and, as proof of the cloud's existence, often brought up the fact that several of her siblings had died before their time. In fact, she did have a hellacious childhood, but her current life was pretty darned good: she had an excellent job, friends, and a loving husband. By letting what she lacked dominate her thinking, she missed out on appreciating the good fortune she had.
* * *
Get Smart!
How do you view problems — as a natural part of life or as major impediments to your pleasure and peace of mind? Is this a view your parents held? What were you taught by them about problems and problem solving? What view would you like to have about problems, and what will you do to alter your attitude?
* * *
A key skill in problem solving is holding a positive, empowering view of what happens to you in life. In fact, the word _problem_ is fraught with a gloomy denotation. According to the 2014 _Oxford Dictionary_ online, it means "a matter or situation regarded as unwelcome or harmful and needing to be dealt with and overcome." Ouch, huh? Embedded in this meaning is the idea that an event has appeared unbidden in order to harm us, and that we'd better fix it ASAP before it does us in. Well, that's a cheery way of looking at circumstances that are part and parcel of life.
People who appear to deal well with problems tend to see them not as impossible situations out to get them but as challenges that can be surmounted. Debris on the garden path isn't a reason to turn around and head home, but rather an occasion to figure out how to clear the path or enjoy the walk anyway. As I said, some people come by this attitude genetically or as a result of their upbringing, and it makes life hugely easier to manage. If, however, you are not one of these people, you can still, at whatever age you are now, learn to be more like them.
How? One way is through an approach called positive psychology. Even the name points you in an uplifting direction. According to Dr. Martin Seligman, director of the Positive Psychology Center, this scientific approach "has three central concerns: positive emotions, positive individual traits, and positive institutions." Here are its key points (which I've reproduced as bullet points to make them easier to absorb):
• Understanding positive emotions entails the study of contentment with the past, happiness in the present, and hope for the future.
• Understanding positive individual traits consists of the study of strengths and virtues, such as the capacity for love and work, courage, compassion, resilience, creativity, curiosity, integrity, self-knowledge, moderation, self-control, and wisdom.
• Understanding positive institutions entails the study of meaning and purpose as well as the strengths that foster better communities, such as justice, responsibility, civility, parenting, nurturance, work ethic, leadership, teamwork, purpose, and tolerance.
The goal of positive psychology is not to make you giddy or euphoric all the time or to blind you to what may harm you. It's one thing to wear rose-tinted glasses and quite another to wear no glasses at all if you need them! Rather, positive psychology maintains that we can be taught to view the world in a more optimistic manner, just as many of us were taught to view it in a pessimistic one. Suffice it to say that what you focus on in life will dictate both your worldview and how successful you are in negotiating the vicissitudes of life.
As I've maintained in this book, there is a host of skills for you to learn that can help resolve your eating problems and improve your life. Toward that end, I heartily encourage you, if you tend to be negative, critical, depressed, or a worrier, to learn more about positive psychology by examining the Positive Psychology Center's website and the many books written about the subject. Remember, one of the first steps in transformation is to change your negative thinking, because it generates pessimistic feelings. Even if you can't do a complete makeover on yourself and become a person who's always chipper and looks on the bright side, I guarantee you'll learn techniques and strategies to alter your mind-set about problems and, therefore, about problem solving.
* * *
Get Smart!
Are you a pessimist or an optimist or somewhere in between? What prevents you from thinking positively and viewing problems as learning experiences or temporary impediments to success? Take a minute to consider your eating problems, and _right now_ say something positive about your ability to recover.
* * *
### If I Look at Problems in a Rosy Light, Will They Bloomin' Go Away?
There are two types of difficulties we all face: those that are temporary and easily resolved, and those that are more permanent and chronic — ones we must learn to live with. Being able to distinguish between the two is a crucial skill. After all, you don't want to emotionally equate getting a scratch on your brand-new car with the fact that your brother was just diagnosed with prostate cancer. But that's what some people do. They just lump all their misfortunes together and get equally upset by them all.
The good news is that some problems do go away: you have a cavity filled, and the pain in your tooth stops; your noisy neighbors move to another state; winter passes and you don't have your heart in your mouth while slipping and sliding on ice every day; and your hellion of a teenager matures into a surprisingly sweet twentysomething. The nature of these examples is that they are temporary. That doesn't mean your toothache won't return at some point, or that you won't develop pain in another tooth; that you are sure never to have troublesome neighbors again; that winter won't return; or that your congenial young adult won't occasionally exhibit the worst traits of adolescence.
It's important, when a problem is over and done with, to recognize it as such and be grateful that it's gone. But you may find yourself spending much of your time worrying that, like the shark in _Jaws_ , the problem is sure to return. This is a great way of ruining good times or, worse, preventing yourself from having them. So make sure that when a problem is resolved — your computer is finally working again after crashing, your eyebrows are growing back after you burnt them off when the grill misfired, you've found a terrific new babysitter to replace the one who went off to college — that you aren't continuing to replay the things that haven't gone swimmingly. Mind that you aren't so afraid old problems will recur that you can't relax and enjoy life. If that happens, you might as well still have your original difficulties.
I see this kind of ongoing fear frequently when clients are attempting to improve their eating habits, lose weight, or generally take better care of themselves. Rather than glory in their bits of progress and feel deep pride in doing what's best for themselves even occasionally, they won't allow themselves positive feelings because they're afraid they'll lose their gains. If this isn't pretzel logic, I don't know what is. What's the point of not enjoying the present because the future might hold pain? Sure, you think you're preparing yourself for disappointment over a return engagement of misery, but what you're actually doing is trashing the here and now. Better to enjoy the pleasure sandwiched in between periods of difficulty. It makes these upbeat moments all the sweeter and gives you something to look forward to when a cloud happens to stall above your head.
Moreover, some dysregulated eaters won't permit themselves pride in taking small problem-solving steps and instead insist on waiting until all their behaviors related to food are perfect before they let themselves feel good. The truth is, they will never be perfect, because that is not what "normal" eating is all about. If you are making any kind of progress, focus on that and not on your failures.
If the good news is that some problems are temporary, the bad news is that some aren't. There are chronic health problems like irritable bowel syndrome, gum disease, migraines, arthritis, sciatica, cancers, and other recurrent conditions. Some people have thinning hair or plantar fasciitis. Others have seasonal allergies, hearing loss, or fibromyalgia. The truth is that everyone has something. True, some people have more, or worse, somethings than others, but life isn't a competition to see who can make it through with the fewest difficulties. We each have what we have.
When problems are ongoing or recurrent, it makes sense to acknowledge that fact rather than think every bout is the last. Maybe you'll be right and it will be, but don't be surprised if it isn't. And I'm not just talking about health problems. Cars and houses get old, friendships fray, and weather takes its toll. We get old and so do the items we own. And often what we call a problem is just nature doing its thing. That is _life_! This sentiment reminds me of what a therapist friend said to me once — and trust me, this woman has had a lot of ups and downs in her life. "Why," she wondered, "can't I ever get it across to my clients that things going wrong for them don't need to be a drama? It's simply the nature of existence." The more you can accept even the most devastating problems as part of what happens to human beings living on earth, the easier it will be for you to handle them.
There are two other kinds of problem categories: those that befall us through no fault of our own, and those we bring on, in whole or in part, by ourselves. And there's a mighty long continuum between the two. Dysregulated eaters tend to heap blame on themselves for their eating and weight challenges, instead of acknowledging that, although they have free will, they also are subject to the laws of heredity and to being socialized in certain ways that can't help but affect their mental and physical health. How can you accept the fact that both sets of grandparents, your parents, and all your siblings have food problems, just as you do, and that it may be more difficult for you than for other people (yet not impossible) to eat "normally"? How will you learn to live with the knowledge that sexual, emotional, or physical abuse and other kinds of trauma that you've suffered put you at risk for eating and weight struggles and how will you improve your relationship with food and your body?
One thing you can do this minute is stop blaming yourself for your eating problems. There's no point in faulting your heritage either, though it does help enormously to recognize genetic and traumatic factors. Once you accept that there is no such thing as an even playing field in life, you can acknowledge what has made you as you are and start taking responsibility for changing it. You may have to work harder than other people, it's true. But they likely will be working harder to solve some difficulty that doesn't even show up on your radar screen as a problem — and that's the way the ball bounces. Remember, even if you brought on a problem yourself, you are as entitled to fix it as you would be if it simply befell you. Who cares how it came about? It doesn't matter. The goal is to get rid of the problem, not to punish yourself by failing to do all you can to resolve it. All you need is an improved skill set. To paraphrase Albert Einstein, to solve our problems we need thinking skills better than the ones we were using when we created the problems to begin with.
* * *
Get Smart!
Do you lump problems together as all awful, or are you able to view them according to their true threat level? Do you tend to have the same overreactions to all problems major and minor, whether they are temporary or chronic? Make a statement about how you will view problems in a more realistic light from now on. Do you feel differently about the problems you've created than about the ones that could happen to just anyone?
* * *
### What Skills Do I Need for Problem Solving?
Well, first you want to have the right attitude — that is, that you will remain hopeful and try your darndest to improve a situation. No telling yourself before you begin that your efforts aren't going to work. As the saying goes, if you think you can't, you can't! So believe you can, and you will!
Even though some people seem like natural-born, terrific problem solvers, there's usually a method to their madness and many years of practice involved. Assume that you will get better at problem solving as you apply yourself, and know that you will have no shortage of difficulties to practice on.
#### Be Willing to Experiment
One of the first traits to develop is a willingness to experiment. Unless you're performing triple bypass surgery or are an assassin who has only one bullet to fell your elusive target, you probably have some latitude in the way you go about resolving a problem. So, the best thing you can do initially is to keep your mind open to solutions. Don't grab onto the first fix you think of, or one that is pushed on you by some well-meaning friend or know-it-all authority figure. Unless your house is on fire or your child needs to be rushed to the hospital, you're better off taking time to come up with your own plan.
Always consider various options before choosing the best of them. Noodle over solutions alone and brainstorm with people. Don't say no to any solution initially. Don't rule out what Dad says because you want to come up with a fix yourself. Conversely, don't acquiesce to what Dad says because you want to show him how much you respect his strategic abilities and garner his approval.
#### Don't Rush In; Step Back and Take Your Time
If you can pull back and use a wide-angle lens to view a problem, you'll see it in context. Stand too close and you may see only a small part of it. For example, if a molar hurts and gets attended to, and then the tooth next to it starts to throb, the problem might be that your bite is misaligned. If your daughter has had difficulty in every math class she's ever taken, don't simply assume she's not trying hard enough or that she has ineffective teachers. She may have significant learning disabilities.
So often we get an idea of how to solve a problem and rush out to do it. I've certainly been guilty of this approach myself. My husband, on the other hand, takes his sweet time and thinks about a problem from all angles, mentally thumbing through a host of solutions before moving forward. His tendency to be methodical and cautious has been known to drive me a mite crazy, but I'm also impressed with and immensely grateful for his effective problem-solving skills.
#### Reassess, Reassess, Reassess
It's useful when you're problem solving to take a break and see how things are going. A few months ago, I was trying to fix a belt buckle and kept whacking it so hard with the hammer that it broke apart. If I'd stopped after gently tapping it a few times, I would have seen that the metal was caving and about to snap. The lesson here: you don't want to approach life by whacking at your problems with a hammer!
Nowhere is it more necessary to keep evaluating how things are going than when you're trying to resolve medical problems. We tend to either go full tilt with what doctors and other medical personnel instruct us to do or insist on making ourselves better without asking for help. Instead, gather as much information as possible and make informed decisions based not on hopes and wishes but on reality and evidence. Some problems will resolve themselves and some won't. Either way, you will have done the best you could with the information you had — and that's the best that any of us can do.
#### Know When You're Done
Here's that pesky issue again: knowing when enough is enough. Sometimes you want to do a perfect or near perfect job when you're resolving a problem. I sure hope that my dentist feels that way when he's making me a new crown, and that my mechanic goes the whole nine yards when he's installing a new carburetor in my car. However, it doesn't always matter that a problem is solved to perfection. For instance, if you decide to throw a last-minute party, but you have barely any food in the pantry, maybe it's no big deal if, instead of preparing everything yourself, you potluck it and guests simply bring whatever they want. Similarly, if your living-room rug is permanently stained in certain areas, must you recarpet the entire room or can you get by with a few well-placed throw rugs?
#### Recognize Your Path to Success
One of the most important skills that expert problem solvers have is the ability to recognize what they did to solve a previous problem, so that if need be they can replicate those steps in the future. Moreover, successful problem solvers are able to generalize the steps they took that led to success. When I sat down to write my first book, _The Rules of "Normal" Eating_ , the endeavor seemed daunting. I had a vague outline to work from, and the rest of the material was still in my head. My agent suggested that I develop a more detailed outline, so that is what I did. The new outline solved the problem of what I was going to write, and now I create a fairly detailed outline whenever I'm planning a book.
* * *
Get Smart!
Which problem-solving approaches have helped you resolve difficulties? Which ones could you improve? Think about the times you've done a great job fixing problems, and identify what you did that led to success. Repeat as needed.
* * *
Unfortunately, there are several common personality traits that may get in the way of being a super-duper problem solver:
#### Denial
"What grinding sound in my car's wheelbase?" "No way that tree's going to fall on the house." "I still have plenty of time to get my taxes done." "It's probably not cancer, so the heck with going to the doctor." "I'm going to tell the boss what I think of her, and I don't care if she fires me." "I can quit drinking myself." "But my Uncle Sal smoked two packs a day and lived to be 103."
Well, you can tell where all these statements are heading, and it's nowhere you'd want to go. When we deny a truth that is obvious to the rest of the world, it's because we're afraid. We're not purposely trying to exacerbate our difficulties and do ourselves in. In a twisted sort of way, we're trying to take care of ourselves by avoiding heightened emotional discomfort. Denial makes a convoluted kind of sense, but it's far from life enhancing. If primitive humans had engaged in denial often enough, we wouldn't be here today.
To complicate matters, there are actually times when denial is useful. A certain amount of denial is necessary so we can live without being constantly depressed. Who wants to think a lot about the possibility of getting Alzheimer's or a terminal illness like ALS? What would happen if you regularly thought about the fact that one day the world will be here and you won't? If every day you gazed in the mirror and said that someday you'll look so old you'd barely recognize yourself, you'd probably stop peering into the looking glass altogether.
Just remember that there's a difference between denying that fixable problems exist and being able to go with the flow when problems are irreparable.
#### Victim Mentality
I've already described in this book the dangers of holding on to a victim mentality, so I'll make only brief mention of it here. I'll go out on a limb and promise that you will never be a skilled problem solver if you insist on seeing yourself as a perpetual victim. Telling yourself what you can't do and why you can't do it, for whatever reason, is nothing less than marinating your mind in hopelessness and despair. It creates anxiety and negativity, which are the antitheses of the creativity you need to resolve problems. The time to see whether success is in the cards is _after_ you've tried to solve the problem, not _before_. I understand that the reason you downplay the possibility of resolving your difficulties is that you don't want to be disappointed if you fail, but that's no excuse for programming yourself for failure. You allow yourself to think either like a victim or like an empowered problem solver, but you can't do both. You choose.
#### Chronic Fantasizing
Many dysregulated eaters fantasize a lot — and I mean a whole lot. Some spend almost as much time in their fantasies as they do in reality. Daydreaming a bit here and there — about how great life's going to be when you graduate from college, about the award ceremony for your agency, or about your upcoming Bahamian vacation with the family — is a pleasant way to pass time and often shifts your energy from negative to positive. Moreover, noodling about problems can be a creative way of dreaming up solutions. I do it often, especially when I'm trying to come up with a book title. Daydreaming in targeted, useful ways can often help solve problems.
On the other hand, chronic daydreaming, on autopilot, which I'd define as daydreaming daily or many times a week for prolonged periods of time, is nothing but an escape from reality. Studies tell us that fantasizing about what we want to happen triggers the release of dopamine in our brains. No wonder losing yourself in happy thoughts can become a habit. How much more pleasant that is than acknowledging being mired in problems. But when you're fantasizing, you're moving away from finding solutions, not toward them. So, do keep the daydreams in check. Break the habit, and put the time you'd use for it into solving your problems by becoming a critical thinker.
### I Thought We Weren't Supposed to Think Critically, but Now You're Saying We Need to Learn Critical Thinking: What's Up with That?
Most people go through a lifetime without hearing or speaking the term _critical thinking_. Or, if they do use it, they're describing being critical of themselves or others. The term means nothing of the sort, but is instead a way of approaching decision making and problem solving that uses your brain to its best advantage. Experts studying critical-thinking skills use different criteria to describe this process. Attributes of critical thinkers include open- and fair-mindedness, rationality, curiosity, a desire to be well informed, flexibility, and respect for differing points of view. Robert H. Ennis offers some useful additions to this list: a person with critical-thinking skills is "capable of taking a position or changing a position as evidence dictates, [can] take the entire situation into account, [can] deal with the components of a complex problem in an orderly manner, and [can] use credible sources."
I'll add the characteristic of skepticism to the criteria listed so far. It's not to your advantage to believe everything you see or hear — such as your best friend touting her success with some weight-loss pill, a diet book on the _New York Times_ bestseller list for two months that makes you want to believe the authors know what they're talking about, or an infomercial citing study after study about the wonders of some new exercise machine.
Am I getting through to you? Where is your healthy skepticism? Out the window is where. Consider what kind of critical-thinking skills your best friend has if she would use weight-loss pills; consider whether, just because a book is popular, that means it speaks the truth; and consider whether the makers of an infomercial would really go out of their way to present studies that don't support their product's success.
If you're still not getting a clear-as-day mental picture of the consummate critical thinker, it may be easier to develop an image from some of the things he or she is not: impulsive, illogical, reactive, rigid, unquestioning, afraid of being wrong, dogmatic, closed-minded, narrow-minded, irrational, a believer in pie in the sky, a Pollyanna, overly trusting, or myopic.
Hopefully, you know people who are critical thinkers, and you may even be one yourself. The truth is that people may use critical-thinking skills in one area, such as their job, but never use them anywhere else. Take a minute to tick off the attributes of critical thinkers, and see if you fit the bill.
My point in talking about this subject is that there are all sorts of ways to solve problems, including being solely intuitive, which in my experience is the wrong direction for dysregulated eaters to take, especially when it comes to their eating problems. They often say things like: "I feel fat" or "I can't stand going to the doctor." Statements like these are the antithesis of critical thinking and are among the barriers keeping troubled eaters from making progress.
If you intend to resolve your eating problems once and for all, you must develop critical-thinking skills. You can't keep wishing and hoping and thinking and praying that you're going to be a "normal" eater. You can't expect to keep making decisions and solving life's big and little difficulties, including eating problems, without them. Face it: unless you're willing to make a huge effort to change your behavior, it's going to stay the same.
Let's look at an example to see how an emotional-eating situation could play out. Say you, as a noncritical thinker, are alone at home with plenty to do and no energy to do it. The thought of munching mindlessly or having a binge has been lurking in the back of your mind all day, and you haven't been able to shake it. Rather than be curious about options you could take other than eat if you're not hungry, you stick your head into the refrigerator to see if you can find anything interesting to eat. Then, instead of taking the entire situation into account — that is, instead of recognizing that your problem has nothing to do with food and everything to do with your mood — you pull out some leftover quiche and take a forkful. Without calculating the consequences and allowing yourself to change your mind, you figure that since you already started on the quiche you should finish it, and then you'll just skip dinner. And there you have a typical scenario of acting without first employing critical-thinking skills.
Dysregulated eaters often fall short on a number of clear-thinking attributes. Rather than being curious about their behavior, they're automatically judgmental. If you've eaten an entire apple pie in a sitting (something I used to do decades ago), the mental place to go to when you're wiping the crumbs off your mouth isn't to judgment but to curiosity. Not "Shame on me for doing that" but "Why the hell did I do that?" If you can develop curiosity, that's an excellent start to critical thinking.
As noted earlier, people with food problems are often short on skepticism. They wish so desperately to lose weight — sadly, more than they want to become healthy — that they will try anything and everything that promises the pounds will fly off. It truly breaks my heart to hear of people wasting their well-earned money and precious time on bogus plans, programs, and products. Instead, look for documented, impartial studies on weight-loss methods, if you must focus on weight at all. Don't just ask your doctor, but do a lengthy and substantive search on the internet to dig up any dirt — such as lawsuits against a program or product — on whatever you intend to do. If a procedure has a low success rate, don't fall into denial and tell yourself that you're going to be one of the rare lucky ones, just because you want to be. Think of skepticism as one of the sharpest tools in your toolbox for recovery.
Please understand that I'm encouraging you to develop critical-thinking skills not only in the food arena but also in every aspect of life. Many troubled eaters agree to take on tasks without first thinking through what work is involved, then feel overwhelmed, get stressed out, and head for a Double Whopper. Remember that you can develop the skills to make informed choices so that you know exactly what you're getting into, and that you're allowed to change your mind if the evidence doesn't stack up the way you believed or were told it did.
* * *
Get Smart!
Be honest, how effective are your critical-thinking skills? Which traits do you possess? Which traits do you lack? Which traits will be the most difficult to acquire? Which will most improve your relationship with food and enhance your life in general?
* * *
You know, it's funny about eating disorders. In order to recover from them, you have to mend so many of the broken parts of your life. They are a gift in that sense. Before reading this chapter, you might not have known that your problem-solving skills could use a bit of polishing up. And you might never have even considered how critical-thinking skills could improve every aspect of living.
Using intuition to help you maneuver through life is a legitimate approach. But remember that you want facts and clearheaded thinking to back up most decisions. One approach is not better than the other. The idea is to be skilled at both and know which application is necessary for which situation, and how to use one to complement the other.
* * *
Skill Boosters
1. Make a list of your irrational beliefs about "problems," and reframe them as rational beliefs. If you need help, use the beliefs section of my _Rules of "Normal" Eating_.
2. Start a pride or gratitude journal and write in it every night. Include behaviors you are proud of (no matter how inconsequential you think they are) and ones that make you glad you're you.
3. Within the next twenty-four hours, be more upbeat in two situations in which you'd normally be critical or hopeless.
4. When someone asks how you are from now on, say something positive about your life before saying something negative (especially if the person asking is your therapist — surprise them!).
5. Listen to how you portray your life to other people. Do you describe yourself in a way that says you're a strong, kick-ass person, or in a way that says you're inadequate or a victim?
6. How did each of your parents view and manage problems? How does this influence your mind-set about your challenges today?
7. Sort through your problems and make a list of minor irritations, as opposed to major difficulties.
8. Sort through your problems and make a list of difficulties that are temporary, as opposed to chronic ones.
9. Describe the kind of attitude you'd like to have about minor, temporary problems, and what kind you wish to have about major, chronic ones.
10. Name three ways you can change in order to be more upbeat about all your problems.
11. Identify the skills you need for effective problem solving, which ones you have, and which ones you will work to acquire.
12. Name three things you enjoy (yes, enjoy) about problem solving.
13. Think of something "bad" that happened to you this week, and write a few sentences about it as if you were a victim. Then write a few sentences about the same event, taking responsibility for and empowering yourself. Do this every day with one incident.
14. Hear and watch the news with skepticism. Look for discrepancies, contradictions, and outright falsehoods. Do the same with stories people tell you that sound fishy but which they insist are true.
15. Ask people why they believe what they do, and what evidence they have to support their opinions. Always ask (nicely, of course), "How do you know that?"
16. Practice one aspect of critical thinking every day — being curious, skeptical, seeing the whole picture, entertaining the opinion of someone with a viewpoint different from yours, changing your mind because you think someone has a better way of managing a difficult situation than you do, or insisting upon seeing or hearing evidence.
* * *
In chapter 8, you'll learn whether goal setting will benefit your unique personality and how to get the most out of goals.
## CHAPTER 8
## Setting and Reaching Goals
## What If I Can't Get There from Here?
Dysregulated eaters are masterful at setting goals. And they're often great at reaching them as well. I've met scores of clients over the decades who've lost a hundred pounds or more several times. Maybe you've done that too. Some have lost and regained the same bleepin' twenty, thirty, or seventy-five pounds more times than they can count. People who don't have eating problems and can maintain their weight, or who lose a couple of pounds fairly easily, have no idea how difficult it is to reduce their body size in a major way and hold on to the weight loss. Most of them are clueless about the heartache and disappointment that ensue after regaining lost weight that took forever to lose.
The fact that you're reading this book suggests you have been on a diet at some point in your life. Or perhaps you have dieted all of your life and just can't seem to get anywhere. You may be close to giving up on trying to change your eating habits or shed pounds, because the time and effort just don't seem worth it, or because you now understand that diets don't work in the long term for the majority of people.
Is that because most people don't know how to set and reach their goals? Of course not. Eating healthfully and weight loss are not merely a matter of wanting something and going after it. Rather, the whole subject of our relationship with food and the number on the scale is enormously complex and depends far more on biology than on the concepts of internal control and self-discipline. According to Gina Kolata, author of _Rethinking Thin: The New Science of Weight Loss — and the Myths and Realities of Dieting_ , researchers concluded from a study of identical twins, published in 1990 in the _New England Journal of Medicine_ , that "70% of the variation in people's weights may be accounted for by inheritance, which means that a tendency toward a certain weight is more strongly inherited than nearly any other tendency." More recent studies, Kolata says, put weight and body-mass-index heritability closer to 65 percent. That means that about 30 percent of weight is determined by socialization, lifestyle, environment, and other nonbiological factors, one of which is life skills.
### Aren't Goals Really Simple, Like, Duh, I Want to Eat Healthfully and Lose Weight?
To answer that question, let's move away from eating and weight concerns and look at goals in general for a minute. What is their purpose, and why are we so terribly attached to having and meeting them? To summarize the definitions of goals from several different dictionaries, let's say a goal is an aim, an end toward which you put effort, something you're trying to achieve. So far, so good. That's easy to understand.
But the plot thickens, because some goals are explicit — we're aware of having them — and some are implicit: we're unaware of having them. It makes sense that, in order to reach goals, we'd make them explicit, because if you don't know that you wish to do something — or how important it is to you — how are you going to motivate yourself to do it? Here's what I mean. Say you tell yourself you're going to the local bar Friday night to catch up with your friends for your weekly meet-up, but deep down your goal is to spend time with that cute redheaded guy who hangs out there, who said he'd call and never did. Well, if you don't acknowledge your desire to pursue Mr. Red Hair, you're less likely to sidle up to him at the bar and start a conversation.
Why? Because implicit goals often remain unpursued and, therefore, unattained. Maybe you don't want to admit to yourself how interested you are in Mr. Red Hair, because, after all, if he were into you, he would have called by now, wouldn't he? You're disappointed but want to pretend you're not, so you tell yourself that maybe he lost your number, has been out of town, or was kidnapped by aliens.
See what I mean? You're wishy-washy about bumping into Mr. Red Hair, can't acknowledge your desire to bump into him, and therefore likely won't make much of an effort to seek him out. But think of what you might accomplish if your goal is explicit. You might ask your friends to keep an eye out for him and let you know if he arrives, so that as naturally and normally as can be, you can mosey over and start up a conversation with him. Similarly, you might amble over to his friends and inquire about him (which would likely get back to him even if he doesn't show that night). Can you see how having an explicit goal would exponentially improve your chances of meeting Mr. Red Hair?
Returning to the subject of eating and weight, you may also have eating and weight goals that are not all that explicit. Remember that humans are very complex, and though we'd like to believe the opposite, things are not always as they seem, even within ourselves. For instance, you may tell everyone you want to lose weight _to be healthier_ , which is your explicit goal, while your implicit goal is to shed twenty pounds _to be more attractive_ now that you've started up a casual flirtation with a colleague. Rather than shy away from implicit goals, it's important to identify them and be honest with ourselves so we can determine whether they are realistic and compatible with our values.
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Get Smart!
Are most of your goals explicit and conscious, or are they implicit and more or less hidden from you? Take your implicit eating goals and make them explicit. Be honest!
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So, along with having conscious goals, we also have unconscious ones that often trump our best intentions. Said another way, we have motivations to avoid reaching our goals as surely as we have motivations to reach them, and we must ferret out those hidden barriers in order to be successful. Clinically speaking, hidden goals are called _latent_ and recognized goals are called _manifest_. As the words imply, we're usually highly conscious of a manifest goal — say, to join a gym and become fit — and not so aware of a latent goal, such as a desire to avoid adding one more activity to our already brimming schedules and stressful lives.
Here are examples of manifest (overt) and latent (covert) motivations in the eating and fitness arenas. The first part of each statement is manifest, and the second part is latent.
• You say you want to eat "normally," but, like many dysregulated eaters, you also want to continue using food to comfort yourself.
• You insist you want to go to the gym, but are ashamed of how your body looks and how little exercise you're able to do, since you're out of shape.
• You want to lose weight to be healthier, but you're anxious about slimming down because every time you've lost weight before you've been hit on sexually to the point where you almost cheated on your partner.
• When you lose weight, your friends think you look great, but you hate how your parents complain that you're too thin and, consequently, constantly try to fatten you up.
• You want to eat more nutritiously, but that means spending time you don't think you have in order to shop for, prepare, and cook foods other than those your family wants to eat.
Can you see how confounding this might be — being aware only of one set of motivations when you actually have two? Of course you have a hard time moving in one direction when you have feelings pulling you in another. No wonder you have difficulty attaining your goals or maintaining them — what you call reaching them, then backsliding. We'll talk more about this subject, along with the subjects of relapsing and self-sabotage, later on in this chapter. If you're eager to gain a greater understanding of conscious and unconscious motivations and goals, my book _Starting Monday: Seven Keys to a Permanent, Positive Relationship with Food_ is all about how to identify and resolve these internal conflicts.
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Get Smart!
In your own words, describe the difference between manifest and latent motivations. Give examples of goals you've failed to reach or have reached but not sustained because you've been hindered by your latent, often conflicting, motivations.
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### How Come Our Culture Makes Goal Setting and Achievement Seem Like a Cinch, with Oodles of Books and Seminars on the Subject?
Well, if it were such a snap, why would we need all those books and seminars? Why would you be reading this book? In theory, setting and maintaining goals is easy-peasy — just follow these steps and you'll be successful — but in practice, it can be anything but easy. I'm not saying it isn't worthwhile to set down your aims, read books on goal setting, or even have a coach spur you along. These are all valuable activities for certain people. All I'm saying is not to expect the process to be all it's cracked up to be. In my personal and professional experience, there's a whole lot more to attaining and maintaining goals than what meets the eye, especially the subtleties, described earlier, involving explicit and implicit goals. There's also what I call goal interruptus owing to latent motivations.
When I searched for "goal setting" books on Amazon.com, I came up with 68,680 results. Whew! Lots of people are certainly reaching their goal if it's to publish a book on goals! That said, let's look at some advice by goal gurus on getting where you want to go and staying there. Michael Hyatt, motivational speaker and bestselling author of _Platform: Get Noticed in a Noisy World_ , a book on successful leadership in business, encourages people to do the following:
• Keep goals few in number.
• Make them "SMART," an acronym for five objectives: specific, measurable, actionable, realistic, and time-bound.
• Write them down.
• Review them frequently.
• Share them selectively.
Sounds pretty easy, doesn't it? Then we have Bradley Foster, life and executive coach, who says you need to do the following:
• Believe and have faith in the process.
• Visualize what you want.
• Get it down in writing.
• Have purpose.
• Commit to the process.
• Stay focused.
• Have a plan of action.
• Do something right now that will get you moving toward fulfilling your goals.
• Be accountable.
• Review your goals and actions taken daily.
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Get Smart!
What has been your experience with setting and reaching eating or weight goals? Is your problem one of attaining or of maintaining goals — or both? What impact have workshops or books had on your success rate?
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I don't know about you, but I can't quibble with either Mr. Foster or Mr. Hyatt. I think they're both right on the money, and that what they say seems mostly like common sense, but it is also likely based on research about goals and success. The thing is, my assessment from thirty years of clinical experience flies in the face of their advice — either these techniques don't work with dysregulated eaters, or this population doesn't seek out these strategies and use them effectively. Frankly, I don't know many people, other than athletes and business executives or colleagues who are life coaches, who work with goals so specifically and comprehensively. But, I have tried assiduously to get clients to set and follow through with goals, and the process, at least as it relates to eating and self-care, simply has not worked.
For example, in my coaching and psychotherapy sessions, and in workshops, I generally ask if clients or participants want homework between sessions and I give them assignments. Often people in these settings seem eager for homework — and occasionally they even complete it. These people are often successful in life and accomplished in their fields, and they zip through a to-do list with little apparent effort. So I know they're capable of setting goals and following through. But when it comes to eating and self-care, honestly, not so much . . .
That is why I suggest caution when throwing yourself into goal setting in the eating, fitness, and self-care arenas, if it hasn't worked for you in the past. That doesn't mean you can't reach your goals; it just means you may want to go about achieving them in a less direct way. If you're going to rely on setting and meeting goals as your primary method of forging ahead, you had better make darn sure you aren't ambivalent about succeeding. If you insist on setting goals, you'll want to be certain you've deeply and thoroughly explored any mixed feelings you might have about being thinner or giving up food dependence.
The truth is that people who succeed with goals are generally unambivalent about reaching them. If you're going the goal route, you'd better make sure that you believe you are 100 percent worthy of success. You'd better shout out an unequivocal, resounding "Yes!" when asked if you deserve to be happy and healthy. Most dysregulated eaters answer that question with a little hemming and hawing followed by hesitant responses.
If you're going to do any goal setting — and I'm not against it — I highly recommend that you make sure you break those suckers down into manageable pieces. The biggest problem I've run into with dysregulated eaters, most of whom happen to be perfectionists, is that they're wildly out of sync with reality when it comes to setting doable goals. For example, here's what I hear: "I'll go to the gym every day after work, shop only at the farmers' market, eat organic, get the family into weekend hikes and bike rides, fit into my size-ten jeans by New Year's Eve, look awesome in my bikini for that cruise, and eat no foods that are white."
Dysregulated eaters seem to fall at either end of the spectrum: their expectations are too high and they're living on Fantasy Island, or they don't want to disappoint themselves and so they shrug off goal setting. But fear not, you can reach your goals if you're willing to go a more nontraditional route, one that is better suited to your personality and clinical issues.
### All the Experts Tell Me I Must Make a Commitment to Eat Better and Exercise, So Why Can't I Just Do That and Be Done with It?
Surprisingly, one of the paths I recommend that you not take is the one where you make a commitment to be fit or eat "normally" or more healthfully. I know, I know, every book ever written about behavioral change states first and foremost that you _must_ have a commitment to success. Every TV health pundit insists that you commit your heart and whatever other organs you'd care to involve in living healthfully. Well, I suggest that you look to your own experience to see if committing to a goal has worked for you. If it has, then, by all means, go the commitment route because you might succeed with it again.
However, my hunch is that your answer to my question about your success will be rather circular: that things went well while you were committed, but then they stopped going well and you were no longer committed — or you stopped being committed, so things stopped going well. So which came first, quitting the behavior or giving up the commitment? Kinda chicken-and-egg, don'tcha think?
My problem with encouraging dysregulated eaters to make commitments to become healthier comes from my decades of experience, my personal recovery from food problems, and from an understanding of why we make commitments. Evidence points to the fact that we push ourselves to commit to goals for eating, exercise, education, financial security, and so on in order _to overcome our valid doubts that we'll succeed_. Commitment portends (or, rather, pretends) certainty of our success. It feels like holding a guarantee in your hand.
But think about the concept of pledging a rigid allegiance to fitness and healthy eating. Why would we need to do that? Why wouldn't we think enough of ourselves to do what's good for us because it makes no sense to live any other way? That's the crux of the problem. My take is that we feel a need to make commitments precisely because we don't value ourselves enough to automatically do what's in our best interest.
Moreover, consider how dysregulated eaters often equate healthy and "normal" eating with restriction and deprivation. So when they commit to a "better" kind of eating, it feels like saying no to themselves a million times over. Frankly, most dysregulated eaters equate the word _commitment_ with doing something arduous and distinctly unpleasant — something they don't really want to do. This association is so strong that a commitment to almost anything healthy becomes both prohibitive and sure to self-destruct sooner or later. Think about it: if eating better or becoming fitter weren't perceived as odious, we'd just do it and wouldn't have to make a commitment to it. Read on and you'll see why making promises hasn't worked for you, even when you thought you were making them for all the right reasons.
There are several reasons why making commitments doesn't work. First, _we don't make commitments to goals that make sense and that we know are easily achievable_. There's no need to. If it's easy and sensible, no big deal, we just go ahead and do it. Instead, _we make commitments to things we don't know are possible but hope or wish were_. For example, custom aside, uncertainty is the reason we may feel a need to "commit" to marriage: Because we can't see into the future and _know_ we'll live happily ever after, we make a pledge in order to convince ourselves that this will be sure to happen. The same goes for diets. We have no assurance that we'll stay on them long enough for them to slim us down, so we believe with all our might (against scientific evidence and our own hard-core experience and better judgment) that our dreams of dieting down to a lower weight will come true.
Second, _the more we sacrifice for a commitment and the harder it is to keep it, the more committed we become_. Counterintuitive, huh? In terms of dieting, the more rigid a diet is — fasting, grapefruit, only carbs, no carbs, ice cream only, and so on — the stronger we cling to the belief that it's a winner. The more we must deny ourselves, the harder we try to do so, and the firmer becomes our resolve to keep on truckin'. It's true: in its own warped way, hardship strengthens commitment. Apparently we believe that whatever comes naturally and easily and makes perfect sense is not worth pursuing; but tell us the odds of success are slim to none, and you can sign us up.
Third, _the more we invest in commitments_ (in the case of diets, this means time, money, energy, and constantly focusing on them), _the more difficultit is to be honest and fess up that whatever we're doing isn't working_. Instead, because we hate to think we've failed, and that we were wrong all along, and because we rue the lost time, energy, and money we've expended, we redouble our efforts and blindly soldier on.
Many dysregulated eaters say they hate dieting, but it's so ingrained in them that they still cling to remnants of a diet mentality, albeit one buried deeply below consciousness. I've worked with people for months or years on "normal" eating, only to have them tell me out of the blue that they've returned to Weight Watchers or resumed counting calories. I understand how difficult it is to give up diet thoughts, and it's precisely for this reason that I recommend not making a commitment to any kind of eating. _The act of committing_ , with all its previous, negative associations, cited earlier, is going to come back and bite you.
Commitment does not forecast success. After all, look at the millions of people (maybe you) who year after year take a pledge on New Year's Day to lose weight — and fail. What does this tell you about commitments and success? Knowing dysregulated eaters as I do, I suspect that you probably think _you're_ the failure. Not true! The truth is that we succeed at healthful living for other reasons; certainly not because we've made a commitment to do so. And sometimes making the commitment itself is what steers us off course.
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Get Smart!
How have you fared when you've made a commitment to eat healthfully or get fit? When commitment hasn't led to permanent success, what has gotten in the way? How do you feel about making health commitments now? What, other than commitment, might lead you to success?
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### Okay, So If I Don't Work toward Goals or Make a Commitment, How the Heck Am I Ever Going to Move Forward?
With baby steps, of course — as if there's any other way. There's a big difference between pushing yourself to meet a series of goals or objectives within a time frame and having a general purpose and inching toward it slowly but surely. For example, which of the following sounds more doable? Setting a goal to be a totally "normal" eater four weeks from now, or reading books on intuitive eating, joining a support group for dysregulated eaters, honestly examining why you might be a bit uneasy about slimming down or giving up food for comfort, and developing skills for living without abusing food?
In the first instance, you're focusing solely on behavior and diving in, fears and conflicting emotions be damned. In the second, you're getting your feet wet and choosing activities that will actually speed your recovery and have a huge payoff down the road. Baby steps involve _thinking_ differently about eating and fitness, not simply _acting_ differently.
Here are some questions to answer regarding your baby-stepping journey.
• What are valid measurements and markers of progress?
• How will I take a positive view of setbacks, mistakes, relapses, and failures?
• How will I prevent perfectionism from getting in the way of feeling good about minor achievements when I stall and plateau for a while?
• How will I progress at my own pace and ignore external pressures that make me want to move faster than is good for me?
• What life skills do I want to improve while also changing my food and fitness attitudes and habits?
• How will I make sure I get support in becoming and staying healthy?
• What functional beliefs must I have in order to take ongoing, excellent care of myself?
All these questions may seem too inconsequential to bother with, and answering them may seem like sidetracking if you are going the goal route, but I can assure you that in order to succeed in permanently caring for your body, you will need to grapple with them all and come out the other side. Let me help you answer some of these questions.
#### What Are Valid Measurements and Markers of Progress?
For starters, here are three concrete, simple ways to mark and measure progress. The first is by the _duration_ of the dysfunctional behavior — that is, how long it goes on. Say, for example, your usual binge lasts for two hours. Or, conversely, let's say that you can starve yourself for several hours, but remain in denial that you are doing so. You're making progress in the first instance if you binge for twenty minutes, catch yourself, then stop, and in the second instance if you feed yourself nourishingly after an hour of self-imposed starvation rather than hold out for several more.
A second way to assess progress is through _intensity_ — that is, how thoroughly absorbed you become in a dysfunctional behavior. Let's say, for example, that your binges are generally ferocious, and only after you've cleaned out all the food in your home do you realize you've been gorging; or that you're so totally in the grip of obsessing about weight loss that you're light-headed from not eating all day. You're making progress in the first case if, during your binge, you remain aware that what you're doing is self-destructive and you don't go totally "unconscious" and deny it. In the second case, you're inching forward if you struggle with whether or not to eat, even if you lose the debate and continue to abstain.
The third way to measure progress is by the _frequency_ of undesired behavior. Maybe you now can go weeks between binges. Maybe you used to count the calories in every morsel that went into your mouth, and now you do it only after you've overeaten. The goal is to lengthen the periods of normalcy and functional behavior between bouts of dysfunction. The more you stretch out the time between incidents of dysregulated eating, the more you're retraining your brain to reregulate your appetite appropriately.
Use these markers to recognize whether you're making progress. Don't let one eating experience that you're unhappy with — or even a few — make you believe that your behavior hasn't changed at all. Look for changes in duration, intensity, and frequency to chart your movement toward recovery.
#### How Will I Take a Positive View of Setbacks, Mistakes, Relapses, and Failures?
One way is through the concept of "failing forward," which means using mistakes or failures in the service of moving ahead. Whereas _relapse_ connotes slipping backward, failing forward is about making progress. Think of every failure to become a functional eater as preparation for success — a practice session, rehearsal, dry run, or experiment. Recognize that locked away in every error (or relapse, if you will) are all the nuggets of wisdom you need in order to not make the same error again. If you analyze why you've failed, you'll know exactly what not to do next time and what you must do to succeed in the future.
By staying with the process and doing what has proven universally effective in learning to eat "normally," you _will_ reach your goals. Ironically, overfocusing on goal attainment is one of the ways you're almost certain to fail. Stop asking yourself, "How come I don't get this yet? Why am I still overeating? What's the matter with me that I'm not succeeding?" It's better to say, "This is a challenging process for everyone; I'm learning as fast as I can." Or: "I'll get there if I pay attention to what I need to learn right this minute, not push myself to recover faster." Please remind yourself that you are not in the recovery Olympics!
Another instructive way to look at progress and change — and mistakes and setbacks — is from a "fixed" versus "growth" mind-set. People with a fixed mind-set see themselves and their attributes or inadequacies as more or less permanent — they're good at some things and bad at others, outgoing or shy, lovable or unlovable, smart or stupid, period. Fixed-mind-set thinkers view their successes as the result of their positive qualities, and their failures as the result of their inherent defects. And dysregulated eaters seem to think they have a great many defects. You likely have a fixed mind-set: you're either this or that, and mostly you don't even feel good enough about yourself at your very best. Desperate to succeed, you're thrilled and relieved when you do, and you feel devastated when you don't, as if failure confirms your worst fear — that there's something permanently defective about you. Nonsense! It's all in your thinking and has nothing to do with inherent gifts or deficits.
Step outside yourself for a moment and slip into a growth mind-set. People with a growth mind-set think in terms of learning and changing to solve problems. They don't believe there's anything especially defective about them that prevents them from happiness or success. They assume that changing their attitude or behavior will resolve most problems. A fixed mind-set considers binges and overeating as by-products of character flaws, while a growth mind-set considers them as less-than-effective ways to solve problems that have nothing to do with food.
Here's another example. If you had a fixed mind-set and skipped out on going to the gym for the second time this week, you'd probably say to yourself, "What's wrong with me? I don't go because I'm lazy or unmotivated." Thinkers with a growth mind-set, on the other hand, would acknowledge the need to develop better strategies for being more active, because the ones they've been using clearly aren't working well. They would never think of attributing a lack of success to _who they are_. Rather, they'd see about changing _what they did_. Get the difference? Not to belabor the point, but a fixed-mind-set thinker would view failure as "proof that there's something wrong with me," while a growth-mind-set thinker would view it as "proof that I have yet to find better ways to succeed." If you believe you can't go from being a fixed- to a growth-mind-set thinker, then tell me: is that thought fixed or growth-oriented?
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Get Smart!
What has prevented you from seeing your instances of progress, however small? What ways will you change so that you can see more of your growth, rather than your lack of it? How can you develop a growth, rather than a fixed, mind-set?
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#### How Will I Prevent Perfectionism from Getting in the Way of Feeling Good about Minor Achievements When I Stall and Plateau for a While?
Examine your beliefs about perfectionism, success, and failure, and they will give you all the information you need to abolish the desire for perfection. Perfectionism is the result of all-or-nothing thinking: if I'm not this, I must be that. But life is rarely all one way or another. Look around and decide: does life really exist only at extremes, or is there a great deal of something in between? Perfectionism, according to my colleague Joanna Poppink, author of _Healing Your Hungry Heart_ , is a way to keep yourself safe and above reproach, as if by staying in the perfect zone you can prevent others from ever faulting you.
But the fact is, chasing perfection is a fool's errand. Maybe as a child you tried to be perfect to please others or to avoid mistreatment, rejection, or abandonment. Now, however, you get to evaluate yourself rather than depend on how others view you. As an adult, you're no longer at the mercy of someone else's critical standards. You can eat however you want (well or poorly) and act however you wish (again, well or poorly), and you are accountable only to yourself.
By forgoing perfectionism and making a big deal out of minor achievements, you spur yourself on. Isn't it a delicious feeling to be your own best friend and head cheerleader? And so what if you stay a while at a plateau without gaining skills or losing weight? We need to rest and consolidate changes once in a while. It's only in fantasy that change happens overnight and marches straightforward without resting. Think of a plateau as a reconsolidation spot and jumping-off place, from which you'll take your next big leap forward.
#### How Will I Progress at My Own Pace and Ignore External Pressures That Make Me Want to Move Faster Than Is Good for Me?
Do you ask someone else when to schedule your annual checkup, or how fast you want to be driving? Then why would you think that any individual could possibly determine your proper pace as you recover from eating problems? Who knows you best? Why, you do, of course. When you feel this kind of pressure, ask yourself why someone else wants you to speed up or slow down. If the thought comes from them, then even if your name is attached to it, how could it possibly be about anyone else but them? Remember that people — even those who profess to love you — have various reasons for wanting you to do or not do things, but that doesn't mean these reasons are in your best interest.
Other pressures may appear to be external, but they're really coming from you. Usually this pressure arises in the form of a date or event at which you want to look a certain way — thinner. You believe that people will like or value you more in a slimmer body, and that they'll be less likely to accept you the way you are. You think you'll be judged by your size. Well, the truth is that maybe you will be, and maybe you won't be; but when it comes down to it, who's the harshest critic of your appearance, hands down? I'd say it's the person whose face you see in the mirror every day. You're on yourself 24/7 about getting that new body as fast as possible. So, forget about external and internal pressures to go faster than you possibly can. Be grateful you're moving slowly. Research tells us repeatedly that gradual learning and change is the best kind. And I'll add that it's the only kind that works!
#### What Life Skills Do I Want to Improve While Also Changing My Food and Fitness Attitudes and Habits?
All the skills taught in this book, and whatever additional ones you want to become more proficient in, will, if practiced, speed you on your way to taking better care of your health. Maybe you want to learn to be more assertive or to take things less personally. Could that possibly hurt? Perhaps you'd like to be more outgoing, or you yearn to uncover your creative side. These skills will complement your desire to become your own person and be more fulfilled.
If I tell you now all the skills you'll need in order to live your best life, I'll be ruining all your fun. I'd rather you enjoy discovering them by yourself!
#### How Will I Make Sure I Get Support in Becoming and Staying Healthy?
These days there's no shortage of ways to gain support. Now that eating disorders are talked about openly in lectures and on talk shows, and binge eating has its own diagnosis in the 2013 _Diagnostic and Statistical Manual of Mental Disorders_ , dysregulated eating has really taken a step out of the closet. Because of this, there's no shortage of places to get support. Of course, you can start by talking about your food problems with your friends and family, but only if you get the feeling they'll be interested and supportive. Test the waters, and don't expect everyone to understand what you're going through. Some will and some won't, and that's fine. Think like an adult and accept that some people will want to know about your food struggles, while others will be totally uninterested or uncomfortable with the subject. Don't push. Try to educate them, and determine how educable they are. You never know if people might change their minds down the road and come around to supporting you. Alternatively, if they're not interested in what you have to say right now, so be it. Find someone who is.
Beyond seeking help from friends and family, use the skill of valuing yourself enough to check your community and online for places to share and continue growing. Be absolutely determined to get help, because you deserve it. Learning to seek help and accepting it can be major obstacles for dysregulated eaters who feel they must cure themselves alone or are too ashamed to ask for assistance. If this has been a problem for you, go out of your way to garner support and join the rest of the world in learning the joys of dependence and interdependence.
#### What Functional Beliefs Must I Have in Order to Take Ongoing, Excellent Care of Myself?
There are far too many such beliefs to list in the space of this book. Fortunately, I wrote _The Rules of "Normal" Eating_ , which contains several chapters on rational and irrational beliefs about food, eating, weight, and body image. Moreover, the book provides instruction on and a template for changing beliefs so that you can easily learn the process for doing so.
By the way, make sure not only to develop a set of functional beliefs about eating and weight but also to pay attention to your core beliefs, which are about who you are, your place in the world, and your views of how life works. Too many people spend time reframing their food- and body-related beliefs, but leave their irrational core beliefs untouched, then wonder why they still have eating problems. The answer is that they continue to have a dysfunctional way of viewing the world and their place in it, and their food and body-image problems won't resolve until their core beliefs are healthy and rational.
A few last words about the "self" skills you'll need in order to reach the finish line. One invaluable skill is self-reflection, which means thinking often about life and everything in it, including yourself. I don't mean being self-absorbed, but wondering about your thoughts, feelings, and behaviors in a curious, nonjudgmental way. Self-reflection is the opposite of self-judgment, at which most dysregulated eaters excel. It's a neutral appraisal that you undertake in order to gather information, like listening to the news. You don't turn on CNN to like or dislike what you hear, do you? You turn it on to learn what's going on. The purpose of reflection is to get to know yourself better and then, if you wish, to apply what you learn to improving yourself.
Another essential skill is self-honesty, which is seeing not only what you've done wrong but also what you've done right. Believe it or not, this is a tall order for most troubled eaters. Most people would rather focus on their strengths, not their weaknesses. Not dysregulated eaters; they're just the opposite. So remember that being honest means it's okay that you did better than someone else, or that you were right when others were wrong, or that you were smarter, or faster, or more clever, or more successful. Excelling often makes dysregulated eaters wildly uncomfortable. While craving praise, they don't feel they deserve it or are convinced that if people really knew them, they wouldn't be praising them. They believe they're not as good as others think they are and feel like imposters. Just remember that being better (like being worse, for that matter) than someone else is almost always a temporary condition. Enjoy it for the moment, then let it go, but please don't miss out on it.
Last but not least is the essential skill of self-talk. People who are skilled in positive self-talk possess one of the greatest gifts in the world. And I don't mean giving yourself praise when you don't deserve praise, but being there for yourself every minute of every day and giving yourself whatever you need: comfort, a small dose of reality, an inner smile of encouragement, a good laugh at yourself, respect, compassion, a pat on the back for a job well done, and so on. Practice positive self-talk, especially with an eye toward compassion, and you'll be amazed that you can still be the same person but feel like a completely different one. Moreover, you will not succeed at permanently taking effective care of your health and your body if you do not become proficient at positive self-talk.
* * *
Get Smart!
Which of the skill areas that I've described will be easiest for you to learn? Which will be most difficult? Are there other life skills you'd like to learn in order to help resolve your eating problems? What are three positive statements that you could make about yourself right now?
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* * *
Skill Boosters
1. What goals have you been successful in reaching? Was that because you did everything right to reach them, or did you reach them in spite of making mistakes along the way?
2. How successful have you been in making — or breaking — commitments?
3. What are the three markers of making progress?
4. Describe a mistake you made that turned out to be fortuitous in the long run.
5. Describe a success that you achieved by taking baby steps.
6. As if you're talking to an eight-year-old, explain why perfection is an illusion.
7. Describe pros and cons of (1) asking for support and (2) receiving support.
8. Is your mind-set fixed or growth-oriented? Explain the difference between the two.
9. Come up with a mantra to use when you or other people pressure you to move faster in your recovery than you can. (Hint: mine is "I'm doing the best I can.")
10. Make a list of irrational beliefs about being successful, needing to be perfect, getting support, needing others' approval of your eating habits or weight, and valuing yourself, then rewrite them all so they're rational (read _The Rules of "Normal" Eating_ for help).
11. What are you afraid you'll see if you're honest with yourself? How can you make it all right to see yourself honestly? (Hint: think curiosity, not judgment.)
12. At the end of every day, look back and identify something you did well and something you didn't.
13. Keep a journal that chronicles your eating progress, with a heavy emphasis on the achievements you're proud of.
14. What are the kindest, most loving and compassionate words you can say to yourself? Say them now aloud in front of a mirror. Repeat as necessary.
* * *
In chapter 9, you'll learn the purpose of work and play and how to value both.
## CHAPTER 9
## Balancing Work and Play
## All Work and No Play Makes Jack . . . Crave a Snack!
Dysregulated eaters too often spend their lives racing against the clock and can't wait until they have time to themselves. But then, more often than not, they don't know what to do with it. Busy bees, they go from dawn 'til dusk ticking off tasks on their voluminous to-do lists; and when they finally do get a chance to sit down and relax, their minds continue to torment them with what they haven't yet done and all the things they must do tomorrow. Relaxation doesn't feel right if there are chores left undone, but then, neither does doing them. Funny how the only thing that fits the bill is — guess what! — eating. After all, the thinking goes, one has to eat, doesn't one?
Is this drudge of a life the one you're living? I understand that money must be made, children demand care, and other various and sundry attentions must be paid to objects animate and inanimate. Cars require tuneups, feet need new shoes, and living quarters require cleaning. Relatives insist on sharing your time, and lawns don't mow themselves. There is always work to do, I'll grant you that. That is, there's always work to do unless you have the financial wherewithal to pay someone to do most of it. The truth is, however, that many fortunate people who do have the money to pay others to take care of what they don't care to do still have difficulty kicking back.
So, if it's not a lack of means that prevents you from turning off your overloaded mind and resting your weary body, what is it? Could it be something within that's like a gun to your head that keeps you on the go? Could it be the guilt that pops up the minute your fanny hits the recliner, or the shame that descends on you because you feel you're being selfish by taking care of yourself? In my experience, guilt and shame are what make it difficult for dysregulated eaters to take a break and engage in what I'd call rest or play.
To play is to be engrossed in activity that has no purpose other than to bring you in-the-moment pleasure. The 2014 _Oxford Dictionary_ online tells us that play is to "engage in activity for enjoyment and recreation rather than for a serious or practical purpose."
_Sounds good to me!_ According to the National Institute for Play (who even knew there was such a thing?), play
is a state of being that is intensely pleasurable. It energizes and enlivens us. It eases our burdens, renews a natural sense of optimism and opens us up to new possibilities. These wonderful, valuable qualities are just the beginning of what play is.
Scientists — neuroscientists, developmental biologists, psychologists, scientists from every point on the scientific compass — have recently begun viewing play as a profound biological process.
They are learning that play sculpts our brain; it makes us smarter and more adaptable. For many animal species it has evolved over eons with the result that the most advanced animals play the most, i.e., play is more central to their development. Humans are the biggest players of all, specially designed by nature to play throughout our long lives.
Who'd have thunk it, huh? If you hadn't read the preceding quote, would you have believed that play is an essential component of this thing we call life? Would you have known it was so valuable and vital to pleasure and happiness? If not, that's okay. By reading this chapter and following its suggestions, you'll learn to bring yourself to a higher level of enjoyment, as well as a deeper one, by developing new habits of play, passion, and pleasure.
Remember how as a child you'd go to the beach and lose yourself for hours building an elaborate sand castle with moats and turrets? Maybe you were so absorbed in your architectural endeavor that you were willing to forgo swimming or collecting shells along the seashore with your siblings. No one was paying you to build castles, but you did it anyway. And then, when you were done with your masterpiece, what did you do — after you showed it off, of course? You knocked it down and started over, building another castle. So, pray tell, was that work or play?
Many troubled eaters have difficulty with both concepts. When they're at work, they're daydreaming about having fun or kicking back, and when they're supposed to be out having a good time or enjoying relaxation, they're feverishly worrying about work. And by that, I don't mean simply that they worry about the job they're paid to do, but also that they worry about the general tasks, chores, errands, and grunt work that make up — and, sadly, take up — much of our lives. They have difficulty keeping their minds on work because they feel undervalued, taken advantage of, disinterested in what they're doing; long to be doing another activity; or resent the time spent working when they want to be playing. This makes for very unhappy and dissatisfied campers.
On the other hand, when they do finally get a chance to play or chill out, they feel guilty and can't quite let themselves go completely, because _not_ enjoying themselves is a kind of penance they believe they deserve to pay for not being busy and productive. Honestly, did you have any idea that the subject of play would be so complex? Dysregulated eaters often have this kind of push-pull, confounding, tension-riddled relationship with work and play. And the reason I want to set the record straight is because I want you to develop the skill of knowing how to distinguish when it's time to work and when it's time to play, so that you can find a better balance between the two. This skill is touched on in chapters 4 and , so it's worth rereading them both if you suspect that being unskilled in balancing work and play is a barrier to enjoying a positive relationship with food.
* * *
Get Smart!
What are your feelings about work and about play? How does my description of play — as necessary in order for you to thrive — change your view of it? Are you starting to understand how feeling guilty about play and relaxing drives you to eat?
* * *
### If It's Natural for Children to Play, How Come I'm So Confused about the Subject?
You're not alone. Many people are unsure how much to work and how much to play. They either want to party, party, party or keep their noses to the grindstone and never look up. Any guesses on how we develop into one type of person or the other?
Although play comes naturally to children, they also enjoy learning; and without a doubt, the desire to move toward mastery is embedded in human DNA. Of course, children don't know that they're often learning while they're playing. That's the magic of play. Remember those sand castles I was talking about earlier? Consider what you learned by trial and error as you were building and rebuilding them. You learned to gauge how far back in the sand to build your castle to make sure a wave didn't roll over it and wash it away. You figured out a workable ratio of sand to water that would enable the sand to hold its shape — too little water and the sand wouldn't firm up and grains would roll or fly off, but too much and the sand would collapse into mush.
I'm sure you didn't realize at the time that you were acquiring the basics of engineering and physics as you built your dream fortress, yet you were. Moreover, even as you were finding pleasure in the mere act of fashioning your castle, you were also feeling joy in improving at what you were doing. Without your realizing it, each castle you made was likely an improvement on the last. In this way, our brains link play to learning; or to put it another way, they link fun to mastery.
But let's back up and see what would have happened if you had been messing with sand and water and having a grand old time and some well-meaning adult (a parent, relative, family friend, or even a stranger) strolled over and started instructing you on how to build a proper castle — add more water, mix in more sand, put in a drawbridge, make those turrets taller, and so on. If this person offered suggestions in a friendly, take-it-or-leave-it manner, you might have been happy that he or she had the time and interest to share ideas, and thrilled that the advice made your castle-building easier and improved the structure.
If, on the other hand, this person told you or implied that you had no idea how to build a castle, you might have felt like burying your head in the sand. If Mr. or Ms. Critical continued to drone on with unsolicited advice as if you were competing for an international architectural prize, you might have wanted to smash the whole edifice down with your fists, give up, and walk away. Worse, you might have begun to feel that play must be done in a certain way or it wasn't worth the effort. And all too likely, the joy of being in the moment and using your imagination would have washed away like the tide.
In order to understand your skill deficit in balancing work and play, you'll have to time-tunnel back to your childhood, and sort out how your parents viewed these activities, what they taught you explicitly and, perhaps more important, implicitly about them. Your responses to the following twelve statements will give you a clearer picture of your family values regarding work and play.
### Exercise
1. Three words Mom might have applied to work or chores are __________, __________, __________.
2. Three words Mom might have applied to play or relaxing are __________, __________, __________.
3. Three words Dad might have applied to work or chores are __________, __________, __________.
4. Three words Dad might have applied to play or relaxing are __________, __________, __________.
5. Other significant relatives had this to say about play or being productive: _______________________________________________________________________________.
6. One or both of my parents were overly focused on getting things done and had difficulty kicking back. ___yes ___no
7. One or both of my parents valued productivity over nonproductivity. ___yes ___no
8. Play and work were equally encouraged by my parents. ___yes ___no
9. One or both of my parents were much more interested in play than work. ___yes ___no
10. My parents enjoyed a healthy balance between work and play. ___yes ___no
11. My parents often interrupted my play or downtime with directives about getting more important things done. ___yes ___no
12. My parents taught me to enjoy a healthy balance between work and play. ___yes ___no
What did you learn from this exercise? Does it help you understand why you're not more skilled at balancing work and play? Please understand that your parents didn't set out to mess with your mind in these areas. They learned from their parents, who learned from their parents, and so on, to the beginning of your lineage. Different historical periods influenced how work and play were viewed, and class, culture, and religion had a bearing on these activities, as well. If your ancestors were landed gentry and able to afford servants, they passed down values to subsequent generations that differ from the values you inherited if you had a family tree of farmers and day laborers.
* * *
Get Smart!
What is your understanding now of why you are unskilled at balancing work and play? Write the adage you've been living by with each of them. Write the adage you wish to live by with each of them.
* * *
### Okay, So I Haven't Gotten Work and Play Exactly Right; How Do I Find a Better Balance?
I'm going to make the assumption here, based on the troubled eaters I've treated for decades, that your difficulty in discerning how to balance work and play stems from the fact that you were taught to value the former over the latter. That is, you learned by word or deed that play is a distraction or a luxury, and that it's better to work and be productive than to slack off and have a good time. This may not be true for all dysregulated eaters, but I bet it's true for many. Of course, the opposite may be true as well, that your parents goofed off a lot and didn't work very hard or teach you to do so.
Since we've been examining how your upbringing shaped your view of work and play, let's do another exercise, one that will bring your dilemma into sharper focus. Because cognitive-behavioral theory tells us that our behaviors are rooted in our beliefs, it's time to put beliefs under a microscope and examine them more closely. What you believe about work and play, even if you're unaware of these beliefs ( _especially_ if you're unaware of them), will dictate how you value spending your time.
First, a word about beliefs in general. Although our feelings and behaviors are based firmly on our beliefs, generally we hardly ever think seriously about them. Sure, you might pause to consider your religious beliefs when the issue of separation of church and state comes up, or ponder your convictions about child rearing when you learn you're pregnant. But for the most part, our cognitions interest us far less than our emotions and behaviors. However, in order to live a conscious life — and have a relationship with food that is positive and healthy — we _must_ examine why we act and feel the way we do. I agree with Socrates who insists, "The unexamined life is not worth living." Not if you're looking to have a successful, happy, meaningful time on earth.
Now is the time to examine your irrational, dysfunctional, or otherwise unhealthy beliefs about work and play. I've corralled a bunch of beliefs on the subject in a list that follows. Some are learned from our culture through school and religion, but most are acquired from our parents and extended family.
Once more, let me assure you that I'm not blaming your parents for teaching you a set of beliefs that aren't functional. Nor am I blaming you for holding these assumptions. We all learn beliefs in childhood that don't do us proud in adulthood. The enlightened among us know this and periodically sift through our cognitions to determine if they're keepers or not. There's nothing wrong with discarding old beliefs, mind you, and selecting ones that make more sense to you nowadays. Being selective about beliefs is a sign of wisdom.
Here's one more caution: please don't judge yourself for having a belief that, said aloud, seems silly, bizarre, or otherwise unacceptable. No judgments, okay? Instead, allow yourself to be curious and completely open to whatever responses you have. Nonjudgmental self-acceptance is an essential life skill in itself and one that is sorely needed by most troubled eaters. So, here are beliefs about _work_ and _play_ , which are also code words for _productivity_ and _relaxation_ :
1. It is important, above all else, to be productive in life.
2. I can't play until I'm finished with all of my work.
3. Play is frivolous and a waste of time.
4. One of the worst things I can do is waste time.
5. If I don't stay busy, I'll get myself into trouble.
6. If I don't have anything to do, I probably left a job undone or poorly done.
7. I can play only if I squeeze it in between periods of work.
8. If I don't have goals and work toward them, I won't amount to anything.
9. I'd better keep my mind busy.
10. Adults don't have time to play; only children do.
11. There is no point in play.
12. Play only distracts me from tasks at hand.
13. To feel proud of myself, I must keep busy and be productive.
14. If I play too much, I'll never be successful.
15. If I take time off to play, people will think I'm lazy.
16. Work builds character, while play leads to shirking responsibility.
17. If there's work to be done, playing is selfish and foolish.
18. People will judge me by how much work I do and how well I do it.
19. Serious people spend time working, not playing.
20. If I haven't worked hard, I don't deserve to play.
Do any of these beliefs resonate with you? Maybe you've known all along that your ideas about work and play were a little off, but you didn't want to "waste time" thinking about them. Maybe your beliefs have made you uncomfortable and angry at your parents for instilling them in you. Perhaps you are angry at yourself for knowing that how you think is killing you little by little, but you're still living as if they're worthwhile. No matter. Set your beliefs aside and reframe them, and then you can choose which ones you want to keep. And remember, you can always change your beliefs if you get more or different information that challenges them. I hope you recall learning this in chapter 7, which lists, as a criterion for critical thinking, the ability to alter your thinking in response to new information.
Here's how I would reframe the preceding list of irrational beliefs. Feel free to "play" with them to make them truer for you.
1. It is important, above all else, to be productive in life. Work and play are equally important.
2. I can't play until I'm finished with all of my work. I can play whenever I want.
3. Play is frivolous and a waste of time. Play is essential to health and well-being.
4. One of the worst things I can do is waste time. Time is mine to do anything I want with it.
5. If I don't stay busy, I'll get myself into trouble. I'll enjoy myself if I'm not busy.
6. If I don't have anything to do, I probably left a job undone or poorly done. If I don't have anything to do, it's time to play.
7. I can play only if I squeeze it in between periods of work. I can play whenever I feel like it.
8. If I don't have goals and work toward them, I won't amount to anything. To be a fully rounded person, I need goals, work, and play.
9. I'd better keep my mind busy. Nothing bad will happen if my mind isn't busy.
10. Adults don't have time to play; only children do. People can enjoy play at any age.
11. There is no point in play. The point of play is to have pleasure and fun and to unwind.
12. Play only distracts me from tasks at hand. Play recharges me so I can return to tasks at hand.
13. To feel proud of myself, I must keep busy and be productive. I am as proud of playing as of working and being productive.
14. If I play too much, I'll never be successful. I can balance work and play and be successful.
15. If I take time off to play, people will think I'm lazy. Taking time off to play is natural, healthy, and essential.
16. Work builds character, while play leads to laziness and shirking responsibility. Play builds pleasure and sometimes new learning.
17. If there's work to be done, playing is selfish and foolish. Taking care of myself with play is never selfish or foolish. 18. People will judge me by how much work I do and how well I do it. I value myself for how well I balance work and play.
19. Serious people spend time working, not playing. I can be serious sometimes and enjoy play at other times.
20. If I haven't worked hard, I don't deserve to play. I deserve to play whether I've worked hard or not.
What did you learn from reframing your beliefs about work and play? I wouldn't be surprised if you were a tad uncomfortable in the process of reading my reframed beliefs. That's because even though you might be drawn to these new ideas, you haven't quite put your old beliefs to rest yet. That's understandable. You've had them for a long, long time, and in fact they are deeply grooved in your neural pathways. However, don't worry: as you read and act on your new beliefs repeatedly — and really think about how they belong to the person you wish to be and are becoming — they'll take root.
* * *
Get Smart!
How did you feel reading over the irrational, unhealthy beliefs? How did you feel reading over the rational, healthy beliefs? What is your next step in creating a functional belief system as a foundation for work and play?
* * *
### Sometimes When I'm Thinking about Food or Eating Mindlessly, I Wonder If It's Because I'm Trying to Feed My Empty Life
Dysregulated eaters are full of intriguing paradoxes. On the one hand, their lives are overflowing and they never seem to have a moment to spare. Then, when they do, they often feel uneasy about what to do with it. Clients will describe eating when they feel overwhelmed and stressed from what they perceive as too much to do, but will also cite instance after instance when they're feeling bored and head for the fridge to do _something_. Mind-boggling indeed.
Part of the problem here has to do with self-regulation gone awry and with getting caught up in doing too much or too little. If having too much on your metaphoric plate triggers overeating, well, then, make sure you do less. And if having nothing to do drives you up a wall — one that happens to have a kitchen cabinet built into it — don't let yourself get into that situation. But doing more or less isn't really the whole story. At the core is your fear of an empty mind and an empty life, bereft of meaning, attachments, and pleasure. It's the void that gets to you and that you try to fill with food. However, you can't fill emotional emptiness with something you chew and swallow.
There is likely a biological underpinning to the feeling of excessive, intense inner emptiness, and you can see from the previous exercise how it can be reinforced by being taught to fear not being busy and productive. On the other hand, everyone feels bored, lonely, and disconnected from the world sometimes. There is nothing at all unusual, wrong, or worrisome about this affective state. The problem is the meaning that dysregulated eaters attach to not having anything to do: it is what causes them to feel certain things — antsy, edgy, empty, frightened, stuck, paralyzed, adrift, or panicky.
* * *
Get Smart!
What meaning do you ascribe to having nothing to do? What feelings arise from the meaning you make? Is your life really empty, or does it only feel that way? How does nonhunger eating make you feel better — and worse?
* * *
There is no brilliant, quick, one-size-fits-all answer for what to do when you feel inner emptiness or boredom. I could remind you that you have friends, and that there are activities you could do if you wanted to do them, both inside and outside your home. I could give you the standard advice of calling someone, getting out, or finding something interesting to engage your mind or body.
However, my instinct and clinical experience tell me that you will need to dig deeper (via self-reflection, talking with intimates, reading, or if these things don't help, getting professional help) to discover and address what's really bothering you. In fact, I'd wager that this is a must-do if you wish to improve your skills at balancing work and play and not turn to food when you're alone and bored. One book I recommend is _Dark Nights of the Soul: A Guide to Finding Your Way through Life's Ordeals_ by Thomas Moore. Essentially, Moore advises readers to learn more about their "dark nights" — in this case your internal emptiness — as a way to understand what these moments are trying to teach them and what is needed so they can live their best possible lives.
### Since We're into Serious Stuff, What's the Secret to Filling My Life with More Than Food?
The secret is that there is no secret, as in: if you come a little closer, I'll whisper it in your ear. What will fill up your life in a meaningful way is something only you can discover. Each of us is tasked with finding that out on our own. And as to secrets, there is no one thing that amounts to a foolproof plan for giving your life meaning. In the same way that there are many individuals who would be a good match for you, there are many avenues that will enrich your life and breathe passion and satisfaction into it.
Sadly, it's so easy to make existence seem meaningful through obsessing about food and weight — dieting, engaging in abusive eating, and making ourselves miserable afterward — that many troubled eaters never get to the root of their difficulties with emptiness and lack of fulfillment. Each of us has a path to take, and each of us has to stay on it until we find ourselves. Maybe you believe you'll be fulfilled only when you're engaged in activities that benefit humankind. But not everyone can be out there on the front lines like Doctors Without Borders, or do public service by running for city council, or advance science by discovering renewable energy sources or miracle vaccines. Maybe you believe that being adored by the public will give you what you crave, but think of all the writers, artists, actors, musicians, and celebrities with mental health or addiction problems.
Now consider some ordinary people you know who seem to love life and life seems to love them back. They're not running around trying to fill up the proverbial hole in their soul; but rather, life seems to burst from them because they're content with whatever they're doing. Some work outside the home and some inside; some care for their own children, and some care for other people's children; some are well known in their communities, and some pass through life preferring to be left alone to follow their own star. What I'm saying is that _there's no one way to feed an empty life_ , and the only way to discover what will nourish you is by trial and error and listening to what calls your name.
I've been privy to a great deal of this kind of seeking since I moved to Sarasota, Florida, in 2005 and came to know many people who are retired or semiretired. Some newcomers like me have simply continued the same work they did wherever they last lived. In my case, that was providing therapy, doing tele-coaching, teaching, and writing. Others had had their fill of former careers and were looking for something new to engage or challenge them. Some found it right away, and some took a long time, while others are still searching.
I see the same dynamics in the twentysomethings I treat for eating disorders. Although many are on a career path, they still seem to be craning their necks and squinting their eyes looking ahead for fulfillment rather than enjoying what's available to them in the moment. These same undercurrents are there for empty nesters and for people who are recently divorced or widowed. Shifts in status like these often generate situational self-doubts and feelings of unfulfillment that generally pass with time.
But many dysregulated eaters appear to have lived with an inner void for most of their lives. If you are one of these people, be careful not to go overboard with activity just to feed the void, because the absence of bustling around is going to make natural downtime seem even emptier. Look inward and ask yourself what is missing, even as you are keeping your eyes and heart wide open for opportunities to make connections to other people and for activities that cement your connection to yourself.
* * *
Get Smart!
Have you felt an inner void ever since you can remember, or is yours attributable to a change in status or venue? How have you tried to fill this void, other than with food or dieting? Think about the times you felt fulfilled, not through busy-ness or achievement, but through a sense of inner peace or passion in your core. How did you make that happen? How can you make it happen now?
* * *
### Don't I Need Passion, Purpose, or Meaning to Feel Well Balanced in My Life?
To feel alive and emotionally engaged with the world and other people, it helps to have a reason to get up in the morning, but that reason need not be something special or unique. The problem with using words like _passion_ , _purpose_ , and _meaning_ is that they sound grandiose and exceptional, as if you have to go through some spiritual awakening or to experience transcendence to find them for yourself. The fact is, some people do need an epiphany to awaken from their walking slumber and realize exactly what nourishes them. Other people seem to stumble upon such a thing, give up what they have for what they've stumbled upon, and never look back. Yet others leap from challenge to challenge, and this string of challenges ensures engagement throughout a lifetime.
You may find your passion, purpose, or meaning in work, especially if it involves creativity; yet there are many clever folks who feel truly inspired only by using their creativity for something larger than themselves. Maybe a choreographer begins an inner-city dance troupe or, as an artist friend of mine has done, becomes an adjunct professor at a community college. Alternatively, if you're in business or are a professional, that may not be where your juice comes from. I have a retired friend who, while in the middle of writing a book, resumed painting, something he hadn't done for decades. Now that's all he wants to do, and his manuscript probably will remain forever unfinished.
A client of mine, a businesswoman and mother of young adult children, makes jewelry and loses herself so entirely in her craft that she forgets to eat. Another client works in a city but drives out to an animal sanctuary in the country every weekend to learn how to engage with and care for wild birds. One friend is a movie buff, living and breathing everything cinema. A dentist I know plays in a rock-and-roll band, and a friend who's an insurance broker is working on reigniting a singing career. I know people who are news junkies and who come alive through reading and discussing politics, and a few folks who've been enthralled for years with developing their family trees. I have another friend who returned to school in her sixties to earn a social-work degree and then wrote a memoir.
What I'm trying to show you is that you can nourish yourself in an infinite number of ways. Of course, even when you're fulfilled by work or play most of the time, it doesn't mean you won't ever feel lonely or experience a kind of ennui, which sometimes settles over people. I went through a difficult period of ennui for about a year before I quit my methadone-clinic job, started my own psychotherapy practice, and began writing seriously. Loneliness and boredom are part of the human condition. They are certainly easier to ride through, however, when you know that they are temporary, that when you come through them you'll find or return to a more engaged, more impassioned life.
* * *
Get Smart!
How have you sought passion, meaning, and purpose in life? How does eating prevent you from finding what you're looking for? The next time you feel bored, lonely, or unfulfilled and have the urge for food, what can you do or say to yourself rather than eat?
* * *
### There Are So Many Things I Feel I Should Do; How Will I Ever Find the Time to Play?
Ah! Therein lies the rub — all those pushy "shoulds." If you ever want to find a balance between work and play, you would do well to give up commands like the S-word. Here's why. People who joyfully and fluidly keep their lives in balance don't run on words like _should_. To get where they want to go or be, they don't order or bully themselves around.
Tell me, when someone says you _should_ or _shouldn't_ do something, what's your reaction? When they insist you _need to be_ or _must not be_ a certain way, how does that make you feel? If you're like most people — me included — commands like these make you bristle and, worse, sometimes make you want to do exactly the opposite of what you're being instructed to do, even though you may not be aware of it.
Let's face it, by the time you're out of knee pants, you don't much want anyone telling you what to do. That doesn't mean they're wrong in doing so, or that you won't benefit from listening to what they have to say. It just means that these words kick up a good deal of dander within us because we associate them with childhood, when we heard — and had to heed — them 24/7. And because you had little or no choice in avoiding what your parents said you had to do back then, now when you hear commands, you often want to rebel. It's a natural instinct.
Think about words like _should_ , _need to_ , _have to_ , _must_ , _ought to_ , and _are supposed to_. Or their opposites: _shouldn't_ , _mustn't_ , and _aren't supposed to_. They're called external motivators and don't work well in the long run. Here's why. Tell yourself, "I have to clean the bathroom," and notice the prickles of resentment or annoyance, the "Yeah, make me" attitude that can spring up from seemingly nowhere. If you frequently use these words to get yourself to do this and not do that, you'll more than likely rebel against them. The reason is that these were the words you heard constantly as a kid. If your parents regularly issued commands, rather than asked or encouraged you to take or not take action, you're probably sick to death of following orders; yet ironically, there you are, ordering yourself around without realizing it.
So ditch the self-directives. You're far better off using internal motivators like _want_ , _would like_ , _desire_ , _wish_ , or _prefer_ to express intention. After all, you do want to clean the bathroom (because you'll feel good when it's done), you _wish_ to take the car in to be fixed on your day off (so that it runs well in the future), you _prefer_ to work overtime (in order to worry less about making ends meet), you _desire_ to take care of your sick child and pass on those free movie tickets (because you'll feel proud of your parenting skills), and you _would like_ to have dinner with your husband's crabby parents (because he does things he doesn't want to do in order to make you happy). You don't _need_ to do any of those things. The only thing we need to do in this life is die; everything else — you make a choice and live with the consequence — is optional.
This concept can be a tough sell to dysregulated eaters. They'll argue with me that they _should_ eat healthfully, _must_ stop overeating, _need to_ lose weight, are _supposed to_ be productive, _have_ _to_ take good care of themselves, and _ought to_ exercise. And I'll answer right back that they need not do any of these things as long as they're willing to live with the results of not doing them. Eventually, most people start to see the light.
All we ever have is desires, which, by their very nature, often compete with other desires. For instance, I want to stop writing now because I'm getting tired, and I also want to complete this chapter tonight. It's not that I _should_ or _need_ to finish it. I really want to — and I really don't. Can you see the benefits of using internal motivators? When you do, you're weighing words of equal value against each other, and they can have a fair fight. Commands simply are fraught with too much negative baggage to be useful in decision making.
"And what," you might ask, "does this have to do with balancing work and play?" Everything, absolutely everything. When you command yourself to work, do chores, or take actions using external motivators, your natural response is to balk, rebel, and not do them. It's as simple as that. Using those words is a setup for ensuring that you won't do what you keep telling yourself you "ought" to do. _Capisce_?
That's one of the reasons you have such difficulty working when it's time to work and playing when it's time to play. When you're working, you would like to be playing because you hardly ever give yourself that kind of break; and, even when you do, you feel so guilty that you don't enjoy yourself. When you're playing, you're convinced you should be working; and that takes all the joy and pleasure out of it. And all this nonsense starts with words like _should_ and _need_ _to_ or their negatives. The way out of this bind is to — duh — stop using these words. If you want to work and you also want to take a break from work, well, that seems reasonable. Hash it out with yourself. Don't bully yourself into deciding you _need_ to do something you _don't_ need to do and, in the process, set the stage for instant rebellion.
It helps to think of using words like _need_ and _have to_ as a finger poking you in the back, telling you to move along. Who would ever want that? On the other hand, words like _want_ and _desire_ are more like a finger inside that beckons you forward. It's clear which is the better motivator, isn't it?
Moreover, when you feel you _ought_ _to_ be working and won't allow yourself a break, how do you oh-so-cleverly manage to take one after all? You don't even need three guesses to answer that. You make sure you cut loose by eating, even if you're not the least bit hungry, even if you're stuffed to the gills. Paradoxically, when you rebel against yourself — for who else are you, as a fully autonomous adult, rebelling against? — you do it in the form of abusing food and your body. It may sound strange, but it's true.
* * *
Get Smart!
Do you often bully yourself with external motivators like _should_ and _have to_ , _shouldn't_ and _mustn't_? Are you willing to swap them for internal motivators like _want_ and _wish_? In your own words, describe how you rebel against doing things you don't want to do by abusing food.
* * *
To sum up, there are several skills to learn regarding balancing work and play:
• Comb through your beliefs and make sure they're not lopsided — heavily prowork and pro-productivity, or heavily proplay — and that they support a sensible balance between the two.
• Make sure you understand exactly what play is, and that you're not doing it with a hidden agenda to get something done.
• Explore and understand your "need" for busy-ness and productivity.
• Fully examine the issue of perceiving your life as empty and trying to feed that void with food. Take time to decide what might bring passion, meaning, and purpose to your life, but don't even think of pushing yourself into or toward anything. You'll know when you find it, and you'll benefit from giving things a chance to engage you.
• Stop bullying yourself with external motivators that make you likely to rebel and abuse food, and start using internal motivators.
• We hear so much talk these days about mindfulness, but remember, please, that there's a time to be mindful and a time to be mindless.
Over time, you'll develop the skill to keep work and play in balance — and you'll find that unwanted eating is less of a problem.
* * *
Skill Boosters
1. Define play according to how you understand it.
2. What three words come to mind to describe how you feel about work and daily-living tasks?
3. What three words come to mind to describe how you feel about play or relaxation?
4. Recall an anecdote that nails down exactly how your family felt or feels about anyone not doing their job or not doing it well.
5. Recall an anecdote that shows how you absorbed your family's message about hard work and productivity or rebelled against it.
6. What play activities do you wish you'd done more of as a child or adolescent?
7. Recall the best times you had at play as a child or adolescent.
8. Read your reframed beliefs about work and play aloud three times a day in front of a mirror, as well as first thing when you awaken and last thing before you go to sleep.
9. During the day, check in with yourself when you're working or playing and see how you're feeling, especially if you're working and desire a break or are playing and are ready to get back to work.
10. Practice sitting while doing nothing for one minute for three days, three minutes for three days, five minutes for three days, and so forth, until you've worked yourself into comfortably sitting and doing nothing for ten minutes a day. (And notice that the world doesn't end!)
11. Whenever you have the urge to eat at work when you're not hungry, recognize it as a desire to take a break; and know that it's about downtime, not chow time.
12. Ask people close to you to point out when you use external motivators, and try to catch yourself using them. Then replace them with internal motivators.
13. When you play or relax, shoo away guilty thoughts.
14. Consider effective ways to deal with people who might have a difficult time seeing you spend more time at play or relaxing.
15. To find out what excites you and what bores you to tears, notice how engaged or unengaged you are in your activities throughout the day. Do more of what excites you and less of what bores you.
16. Review your day, each day, to assess whether your work and play felt balanced and what adjustments you might have made to enjoy a better balance.
* * *
After completing the Life Skills Postassessment, you'll reach chapter 10, which will pull all the threads in this book together for you.
## Life Skills Postassessment
It's time to do your postassessment, which I've tucked between the preceding chapters on individual skill sets and the final chapter on integrating these skills into your life. Take a few deep breaths and make sure you're wearing your curiosity cap and not your Wicked Witch of the West hat. This questionnaire is designed to help you determine what you've learned and which specific, challenging skill sets need more attention. No judging yourself, pretty please, if you're not as competent in a skill area as you'd like to be. Instead of being hard on yourself, practice curiosity and compassion and maintain the growth-oriented mind-set discussed earlier: recognize that, with practice, you'll feel a good deal better about your skills as time goes on.
Remember that there are no right or wrong responses to the statements in this questionnaire, and give each one some thought before answering as honestly as you can. This is not a final exam. In fact, I encourage you to run through this assessment again one month, three months, six months, and a year from now. You'll be surprised at how quickly you acquire skills when you put your mind to it.
**Instructions:** Circle the number that best describes your response to each statement, with the number 1 representing _least true_ and 10 representing _most true_.
1. Overall, I have effective life skills.
1 2 3 4 5 6 7 8 9 10
2. I surround myself with people who have effective life skills.
1 2 3 4 5 6 7 8 9 10
3. Improved life skills would help me eat "normally" and attain and maintain a healthy weight.
1 2 3 4 5 6 7 8 9 10
4. I take excellent care of my health.
1 2 3 4 5 6 7 8 9 10
5. I have routine medical tests, follow doctors' orders, and take care of emergency medical concerns right away.
1 2 3 4 5 6 7 8 9 10
6. I get sufficient sleep most nights.
1 2 3 4 5 6 7 8 9 10
7. I take vitamins and supplements or medication consistently.
1 2 3 4 5 6 7 8 9 10
8. I get exercise (formal or informal) on a regular basis.
1 2 3 4 5 6 7 8 9 10
9. I am generally in touch with and can identify my feelings.
1 2 3 4 5 6 7 8 9 10
10. I value and am willing to experience all my feelings.
1 2 3 4 5 6 7 8 9 10
11. Rather than judging them, I am curious about my feelings.
1 2 3 4 5 6 7 8 9 10
12. I can tolerate intense and uncomfortable or conflicting feelings.
1 2 3 4 5 6 7 8 9 10
13. I express emotions appropriately and effectively.
1 2 3 4 5 6 7 8 9 10
14. I comfort and calm myself effectively.
1 2 3 4 5 6 7 8 9 10
15. For the most part, I live consciously and in the present.
1 2 3 4 5 6 7 8 9 10
16. I don't spend time unnecessarily worrying about the past.
1 2 3 4 5 6 7 8 9 10
17. I don't spend time unnecessarily worrying about the future.
1 2 3 4 5 6 7 8 9 10
18. I know that whatever befalls me in life, I will manage.
1 2 3 4 5 6 7 8 9 10
19. I plan for the future, then let it take care of itself.
1 2 3 4 5 6 7 8 9 10
20. I am comfortable in most social situations.
1 2 3 4 5 6 7 8 9 10
21. I am generally honest and share my feelings with intimates.
1 2 3 4 5 6 7 8 9 10
22. Intimates care about me as much as I care about them.
1 2 3 4 5 6 7 8 9 10
23. I set boundaries with intimates, and they respect them.
1 2 3 4 5 6 7 8 9 10
24. I take care of myself as well as I take care of others.
1 2 3 4 5 6 7 8 9 10
25. I'm good at knowing who to trust and who not to trust.
1 2 3 4 5 6 7 8 9 10
26. I know how to make and keep wonderful friends.
1 2 3 4 5 6 7 8 9 10
27. I'm pretty even emotionally and avoid emotional extremes.
1 2 3 4 5 6 7 8 9 10
28. I generally know when enough is enough.
1 2 3 4 5 6 7 8 9 10
29. I don't tend to frequently overdo or underdo.
1 2 3 4 5 6 7 8 9 10
30. I don't depend on others to regulate me emotionally.
1 2 3 4 5 6 7 8 9 10
31. When I'm with difficult people, I stay on an even keel.
1 2 3 4 5 6 7 8 9 10
32. I don't think in all-or-nothing, black-or-white terms.
1 2 3 4 5 6 7 8 9 10
33. I value both structure and freedom.
1 2 3 4 5 6 7 8 9 10
34. I'm skilled at troubleshooting and problem solving.
1 2 3 4 5 6 7 8 9 10
35. I'm neither too cautious nor impulsive in decision making.
1 2 3 4 5 6 7 8 9 10
36. I don't second-guess myself after making a decision.
1 2 3 4 5 6 7 8 9 10
37. I don't put off decisions because they're hard to make.
1 2 3 4 5 6 7 8 9 10
38. I have confidence in my critical-thinking skills.
1 2 3 4 5 6 7 8 9 10
39. I generally reach my goals.
1 2 3 4 5 6 7 8 9 10
40. I'm good at creating concrete, realistic goals for myself.
1 2 3 4 5 6 7 8 9 10
41. I know how to divide big goals into smaller ones.
1 2 3 4 5 6 7 8 9 10
42. I'm skilled at sustaining the effort needed to reach my goals.
1 2 3 4 5 6 7 8 9 10
43. I'm good at maintaining my hard-won achievements.
1 2 3 4 5 6 7 8 9 10
44. I don't sabotage my progress or achievements.
1 2 3 4 5 6 7 8 9 10
45. I recognize that all progress consists of baby steps.
1 2 3 4 5 6 7 8 9 10
46. My goals are composed of my wants, not "shoulds."
1 2 3 4 5 6 7 8 9 10
47. I can easily ask for or accept help in reaching my goals.
1 2 3 4 5 6 7 8 9 10
48. I always speak kindly to myself about myself.
1 2 3 4 5 6 7 8 9 10
49. I don't let other people be unkind or hurtful to me.
1 2 3 4 5 6 7 8 9 10
50. I don't have to be perfect.
1 2 3 4 5 6 7 8 9 10
51. I value myself and expect others to value me too.
1 2 3 4 5 6 7 8 9 10
52. I both love myself unconditionally and strive to do better.
1 2 3 4 5 6 7 8 9 10
53. I'm generally more curious about, than critical of, my mistakes.
1 2 3 4 5 6 7 8 9 10
54. I take failure in stride and learn from it.
1 2 3 4 5 6 7 8 9 10
55. I have a stable sense of myself.
1 2 3 4 5 6 7 8 9 10
56. I live a life balanced between obligations and play.
1 2 3 4 5 6 7 8 9 10
57. I can relax and let go fairly easily in healthy ways.
1 2 3 4 5 6 7 8 9 10
58. I know when I'm ready to work or play or relax.
1 2 3 4 5 6 7 8 9 10
59. I rarely procrastinate or rebel against commitments.
1 2 3 4 5 6 7 8 9 10
60. I live purposely and know the meaning of my life.
1 2 3 4 5 6 7 8 9 10
Okay, have you recovered from doing your postassessment? Please, if you're a perfectionist, listen up. Now that you've done your assessing and approach the final chapter of this book, I hope you're not expecting to be proficient in all the life skills you've been reading about. I assume that, like the rest of us muddling along on this planet, you've been working on improving at least some of these skills for quite a while, and that this book has given you the encouragement and guidance to keep on improving them.
## CHAPTER 10
## Integrating Life Skills into Eating "Normally"
## I Get It — Gain the Life Skills, Lose the Food Problem!
We all need essential life skills for health, happiness, and well-being. In addition, you happen to need them in order to end your eating problems. Perhaps you're even saying to yourself right now, "Gee, maybe my problems were never really about food in the first place, but instead were the result of not having effective life skills." I wouldn't have written _Outsmarting Overeating_ if I'd thought otherwise!
Now that you're almost finished with this book, the question is how to move forward. My fervent wish is that you will use compassion, pace yourself, view difficult experiences with food and other disappointments as learning experiences, and avoid trying to be perfect at anything except being imperfect. Remember, the goal of life is not to be flawless but to take in all it has to offer and live each day to its fullest. Isn't that a lot easier than always pushing yourself to go for the gold? I guarantee that you'll do better at some skill sets than others, and that life will sorely test your abilities with all of them. That's been my experience; and when you think of it, how could it be otherwise?
Now it's time to consider how to apply the skill sets you've been learning and to improve your relationship with food. I've talked a bit about eating in each chapter, but intentionally less than you may have hoped, because I wanted you to put your full attention on skills, not on your perceived feeding failures. The truth is, getting troubled eaters to focus on their eating is as easy as pie. But it's heavy lifting to get them to shift gears and put attention on other aspects of their lives that aren't going so swimmingly. And now it's time to combine the two.
So, let's put some of those skills — say, the skills for living consciously and staying regulated — to work right away. Now that you've done the postassessment, what are you thinking and feeling? Be sure to stay in the present rather than drifting toward thoughts about how you've done with food in the past or jumping ahead to how you might manage it better in the future. Anchor yourself to current pinpoints of thought and emotion, and see where they take you as you express compassion for, rather than judgment about, yourself.
If you start to feel upset that you're not where you want to be, simply note that your response to the assessment triggered some emotional dysregulation. Change your thinking and come back to a centered, neutral position, take a few deep breaths, and use positive self-talk to reregulate yourself. If you want to handle, by yourself, what's come up, dig deep for your inner strengths. If not, could you share your feelings with someone, write about them in a journal, or post your distress on a message board?
In terms of problem solving and using critical-thinking skills, consider what you're feeling. What can you do right now to feel more encouraged and hopeful? If you have ideas in mind, write them down. Use your critical-thinking skills to test your feelings against rational thinking and your experience. For example, if you feel like eating because you're bummed that you haven't made the progress you hoped to make, ask yourself if food will truly help. Make a list of how you want to spend the rest of the day foodwise and otherwise. Tap into your wisest self. Because you've worked hard reading this book, go ahead and engage in something fun that will allow you to lose yourself and not have to use your brain to think or reflect.
### How Do I Put These New Skills I'm Learning into Action in Real Life?
Consider how you felt learning a new job. Every day was full of challenges, and perhaps you often felt that you weren't picking up skills and information as quickly as you wished. Now think about having learned that job and how you felt about it six months or a year or more later. You could probably do some of it in your sleep by now. Remember that you've chosen, or been thrown into, many situations — work, kids, taking care of aging or ill loved ones — in which you had to acquire skills on the proverbial job.
And you did it. Skill-building for better eating is no different. What you want to avoid is focusing on how poorly you did in the past, especially on your "mistakes." Make your mantra: "That was then, and this is now." And don't keep worrying about how you're going to do. The worry takes you away from the present, which is _the only place where learning and change can occur_. You can learn from the past and apply that wisdom to the future, but the actual, hard-core doing-things-differently can happen only in the moment. You can think about having done something, and about what will be done, but the only time you actually can _do_ is in the here and now.
That noted, let me say that replaying an eating situation by purposefully spinning out your story with a different ending can be helpful. So, it's okay to consciously rerun the past and change your actions to give yourself a happier ending to an eating story. You also may find it valuable to spend time visualizing — called rehearsing — how you want to respond to difficult future situations in order to eat "normally." Notice that the life skill used in both situations is "living consciously." Your mind isn't drifting into the past without your awareness and making you feel hopeless, and it isn't anxiously rushing ahead trying to make the future turn out right for you. Rather, you're intentionally replaying a memory to analyze how you could have made a situation come out more successfully by using problem-solving and critical-thinking skills. Ditto when you're rehearsing: you're not unconsciously loading yourself up with pressure to do better but are intentionally imagining a future circumstance in order to plan out how you want to respond to it.
I've come up with a set of essential questions to ask yourself that will help you integrate into the real world the eight life skills discussed in this book. This protocol is similar to any ritual you follow. For example, when you want to drive, you automatically unlock your car and start up your engine, and maybe you shove in a CD or turn on the radio. As you start to pick up speed, you may even set your car on cruise control. Similarly, you follow a set of actions you barely think about when you reenter your house after vacation, are diapering your infant, or have to reset anything electrical in your house when the electricity goes out. Every time I leave the house, I unconsciously run through a list: Did I take my purse (because I have on more than one occasion caught myself halfway to somewhere without it)? Do I have my cell phone and sunglasses? Can I get a sighting of the cat so I know I haven't unintentionally closed her in a room? Are all the windows shut so I can set the house alarm? Think of a ritual or protocol that you engage in daily or weekly, one that is second nature by now, so that you barely notice going through it.
Following are the questions you'll want to get in the habit of asking yourself in order to integrate life skills into your world, with a special focus on those that will guide you toward "normal" eating. You can apply this eight-step protocol to any situation, whether it's one that happens down the road or one that is in your face right now. And don't worry, I'm going to take you through a whole bunch of examples to help you see how the questions are applicable and useful. Remember, it's all about enhancing your life skills!
1. Wellness and Physical Self-Care: How will this situation, advice, request, demand, interaction, occasion, or other circumstance affect my health and ability to take good care of myself physically, especially regarding eating?
2. Handling Emotions: How will I manage my emotions effectively in this situation or interaction or on this occasion?
3. Living Consciously: Rather than react automatically, how will I make conscious choices in this situation, interaction, or occasion or respond to this advice, request, or demand?
4. Building and Maintaining Relationships: How can I respond appropriately to this situation, advice, request, demand, interaction, or occasion in a way that honors myself and others?
5. Self-Regulation: What steps will I take to remain emotionally self-regulated regarding this situation, advice, request, demand, interaction, or occasion?
6. Problem Solving and Critical Thinking: How will I respond to this situation, advice, request, demand, interaction, or occasion in a way that is based on effective problem solving and critical thinking?
7. Setting and Reaching Goals: What are my goals regarding this situation, advice, request, demand, interaction, or occasion, and what steps will I use to reach them?
8. Balancing Work and Play: How can I respond to this situation, advice, request, demand, interaction, or occasion in a way that maintains a balance between work and play?
Now that you know the questions, it's time to try some hypothetical scenarios and practice using this eight-step protocol. I'll start you off with my ideas for how you might think through your responses, then I'll let you try the rest on your own. Notice that I'm not going to make decisions for you, that I will only show you how you might make effective choices for yourself. Moreover, there are no right answers, and that's an important caveat. There are numerous ways to respond to each scenario. The point is to get you skilled in thinking about how to make effective choices, not how to find the right choice, because often there is no right choice, just one that is better or worse for you. Okay, here goes.
Scenario 1: It's Friday near closing time at work, and your boss asks you to stay late. All day you've been sensing that you're coming down with a cold, and you have tentative plans to attend a movie with a friend but nothing else scheduled for the weekend. Your boss rarely asks for anything at the last minute, but seems anxious to clear out her in-basket. What do you do?
According to the protocol, you would ask yourself:
1. Wellness and Physical Self-Care: How might staying late affect my impending cold? Might I avoid letting it become full blown by leaving on time and chilling out for the evening, even canceling the movie with my friend? Alternatively, might I work late and rest up over the weekend? If I stay, will I arrive home hungry and be more tempted to eat quickly or choose nonnutritious food; or can I run out and grab something now that will tide me over until I get home? How am I going to take care of feeding myself?
2. Handling Emotions: How will I feel emotionally if I work late or leave on time? Will I feel taken advantage of and resentful because my boss has sprung a last-minute demand on me? Will I feel guilty if I refuse her request and leave for home on time? Will my feelings be so out of control either way that they'll drive me to eat?
3. Living Consciously: Am I staying in the present while making this decision, or is my mind slipping backward, to a memory of my old boss who blew up at me when I refused to cancel my vacation and fill in for an employee who quit? Am I getting ahead of myself by focusing on a rumor that the company might be downsizing, and by worrying that a refusal of my boss's request might be a reason to can me?
4. Building and Maintaining Relationships: How will refusing my boss's request affect our relationship? If it will cause friction between us, is it still worth saying no? If I cancel going to the movie, how will that affect my friendship? Will my friend understand or be upset with me for backing out at the last minute? How will I handle her upset?
5. Self-Regulation: How has my boss's request affected my ability to self-regulate emotionally? Is it triggering memories of how my demanding father used to insist I stop playing and finish my chores? Am I getting so angry at my boss that I can't think straight? Whatever I decide, how will I feel better emotionally without turning to food?
6. Problem Solving and Critical Thinking: Is there evidence that my boss's request is or might become part of a pattern of asking me to work late frequently? Is there a way to create a win-win situation here? What are my options? Stay late and cancel my outing with my friend or meet her for a late movie? Leave on time and take work home? Leave on time and come in over the weekend to finish up? Any other options?
7. Setting and Reaching Goals: What are my goals with this job: Staying with it over the long term and moving up the ladder? Or do I plan on leaving soon, so that tonight's decision doesn't mean a great deal to my career? Will saying no affect my job security?
8. Balancing Work and Play: Is there a way to respond so I'll feel that work and play are in balance? Maybe stay late at work and do nothing else over the weekend? Maybe go home and come back tomorrow? Do I want to forgo the pleasure of seeing a movie with my friend, after working hard all week, to spend more time on the job?
How'd that go? Is that how you would have sussed out the situation? Or would you have panicked and said yes without thinking things through or declined immediately because you hate it when people just assume you'll do what they ask or pressure you to do things? Even though it looks as if you might need to spend a great deal of time running through this protocol, these eight quick steps will get you in the habit of thoroughly analyzing your response options until your skills automatically become razor sharp and kick right in.
Let's try a couple more scenarios.
Scenario 2: You're remarried and your spouse's kids from a first marriage are coming to stay with you and your three kids over the holidays. Your spouse didn't exactly ask you first, but you know how much it means for him or her to have parent-child time together, so you didn't make a fuss when you weren't told that the kids decided to take an earlier flight than you'd expected. Holidays are always a busy time, and you like to make everything as perfect as possible for everyone. That's how you function in holiday mode, and you're usually exhausted and about five pounds heavier by the end of the year.
1. Wellness and Physical Self-Care: How will I take care of myself physically with so much going on and so many people to care for? When can I block out time on the calendar to exercise and be active? What strategies can I develop to manage food so that I make healthy choices, and where can I post a list of eating guidelines for myself so that I will read them over often? How can I find time every day to be alone, unwind, and care for my body and mind? How will I ensure that I get enough sleep every night no matter what's going on? Who must I talk with beforehand to make sure my self-care plans are clear and firmly in place? How will I approach my family about them?
2. Handling Emotions: How do I really feel about my spouse's kids coming to visit? How do I feel about not being asked? Do I have leftover anger or resentment? Would I have said yes or no if asked? How am I feeling about spending time with my stepchildren? Do I want to eat to lessen the intensity of my feelings? What will I do if I start to feel out of control emotionally? What's my plan to keep myself from abusing food?
3. Living Consciously: How can I not drive myself crazy anxious about making things just right? How will I remind myself to stay in the moment and not try to micromanage everyone's time? How will I make sure to have some fun myself over the holidays without turning to food for it? What steps will help me eat mindfully at every meal?
4. Building and Maintaining Relationships: What can I do to help my spouse's kids and our kids have a good time separately and together? How will having stepkids here affect my relationship with my own children? Will I promise to be honest with my spouse about how things are going, or will I say that things are fine when they're not? Will I request beforehand that my spouse be the one to take special care of his or her kids and not dump all responsibility on me?
5. Self-Regulation: How will being thrown off my usual routine affect me? If ever there were a time for me to abuse food, this would be it, when I'll be feeling so discombobulated, so how can I stay centered? What, other than food, will help me reregulate myself when I'm knocked off course emotionally, which is likely to happen no matter how well I plan?
6. Problem Solving and Critical Thinking: Are there any problems in making arrangements that need to be taken care of ahead of time? Who will help me? Might it be a good idea to talk with my spouse ASAP and divvy up tasks as well as arrange some preplanned activities? Would it be a good idea for my spouse and me to have time each day to check in with each other to assess how things are going between us and with the kids? In what ways have I avoided overeating during previous holidays, or in other family situations, that I can apply to this situation this year? How have I fallen into nonhunger or mindless eating during past holidays, and what steps can I take to do better this year?
7. Setting and Reaching Goals: What are my goals for this holiday — to be the world's greatest spouse, parent, stepparent, or host or hostess? To make sure I don't abuse food? To enjoy myself? To get through the holidays without losing my mind?
8. Balancing Work and Play: How can I make sure I get done what I need to and still have sufficient time to relax? What will my spouse and both sets of children need to do so that everything doesn't rest on my shoulders? How will I make it clear to everyone that I will take time for myself whether it pleases them or not? How will I follow through with my plan to take play or alone time even if it bothers or inconveniences others?
Perhaps you see how this scenario is more complicated than the last one and requires putting greater skill into action. There's a good deal more going on that might precipitate opportunities for stress eating, and as a result there's an increased need to focus on yourself. If you practice life skills every chance you get, you can't help but exponentially elevate the likelihood that they'll emerge naturally when the pressure's on. Can you see how honing your life skills would make even the previous scenario a great deal easier on you and your appetite?
Okay, let's run through one final scenario together.
Scenario 3: You and a bunch of your old high school friends think it would be fun to get away together for a three-night cruise. You're still crazy about two of them, one you can take or leave, and the other is a riot to hang with but a spoilsport unless she gets her way. The last time you were together, she went from ignoring you to relentlessly picking on you in her hey-I'm-just-teasing way. Plus, you're terrified that all that yummy cruise food and the hyperfocus on eating will put too much stress on your desire to eat "normally" when most of your friends have crummy food habits. You sorely need a vacation, and this cruise has ports you're dying to visit, but you're really torn about whether to go.
1. Wellness and Physical Self-Care: How will this trip enhance or impair my health? If I intend to eat "normally," what will I have to think, feel, and do every day for that to happen? Do I tend to get seasick? How will my friends' careless eating habits affect me, and will my best intentions get steamrolled by groupthink?
2. Handling Emotions: If I'm already anxious about the trip, how am I going to feel while on it? How can I ignore my friend if she acts up and yet still have a good time? How much of my reaction to her is triggered by her resemblance to my bossy older sister, who used to tease me and make jokes at my expense? What attitude do I want to have in order to be emotionally mature on the trip? Will I get so upset that I abuse food to handle my emotions?
3. Living Consciously: Will I worry the whole time I'm away that I'm not going to enjoy myself or that I'll eat crazily? Even if things are fine with that one friend, will I be on edge while waiting for her to zap me? How will I eat mindfully at every meal? How can I stay present at every moment of the trip so that I enjoy myself?
4. Building and Maintaining Relationships: How can I use my positive friendships to buffer my relationship with the one friend I don't care for so much? If I don't go on the cruise, will it upset my friends or will they be relieved? Will they invite me on a group outing again? Will my forgoing the trip ruin individual relationships with my friends?
5. Self-Regulation: Will my friends want to do everything together as a group, or will I have time to myself? How will I let them know how important self-time is to me? How will I follow through on my plan to go off on my own if need be, even if it angers or disappoints my friends? What will I do to reregulate myself, other than eat, if I feel pressured or upset?
6. Problem Solving and Critical Thinking: What are the pros and cons of going? Is there any other available info about the trip that would help me make a decision? How might considering my previous experience with these friends help me decide? Who can I talk with to help me decide? Am I in denial about any factors that I would like to pay attention to? Is it wishful thinking to believe that I can go on the cruise and eat "normally," or is there evidence that this is highly unlikely?
7. Setting and Reaching Goals: What are my goals for the trip — to hang out with friends, sightsee, get away from work and routine, or have adventures? When will my friends and I talk about our goals for the trip to make sure they mesh? Do I want to make it a goal to eat "normally" while on the cruise, or just give myself a vacation from thinking about everything I put in my mouth? What goals will help me stay active on the cruise?
8. Balancing Work and Play: Is the timing right to take a vacation, and is this cruise the right activity? Would I be better off passing on the invitation and planning another getaway more to my liking, or will I regret missing this cruise and having fun with my friends?
Are you getting the hang of it? It's easy to apply life skills to any problem, although not every situation will need all your skills. You get to pick and choose, and after a while you'll know which ones are required. Mainly, what using life skills does is slow down the process of automatically reacting to what's going on inside of and around you. It helps you identify your emotions but not rely on them for decision making. It takes you off Fantasy Island, especially in response to food, exercise, and self-care, and firmly plants your feet on this earth. It brings many of the same abilities you use at work into your personal life: evaluating pros and cons, taking history into account, and choosing options based on rationality and not your whim of the moment.
To give yourself practice employing your newly acquired life skills in the eating arena, use the eight-point protocol to consider what to do under the following circumstances:
• Scenario: For years, your mother, now ninety, made twenty kinds of Christmas cookies and stored them in the freezer for holiday gifts and gatherings. Recently, your sister-in-law had the idea of making a "recipe book of Mom's cookies," insisting that everyone in the family bake a few kinds, take pictures of them for a book for Mom, and keep them frozen until the holidays. You've been learning to be a "normal" eater and practicing life skills for about six months, and you aren't sure how you feel about making and storing the holiday cookies for the recipe book. Having all these delicacies around might be too tempting for you at this stage of learning to eat "normally."
• Scenario: Your younger brother just lost his job and wants to move in with you "for just a little while." Last time he did this, he failed to clean up after himself and ate a constant diet of junk food no matter what you cooked for him. Because you live alone, it is completely your decision whether you deny or accede to his request; but you're not sure what to do, mostly because your brother has pulled you out of several jams and you want to return the favor. Also, recently he seems to be making better decisions and finally maturing. You've just started reading books on "normal" eating and are intrigued by the idea of learning how to eat healthfully, and you wonder if his living with you might derail your plans.
• Scenario: Your entire office is starting a diet contest, to be accompanied by a weekly weigh-in. You've gained and lost hundreds of pounds, and dieting is the last thing you want to do, although you're unhappy with your current weight. The office manager is asking staff to choose up teams. Everyone seems gung ho. You don't want to be the odd person out, but you worry that dieting will trigger binge eating as it always has in the past. Plus, you're trying not to weigh yourself, because you'd rather focus exclusively on trying to eat well without the pressure of the scale.
• Scenario: Your mother has been nagging you to join her Saturday morning walking club. You enjoy the men and women in it, you could use structured exercise, and Saturday morning is a perfect time for you. But you haven't gotten along well with your mother since you were young enough to sit on her lap, and you don't want to make the relationship more strained than it already is. You can't decide whether to join the walking club.
• Scenario: You've been noticing that your aunt, who lives with her brother, your widowed father, is getting more forgetful. She's like a second mother to you because, when you were a child, your mother was sick a great deal of the time and your aunt moved in with your family to help raise you while your father worked. When you were old enough, she moved out, and you took care of your mother and moved out only after she died. Although Dad refuses to get his sister checked out medically, you believe she has Alzheimer's. Dad wants you to move in with them when your apartment lease is up — you can have your old room back — and you're not sure what to do. You've been enjoying taking care of just little old you — eating healthfully and exercising — for the first time in your life, but you feel the call of duty. You're afraid that having more people to care for will make it difficult to care for yourself.
• Scenario: You have what feels like a once-in-a-lifetime opportunity, a scholarship to study abroad. Your parents and college professors and adviser are all for it, but you're scared of being off on your own and you fear how a new environment will influence your relationship with food. Eating goes best for you when you're alone in your studio apartment and have control over what you buy, cook, and eat. You're fearful of being in a new place with new foods, so you're thinking of turning down the scholarship.
How did you do while using your new life skills in these hypothetical situations? Are you getting the knack of running through the protocol more quickly and adeptly? What I'd like you to pay attention to is whether you had previously used these skills in decision making, and what you learned from doing so in these scenarios that you can apply in your life. After all, it's not that hard to decide what's best for an imaginary person. But as you acquire life skills, you'll find it easier and easier to keep them at the ready for the real world you live in.
Here's one other way you can practice life skills without any consequence to you. Every time you hear of a problem situation — just listen to friends, coworkers, and family, and you'll hear plenty of them — use the eight-point protocol to consider how you would suss things out. Consider what emotions you'd have. Maybe they'd be the same as someone else's, but maybe not. Think about how you would stay rational and present while resolving whatever is troublesome rather than reactively jumping to a decision. Take time to come up with a solution that reflects most, if not all, of the eight-point protocol. Notice if others use similar strategies or simply dive into solutions headfirst without engaging in much rational thinking. If you continue to imagine how you'd solve other people's problems, similar problems will be a great deal easier when they're your own.
Well, here we are at the end of the book. But for you, I hope, it's not an ending but a beginning. My wish is that your growing competencies open up a brand-new chapter in your life, one that is skill filled, more satisfying and pleasurable, and that your new abilities keep moving you along toward "normal" eating. I promise that, as you slowly acquire new skills, your self-care will improve exponentially. Everything will come together in good time.
## Acknowledgments
Many thanks to Georgia Hughes, New World Library's editorial director, and to the rest of the editorial staff for their gentle guidance and expertise. And oodles of gratitude to Janice M. Pieroni, Esq., my literary agent and friend, who is always a pleasure to work with, from shaping up my manuscript to finding it a happy home.
## Notes
### Introduction
3 _"abilities for adaptive and positive behaviour"_: Skills for Health, World Health Organization's Information Series on School Health, Document 9 (Geneva: WHO, 2003), p. 8, www.who.int/school_youth_health/media/en/sch_skills4health_03.pdf, accessed July 27, 2014.
4 _"You did the best that you knew how"_: Maya Angelou, Maya Angelou Quotes website, www.mayaangelouquotes.org/page/4/, accessed July 19, 2014.
### Chapter 1. The Definition and Purpose of Life Skills
12 _five basic life skills that span every culture_: World Health Organization, Mental Health Promotion, Partners in Life Skills Education, "Conclusions from a United Nations Inter-Agency Meeting," Geneva, 1999, five basic life skills that are relevant across cultures, www.who.int/mental_health/media/en/30.pdf, accessed August 14, 2014.
19 _ten thousand hours is the average number of hours_: Malcolm Gladwell, _Outliers: The Story of Success_ (New York: Little, Brown, 2007), 39.
24 _"sophisticated methods of assessing beliefs, opinions"_: Ronald J. Massey, "Glossary of Common Psychological Terms," s.v. "Critical Thinking Skills," Ronald J. Massey, PhD, and Associates website, accessed July 27, 2014, www.drronmassey.com/Glossary.html.
### Chapter 3. Handling Emotions
47 _"In my view, if you forget everything else"_: Robert E. Thayer, _Calm Energy: How People Regulate Mood with Food and Exercise_ (New York: Oxford University Press, 2001), 8.
48 _"less than 90 seconds"_: Jill Bolte Taylor, _My Stroke of Insight: A Brain Scientist's Personal Journey_ (New York: Viking, 2006), 146.
56 _Most of the time when we're reacting strongly_ : Jon Connelly, PhD, LCSW, "Clinical Hypnosis with Rapid Trauma Resolution," clinical training manual (revised March 16, 2010), 37.
59 _"emotional intelligence"_: Daniel Goleman, _Emotional Intelligence: Why It Can Matter More Than IQ_ (New York: Bantam, 1995).
### Chapter 4. Living Consciously
70 _how little attention we pay to everyday life_: Thomas Moore, _Care of the Soul: A Guide for Cultivating Depth and Sacredness in Everyday Life_ (New York: HarperCollins, 1992).
### Chapter 7. Problem Solving and Critical Thinking
126 _"has three central concerns: positive emotions"_: "Frequently Asked Questions," Positive Psychology Center, University of Pennsylvania, www.ppc.sas.upenn.edu/faqs.htm, accessed August 8, 2013.
135 _Attributes of critical thinkers_: Emily R. Lai, _Critical Thinking: A Literature Review_ , June 2011, Pearson Assessments, www.pearsonassessments.com/hai/images/tmrs/criticalthinkingreviewfinal.pdf.
135 _"capable of taking a position or changing a position"_: Robert H. Ennis quoted in "Overview of Critical Thinking Skills," American Dental Education Association, accessed August 27, 2013, www.adea.org/adeacci/Resources/Critical-Thinking-Skills-Toolkit/Pages/Overview-of-Critical-Thinking-Skills.aspx.
### Chapter 8. Setting and Reaching Goals
142 _"70% of the variation in people's weights"_: Gina Kolata, _Rethinking Thin: The New Science of Weight Loss — and the Myths and Realities of Dieting_ (New York: Picador/Farrar, Straus and Giroux, 2007), 123.
145 _Keep goals few in number_: This list is adapted from Michael Hyatt, "The Beginner's Guide to Goal Setting," Michael Hyatt website, accessed June 14, 2013, <http://michaelhyatt.com/goal-setting.html>. See also Hyatt's book _Platform: Get Noticed in a Noisy World_.
146 _Believe and have faith in the process_: The list is adapted from Bradley Foster, "10 Steps to Successful Goal Setting," _Huffington Post_ , May 7, 2013, www.huffingtonpost.com/bradley-foster/how-to-set-goals_b_3226083.html.
149 _several reasons why making commitments doesn't work_: Leslie L. Downing, "Fragile Realities: Conversion and Commitment in Cults and Other Powerful Groups" (unpublished manuscript, 2010).
155 _Perfectionism . . . is a way to keep yourself safe_: Joanna Poppink, _Healing Your Hungry Heart: Recovering from Your Eating Disorder_ (San Francisco: Conari Press, 2011).
### Chapter 9. Balancing Work and Play
162 _play "is a state of being that is intensely pleasurable"_: "What Is Play," National Institute for Play, archived at <http://archive.today/vcMgC>, accessed July 27, 2014.
172 _learn more about their "dark nights"_: Thomas Moore, _Dark Nights of the Soul: A Guide to Finding Your Way through Life's Ordeals_ (New York: Dover, 2003).
## Index
### A
Alcott, Louisa May: _Little Women_ ,
all-or-nothing thinking. _See_ self-regulation
amygdala, ,
Angelou, Maya,
anxiety,
APPetite,
aunt-with-Alzheimer's scenario,
awe,
### B
balance. _See_ work and play, balancing
bingeing, , ,
boredom, ,
boundaries, 94–96
breathing,
brother's-lost-job scenario,
### C
_Calm Energy_ (Thayer), 47–48
carbohydrates,
_Care of the Soul_ (Moore),
caring distance,
children, hitting, 30–31
clothing and shoes, ,
commitment, 148–51
conscious living, 67–83,
vs. autopilot, , 69–70
via breathing/meditating,
and eating, , , 80–81
exercises, , , , ,
overview, 67–68,
past/future, living in, 70–73
present, learning to live in, 73–74
reality, consciously returning to, 75–79
scenarios for practice, , ,
skill boosters, 82–83
time out from, , 79–81
cookie scenario,
cooking, , 41–43
cortisol, ,
critical thinking. _See_ problem solving/critical thinking
### D
_Dark Night of the Soul_ (Moore),
daydreaming, 134–35
denial, 133–34
_Diagnostic and Statistical Manual of Mental Disorders_ ,
diet-contest scenario, 200–201
dieting, , 149–50
doctors/dentists, 33–36
dopamine, , ,
duration of dysfunctional behavior,
### E
eating disorders,
eating "normally," 189–202
via conscious living,
via critical thinking, ,
difficulty of,
eight-step protocol for, 192–93,
overview, 189–90
as restriction/deprivation, 148–49
scenarios to practice the protocol, 193–201
via self-talk to reregulate,
via staying in the present, ,
via visualizing/rehearsing,
emotions, 47–66,
acknowledging, , ,
avoiding,
as chemical surges, 48–49
crisis management, 21–22
discombobulation, 57–58
and eating, 21–22, 47–48, 60–62
emotional dysregulation, 51–52
emotional intelligence, 59–60
exercises, , , , , ,
feelings about feelings, 58–60
good vs. bad, ,
how to react to them, 49–50
in men vs. women,
overview, 47–49
pain long after the wound, , 54–57
positive,
in the present, 78–79
purpose of pain/pleasure, 53–54
scenarios for practice, , ,
seeking help,
self-soothing, , 60–64
sensitivity differences, 60–61
skill boosters, 64–66
skills for emotional management, 59–60
suppressing,
triggered by memories, 55–57
what to do with, 50–54
empathy, 93–94
emptiness, 171–74
Ennis, Robert H.,
Ericsson, Anders,
### F
failing forward,
fantasizing, 134–35
fear
and denial,
of doctors and dentists, 33–34
of failure, ,
of life not turning out well,
of problems returning,
triggered by memories,
feelings. _See_ emotions
fixed vs. growth mind-set people, goals of, 153–54
_The Food and Feelings Workbook_ (Koenig), ,
Foster, Bradley,
frequency of dysfunctional behavior,
friends-on-a-cruise scenario, 197–99
frustration,
fulfillment, 171–74
### G
Gladwell, Malcolm: _Outliers_ ,
goals, 141–60,
ambivalence toward, 147–48
baby steps toward, 150–51
books/advice on setting, 145–46
commitment to, 148–50
defined,
dieting, , 149–50
and eating, 24–25
exercises, , , , , ,
and failing forward,
of fixed vs. growth mind-set people, 153–54
functional beliefs, 157–58
implicit vs. explicit, 142–43
latent (hidden) vs. manifest (recognized), 143–44
life skills to improve along the way,
measuring progress,
overview, 141–42
and perfectionism, 154–55
scenarios for practice, 194–95, ,
setbacks, mistakes, relapses, and failures, , 153–54
setting your own pace, 155–56
skill boosters, 159–60
support, 156–57
Goleman, Daniel,
grocery shopping, , 41–43
growth vs. fixed mind-set people, goals of, 153–54
guilt,
### H
habits, acquiring, ,
hair care,
health insurance,
heredity,
hitting children, 30–31
holiday-cookie scenario,
holiday scenario, 195–97
Hyatt, Michael: _Platform_ ,
### I
impatience,
intensity of dysfunctional behavior,
### K
Koenig, Karen R.
_The Food and Feelings Workbook_ , ,
_The Rules of "Normal" Eating_ ,
_Starting Monday_ , ,
Kolata, Gina,
### L
life skills, 11–27
defined, ,
and eating, 20–26
eight-step protocol, 192–93,
exercises, , , , , ,
heredity's role,
identifying which skills to learn,
importance of, 3–4
length of time to learn, 18–20
motivation's role, 16–17, ,
optimism vs. pessimism about, 12–13,
origins, 13–16
overview, 11–12
postassessment questionnaire, 182–87
preassessment questionnaire, 5–10,
scenarios to practice, 193–201
skill boosters, 26–27
_See also_ conscious living; emotions; goals; problem solving/critical thinking; relationships; self-regulation; wellness/self-care; work and play, balancing
_Little Women_ (Alcott),
loneliness, , 63–64, ,
### M
makeup, 38–39
Massey, Ronald J.,
meal planning/preparation, , 41–43
meaning, 174–75
meditation,
memory, 71–72, 76–77, ,
mindfulness, , , ,
_See also_ conscious living
Moore, Thomas
_Care of the Soul_ ,
_Dark Night of the Soul_ ,
motivation
hidden,
role in life-skills proficiency, 16–17, ,
"shoulds" (self-commands) vs. needs/desires, 176–78
_See also_ goals
_My Stroke of Insight_ (Taylor), 48–49
### N
National Institute for Play,
nature (genetics) vs. nurture (environment),
negativity,
neural pathways, new,
neurotransmitter imbalances, ,
### O
optimism vs. pessimism, 124–27,
_Outliers_ (Gladwell),
### P
pain. _See_ emotions
panic attacks,
parallel play,
passion, 174–75
peace,
perfectionism, , 154–55,
_Platform_ (Hyatt),
Poppink, Joanna,
Positive Psychology Center, 126–27
positive self-talk,
_The Power of Now_ (Tolle), 73–74
problem solving/critical thinking, 123–40, 192–93
attributes of critical thinkers, 135–36
critical thinking vs. being critical, 135–38
defined,
denial, 133–34
and eating, , , 128–30, 136–37
enjoying progress, 128–29
exercises, , , , ,
fantasizing, 134–35
vs. feelings, 33–34
knowing when you're done,
lack of,
optimism vs. pessimism, 124–27,
overview, 123–24
paths to success, 132–33
via positive psychology, 126–27
problem, defined, 125–26
reassessment,
rushing vs. taking your time, 131–32
scenarios for practice, , 196–97,
skepticism, 135–37
skill boosters, 138–40
temporary vs. permanent/chronic, 127–30
victim mentality,
willingness to experiment,
projection, 98–99
purpose, 174–75
### R
rationality. _See_ problem solving/critical thinking
relationships, 85–101,
boundaries, 94–96
closeness/distance, 87–88
difficulties building/maintaining, 86–89
dynamics of, , 97–98
and eating, 22–23, 85–86,
empathy, 93–94
exercises, , , , , , , ,
expectations, 96–97
overview, 85–86, , 99–100
remarks by others, 98–99
role models,
scenarios for practice, , ,
sharing, 91–93
skill boosters, 100–101
suppressing emotions,
sympathy,
trust, 89–91
with VDPs (Very Difficult People), 99–100
responsibility vs. blaming,
Roth, Geneen,
_The Rules of "Normal" Eating_ (Koenig),
### S
scholarship scenario,
self-care. _See_ wellness/self-care
self-honesty,
self-reflection, 157–58
self-regulation, 103–21,
by avoiding VDPs (Very Difficult People), 117–18
balancing work and play via,
and crises,
dysregulation triggers, ,
and eating, ,
emotional dysregulation, 51–52, 113–14
examples of dysregulation, 105–7
exercises, , , , , , ,
learning enoughness (sufficiency), 107–13
overview, 103–4
parents' influence on, 109–10
preventing dysregulation, 116–19
recognizing dysregulation, 115–16
scenarios for practice, , , 198–99
self-soothing,
skill boosters, 120–21
structure vs. freedom, 103–4
vs. yo-yo cycles, ,
self-soothing, , 60–64,
self-worth, ambivalence about, 43–44
Seligman, Martin,
sensitivity, 60–61,
sharing, 91–93
skepticism, 135–37
skin, care of,
sleep,
smoking,
Socrates,
_Starting Monday_ (Koenig), ,
stress, , 63–64
sufficiency. _See_ self-regulation
sympathy,
### T
Taylor, Jill Bolte, , ,
_My Stroke of Insight_ , 48–49
teeth, care of,
_See also_ doctors/dentists
Thayer, Robert E.: _Calm Energy_ , 47–48
Tolle, Eckhart: _The Power of Now_ , 73–74
trust, 89–91
### V
victim mentality,
### W
walking-club scenario,
weight/body mass index, determiners of,
Weight Watchers,
wellness/self-care, 29–45,
ambivalence about self-worth, 43–44
buying and making food, , 41–43
clothing and shoes, ,
developing better skills, 40–43
dislike of doctors and dentists, 33–34
and eating, ,
everyday care, 36–40
exercises, , , ,
health insurance,
hitting children, 30–31
how to start, 34–36
living space, ,
makeup, 38–39
parents' care of their children, 29–32
parents' care of themselves, 32–33
scenarios for practice, , 195–96,
skill boosters, 44–45
sleep,
teeth, skin, and hair,
WHO (World Health Organization), ,
work and play, balancing, 161–87,
confusion over, 164–66
and eating, , , 171–72
exercises, , 165–66, , , , ,
and feeding an empty life (seeking fulfillment), 171–74
finding time for play, 176–79
and giving life meaning, 172–74
and guilt,
improving, 166–70
irrational beliefs about, 167–70
learning and play,
overview, 161–63,
play, defined,
role of passion/purpose/meaning, 174–75
scenarios for practice, , ,
by self-regulating,
"shoulds" (self-commands) vs. needs/desires, 176–78
skill boosters, 179–81
working-late scenario, 193–95
World Health Organization (WHO), ,
### Y
Young, Lester,
## About the Author
Karen R. Koenig, LCSW, MEd, is a psychotherapist, educator, eating coach, and expert on the _psychology of eating_ — the why and how, not the what, of it — with thirty years of experience teaching overeaters and undereaters how to eat "normally" and maintain a comfortable, healthy weight for life without dieting and deprivation.
She is the author of five books: _Starting Monday_ , _Nice Girls Finish Fat_ , _What Every Therapist Needs to Know about Treating Eating and Weight Issues_ , _The Food and Feelings Workbook_ , and _The Rules of "Normal" Eating_. Three of her books are available in multiple foreign languages.
Her articles and essays have appeared in _Social Work Focus_ , _Social Work Today_ , _Eating Disorders Today_ , the _Boston Globe_ , the _Boston Herald_ , and the _Sarasota-Herald Tribune_. She has been quoted in _Ladies Home Journal_ , _Berner Zeitung_ , the _Wall Street Journal_ , _Women's Health_ , _Self_ , _Shape_ , _Weight Watchers_ , _In Touch_ , and _OK_ magazines. She has been interviewed on TV networks and programs including ABC, FOX, WHDH, SNN (Brookline, Massachusetts), and Manatee, Florida, cable, as well as on scores of radio and internet shows and podcasts.
Among other venues, she has taught seminars for Simmons College School of Social Work, Boston University School of Social Work, Massachusetts School of Professional Psychology, National Association of Social Work (Massachusetts and Florida), Massachusetts Dietetic Association, National Organization for Women, Girl Scouts of America, and the Breast Cancer Awareness Association of Minnesota.
A graduate of Simmons College School of Social Work, Koenig practices and teaches in Sarasota, Florida. She blogs weekly at www.eatingdisordersblogs.com, and her website is www.karenrkoenig.com.
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La chiesa di Santa Maria Maddalena è un luogo di culto cattolico situato nel comune di Bordighera, in piazza del Popolo, in provincia di Imperia. La chiesa è sede della parrocchia omonima del vicariato di Bordighera e Valle Nervia della diocesi di Ventimiglia-San Remo. La chiesa fa parte degli immobili tutelati dalla Soprintendenza per i Beni Architettonici e Paesaggistici della Liguria.
Storia e descrizione
Eretta nel XVII secolo antistante la piazza del Popolo, fu consacrata nel 1617 e in seguito ristrutturata nel 1866 abbellendola con nuove decorazioni in stucco e oro zecchino. Secondo i racconti locali i lavori di ristrutturazione furono pagati dagli stessi abitanti, grazie alla donazione spontanea di vari monili consegnati al parroco e successivamente fusi.
Tra il 1881 e il 1883 vi fu un secondo restauro e sembra che il parroco, padre Giacomo Viale, a causa delle ridottissime disponibilità economiche, sollecitò impresari, artisti e uomini di cultura a partecipare, in vario modo, ai lavori di restauro dell'edificio religioso. È grazie a una lettera degli impresari Mombelli e Bulgheroni, che si è appreso dell'invio delle piantine della chiesa a Charles Garnier. Senza dati certi si può forse supporre che Garnier partecipò con dei semplici consigli al restauro della chiesa, ma non si conoscono con certezza i dettagli.
La facciata, decorata con stucchi in stile rococò, è del 1906.
Al di sopra del portale è presente un affresco raffigurante la Maria Maddalena, opera del 1742 di Giacomo Raimondo e in seguito rifatto nel 1922 da Luigi Morgari. Gli stucchi, datati al 1670, sono opera di Francesco Marvaldi membro della famiglia dei Marvaldi di Candeasco presso Oneglia, architetti di grande fama all'epoca. Il lampadario posto al centro della volta è dono della regina Margherita di Savoia.
Sull'altare maggiore è presente una statua in marmo del XVIII secolo, la Maddalena in Gloria, disegnata da Filippo Parodi e forse scolpita dal figlio Domenico Parodi tra il 1714 e il 1717. La statua lignea della Madonna del Rosario del 1702 è invece collocata sull'altare del Rosario, mentre la celebre statua in cera in cui sono poste le reliquie di sant'Ampelio - patrono di Bordighera - è posizionata nel secondo altare sulla destra della chiesa.
Il campanile, separato dalla chiesa, svetta sopra una loggia del tardo medioevo, dominando il profilo della Città Vecchia con la guglia decorata con tegole di maiolica dipinta. Anticamente fu una torre di avvistamento medievale, usata dagli abitanti per scorgere nel mare eventuali imbarcazioni piratesche, frequenti nel periodo. La torre fu in seguito sopraelevata e trasformata nel XVIII secolo nell'attuale campanile.
Note
Voci correlate
Bordighera
Chiesa (architettura)
Diocesi di Ventimiglia-San Remo
Maria Maddalena
Altri progetti
Collegamenti esterni
Maria Maddalena
Bordighera
Maria Maddalena | {
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package org.apache.hadoop.hbase.filter;
import java.io.IOException;
import java.util.ArrayList;
import java.util.List;
import org.apache.hadoop.hbase.Cell;
import org.apache.hadoop.hbase.CellComparator;
import org.apache.yetus.audience.InterfaceAudience;
/**
* Base class for FilterList. Currently, we have two sub-classes to extend this class:
* {@link FilterListWithOR}, {@link FilterListWithAND}.
*/
@InterfaceAudience.Private
public abstract class FilterListBase extends FilterBase {
private static final int MAX_LOG_FILTERS = 5;
protected final ArrayList<Filter> filters;
/**
* For each sub-filter in filter list, we save a boolean flag to indicate that whether the return
* code of filterCell(c) for sub-filter is INCLUDE* (INCLUDE, INCLUDE_AND_NEXT_COL,
* INCLUDE_AND_SEEK_NEXT_ROW) case. if true, we need to transform cell for the sub-filter.
*/
protected ArrayList<Boolean> subFiltersIncludedCell;
public FilterListBase(List<Filter> filters) {
reversed = checkAndGetReversed(filters, reversed);
this.filters = new ArrayList<>(filters);
}
protected static boolean isInReturnCodes(ReturnCode testRC, ReturnCode... returnCodes) {
for (ReturnCode rc : returnCodes) {
if (testRC == rc) {
return true;
}
}
return false;
}
protected static boolean checkAndGetReversed(List<Filter> rowFilters, boolean defaultValue) {
if (rowFilters.isEmpty()) {
return defaultValue;
}
boolean retValue = rowFilters.get(0).isReversed();
for (int i = 1, n = rowFilters.size(); i < n; i++) {
if (rowFilters.get(i).isReversed() != retValue) {
throw new IllegalArgumentException("Filters in the list must have the same reversed flag");
}
}
return retValue;
}
public abstract void addFilterLists(List<Filter> filters);
public int size() {
return this.filters.size();
}
public boolean isEmpty() {
return this.filters.isEmpty();
}
public ArrayList<Filter> getFilters() {
return this.filters;
}
protected int compareCell(Cell a, Cell b) {
int cmp = CellComparator.getInstance().compare(a, b);
return reversed ? -1 * cmp : cmp;
}
/**
* For FilterList, we can consider a filter list as a node in a tree. sub-filters of the filter
* list are children of the relative node. The logic of transforming cell of a filter list, well,
* we can consider it as the process of post-order tree traverse. For a node , before we traverse
* the current child, we should set the traverse result (transformed cell) of previous node(s) as
* the initial value. (HBASE-18879).
* @param c The cell in question.
* @return the transformed cell. n
*/
@Override
public Cell transformCell(Cell c) throws IOException {
if (isEmpty()) {
return super.transformCell(c);
}
Cell transformed = c;
for (int i = 0, n = filters.size(); i < n; i++) {
if (subFiltersIncludedCell.get(i)) {
transformed = filters.get(i).transformCell(transformed);
}
}
return transformed;
}
/**
* Filters that never filter by modifying the returned List of Cells can inherit this
* implementation that does nothing. {@inheritDoc}
*/
@Override
public void filterRowCells(List<Cell> cells) throws IOException {
for (int i = 0, n = filters.size(); i < n; i++) {
filters.get(i).filterRowCells(cells);
}
}
@Override
public boolean hasFilterRow() {
for (int i = 0, n = filters.size(); i < n; i++) {
if (filters.get(i).hasFilterRow()) {
return true;
}
}
return false;
}
@Override
public boolean isFamilyEssential(byte[] name) throws IOException {
if (this.filters.isEmpty()) {
return super.isFamilyEssential(name);
}
for (int i = 0, n = filters.size(); i < n; i++) {
if (filters.get(i).isFamilyEssential(name)) {
return true;
}
}
return false;
}
@Override
public void setReversed(boolean reversed) {
for (int i = 0, n = filters.size(); i < n; i++) {
filters.get(i).setReversed(reversed);
}
this.reversed = reversed;
}
@Override
public String toString() {
int endIndex = this.size() < MAX_LOG_FILTERS ? this.size() : MAX_LOG_FILTERS;
return formatLogFilters(filters.subList(0, endIndex));
}
protected abstract String formatLogFilters(List<Filter> logFilters);
}
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{"url":"https:\/\/www.physicsforums.com\/threads\/eulers-method-for-coupled-odes.766031\/","text":"# Euler's method for coupled ODE's\n\n1. Aug 14, 2014\n\n### Maylis\n\n1. The problem statement, all variables and given\/known data\nConsider the following pair of coupled first order ODEs\n\n$\\dot{y_{1}} = y_{2}$ with $y_{1}(0) = 1$\n$\\dot{y_{2}} = -y_{1}$ with $y_{2}(0) = 1$\n\nUse the Euler integration method with a step-size $h = 1$ and fill out the entries in the table below\n\n$\\begin{bmatrix} t_{k}&y_{1}(t_{k})&y_{2}(t_{k})\\\\ 0 & &\\\\ 1 & &\\\\ 2 & &\\\\ 3 & & -4\\\\ \\end{bmatrix}$\n\n2. Relevant equations\n\n3. The attempt at a solution\nNormally I understand how to do Euler's method, but of course now it's a coupled ODE so I am very confused how to do it. I know in general you do\n\n$y_{k+1} = y_{k} + f(t_{k},y_{k})h$\n\nBut with the coupled ODE I am lost\n\nLast edited: Aug 14, 2014\n2. Aug 14, 2014\n\n### pasmith\n\nIn this equation:\n$y$ can be (and normally is) a vector, and $f$ can be (and normally is) a vector-valued function.\n\n3. Aug 14, 2014\n\n### Maylis\n\nHow does it look?\n\n$\\begin{bmatrix} t_{k}&y_{1}(t_{k})&y_{2}(t_{k})\\\\ 0 & 1 &1\\\\ 1 & 2 &0\\\\ 2 & 2 & -2\\\\ 3 & 0 & -4\\\\ \\end{bmatrix}$\n\n$\\begin{bmatrix} y_{{1},t_{k}=1} \\\\ y_{{2},t_{k}=1}\\\\ \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 1 \\\\ \\end{bmatrix} + \\begin{bmatrix}1\\\\-1\\\\ \\end{bmatrix}*1 = \\begin{bmatrix} 2 \\\\ 0 \\\\ \\end{bmatrix}$\n\n$\\begin{bmatrix} y_{{1},t_{k}=2} \\\\ y_{{2},t_{k}=2}\\\\ \\end{bmatrix} = \\begin{bmatrix} 2 \\\\ 0 \\\\ \\end{bmatrix} + \\begin{bmatrix}0\\\\-2\\\\ \\end{bmatrix}*1 = \\begin{bmatrix} 2\\\\ -2 \\\\ \\end{bmatrix}$\n\n$\\begin{bmatrix} y_{{1},t_{k}=3} \\\\ y_{{2},t_{k}=3}\\\\ \\end{bmatrix} = \\begin{bmatrix} 2 \\\\ -2 \\\\ \\end{bmatrix} + \\begin{bmatrix}-2\\\\-2\\\\ \\end{bmatrix}*1 = \\begin{bmatrix} 0\\\\ -4 \\\\ \\end{bmatrix}$\n\nLast edited: Aug 14, 2014","date":"2017-08-21 03:20:56","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\": 2, \"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\": 0.8344653844833374, \"perplexity\": 1525.5526633939617}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886107487.10\/warc\/CC-MAIN-20170821022354-20170821042354-00275.warc.gz\"}"} | null | null |
Alshon Jeffery (born February 14, 1990) is an American former professional football player who was a wide receiver in the National Football League (NFL). He played college football for the South Carolina Gamecocks, and was drafted by the Chicago Bears in the second round of the 2012 NFL Draft. As a member of the Philadelphia Eagles, he won Super Bowl LII.
Early years
Jeffery attended Calhoun County High School in St. Matthews, South Carolina, where he was part of a four-time state championship winning basketball team that recorded an 84–1 record. He also played two years of football and was widely recruited, giving a verbal commitment to the University of Southern California before switching his commitment to the University of South Carolina. Jeffery's younger brother, Shamier Jeffery, committed to play football for the Gamecocks in 2011, following in the footsteps of both of Jeffery's two older brothers Charles Ben and Darren Ben; both of whom also played basketball and football for Calhoun County High School.
College career
2009 season
As a freshman, Jeffery caught five passes for 61 yards in his first five games before making seven catches for 138 yards and three touchdowns against the University of Kentucky. The performance was the second-best single-game performance in school history and Jeffery became the third Gamecock in 2009 to earn SEC Freshman of the Week, following defensive end Devin Taylor and running back Jarvis Giles. He was a consensus first-team Freshman All-SEC and first-team Freshman All-American in recognition of his successful freshman season.
2010 season
In a Week 4 game against Auburn, Jeffery had a breakout performance in the 35–27 loss. He was only a few yards shy of 200 receiving yards and added two touchdowns for the Gamecocks. Another one of his best games in 2010 came against in-state rival Clemson where he had five catches for 141 yards and a touchdown. Through 14 games, Jeffery made 88 catches totaling 1,517 yards and nine touchdowns including eight games with 100 yards or more receiving, and a 108.4 yd/game average. Jeffery was named a Biletnikoff Award finalist, the award given to the nation's top wide receiver. Because of his performances, helping lead the Gamecocks to their first SEC Championship Game appearance in school history, Jeffery was selected as an All-American.
2011 season
At the beginning of the 2011 season, Jeffery was named by ESPN as the best overall player in the SEC. In the season opener against ECU in which he caught five passes for 92 yards, Jeffery extended his active streak to 24 consecutive games with a reception. In the 2012 Capital One Bowl, Jeffery caught four passes for 148 yards and a touchdown but was ejected in the third quarter for fighting with Nebraska player cornerback Alfonzo Dennard. Despite this, Jeffery was named the Capital One Bowl MVP.
Collegiate statistics
Professional career
Chicago Bears
Jeffery was selected in second round of the 2012 NFL Draft with the 45th overall pick by the Chicago Bears.
2012 season
On May 2, Jeffery and the Bears agreed to a four-year contract, making him the first player in the 2012 draft class to sign, and the earliest second-rounder to sign since Dallas Cowboys receiver Kevin Williams in 1993. Jeffery made his debut against the Denver Broncos in week one of the preseason, and caught a game-high four passes for 35 yards, though the Bears lost 31–3. In the second preseason game, Jeffery was penalized for unnecessary roughness for fighting Washington Redskins cornerback DeAngelo Hall. Jeffery was later told by coaches to "play smarter". In his regular season debut against the Indianapolis Colts, Jeffery caught a Jay Cutler pass for a 42-yard touchdown. In Chicago's Week 5 away game against the Jacksonville Jaguars, Jeffery left the game after catching a touchdown thrown by Cutler. Although the injury did not appear to be significant, x-rays done on the following Monday determined Jeffery broke his right hand, which led to him missing a few games. Though the injury did not require surgery, Jeffery still had to wear a cast. Jeffery eventually practiced for the first time on November 9. Jeffery returned in Week 11 against the San Francisco 49ers, but would injure his knee, and will be sidelined for 2–4 weeks. In Week 15 against the Green Bay Packers, Jeffery was penalized three times for pass interference, nullifying a touchdown and two receptions of 36 and seven yards, as the Bears lost 21–13. Jeffery ended his rookie season catching 24 passes for 367 yards and three touchdowns.
2013 season
In 2013's Week 4 loss to the Detroit Lions, Jeffery recorded his first career 100-yard receiving game. The following week against the New Orleans Saints, Jeffery broke the Bears single-game receiving yards record with 218 yards, along with recording a career-high 10 receptions. Jeffery broke the record on the final play of the game, a 21-yard pass from Jay Cutler, passing Harlon Hill's record set in 1952 against the 49ers. In Week 13 against the Minnesota Vikings, Jeffery became the first player in franchise history to record two 200-yard games in the same season. In that game, he surpassed his own franchise record mark set earlier in the season. Jeffery ended the 2013 season with 89 receptions for 1,421 yards and seven touchdowns. The 1,421 yards are the second-most in team history, behind Brandon Marshall's 1,508 in 2012. His 89 receptions are the sixth-highest in franchise history. After an injury to Lions receiver Calvin Johnson, Jeffery was named to the 2014 Pro Bowl. On January 17, 2014, Jeffery was named the Pro Football Writers Association's Most Improved Player. He was ranked 54th by his fellow players on the NFL Top 100 Players of 2014.
2014 season
On September 7, Jeffery caught five passes for 71 yards in the season opener against the Buffalo Bills. Jeffery recorded a 44-yard reception. On September 22, Jeffery caught eight passes for 105 yards against the New York Jets. On September 28, Jeffery caught his first touchdown of the season against the Green Bay Packers. On October 5, Jeffery caught six passes for 97 yards and a touchdown against the Carolina Panthers. On October 12, Jeffery caught five passes for a season-high 136 yards against the Atlanta Falcons. On October 26, Jeffery caught his third touchdown of the season against the New England Patriots. On November 16, Jeffery caught a season-high eleven passes for 135 yards against the Minnesota Vikings. On November 23, Jeffery caught his fourth touchdown of the season against the Tampa Bay Buccaneers. On November 27, Jeffery caught nine passes for 71 yards and a season-high two touchdowns against the Detroit Lions. On December 4, Jeffery caught six passes for 95 yards and a touchdown against the Dallas Cowboys. On December 15, Jeffery caught his ninth touchdown of the season against the New Orleans Saints. On December 21, Jeffery caught six passes for 72 yards and a touchdown against the Detroit Lions. Jeffery caught a 20-yard touchdown.
In Jeffery's third season in the NFL, he recorded over 1,000 yards for the second time in his career with 1,133 yards with 85 receptions for 10 touchdowns on 145 targets.
2015 season
On September 13, Jeffery caught five passes for 78 yards in the season opener against the Green Bay Packers. On October 18, Jeffery caught eight passes for 147 yards and a touchdown against the Detroit Lions. On November 1, Jeffery caught a season-high ten passes for 116 yards and a touchdown against the Minnesota Vikings. On November 9, Jeffery caught ten passes for a season-high 151 yards against the San Diego Chargers. On November 26, Jeffery caught seven passes for 90 yards against the Green Bay Packers. On December 13, Jeffery caught six passes for 107 yards against the Washington Redskins. On December 20, Jeffery recorded a 10-yard touchdown reception against the Minnesota Vikings. On December 30, Jeffery was placed on injured reserve with a hamstring injury, ending his season.
Jeffery's season ended with 54 receptions for 807 yards and four touchdowns on 93 targets. Jeffery would end up leading the Bears in receptions in the 2015 season with 54.
2016 season
On February 26, 2016, it was announced that the Bears placed the franchise tag on Jeffery. On September 11, Jeffery caught four passes for a season-high 105 yards in the season opener against the Houston Texans. On September 19, Jeffery caught five passes for 96 yards against the Philadelphia Eagles. On October 16, Jeffery caught a season-high seven passes for 93 yards against the Jacksonville Jaguars. On October 31, Jeffery caught four passes for 63 yards and his first touchdown of the season against the Minnesota Vikings. On November 14, the NFL suspended Jeffery for four games for violating the NFL's performance-enhancing drug policy. On December 18, Jeffery caught six passes for 89 yards and a touchdown against the Green Bay Packers.
Jeffery amassed 4,549 receiving yards and 304 receptions during his five-year career with the Bears. He possesses the third most receiving yards in Bears franchise history.
Philadelphia Eagles
2017 season
On March 9, 2017, Jeffery signed a one-year, $14 million contract with the Philadelphia Eagles. Jeffery made his Eagles debut on September 10, 2017, in the season opener against the Washington Redskins. He finished the game with 38 yards on three catches. In Week 2 on September 17 at Kansas City, Jeffery had 92 yards on 7 catches and one touchdown, his first as an Eagle, in a 27–20 loss to the Chiefs. On December 2, 2017, Jeffrey signed a four-year extension worth $52 million with $27 million guaranteed.
The Eagles finished the season 13–3 and earned a first round bye. In the Divisional Round against the Atlanta Falcons, Jeffery recorded 4 catches for 61 yards in a 15–10 victory. In the NFC Championship against the Minnesota Vikings, Jeffery caught five passes for 85 yards and two touchdowns in 38–7 victory to advance to Super Bowl LII. In the Super Bowl, Jeffery caught 3 passes for 73 yards and scored the first touchdown of the game. The Eagles went on to defeat the New England Patriots 41–33 to give Jeffery and the franchise its first Super Bowl championship. On February 21, 2018, it was revealed that Jeffery underwent surgery for a torn rotator cuff.
2018 season
Jeffery underwent rotator cuff surgery in the offseason. He was kept out of action going into the regular season. In Week 4 of the 2018 season, Jeffery returned from his injury and recorded eight receptions for 105 receiving yards and a touchdown against the Tennessee Titans. In Week 6, against the New York Giants, he had eight receptions for 74 receiving yards and two touchdowns. The Eagles ended up making the 2018–19 NFL playoffs with a 9–7 record. In the Wild Card round, Jeffery made six receptions for 82 receiving yards against his former team, the Chicago Bears. The Eagles won 16–15. In the Divisional Round matchup against the New Orleans Saints, Jeffery had 5 catches for 63 yards. As the Eagles were on a successful drive late in the 4th quarter, a wide open Jeffery dropped a pass from Nick Foles that fell into the hands of Saints' cornerback Marshon Lattimore. After the interception, Jeffery and the Eagles' offense never got the ball back as the Saints won 20–14.
In the postgame interview, Jeffery stated that "It sucks right now. Everyone in the locker room, we're all down. Like I said, I let my teammates down, I let the city of Philadelphia down. That's on me. We'll be back next year for sure. One play don't define me, it happens. It's part of football. I just hated the way it happened in the playoffs, and it was the final moment."
2019 season
In Week 1 against the Washington Redskins, Jeffery caught five passes for 49 yards and the first receiving touchdown of the season as the Eagles won 32–27. In Week 2 against the Atlanta Falcons, Jeffery left early in the game due to a calf injury. He was listed as questionable to return to the game, but he did not play for the remainder of the game The Eagles lost their first game of the season 20–24. Jeffery made his return from injury in Week 4 against the Green Bay Packers. In the game, he caught three passes for 38 yards and one touchdown in the 34–27 win. Jeffery suffered a hip injury in Week 9 against the Chicago Bears and was forced to miss the next two games. He made his return in Week 13 against the Miami Dolphins. In the game, Jeffery caught nine passes for 137 yards and a touchdown in the 37–31 loss. He was placed on injured reserve with a foot injury on December 12, 2019. Overall, Jeffery finished the 2019 season with 43 receptions for 490 receiving yards and four receiving touchdowns.
2020 season
On July 28, 2020, the Eagles placed Jeffery on the active/physically unable to perform list to begin training camp. He was activated on September 5, 2020. He made his season debut against the New York Giants in week 10. He was targeted once with no catches. He made his first reception in week 12 from Jalen Hurts against the Seattle Seahawks. He had his first touchdown of the season against the New Orleans Saints in Week 14.
Jeffery was released by the Eagles on March 17, 2021.
NFL career statistics
References
External links
South Carolina Gamecocks bio
1990 births
Living people
Players of American football from South Carolina
People from St. Matthews, South Carolina
American football wide receivers
South Carolina Gamecocks football players
Chicago Bears players
Philadelphia Eagles players
Unconferenced Pro Bowl players
Under Armour All-American football players | {
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Q: MYSQL SELECT with custom Function I want to be able to select records using custom function in mysql.
I tried the following code but I got null result where there is data:
SELECT id, getAmountS(id) FROM product WHERE id=2
DELIMITER $$
CREATE FUNCTION getAmountS(pid INT(11)) RETURNS DOUBLE(12, 2) DETERMINISTIC BEGIN
DECLARE sellvalue DOUBLE(12, 2) ;
SELECT
SUM( CASE WHEN s.invoicestatus = 'Active' THEN ROUND(
(r.rate * 1 -((r.rate * 1) * r.discount / 100))+((
r.rate * 1 -( (r.rate * 1) * r.discount / 100)) * r.gstrate / 100),
2 ) ELSE 0 END) INTO sellvalue FROM morder r
LEFT JOIN sale s ON r.saleid = s.id
WHERE r.productid = pid;
RETURN sellvalue;
END $$
Please help.
Table: product
id(INT), productname(VARCHAR), manufacturer(INT)
Table: morder
id(INT), saleid(INT), gstrate(INT), rate(DOUBLE), productid(INT), discount(DOUBLE)
Table: sale
id(INT), invoiceno(VARCHAR), invoicestatus(ENUM), invoiceDate(DATE)
The following statement works fine:
SELECT SUM( CASE WHEN s.invoicestatus = 'Active' THEN ROUND(
(r.rate * 1 -((r.rate * 1) * r.discount / 100))+((
r.rate * 1 -( (r.rate * 1) * r.discount / 100)) * r.gstrate / 100), 2 ) ELSE 0 END) sellvalue FROM morder r
LEFT JOIN sale s ON r.saleid = s.id
WHERE r.productid = 30340
A: You need test your code from basic to complex. Add one thing each time until you found the problem.
Starting with this basic function. I can return the value so the function structure is ok. the problem is the query
SQL DEMO
CREATE FUNCTION getAmountS(pid INT(11)) RETURNS DOUBLE(12, 2) DETERMINISTIC BEGIN
DECLARE sellvalue DOUBLE(12, 2) ;
SELECT 2*pid INTO sellvalue FROM Table1
WHERE id = pid;
RETURN sellvalue;
END;
If you add your own data to the rextexter demo we can dig deeper and see where is the problem.
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Wide rims are here to stay, and if you still aren't sure what all the fuss is about, let us introduce you to the HED Ardennes Plus LT Clincher Road Wheelset. Some things are best left unchanged, and accordingly, the Ardennes LT keeps its aluminum rims and HED Sonic hubs. But change can be a good thing too, which is why HED bumped the width up to 25mm. And that means more of the fast rolling and superior grip that you've come to expect.
Wide rims have become standard for road use over the past few seasons, and HED's C2 rim shape has been at the forefront of this. The Plus LT takes this trend a step further by upping the width of the scandium alloy rim by two millimeters, to 25mm. The upshot is that you get more of the same traits that are relegating narrow rims to the antique bin. Namely, better grip and faster rolling through an improved tire profile. And, you can rest assured that these aren't empty promises. All things being equal, a wider rim puts more of the tire on the road, and that increased contact patch means more grip, especially while cornering. Better yet, by widening the tire's base, more of the sidewall is oriented vertically, and that means that the tire is better able to deform around irregularities in the road surface. So, you'll roll through them, not over them. And just in case you're looking for the fastest rolling setup around, the Plus LT is compatible with tubeless setups, in addition to traditional clincher tires with tubes.
The Plus LT wheelset rolls on HED's Sonic hubs, much like its FR and Plus FR brethren. The spokes are stainless items, of which the front wheel has 18 and the rear has 24. And they're bladed to reduce aerodynamic drag. To keep the wheelset in place, HED includes its steel quick-release skewers. The result is a wheelset that weighs just around 1564 grams.
The HED Ardennes Plus LT Clincher Road Wheelset is available with either a Shimano/SRAM or Campagnolo freehub compatibility and in the color Black. Please note that rim tape is not included and HED's suggested rider weight limit for these wheels is 225lb. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,977 |
{"url":"https:\/\/code.tutsplus.com\/tutorials\/how-to-create-a-simple-event-calendar-for-your-php-website--cms-33738","text":"# How to Create a Simple Event Calendar for Your PHP Website\n\nIn this post, we\u2019re going to review\u00a0Events Calendar\u00a0from CodeCanyon (no longer available). This script\u00a0allows you to add calendars to your PHP website. It\u2019s a ready-to-use tool which you can integrate in your existing PHP website and start using today!\n\nIf you\u2019re building a website which is related to events or\u00a0would like to present your site events in an interactive and interesting way, then you've come to the right place!\n\nThere are plenty of premium calendar extensions and scripts available for your PHP website. For each of these premium scripts and extensions, you can expect quality code, bug fixes, support, and new enhancements.\n\nIn this post, we\u2019re going to discuss the Events Calendar tool, available at\u00a0CodeCanyon for purchase at a very reasonable price. It allows you to create calendars from an admin page and add events to each calendar. Each calendar is capable of displaying an unlimited number of events.\n\nSo if you\u2019re looking for a script which can provide you both calendars and events, the Events Calendar tool is a perfect solution for you.\n\nLet\u2019s quickly go through the important features it provides:\n\n\u2022 unlimited number of calendars\n\u2022 unlimited number of events with each calendar\n\u2022 single and multiple day events\n\u2022 media support for events\n\u2022 import photos from Flickr\n\u2022 multi-language support\n\u2022 calendar customization options\n\u2022 and more\n\nAs you can see, the Events Calendar tool provides a lot of useful features that allow you to set up event calendars quickly. Throughout the course of this tutorial, we\u2019ll explore the different features provided by this plugin. To start with, we\u2019ll see how to download and install the Events Calendar script from the CodeCanyon marketplace.\n\n## Installation and Configuration\n\nIn this section, I'll show you\u00a0how to install and configure the\u00a0Events Calendar tool once you\u2019ve purchased and downloaded it from CodeCanyon.\n\nAs soon as you purchase it, you\u2019ll be able to download the zip file. Extract this, and you will find a directory with the main script code: events. It provides two different versions of event calendars you could choose from.\n\n#### Classic Embeddable Events Calendar\n\nIn this version, you can embed a calendar in a PHP page of your website within a div or an iframe.\n\n#### New SEO-Compliant Events Calendar\n\nThis is a\u00a0new version of the calendar\u00a0with social buttons and meta tags for every event page to improve indexing of the page. In this version, the calendar is not fully embeddable; you can only embed it within an iframe.\n\nIn this post, we\u2019ll discuss the classic embeddable version. Of course, there\u2019s nothing stopping you if you would like to use the new\u00a0SEO-compliant version. If you have any queries while setting up the SEO-compliant version, feel free to ask me in the comments below.\n\nIn our case, as we\u2019ve selected the classic embeddable version, it\u2019s the events\/embeddable directory which we\u2019re going to use. Copy this directory to your PHP application. For example, if your project is configured at \/web\/demo-app\/public_html, you should copy the embeddable directory to \/web\/demo-app\/public_html\/embeddable.\n\nNext, you need to import the necessary database tables. You can find the database SQL file in the events_DB directory which you need to import in your database: events.sql. Import this script to your database, for example with phpMyAdmin,\u00a0and you're done with the database setup!\n\nFinally, you need to update the embeddable\/admin\/include\/db_conn.php file to reflect your database connection settings. Update the following snippet as per your connection settings.\n\n### Configure Global Settings\n\nNow that we're done with the Events Calendar setup, you should be able to access the admin control panel at http:\/\/your-site-domain\/embeddable\/admin with the demo credentials.\n\nOnce you\u2019re logged in to\u00a0the admin\u00a0back-end, access the Settings > General Settings\u00a0link in the main navigation menu. Let\u2019s go through a couple of important settings in this section.\n\n#### Absolute Path of Events Calendar Installation\n\nThis is the path where you\u2019ve set up your event calendar application. As we\u2019ve placed the event calendar application in the embeddable directory, it should be http:\/\/your-site-domain\/embeddable.\n\n#### Timezone\n\nIt allows you to set the time zone for your event calendar.\n\n#### Choose Calendar Date Format\n\nIf you would like to switch between US, UK and EU date formats, you can set it here.\n\n#### Choose Calendar Time Format\n\nThis allows you to switch between 12-hour and 24-hour time formats.\n\n#### Default View\n\nThis setting allows you to select the default format of events in the front-end. There are two views to choose from: calendar view and\u00a0list view.\n\nAnd with that, we\u2019ve done with the necessary setup to create calendars and events. In the next couple of sections, we\u2019ll see how to create calendars and add events to it.\n\n## How to Create a Calendar\n\nIn this section, we\u2019ll see how to create a calendar from the back-end. Go ahead and access the Calendars link in the top navigation. By default, this tab lists all the calendars available.\n\nAt the top of the listing, you can see the Create a new calendar section, which allows you to create a new calendar right away. You just need to enter the calendar name to create a new calendar.\n\nGo ahead and create a new calendar. I\u2019ve created the Seminars calendar, as you can see in the above screenshot. Once you create a calendar, you\u2019ll be able to see it in the calendars listing.\n\nOnce you\u2019ve created the calendar, you are now ready to add events to the calendar. In the next section, we\u2019ll see how to add events to the calendar we\u2019ve created in this section.\n\nIn the previous section, we created the calendar which we\u2019ll use to display events. In this section, we\u2019ll see how to add events to the calendar.\n\nOn the calendar listing page, you can see the Manage Events link for each calendar you\u2019ve created, as shown in the following screenshot.\n\nClick on that link and it\u2019ll take you to the events listing page.\n\nAs you can see, the event form contains a lot of fields that you can fill in. Most of the fields are self-explanatory. Go ahead and fill in all the necessary fields for your event and save it. You\u2019ll be redirected back to the events listing page.\n\nOf course, you can add as many events you want. And if you want to categorize your events, you could create multiple calendars.\n\nSo with that, we\u2019re done with creating a calendar and adding events to it. In the next section, I'll show you how to integrate the calendar which we\u2019ve created in a PHP web page.\n\n## How to Embed the Calendar in Your PHP Site\n\nThere are different ways to\u00a0embed the calendar in your PHP web page. The embeddable\/index.php file contains comments that explain how to embed the calendar into your PHP web page without an iframe. I'll show you\u00a0how to embed the calendar with an iframe.\n\nIt's easy\u2014you just need to create an iframe element and point its\u00a0src\u00a0attribute to the\u00a0http:\/\/your-site-domain\/embeddable URL. The Event Calendar script\u00a0will do the rest.\n\nThat should load a calendar which should look like this:\n\nWhen you click on any date with events, you\u2019ll be taken to an events listing page.\n\nFinally, if you click on the Read More link on any event, you\u2019ll be taken to the event detail page.\n\nSo as you can see, Event Calendar\u00a0provides a complete solution to creating, managing, and displaying\u00a0calendars of events. With this script, you can create a number of different calendars and present your events with a nice front-end interface.\n\nFeel free to try other options available in the script, and don\u2019t hesitate to ask in the comments if you have any queries.\n\n## The Next Step: A Quick Look at a Couple of Other Events Calendar Scripts\n\nIf you're looking for more advanced calendar scripts that you could use right away, I recommend that you check out the following post, which summarizes some excellent scripts that are available for a low cost.\n\nHere are a couple of interesting ones:\n\n## Cleanto\n\nCleanto\u00a0is ideal for many different types of service companies looking for a reliable way to provide clients with full-featured online booking.\n\n## Ajax Calendar 2\n\nAjax Calendar 2\u00a0is a highly customisable personal calendar designed to help you keep organised. This is a best-selling update of another popular script, the Ajax Full Featured Calendar.\n\n## Tiva Timetable\n\nFully responsive, easy to set up, and very customizable, the\u00a0Tiva Timetable\u00a0is a good choice for those who are looking for a calendar with a clean and simple modern design.\n\n## Conclusion\n\nToday, we reviewed the Events Calendar script available from\u00a0CodeCanyon. It allows you to create events and calendars on your PHP website. In this post, we discussed how to download, install and embed this script in your PHP web page. Considering the features it provides, I believe it\u2019s reasonably priced and it\u2019s worth giving a try.","date":"2022-07-05 10:13:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.19153223931789398, \"perplexity\": 1721.0988609607057}, \"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-2022-27\/segments\/1656104542759.82\/warc\/CC-MAIN-20220705083545-20220705113545-00372.warc.gz\"}"} | null | null |
/**
*
* Process Editor
*
* (C) 2009, 2010 inubit AG
* (C) 2014 the authors
*
*/
package com.inubit.research.server.config;
import com.inubit.research.server.ProcessEditorServerHelper;
import com.inubit.research.server.ProcessEditorServerUtils;
import com.inubit.research.server.manager.ISLocation;
import com.inubit.research.server.persistence.DatabaseConnector;
import com.inubit.research.server.persistence.DatabaseSchema;
import com.inubit.research.server.user.Group;
import com.inubit.research.server.user.GroupProxy;
import com.inubit.research.server.user.SingleUser;
import com.inubit.research.server.user.SingleUserProxy;
import java.sql.PreparedStatement;
import java.sql.ResultSet;
import java.util.HashMap;
import java.util.Map;
import java.util.Set;
/**
*
* @author fel
*/
public class DatabaseUsersConfig implements UsersConfig {
private DatabaseConnector db;
private PreparedStatement singleUserSelect;
private Map<String, SingleUserProxy> users = new HashMap<String, SingleUserProxy>();
private Map<String, GroupProxy> groups = new HashMap<String, GroupProxy>();
public DatabaseUsersConfig() {
this( ProcessEditorServerHelper.getDatabaseConnector() );
}
public DatabaseUsersConfig( DatabaseConnector dc ) {
this.db = dc;
loadAllUserProxys();
checkForExistingUsers();
}
public Set<String> getUserNames() {
loadAllUserProxys();
return this.users.keySet();
}
public Set<String> getGroupNames() {
this.groups = new HashMap<String, GroupProxy>();
Set<Object> groupNames = db.selectSingleAttribute(DatabaseSchema.Attribute.GROUP_NAME, DatabaseConnector.EntityType.GROUP);
groupNames.addAll(db.selectSingleAttribute(DatabaseSchema.Attribute.SUBGROUP_NAME, DatabaseConnector.EntityType.SUBGROUP));
for ( Object group : groupNames )
groups.put( (String) group, new GroupProxy((String) group, db) );
return groups.keySet();
}
public Group getGroup(String name) {
if ( !this.groups.containsKey(name) )
this.getGroupNames();
return this.groups.get(name);
}
public SingleUser getUser(String name) {
if ( this.users.containsKey(name) )
return this.users.get(name);
//if user is not listed, try fetching it from the database
try {
this.singleUserSelect.setString(1, name);
ResultSet result = this.singleUserSelect.executeQuery();
if ( result.next() ) {
SingleUserProxy newUser = new SingleUserProxy(name, db);
this.users.put( name, newUser );
return newUser;
} else return null;
} catch ( Exception ex ) {
ex.printStackTrace();
return null;
}
}
public boolean addUser(String name, String pwd) {
pwd = ProcessEditorServerUtils.getMD5Hash(pwd);
SingleUser su = new SingleUser(name, pwd);
if ( db.addUser(su) ) {
users.put(name, new SingleUserProxy(name, db));
return true;
} else {
return false;
}
}
public boolean addGroup(String name) {
if ( !this.groups.containsKey(name) ) {
this.groups.put( name, new GroupProxy(name, db));
return true;
}
return false;
}
public void setMail(String userName, String mail, boolean deferWrite) {
SingleUser su = this.getUser(userName);
if ( su != null ) su.setMail(mail);
}
public void setPictureId(String userName, String id, boolean deferWrite) {
SingleUser su = this.getUser(userName);
if ( su != null ) su.setPictureId(id);
}
public void setRealName(String userName, String name, boolean deferWrite) {
SingleUser su = this.getUser(userName);
if ( su != null ) su.setRealName(name);
}
public void setAdmin(String userName, boolean isAdmin) {
SingleUser su = this.getUser(userName);
if ( su != null ) su.setIsAdmin(isAdmin);
}
public void setGroupMembers(String name, Set<String> members) {
this.getGroup(name).setMembers(members);
}
public void setSubgroups(String name, Set<String> subgroups) {
this.getGroup(name).setSubGroups(subgroups);
}
public boolean addISConnection(ISLocation ism, SingleUser user) {
if (ism.checkConnection()) {
ism.setOwner(null, user, null);
user.addISConnection(ism);
return true;
}
return false;
}
public void removeISConnection(ISLocation ism, SingleUser user) {
user.removeISConnection(ism);
}
private void checkForExistingUsers() {
if ( this.getUserNames().isEmpty() ) {
this.addUser("root", "inubit");
users.get("root").setIsAdmin(true);
}
}
private void loadAllUserProxys() {
this.users = new HashMap<String, SingleUserProxy>();
Set<Object> userNames = db.selectSingleAttribute( DatabaseSchema.Attribute.USER_NAME, DatabaseConnector.EntityType.USER);
for ( Object userName : userNames ) {
users.put( (String) userName, new SingleUserProxy( (String) userName, db));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,720 |
La mosquée de Wazir-Khan est une mosquée de Lahore. Elle fut construite sous le règne de Shah Jahan, en 1634, par le gouverneur de la province du Pendjab, qui répondait au nom de Nawab Wazir Khan.
Cette mosquée, l'une des plus belles du Pakistan, est célèbre pour l'exceptionnelle qualité de ses mosaïques, mélange de motifs floraux et de compositions calligraphiques.
La salle des prières se compose de deux séries de deux espaces qui encadrent la salle principale. Chacun est surmonté d'un dôme richement décoré.
Galerie
Mosquée au Pakistan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,309 |
Bloomsbury Egyptology
_Series editor: Nicholas Reeves_
_Ancient Egyptian Technology and Innovation_ , Ian Shaw
_Archaeologists, Tourists, Interpreters_ , Rachel Mairs and Maya Muratov
_Asiatics in Middle Kingdom Egypt_ , Phyllis Saretta
_Burial Customs in Ancient Egypt_ , Wolfram Grajetzki
_Court Officials of the Egyptian Middle Kingdom_ , Wolfram Grajetzki
_The Egyptian Oracle Project_ , edited by Robyn Gillam and Jeffrey Jacobson
_Hidden Hands_ , Stephen Quirke
_The Middle Kingdom of Ancient Egypt_ , Wolfram Grajetzki
_Performance and Drama in Ancient Egypt_ , Robyn Gillam
## Contents
List of Illustrations
Acknowledgments
Egyptian Chronology
1 Ancient Egyptians at Play: An Introduction
A range of casting devices
Board games across borders: Identifying Egyptian games
2 _Mehen_ and _men_ : The First Signs of Egyptian Board Games
_Mehen_ : The game of the coiled serpent
_Mehen_ boards
_Mehen_ pieces
Rules of _mehen_
Pictorial evidence
Textual evidence
Archaeological evidence and social context
The demise of _mehen_ in Egypt
_Mehen_ in Nubia
_Mehen_ in the Levant and Mesopotamia
_Mehen_ in Cyprus and the Aegean
_Men_
_Two rows of thirteen_ and _forty-two and pool_
3 _Senet_ across Borders
Early evidence for _senet_
Old Kingdom: Ritual use and graffiti
Middle Kingdom: Changes and consistency
New Kingdom: Religious meaning
Later history of _senet_
Playing pieces
_Senet_ in Nubia
_Senet_ in the Levant
_Senet_ in Cyprus
The _game of thirty-three_
4 The _Game of Twenty_ : A Foreign Acquisition
Origins and chronological distribution
Beni Hasan playing scenes
Ancient names
Boards for the _game of twenty_
Special squares and decorations
Archaeological contexts
Rules for the _game of twenty_
"Uniting of the twenty squares" or _thirty-one_
The Levant and Cyprus: Games as heirlooms
5 The Game of _Hounds and Jackals_ : From Thebes to Susa
Boards for _hounds and jackals_
Gaming pegs
Beni Hasan playing scenes
The game outside of Egypt
Reconstructed rules
Symbolism of the game
The Coptic board game
6 Roman Board Games Crossing the Borders of Egypt
The sources
The game of _five lines_ or _πέντε γραμμαί_
_Duodecim scripta_ or _ludus duodecim scriptorum_
_Latrunculi_ or _ludus latrunculorum_
_Merels_ or _mill_ game
_Marbles_
Remaining configurations
The borders of Egypt
7 Arab and Ottoman Invaders Scratching the Surface
Graffiti games
_Seeja_ or _siga_
_Seeja_ playing rules
_Tâb_
_Mancala_
8 The Role of Board Games in Understanding Antiquity
Spread of board games
Religiosity of board games
Site use
Unidentified board games and new approaches
References
Author Index
Subject Index
## Illustrations
### Figures
All drawings, unless otherwise indicated, were made by Jennifer Steffey at the American Museum of Natural History in New York, who also designed the maps for each of the chapters. Dimensions of objects are indicated when available either as a scale on the photograph or in the caption.
1.1Casting devices from Egypt and Sudan.
2.1Map of sites mentioned in Chapter 2.
2.2 _Mehen_ board demonstrating a Predynastic rendering of the game.
2.3 _Mehen_ board bearing the name of Hor Aha.
2.4Second Dynasty _mehen_ board accompanied by spherical playing pieces.
2.5Painting from the tomb of Hesy-Re showing _mehen_ , _senet_ and _men_.
2.6Ivory _mehen_ piece from Abydos in the form of a couchant lion.
2.7 _Mehen_ game from Episkopi _Phaneromeni_ , Cyprus.
3.1Map of sites mentioned in Chapter 3.
3.2Drawing of _senet_ patterns, showing markings common during different periods.
3.3Drawing of scene from the mastaba of Nikauhor showing _senet_ playing.
3.4Facsimile of a painting from the tomb of Nefertari.
3.5 _Senet_ boards on terracotta platters from Twenty-Sixth Dynasty fortress at Tell Defenneh.
3.6Playing pieces collected by F.G. Hilton Price.
3.7Drawing of Merenptah playing _senet_.
3.8 _Senet_ game from Arad, with drilled depressions as the playing spaces.
3.9 _Senet_ game from Hazor, with the _game of twenty_ on the opposite side.
3.10 _Senet_ games from the Episkopi region, Cyprus.
3.11 _Senet_ game of Late Period date, with the _game of thirty-three_ on the opposite face.
4.1Map of the _game of twenty_ from the mid-third to the first millennium BCE.
4.2The _royal game of Ur_ with gaming pieces and tetrahedrons.
4.3The _game of twenty_ in the second and first millennia BCE, and the route of play.
4.4Playing scenes A, B and C in the tombs of Baqet III and Khety, Beni Hasan.
4.5 _Game of twenty_ from Thebes with _senet_ on the opposite side.
4.6Ostracon from Deir el-Medina.
4.7Drawing of the Turin Papyrus with four games.
5.1Map of the game of _hounds and jackals_ during the second and first millennia BCE.
5.2Game of _hounds and jackals_ from Thebes.
5.3Violin-shaped game board.
5.4Drawing of hippopotamus-shaped game board.
5.5Turtle stand or simulacra from Dra Abu el-Naga.
5.6Playing scene in Beni Hasan, probably from Tomb 17.
5.7Boards from the deposit of the temple of Inshushinak, Susa.
5.8Reconstruction of the numbering sequence of the holes.
5.9Coptic board game.
6.1Map of sites mentioned in Chapters 6 and .
6.2Example of a Roman games context at Palmyra, Syria.
6.3 _Five lines_ at the Luxor temple and at Qasr al Ghweita.
6.4Drawings of _duodecim scripta_ boards from Qustul and Dawwi.
6.5 _Duodecim scripta_ boards at Kom Ombo.
6.6Drawing of a terracotta game board.
6.7 _Latrunculi_ board at Kom Ombo.
6.8 _Merels_ boards at Dendera, Kom el-Dekka, Kom Ombo and Silsila.
6.9Drawing of a _nine-men's-morris_ board as found on a column at the Ramesseum.
6.10Drawing of a marble lane.
6.11Two unidentified game boards at Beni Hasan and Silsila.
6.12Possible _seeja_ board at the Kharga Oasis.
7.1Example of an Arab or Ottoman games context at Petra, Jordan and on Sai Island, Sudan.
7.2Placement of first pieces on a _seeja_ board.
7.3Examples of _seeja_ boards at Silsila, Medamoud, and el-Kab.
7.4Examples of _tâb_ boards at el-Kab.
7.5Drawing of a _mancala_ board at the "Third Pyramid, Gizeh."
7.6Sudanese _mancala_ boards carved in the temple of Tiye, Sedeinga, and on the temple of Soleb.
### Table
4.1Table with types of marking and distinct shapes for special squares in Egypt.
## Acknowledgments
First of all, we would like to thank Nick Reeves for initiating this book, Anna MacDiarmid and the staff at Bloomsbury Academic Press for their support in the process as well as two anonymous reviewers for their constructive and valuable comments.
At the Metropolitan Museum of Art, we are indebted to the Department of Egyptian Art, led by Lila Acheson Wallace Curator in Charge Diana Craig Patch, for allowing us to study games in the collection, and in particular to Niv Allon, Elizabeth Fiorentino, Janice Kamrin, Marsha Hill, Adela Oppenheim, Catharine Roehrig, Morena Stefanova, as well as Ann Heywood from the Sherman Fairchild Center for Objects Conservation for their assistance in object or text studies and/or image requests. We wish to acknowledge the Department of Ancient Near Eastern Art, and Curator in Charge Joan Aruz, for their considerable support, and especially Blair Fowlkes-Childs, Elizabeth Knott and Michael Seymour for their contributions.
Ben Haring and Olaf Kaper (Leiden University), Pavel Onderka (Narodni Muzeum, Prague), Rudolf Haensch (Kommision für Alte Geschichte und Epigraphik des Deutschen Archaeologischen Instituts), Dennys Frenez (Università di Bologna), Mark Kenoyer (University of Wisconsin), Jennifer Webb and David Frankel provided us with helpful information, and special mention should be made of our colleagues at Arizona State University, Nancy Serwint, Leif Jonsson and Kostalena Michelaki, as well as Annie Caubet (Musée du Louvre) and Francesco Tiradritti (Università di Enna Unikore) for their invaluable contribution to the progress of our research.
Thanks to Dr. Abdel-Hamid (Egyptian Museum, Cairo), Sara Al-Ashmawi (Egyptian Museum, Cairo), Nadine Cherpion (Institut Français d'Archéologie Orientale, Cairo), Nevine Kamal (Institut Français d'Archéologie Orientale, Cairo), Yael Barschak (Israel Antiquities Authority), Christopher Sutherns (British Museum), Tracey Golding (Petrie Museum), Vincent Rondot (Musée du Louvre) and Isabelle Bardiès-Fronty (Musée de Cluny) for facilitating access to images and granting permission.
Special thanks to Maria Nilsson (Lund University) and John Ward for their generosity in supplying multiple images for this publication, to Paul Whelan for his illustration, and also to Stuart Swiny for photographs, insightful discussion and access to unpublished material. We are grateful to Vincent Francigny (Section Française de la Direction des Antiquités du Soudan), Alexandra Christopoulou (National Archaeological Museum, Athens), Eleni Tourna (National Archaeological Museum, Athens), Marie-Noël Bellessort (École du Louvre), Olga Fast (Ägyptisches Museum, Bonn), Irving Finkel (British Museum), Stephen Quirke (Petrie Museum), Rachael Sparks (Institute of Archaeology, London), Ashley Cook (World Museum, Liverpool), John Wyatt (Griffith Institute and Bodleian Library, Oxford), Christopher Dobbs (University of Missouri), Jay VanRensselaer (Johns Hopkins University, Baltimore), Jack D.M. Green (Oriental Institute Museum, Chicago), Linda Evans (Macquarie University, Sydney), Naguib Kanawati (Australian Centre for Egyptology), María Antonia García Martínez (Tamkang University of Taipei), Christine Lilyquist and Peter Michaelsen for answering our inquiries. We owe particular thanks to Rozenn Bailleul-LeSuer (University of Chicago) for valuable advice, Ulrich Schädler (Swiss Museum of Games) for his insights and Constance Dickmeyer for commenting on our text. Finally, we are most grateful for the many drawings provided by Jennifer Steffey at the American Museum of Natural History in New York.
## Egyptian Chronology
This chronology has been compiled on the basis of Shaw (2000:xiii–xiv) and Francigny et al. (2014:7–8). Dates for the Late Period and onwards are absolute. All other dates are approximate. During the Intermediate Periods, dynasties may overlap. Dates relevant for regions outside of Egypt are included in the text with those for Nubia taken from Rilly and de Voogt (2012:187), those for the Levant from Sharon (2014:62) and those for Cyprus from Knapp (2013:27).
PREDYNASTIC PERIOD 5300–3100 BCE
Naqada
EARLY DYNASTIC PERIOD 3100–2686 BCE
First–Second Dynasties
OLD KINGDOM 2686–2181 BCE
Third–Sixth Dynasties
FIRST INTERMEDIATE PERIOD 2181–2055 BCE
Seventh–Eleventh Dynasties
MIDDLE KINGDOM 2055–1650 BCE
Eleventh–Fourteenth Dynasties
SECOND INTERMEDIATE PERIOD 1650–1550 BCE
Fifteenth–Seventeenth Dynasties
NEW KINGDOM 1550–1069 BCE
Eighteenth–Twentieth Dynasties
THIRD INTERMEDIATE PERIOD 1069–747 BCE
Twenty-First–Twenty-Fourth Dynasties 1069–715
LATE PERIOD 747–332 BCE
Twenty-Fifth Dynasty (Kushite Period)
Twenty-Sixth Dynasty (Saite Period)
Twenty-Seventh Dynasty (First Persian Period)
Twenty-Eighth–Thirtieth Dynasties
Second Persian Period
PTOLEMAIC PERIOD 332–30 BCE
ROMAN PERIOD 30 BCE–395 CE
BYZANTINE PERIOD 395–640 CE
ISLAMIC PERIOD 640 CE–present
##
## Ancient Egyptians at Play: An Introduction
The material culture of board games in Egypt has long been a topic of interest for archaeologists, ethnographers and lay people alike. The climatic conditions of the Nile Valley afford for the preservation of perishable materials, and thus a multitude of evidence to sustain this interest. Game boards and their paraphernalia have been identified among the material culture of ancient Egypt since the early days of archaeology (e.g., Prisse d'Avennes 1847:9). The number of well made, easily identifiable boards and pieces, as well as the variety of games represented in the archaeological record is striking in comparison to other ancient cultures. As early as the 1910s, Georges Bénédite compiled the _Catalogue Général des Antiquités Égyptiennes du Musée du Caire_ for games, but his manuscript remains unpublished (Drioton 1940:182, note 2). In this book, we synthesize the material evidence in Egypt from Predynastic through Islamic times, in order to aid in the identification of board games in archaeological contexts at sites within Egypt as well as in the neighboring regions. This scope also provides evidence for a wider discussion on how games are transferred across cultural boundaries, as the long history of Egypt demonstrates the facility with which board games are able to cross borders both real and imagined.
Previously, scholars have made an invaluable contribution to the study of Egyptian board games by compiling the material, textual and artistic evidence for the various board games that existed in Egypt. In particular, Pusch (1979), Piccione (1990b), Decker and Herb (1994) and Rothöhler (1996) accomplished the monumental task of compiling catalogs of the known surviving game boards from Predynastic and Pharaonic Egypt, as well as texts about games and representations of game playing. Reproducing their work here would be unnecessarily repetitive as few games have been found since, but they prove to be invaluable resources for any researcher interested in the topic. Instead, this volume seeks to build from their work by expanding the scope spatially and chronologically. Archaeological evidence in the past thirty years increasingly suggests that Egyptian games may have spread to neighboring local populations where Egyptians were economically or politically active, and in some cases even beyond Egypt's sphere of influence. Furthermore, the presence of Greco-Roman and Islamic games on Pharaonic monuments demonstrates the contextualization of those structures within succeeding cultures.
Egyptian texts have helped to describe the religious significance of board games, while shorter inscriptions have given snapshots of gameplay. Much of the early scholarship on board games in Egypt focused on possible game rules. Evidence from the boards themselves, paraphernalia found with games, captions accompanying playing scenes and longer religious texts led to multiple theories on the modes of play (Ranke 1920; Murray 1951; Vandier 1964:486–512; Kendall 1978; Piccione 1990a, 1990b), though some are particularly speculative (e.g., Falkener 1892; Breyer 2010). Even when board games are depicted in art, it is important to note that it can be difficult to identify a painting or relief as a particular game, as the board surface is commonly not shown, according to Egyptian artistic convention. As a result, inferences must be made from textual evidence or based on artistic conventions. Apart from rules used for games in Roman Egypt and the period thereafter, the playing rules of ancient Egyptian games remain largely unknown. The changes in rules or the variations of play are also beyond what we can currently learn from the archaeological record. There is one exception for the _game of twenty_ , for which a Babylonian tablet from the Seleucid period has provided relevant details. These rules written in a region outside of Egypt and in a time period postdating the game's existence in the Nile Valley remain the main advance in game rule research (Finkel 2007).
As with most aspects of Egyptian life, board games were imbued with meaning connected with the journey into the afterlife. This is made explicit in art and texts from Pharaonic times, and is particularly striking for the games _senet_ and _mehen_. Board games, particularly _senet_ and to a lesser degree _mehen_ , appear in the religious literature of ancient Egypt, including the Pyramid Texts, Coffin Texts and the Book of the Dead. The religious nature of _senet_ is particularly prominent in the Great Game Text, which also provides some evidence for the nature of play (Piccione 1990b:191–241). New texts on this subject in the last decades are largely absent, and a reinterpretation of well-documented and sometimes contentious texts is outside the scope of this work. Often, interpretations of texts by Egyptologists interested in games may differ from canonical translations as they may see aspects of gaming in the texts that may be overlooked by other epigraphers. These texts offer crucial insight into the meaning of these games, but offering new interpretations of texts is a task that would fill a volume on its own. Likewise, depictions of games in tomb paintings and reliefs often depict the play of games in ritual contexts. Social historians have pointed out the similarity of ritual and play (Huizinga 1950:15–27; Sutton-Smith 1997:169). As a consequence, and because tomb assemblages make up the majority of the archaeological evidence in Egypt, much of what is known about games in Egypt is limited to their ritual use. The works of Kendall (1978, 2007) and Piccione (1990a, 1990b, 2007) demonstrate the connections of game playing and mortuary symbolism. Piccione's works in particular collected both material and textual evidence to comprehensively discuss the religious implications for the games of _senet_ and _mehen_ for the first time. Since their extensive explorations of the topic there has been little advancement in the discussion of this aspect of Egyptian board games, and for this reason we summarize earlier arguments made from textual evidence while drawing new conclusions and parallels by expanding the scope of material evidence considered.
The problems with understanding game rules and the detailed work on the ritual use of board games cannot be overcome by simply reexamining board games in the archaeological record of Egypt. New discoveries of game boards or texts about games from Egypt have been rare since the 1990s, and thus focusing on gaming within Egypt itself is limited to reexaminations of existing evidence. This stands in contrast with the number of games that have been discovered outside of Egypt, which has increased dramatically over the past thirty years. Since 1980, when Stuart Swiny first published his article identifying _senet_ and _mehen_ in Cyprus (Swiny 1980), analogous games have been found throughout the island as well as in the Levant, which have greatly increased our knowledge of these games outside of Egypt. The rate at which these games have been found has been so high that the number of _senet_ and _mehen_ games found in the Levant and Cyprus is now greater than that in Egypt itself. Similarly, Roman and Arab/Ottoman games have been discovered in Egypt, but are commonly excluded in literature on Egyptian games. Archaeological evidence from regions outside of Egypt provide significant insight on these board games, and understanding their morphology and chronology can help to prevent misidentification and provide interesting layers to discussions of site histories. In other words, the study of Egyptian board games benefits from a comparative approach that includes insight gained outside of Egypt and allows a context of these games across borders.
In recent years there is also a shift in Egyptological research as games have been identified carved into pavements at Egyptian monuments or traced on limestone ostraca, which point to non-elite Egyptians playing these games just as much as royalty and the nobility. Much like the Cypriot and Levantine boards, for instance, these artifacts are cruder in their construction than the manufactured Egyptian game boards found in royal tombs. The differences in layout and markings on boards made of different materials add a new layer of understanding the use and variation of board games.
While archaeological research in Cyprus and the Levant has thus far produced more game boards than Egypt, the Nile Valley has greater variety of types as well as a longer history of board games, allowing for an examination of how they change through time. Similar artifacts found in neighboring regions help attest to foreign interactions with Egypt, including Nubia, the Levant and Cyprus. This process includes games introduced into Egypt as well as from Egypt to nearby lands. The variety of games preserved in Egypt's archaeological record presents an ideal setting in which to examine games and exchanges in the ancient world.
Since patterns carved on rock faces and ostraca have, so far, received little attention in the literature, the following chapters will focus on boards made of a variety of materials and the process of identifying board games based on those types. Graffiti games are often not datable to the same time the monument was built, which makes it important to understand the possibilities and to be able to identify whether such games are Pharaonic, Greco-Roman or Islamic in origin. Collated material of this kind is presented in a roughly chronological manner, focusing on the major games of Pharaonic Egypt, followed by Greco-Roman and later games to highlight chronological changes, facilitate game identification and inform a site's history.
Playing pieces, accompanying a board or found in isolation, may also be diagnostic and aid in the identification of games where the board may not have survived. Casting devices that accompany a board add detail both in the possible age and the variation of play. Advances in understanding of these game paraphernalia outside of Egypt assist in contextualizing Egyptian game practices. Introductions to these elements of board games present a first understanding of the complexity of the material and the possibilities of comparing Egypt with its neighbors. Furthermore, types of casting devices are often not specific to certain games, and are best introduced before the games themselves in order to facilitate discussion of the individual games without digression. They also provide a microcosm of the variation seen in the types of games found in this book, with a long chronological range as well as foreign types mirroring the dispersal of games throughout the ancient world. In some cases, casting devices comprised games on their own, that remain outside the scope of our survey. They may have inspired some of the earliest forms of board games as has been suggested in the New World, where evidence may indicate that early board games were essentially counting mechanisms for a game involving casting sticks (Voorhies 2013). While it is not possible with current evidence to suggest that Egyptian board games evolved from games where the casting devices were the main element, these randomization tools appear early in the archaeological record and are common paraphernalia accompanying games throughout Egyptian history.
### A range of casting devices
Ancient Egyptians used different kinds of casting devices to determine the number of squares on a gaming board on which they would move playing pieces including sticks, astragali, teetotums and cubic dice (see fig. 1.1). The latter is the only accessory that has not been found in direct association with any game board from the Pharaonic period while the three others were discovered with _senet_ and the _game of twenty_. Throwing sticks may have been used for _mehen_ but this is not certain and no _hounds and jackals_ game was preserved with casting equipment.
Casting or throwing sticks functioned as the principal randomizing agent from the Predynastic period through the New Kingdom and are still used today in Egypt and Sudan (Kendall 1982:271). Sticks appear in archaeological assemblages in sets of two, three or more. Sometimes two sets of distinct throwing sticks were found together, which may indicate that each player used their own set. The players would give values to the different sides as one side is marked, and/or tinted, the other one unmarked. The sticks would have been thrown together, mikado like, and the player would count the number obtained depending on the sides facing up.
At Predynastic Ballas, Petrie and Quibell found ivory rods decorated on one side with incised lines or with motifs imitating reed joints (Petrie & Quibell 1896:14, pl. VII). From the First Dynasty onwards, throwing sticks took the form of semi-cylindrical strips of wood, bone or ivory corresponding to split reeds or palm branches; in cross section one side was rounded, the other flat. The sides could also be distinguished by the color or the decoration. The size of the sticks can vary greatly: in First Dynasty tombs at Saqqara, ivory sticks more than 28 cm long with incised decoration were discovered with gaming pieces in Tomb 3504 (Emery 1954:56–7, fig. 61, 59, fig. 67), while three strips of ivory measuring between 4.7 cm and 9.4 cm long were found with tall cylindrical and semi-circular gaming pieces in Tomb 3471 (Emery 1949:62). A painting in the Third Dynasty tomb of Hesy-Re at Saqqara depicts two pairs of sticks and two sets of seven playing pieces accompanying the _senet_ game (see fig. 2.5). The sticks are decorated with three cross-bands consisting of two lines, and each pair is distinguished by the color of the lines, which are either red or black. Throughout the period of their use, incised lines were the main decoration on sticks.
During the Second Intermediate Period and New Kingdom, the ends of sticks were often carved in the form of human fingertips or the head of a canine, and this imagery certainly contributed to the symbolism of _senet_ (Hayes 1959:26, 200) (fig. 1.1, top left). In the Great Game Text, the player whose "seven pieces are in front of his] fingers like the jackals that tow the solar bark" may refer to such throwing sticks (Kendall 1982:272) or a square called the "House of Towing" (Piccione 1990b:149). The animal with a pointed snout and long ears lying back along the sides of the stick is interpreted as a jackal, a fox or a fennec (Tait 1982:32–6). There is a similarity with animal heads found on opposite ends of magical wands from the Middle Kingdom (Tait 1982:33). We may explain the appearance of this motif on throwing sticks as a shift from the magical wand, a device that disappeared in the New Kingdom. A unique set of two pairs of ivory sticks representing foreign captives was found in the tomb of Tutankhamun and is now in the Egyptian Museum, Cairo, henceforth Egyptian Museum (JE 62059) (Tait 1982:36, pl. XIII). Small figurines of captives were also used as playing pieces (see [Chapter 3).
**Figure 1.1** Egyptian casting devices dating to the New Kingdom and a set of cubic dice from the third century CE, Sudan. Top left: throwing sticks, max. 22.8 × 1.4 cm. The Metropolitan Museum of Art, Rogers Fund, 1919, 19.2.19–27a,b. Top right: four sides of an astragal from Thebes, 2.5 × 1.6 cm. The Metropolitan Museum of Art, Rogers Fund, 1916, 16.10.505c. Image © The Metropolitan Museum of Art. Bottom left: teetotum from Qau, 2.4 × 2 × 1.4 cm. © Petrie Museum of Egyptian Archaeology, University College London, UC26284. Bottom right: cubic dice from Sedeinga, Sudan, 2.5 cm. Photograph courtesy of Vincent Francigny.
The relationship between fingers and throwing sticks is traced back to the Old Kingdom at Giza, both in archaeological and epigraphic material (Kendall 1982:271–2). A wooden finger measuring 14.7 cm long was excavated by the Harvard University-Boston Museum of Fine Arts Expedition in the Western Cemetery at Giza from Pit G 2385 A dated to the Sixth Dynasty (Museum of Fine Arts, Boston, 13.3441). It could be an early example of the throwing stick with fingertip. It has been suggested that _ḏb w_, fingers, served to indicate throwing sticks, also used as a counting device (Kendall 1982:271). The word _ḏb _ is often described in _senet_ -related inscriptions and it is translated as "finger" since we are not sure about its secondary meaning (Piccione 1990b:60, note 114). In the Sixth Dynasty tomb of Idu at Giza, a player warns his opponent about the decisive role of _ḏb _ : _rḏỉ.(i) sšm ḏb _. _(i) r pr hb_ translated as "I cause my finger to be led to the house of the plough... [or] the house of the ibis" (Simpson 1976:25) or "I will cause my finger to lead the way... to the house of penetration or humiliation" (Piccione 1990b:60). As the different translations show, the name of the field—referred to as a house—is uncertain but it could be the unlucky square twenty-seven where the opponent is supposedly drowned according to the Great Game Text (Kendall 1982:272).
Throwing sticks were excavated in Sudan in the 1910s by the Harvard University-Boston Museum of Fine Arts Expedition at the capital city of the Kingdom of Kerma. Five ivory sticks decorated with black-filled incised chevrons were discovered with nine cylindrical and ten conical playing pieces in Tomb K6002:1 (Museum of Fine Arts, Boston, 15-3-281). Another lot decorated with black-filled incised lines comes from Tomb K2100 (Museum of Fine Arts, Boston, 20.1522). Both groups date to the Kerma period ( _c._ 2500–1500 BCE).
The French excavations at the coastal site of Ugarit in Syria have brought to light three ivory throwing sticks with incised decoration of unknown context and an undecorated one from a Late Bronze Age III level, dating to the thirteenth century BCE (Gachet-Bizollon 2007:212, 307, pl. 50). While Egyptian-type playing pieces are found at several sites in the Levant, including at Ugarit (Gachet-Bizollon 2007:212), these sticks are the only ones known to date for that region, confirming the significant influence of Egyptian culture at Ugarit, an important trading center during the second millennium BCE. The presence of playing pieces and throwing sticks at Ugarit attests to the practice of board games. Ivory plaques have been retrieved at the site but so far no gaming board has been identified definitively.
Another method to determine the number of spaces moved by the pieces was the use of knucklebones. "Knucklebone" is an inaccurate term for the astragalus, a small bone found within the tarsal joint of hooved animals more commonly referred to as the talus in anatomical terminology. Mainly sheep and goat bones were used. This bone is useful as a randomizing device because it has four sides on which it could land when cast, with none of these sides alike. Two long sides are noticeably broader: one is concave, while the other is convex. One narrower long side is indented and the other is flat. The astragal can land on one of the four long sides with the opposite side facing up, so each one was assigned a name and a different numerical value. Figure 1.1 shows the four sides from top left to bottom right, followed here by the value applied in Classical times: dorsal (three), plantar (four), lateral (six) and medial (one) (Amandry 1984: 348–9). Sets of astragali of different size may have functioned for distinctive throws (Finkel 2007:21, note 12). Imitations of astragalus bones were also made from wood, clay, ivory, stone or metal. In Egypt, artificially produced astragali seem to occur more often than natural bones (Gilmour 1997:170).
Astragali are frequently found in the Near East as early as the Chalcolithic period in Anatolia (5500–3000 BCE) (Muscarella 1974:80–1, note 21). Astragali were used also for games of skill as well as divination. Although an astragal was found at Abydos, possibly in the tomb of Den, king of the First Dynasty (Hayes 1953:46; Kendall 1982:270) (Metropolitan Museum of Art, 01.4.92), their use by the Egyptians is better attested from the Seventeenth Dynasty on and is likely related to the introduction of the _game of twenty_ (Finkel 2008). Astragali have been photographed with the Theban game of _hounds and jackals_ (Pritchard 1954:fig. 213), but in this case they are not likely to belong to the game board as they are not listed in the excavation report along with the itemized accoutrements of the game. Egyptians adopted the astragali to play _senet_ and they are the only casting devices represented in New Kingdom playing scenes (Pusch 1979:pls. 18, 28, 30) (see cover image).
Astragali are normally found in pairs with game sets. A pair of ivory astragali was found at el-Asasif (Lansing 1917:26), pairs of ivory and resin—of notably different size—in the tomb of Tutankhamun (Tait 1982:38–41), and a pair of wooden astragali at Saqqara (Quibell 1909:114). While most of the archaeological evidence comes from burials, a few are reported from domestic contexts at Deir el-Medina (Dunn-Vaturi 2012a) and el-Amarna (Frankfort & Pendlebury 1933:25, pl. 29.2).
During the Greco-Roman period, cubic dice become more common and gradually replaced astragali for use with board games. Despite this phenomenon, playing pieces dating to this period show a continued connection between astragali and board games. Two baboon-shaped playing pieces carefully integrate the contour of an astragalus to the body of the animal as if it had been carved from a natural knucklebone (Arnold 1995:60) (Metropolitan Museum of Art, 66.99.75; Walters Art Museum, 71.512). These pieces are stylistically attributed to an Alexandrian workshop active around the third century BCE.
A group of stone astragali carved with enigmatic scenes was not meant for an ordinary game but imbued with magical powers (Dandoy 2006:133; Piccione 2007:58). Three examples, said to be from Egypt and attributed to the Ptolemaic or Roman period, have been identified in museum collections (Musée du Louvre, E 11171; Metropolitan Museum of Art, O.C. 428; Petrie Museum, UC44997). The worn reliefs on the Louvre example are difficult to interpret (two standing figures, scorpion?). The steatite astragal now in London shows an erotic scene on one side (Petrie 1927:57, no. 227, pl. 49) while the gabbro astragal in New York shows a nude crouching female on the plantar side—possibly Omphale—and standing figures on the narrow sides (for the iconography of Omphale and magical gems see Dasen 2008).
During the New Kingdom, a four-sided teetotum was also used as dice equipment. A teetotum is a truncated four-sided pyramidal die, its faces numbered with one to four dots and pierced with a plug or rod allowing it to be spun. This new shape came to Egypt from the Levant where examples dating to the early second millennium BCE are attested at Beth Shean (Albright 1938:48), Tell el-Ajjul (Petrie 1934:pl. XXIV, XXXVI, 25) and Tell Beit Mirsim (Albright 1938:48, pl. 21,b; Pritchard 1954:fig. 214). The ivory teetotum from Stratum D at Tell Beit Mirsim measures 1.7 cm in height, 1.75 cm in width at the base, 1.3 cm in width at the top; it still has the ivory plug running through it. Albright (1938:48, note 48) estimated the examples from Tell el-Ajjul were cruder than his find at Tell Beit Mirsim but stated that they "all follow the same principle, with spots only on the four sides."
Kendall (1982:267) described the teetotum as possibly "the earliest prototype of the cubic die known in Egypt." Two ivory examples from Egypt are known to the authors so far. One was found with a double-sided game box in a tomb dating to the mid-Eighteenth Dynasty at Zawiyet el-Aryan (Dunham 1978:72–3, pl. LIX) (Museum of Fine Arts, Boston, 11.3096). The other one, found at Qau, is assigned to the New Kingdom (Petrie Museum, UC26284) (fig. 1.1, bottom left).
Cube-shaped dice represent "an invaluable mental and artistic performance. It functions as a randomizing agent only if its body... is manufactured with great geometrical precisions." (Schädler 2007:13). Schädler considers the mid-third millennium BCE dice from Tepe Gawra in Mesopotamia to be the earliest evidence of cubic dice. Others trace its origin to the Indus Valley, where many terracotta, stone and faience examples have been discovered at the sites of Mohenjo-daro, Harappa, Alamgirpur and Lothal (Dales 1968:18–19). The use of both cubed and oblong dice is attested at these Harappan sites around 2300 BCE (Brown 1964:34). The opposite sides are numbered consecutively, so that one is opposite two, three opposite four, and five opposite six. Cubic dice with the number dots arranged like the Indian dice are attested in Mesopotamia since the mid-third millennium BCE, but their use in the Near East remained rare (Dales 1968:18) and they have never been found clearly associated with any board game, in contrast to the more popular tetrahedral and oblong dice.
Cubic dice were not common in Egypt until the Ptolemaic period (Petrie 1927:57, pl. XLIX), although isolated finds dating to the New Kingdom are reported from el-Amarna and Lisht (Hayes 1959:405) as well as Deir el-Bahari (Carnarvon & Carter 1912:58). During the Ptolemaic period, the numbering of the sides is standardized, with opposite sides always adding up to seven, as in modern dice: 1+6, 2+5 and 3+4. During Roman times, the proliferation of dice games was abetted by the movements of soldiers throughout the empire. Dice were thrown with the hand or small boxes were used as throwing cups to prevent manipulations.
Most of the dice are numbered from one to six but some sides may have letters or words. A cubical limestone die, dating from the Ptolemaic period, is inscribed with hieroglyphs. Names of gods appear on each side: Osiris, Horus, Isis, Nephthys, Hathor and Har-Behdety (Petrie 1927:57, pl. XLIX.233; Tait 1998) (Petrie Museum, UC38176). A large faience die dating to the Byzantine period, in the fifth century CE, and now in the Swiss Museum of Games (2762), combines numbers one to six and Greek letters: alpha (one), delta (four), beta (two), eta (eight), sigma (six), iota (ten). Questions are still raised whether this type of die was used to play or was involved in magical/divinatory rituals (Schädler 2007:13–14).
A similar problem is found with twenty-sided dice as it is unclear if and how these dice were used for games. They are exclusively found in Greco-Roman contexts with most examples in the literature attested in Egypt. They normally feature the Greek letters alpha through upsilon and Pedrizet (1931) has shown that examples found in Alexandria point to the end of the Hellenistic period in Egypt. Their occurrence is part of the meager evidence that exists of direct Greek influence on possible gaming materials. At the same time they are a helpful tool to date archaeological sites (e.g., Francigny et al. 2014).
### Board games across borders: Identifying Egyptian games
While divination and religious ritual have been a focus of discussion with regard to ancient board games, their role in facilitating interactions across social boundaries is more useful for studying the social implications of gaming (Crist 2015; Crist et al. in press). In order to identify games of Egyptian origin, as well as foreign games within Egypt, it is necessary to first examine what is known about them and which games existed where and at what times. Furthermore, descriptions of the different forms these games assume will aid archaeologists in identifying them in surveys and excavations. Recent work (de Voogt 2010, 2012; Jacquet-Gordon 2003) has discovered graffiti games at previously excavated sites, indicating that games may exist at other sites awaiting discovery and identification. In the following chapters, we discuss the major games that existed from the Predynastic through the Islamic periods, in order to aid scholars in recognizing what games belong to antiquity and which are more recent.
The early history of Egyptian board games is discussed in Chapter 2, where _mehen_ and other, lesser-known games are first introduced. _Mehen_ takes prominence as the most popular of the Predynastic board games in Egypt, and indeed it was the first that is known to have crossed into other regions. It also sets the trend of board games in religious and mortuary contexts, which can be seen throughout Pharaonic Egypt. After discussing the forms and contexts of _mehen_ in Egypt, its appearance in the Levant and Cyprus is discussed, along with an examination of the cultural circumstances under which the games arrived in these lands. Finally, the game known as _men_ , as well as two other early but poorly understood board games are briefly discussed, as they are not as well known as most of the other games discussed here.
Chapter 3 focuses on the most famous of Egyptian board games, _senet_. Its development into the most prominent and religiously important board game in Egypt shows that it may have changed through time, which is to be expected for any game that was played over roughly three thousand years. The transmission of _senet_ to the Levant, Cyprus and Nubia is also not surprising in consideration of its popularity in the Nile Valley, but it is clear that it took on a different kind of social context than it did in Egypt. At the end of the chapter, we also discuss the enigmatic _game of thirty-three_ , found on the opposite side of _senet_ boards during the Late Period. The playing pieces for _senet_ and the _game of twenty_ are also discussed in this chapter, since both games used pieces of the same type.
The _game of twenty_ , which is the focus of Chapter 4, was the first foreign game to be widely played in Pharaonic times. This game is originally attested at the Royal Cemetery of Ur during the third millennium BCE and only arrived in Egypt much later and in a different form. It had been popular in Mesopotamia and the Levant, and was introduced to Egypt perhaps as early as the Middle Kingdom, reaching the pinnacle of its popularity during the New Kingdom. During this period it was a popular game seemingly related to interregional elite gift exchange and its boards were likely kept as heirlooms throughout the Eastern Mediterranean. It was never fully Egyptianized though, as can be seen in its decoration and its abandonment after the interaction networks of the Late Bronze Age collapsed ( _c._ 1050 BCE). A more complete discussion of the rules of play is allowed by the aforementioned tablet found in Babylon.
_Hounds and jackals_ is covered in Chapter 5. The different types of boards are discussed, as this game was manufactured in a number of varying styles. It is another game that originated in Egypt and spread abroad, but these mechanisms are less well understood because it has been found as far away as Anatolia without strong direct connections. The unique playing pieces in the form of pegs may point to the existence of the game in places where the boards have not survived. New interpretations of pictorial evidence suggests that the game may have been depicted in playing-scene contexts during the Middle Kingdom.
Chapters 6 and focus on later games that were introduced to Egypt from foreign lands. Understanding the presence of these games on Pharaonic monuments will help to correctly identify them and to understand site histories. Chapter 6 focuses on Greco-Roman games, particularly _five lines_ , _latrunculi_ and _duodecim scripta_ , while Chapter 7 focuses on Arab and Ottoman games, including _seeja, mancala and tâb_ , which are still played today. These later games have rarely been discussed in conjunction with Pharaonic games, and by emphasizing them in these chapters we present mounting evidence for their existence in Egypt alongside an understanding of how they were played and by whom.
The final chapter summarizes some of the main themes that are apparent from the study of games in Egypt. Focusing on the roles games played in ancient Egypt, the process of how games transcended borders in Egypt and the ancient Near East is highlighted, and the often over-emphasis of religion with relation to games is discussed. Finally, there is an examination of the broad interest board games have received in anthropological and Classical archaeology and how this volume may assist these fields and Egyptology in particular.
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## _Mehen_ and _Men:_ The First Signs of Egyptian Board Games
### _Mehen_ : The game of the coiled serpent
The Fourth Dynasty tomb of Prince Rahotep at Meidum provides the earliest known names associated with specific board games anywhere in the world. In a list of offerings for the king to use in the afterlife are three different games: _mn_ , " _men,_ " _zn_. _t_ , " _senet_ " and _mḥn_ , " _mehen_." Translated as "to coil" or "the Coiled One" (Ranke 1920:372–5), _mehen_ shares its name with the god Mehen, an immense coiled serpent associated with the netherworld, whose primary function was to envelope the sun god Ra in his many coils, thus protecting him from all evil (Piccione 1990a:43; Rothöhler 1999:12–19). Accordingly, _mehen_ boards used in the game reflect the typical coiled serpent form of this god. The playing field is laid out in the form of a spiral, sometimes with the tail and head of a serpent depicted on the board itself (Ranke 1920:4).
Of the three games appearing in the offering lists of Rahotep, _mehen_ seems to be the oldest, judging from the archaeological record. _Mehen_ boards vary in their morphology in ways the other more standardized Egyptian games do not. First of all, the playing spaces do not always take the form of bounded squares as they often do in _senet_ , rather the playing spaces can be delineated by a number of alternating bosses and recesses, with each boss or each recess apparently being one playing space (Vandier 1964:486–7). Another method of delineating the playing spaces was to deeply incise the divisions between the spaces (Vandier 1964:487). In addition to this, the number of playing spaces represented on the boards and in the pictorial evidence is not standardized. Unlike _senet_ , in which every known example of the game has three rows of ten spaces, _mehen_ has a wide variation of possible numbers of spaces. Boards have been found with as few as forty-nine spaces, and some had as many as four hundred (Swiny 1986:57). There also was not a predetermined number of circles in the spiral, with as few as two (e.g., Petrie & Quibell 1896:Pl. 43) and as many as seven (e.g., Quibell 1913:Pl. XI, see fig. 2.5) known from archaeological contexts. The serpent could also be coiled in either a clockwise or counterclockwise direction (Piccione 1990a:47).
**Figure 2.1** Map of sites mentioned in Chapter 2.
### _Mehen_ boards
Of the few boards that have good archaeological provenance, the oldest comes from the late Naqada Tomb 19 at Ballas, now in the Ashmolean Museum (Petrie & Quibell 1896:42), and dates to the end of the fourth millennium BCE (Rothöhler 1999:11). Found covering a pot, it was likely a votive representation of a _mehen_ board rather than one used for play, since it was only 10.5 cm in diameter (Kendall 2007:37), but appears in the form of other full-sized game boards. There is a pierced rectangular protrusion extending from the circular board, a feature paralleled in many other _mehen_ games. The body of the serpent is divided into squares, with the head and tail of the snake differentiated from the other segments.
Three other boards have been compared to this artifact and, because of their similar execution, may be dated around the same time as the Ballas _mehen_. While their provenance is incomplete, one is rumored to have been found at Quft (Petrie & Quibell 1896:42; Vandier 1964:488), now in the Egyptian Museum in Cairo (JE 27354), and another said to have been found at Abydos now is in the Bode Museum (13868) in Berlin (Scharff 1926:145). The third game is in the Petrie Museum in London, but its provenance is unknown (UC20453, see fig. 2.2) (Petrie 1914:25).
They all contain a similar execution of the snake motif, with segmented bodies making up the individual playing spaces and differentiated heads and tails. They also exhibit a protrusion on one end, though these are smaller and more rounded in comparison to that on the Ballas _mehen_. It has been suggested the type of protrusion shared by these three games is a representation of a turtle's head (Fischer 1968:17). This connection may be corroborated with a turtle in the Metropolitan Museum of Art (61.33) that has a series of concentric circles carved on it, and interpreted as a votive _mehen_ board. The Petrie and Bode boards also are pierced, which suggests they may also have had some sort of amuletic function (Kendall 2007:36) which Petrie suggested was protection from serpents (Petrie 1914:25), an interpretation which he even admitted is questionable.
**Figure 2.2** _Mehen_ board demonstrating a Predynastic rendering of the game. 28.8 × 3.4 cm. © Petrie Museum of Egyptian Archaeology, University College London, UC20453.
A _mehen_ board in the Metropolitan Museum of Art (58.125.1, see fig. 2.3) is incised with the name of the First Dynasty pharaoh Hor Aha on its rectangular protrusion. It has been argued that this game may be a forgery (Kendall 2007:38), based on presumed increasing realism in the execution of _mehen_ boards continuing from the Second Dynasty. Indeed, while the manner in which the serpent is incised is unusual when compared to the rest of the corpus of games, the number of _mehen_ boards we have is small, with only fifteen examples, and so all of the variation in style that may have existed in antiquity may not be reflected in the games that have been preserved. In any case, the execution of the snake with its head and tail, along with the presence of a rectangular protrusion, is reminiscent of the boards already discussed, though it is not pierced. Results of an examination of the board by Ann Heywood at the Metropolitan Museum of Art suggests the artifact is ancient rather than modern as Heywood found calcium carbonate accretions and what appears to be soil deposits in the hieroglyphic and _mehen_ carvings. The hieroglyph—the authenticity of which has been doubted by I.E.S. Edwards and Gunter Dreyer (Diana Craig Patch 2014, personal communication)—appears to have been made around the same time as the _mehen_ board, but probably by two different individuals based on the manner in which they are carved (Ann Heywood 2014, personal communication).
**Figure 2.3** _Mehen_ board bearing the name of Hor Aha, shown before restoration, 32.1 × 27 × 2.2 cm. The Metropolitan Museum of Art, Dick Fund, 1958, 58.125.1. Image © The Metropolitan Museum of Art.
Another _mehen_ game without provenance, now in a private collection, may have been a jar lid much like the Ballas game as it is only 4.5 cm in diameter and unlikely to have been used for play considering it contains 336 very small spaces (Kendall 2007:37). It depicts a coiled serpent, with a tail and head, much like the others already discussed, but with an important distinction: four holes were drilled on different parts of the serpent and may have been filled with a colored paste or some kind of inlay, distinguishing those spaces from the others (Kendall 2007:37). The protrusion on this board is broken, so it is uncertain whether it was rectangular or of the "turtle head" type.
The Predynastic and First Dynasty boards stand in marked contrast to those dating to the Second Dynasty, all of which depict a more abstract rendering of the game (see fig. 2.4). Five are known, four from the tomb of Pharaoh Peribsen at Abydos (Amélineau 1899:494–5, plate 47) and another with unknown provenance in the British Museum (1961,0408.1) (Shore 1963:Plate 33A). All of these artifacts are made of faience. In these examples the representation of the serpent becomes more abstract; the board in the Musée Royal de Mariemont (B.102.0) has a coil in its center resembling a snake's body, and the board in the British Museum has a stylized serpent's head in the center while in the examples from Peribsen's tomb (Musées Royaux d'Art et d'Histoire, Brussels, E.4180; Musée du Louvre, E 29891) the serpent's head and tail are not well executed (Kendall 2007:35–6). During this period, spaces were no longer delineated by incised lines. Instead, the playing spaces consist of alternating rectangular bosses and recesses producing a checkerboard pattern on the playing field.
Further diverging from the standards of earlier _mehen_ boards, the spaces on the Second Dynasty boards were no longer always arranged in a spiral pattern. The spaces on one of the Peribsen _mehens_ were positioned in concentric circles, while another had four spiraling tracks, and a third had spaces in concentric circles that connected with each other, thus allowing for a spiral-like movement along the board (Kendall 2007:35). Swiny (1986:55) suggests the arrangements of the Peribsen games make their identification as _mehen_ suspect, but the Mariemont _mehen_ also has a pattern of concentric circles rather than a spiral, and there is a definite, though abstract, depiction of a serpent's head at the center of this board. Rather than arguing against its use as a _mehen_ board, Kendall (2007:35) suggests certain spaces on this artifact may have originally been marked to indicate when a player would advance to the next circle, thus effectively allowing it to function as a spiral. Whatever the case, these boards are markedly different from those known from any other _mehen_ boards currently known, and suggest the Second Dynasty was a time in which some experimentation was underway in the design of the boards.
**Figure 2.4** _Mehen_ board of unknown provenance demonstrating a Second Dynasty type accompanied by spherical playing pieces, 37 × 7 cm. The British Museum, 1961,0408.1. © The Trustees of the British Museum.
The next datable _mehen_ is not a board that can be played, but a pictorial representation (see fig. 2.5). It is the earliest known depiction of a _mehen_ board and was found in the Third Dynasty mastaba of Hesy-Re at Saqqara (Quibell 1913:18–21; Emery 1961:248–51; Kendall 2007:33). The paintings on the walls of this mastaba depict different types of equipment considered to be necessary for the deceased in the afterlife, and include not only _mehen_ but also the first known depictions of the games of _senet_ and _men_ , accompanied by sets of gaming pieces (see below). This representation is the only painting or relief of a _mehen_ board in which the individual spaces and the head and tail of the snake are shown. Some of the divisions of the spaces are not well preserved, although it does seem that the seven spirals of the snake were divided evenly, and, if the number of spaces is extrapolated considering the size of those preserved, the total number of spaces is roughly four hundred (Swiny 1986:57). The trapezoidal appendage, always shown when _mehen_ is represented in art (in contrast to the _mehen_ boards themselves that do not always have a protrusion), is depicted for the first time. If this _mehen_ board is accurate, it could represent a new type, since it is markedly different from the Second Dynasty boards and seems to return to many of the conventions seen before in the Predynastic and First Dynasty _mehens_.
The Louvre (E 25430), the Oriental Institute Museum (16950) in Chicago and the Fitzwilliam Museum (e.g. A. 4464.1943) in Cambridge each have a _mehen_ board without provenance (Vandier 1964:488; Swiny 1980:69; Piccione 1990a:46–7). These boards are all morphologically quite similar to _mehen_ boards predating the Second Dynasty, but exclude any kind of protrusion. The segments of the snake's body are all carefully delineated, and the head of the snakes are all carefully executed on these three boards. None has a protrusion, though Shore (1963:91) argues there is evidence one did exist on the Fitzwilliam game. In the center of the spiral, the head of the serpent takes up more space than on the games predating the Second Dynasty, leaving considerably more negative space in the center.
The example from the Louvre is supported by a foot, a feature that has led to questions regarding its authenticity (Swiny 1986:55, note 482); however, the Quft and British Museum _mehens_ also have feet to support them, so its presence on the Louvre game does not preclude it from being an authentic artifact. Kendall's (2007:38–9) suggestion that these boards date to the Third Dynasty or later is probably correct, since they deviate from the earlier types only slightly.
**Figure 2.5** Drawing of a painting from the tomb of Hesy-Re, dating to the Third Dynasty and showing the games _mehen_ (left), _senet_ (top right) and _men_ (bottom right) along with their playing pieces and throwing sticks. Drawing reproduced from Quibell (1913:Pl. XI).
Another way the Louvre and Chicago games differ is that, instead of a clearly defined tail, the head of some kind of waterfowl (probably a duck or goose) is carved on the outer edge of the board. There is probably a connection here with the "Great Cackler," the primordial goose that, in one early creation myth, laid the egg from which Ra was hatched (Rundle Clark 1991:55–6, 213; Kendall 2007:40), which, in turn, is related to the solar connections of the god Mehen (see below). The Fitzwilliam _mehen_ differs in one other way in that five of the spaces are crosshatched, perhaps marking them as special spaces important for the play of the game (Swiny 1980:69; Kendall 2007:39). This is similar to the drilled spaces in the jar lid discussed above in the New York private collection _mehen_.
The materials out of which _mehen_ boards were made—stone, ivory or faience—are all materials that are well preserved in the archaeological record. It is possible that _mehen_ boards may also have once been made of wood, but did not survive. There have been multiple tombs that have produced sets of _mehen_ pieces of the type depicted in the tomb of Hesy-Re without an accompanying board, which argues the board itself has decomposed. Indeed, no _mehen_ pieces have been found associated with a board (Kendall 2007:34–5), so it is very likely there were boards made out of perishable materials that have not been preserved; however, decomposed boards should not be inferred in every case, and such conclusions should only be made when complete or near-complete sets are found.
### _Mehen_ pieces
_Mehen_ pieces are depicted next to the _mehen_ board in the painting in the tomb of Hesy-Re, the only complete depiction of a set of _mehen_ pieces anywhere (Quibell 1913:Pl. XI, see fig. 2.5). The set pictured there contains thirty-six marbles, grouped by color into six groups of six, as well as six white couchant lions. The six sets of pieces may suggest the game could be played with up to six players, which could be supported by the pictorial evidence from later in the Old Kingdom (see below). It is important to note that no dice are pictured with the _mehen_ pieces on the Hesy-Re painting, nor are they depicted on playing scenes. Throwing sticks have been found with _mehen_ pieces, but only in tombs where _senet_ and/or _men_ pieces have also been found, so it is unclear whether they were associated with the _mehen_ pieces as well. When found alone, _mehen_ pieces have never been found along with throwing sticks. While it has been suggested that _mehen_ was a race game (Kendall 2007:35), the absence of dice or other randomization devices may mean _mehen_ was less likely to have been a race game, but may have been some kind of strategy game instead.
_Mehen_ pieces have typically been found in graves in Egypt, mostly from the Predynastic period through the Second Dynasty (see fig. 2.6). A full set has never been preserved, but the closest was found in Tomb 3504 at Saqqara, in the same cache that contained a complete set of _senet_ pieces (Emery 1954:58). The couchant lions found there were made of ivory, and the spherical pieces, thirty-nine in number, were made of limestone. Other sets of _mehen_ pieces are incomplete. At Abu Rawash, Tombs I and VIII both produced _mehen_ pieces. Tomb I produced three lions and three lionesses made of ivory, and Tomb VIII produced a nearly complete set, with a full complement of lions and lionesses and a series of spherical pieces in two colors: white and red (Montet 1946:189). Another incomplete set comes from Tomb 100 at Naqada, where four lions, eight spherical pieces and a couchant hare were found (Petrie & Quibell 1896:35, pl. 7). Tomb 3507 at Saqqara produced a fragment of an ivory lion piece, and also a number of spherical pieces made of amethyst and one of rock crystal (Emery 1958:84). It is possible, though not conclusive, these objects form part of a set of _mehen_ pieces. A First Dynasty private tomb near that of Zer-Ta at Abydos produced four couchant lion pieces (Petrie 1901:23). Petrie notes that the bottom surfaces of the lions are worn smooth, probably through extensive use, and also that the details of the lions' physique has been partially worn through extensive handling, probably in the course of gameplay. In some cases, hounds appear to have been substituted for lions (Scharff 1926:56, 63; Kendall 2007:34). Other marbles and couchant lions called _mehen_ pieces have been found in tombs throughout Egypt (de Morgan 1896–7:192–4; Amélineau 1899:Pl. 31; Capart 1905:140, 178–9; Petrie 1920:11, 1925:6–7; Schweitzer 1948:12; Saad 1969:45; Kaiser et al. 1976:86; Spencer 1980:69–70), but it is nearly impossible to ascribe them this function without boards as the lions could be votive objects and the marbles could have served multiple purposes. One set of pieces from Naqada, in which only four marbles remain, may have been part of a _mehen_ set, because they were found alongside pieces of types consistent with _senet_ and _men_ pieces, as shown on the walls of Hesy-Re's tomb. Interpreted by Petrie as a game of _skittles_ (Petrie & Quibell 1896:Pl. 7; Vandier 1952:405), it seems more likely the pieces did not make up one game together, but rather were incomplete sets for the games _senet_ , _mehen_ and _men_.
**Figure 2.6** Ivory _mehen_ piece from Abydos in the form of a couchant lion, 8.7 × 4.3 × 3.2 cm. The Metropolitan Museum of Art, Gift of Egypt Exploration Fund, 1903, 03.4.13. Image © The Metropolitan Museum of Art.
### Rules of _mehen_
Much like _senet_ , the rules for playing _mehen_ are unknown, although some aspects of the course of play can be determined through textual evidence and the artifacts themselves. As mentioned above, _mehen_ appears to have been played with a number of playing pieces in the shape of couchant lions or lionesses, or in the shape of small spheres (Quibell 1913:Pl. XI, see fig. 2.5). Many have classified this game as a hunting-type game (Vandier 1964:492), and Montet (1955:195–7) constructed an elaborate set of rules based on representations of hunting in Predynastic Egypt, which, Vandier (1964:492) admitted, was going too far. Indeed, the only way to reconstruct a convincing set of rules for the game is to find textual evidence suggesting a sequence of play, as has been somewhat possible for _senet_ , but not for _mehen_ (see Chapter 3).
Some evidence for the method of playing _mehen_ has been found on one of the known _mehen_ boards. The board in the Oriental Institute appears to have once been painted with brown pigment and is worn away on the long spiraling grooves outlining the body of the snake, while the paint in the grooves dividing the individual spaces is still thickly applied. Piccione (1990a:47) believes this suggests the spherical playing pieces associated with _mehen_ were moved along this spiraling channel.
### Pictorial evidence
Aside from the depiction of _mehen_ on the wall of the tomb of Hesy-Re, other scenes from Old Kingdom tombs depict the playing of _mehen_. Five _mehen_ playing scenes come from the tombs of the nobility of the Fifth and Sixth Dynasties (Decker & Herb 1994:634–6). These scenes typically show two men sitting opposite each other across the _mehen_ board, with the board shown as it is seen from above. One feature of the _mehen_ boards as depicted in these scenes that has caused some degree of discussion in the literature is that of the so-called trapezoidal appendage, ubiquitous in the pictorial evidence, but not clearly attributable to a feature of the actual game boards known archaeologically (Montet 1955; Vandier 1964; Swiny 1986). The debate stems from a disagreement as to whether this appendage represents some form of foot on which the game board rested as Swiny (1986:56) seems to be suggesting or if it was an extension of the game board itself. A foot would make it into a sort of table. The extension is typically interpreted as a sort of "garage" for the playing pieces, which played a part in the course of the game, i.e., in the form of a "turtle head" or rectangular protrusion (Montet 1955:196). Usually the appendage is rather vestigial and probably incapable of functioning as the type of "garage" suggested. Without more conclusive archaeological evidence, the function of the trapezoidal appendage depicted in the tomb paintings cannot be identified. It also seems unlikely to have been a foot, since only one of the scenes depicts the appendage facing down, and in this case it is depicted resting on a table (Simpson 1976:25, fig. 38).
In one case the spherical gaming pieces are shown being moved on the board (Vandier 1964:490). The individual spaces are never depicted on the board, and likewise there is no known depiction of the lion or lioness pieces being used on one (Montet 1955:139).
Judging from the scenes, it is apparent _mehen_ was typically played with the board on the ground, while the opponents likewise sat on the ground (or floor) opposite one another while they played. One scene shows the game resting on a table, but the players are still depicted as sitting on the floor, rather than seated. It is perhaps important to note that all of the _mehen_ playing scenes that have been found accompany _senet_ playing.
The earliest scenes come from the tomb of Rashepses, vizier of the pharaoh Isesi of the Fifth Dynasty, at Saqqara (Lepsius 1849:Fig. 61b). These two scenes are similar, and both depict what appears to be four players, or two players with two spectators, engaged in a game of _mehen_. These scenes are also the only ones depicting pieces on the board as what appear to be marbles, which are found toward the center of the snake, though the lion or hound pieces are nowhere to be seen. The caption accompanying the scene reads _ḥ b mḥn_, "playing _mehen_." They appear among _senet_ playing scenes, as well as depictions of singers and dancers, a common placement of games throughout the Fifth and Sixth Dynasty reliefs.
In the Sixth Dynasty tomb of Kaiemankh at Giza, two _mehen_ players can be seen next to a game of _senet_ the deceased is playing (Junker 1940:35–8). Much like the depiction of _mehen_ from the tomb of Rashepses, this scene is accompanied by images of musicians and dancers, as is a poorly preserved scene from the Fifth–Sixth Dynasty tomb of Isesi-mery netjer at Giza (Pusch 1979:29–32). Music and dancing are even more apparent in the reliefs from the Sixth Dynasty tomb of Idu at Giza, where _mehen_ is played alongside two _senet_ games, and musicians, dancers, as well as sports are depicted. The text introducing the scene reads: _ỉnḏ ḥr_. ṯ _m nḫ Ḥwt_- _Ḥr swt k _. _t ḥtptỉ nbỉ_. ṯ _imrṯ nfrw_ , "Hail, to you in life, O Hathor, the places of your _ka_ are propitiated, that you should glow is what the _nfrw_ desire" (Simpson 1976:25), and suggests these games are funerary games and celebrations in honor of that goddess (Kendall 2007:40).
There is one enigmatic depiction from the First Intermediate Period that could possibly represent a _mehen_ board. It comes from the tomb of Ankhtifi at Moalla and depicts a scene of woodworking (Vandier 1952:81–2). In this scene, there is an object that could be interpreted as a _mehen_ board viewed from the side, with two (one incompletely preserved) couchant lions placed atop it. This relief is far from definitive as to whether it depicts a _mehen_ board. If it is, it is the only known representation of the board from a side view. It also is the only evidence _mehen_ may have survived the end of the Old Kingdom, which was not thought to have been the case previously (Vandier 1964:489; Swiny 1986:56; Piccione 1990a:47).
In spite of this, two Twenty-Sixth Dynasty reliefs appear to depict _mehen_ in very much the same way the Old Kingdom reliefs do. One was found in the tomb of Ankhefensakhmet at Memphis, and is now in the Walters Art Museum in Baltimore (Capart 1938:13–18), while the other was found in the tomb of Aba in Thebes (Wilkinson 1878:55). Both scenes show the game flanked by four men, two on each side, sitting on the floor and interacting with each other, facing away from the board. The inscription accompanying these scenes describe the players as playing _senet_ , _mehen_ and the enigmatic game _ṯ w_ (Pusch 2007), though only two game boards are shown in each relief. Vandier (1964:489) suggests the scenes are so alike they shared a common model, most likely the tomb of Iby at Deir el-Gabrawi, which dates to the Fifth Dynasty, and is known to have been the inspiration for the decoration in the tomb of his Twenty-Sixth Dynasty namesake. Ranke (1920:13–14) agrees that tomb scenes from the Saite period were often inspired by Old Kingdom prototypes. While, unfortunately, no _mehen_ scene was preserved in the tomb of Iby, there were a considerable number of reliefs that did not survive (Davies 1902:11), and it is entirely possible the original _mehen_ scene once existed in this tomb.
### Textual evidence
There is little textual evidence on _mehen_ to provide clues as to the rules of play for the game, or the settings in which it was played. Most of the captions accompanying the _mehen_ playing scenes are statements such as _ḥ b mḥn_, "playing _mehen_ " or, _ḥ b k_( _wỉ_ ) _mnḥ_ ( _sic_ ) _r_. _k_ , "I am playing _mehen_ against you" (Kendall 2007:40). In the tomb of Kaiemankh, the caption may indicate some events in the play of the game. The caption above the player to the right reads _iṯ_. _t mḥn_ , "seizing _mehen_ ," which Kendall (2007:40) interprets as possibly meaning, "gaining advantage in _mehen_ ," "seizing the lead in _mehen_ ," or "my turn," while the other responds _ms_. _(i) r_. _k_. _ḥ b_. _(i) r_. _k_ , "I take aim at you and play towards you." While these passages do not further enlighten the rules of the game, they do indicate it certainly was competitive in nature.
Other texts give indications that may reveal the religious connotations the game had during the Fifth and Sixth Dynasties. In the Pyramid Texts, the name of the god Mehen contains the determinative for the game _mehen_ for the first time, suggesting that by the Fifth Dynasty there was a connection between the god and the game that extended beyond a shared iconography. Pyramid Text Utterance 332, in the pyramid of Teti, reads: "Teti is that one who emerged from the Coiled One: Teti has emerged from his fiery blast, while he is turned away" (Allen 2005:69). Kendall (2007:41) interprets this as the king passing along the coils of the game to the head and back, reflecting an ascension to heaven and resurrection on earth. Rothöhler (1999:14–15) interprets this text differently, suggesting an alternate translation in which the king ascends to heaven _as mehen_. Allen (2005:69) offers a better interpretation, asserting the text refers to the successful conclusion of the game that symbolizes the escape from Mehen, who tries to inhibit the passage of the sun through the _duat_. Utterance 659, from the pyramid of Pepi II (also appearing in the pyramid of Pepi I), commands the king to move his "teeth" along the board, with "teeth" as a metaphor for white ivory playing pieces (Piccione 1990a:48). Allen's translation (2005:268) "You have received your white teeth, Pepi Neferkare, and the coils that go around them, as an arrow, in their identity of an arrow." is less clear. From the pyramid of Neith, Utterance 758 describes the birth of the queen into the afterlife, and her wishes to be with Re-Horakhty and in the coils of _mehen_ (Rothöhler 1999:16). Allen's translation (2005:329) contains no mention of _mehen_.
These texts suggest a connection between the protection Mehen gives within his coils and the resurrection of the soul. Indeed, Middle and New Kingdom texts concur that Mehen was instrumental in protecting Ra in his nightly journey through the underworld (Piccione 1990a; Rothöhler 1999; Kendall 2007:41–2). It is also generally accepted that the goal of the game was to reach the center of the coiled serpent, to achieve either protection or union with Ra (Piccione 1990a:52; Rothöhler 1999:16; Kendall 2007:42). In any case, the symbolism contained within the boards (coiled serpent, primordial goose, turtle-head appendages) is consistent with solar ideology, particularly in relation to the journey of Ra through the netherworld, with obvious connotations of rebirth into the afterlife for the deceased. Later Egyptian texts suggest the "Roads of Mehen" were a circuit of nine concentric roads in the netherworld, at the center of which the god Ra sat on his "throne of millions of years." The deceased was required to journey through this path and approach Ra without succumbing to the dangers of the roads (Piccione 1990a:44).
### Archaeological evidence and social context
_Mehen_ boards are comparatively homogenous in their manufacture, but show variation through time. Some of the boards, as well as the playing pieces, are made out of luxury materials such as faience and ivory, suggesting their use by the upper class of Egyptian society. Indeed, the presence of game boards in the tomb of Pharaoh Peribsen, as well as its listing in the tomb of Prince Rahotep as funerary equipment, and its depiction in the mastabas of other nobility such as Hesy-Re, Idu, Kaiemankh and Rashepses indicate wealthy Egyptians played the game, but there is less evidence that lower classes of Egyptians played the game, unless it is inferred that the players depicted in the playing scenes were of lower class. It should be noted, though, that the children of Idu are all depicted among the dancers and players in the scene containing _mehen_ play, and are not distinguished by size, thus indicating relatively equal status with those around them. Indeed, Idu's son, Qar, is depicted playing _senet_ with the scribe Isi in the relief in Idu's mastaba, and they are depicted on the same scale as the _mehen_ players. It is not impossible that Egyptians of lower status also played the game, but their gaming equipment has not survived because they were likely made of more perishable materials, such as wood, or were made into the sand, as is common practice for board games (Merriam 1953:170; Pankhurst 1971; Tournay & Tournay 1971). It should be noted, however, that graffiti games, like the _senet_ games found in Fifth and Sixth Dynasty mortuary temples, which were crudely scratched into pavements, have never been found in a _mehen_ pattern. Since all of the evidence for _mehen_ in Egypt comes from elite burial contexts, as is most of the archaeological material discovered in Egypt, a clear bias exists in the data for the interpretation of social aspects of gaming.
The players shown with _mehen_ on tomb reliefs are exclusively men. _Mehen_ is mentioned in the Pyramid Texts of Queen Neith, but there is no indication that this means she played the game. In the mastabas that show gaming in conjunction with dancing and music, the dancers and musicians may be men or women, but the _mehen_ players (or, indeed, _senet_ players that accompany them) are never shown to be women. Cross-culturally, games are often restricted, at least officially, to a certain sex, and it can be considered inappropriate for certain genders to play certain games (e.g. Pankhurst 1971; Popova 1976:439–40). It is entirely possible that such social rules existed in Old Kingdom Egypt, though this may also be a reflection of the limited amount of information that has survived regarding _mehen_.
It is clear that one social context of _mehen_ was its inclusion in religious celebrations dedicated to Hathor as part of mortuary ritual, as can be inferred from the playing scenes of Fifth and Sixth Dynasty tombs, which is logical considering the funerary symbolism evoked by Mehen as represented on the game. Further archaeological evidence for a ritual use of the game comes from a set of eight lion and lioness gaming pieces which were found in a First Dynasty temple at Abydos (Petrie 1903:24, pl. III). The play of the game outside of these ritual contexts, for example for everyday leisure, is less clear, as no evidence has appeared to suggest this—though its appearance outside of Egypt (see below) argues against its sole use as a ritual game.
### The demise of _mehen_ in Egypt
Aside from Saite reliefs—which are likely copied from Old Kingdom originals and thus not evidence for the presence of _mehen_ in that period—the latest appearance of the game _mehen_ is the aforementioned painting on the wall of the First Intermediate Period tomb of Ankhtifi at Moalla. There are no boards, representations, or textual references to this board game from Middle or New Kingdom Egypt. Kendall (2007:42–3) argues it may have been forbidden by religious authorities as the segmentation of the snake's body into squares was actually a symbolic killing of the god Mehen, which would have been a serious taboo. Several coiled serpent boards resembling a _mehen_ game, but with uncut serpents, are known and may reinforce Kendall's argument (e.g., Ranke 1920:Fig. 8; Piccione 1990a:Fig. 4; Kendall 2007:Fig. 41.9).
The demise of _mehen_ is concurrent with the increasing popularity of _senet_. During the Predynastic and Early Dynastic Periods, _mehen_ appears to have been the more popular of the two games as there are more intact boards and pieces of _mehen_ type than _senet_ before the Old Kingdom. _Senet_ was more popular in the later years of the Old Kingdom since it appears more frequently than _mehen_ in Fifth and Sixth Dynasty tomb reliefs. Furthermore, after the disappearance of _mehen_ , _senet_ increased dramatically in popularity through the Middle and New Kingdoms, and by the New Kingdom explicitly had taken on the netherworld associations that _mehen_ had during the Old Kingdom. It is possible that, rather than a top-down moratorium imposed on _mehen_ play, its abandonment was more a reflection of changing popular tastes in gaming.
### _Mehen_ in Nubia
Before _mehen_ fell out of popularity in Egypt, it may have spread to the regions bordering it where there is some evidence suggesting that _mehen_ was played in Nubia. Excavations at the A-Group ( _c._ 3100–2800 BCE) Royal Cemetery at Qustul produced lions and lionesses, as well as hippopotami and marbles of different colors that closely parallel the lions and lionesses and marbles typically used in Egyptian _mehen_ games (Williams 1986:130).
An interpretation put forth by Kendall (1989, 2007:43–4) suggests a game observed in 1921 by Davies (1925:145–6) is a descendant of _mehen_. The game, which was played in a series of holes scooped out of the sand in a spiral pattern, was called _li'b el merafib_ ("Hyena Game") and was observed being played by the Kababish nomads in central Kordofan, in Sudan. Other spiral games are known throughout Saharan Africa, but their connection to _mehen_ , if any, is unclear (Depaulis 2001:55). Drawing connections to the equipment and presumed rules of play for ancient _mehen_ , Kendall (1989) concludes it is the ancient ancestor of this game among the Sudanese. While it is certainly a possibility, without historical and archaeological continuity of the game from antiquity it is impossible to conclude with any amount of certainty this game does represent a modern version of _mehen_.
### _Mehen_ in the Levant and Mesopotamia
_Mehen_ may also be represented in the archaeological record of the Levant and Mesopotamia, particularly in Canaan, perhaps not surprising considering the level of activity the Egyptians undertook in the Levant, discussed further in the section about _senet_ in the Levant (see Chapter 3). There are three games from the third millennium BCE that are analogous to _mehen_. These games are more crudely fashioned than those from Egypt, and consist of a series of depressions pecked into limestone blocks in a spiral pattern. Only one of these comes from a clearly datable archaeological context. This example is from Bâb edh-Dhrâ' and was found in the destruction debris within the walls of the town, datable to the Early Bronze Age II–III ( _c._ 3300–2500 BCE) (Rast & Schaub 2003:637). This game may be described as _mehen_ since it appears at the site alongside apparent _senet_ games in similar style (see Chapter 3). The other two stones come from the cave dwellings at Lachish that were occupied throughout the third millennium BCE (Tufnell 1958:40), placing them within the period that _mehen_ was used in Egypt. One of these has a clear spiral pattern, while the other may have a spiral pattern, although it is less certain (Tufnell 1958:Plate 21).
Also questionably identified as _mehen_ is a large limestone object with a spiral of pecked depressions found at Tell Brak in Mesopotamia and dated to the third millennium BCE (Oates et al. 2001:266). Its location on a brick pavement outside a building is consistent with the typical placement of games in open air spaces known from antiquity. It is possible that these are _mehen_ games, but without contemporary _senet_ boards from Lachish or Tell Brak (though another fragmentary board at Tell Brak could be identified as _senet_ , see Chapter 3) the identification is tentative, even if they are all contemporary with _mehen_ in Egypt.
### _Mehen_ in Cyprus and the Aegean
In Cyprus, _mehen_ appears with similar morphology to that found in the Levant in limestone blocks with pecked depressions laid out in a spiral formation (see fig. 2.7). First identified as such by Swiny (1980; see Crist 2015 for a reassessment), because some of these patterns appear on the opposite side of apparent _senet_ games, thirty-nine _mehen_ games have since been found at fourteen sites, intriguingly producing more games than have been preserved in Egypt itself. A fragmentary _mehen_ game was found at Lemba _Lakkous_ , and is the earliest game of any type to have been found on the island ( _c._ 2700–2500 BCE) (Swiny 1982). Often, there is a larger depression at the center or at the beginning of the outer ring of the spiral (sometimes both), which seems to mimic the differentiation of these places on the Egyptian boards (as the head and tail of the snake), though the serpent imagery is absent from the Cypriot games. Like the Egyptian examples of the game, the number of depressions varies, with twenty to ninety depressions present on the known examples, and the spiral can run either clockwise or counterclockwise.
**Figure 2.7** _Mehen_ game from Episkopi _Phaneromeni_ , Cyprus. Kourion Museum, CHM RR325. Photograph by Walter Crist, 2012.
What is remarkable about the _mehen_ games found on Cyprus is the degree to which they were adopted wholesale into Cypriot society. While the same is true for _senet_ (discussed below), _mehen_ was the first game adopted onto the island and incorporated into public and domestic life as it played a part in public feasting events as well as domestic and perhaps mortuary games (Crist 2015). It continued to be a part of Cypriot life well after the game was no longer played in Egypt; the latest _mehen_ game found on Cyprus was found at Kition _Kathari,_ and dates to the very end of the second millennium BCE (Karageorghis 1976:880, 1985:242), nearly a thousand years after it was abandoned in Egypt. This stands in stark contrast to the appearance of _mehen_ in the Levant, where it appears during the first half of the third millennium BCE, and disappears after Egypt became less reliant on middlemen in Canaan after it seized control of the copper mines in Sinai (see below with regard to _senet_ in the Levant), never to reappear again. It is possible Canaanites held a close association between the game and its Egyptian origin that was not shared in Cyprus. Canaanites would have seen Egyptians playing _mehen_ more frequently than Cypriots, who may have first acquired the game at Byblos, where there was a sizable Egyptian colony. A similar process is discussed in further detail in Chapter 3 with regard to _senet,_ for which there is better evidence (Dunand 1954:310, 1958:531, 573, 661). Only certain individuals would have made the journey to Byblos, and when they brought the game back to the island it lost the Egyptian connotation when played with other Cypriots since they likely never encountered an Egyptian playing the game. It may have still held an esoteric meaning originally as a foreign practice (Crist 2015, in press). It seems that after the upheavals that brought about the end of the Bronze Age throughout the Eastern Mediterranean also brought a final end to _mehen_ in the region as it was no longer played in Cyprus thereafter.
Similar stones have been identified in the Aegean, but there is little agreement on their function. They have been interpreted variously as astronomical devices, offering tables or games (van Effenterre 1955; Bardanis 1966, 1988/89; Hood 1995; Hillbom 2003, 2011). Few of these patterns appear to be spirals. Most of these come from Naxos, but many of them are on vertical rock faces, precluding them from use as a game, so their identification as such is unlikely. Only one spiral has been found on Crete, from Zakros (Hillbom 2011:172). Hillbom (2011:114–15) hesitates to call this spiral a _mehen_ game, and, as the single instance of this pattern on the island, it is impossible to conclude that it is a _mehen_ game. It dates to the Middle Minoan period ( _c._ 2000–1700 BCE), which is outside the time period when it was played in Egypt, so if it was in fact played as a game, it was more likely to have been inspired by the Cypriot version of _mehen_. Regardless, considering the level of contact with Egypt, the scarcity of spiral patterns, and the date of the game, it is impossible to call it a _mehen_ board without further evidence.
### _Men_
Also listed among the games offered to Prince Rahotep is the game of _men_ , which is one of the oldest games of Ancient Egypt, but one of the least understood. It is never shown in playing scenes, and only a few examples of the game or its pieces have been found from archaeological contexts. _Men_ may also appear in the painting in the tomb of Hesy-Re. While the three games appearing in this painting are not identified by name, _mehen_ and _senet_ are immediately recognizable from later boards and inscriptions, but the third is never depicted elsewhere, so it is inferred to be a board for the game _men_ based on its inclusion in the list of Rahotep. No other games dating to the Old Kingdom have been identified by name, and the archaeological evidence suggests _mehen_ , _senet_ and _men_ were sometimes included together in funerary equipment, and so it is likely the third game depicted in the tomb of Hesy-Re is _men_. The painting shows a long rectangular board with sixteen lines transecting its width. Accompanying the board is a set of playing pieces, which consists of two sets of five rectangular or possibly cubic pieces.
_Men_ apparently predates the Old Kingdom as it was found in Mastaba 3504 at Saqqara, dated to the reign of Djet, third king of the First Dynasty (Emery 1954:1). In one of the sub-magazines of the mastaba itself, a slate palette with ten incised lines was found (Emery 1954:66, pl. 30), which is very reminiscent of the painting of the _men_ board in the tomb of Hesy-Re. No pieces were found associated with it, so it is impossible to definitively say whether this is an early version of _men_ , or if it was a game at all. Playing pieces were found scattered throughout the tomb and subsidiary burials (Emery 1954:31, 58), so it would not be unusual that _men_ should appear in this mortuary complex. One grouping seems to be a complete game set of board and pieces, and was found with one of the subsidiary burials at this mastaba (see below).
Perhaps the best-preserved and most convincing example of a _men_ board was actually found in the Royal Cemetery of the A-Group at Qustul in Nubia, where one intact and one fragmentary board were found (Williams 1986:130). The intact board, found in Tomb L-23, is a limestone plaque with sixteen grooves, exactly replicating the configuration seen in the Hesy-Re painting. While there were no other apparent gaming paraphernalia associated with this board, playing pieces associated with _mehen_ , as well as ivory plaques that resemble the _men_ pieces seen in the tomb of Hesy-Re, were found in Tomb L-24, which produced a fragmentary _men_ board. This board, roughly half of which is preserved judging from the eight grooves that remain, was pierced in two places on one end, which Williams suggests could be for suspension or for mounting legs. Considering that _mehen_ boards were often pierced for suspension, it is possible _men_ boards were similarly pierced in order to be hung on the wall for storage. While there is a suggestion that _mehen_ boards could have been hung for amuletic purposes, such an indication for _men_ is not apparent since its ritual or symbolic significance is unknown due to its absence in the Egyptian textual record.
_Men_ seems to have never been as popular as _mehen_ or _senet_ , and it is unclear if it was played after the Old Kingdom since all boards, pieces and textual evidence for _men_ date to the Fourth Dynasty or earlier. An example of _men_ from the Middle Kingdom may exist on the ship model found in the tomb of Nefwa, now in the Ashmolean Museum (E2301). This scene, which has often been assumed to be a representation of _senet_ playing in the literature (May 1992:141), shows two sailors playing a board game on a warship (Garstang 1905:220). Closer examination of the board itself, however, shows the playing surface of the board is marked with eight transverse lines. While not an exact replication of the sixteen grooves seen on _men_ boards, it is a closer approximation in miniature to that game than a _senet_ board, which would be expected to have crossed lines forming rows of squares. Such identification can only be tentative due to the rarity of evidence for _men_ playing in Egypt.
### _Two rows of thirteen_ and _forty-two and pool_
Not attested in the textual or pictorial record, a sole example of _two rows of thirteen_ is known from subsidiary Grave 16 in Mastaba 3504 at Saqqara, dated to the First Dynasty (Emery 1954:31, pl. 29). The board was wooden, inlaid with ivory, forming two rows of thirteen playing squares. It had been placed lengthwise along the spine of the interred. The playing pieces were found in a tight cluster, and appear to have been contained in some kind of bag, which had since decomposed. Thirteen tall and thirteen short pieces were found, corresponding to the number of spaces in the board found in this tomb, and thus likely belonging to the same set. The bag containing the pieces was placed where the missing skull of the burial should have been, which likely had some kind of symbolic meaning. Kendall (2007:33, note 8) suggests this game may be a version of _men_ , but there is no evidence suggesting _men_ was played with this type of domed piece, or that the board ever took this form. It is more likely a less-popular game (at least among the upper classes of Egyptian society) than its contemporaries.
Another game is named _forty-two and pool_ by Petrie (1927:55). Found on a block of limestone at Memphis, this game board consists of three rows of fourteen drilled depressions, with one larger cup hole located to one side. Petrie interprets this game as having been played in a similar fashion to _senet_ , and that the gameplay involved the taking of pieces, which were then stored in the large cup hole, though there is no evidence of the rules of this game. He goes on to suggest beans or chips of pottery were utilized as playing pieces, due to the small size of the depressions. Petrie does not offer a date for this artifact, and so without further evidence it cannot be dated securely. Nevertheless, it is presented here with the range of unique games from early Egypt, as later Egyptian periods have produced a different range of games most of which are well documented and appear multiple times in the archaeological record.
##
## _Senet_ across Borders
Perhaps the best-known board game from ancient Egypt, _senet,_ also appears in the offering list in Prince Rahotep's tomb (see Chapter 2). As it was for _mehen_ , this is the oldest known inscription offering the name of this game that is well attested in New Kingdom contexts. Much like _mehen_ , this game also appears to have held strong connotations with the afterlife. The word _zn_. _t_ means "passing" in Egyptian, and though any specific religious meaning the game may have held is unclear before the New Kingdom, its name does suggest at least a similar connection with the passing of the _ba_ through the _duat_ that is made explicit in the Book of the Dead (Kendall 1978:28–31). Piccione (2007:54) believes the full name of the game, _zn_. _t n_. _t h b_, "the passing game," comes from the nature of gameplay where the pieces pass each other on the board. A canonical _senet_ board is between 12 and 55 cm long, laid out in three rows of ten playing spaces, often with certain spaces marked, likely indicating a special outcome during the play of the game, as is shown below. Piccione (1990b:1–2) points out that the name _senet_ is only directly attributed to this pattern of spaces in the New Kingdom, and that Egyptologists have assumed that, when it appears in earlier texts, it refers to the same game. Without any evidence to suggest the name was once used for a different game, it is Piccione's judgment the name _senet_ referred to the same game in the Old and Middle Kingdoms as it did in the New Kingdom.
### Early evidence for _senet_
The origins of _senet_ in Egypt are confusing as there are no intact game boards that date before the Fifth Dynasty, though fragmentary boards, playing pieces and textual and representational art have been found to suggest its earlier existence. One of the more controversial and earliest game boards known from Egypt is a clay gaming table found in a Predynastic grave at el-Mahasna (Ayrton & Loat 1911:30). This artifact has been described by some as the earliest known _senet_ game (Vandier 1952:406; Kendall 1978:7; Pusch 1979:156–7), but the game displayed is not in the canonical _senet_ form because it displays three rows of six, rather than three rows of ten playing fields. While the table itself and the pieces associated with it anticipate later _senet_ gaming sets, it cannot be said that this artifact is definitively an early version of the game. There are patterns found in Pharaonic Egypt that could have been games, but which only occur once in the archaeological record. A 3 × 6 game may have been one of them rather than a game that morphed into _senet_ at a later date. A recent interpretation of the board and pieces has been offered describing them as a potential offering table with votive garlic bulbs (Eyckerman & Hendrickx 2011:414).
**Figure 3.1** Map of sites mentioned in Chapter 3.
Even before the appearance of fragmentary _senet_ boards in the archaeological record, the appearance of a game board with three rows of spaces appears in the hieroglyphic writing system. The hieroglyph _mn_ appears in early sealings, particularly those associated with Narmer (Petrie 1901:Plates XIII, XIV, XXIII), as well as in other sealings dating to Dynasty 0 and the First Dynasty. The hieroglyph typically depicts three rows of squares with a series of pieces placed on the board, and should not be confused with the game _men_ , which uses a different determinative in Rahotep's list.
Fragmentary game boards that resemble later _senet_ boards appear in the First Dynasty. One, found by Petrie at Abydos (Petrie 1901:36) and now in the University of Pennsylvania Museum (E9368), still has two rows of squares, but is likely to originally have had three, based on the breakage pattern (Piccione 1990b:382). Another likely candidate for an early _senet_ game was found in Tomb M1 at Abu Rawash and dates to the reign of Den (Montet 1946:185). All that remained of this board were twenty-five ivory squares, three of which were marked in ink in a fashion similar to later _senet_ boards (Piccione 1990b:37). Another similarly marked set of an undetermined number of ivory squares was found in the same tomb, and may be another _senet_ board (Montet 1946:184; Piccione 1990b:38). A final candidate for an early _senet_ board, the present location of which is unknown (Piccione 1990b:382), was discovered at Saqqara, and was a fragmentary alabaster board, like the one discovered by Petrie, "engraved with three rows of squares" (Emery 1958:14).
Sets of gaming pieces are also known from the First Dynasty and later and may have accompanied gaming boards that have not been preserved. Piccione (1990b:17–18, note 51) enumerates many references to Early Dynastic Period playing pieces in excavation reports. A set of seven tall and seven short conical pieces, found together with a set of _mehen_ pieces and some throwing sticks, were found in a cluster at Abydos (Emery 1954:58). Similarly, conical pieces that were likely used for _senet_ were found alongside _mehen_ and _men_ pieces at Naqada (Petrie & Quibell 1896:Pl. 7; Vandier 1952:405). The set at Abydos, if it is complete, suggests that a full complement of seven pieces per player was typical for _senet_ during the Early Dynastic Period, though this would later change (Piccione 1990b:18). The set of domed pieces associated with the 2 × 13 game found at Saqqara is similar to those typically attributed to _senet_ , so one must be careful in attributing partial sets of pieces to a certain game since different games may have used pieces of similar types.
### Old Kingdom: Ritual use and graffiti
During the Third Dynasty, a game that took the form of three rows of ten squares was important enough to be included in the painting of funerary equipment in the tomb of Hesy-Re (see fig. 2.5), alongside _mehen_ and _men_ (Quibell 1913:18–21). As with those two games, _senet_ is accompanied in this painting by gaming pieces, i.e., two sets of seven domed pieces (one short and one tall set) along with four throwing sticks, which seem to have been the favored type of randomizing agent. Throwing sticks are known from First Dynasty and later tombs (Piccione 2007:56) and appear as markings on the aforementioned Abu Rawash playing spaces as well as on the twenty-sixth square of the Hesy-Re painting. In this painting, several squares are marked, including square one with _sb _ , "star," squares eleven and twenty-one with "ten," square twenty-seven with "four" and square thirty with "one." The numbers "four" and "one" are rendered as throwing sticks rather than strokes. This pattern anticipates, but differs significantly from, the canonical sets of markings that can be seen from at least the Middle Kingdom onwards.
The existence of _senet_ is also attested during the Fourth Dynasty by its inclusion in Rahotep's list of funerary offerings. Actual game boards are rare from Old Kingdom contexts, and those that do exist are not definitively datable to this period. All extant boards are graffiti on Fifth and Sixth Dynasty funerary monuments (Piccione 1990b:45–8; Pusch 1979:169–77), or surface finds associated with Late Old Kingdom-First Intermediate Period finds at Abu Ballas and Muhattah Jaqub in the Dakhleh Oasis (Förster 2007:4, 22). It is important to note that graffiti games are notoriously difficult to date, and their location in Fifth and Sixth Dynasty monuments only provides a _terminus post quem_ for those games, rather than the date at which they were used. Furthermore, the finds from the Dakhleh Oasis were found on the surface, and although associated finds were of late Old Kingdom or First Intermediate Period date, the game could also have been later in date. Despite this, if the game found at Abu Ballas (Förster 2007:22) is of Old Kingdom or First Intermediate Period date as the excavators suggest, it would be the oldest intact _senet_ game preserved.
The markings in some of the squares on the graffiti games appear in patterns that are commonly known from games that are datable to the Middle Kingdom and later, but it is an open question as to whether they represent the earliest appearance of the canonical pattern, or if they allow those games to be dated to a later period. For example, the game at the pyramid temple of Userkaf shows _nfr,_ "good" or "beautiful," in square twenty-six, "X" in square twenty-seven, "three," in square twenty-eight and "two" in square twenty-nine (Pusch 1979:169). This pattern also appears twice in the mastaba of Ptahshepses (Pusch 1979:171–3; Piccione 1990b:384–6). The board scratched into the vestibule of the mastaba of Seshemnefer IV contains "three," "two," and "one" in squares twenty-eight to thirty. Only the board pictured in the tomb of Hesy-Re and a New Kingdom board from Zawiyet el-Aryan (Dunham 1978:72) depict "one" in square thirty. The pattern of markings in the _senet_ from Seshemnefer IV's mastaba more closely resembles that seen in the tomb of Hesy-Re, as they both lack important elements of the later canonical markings (i.e., _nfr_ in square twenty-six and "X" in square twenty-seven), which seems to suggest that this graffiti is early in date, probably not later than the Old Kingdom.
_Senet_ is better documented in pictorial evidence, as it is attested already fifteen times in tomb paintings and reliefs, particularly during the Fifth and Sixth Dynasties. These scenes are rather similar in their composition, in that they typically depict two players facing each other across a _senet_ board either in the act of moving a piece or about to move it. Often, a brief caption of dialog accompanies the scene, usually one line from each player.
**Figure 3.2** Drawing of _senet_ patterns, showing markings common during the Old Kingdom (A), Middle Kingdom (B, after Egyptian Museum CG 37794), New Kingdom (C, after Egyptian Museum JE 62058), Late Period (D, after British Museum 102396) and an ostracon (E, after Egyptian Museum CG 25183).
The majority of the known Old Kingdom depictions of _senet_ are playing scenes. Most of these, nine in all, show at least one pair of players sitting opposite one another with a _senet_ board allayed with pieces in between, but in the context of funeral celebrations dedicated to Hathor (Piccione 1990b:79–82), as is also true for _mehen_ , which accompanies _senet_ in some of these reliefs. These celebrations frequently included musicians, dancers and sports (see fig. 3.3). Singers and musicians are the most common accompanying groups, who, along with the presentation of offerings to the deceased, are frequently depicted adjacent to scenes of _senet_ playing, as in the mastabas of Ankhmare (Piccione 1990a:49), Nikauhor (Piccione 1990a:51), Isesi-merynetjer (Pusch 1979:29–32), Pepiankh-heryib (Blackman 1924:31) and Nebkauhor (Hassan 1975:5).
The addition of dancers is notable in the scenes known from the mastabas of Neferiretenef (Mariette 1889:324–8), Rashepses (Quibell 1909:79–82), Idu (Simpson 1976:19) and Kaiemankh (Junker 1940:35–40). A unique context for a _senet_ game is found in the wall paintings of the mastaba of Mereruka. This playing scene is shown in the context of agricultural activities and the harvesting and presentation of funerary offerings to the deceased, who is depicted on an enlarged scale (Saqqara Expedition 1938:2). The context of the funerary ritual remains the same as those discussed above, yet the clearly celebratory nature of those scenes is absent from this example.
The depictions of _senet_ in the Sixth Dynasty tombs of Kahep and Kheni at el-Hawawish in Upper Egypt (Kanawati 1980:21, 1981:22) do not depict the playing of _senet_ , but rather show _senet_ boards included among mortuary equipment in a carpenter's workshop (Piccione 1990a:43). The remaining scenes, namely those from the mastabas of Metjetji (Goedicke 1958:18) and Kairere (Lauer 1976:77), are without context with regard to their place in the mortuary rituals of the deceased as they are fragmentary and much of the adjacent scenes is not preserved.
The inclusion of _senet_ in these rituals may be indicative of a relationship between the game and the afterlife, but its presence in the ritual celebrations of Hathor is the only indication of this use. _Senet_ is not mentioned in Old Kingdom texts in relationship to the afterlife in the same way that _mehen_ is. The markings on the _senet_ boards of this time period also do not provide a religious meaning as they do in the New Kingdom. Speaking to the state of the evidence, Piccione (1990b:79) argues that _senet_ had an underlying religious meaning, though for the most part it was a secular device used in a religious setting. It is interesting to note that after _mehen_ fell out of fashion after the Old Kingdom, _senet_ appears to have taken on at least some of its religious connotations since during the Middle Kingdom its symbolism with respect to the afterlife becomes clearer. It remains an open question as to whether these meanings became more prominent for _senet_ in the Middle Kingdom where _mehen_ is no longer attested. One can imagine a scenario in which _senet_ and _mehen_ both held similar religious meanings, and in which _senet_ 's increasing popularity eventually eclipsed _mehen_ 's importance in ritual activity, forcing the game out of favor. It is apparent from the archaeological and pictorial evidence that _mehen_ was more popular at the beginning of the third millennium BCE, while _senet_ was more popular towards the end. Regardless of the possible relationship between _senet_ and _mehen_ , _senet_ survived for two thousand years longer than _mehen_ , and its place in Egyptian society continued to change through those years.
**Figure 3.3** Drawing of scene from the mastaba of Nikauhor showing _senet_ playing alongside singers and musicians. The Metropolitan Museum of Art, Rogers Fund, 1908, 08.201 (Hayes 1953:Fig. 59).
### Middle Kingdom: Changes and consistency
Beginning in the First Intermediate Period and extending throughout the Middle Kingdom, it is apparent that _senet_ began to be represented differently in art, in the execution of the game boards, and, for the first time, in textual sources. The first representation of _senet_ to post-date the Old Kingdom is a painted scene on the side of a wooden model granary depicting Antef playing against Mery in a more quotidian setting than earlier depictions, with a servant girl fanning Antef with a palm frond and holding a jar of (presumably) beer (Blackman 1920:206; Kendall 1978:15; Piccione 1990b:65–6).
Game boards of the Middle Kingdom, most of which are datable to the Twelfth Dynasty, contain more consistent decoration with features that may be diagnostic of this time period (see fig. 3.2B). When marked, the spaces on the boards consistently contain an "X" in spaces fifteen and twenty-seven, _nfr_ in space twenty-six, "three" in space twenty-eight and "two" in space twenty-nine. This pattern of markings has its origins in the Old Kingdom (see fig. 3.2A), but is more consistently applied in Middle Kingdom game boards. The orientation of _nfr_ indicates a change in the orientation of the boards of Middle Kingdom date. In earlier and later boards, the decoration of _senet_ boards is always oriented in such a way that the decorated squares twenty-six through twenty-nine are on the bottom right, but in the Middle Kingdom _nfr_ would be upside down in this position, therefore indicating that the preferred placement of the decorated spaces was at the top left, a 180 degree difference compared to _senet_ boards of other time periods (Piccione 1990b:3).
Another feature of Middle Kingdom boards that does not appear on _senet_ boards of other periods are curved or pointed projections drawn above square twenty-six, twenty-eight and twenty-nine. It is unclear what this new feature of the boards indicates, but Piccione (1990b:48) argues it was not a regional variation, since boards laid out in this manner have been found in different regions of Egypt.
So far there are nine boards dating to the Middle Kingdom in the archaeological record. Two of these come from Abydos, one from the temenos of the tomb of Senusret II (Petrie 1927:53), and another from the cenotaph of Senusret III (Pusch 1979:189–90). The second example is the oldest known game outlined in ink on a limestone ostracon, more of which are known from the New Kingdom, particularly in the Valley of the Kings and Deir el-Medina, as is discussed below. These games are typically interpreted as having been played by the builders of the structures/tombs with which they were associated (Piccione 1990b:390).
One _senet_ board was drawn in red ink on the inside of a box lid from Lahun (Manchester Museum, no. 73), in the type of box that was often used for infant burials in the houses at this site (Petrie 1890:24, 30; Piccione 1990b:391). It shows a typical Middle Kingdom orientation of the board. Another _senet_ game, now in the Narodni Muzeum in Prague, was painted in black ink on the inside of a box lid of New Kingdom date (Narodni Muzeum n.d.). The markings on this board, though not well preserved, indicate its orientation was not of the Middle Kingdom type. Instead, the markings are typical of boards dating to the New Kingdom.
A graffiti game, found at Shatt el-Rigal, north of Silsila, was carved on the floor of the valley (Petrie 1888:15; Pusch 1979:184–5; Piccione 1990b:393–4), and is datable to the Middle Kingdom based on the presence of curved projections over squares twenty-six, twenty-eight and twenty-nine. At the fortress of Buhen, a game was discovered on a building block, which may have been either a pavement or reused block, and displays the diagnostic Middle Kingdom _senet_ pattern (Emery 1979:146, 220; Piccione 1990b:394–5). Another graffiti game at the tomb of Baqet III at Beni Hasan was found perpendicular to the wall, which seems to indicate that players would sit across from each other adjacent to the longer sides of the board (Piccione 1990b:394), though they are typically shown in paintings and reliefs, in all periods, to be sitting on the shorter ends of the board. Other boards from this period came from Lahun (Manchester Museum, no. 262), Tell el-Hisn (Egyptian Museum, CG 37794) and one with an unknown provenance now in the Brooklyn Museum (36.2) (David 1979:13, 15; Pusch 1979:183, 191; Piccione 1990b:392–3).
Middle Kingdom scenes of _senet_ playing are rare, and only two are known, from the tombs of Baqet III and Khety at Beni Hasan. The painting in Baqet III's tomb is discussed in great detail with relation to the _game of twenty_ (see Chapter 4) together with a painting found in the tomb of his son, Khety, being very similar. In both tombs, _senet_ is played by two men with a caption reading _ḥ b 5_, "playing five," likely an allusion to some aspect of gameplay (Piccione 1990b:68–73). What is most interesting about these reliefs is that they show _senet_ being played on gaming tables with zoomorphic feet, which have not been found archaeologically on _senet_ gaming tables earlier than the New Kingdom. Piccione (1990b:393) suggests that the well-made _senet_ board from Tell el-Hisn may have once been set in a gaming table similar to those seen here.
The appearance of _senet_ in the Coffin Texts marks the first of rare discussions of _senet_ in Egyptian religious literature. Only three spells, 335, 405 and 1019, mention the game, and even these spells are exceedingly unusual, with 405 attested only twice, 1019 only once, and the relevant passage in 335 only as an epilogue that appears once. Spell 405 seems to reflect a non-ritual context for the play of _senet_. This spell appears on only two coffins: Egyptian Museum JE 42909 and Metropolitan Museum of Art 12.182.132, the latter of which does not contain the line that refers to _senet_. Both were found in a necropolis at Meir. The coffin in the Egyptian Museum, containing the only complete version of the spell, belonged to the daughter of the nomarch of the Fourteenth Nome during the Twelfth Dynasty (Piccione 1990b:83). One line in the spell is from the tribunal of the gods, saying "Let him play _senet_ with those who are on earth." (Piccione 1990b:84), just as in the tomb reliefs of Kaiemankh, Isesi-merynetjer and Mereruka, where the deceased play _senet_ with still-living Egyptians. The subject is also bestowed with various other abilities, including the capability to sing and dance, to inspect his children and to go to his house (Faulkner 1977:56). The everyday nature of these activities suggests that _senet_ not only held religious significance during the Middle Kingdom, but also was used for pleasure.
Coffin Text 1019 is more problematic as it is fragmentary, and only exists in one example, written on papyrus and now in the British Museum (EA 10676). It reads in one section "when you shall be removed from the _senet_ board," an apparent allusion to the conception of the afterlife as a journey through the _senet_ game (Piccione 1990b:86–8). If this interpretation is correct, it represents the earliest attestation of this belief, which is well known from the New Kingdom and later. Faulkner (1978:121) offers a different translation, suggesting "you are saved from the cutting-table." The fragmentary nature of the text does not offer contextual evidence for either translation.
An epilogue to Coffin Text 335 on the coffin of Mentuhotep in the Ägyptisches Museum und Papyrussammlung in Berlin (coffin 9), appears only once and refers to the deceased "mooring happily with Osiris... going out into the day, playing at draughts (i.e., _senet_ ) and sitting in a booth after death" (Faulkner 1973:269). This passage is analogous to Chapter Seventeen of the Book of the Dead in which the _ba_ plays _senet_ in a pavilion, allowing passage between the land of the living and the dead. This suggests some continuity of belief in the importance of _senet_ from the Middle to New Kingdom, when a connection with the afterlife comes to the forefront in the textual and pictorial evidence.
### New Kingdom: Religious meaning
Toward the end of the Second Intermediate Period, _senet_ game boxes appear in the archaeological record, though they probably existed earlier. While _senet_ only appears in the material record as slabs or graffiti prior to the Second Intermediate Period (Piccione 1990b:3–11), beginning in the Seventeenth Dynasty game boxes start to appear in Egypt. This type of board consists of a box, typically with a drawer that usually features a sliding bolt that locks the drawer shut (Piccione 1990b:11–12). The two boxes from the Seventeenth Dynasty are also the earliest known examples of the _game of twenty_ to have been found in Egypt, and, as such, will be discussed in greater detail in Chapter 4. Throughout the New Kingdom, these games appeared together on opposite sides of game boxes and boards, thirty of which have survived. Considering that forty-one _senet_ boards have been found that can be dated securely to the New Kingdom, it is significant that so many of them also exhibited the _game of twenty_. Game boards and boxes were found made out of a variety of materials, including wood (often with faience, ivory or glass inlays), ivory, faience and limestone as well as graffiti in limestone pavements and drawn in ink on limestone ostraca.
The orientation of the New Kingdom boards changed back to the Old Kingdom variation, i.e., with the decorated squares placed to the bottom left (with two notable exceptions: Hall 1925:203–5; Needler 1953:Fig. 3), and the projections above spaces twenty-six, twenty-eight and twenty-nine disappeared. One board, found in a Seventeenth Dynasty tomb at el-Asasif (Carnarvon & Carter 1912:36), was drawn in ink on a writing tablet and maintained the Middle Kingdom orientation of the board, but without projections (Egyptian Museum, JE 41790). It may indicate that this orientation survived into the Second Intermediate Period, as the other two _senet_ boards from this period, another from el-Asasif (Metropolitan Museum of Art, 16.10.475) (Hayes 1959:24–5), the spaces of which are unmarked, and one from Dra Abu el-Naga (Egyptian Museum, JE 21462), which is reconstructed (Pusch 1979:195–8), cannot speak definitively to the orientation of Seventeenth Dynasty _senet_ boards. It is certain though that by the early Eighteenth Dynasty the orientation reverted back to the way it was during the Old Kingdom.
The decoration of squares during the New Kingdom was more elaborate on some boards than it was during the Old and Middle Kingdoms, while maintaining the key elements of the decoration ( _nfr_ in square twenty-six, "X" or some kind of hazard in square twenty-seven, "three" in square twenty-eight and "two" in square twenty-nine). Piccione (1990b:242–8) lays out a chronology of the changes in the decoration of the final four spaces of the board. While he provides a general guide to the changes in style through time, it should be noted that some boards are difficult to date, and the decoration does not necessarily determine that a board was made during a certain dynasty as these were styles that came and went throughout time, and were likely chosen based on the function of a board. Those placed in tombs are more likely to carry religious iconography than those made as graffiti boards, which were likely more for leisure than ritual.
According to Piccione's (1990b:242–7) ordering, in the early Eighteenth Dynasty the preferred markings were _nfrw_ in square twenty-six, _mw_ (unanimously interpreted as some kind of hazard related to water) in square twenty-seven, _b w_, in square twenty-eight and two men in square twenty-nine (see fig. 3.2C). During the reign of Hatshepsut, a new preference replaced this, with three men appearing instead of _b w_ as well as the deification of some of the symbols, e.g., Ma'at holding _nfr_ , Hapy in square twenty-seven (Daressy 1902:Pl. 9; Hayes 1935:32–4). Hippopotami could also appear as the hazard instead of _mw_. Only one board is known from the Eighteenth Dynasty in which square thirty is decorated (birds among papyrus). In the Nineteenth Dynasty, _nṯrwy,_ "two gods," appears in square twenty-nine and square thirty is decorated with the sun disk, and henceforth is almost always decorated whereas previously it rarely was marked. By the Twentieth Dynasty, deities became the focus of the final five squares, with the preferred arrangement being _nfrw_ , Hapy, _nṯrw_ "three gods," _nṯrwy,_ and Horus (see fig. 3.2D). The "gods" in squares twenty-eight and twenty-nine could be represented by specific deities, which were somewhat standardized, with Thoth, Shu and Ma'at sometimes appearing together in place of _nṯrw_ and Ra and Atum appearing in place of _nṯrwy._
Other squares aside from the final five were occasionally decorated. The most common was square fifteen (as in fig. 3.2C), which could be marked in a checkered pattern (Hayes 1935:Fig.18), a rosette (Pieper 1909:Fig. 5), or with _ nḫ w st_ (Tait 1982:Fig. 1). Squares other than these were sometimes marked after the late Nineteenth Dynasty, but without an obvious pattern.
Two _senet_ games represented on the Twentieth Dynasty Turin Papyrus 1775, henceforth Turin Papyrus, appear to have once contained symbols in every square, though not all of them are preserved (see fig. 4.7). Piccione (1990b:311–17) provides a progression of symbols from squares one through thirty, interpreting the hieroglyphs in the two fragmentary depictions along with relevant passages from the Great Game Text. This system by which the squares are named corresponds to the text of the Great Game Text, and is a reflection of the passage of the deceased in the _duat_ , recreating the passage of Ra under the protection of the god Mehen (Piccione 1980, 1990b). The markings on the boards themselves are also reflective of those found on the Turin Papyrus game and in passages from the Great Game Text, which suggests a consistent meaning to the marked squares.
The Great Game Text—versions of which appear in the Cairo Papyrus 58037 (Wiedemann 1897; Pieper 1931; Röder 1961:258–61; Piankoff & Jacquet-Gordon 1972:119–20), on the opposite side of the depictions of game boards on the Turin Papyrus, and in the tomb of Inherkau at Deir el-Medina (Bruyère 1930:42)—dates to the Twentieth Dynasty, and Piccione (1990b:97–8, 105–6) suggests the different versions are within two generations of each other, with the Cairo text probably dating to the reign of Rameses III. He provides an in-depth analysis of the texts (Piccione 1990b), as they relate to other evidence for _senet_ , and discusses the many ways in which this text encompasses most of the beliefs and practices suggested in other texts, representations and on the boards themselves. A comprehensive discussion of the Great Game Text is outside the scope of this volume, but these texts essentially describe the course of the game as a struggle between the deceased and an unnamed opponent for the life of the _ba,_ and, upon winning the game, the deceased is declared justified, whereas the opponent is drowned in the water (Piankoff & Jacquet-Gordon 1972:118). It depicts the journey of the player as a religious process, which is clearly paralleled in the Egyptian literature relating to the passage through the _duat_ (Piccione 1990b:197–241). The correspondence of spaces with events highlighted in the Great Game Text has led to some suggestions about the course of play of the game (e.g., Kendall 1978:59–67).
Similarly, Chapter Seventeen of the Book of the Dead describes the _ba_ playing _senet_ in a pavilion in the _duat_ (see fig. 3.4), and performance of this spell allows the _ba_ to move back and forth between the world of the living and that of the dead (Piccione 1990b:292–302). This particular passage is a derivative of Coffin Text 405, which mentioned _senet_ in a similar light (Piccione 1990b:290–1), and is directly presaged by Coffin Text 335, which describes the deceased playing _senet_ in a booth (Faulkner 1973:269). Vignettes of this scene depict the deceased playing _senet_ without a visible opponent inside a pavilion, the _ba_ perched on the tomb, indicating that, "through _senet_ , the deceased in the pavilion has 'gone forth' as a living _ba_ " (Piccione 1990b:303). Twenty-one of these scenes appear in various media, including papyri, tomb reliefs/paintings and coffin paintings (Piccione 1990b:259–60), most of them from the Nineteenth and Twentieth Dynasties.
**Figure 3.4** Facsimile of a painting from the tomb of Nefertari, depicting the queen in the _senet_ playing vignette from Chapter Seventeen of the Book of the Dead. The Metropolitan Museum of Art, Rogers Fund, 1930, 30.4.145. Image © The Metropolitan Museum of Art.
Five scenes that include _senet_ games are known from the Eighteenth Dynasty, and they do not portray this vignette from the Book of the Dead. Two of them, that from the tomb of Amenemhat (Davies & Gardiner 1915:70, pl. 26) and that of Neferhotep (Bénédite 1894:Pl. 2) depict the game played between two players, and in the Neferhotep scene in the context of funerary offerings and musicians (Piccione 1990b:266). The third, from the tomb of Sennefer, depicts the game next to the chair of the deceased as he receives funerary offerings (Piccione 1984:178). Two other depictions of _senet_ boards, where they are not in play, were painted on walls in the tomb of Rekhmire (de Garis Davies 1963:Pl. 79, 93). These depict a gaming table being carried as funerary equipment, and sitting in a garden, near a pool among a grove of trees.
Some later scenes that do not invoke Chapter Seventeen of the Book of the Dead still show the influence of the scene in depicting the playing of _senet_ , as in the tomb of Piay, where the deceased and his wife sit beside each other in the pavilion, though they are typically not interpreted as playing against each other (Pusch 1979:Pl. 23; Piccione 1990b:261). In the tomb of Nebenma'at (Maystre 1936:Pl. 6), he is shown playing _senet_ against his wife Meretseger, who is depicted as being of greater or equal status as her husband (Piccione 1990b:279–80), but not in a pavilion.
Other scenes unrelated to the Book of the Dead (Piccione 1990b:275–6, 280, 282) illustrate the player facing another person, who stands on the opposite side of the board, as exemplified by the scene on the game box of Imenmes (see cover image). These scenes are typically in the context of drinking as the person opposite holds a drinking cup in the scenes of Imenmes, Neferhotep and one of the scenes on the Eastern High Gateway of the temple of Medinet Habu, of Rameses III playing _senet_ in his harem (Pusch 1979:Pl. 31), while a large drinking jug is included in the scene in the tomb of Khonsumose (Pusch 1979:Pl. 29). These scenes appear to demonstrate the game in a more everyday setting, and may indicate that it was accompanied by drinking, though this is never shown in scenes with a religious meaning. A second _senet_ scene accompanying the previous one at Medinet Habu has no evidence of drinking, but the relief is in fragmentary condition. Alternatively, Piccione (1990b:424–5) also sees a similarity to offering scenes in the one depicted on the Imenmes box.
It is important to note that the nature of archaeological evidence from Egypt, which is primarily funerary, biases our understanding of the connotations games had in different socio-economic classes. While most of the games dating to the New Kingdom were placed in tombs of the nobility and the pharaohs (including four from the tomb of Tutankhamun, and one each from the tombs of Hatshepsut and Thutmose IV), it is still apparent that lower classes on the socio-economic scale also played _senet_ , as evidenced by the appearance of games on limestone ostraca at Deir el-Medina and in the Valley of the Kings. These games seem to indicate a divergence from the canonical representation of _senet_ as seen among the upper classes of Egyptian society.
Two ostraca from the Valley of the Kings, which Dorn (2011:322, pl. 314–15) dates to the middle of the Twentieth Dynasty, depict _senet_ boards in a dramatically different fashion from that which is seen on manufactured boards and game boxes. One ostracon has _senet_ patterns painted on both sides which, when viewed so the three rows of squares run vertically, have dome-shaped projections above the final squares on one side. Pusch (1979:362) interprets such extra spaces as representations of gaming pieces seen above game boards in pictorial representations, but it is not consistent with the iconography of game boards from any time period in Egypt where the pieces are shown arrayed along the long side of the board, when depicted.
The other ostracon, which shows an incomplete pattern of squares, shows decoration in what appears to be the middle row. Dorn (2011:322) recognizes this is not typical for a _senet_ pattern, and all of the squares should be marked when squares in the central row are marked in this fashion. These markings are even more divergent from the typical pattern in the _game of twenty_ (see Chapter 4), leaving _senet_ as the most likely candidate for identifying this game.
Another ostracon, found in the tomb of Rameses V and VI again shows _senet_ with dome-shaped projections (Pusch 1979:361, pl. 97b) (Egyptian Museum, CG 25183), this time identifiable as being above squares ten, twenty and thirty (see fig. 3.2E). Squares twenty-seven, twenty-eight and thirty are marked with _nfrw_ , the only time this symbol appears in these positions. It does not conform to the interpretation of these spaces on other boards, namely that square twenty-seven is a hazard. Furthermore, _nfrw_ is oriented so that the board is to be viewed vertically, suggesting the players would sit across from each other on the short ends of the board, something never seen in manufactured _senet_ boards.
A final ostracon, found in more recent excavations in the Valley of the Kings, has markings in squares twenty-six, twenty-eight and twenty-nine, but it is impossible to determine what the markings are or their orientation from the photograph (Hawass 2011:70).
The extensive graffiti on the roof of the temple of Khonsu at Karnak include three _senet_ games. One of these presents a _senet_ pattern arrayed vertically with domed projections that must have been used as playing spaces on this board, since there are only three rows of nine squares if they are not included (Jacquet-Gordon 2003:78, pl. 84). Moreover, the decoration in the squares only conforms to the typical pattern ( _nfrw_ , _mw_ , "three," "two") if the domed projection is an unmarked square thirty. The placement of the marked squares on the left-hand side is particularly unusual, and is unknown on any other _senet_ board. Another board, apparently unfinished, was marked with "three" and "two" (Jacquet-Gordon 2003:84, pl. 92).
The decoration and orientation of these boards may indicate that those who made them were not familiar with the canonical representation of the _senet_ board, or that the preferences shown in funerary board games was not necessary for everyday gaming. Nevertheless, one graffiti game from the Temple of Khonsu demonstrates that the person who scratched it into the surface had the requisite knowledge of the canonical representation to produce it for their personal use (Jacquet-Gordon 2003:30, pl. 22). This game reflects the decoration that was most common during the Twentieth Dynasty, but it remains difficult to date any graffiti games, as the structures in which they were carved merely provide a _terminus post quem_.
An example of graffiti games which could be significantly later than their host structure are two graffiti boards found in the north colonnade of the First Court of the temple of Medinet Habu (Pusch 1979:320–1). These two _senet_ patterns were found parallel to each other roughly twelve centimeters apart, and set perpendicularly to the wall. Because of this arrangement, the players would have had to sit along the long sides of the board with both _senet_ games between them. This arrangement is also seen at the small boat ramp of Taharqo (r. 690–664 BCE) at the temple of Amun at Karnak (Piccione 1990b:436–7), and so must date to later than the Twenty-Fifth Dynasty. Since this is the only other example in Egypt where players used two _senet_ games at the same time, the Medinet Habu patterns are possibly close in date to those on the Taharqo ramp, and not necessarily earlier than the Twenty-Fifth Dynasty.
### Later history of _senet_
After the end of the Twentieth Dynasty, the material evidence for _senet_ is less frequent since only sixteen boards and four scenes are known. Literary evidence may suggest that _senet_ maintained some form of religious connection during this period, but it appears to differ in some ways from that of the New Kingdom. It is likely that increasing foreign influence during the Late Period had an effect on the cultural contexts of gaming in Egypt due to incursions of Libyans, Kushites, Assyrians, Persians, Greeks and eventually Romans.
None of the surviving boards from the Third Intermediate Period onwards were on game boxes of the type known from the New Kingdom. Furthermore, none of them contain the _game of twenty_ on the opposite face because this game appears to have disappeared from Egypt after the end of the New Kingdom (see Chapter 4). Instead, a new game is preserved on the opposite side of _senet_ , the _game of thirty-three_ that appears on the verso of five _senet_ game boards. This game is not well understood, but its origins may lie in the Near East, as discussed at the end of this chapter.
The decoration in the squares of late _senet_ boards appears to continue the tradition observed in the New Kingdom, with _nfr_ or _nfrw_ , often seen in square twenty-six, _mw_ , "X" or Hapy in square twenty-seven, "three" or _nṯrw_ in square twenty-eight, "two" or _nṯrwy_ in square twenty-nine and Horus in square thirty. One board from the Twenty-Sixth to Thirtieth Dynasty contains _s _ , "protection" (Petrie Museum, UC2317), though this is not necessarily a new phase as interpreted by Piccione (1990b:247–8). One might interpret a wider trend toward the dismantling of the canonization of the iconography of _senet_ in the later periods as the religious meaning trended away from that which it held in the New Kingdom.
A recently discovered game from Heliopolis, likely dating between the Twentieth and Twenty-Second Dynasties (Iskander 2010), has elaborate inscriptions in the final five spaces that include the typical pattern of _nfr_ , _mw_ , _b w,_ two (ladies in this case) and Horus. It differs from earlier games in that the playing spaces are not laid out as squares, but as rectangles, and also that spaces ten and eleven are marked. The inscription in these squares, probably meant to be read together, invokes the name of the board's owner, Keramit, and her title as priestess of Mut. This reflects the invocation of Mut in the Great Game Text (Iskander 2010:125) and depiction of the same goddess on the Turin Papyrus in square eleven (fig. 4.7).
Three _senet_ boards scratched into circular terracotta platters, the locations of which were unknown to Piccione (1990b:444–5) and Pusch (1979:370–1), are now located in the British Museum (EA 22323, EA 23802 and EA 23803, see fig. 3.5). These were found in Chamber 9 in the fortress at Tell Defenneh, and are dated to the Twenty-Sixth Dynasty (Petrie 1888:74). EA 23803 shows decorations in the final squares, the only one of which that can be identified is Horus in square thirty. The appearance of these games on circular terracotta platters is unknown during any other period of Egyptian history. They were found alongside other patterns of squares carved into stone slabs, though Petrie did not describe or illustrate these patterns (Petrie 1888:74), but the possibility that they were games found near these ceramic platters at a fortress points to a later pattern seen at the Roman fort at Abu Sha'ar, where several Roman games were found (see Chapter 6).
Piccione (1990b:449–51) has identified, but did not illustrate, three _senet_ boards on the roof of the temple of Dendera, dated to the Ptolemaic or Roman period. The patterns exhibited on this structure are different from all other _senet_ games from Egypt in that the board is laid out as a series of depressions rather than rows of squares. This manner of delineating a _senet_ pattern is well known from contexts in the Levant and Cyprus, but is rarely documented within Egypt. It is difficult to verify these patterns as _senet_ without further documentation. Other patterns are known from this roof that were manufactured in this manner but that are not arranged in three rows of ten and therefore were meant for some other use.
_Senet_ scenes are rare from later Egyptian history. Two are known from the Saite period, and seem to be copies of Old Kingdom reliefs (Piccione 1990b:286–7). These two were from the tomb of Ankhefensakhmet, now in the Walters Art Museum in Baltimore, and the tomb of Ibi, both also mentioned earlier with reference to _mehen_. With these reliefs alone it would be difficult to place _senet_ within the Late Period, but the presence of _senet_ graffiti on the boat ramp of Taharqo does indicate that the game survived at least that long. A later scene, in the tomb of Petosiris and dating to the reign of Darius III (r. 336–332 BCE) or Alexander the Great (r. 332–323 BCE), depicts two men playing a game (Kendall 1978:40; Piccione 1990b:288–9) that has three rows of eleven squares. It is possible this game was not meant to be _senet_ , but rather the game of _thirty-three_ , but it is important to note that representations of games in reliefs did not always depict the correct number of squares on _senet_ boards. These reliefs offer little in the way of interpretation of the use of _senet_ during the Late and Ptolemaic periods.
**Figure 3.5** _Senet_ boards on terracotta platters from Twenty-Sixth Dynasty fortress at Tell Defenneh, one showing the thirtieth square marked with Horus. Top: 31.5 × 2.5 cm. Bottom: Original 27 × 3.1 cm. The British Museum, EA 22323 (top) and EA 23803 (bottom). © The Trustees of the British Museum.
Textual evidence provides some hints that at least some of the understanding of _senet's_ earlier use in connection with mortuary ritual may have continued into the Late Period. Herodotus, writing in the fifth century BCE, recounts the legend of King Rhampsinitus (Rameses III), told to him by Egyptian priests, that describes his descent into the underworld, where he played dice with Demeter/Isis ( _Histories_ 2.122). This may have been an altered telling of Chapter Seventeen of the Book of the Dead (Piccione 1990b:333–6). A third century BCE Demotic text tells the tale of Setne who played three games of _senet_ against Nineferkaptah in an attempt to gamble for the lost book of Thoth (Piccione 1990b:336). This story depicts gambling by _senet_ for the first time, though gambling is likely to have existed in other forms (Tait 2007:49–51). The passage refers to the playing pieces as _ỉwỉw_ , "hounds," which Piccione (1990b:337) relates to a Nineteenth Dynasty scene of Merenptah at the Osireion playing _senet_ with dog-shaped playing pieces (see fig. 3.7). Considering this text comes from the Ptolemaic period, as well as the Hellenic custom of referring to playing pieces as dogs (Schädler 2013a:2844), it is probably more likely interpreted as a Hellenizing influence on Egyptian gaming vocabulary.
The final text referring to _senet_ is very late, dating to the third century CE. Oxyrhynchus Papyrus 470 describes a device called a _πεσσευτήριον_. Eustathius ( _Commentarii in Homeri Odysseam_ 1,28.1. line 23) states Plato knew of such a device, which he claims the Egyptians used for astronomical measurements. More likely, the name of this device is related to the Greek _πέσσοι_ or _πεττεία,_ which may be translated as "board game" (Kurke 2002:15–30). In the text, certain aspects of the description of this device suggest that it is, indeed, a _senet_ board that is being described (Piccione 1990b:344–6). There are thirty squares in the device, and the counters used on it are referred to as _κυῶν_ , "dogs." The twenty-sixth square is called in Greek _φερνούσι_ , which is likely a Hellenization of the Egyptian _pr nfr_ , "good house," while the thirtieth square is called _φόρωρ Ὥρου ὄικος_. _Φόρωρ_ is a Hellenized transcription of the Egyptian word _pr_ Ḥ _r_ , "House of Horus," which is then translated into Greek " _Ὥρου ὄικος_." Therefore, it seems, even at this late date, squares twenty-six and thirty held the same meaning as they did at least as early as the New Kingdom, as attested on _senet_ boards containing analogous decoration in those squares. Piccione (1990b:346–68) goes on to describe potential astronomical and astrological meanings behind _senet_ boards. Whether these connections were truly Egyptian ideas relating to the game, or were misinterpreted by the Greeks, this debate is outside the scope of this overview. Regardless, this late text is the final reference to _senet_ in the historical or archaeological record, and represents the end point of a game that can be documented for three millennia, longer than any other known board game in human history.
### Playing pieces
For _senet_ and the _game of twenty,_ each player controlled a set of uniform pieces whose number could vary. Playing pieces are well known from visual representations as well as archaeological finds, and both help the attribution of hundreds of pieces recorded as pawns or draughtsmen in museums and private collections (fig. 3.6). They are sometimes the only evidence of the practice of board games because—apart from rare wooden examples—most of them were made of non-perishable materials such as clay, faience, stone, ivory, bone and bronze. Isolated finds of playing pieces point to the possible deterioration of boards. Opposite teams were differentiated by the shape, size and/or color of the pieces. On the inner side of the wooden door from the tomb of Sennedjem at Deir el-Medina, the excellent state of preservation of the playing scene shows the distinction of the light- and dark-colored playing pieces (Bruyère 1959:2, pl. XVII) (Egyptian Museum, JE 27303). Pawns could be distinguished by different arrangements of colored bands, with, for example, alternating natural ivory and red painted bands on pieces from the tomb of Tutankhamun (Tait 1982:30, pl. XII). Faience allowed many variants of colored glazes and striped motifs (Hayes 1959:198, fig. 113).
The earliest form of _senet_ included two sets of seven domed playing pieces (see fig. 3.3). The sets were distinguished by the size of the pieces or by the addition of a knob at the top. During the New Kingdom, _senet_ and the _game of twenty_ shared their equipment as they were associated on double-sided boxes. The most common game sets consisted of geometrically simple forms like in the earlier periods. The two opposite types were conical pieces often with a knob at the top, and spool-shaped pieces (see fig. 4.5). The latter would sometimes be mushroom-shaped (Nash 1902:345, pl. IV, 10–11). The conical shape is referred to as halma type, after similar pawns from the modern game called Halma. This is the model for the hieroglyph used to designate a playing piece. This sign is related to the word _ỉb ,_ "dance," so playing pieces are called "dancers." A link to the word _ b,_ "ivory," had initially been proposed because many pieces are made of ivory (Birch 1865:59).
**Figure 3.6** Playing pieces collected by F.G. Hilton Price and now in museum collections, such as the Musées Royaux d'Art et d'Histoire, Brussels, E.06165 c, f, h, i (nos. 1, 3, 4, 6: max. 3.5 × 2.2 cm), and the Metropolitan Museum of Art, 26.7.1452 (no. 10: Head of a leopard with the name of Hatshepsut, 3.2 × 3 × 3.5 cm). Plate reproduced from Towry-Whyte (1902:Pl. I).
Playing pieces could also be figurative. They would be given elaborate shapes like prisoners with Nubian or Asiatic features (Kendall 1982:269; Franco 2004:229). The captives, with their elbows tied behind their backs, are represented naked or wearing a flaring skirt that molds the curve of the base. Music and board games were part of the same sphere as illustrated by the juxtaposition of musical and playing scenes in tomb decorations. The Hilton Price collection holds a flute player and a figurine with an instrument described as a musician or an archer (Towry-Whyte 1902:262, pl. I, 6; May 1992:146–7, figs. 134–5) (Musées Royaux d'Art et d'Histoire, Brussels, E.06165i–j) (fig. 3.6, no. 6). An extraordinary game set of ten Bes-headed figures is interpreted as the underworld deities, called _Ahau_ , the "fighters" who defended the sun god in the netherworld (British Museum, 1893,0514.42-57). These images would protect the player on the board (Kendall 1982:269). This group, made of molded green faience, is dated according to the glaze technique to the Nineteenth or Twentieth Dynasty. The jackal was also a designated representative for players, in his capacity of protector of the dead and the sun god (Kendall 1982:269). Jackal figurines were popular as well as representations of a recumbent or seated jackal on disc-shaped and conical pieces (Nash 1902:345, pl. III, 10–11). As seen in fig. 3.7, Merenptah in the Osireion at Abydos is represented playing _senet_ with dog/jackal-shaped playing pieces (Kendall 1978:36–7, fig. 28). By the Ptolemaic period, gaming pieces were actually called dogs/hounds, for example in the Tale of Setne, when the gaming episode refers to a board with _ỉwỉw_ , its "hounds" (Piccione 1994:199). Echoes of this tradition may be found today in playing pieces called _kelb_ in Arabic (Wilkinson 1878:57, note 1). The name of the Babylonian game _pack of hounds_ or _pack of dogs_ might be derived from its playing pieces (Kendall 1982:269) although no such pieces are known from Mesopotamia.
Animal forms include other species such as lion, baboon, bird, ram, horse and cat (Towry-Whyte 1902:262; Petrie 1927:54). Most of the figurative pieces are isolated finds that entered museum collections without their original board so their attribution to a specific game is difficult and their identification as gaming pieces is even sometimes doubtful. Baboon-shaped figurines from the Ptolemaic period are identified as playing pieces because they carefully integrate the contour of an astragalus to the body of the animal (Arnold 1995:60, no. 82) (Metropolitan Museum of Art, 66.99.75; Walters Art Museum, 71.512). A few playing pieces are inscribed with the name of a pharaoh—Hatshepsut (Dunn-Vaturi 2012b) (Metropolitan Museum of Art, 26.7.1452) and Nekau (Breyer 2010:28–9) (Musée du Louvre, E 5115)—as well as private names (fig. 3.6, nos. 8, 10). Finally, floral motifs are also represented on playing pieces (fig. 3.6, no. 9).
The number of pieces for each player seems to vary across time and probably decrease from seven to five. Representations of the number of pieces—like the number of squares—is not standard so they are not reliable. Variations may even occur within scenes dating from the same period. The pieces are usually aligned on the board either with alternating opposing gaming pieces, or with opposing gaming pieces arranged in two camps (see fig. 4.4). Only one representation, on the Imenmes gaming box, shows one spool-shaped piece on top of another one (see cover image).
### _Senet_ in Nubia
With the great popularity of _senet_ in Egypt, it is no surprise that it has been found in other regions that have had contact with Egypt. Considering the history of Egyptian activity in Nubia, it is also expected that _senet_ has been found there. The earliest example of a _senet_ game in Nubia comes from the fortress at Buhen. It was inscribed on a building block that was reused (Emery 1979: 146, 220). Since the context in which it was found was mixed (ranging from the Middle Kingdom to the Seventeenth Dynasty), it is difficult to date this artifact stratigraphically. The gaming surface is rendered in the typical Middle Kingdom pattern, though this pattern is known in the Seventeenth Dynasty in Egypt (Piccione 1990b:394–5). It is possible that the block on which the game was carved once served as a paving block, and the game was carved on it while it was still part of the pavement.
A fragmentary _senet_ game was found in Tomb 109 at Kubban Cemetery 110 (Firth 1927:49, 83). Thirty-six faience inlay squares were found and were reconstructed into three rows of twelve. Three of the squares are decorated: one with _mw_ , one with _b w_ and one with three standing men. Considering the conventions of New Kingdom Egyptian boards, Piccione (1990b:420–1) and Pusch (1979:223–9) both interpret these squares as belonging to two _senet_ boards, which are incompletely preserved, since the three standing men and _b w_ are two representations of square twenty-eight.
During the Kushite Twenty-Fifth Dynasty when Egypt was ruled by Nubian kings, the desire of these foreign pharaohs to emulate Egyptian practices was strong, and thus it is no surprise it was during this period that there was pictorial and archaeological evidence for _senet_ far up the Nile Valley. Ivory playing pieces in the shape of sitting lions and lions with rams' heads were found in the tomb of Shabaqo (Dunham 1950:Pl. 24), reminiscent of the sitting animal pieces shown in the _senet_ scene of Merenptah at the Osireion (see fig. 3.7, Kendall 1978:36).
Ivory inlays were found in the tombs of Neferukekashta, wife of Piankhy, and an unnamed wife of Shabataqo at el-Kurru (Dunham 1950:85, 108; Kendall 1978:37). Kendall interprets these belonging to fragmentary _senet_ boards, though only two squares were preserved from Neferukekashta's tomb, and eight from the other. The decoration on the plaques does not adhere exactly to that known on well-preserved Egyptian boards, but there are some, such as Horus, papyrus marsh, and potentially the god figures that have parallels in the Egyptian corpus. The Kushite kings may not have followed the canonical decoration since at this time it was not always followed in Egypt itself.
Even after the Kushite Dynasty was expelled from Egypt, _senet_ appeared in a tomb relief of the deposed dynasty, which continued to rule Nubia. A relief in Napatan style, now in the Museum of Fine Arts in Boston, attributed to the pyramid of King Aramatelqo (Piccione 1984:173), depicts a _senet_ scene that appears to be of Old Kingdom inspiration, with perhaps two pairs of opponents playing _senet_ (Piccione 1990b:288), much like the aforementioned _senet_ scenes of the contemporary Twenty-Sixth Dynasty. This is the latest evidence for _senet_ in Nubia.
**Figure 3.7** Drawing of Merenptah playing _senet_ with dog- or jackal-shaped pieces at the Osireion, Abydos (after Naville 1911–12:pl. II.6).
### _Senet_ in the Levant
As in Nubia, _senet_ appears in the Levant during periods when the Egyptians were most involved in the region. It appears to have arrived in Canaan early in the third millennium BCE, roughly contemporary with the appearance of _mehen_ there. Games that may be versions of _senet_ have been found at sites stretching from the edge of the Negev desert in the south to Syria in the north.
The site that has produced the greatest number of _senet_ games so far is Tel Arad in southern Canaan. Dating to the Early Bronze Age II ( _c._ 3000–2850 BCE), thirty-five _senet_ gaming boards were found at this site (see fig. 3.8). The boards are strikingly different from those found in Egypt, and are most analogous to graffiti games found there, though they are generally portable objects, typically made of locally available limestone (Sebbane 2001). The _senet_ patterns are executed in three different ways: by the incision of squares, by the drilling of holes and by a combination of these techniques where the holes are drilled in the center of the incised squares. The preference seems to have been for the drilled type, which makes up sixty percent of the assemblage, probably because pebbles (or seeds) were used as playing pieces (Sebbane 2001:218). It is important to note that none of the _senet_ games found at Arad had playing spaces that were specially marked in any way.
**Figure 3.8** _Senet_ game from Arad, with drilled depressions as the playing spaces. Israel Museum, 1989-422. Photograph by Marlana Salzberger. Courtesy of the Israel Antiquities Authority.
Sebbane states that the games were found in domestic contexts as well as public spaces, and some of those in open public areas were on stones that appear to have been stationary, due to their large size (Sebbane 2001:219). The excavations remain incompletely published, so other possible game board patterns near these games are unknown at Arad.
Roughly contemporary with the games at Arad were nine similar patterns at Bâb edh-Dhrâ' (Lee 1982; Rast & Schaub 2003:636–7). They differ from the boards at Arad in that the _senet_ patterns are all delineated by the pecking of small cupules into flat limestone slabs. None of the Bâb edh-Dhrâ' games have incised squares. Though these games were surface finds, they are likely to be from the Early Bronze Age II, during which the site was at its greatest extent, and for historical reasons discussed below.
The existence of these games during the Early Bronze Age II is likely linked to increased Egyptian activity in the area during this period. Roughly contemporary with the First and Second Dynasties, this period saw an influx of Egyptian artifacts at Arad and other sites (Amiran 1978:51; Ilan & Sebbane 1989:153; Schulman 1989:443; Brandl 1992; Porat 1992:437; Kaplony 2002:487; Sowada 2009). Arad itself appears to have been attractive to the Egyptians as it functioned as a terminus of the trade in copper out of Sinai (Ilan & Sebbane 1989). The inhabitants of Bâb edh-Dhrâ', further afield, probably only rarely interacted with Egyptians, explaining why their games are morphologically much different from the Egyptian games than those found at Arad, where the inhabitants would have regularly interacted with Egyptians and probably played on Egyptian boards (i.e., with squares). Once the pharaohs of the Third Dynasty seized the Sinai copper mines, Egyptian activity in the Levant turned toward the coast (Stager 2001), and _senet_ disappeared from Arad and presumably Bâb edh-Dhrâ'.
_Senet_ games from Tell es-Safi, i.e., ancient Gath, exhibit markings in their squares, though they do not exactly fit the Egyptian canon. One of these (Shai et al. 2014:Fig. 11.1a) has an "X" in two squares, which appear to be squares fifteen and twenty-nine. It could be a simpler way of marking two spaces important for the play of the game rather than using potentially unfamiliar Egyptian conventions. This game and another found at the site are both fragmentary, and therefore, the possibility exists that they are not _senet_ games. They display a pattern of incised squares on the opposite side (Shai et al. 2014:38) that could also be some form of game. These games were found in Stratum E5, dated by the excavators to the end of the Early Bronze Age III ( _c._ 2850–2500 BCE) (Shai et al. 2014:28). The appearance of potential Egyptian games here parallels the shifting trading activity toward the coast.
Another artifact, found while dismantling a wall at Megiddo, displays three rows of nine depressions, though it is damaged and may have originally displayed three rows of ten, and therefore possibly an example of _senet_ (Guillaume 2013:1106). This game is difficult to place chronologically, but is probably Early Bronze Age in date.
Further up the coast in modern Lebanon, _senet_ games were found at Byblos, which has long been known to host an Egyptian trading colony, particularly for the procurement of cedar (Gale et al. 2000:349; Stager 2001:629). While no _senet_ board of Egyptian type has been found here, four games of similar type to those from other Levantine sites were discovered at the site (Dunand 1954:310, 1958:531, 573, 661). These games are most like the ones from Arad as they reflect both incised, drilled and a combination of both methods (Sebbane 2001:218), which may reflect that at both sites there was a greater familiarity with the Egyptian games than in other sites. Despite this, one of these games, excavation number 12202, has each square in the outer two rows marked, one row with "X" in each space and the other row with either a box with an "X" or a mark similar to a Minoan double axe (Dunand 1958:505). A pattern may be seen here similar to that in Nubia, where some license in the manner of decoration of the squares may have been taken, as a result of cultural unfamiliarity of some of the religious aspects of the game, if indeed they existed in this early period. Dunand suggests these games may date to the Middle Bronze Age ( _c._ 2200–1540 BCE) (Swiny 1986:41 note 314), though the manner of excavation of this site makes it nearly impossible to date them stratigraphically.
Other games outside of these sites are isolated and more difficult to interpret. One was found at Har Yeroham (Kochavi 1967: 120), two at Mashabei Sade (Cohen 1986:56, 301; 1999:266) and one at Khirbet Iskander (Richard & Boraas 1984:83), all of which date to the Early Bronze Age IV/Intermediate Bronze Age ( _c._ 2500–2200 BCE). One example comes from Hama in Syria and is a well-made example of a Levantine type _senet_ with squares and depressions (Fugmann 1958:76, 80). A final artifact from Tell Brak in Mesopotamia (Oates 2012) is fragmentary, but exhibits the same morphology. These sites are all well away from the coastal areas of Egyptian mercantile activity, so their presence is more difficult to explain. They all either contain or incorporate the drilled method of manufacturing games, suggesting an interpretation similar to that offered for Bâb edh-Dhrâ', whereby those who were playing them were probably not playing _senet_ with Egyptians, but more likely other Levantine peoples, thus they were not mimicking the Egyptian prototypes. In this way, _senet_ may have spread as far as Mesopotamia during the third millennium BCE, likely due to the intense Egyptian activity in Byblos.
Four later games come from the Levant, two from Kamid el-Loz in Lebanon (Meyer 1986:126–36), dating to the Late Bronze Age ( _c._ 1640–1110 BCE), while two come from later periods at Hazor (see fig. 3.9), dating to the ninth century BCE (Yadin 1960:6, 34) and Lachish, likely dating to the eighth century BCE (Sebbane 2004).
**Figure 3.9** _Senet_ game from Hazor (top), with the _game of twenty_ on the opposite side (bottom). Israel Museum, 1995-1112. Photograph by Marlana Salzberger. Courtesy of the Israel Antiquities Authority.
Three of these games were found on the opposite side of a _game of twenty_. Those from Kamid el-Loz were likely exchanged as greeting gifts (see Chapter 4 for _game of twenty_ ), since they were found in the treasury at the site. The games from Hazor and Lachish suggest that during the Iron Age ( _c._ 1200–550 BCE), non-elites may have played _senet_. The game from Hazor was found in the "Pillared Building," interpreted as a storehouse (Yadin 1960:6). That from Lachish was found in debris in the inner gatehouse of the city, probably as graffiti on steps (Sebbane 2004:690). The game from Hazor has no markings, but the Lachish game has an "X" in squares fifteen and twenty, as well as a depression in square thirty. Squares ten, eleven and thirty are dome-shaped, though they are not separate domed markings like those seen in New Kingdom Egyptian graffiti, thus suggesting either an evolution from this earlier type not yet attested in Late Period Egypt, or a misrepresentation or adaptation of the Egyptian pattern by Israelites attempting to reproduce this type of board.
### _Senet_ in Cyprus
The island of Cyprus has produced more _senet_ games than any other region, including Egypt (see fig. 3.10). The current count of games is nearly four hundred (Crist 2015), even though this artifact type was only first identified in 1976 (Swiny 1976, 1980, 1986; Buchholz 1981, 1982). These games appear with surprising regularity, as they have been found at nearly every Bronze Age site excavated on the island since the 1980s. They are also morphologically similar to one another because they are of the pecked type and almost always made of limestone, analogous to those found at Bâb edh-Dhrâ'. No _senet_ game from Cyprus has markings in any of the playing spaces, also because they were depressions and thus not easy to mark.
The identification of _senet_ on Cyprus is contingent on its appearance on the opposite side of a pattern of cupules in a spiral pattern, a Cypriot manifestation of _mehen_ (Crist et al. in press). The earliest _senet_ games appear on the island toward the end of the third millennium BCE at Sotira _Kaminoudhia_ and Marki _Alonia_ (Swiny et al. 2003:231–3; Frankel and Webb 2006:246; Crist 2015), later than the appearance of _mehen_ , which appeared at the beginning of the third millennium BCE. The timing of the arrival of these games parallels the popularity of the games in Egypt, where _mehen_ was more popular during the Early Dynastic Period, while _senet_ was more common during the late Old Kingdom.
**Figure 3.10** _Senet_ games from the Episkopi region, Cyprus. Photograph courtesy of Stuart Swiny.
It seems likely that the Cypriots learned the game from Levantines, rather than Egyptians, since there are no examples of games where the Cypriots were reproducing Egyptian boards with squares and markings. This likelihood is also in agreement with the material record of Prehistoric Bronze Age ( _c._ 2400–1700 BCE) Cyprus, where direct contact with the Levant is well documented (Philip et al. 2003; Webb et al. 2006; Knapp 2008:119–29, 2013:309), whereas contact with Egypt itself is less common (Merrillees 2009). Evidence for foreign contact is concentrated in sites on the north coast of the island, and indeed a _senet_ game was found on the surface at Bellapais _Vounous_ (Swiny 1986:35). Nevertheless, because games were only identified on Cyprus in 1976, the current political status of the island has prevented archaeological research in the northern part of the island, and so evidence for games at Bronze Age sites is concentrated in the Republic of Cyprus. The question of whether Levantines brought the game to Cyprus, or if Cypriots traveled to the Levant and brought the games back is an open question, and cannot be resolved with current evidence.
It is apparent that, once adopted on the island, _senet_ was incorporated into many aspects of Cypriot life as it has been found in domestic, mortuary, public and ceremonial spaces. In some cases, clusters of portable games were found associated with drinking and pouring vessels, which likely indicates that _senet_ playing accompanied drinking and/or feasting events (Crist 2015, in press). One _senet_ game was found pecked into the bedrock at the cemetery of Deneia _Kafkalla_ , though the extent to which it is connected to mortuary ritual is unclear (Herscher 1998:320; Frankel & Webb 2007:149). _Senet_ games have been found at cemeteries in the island, but two in particular stand out since they appear to have been placed intentionally in tombs. One, found in the burial chamber of a looted tomb at Marki _Kappara_ (Frankel & Webb 1996:86, 102) contained _senet_ patterns on both faces, but the game was obstructed by large pecked depressions on each face, rendering the game unplayable. It is possible that this was a form of ritual "killing," which has been seen in other artifacts on the island when placed in burials (Keswani 2004:75; Crist 2015).
The other _senet_ deliberately placed in a tomb was a unique terracotta votive object, claimed to have been found at Kotchati (Swiny 1986:33) but likely to have been found at Marki (Jennifer Webb 2012, personal communication), last known to be in the Hadjiprodomou Collection in Famagusta. It clearly shows three rows of well-made depressions, and it appears to be a rendering of a limestone game in terracotta, and was not likely to have been used for playing.
_Senet_ appears to have been popular at Cypriot sites through the Protohistoric Bronze Age ( _c._ 1700–1050 BCE), though seemingly less popular than during the Prehistoric Bronze Age (Crist 2015). Despite this, game boxes appear on the island for the first time during the Protohistoric Bronze Age, including the famous Enkomi game box (Murray et al. 1900:12). Piccione (1990b:430) incorrectly identifies this game box as double sided with the _game of twenty_ on the upper surface and _senet_ on the bottom. There currently is no _senet_ pattern on the box, though it was incompletely preserved and has been reconstructed. It is possible that there was once a _senet_ pattern on this artifact, but there is no evidence for it. Tombs at Morphou _Toumba tou Skourou_ and Kalavasos _Ayios Dhimitrios_ have produced square inlays that were likely squares on game boards (Vermeule & Wolsky 1990:221, 240, 332; South 1996:167), but these are likely to have been for the _game of twenty_ rather than _senet_ (Crist et al. in press).
There is little evidence that _senet_ was played in Cyprus after the Bronze Age. Artifacts similar to Bronze Age games have been found at Iron Age sites, particularly at Amathus (Fourrier 2003), but only surface finds and those that have been built into walls contain a distinctive _senet_ pattern. One game, found in the dromos of a Cypro-Geometric tomb ( _c._ 1050–750 BCE) at Kouklia _Skales_ , contains a _senet_ board delineated with etched squares, along with examples of the _game of twenty_ (Karageorghis 1983:122). Because this game is the only example of its kind, and dates to after the disappearance of the more typically Cypriot _senet_ pattern with depressions, it seems likely that this is a new, albeit brief, reintroduction of the game on the island. Since there is no apparent tradition of playing _senet_ on a field of squares in Bronze Age Cyprus (although, it must always be kept in mind that games made of wood or other perishable materials may have once existed), this new rendering may be evidence for Iron Age Cypriots imitating contemporary Egyptian _senet_ , rather than Cypriots playing the traditional local version of the game. This would further suggest that the societal changes that brought about the end of the Bronze Age affected the ludic sphere in Cyprus, and a game that had been played for roughly two thousand years in Cyprus became forgotten.
### The _game of thirty-three_
During the Late Period, another game appeared in Egypt that seems to have achieved a certain level of popularity (see fig. 3.11). Typically found on the opposite side of _senet_ boards, this game contains three rows of eleven circles, and, since its ancient name is unknown, it is most often referred to as the _game of thirty-three circles_ , or simply the _game of thirty-three_ (Pusch 1979:377; Piccione 1990b:441–8).
Nine examples of this game are known, and all but one of them appear on the opposite side of _senet_ boards. Only three of these boards have known provenance. Two are from the Sacred Animal Necropolis at Saqqara (Martin 1981:45, 55) and one was found near the pyramid of Senusret I at Lisht, but dated to the Nineteenth Dynasty (Pusch 1979:365–7). The two from Saqqara are firmly dated to the Late Period, which corresponds to Petrie's dating of four unprovenanced boards to that era (Petrie 1927:53; Piccione 1990b:446–8). Two boards are dated to the Nineteenth or Twentieth Dynasty by Piccione and Pusch. One is currently in the British Museum (102396) (Pusch 1979:309; Piccione 1990b:427) and the other is now in the Yale University Art Gallery (1937.161) ( Pusch 1979:311; Piccione 1990b:429). The reasoning for the early dating of these boards is unclear, and, due to the lack of documentation on their archaeological provenance, it seems more likely that they date to the Late Period, based on the presence of the _game of thirty-three_ on one face.
**Figure 3.11** _Senet_ game of Late Period date (top) with the _game of thirty-three_ on the opposite face (bottom). 25.4 × 8.7 × 0.7 cm. © Petrie Museum of Egyptian Archaeology, University College London, UC2317.
The manner in which the game is rendered on the board's surface is fairly standardized across the corpus. Four of the games have the gaming spaces depicted with three rows of eleven circles inside squares, while three have just circles in the same arrangement. Sometimes the circles have a depression in the center, which led Petrie (1927:55) to call it " _the Game of Thirty-Three Holes_." Four of the games have spaces marked with rosettes, but perhaps others contained markings as well since they are incompletely preserved. The most commonly marked space is the sixth space of the central row, and a rosette appears in this position on four of the marked examples. The Yale game is the only _game of thirty-three_ board that has other marked spaces: it contains rosettes in spaces three and nine of the central row in addition to that in the sixth space.
Rosettes are an unusual motif to appear on Egyptian board games, as only the _game of twenty_ is marked in this way occasionally in Egypt. Rosettes are commonly used to mark the _game of twenty_ , and examples from the Levant, Cyprus and Mesopotamia contain this form of decoration, though none of the boards found in Egypt contain it. Based on the presence of the motif (which also features prominently in the bands dividing the three rows of spaces in the British Museum game), it is possible that the game has its origin in Western Asia, perhaps Mesopotamia or Persia. Though no examples of this game have been found elsewhere, it is possible that they have not survived, and only the Egyptian games remain due to the preservation afforded by the climate in the Nile Valley. The Late Period, during which the _game of thirty-three_ first appeared, was a period during which the Assyrians and Persians conquered Egypt, and these foreigners could have brought the game to Egypt. It is important to note that the rosettes are particular to the _game of thirty-three_ as _senet_ games on the opposite face of these boards contain hieroglyphs in the marked spaces. The rosettes, therefore, do not appear to be a stylistic choice adopted for use in games more generally.
Whether the game was an Egyptian invention or a foreign introduction cannot be determined at this point in time, but its presence in Egypt appears to have been confined to the Late Period, with no boards known from the Ptolemaic period or later, though of course the unprovenanced status of much of the corpus makes chronological inferences difficult. Nevertheless, the _game of thirty-three_ appears to have been the latest board game introduced into Pharaonic Egypt before the Greco-Roman period.
##
## The _Game of Twenty_ : A Foreign Acquisition
**Figure 4.1** Distribution of the _game of twenty_ in Egypt and the Near East from the mid-third to the first millennium BCE.
The _game of twenty_ , named after the number of playing squares it contains, was one of the most popular games in the ancient Near East. From the mid-third to the mid-first millennium BCE, the _game of twenty_ was distributed in far-flung regions, from Iran to the Levant (Finkel 2007:17; de Voogt et al. 2013) (fig. 4.1). The game appears in the Egyptian archaeological record for a shorter period, from the Seventeenth to the Nineteenth Dynasty, where it was combined with _senet_ on reversible boxes. Egyptian carpenters were already accustomed to manufacturing game boxes with drawers to keep the playing pieces and dice for the game of _hounds and jackals_ (see Chapter 5). We may attribute to Egypt the invention of the bifacial game box with a drawer for the shared accessories but this innovation could also have been elaborated in the Levant where both _senet_ and the _game of twenty_ were known by the early second millennium BCE. The Egyptian climate may explain the large number of better-preserved examples from the game box category.
### Origins and chronological distribution
The game is also known as the _royal game of Ur_ , after the famous boards from Sumer in southern Mesopotamia (modern Iraq) (fig. 4.2). Several boards made of wood, inlaid with shell, red limestone and lapis lazuli, were discovered in the 1920s by Leonard Woolley at Ur in the Royal Cemetery (Woolley 1934:274–9, pls. 95–8; Becker 2007). They were found with sets of seven round gaming pieces as well as two kinds of dice: four-sided sticks and tetrahedrons. The Sumerian board, dating to about 2600–2400 BCE, consists of a rectangle of twelve squares (4 × 3) joined by two squares to a smaller rectangle of six squares (2 × 3). It has been claimed that the game originated in the Indus Valley on the basis of finds from sites that flourished during the Harappan civilization (2600–1900 BCE). Possible fragments of the _game of twenty_ were discovered at Mohenjo-daro (Mackay 1938:575-6, pl. CXLII.82) and Lothal (Rao 1985:504, pl. CCXIX.1) whereas a characteristic block of 3 × 4 squares, with one additional square from the broken middle row, is recorded at Dholavira (Soni & Bagchi 2011:75–6; Bisht 2015:594–6, figs. 8.308–11). The possible Indian origin of the _game of twenty_ is often accompanied by its comparison to _pachisi_ , a traditional race game played in India (Parlett 1999:65). It would be rash to rely on such evidence because the origins of _pachisi_ remain uncertain. Its earliest testimony in India dates to the sixteenth century CE (Finkel 2004:47).
The emergence of the _game of twenty_ in widely separated areas attests to the cultural contacts across the vast region that stretches between the Euphrates and Indus rivers during the mid-third millennium BCE. The Indus cities were involved in the long-distance trade of semi-precious stones such as lapis lazuli and carnelian (Aruz 2003:243). Sites like Shahr-i Sokhta, in the eastern Iranian region called Seistan, were part of this commercial network (Tosi & Lamberg-Karlovsky 2003:349). A wooden board with the twenty fields outlined by the coils of a snake, carved in relief, was found by the Italian expedition at that site, again in a funerary context (Piperno & Salvatori 1983).
**Figure 4.2** The _royal game of Ur_ with gaming pieces and tetrahedrons, board: 30.1 × 11 × 2.4 cm. The British Museum, 120834, 1928,1009.379a-n. © The Trustees of the British Museum.
At the turn of the third to the second millennium BCE, the playing surface underwent a slight change. It unfolded the rightmost squares of the _royal game of Ur_ into a straight tail (fig. 4.3). The middle row extending a further eight squares lead to the appellation "head-and-tail" type (Dunn-Vaturi & Schädler 2009). The reduction of the number of playing pieces from seven to five is attributed to the same period. Special squares can be marked every fourth square. The new arrangement of the track was first observed at two sites, Tepe Yahya and Jiroft, in the Kerman region in southern Iran (Finkel 2004:95; Dunn-Vaturi & Schädler 2006). At Jiroft, illicit "excavations" in the early 2000s brought to light several stone gaming boards in the shape of birds of prey, scorpions and fantastic creatures (Dunn-Vaturi & Schädler 2006:4–6, pls. 1–3). These figures are characteristic of the so-called Intercultural Style visible on carved chlorite vessels and handle-weights. Kerman, Tepe Yahya in particular, is an important production center of these objects distributed from Central Asia to Syria (Aruz 2003:325). The active network of trading routes facilitated the development of the evolved form of the game elsewhere in the early second millennium BCE, especially in Mesopotamia. Several games scratched on pavement bricks attest that the game was played at the time of the Amorite king Zimri-Lim ( _c._ 1775 BCE) in the palace of Mari, an important trading center on the Euphrates banks (Parrot 1958:12–13, 47, 182, 247, 275; Sauvage 1991). The Mariote grid usually bears three special squares marked with "X", i.e., squares eight, twelve and sixteen. The game, probably known a few centuries earlier as a graffito, was found on a brick inscribed with the name of Ilum-ishar, Shakkanakku of Mari, _c._ 2064 BCE (Dunn-Vaturi 2012c), unless the game was added later.
**Figure 4.3** The _game of twenty_ in the second and first millennia BCE, and the route of play.
The spread of the Amorite culture throughout the Levant from 1900 to 1700 BCE resulted in the foundation of Amorite dynasties (Burke 2014:405). Mariote texts at the beginning of the Middle Bronze Age II ( _c._ 1750 BCE) mention long-distance trading caravans traveling to the Eastern Mediterranean regions (Burke 2014:407). Sites in the southern Levant—Tell el-Ajjul, Tell Beit Mirsim—have yielded examples of the _game of twenty_ from contexts dating to the Middle Bronze Age II and III ( _c._ 1750–1640/1540 BCE) (Petrie 1933:Pl. 28, 25–9; Albright 1938:49, pl. 37, a). From there, the game would soon be transmitted to Egypt.
### Beni Hasan playing scenes
Some scholars think that the _game of twenty_ may have been introduced in Egypt as early as the Middle Kingdom and is depicted at Beni Hasan (Jéquier 1922:18; Kendall 1982:265). Their theory is based on the association of two playing scenes painted side by side in the tomb of the nomarch Baqet III (Tomb 15), and appearing in reverse order in the tomb of his son Khety (Tomb 17) (Newberry 1893:Pls. 7, 13; Needler 1953:64–5) (fig. 4.4). The tombs of Baqet III and Khety, generally dated to the Twelfth Dynasty, have been re-dated to the Eleventh Dynasty on the basis of textual, archaeological and artistic evidence (Spanel 2001:176). The two scenes, indicated as A and B, show two players engaged in games occurring on tables. Scene A has two sets of seven alternating opposing gaming pieces and scene B has five opposing gaming pieces arranged in two camps. The captions of scene B, read _ỉsb_ or _ỉ sb_, are unknown in other game contexts, whereas the legend _ḥ b 5_, "playing five" above scene A is attested in Old Kingdom scenes related to _senet_ (Jéquier 1922:18; Kendall 1982:265). Jéquier suggests that scene B is the _game of twenty_ because it is associated with _senet_ just as these games are found together on double-sided games in later times. The paintings in the tombs of Baqet III and Khety could prefigure the later connection between the two games. However, the inscription _ỉsb_ was tentatively translated "consumed"—implying that the game was finished—by Birch (1865:62) so that some scholars are inclined towards interpreting the scenes as two stages of _senet_ (Nash 1902:347; Petrie 1927:51). Hoerth (1961:4, 85) argues that the scenes describe a second form of _senet_ because the pieces do not fit the reconstruction of the _game of twenty_ whereas Kendall (1982:265) in contrast states they correspond "precisely to what one would expect in a view of 'twenty squares.' "
**Figure 4.4** Playing scenes A, B and C in the tombs of Baqet III and Khety, Beni Hasan. Drawing reproduced from Newberry (1893:Pls. 7, 13).
If the _game of twenty_ is described at Beni Hasan, it means that the game was known much earlier than the Hyksos period, which is the date generally considered for the introduction of the game in Egypt (Finkel 2008:152). One possible source for the transmission of the game is immigrants coming to settle in Egypt, such as foreigners employed in the Egyptian army or in the mines. Long-distance caravans of Western Asiatic peoples, identified as a group of migrant workers, are depicted carrying foreign objects at Beni Hasan in the tomb of Khnumhotep II (Tomb 3) dated to the Twelfth Dynasty (Kamrin 1999:94–5). The _game of twenty_ could have been brought to Egypt on such an occasion. The grid may have been on perishable material (such as textile) or simply reproduced on the ground, which would explain the absence of evidence for three hundred years, until it became standard in New Kingdom burial equipment. Although this interpretation is conceivable, there is to date no archaeological or physical evidence for the _game of twenty_ in Egypt during the Middle Kingdom. Nash (1902:343) reports the presence in the British Museum of a wooden board "marked out with 20 squares only" from a Twelfth Dynasty tomb at Deir el-Bersha. This information is problematic since this board could not be located at the time Pusch worked on the British Museum boards (Pusch 2007:70, note 19).
### Ancient names
The ancient name of the _game of twenty_ is still a subject for debate. In the nineteenth century, Egyptologists thought it to be called the _game of tjau,_ which translates as " _game of the robbers,_ " because the name _ṯ w_ is associated with playing scenes in the Late Period tomb of Aba at Thebes (Pusch 2007:69). This incorrect identification is often repeated although Pusch (2007:84) suggested that the term _ṯ w_, previously translated "robbers," meant in fact "marbles" and described a game "combining dexterity and guessing" (Decker 1992:133).
The game may have been known as _ỉsb_ or _ỉ sb_, if scene B in Beni Hasan proves to represent the _game of twenty_. The difference of spelling between the two captions in Tomb 15 and Tomb 17 is not meaningful as the interchange of _ỉ_ and _ _is common, particularly at the beginning of words. They may be written with either of the two, or a combination of both, and has no further significance. The indefinite term _ỉsb_ has been compared to the Babylonian name, _patti apsu_ ("canal of the deep"), listed among the gifts sent by Tushratta, King of Mitanni ( _c._ 1365–1330 BCE), to Amenhotep III on the occasion of his marriage to a Mitanian princess (Kendall 1982:265). The objects described in the Amarna letter EA 22 (Column II, line 54) have been interpreted as games because astragali are associated but some authors suggest that, in fact, two bows with astragal-ornaments were listed (Cochavi-Rainey 1999:65). Finally the root has been linked to the Egyptian word _ỉsb_. _t_ "beam/throne," also used with the form _ỉsb_ , as it could refer to the block of wood used to carve the game (Jéquier 1922:19, 219).
On the Turin Papyrus, described below, it is inscribed: "Uniting of the twenty squares" (see fig. 4.7) above the representations of the game of _thirty-one_ , which is a combination of two _game of twenty_ tracks. This inscription suggests that this game was simply known as " _twenty squares_ " (Pusch 1977:209–11).
### Boards for the _game of twenty_
Due to its Near Eastern origins, the _game of twenty_ is generally believed to have been introduced to Egypt by the Hyksos, Asiatics who ruled the eastern Nile Delta in the Second Intermediate Period, because it is first attested archaeologically during this period (Pusch 2007:70; Finkel 2008:152). But it should be noted that to date no evidence of that game is reported from Hyksos levels at sites in the Delta.
Excavations in Egypt have revealed a great number of _games of twenty_ thanks to the climate in the Nile Valley. The reversible game box is the most represented type in Egypt with at least thirty-two examples, dating from the Seventeenth to the Nineteenth Dynasty. A wooden box with the _game of twenty_ on its top surface, said to be from Akhmim, was purchased by Breasted in the 1890s for the Oriental Institute Museum in Chicago (E 371A) (Hoerth 1961:33). It could be a precursor because it has no _senet_ on its reverse. Some games are too fragmentary to be assigned to a specific category. Three ostraca with a _game of twenty_ have been identified so far and several ostraca with _senet_ are known too (see Chapter 3) but no double-sided ostraca are recorded. Boxes are typically made of solid wood, solid faience or wood with ivory, bone or faience inlays. They are usually provided with a drawer locked by means of a bolt sliding in three staples. In most of the cases, the _game of twenty_ is on the uppermost side, with _senet_ on the reverse, according to the orientation of the hieroglyphs or scenes on side panels. The playing surface consists of twenty rectangular (rather than square) spaces because the twelve fields of the middle row have to fit within the same length as the ten squares fields of the _senet_ game on the opposite side (Pusch 2007:70, note 10).
The earliest examples of the _game of twenty_ come from Thebes and are dated to the late Second Intermediate Period, when the Theban rulers of the Seventeenth Dynasty began to drive the Hyksos kings from the Delta. The oldest excavated example is a wooden and ivory (or bone according to Tiradritti 2010:338) box found in the tomb of Hornakht at Dra Abu el-Naga North and now in the Egyptian Museum (JE 21462, CG 68005). The tomb was excavated by Luigi Vassalli while working for Auguste Mariette, during the 1862–3 season on the West Bank of Thebes. The name Hornakht inscribed on the coffin was initially misread Aqhor (Tiradritti 2010:336, note 44). Mariette (1889:17, pl. LI) described the box as a double board with twenty squares and thirty-six squares but only illustrated one surface with thirty-six squares, according to the Italian excavator's sketch and notes ("schizzo della tavola da gioco con le case numerate da 1 a 12") (Tiradritti 2010:Pl. 117). Therefore this board has been repeatedly misinterpreted as a single playing surface with thirty-six squares including a track for the _game of twenty_ (Murray 1951:17, fig. 6; Hoerth 1961:23; Parlett 1999:68, fig. 4.7).
The reconstructed box (26 × 7.6 × 4.7 cm) has, in fact, a _senet_ on the verso and Pusch suggested a reconstruction for the _game of twenty_ with eight panels along the middle row (Pusch 1979:197, pl. 45; Decker & Herb 1994:665–6; Rothöhler 1996:102–3). The object has been assigned to the Sixteenth Dynasty (Allen 2002) but the tomb is generally dated to the reign of Taa, ruler of the Seventeenth Dynasty, after a cartouche carved on a throwing club. Ivory carved reliefs are divided into three panels on the long side. They show a wild goat, a pair originally, eating from a tree in the central panel framed by a pair of Egyptian couchant sphinxes. Only the sphinx on the left panel is preserved. It is a typical figure of the International Style, a set of artistic conventions common throughout the Near East and Aegean worlds (Caubet 1998:109). In its protective appearance, the sphinx is here associated with symbols of life and renewal. This is the earliest _game of twenty_ attested in Egypt so it is not surprising to see the influence of the Near East in the decoration of the imported game. The motif of a pair of ibex or goats flanking a tree belongs to a long Near Eastern tradition (Ornan 2005:155). Known since the third millennium BCE in Mesopotamia, the motif spread in the Levant during the second millennium BCE. It is attested on two other ivory game boards from the Levant (Megiddo) and Cyprus (Enkomi) dating to the thirteenth and twelfth century BCE. The Hornakht board shows the adoption of the Horus hieroglyph to mark the special squares usually indicated by "X" on the foreign playing surfaces. Horus falcons were incorporated like other Egyptian motifs into the Syro-Levantine glyptic repertoire during the Middle Bronze Age (Teissier 1996:90). Different orderings of the three squares marked with Horus have been proposed (Mariette 1889:Pl. LI; Falkener 1892:97; Wiedemann 1897:40). Mariette's placing of birds in the eighth and twelfth squares conforms better with the majority of the other known boards (Hoerth 1961:23–4). According to Mariette, the game box was accompanied by seven playing pieces, spool-shaped and conical, made of different material (Mariette 1889:17, pl. LI), but Vassalli referred to only one conical "pawn" in the drawer (Tiradritti 1994:70). Finally, Falkener is the only one to mention an oblong die with sides numbered one to four preserved with this game, but his description cannot be verified (Falkener 1892:97–8). Hornakht's burial held several objects inscribed with prominent individuals' names. He must have played a key role at the Theban court (Tiradritti 2010:340) and this could explain the presence of the game box, a newly introduced element in elite burials.
**Figure 4.5** _Game of twenty_ from Thebes with _senet_ on the opposite side, board: 25 × 6.7 × 5 cm. The Metropolitan Museum of Art, Rogers Fund, 1916, 16.10.475. Image © The Metropolitan Museum of Art.
The next datable board was discovered during the Metropolitan Museum of Art expedition in 1915–16 in a late Seventeenth–early Eighteenth Dynasty burial at el-Asasif in western Thebes (Lansing 1917:26; Finkel 2008:152–3) (fig. 4.5). It is a double-sided game box restored in modern wood overlaid with ivory squares and panels acquired by the Metropolitan Museum of Art in the division of finds (16.10.475). None of the squares are marked but the panels flanking the middle row are decorated with animal scenes. One panel shows a lion facing two gazelles, while on the other panel is a lion facing a hound. The animals are rendered in a manner called the "flying gallop," characteristic of the International Style. This object is not only remarkable for its decorations but also for its extensive assemblage in ivory, comprising six conical pieces, six spool-shaped pieces and two knucklebones. The six wands from Pit 3 associated with this set (Hayes 1959:25–6) were not found with it in chamber E but in chamber B (Christine Lilyquist, personal communication, June 4, 2015) (see Lilyquist forthcoming).
Ivory panels with pastoral scenes depicted in the International Style were similarly placed on either side of the tail of a _game of twenty_ whose original wood structure did not survive (Egyptian Museum, JE 40680, CG 68183). They depict bulls, hunting lions, feeding antelopes, brush, bushes and rosettes (Piccione 1990b:398). Parallels in the Levant and Cyprus are discussed below.
The bulk of the game boxes in the archaeological record date to the New Kingdom (Eighteenth to Nineteenth Dynasty), with a peak during the Eighteenth Dynasty. The majority was excavated in Upper Egypt (fifteen from the Theban region, two from Abydos, one from Qau) while only three examples were found in Lower Egypt (Zawiyet el-Aryan, Saqqara and el-Qantir). Fifteen examples, now in museum collections and often said to be from the Theban region, are missing a precise provenance. The southernmost example of a box with the _game of twenty_ and _senet_ on opposite sides was found in the cemetery of Kubban in Nubia (Firth 1927:49, 83) (Nubia Museum Aswan, 664). It is dated between the sixteenth and thirteenth centuries BCE when this region was part of Egypt. The structure of the game boxes remained the same throughout the period and the changes mainly affected the special squares and/or the decoration outside the playing surface.
### Special squares and decorations
Special squares, which are placed every fourth square, are attested on fifteen boards and three boards have their final square made in a distinct shape (table 4.1). Marks are found in all five positions (four a, four b, eight, twelve and sixteen) or only in three (eight, twelve and sixteen) (Pusch 2007:71). Sixteen boards are unmarked and eight examples are either too fragmentary or not adequately described to determine whether they had plain squares or not. They may originally have been emphasized, but motifs have disappeared; others were probably left blank because the squares did not always need to be differentiated. In the Near East, special squares are either marked with a rosette or "X", whereas in Egypt more variants are attested. The Egyptian labels have been assigned to three groups that include "X" and geometric symbols (Group I), sacred symbols (Group II), titles and owner's names (Group III) (Pusch 2007:71–3, fig. 8.3).
The use of "X", widespread in the Near East, is attested on two Egyptian boards. One was found at Abydos in Tomb D99 of the scribe Merymaat dated to the reign of Thutmose III by a scarab bearing a cartouche of this pharaoh (Metropolitan Museum of Art, 01.4.1). The board was previously restored with three crossed lines in positions eight, twelve and sixteen (Pusch 2007:Fig. 8.4). Only one complete faience square with a painted "X" was retrieved from the tomb of the scribe Merymaat. Therefore the new reconstruction of the wooden box displays the unique crossed square at the twelfth square while the other special squares were left plain (Metropolitan Museum of Art 2015). According to Hoerth (1961:29): "It is uncertain where, or even whether, this crossed square fits into the twenty-square surface." This symbol could belong to the _senet_ track since it is also used to mark the _senet_ squares fifteen and twenty-seven on a board from Abydos dating to the Twelfth Dynasty (Piccione 1990b:390). On square fifteen it is a boon, according to the Great Game Text of the New Kingdom (Piccione 1990b:391), whereas on square twenty-seven, it designates a hazard or a pitfall (Kendall 1982:264; Piccione 1990b:244). The other board with an "X" on squares eight, twelve and sixteen and not on every special square as described by Pusch (2007:72–3), has an unknown provenance (Ägyptisches Museum, Bonn, 941).
Other geometric patterns indicated special squares on boards dating to the Eighteenth Dynasty. Every fourth square is filled with a slightly smaller and darker rectangle on the Hatshepsut board at the Louvre (E 913) (May 1992:145, fig. 137b). A checker pattern is preserved on three squares on a Theban board now in the Egyptian Museum (JE 65372) (Hayes 1935:33, fig. 18) as well as on square twelve on the fragmentary faience board from Tomb 499 A'08 at Abydos (World Museum, Liverpool, 55.82.9). Square sixteen, the final square, on game boards 393 and 585 from the tomb of Tutankhamun is precisely square whereas the other fields are, as usual, slightly oblong (Tait 1982:18). Apart from its shape, Tait did not describe any difference in the appearance of this square, so Hoerth's comment about the square of board 585 being darker might just be due to the shade on the photograph (Hoerth 1961:30). The final square on a limestone board dated to the Nineteenth Dynasty has a semi-circle (Pusch 2007:Fig. 8.4) (Egyptian Museum, CG 68007).
The oldest example of the _game of twenty_ in Egypt, found in the tomb of Hornakht and already discussed above, illustrates the adoption of the Horus falcon, an Egyptian sacred symbol, to mark at least three positions. During the New Kingdom, other symbols are attested such as _ nḫ, " _life_ ," nfr_, "good," double _nfr_ and _wḏ t_. Some of these signs were also used for _senet_ as well as the game of _hounds and jackals_ to mark special positions. Board 345 (JE 62058) from the tomb of Tutankhamun has sacred symbols incised in three positions that could be read as a short formula: _ḥḥ ḥbw-sd nḫ-ḏdw s,_ "Millions of _sed_ -festivals, life, eternity and well-being." (Pusch 2007:72) (see table 4.1). Five boards from the Eighteenth and Nineteenth Dynasties have the titles and names of the board's owner marked, broken into five groups corresponding to the total of special squares (Pusch 2007:72–3, fig. 8.4).
Markings were painted, and when incised they could be filled with pigment or stone inlays. The special squares of the game 593 from the tomb of Tutankhamun were, according to Carter's records, "faintly marked upon the ivory in white paint" so Tait (1982:25) suggested "that a pigment contained in this paint has faded, or possibly that a gold leaf laid over it has entirely disappeared."
During the New Kingdom, boards bore decorations outside the playing surface that differ from the pastoral scenes seen on the earliest boards. Decorations and offering formulae were along the tail, on each panel bordering the central row, as well as on the long and short sides of the box. One theme was "tied to the meaning of the _senet_ game as offering ritual" (Piccione 1990b:406). _Senet_ -playing scenes are accompanied with food offerings in vignettes of Chapter Seventeen of the Book of the Dead. This episode is often represented on the tomb walls from the Eighteenth and Nineteenth Dynasties. Boxes are adorned with offering formulae as well as acts of gift scenes (food, lotus and wine). Two wooden boxes with unknown provenance, now in Berlin and New York, show a banquet scene on the end opposite the drawer. The owner of the box is seated in front of a table with offerings. On the box from the Ägyptisches Museum und Papyrussammlung in Berlin (10756), Sennefer is accompanied by his wife and they both sniff lotus blossoms (Piccione 1990b:406–7). The box from the Metropolitan Museum of Art (12.182.72) shows Taya and his wife seated opposite her mother who is sniffing a lotus (Piccione 1990b:418–19). Both boxes are dated to the Eighteenth Dynasty. The Berlin box is dated to the reign of Thutmose III or Amenhotep II after the style of the incised scene (Kendall 1982:267) whereas the Metropolitan Museum of Art box is dated to the reign of Thutmose IV or Amenhotep III. The back end of box 585 from the tomb of Tutankhamun shows "the King seated upon a throne with the Queen standing before him offering a lotus flower" (Tait 1982:18, note 4, pls. IX, XXIV). On a box dated to the Nineteenth Dynasty a playing scene is depicted (Dunn-Vaturi 2012d) (Musée du Louvre, E 2710). The deceased Imenmes is probably playing with an invisible opponent as the person, standing on the other side of the table and offering him a drinking cup, does not seem to engage in the game (see cover image and discussion in Chapter 3). The conic pawns are differentiated from the reel-shaped ones and two knucklebones are also represented.
The back end of game boxes can hold apotropaic motifs such as Bes on the faience box inscribed with the name of Hatshepsut (Deveria 1897:88). The image is not clear so it was identified as a _sema tawy_ , a stylized representation of the windpipe flanked by lungs and tied with the lotus and the papyrus, representing the unification of Upper and Lower Egypt (Pusch 1979:207). In fact, the figure of Bes holding snakes has been confirmed by the Louvre curators (see object file E 913). The decoration of the small side on the board of Ptahmay, "overseer of the craftsmen" during the reign of Amenhotep IV, is described as funerary with _nfrw_ centered between two _wḏ t_ eyes, all surmounting _nbw_ , "gold" (Sammlung Nassau Altertümer, Wiesbaden, 2308). A fragment of a limestone board from Deir el-Medina shows, on one of the long sides, a kneeling man probably with his hands in front of his mouth as a blessing gesture, but most of the object is missing so the scene cannot be fully interpreted (Pusch 1979:297–8, pl. 98). Other boxes have repeated motifs on the long sides. The ivory plaques of box 593 from the tomb of Tutankhamun are carved and stained with a floral pattern (Tait 1982:20, pls. X, XI).
Hieroglyphic offering formulae are attested on about ten game boxes in the New Kingdom whereas the previous examples did not have traces of such inscription. The board from the tomb of Hornakht may have been inscribed but the inscription was not preserved. Maspero (1871:78) wrote that this object bore the name of "Tûaû"/Thuyu (Winlock 1924:258, note 3) so it is repeated by Piccione (1990b:395) but no indication of the board being inscribed with its owner's name appears in the archaeological report (Mariette 1889:17). Plain panels of game boxes may have originally held painted inscriptions or motifs.
### Archaeological contexts
_Game of twenty_ boards were mostly found in funerary contexts. Smith's studies of Theban funerary assemblages point to the presence of double-sided game boxes in middle- to high-status tombs (Smith 1992:204, 218–19). The tomb of Tutankhamun housed at least four double-sided game boards, including one prestigious ebony and ivory example on a sledge-stand identical to those described in the tomb of Sennefer and Rekhmire (Piccione 1990b:412) as well as "pocket-size" editions (Tait 1982; Reeves 1995:160–2) (Egyptian Museum, JE 62058-62061).
The range of appearances, from ornate boxes made of exotic material to graffiti on slabs, show that the game was popular with Egyptians of different social classes. To date only one example in a domestic context is known. It is a slab of limestone probably hastily engraved with the game by an inhabitant of the village of Deir el-Medina, home to the craftsmen who decorated the tombs in the nearby Valley of the Kings (Dunn-Vaturi 2012a). Other ostraca were found at Deir el-Medina but they are fragmentary and no information about their findspot is known to the authors. One ostracon bears the _game of twenty_ painted in red, its eighth square marked with double _nfr_ (Vandier D'Abbadie 1959:231, pl. CLIX) (fig. 4.6). Originally described as an architectural plan, the configuration of the game can be recognized despite its fragmentary state of preservation. It is now in the collection of the Institut Français d'Archéologie Orientale, Cairo (inv. 4116). Another grid for the _game of twenty_ (of which only the last seven squares of the "tail" remain) is visible on a delicately painted ostracon depicting a nude dancer with a long flute that is stylistically dated to the Nineteenth Dynasty. It was found by the German expedition in 1913 and is now in the Ägyptisches Museum und Papyrussammlung in Berlin (21445) (Brunner-Traut 1956:63–4, no. 59, pl. XXIII).
The faience _game of twenty_ from the private apartments in the palace of Rameses II at el-Qantir was part of a vast program of glazed decoration (Hayes 1959:338; Pusch 1979:303) (Metropolitan Museum of Art, 35.1.140). The board, which is very fragmentary—only one complete and three broken squares are preserved—may have been for a game table. It is inscribed with the name of Rameses II on the side and, on the top surface, on a panel bordering the middle row.
**Figure 4.6** Ostracon from Deir el-Medina. Institut Français d'Archéologie Orientale, Cairo, inv. 4116.
### Rules for the _game of twenty_
Unlike the descriptions of _senet_ there is not much information as to the rules of the _game of twenty_ in Egyptian sources. It is generally assumed that the two players started on each of the opposite sides of the board (Kendall 1982:265; Finkel 2007:27) (see fig. 4.3). Then they moved their pieces down the central aisle until the final field and then off the board to win the game. The new version, the "head-and-tail" type, with a single path all the way to the end increased the difficulty of the game as a player about to win could be overtaken at the last minute (Finkel 2007:18). Special squares, signaled by a rosette, another symbol or inscriptions, are marked every fourth square. These squares may have functioned as lucky fields, so that a playing piece was safe from being captured or that the player was given another throw.
A late Babylonian cuneiform tablet dating to 177–176 BCE provides significant detail on how the game proceeds (Finkel 2007) (British Museum, 33333B). The game is called "a pack of dogs," whereas the pieces, five on each side, bear the names of birds. They are moved according to the throw of two astragali, one belonging to a sheep and another from an ox, and special results are needed to enter each of the bird-counters into the game. The obverse of the tablet is inscribed with a zodiacal divinatory diagram, and one would be tempted to interpret the five birds of the game and its central row of twelve squares in an astrological context. The foretelling aspect of the _game of twenty_ is corroborated by Late Bronze Age liver-shaped boards from the Levant, which refers to hepatoscopy or divination by the liver of an animal or bird (Meyer 1982; Finkel 2007:25–6). Finally, a link has been proposed between the first Book of the prophet Nahum ( _c._ 615 BCE), and the _game of twenty_ (Guillaume 2009). Guillaume's concordance of the Psalm verses of Nahum 1, interpreted as a set of rules, to the squares of the board, is based on alphabetic acrostics. The use of sacred texts for biblical divination supports the possible connection established between the writings of Nahum and the _game of twenty_ , but with the current state of evidence only speculations about the practice of this particular prophetic scripture remain.
### "Uniting of the twenty squares" or _thirty-one_
Most authors refer to the existence of the _game of twenty_ in Egypt up to the Twentieth Dynasty because they consider that the game was replaced by a variant of the _game of twenty_ with "folded symmetry" (Decker 1992:131). Two blocks of 3 × 4 squares are linked by a bridge of seven squares. This new version, with thirty-one squares and every fourth square marked by a rosette, is known to us in Egypt through three examples dating to the Twentieth and the Twenty-First Dynasties: one drawing annotated with the inscription "Uniting of the twenty squares" upon the Turin Papyrus (fig. 4.7) and two boards with _senet_ on the opposite side (Pusch 1977:199–212) (Egyptian Museum, JE 88006; British Museum, EA 38429). A similar layout was seen by E.J. Banks on a baked clay board in a shop in Baghdad in 1904–15 (Banks 1912:355–6). Thus, it has been suggested that this game with rosettes, a typical motif for special squares on Near Eastern game boards, could have a Babylonian origin (Kendall 1982:265). Unfortunately, we lack information about the provenance and date of the Iraqi board to relate it to the Egyptian examples.
**Figure 4.7** Drawing of the Turin Papyrus with four games: Two times _senet, hounds and jackals_ and _thirty-one_ , _c._ 70 × 13 cm. Museo Egizio di Torino, 1775. Drawing after Decker and Herb (1994:pl. CCCLXXXV).
The track may have been modified in the late New Kingdom to include thirty-one squares for a religious meaning (Pusch 2007:70). The Great Game Text refers to the deceased's desire to be admitted into the divine pantheon as the thirty-first god: "[that they might permit] me to enter the Council Chamber of the Thirty, [so that I may become a god, as the thirty-first]" (Piccione 1990b:123). For this wish to be fulfilled, one may have had to win this new version of the game.
### The Levant and Cyprus: Games as heirlooms
Among the great variety of game boards that existed in the Levant during the Bronze Age and the Iron Age (Macalister 1912:299, pl. CCI), the _game of twenty_ is the best represented. A tradition of game boards with ivory or bone inlays existed in the Levant from the Middle Bronze Age until the Iron Age. Despite the fragmentary state of preservation of the earliest examples, dating to the Middle Bronze Age II and III ( _c._ 1750–1640/1540 BCE), it is more likely that they were for the _game of twenty_ and that they may have inspired the Egyptian production. An ivory fragment of 3 × 4 squares, which could correspond to the block of twelve squares, was found with two kinds of playing pieces—one with rounded tops, the other with pointed tops—and a teetotum die at Tell el-Ajjul in Tomb 364 dated to the Fifteenth Dynasty (Petrie 1933:Pl. 28, 25–9). At Tell Beit Mirsim, twenty ivory squares originally from a wooden game board were retrieved in the Palace Stratum D, dated by the excavator to the Middle Bronze Age III, late seventeenth to early sixteenth century BCE (Albright 1938:49, pl. 37a). Among the square inlays was only one field marked with "X". Albright estimated that the board measured at least 26 cm long. Thin ivory strips were inserted between the rows of squares and wider inlays may have been set on both sides of the middle row of squares. Ten faience playing pieces—five cones and five three-cornered pyramids—and an ivory teetotum were excavated in the adjacent room (Albright 1938:48, pl. 21, b).
The double-sided ivory game box from Thebes decorated in the International Style finds parallels in the Levant and Cyprus in the Late Bronze Age (Finkel 2008:152–3). Such artifacts were important components of the social practices and exchanges taking place in the Eastern Mediterranean in the second millennium BCE. Moreover, game boxes (e.g., Kamid el-Loz, Enkomi) and containers (e.g., Hazor, Morphou _Toumba tou Skourou_ ) with incised inlays, produced sometime in the sixteenth to early fifteenth century BCE, were passed on from one generation to the next resulting mainly in the discovery of such boxes in a much later context dating to the thirteenth to twelfth century BCE (Ben-Tor 2009:52–4).
Cypriot examples of the _game of twenty_ represent the westernmost evidence of this game (Crist et. al. in press). The practice of the _game of twenty_ in Cyprus seems restricted to Protohistoric Bronze Age ( _c._ 1700–1050 BCE) contexts and is therefore connected to the exchanges between elites, whereas it remains vivid during the early first millennium BCE in the Levant (Bieliński & Taracha 1992:50, fig. 7). There are three examples of the _game of twenty_ etched on a limestone block from the cemetery of Kouklia _Skales_ (Karageorghis 1983:122), which may demonstrate a brief reintroduction during the Cypro-Geometric period ( _c._ 1050–750 BCE). Crete is sometimes suggested as a possible destination for this game (Finkel 2007:17) but no firm archaeological evidence from a Minoan context is known to the authors. This statement is made by analogy with the unique so-called _royal draughtboard_ discovered by Evans at Knossos and compared to the boards from the Royal Cemetery of Ur (Hillbom 2011:255–8) (Archaeological Museum of Heraklion, YE 46). Yet, the layout of the cells on the Knossos game board is different from the _game of twenty_ so comparisons can only remain general.
In Egypt and the Levant, the _game of twenty_ accompanied _senet_ on double-sided boxes and stone slabs as described above (see fig. 3.9) from the second to the early first millennium BCE. Physical associations of the _game of twenty_ to other race games are not attested in Egypt. In contrast, during the second millennium BCE in the Near East, it has been linked to the game of _hounds and jackals_ , a discussion of which follows in Chapter 5, as illustrated by the prestigious boards from Megiddo and Susa.
##
## The Game of _Hounds and Jackals_ : From Thebes to Susa
**Figure 5.1** Distribution of the game of _hounds and jackals_ in Egypt and the Near East during the second and first millennia BCE.
The archaeological evidence for the game of _hounds and jackals_ points to its origin in Egypt at the turn of the second millennium BCE. The oldest material for this game was found in the necropolis of the capital Thebes and at provincial sites in the region of the Fayum (Sedment, Lahun and Lisht). _Hounds and jackals_ is almost exclusively attested during the Middle Kingdom within Egypt, and later boards are dated stylistically but lack solid archaeological provenance. Military campaigns and trade relations facilitated the game's spread to Nubia and to central Anatolia as early as the beginning of the second millennium BCE. The game was popular throughout the Near East, including the Iranian plateau, until the mid-first millennium BCE (fig. 5.1). There are about seventy known examples of boards. Seven are from Egypt, three from Nubia, eight from Anatolia, thirty from Mesopotamia, eight from Iran, ten from the Levant and two from Syria (de Voogt et al. 2013:1718–19, fig. 3, table 2).
The ancient name of the game is not known but many modern descriptive names have proliferated such as _hounds contra jackals_ or _palm tree game_ —after a unique set from Thebes—, _pegs and holes_ and _shield game_ (Parlett 1999:68–9). Other designations derive from the track of peg holes that characterizes the game. Petrie (1927:55) was the first to use the term _game of fifty-eight holes_ , which refers to the two parallel rows of twenty-nine holes. The field is completed at the end by a thirtieth hole, larger or at least specially marked. This led to the appellation _game of thirty points_ (Drioton 1940:186).
### Boards for _hounds and jackals_
Early examples were found in funerary and domestic contexts dating from the First Intermediate Period and the Middle Kingdom. The shape of these boards is compared to an axe-blade (Carnarvon & Carter 1912:56) or a shield (Drioton 1940:188, fig. 4). The boards measure from 6.2 to 19.7 cm long, 3.7 to 14.3 cm wide, and the tables are between 5 and 8.4 cm high. They follow the same arrangement but two boards are adorned with additional holes, interpreted as starting posts. Six pairs of special holes plus the goal are made distinct by their larger size and/or a mark. A _nfr_ sign, meaning "good" or "beautiful," marks holes fifteen and twenty-five whereas lines connect holes six and twenty and holes eight and ten.
The game found in 1921 at Sedment in a disturbed shaft tomb, Grave 2122, has been assigned different dates from the Ninth to Eleventh Dynasty (Petrie & Brunton 1924:7–8, pls. XXI,14, XXII, 8–9; Petrie 1928:18) to the early (Petrie Collection Online Catalogue) or late Middle Kingdom (Miniaci & Quirke 2009:349). The board is resting on three legs (Petrie Museum, UC31348). A square storage space was carved on the reverse and closed by a door fastened by a bolt. The loops, bolt and hinge pins were made of horn.
A board found at Deir el-Bahari, Thebes, in the cemetery dated to the Eleventh Dynasty is considered the oldest known (Winlock 1928:10) (Metropolitan Museum of Art, 26.3.154). It is a small wooden table resting on four animal legs with only one original leg left. The drawer is missing but one metal loop remains. Its small size could argue for a votive function, but Winlock, on the contrary, thought that the patch in the outer right row indicates that the holes were worn out because it was played often (Winlock 1928:10). The repair resulted in the loss of a hole between numbers ten and fifteen that "was overlooked by Drioton who included the missing hole in his drawing" (Hoerth 1961:61, note 4). The engraved line at the bottom of the board may imitate the cutout visible on the Sedment board.
In 1910, a unique game set was found in the tomb of Reniseneb (CC 25) at el-Asasif, Thebes (Carnarvon & Carter 1912:56, pl. L) (fig. 5.2). The tomb is dated to the Twelfth Dynasty after a toilet box with the name of Amenemhat IV that was found there. The board of sycamore wood overlaid with ivory and ebony rests on four bull legs (Metropolitan Museum of Art, 26.7.1287a–k). The course of play took place around an incised palm tree. The track has no starting holes, in contrast to the two previous examples. The final position is surrounded by the hieroglyph _šn_ , commonly interpreted as meaning "eternity." A bolted drawer was built-in for storing the playing pieces. This game set has become iconic not only due to its elegance, but also because it is the only example that retains all of its pegs, five per side. The pegs are described below.
Despite the small number of examples known to date, the importance of the game of _hounds and jackals_ as a burial gift is evident. The game table represented in the object frieze on the coffin of It from the Twelfth Dynasty could be a game of _hounds and jackals_ (Lieblein 1873:55, pl. 11) (State Hermitage Museum, 769). The fragmentary inscription accompanying the object, _ḫntš.f m b prw nj sn.t_ "He delights in the senet (?) game board of the thirty fields" (Pusch 1979:55–6), could refer to the thirty holes of each track rather than the thirty squares for _senet_. Moreover, rather than _senet_ , the appearance of the game leans more in favor of _hounds and jackals_. The game is resting on lion feet turned inwards and eight dots are drawn on the upper surface of the board. _Senet_ games on animal feet are represented in playing scenes but no archaeological evidence is known prior to the game from the tomb of Tutankhamun, whereas all the games of _hounds and jackals_ discovered in a Middle Kingdom funerary context are game tables. Playing pieces represented with _senet_ are usually dome shaped or conical, so the dots on the game in St. Petersburg could correspond to the track of holes.
**Figure 5.2** Game of _hounds and jackals_ from Thebes, board: 6.8 × 10.1 × 15.6 cm. The Metropolitan Museum of Art, Purchase, Edward S. Harkness Gift, 1926, 26.7.1287a–k. Image © The Metropolitan Museum of Art.
Among the finds illustrative of daily life during the Middle Kingdom in the town of Lahun is a rough clay board measuring almost 20 cm long and dating to the Twelfth Dynasty (Petrie 1890:30, pl. 16) (Petrie Museum, UC16722). Also, a wooden board with a drawer is stylistically assigned to the same period because its track is similar to the earliest examples (Hoerth 1961:63). It is now in the Egyptian Museum (CG 68128) but its provenance is unknown. The _nfr_ signs indicated next to the twenty-fifth holes (Petrie & Brunton 1924:Pl. XXII, 12) are missing on Drioton's drawing (Drioton 1940:Fig. 7). Drioton may have omitted this detail as he worked from a photograph of the board because it was not available for study at the time of his research (Hoerth 1961:63, note 2). A pair of eyes incised above the central rows may have indicated a starting point, whereas a "butterfly" motif is located at the bottom of the track and pierced. Drioton (1940:190–1) suggested a later date for this example because it does not rest on legs and he compares it therefore to the _senet_ boxes of the New Kingdom.
After the Middle Kingdom, Egyptian examples are scanty and chronologically disparate. The increasing importance of _senet_ in funerary rituals during the New Kingdom may explain this phenomenon. The older practice of _senet_ would resume during the early New Kingdom (Miniaci & Quirke 2009:361, note 127). The late examples of _hounds and jackals_ in Egypt belong to a category of ellipsoidal boards, described as violin/fiddle-shaped, with a round appendage added at the top of the board. It has been suggested that this appendix, referred to as "the labyrinth" by Drioton (1940:193), represents the goal. The additional holes, varying from six to eight around a centrally marked hole, would receive the winning pieces at the end of the game. The connecting lines and the hieroglyphs no longer need to mark the special positions.
A wooden plank, cut in the shape of a violin, was brought to the Egyptian Museum by Eugène Grebaut from his trip to Upper Egypt in 1888 (CG 68127) (Drioton 1940:193, fig. 9) (fig. 5.3). It resulted from a purchase (Miniaci & Quirke 2009:350), not from the excavations at Dra Abu el-Naga (DuQuesne 2002). The special positions are surrounded by incised circles. The numbering of the holes is not evident because there is no distinction or space between the central rows and the finishing holes. We may deduce the numbering from the tenth holes that are marked at the bottom of the middle rows. An incision in the center of the board frames holes three to seven. The fourth holes are placed side by side with the twentieth holes. This is atypical as it is the sixth hole that is usually connected to the twentieth. This board is morphologically dated to the New Kingdom by comparing it to similar excavated examples outside Egypt.
The next datable evidence of _hounds and jackals_ in Egypt belongs to the Twentieth or Twenty-First Dynasty. It appears as a fragmentary drawing on the Turin Papyrus, also illustrated by two detailed _senet_ boards and a _thirty-one_ (Decker & Herb 1994:686–7, pl. CCCLXXXV) (see fig. 4.7). The presence of the round appendix—seven plain holes around a special hole—indicates that the board is of the violin-shaped type. The ellipsoidal board is inscribed in a rectangular frame. Rosettes placed in the corner of the frame may indicate an influence from the Near Eastern boards, whose special positions are usually marked by rosettes. The German scholar Seyffarth examined the papyrus in the Turin museum in the early nineteenth century, during a time in which knowledge concerning Egyptian games was nascent, so he produced two fanciful circular reconstructions with a celestial interpretation of the dots (Seyffarth 1833:Pl. III).
**Figure 5.3** Violin-shaped game board, 17.5 × 9.5 cm. Egyptian Museum, Cairo, CG 68127. Photograph by James VanRensselaer.
When Petrie found the board in Lahun in the late nineteenth century, he wrote, "no such game is known in Egypt as yet" (Petrie 1890:30, pl. 16). In fact, a unique piece was already in the Louvre collection since 1827 but was not identified as a game until Petrie and Brunton (1924:7, pl. XXII, 25) published it with other games a century later (fig. 5.4). This object was part of the vast collection of the British consul Henry Salt (no. 832) purchased by Jean-François Champollion for the Louvre (N 3043). No further information about its provenance is known. It had been cataloged in the inventory as a faience frog-shaped writing case, the larger holes being interpreted as ink-cups and the smaller holes as holders for the brushes (Guichard 2013:228). This game board stands out from the others as much for its large size (21.5 cm × 18 cm) as for its zoomorphic shape, that of a hippopotamus not a frog. Drioton (1940:198) dated this object to the Twenty-Second Dynasty based on the style of the rosette motif in the border. This dating has been revised to a later date (Pierrat-Bonnefois 1998:219). The turquoise faience (fine and matte) and the colorful glass decoration indicate a date during the Late Period, between the Twenty-Seventh and Thirtieth Dynasties, around the sixth to fourth century BCE. During the Twenty-Seventh Dynasty, Egypt was under Persian rule. This exquisite piece may have been manufactured for the elite close to the Persian satrap who came from a region where this type of game was still played after it had declined in Egypt. The arrangement of identically-sized holes on the back of the animal follows the elliptic track of the violin-shaped boards. Holes six and twenty, which are connected by lines on the early examples, are here placed side by side on the same horizontal bar painted in red. Additional holes on the head of the animal could correspond to the labyrinth. The open mouth of the hippopotamus is broken so that two extra holes can be imagined to correspond to the nostrils.
**Figure 5.4** Drawing of hippopotamus-shaped game board based on an illustration by Christian Décamps and Nathalie Couton-Perche, Musée du Louvre, N 3043.
### Gaming pegs
The ivory board from Thebes is the only extant set found with its gaming pieces (fig. 5.2). Many pegs, being made of wood, have perished. Others have probably been ignored or erroneously catalogued as hairpins. The opposing pieces in the ivory sets from Thebes were differentiated by their decoration and their size. Five pegs end with the head of a lop-eared hound (6 cm to 6.8 cm in height) and five taller pegs end with the head of a jackal with pricked ears (7 cm to 8.5 cm in height). Residues of red pigment indicate that color might have contributed to their distinction. Thanks to the Theban assemblage we know what the pieces look like and isolated finds could be identified rightly as gaming pieces when boards are missing. Some pegs attributed to later periods are evidence of the continuity of the game in Egypt, which is questionable after the Middle Kingdom (Hayes 1959:38, 199–200; Hoerth 2007:65). The presence of pegs with a jackal head as well as pegs with disk-shaped heads or plain knobs in the North Cemetery at Lisht points to the existence of the game at this site (Metropolitan Museum of Art, 22.1.721, 15.3.946–50). Such wooden and ivory pins are known in other museum collections (May 1992:156–9, figs. 153, 155) (Musée du Louvre, N4265 A; British Museum, EA 13594; Museo Egizio di Torino, cat. 6934). The canine animals described as hounds and jackals are sometimes referred as dogs and foxes, or as two kinds of dogs. During the Middle Kingdom, dogs are portrayed frequently and with a great variety, notably in the Beni Hasan tombs. The creature with upstanding, pointed ears is "frequently seen as the sacred animal of the god Anubis" (Hayes 1959:250). In any case, the choice of fast-running animals for playing pieces is appropriate for a race game.
A group of six ivory pegs, five with the head of a hound and one with a head of a jackal (between 5.4 and 7.9 cm long), was discovered in the tomb of Neferhotep at Dra Abu el-Naga (fig. 5.5). This burial previously assigned to the Sixteenth or Seventeenth Dynasty has been re-examined and dated to the mid-Thirteenth Dynasty (Miniaci & Quirke 2009:357). The sticks were originally described as hairpins and associated with a wooden turtle-shaped "pincushion" having four rows of five and one row of three holes, found in the same tomb (Bénédite 1911:19, pl. X) (Egyptian Museum, JE 6146–6152, CG 44414). The function of the pegs as gaming pieces was later established by Bénédite but the identification of the turtle as a gaming board has been questioned due to the abnormal number and configuration of the holes (Fischer 1968:33–4). The turtle, measuring only 5.5 cm long, is considered, rather, as a holder for gaming pieces (DuQuesne 2002; Miniaci & Quirke 2009:349–50). No parallel for a peg stand is known so another interpretation is conceivable. Amulets of the game of _fifty-eight holes_ with an irrelevant number of holes bored into them were deposited in tombs in Iran (Dunn-Vaturi 2012f:22–3). The turtle could be a miniature board imitating the game to function as simulacra for the next world. Its presence in the tomb of Neferhotep may be related to a birth theme like other burial goods deposited there (Miniaci & Quirke 2009:361).
The Drovetti collection in Turin holds bronze pieces with jackal or sparrow-hawk heads and pointed ends (Museo Egizio di Torino, cat. 6327, 6328). The Turin sticks were included in the _hounds and jackals_ section in the exhibition _Jouer dans l'Antiquité_ held in Marseille but they measure 32 cm high so they would be rather tall for this game (May 1992:156, fig. 154). They may have been used as stakes for the game known today as _horseshoes_ (Donadoni Roveri 1988:247, fig. 350).
**Figure 5.5** Turtle stand or simulacra from Dra Abu el-Naga. Egyptian Museum, Cairo, CG 44414. Drawing by Paul Whelan.
### Beni Hasan playing scenes
_Mehen_ and _senet_ playing scenes are extensively described in ancient Egypt, whereas the depiction of _hounds and jackals_ is uncertain—apart from the drawing on the Turin Papyrus—and remains absent from the game literature. In fact, it may have been associated with the double board games scenes at Beni Hasan discussed in Chapter 4, relative to the possible representation of the _game of twenty_ in the tombs of the nomarch Baqet III (Tomb 15) and his son Khety (Tomb 17). The scenes painted side by side in both tombs but in a different order are indicated here with letters A, B and C (Newberry 1893:Pls. 7, 13) (see fig. 4.4). Scene C, on the left in Tomb 15 and on the right in Tomb 17, could represent the same or a similar game. These scenes, which are situated in the lower part of the decorated walls, are fragmentary. They have been reconstructed differently by Rosellini (1834:Pl. CIII,C), Wilkinson (1853:194, fig. 211, right) and Newberry (1893:Pls. 7, 13), because by the time Newberry recorded these tombs in 1890, the paintings had been damaged.
Rosellini, who visited the site in 1828, published a drawing without indicating its provenance that is probably more complete than Newberry's version of Tomb 17 (fig. 5.6). The drawing shows two players engaged around a large bowl sitting on the floor with a zoomorphic table above. The table imitates a long-snouted animal unless the orientation of the legs implies that the appendix represents the tail. The five sticks mounted on the table are missing on Newberry's plate (see fig. 4.4, scene C). This scene could be a representation of _hounds and jackals,_ which would perfectly fit a Middle Kingdom context of playing scenes. Two examples of game tables with animal feet for the game of _hounds and jackals_ from Thebes were described above, whereas the number of sticks could refer to one set of playing pegs. The bowl may have been used by the players in which to throw the equivalent of dice and can be compared to the cut boulders used as a rolling device discovered at Megiddo in the southern Levant (Guillaume 2013:1111–12).
**Figure 5.6** Playing scene in Beni Hasan, probably from Tomb 17. Drawing reproduced from Rosellini (1834:Pl. CIII, C).
Scene C in Tomb 15 shows four sticks mounted on a base—the lower part of this element is missing—between two players. The caption above is rendered _imby_. Decker and Herb (1994:631, pl. CCCLII, P.8.3) include this scene in their group of guessing/counting games whereas Vandier (1964:511–12, fig. 280, right) records it in an undetermined category. A similar scene with two players, a bowl and four sticks on some sort of zoomorphic base—resembling a hippopotamus with an open mouth—has been published by Wilkinson with games of guessing/counting (Wilkinson 1853:194, fig. 211, right). No indication about the tomb is given but Wilkinson sketched this scene next to a type B scene so it could well correspond to the scenes in Tomb 15 (MS. Wilkinson dep. a. 21, fol. 292, Bodleian Library, Oxford). The pegs in Wilkinson's drawing have flat horizontal heads possibly imitating the jackal's long muzzle. The caption is rendered _i bi_ with a possible playing piece determinative. Elements from different scenes may have been mixed in the reconstruction. Future campaigns _in situ_ might help understand what was originally described.
### The game outside of Egypt
The military, diplomatic and commercial relations that connected Egypt and its neighbors facilitated the spread of board games. The game of _hounds and jackals_ crossed more borders than any other Egyptian game. It traveled to Nubia and had a wide diffusion in Western Asia as far as Anatolia northward, and Iran eastward.
In the Middle Kingdom, Lower Nubia was conquered by the Egyptians, who built a chain of fortresses along the Nile to guard their frontiers. During the Twelfth Dynasty, under Senusret I a huge fortress was built at Buhen and restored by Senusret III. Buhen was not only a military stronghold, but also a trading and dispatch post of economic importance. Egyptian soldiers lived and played here. Clay fragments of gaming boards are attested from Buhen but unfortunately not from stratigraphically sealed contexts (Emery 1979:145–6). They were found in the debris of the east inner ramparts, and at the south inner ditch between Towers 1 and 2. Middle Kingdom and New Kingdom pottery and objects were found in the debris from the town site so it is difficult to ascertain a date other than stylistically. One fragment (932) is similar to the rough clay version from Lahun so it probably dates to the Middle Kingdom. Two other fragments (I9 and J9), which may belong to the same board, probably date to the New Kingdom as they recall the violin-shaped board with an appendix (Petrie Museum, UC21182 and Sudan National Museum,14140).
No playing pieces were recorded at Buhen but finds from Serra East, Amara West and el-Kurru give an idea of the pieces used in Nubia. An undated bone peg carved with a dog or jackal head was retrieved at Serra East (Oriental Institute Museum, E 19794). Recent British Museum excavations in the cemetery of the New Kingdom town of Amara West brought to light twelve ivory sticks in grave 244 (F9835) (Vandenbeusch & Salvador 2014). The sticks are of two distinct lengths, i.e., seven shorter (about 11 cm) and five taller ones (about 14 cm). The tops of the sticks are decorated with red painted patterns. Six of them have a horizontal band and a vertical band while four of them have a zigzag motif. Finally, the continuity of the game in Nubia during the first millennium BCE is deduced from an ivory set of opposing pegs from the site of el-Kurru. Three pegs carved with the heads of hounds and three with the heads of horses were discovered in Tomb Ku.72 dated to about 698–690 BCE (Dunham 1950:81, fig. 28f; 102, fig. 35h, pls. 35D–E, 36D) (Museum of Fine Arts, Boston, 24.1037a–f).
Although examples with additional posts have a total of holes exceeding sixty, the appellation _fifty-eight holes_ is preferred when talking about the Near East because no pegs with hounds and jackals were found at Near Eastern sites. The game was transmitted to the Near East during the beginning of the second millennium BCE. It is attested during the nineteenth to eighteenth centuries BCE in Anatolia. At this time the Old Assyrian merchant colonies were active and traders from the northern Mesopotamian city of Ashur were established at a number of central Anatolian cities. It suggests that Anatolians learned this game from the Assyrian merchants responsible for the introduction of writing and cylinder-seals in Anatolia. Central Anatolia was part of a vast trading network that took place in the Eastern Mediterranean in the second millennium BCE, so the game also could have been passed to that region without the Mesopotamians playing a role. A trade route connected Cappadocia and Egypt overland and, when maritime connections were favorable, by sea via ports of Ugarit, Tell Sukas, Byblos, Ashkelon and ancient Gaza, among others (Collon 2008:99). The Middle Bronze Age is a time during which, not only at Byblos but also further north at Syrian Ebla, we find an imposing Egyptian artistic presence but no evidence of the game is attested at these sites. Only one clay board, said to be from Tyre in Lebanon, is attributed to this period, dating to about 2000–1500 BCE (Museum of Fine Arts, Boston, 1996.62). Middle Kingdom imports and examples of Egyptian influence are known from sites in central Anatolia. The growth of the Anatolian corpus since the 1960s raises the question of the dispersal of the game. Seven examples of games of _fifty-eight holes_ are recorded from this region. Two complete boards were found in merchant houses at Kültepe, ancient Kanesh, four fragmentary gaming boards—one in Egyptian blue—in a palace at Acemhoyük as well as one fragment at Karahoyük (Dunn-Vaturi 2012e). The oldest example, from level II ( _c_. 1919–1840 BCE) at ancient Kanesh, shows linkages between unusual holes whereas the special holes on the other boards, dated to the eighteenth century BCE, are distinguished by their size and/or inlays. No evidence was found in the home city Ashur where the levels corresponding to the Assyrian colonies in Cappadocia were barely brought to light. In the end, the Mesopotamians may not be responsible for the diffusion of the game in Anatolia.
Games of _fifty-eight holes_ have been circulating in the rest of the Near East during the beginning of second millennium BCE but the dating of early examples from Mesopotamia and Iran is difficult. Boards attributed to this period lack clear archaeological contexts. Further discoveries may help to better understand how these games spread in the Near East. Five fragmentary stone gaming boards and astragals have been discovered in the deposit of the temple of Inshushinak, a group of objects found in the temple precinct of the city god of Susa (fig. 5.7). This deposit, buried at the end of the Middle Elamite period, in the late twelfth century BCE, gathers objects from different periods. The boards have been assigned to the early second millennium BCE after an unprovenanced game with a carved scene stylistically attributed to the Old Babylonian period, around the mid-nineteenth century BCE (Ellis & Buchanan 1966:199) (Yale Babylonian collection, YBC 2439). One of the Susa boards, Sb 2911, has preserved a contrasting inter-hole link that physically shows that pieces could cross from side to side, involving interaction between the players. This element is also repeated on later Iranian boards dating to the beginning of the Iron Age, at the turn of the second to the first millennium BCE (Musée du Louvre, AO 19438; British Museum, 1991,0720.1 and 2003,1201.1).
**Figure 5.7** Boards from the deposit of the temple of Inshushinak, Susa, max. 10.5 × 10.5 × 1.5 cm. Musée du Louvre, Sb 2911, Sb 10190, Sb 10189 (top row, left to right), Sb 2912 (bottom row, first two on left), Sb 10191 (bottom row, right). Drawings reproduced from Mecquenem (1905: Figs. 345–50).
During the New Kingdom, the Egyptian empire embraced Nubia and parts of the southern Levant, and these regions shared the same type of violin-shaped board, also referred to as Palestinian-type boards (Hoerth 2007:65). The elaborate boards found in the Ivory Hoard of the palace at Megiddo are the most famous (Loud 1939:9–10, 19, pls. 47–50). Special holes are indicated by a rosette as well as blue paste and gold inlays. Other boards were found in the southern Levant, at Gezer (Macalister 1912:416, fig. 501), Tell Jemmeh, previously identified with ancient Gerar (Petrie 1928:18, pl. 39, 22) and Beth Shean (Oren 1973:Figs. 41, 37 and 45, 23). They are dated from the fifteenth to the twelfth centuries BCE.
As stated earlier, no pegs with hounds and jackals were discovered in the Near East but new interpretations of Mesopotamian texts suggest that such zoomorphic playing pieces existed. Vermaak (2011:124) connects the Sumerian words gishellag and gishillar, generally translated "wooden throw stick," to a game board. Vermaak's new translation is based on a Sumerian proverb "a fox/jackal walked around the game board" (SP 8 Sec B 34). In other proverbs, dogs and foxes are defined as opposites and could reflect the gaming pieces carved in the shape of two distinct animals. These Sumerian proverbs fit well in the context of our game. Most of the proverbs date from the first half of the second millennium BCE, a time when the game of _hounds and jackals_ travels out of Egypt.
Tapered pegs with notched heads possibly imitating the pointed ears of the jackals were used at Megiddo (Palestine Archaeological Museum, 38797; Oriental Institute Museum, A22316, A 22364, A 22367). Erdös (1986:83) might refer to these when she mentions ivory pins with a dog or jackal head from Megiddo. Some pins identified as hairpins in the archaeological literature may in fact be game pegs, like those found in Neferhotep's tomb discussed above. It has been suggested that ivory and metal pins with animals at the top found at Kültepe/Kanesh in central Anatolia were used to play (Michel 2008:359). Finally, monkey-headed pins found in the deposit of the temple of Inshushinak at Susa may have been used with the fragmentary boards from the same hoard (Dunn-Vaturi 2000:109–10). Examination of traces of use at the end of pins may help identify their real function.
### Reconstructed rules
A description of the rules has not survived, but several attempts have been made to reconstruct them (Carnarvon & Carter 1912:58–9; Petrie & Brunton 1924:7; Murray 1951:15–16; Bell 1979:21; Hoerth 2007:66–8). Archaeologists and experts on the history of games unanimously interpreted it as a race game in consideration of the arrangement of the board, despite the absence of casting implements directly associated with any board in Egypt. Disks with one side marked, accompanying boards at Megiddo in the southern Levant and Nippur in Mesopotamia, are presumably counting devices (Hoerth 2007:64). Astragali were found in the same contexts as boards at Susa and Tepe Sialk in Iran and their use with the Theban ivory board has been suggested, but in Egypt they are better attested from the Seventeenth Dynasty and later so we are not sure how Egyptians were induced to move their pegs along the board.
Carter's reconstruction shown in fig. 5.8 is the most likely route of play because the middle rows of holes are separated from the central position at the top that clearly stands out as the goal. One side of the board is attributed to each player. The players competed by moving their pieces down the center of the board, of either ten or eleven holes, around the outer edge, of nineteen holes, and up to the thirtieth and ending position. A separation between the two tracks is indicated on some boards so it is suggested that no interaction was happening between the opposing pieces. Connecting lines incised on some Near Eastern boards show that, at least in this region, pieces did come into conflict. Each player had five pegs according to the number of pieces found with the Theban board. Carter's rule about each opponent having only one piece at a time on the board may not be valid. If players have more than one piece on the board at a time, a piece may simply not be moved to an occupied space. Other aspects remain unknowable, such as how a piece enters the board and how its final move to the goal was determined if the result of the dice was higher than the spaces to be traversed. Some boards bear additional holes that are considered as starting posts. The progress of the race was affected by inter-hole links and hieroglyphs. Two pairs of holes (six and twenty, eight and ten) are linked by lines. The links could be shortcuts or penalties, like in the modern game _snakes and ladders_ , with a playing piece sent forward or backwards. From ten onwards, every fifth hole is emphasized. It is assumed that _nfr_ , marked on two positions, i.e., fifteen and twenty-five, was beneficial to players that landed on them, similar to square twenty-six on _senet_ boards. The goal is a larger hole clearly surrounded on the example from Thebes by the hieroglyph _šn_ , "eternity," that may have meant "end of the track." This ultimate destination counts as space number thirty, an important number in ancient Egypt. The player who reached this sign won the game and possibly succeeded in his quest for immortality.
**Figure 5.8** Reconstruction of the numbering sequence of the holes. Drawing reproduced from Carnarvon & Carter (1912:57, fig. 14).
### Symbolism of the game
Playing accessories, as well as objects imitating games, are linked to the deceased's journey to the afterlife and the game of _hounds and jackals_ often found in funerary contexts seem to be no exception. The jackal deity, Anubis, whose imagery was part of the gaming pegs, was symbolically responsible for mummification and protection of the dead through the netherworld. His connection to games used as funerary devices is relevant and may also be illustrated by the throwing sticks decorated by a canine head and the action they command (DuQuesne 2002; for passage about jackals in the Great Game Text, see Chapter 1).
The development of the game board into an anthropoid format occurs in the course of the second millennium BCE (Vermaak 2011:118). The connection between the game and a human representation with the additions of anatomical details, such as eyes, was made relatively early if we consider the Cairo board CG 68128 to be from the Middle Kingdom. Anatomical vocabulary is used to describe the head and the feet of the boards from Susa (Ellis & Buchanan 1966:195). The Gezer board initially identified as a "degenerated Ashtoreth plaque" by the excavator (Macalister 1912:416, fig. 501) may have reinforced the anthropomorphic and religious implications for the boards with a round projection.
The anthropomorphism of the board supports the idea of rebirth, essential in ancient funerary rituals. Erdös (1986:118–19) suggests that the insertion of pegs into the sequence of holes would allow the deceased to be reborn. Anubis pegs could perform the necessary ritual on the board, which would symbolize the deceased's body. Schuster and Carpenter (1996:665) also include the game of _hounds and jackals_ in the "Rebirth Gaming board" category and make comparisons with a traditional Indian game, an antecessor of _snakes and ladders_ , which functions like a genealogical chart, a path to be followed into the afterworld. Furthermore, Schuster and Carpenter (1996:671–5) compare the shield-shaped boards to a drawing made by the Malekulans of Vanuatu, located in the otherwise unconnected region of Melanesia in Oceania, to achieve reunion with one's ancestry by connection with joint-marks: "The Malekulan artist first punched a framework of dots in the sand, then drew an unbroken, never-ending line around these dots to form a figure identified as, simultaneously, Human Ancestor and Cosmic Turtle." The special holes of our game would correspond to joint-marks. Twelve marks—a complete set with shoulders, elbows, wrists, hips, knees and ankles—are observed on certain boards. It is interesting to note that the ellipsoidal board with an appendix has been interpreted as a human form, but has also been compared to a turtle (DuQuesne 2002) whereas the stand or votive game in Neferhotep's tomb adopts the shape of a reptile.
The wooden turtle from Dra Abu el-Naga as well as the miniature game table from Deir el-Bahari are votive artifacts deposited in the tomb for the next world. Amulets imitating the game of _fifty-eight holes_ with an irrelevant number of holes bored into them were found among burial offerings at Tepe Sialk and Dinkha Tepe, in Iran (Dunn-Vaturi 2012f:22–3). At Susa, games had an important place in the foundation deposit of the temple of Inshushinak. Five broken boards, some double-sided with the _game of twenty_ or a _game of thirty_ squares, astragali and possible pegs were dedicated to the god Inshushinak, the Lord of the city of Susa, who had among other attributes, that of the judge responsible for the last judgments of the deceased.
The recreational function of games is a subject of debate, especially in the absence of casting devices and textual evidence. Some board games offer astronomical and astrological references. The sequences of twenty-nine or thirty holes equivalent to the number of days in a lunar month have been considered as a device to record the phases of the moon (Novacek 2011:50). It has also been suggested recently that the fifty-nine-hole board was a calendrical tool for lunar-solar synchronism, like the Greek calendar with holes that would develop into a Coptic version (García Martínez 2014).
### The Coptic board game
A board with peg holes, attested by five examples, is designated as the Coptic board game (Decker & Herb 1994:687–8, pl. CCCLXXXVII). Two game boxes, measuring 18.1 cm and 19.5 cm long, are made of wood and ivory plaques (Musée du Louvre, E 11717, E 21047). Three smaller examples, between 11.5 cm and 13.2 cm long, are carved out of solid bone (Egyptian Museum, JE 78126; Musées Royaux d'Art et d'Histoire, Brussels, E 2667; Swiss Museum of Games, 2282) (fig. 5.9). Drioton (1940) insisted on shared characteristics with the game of _hounds and jackals_ and drew up the inventory of the pieces known at that time. The game boxes have a step-type structure divided in three parts. The lower landing is the longest one; it has two groups of twenty holes. The intermediate landing has ten holes on the outside and a varying number of positions in the center. This part is described as a labyrinth and compared to the projection of the violin-shaped boards (Drioton 1940:185, 192–3). The upper part is the smallest one. There is a central hole, considered the goal, surrounded by other holes. Some perforations, not fully formed, served as decorations. Drioton (1940:184–6) reconstructed a track of twenty-nine holes—ten downward and nineteen upward—for each player plus the goal, and suggested calling it the game of the _thirty points_. Despite the similarity of their structure, nothing can relate this game to a late version of _hounds and jackals_ (May 1992:164; Finkel 2008:154). The playing pieces were stored in a drawer or a hollow in the reverse of the board but none have an archaeological context so it cannot be known what a complete assemblage would be. Bone pegs associated with the game are decorated with human heads or with moldings and knobs (Drioton 1940:180–1, 183, fig. 2, pl. II). Rounded disks made of bone with two different types of markings were probably used as dice (Drioton 1940:181, 183).
**Figure 5.9** Coptic board game, 12.4 × 2.9 × 3.1 cm. Swiss Museum of Games, 2282.
Only one such game, previously in the Guimet museum and now in the Louvre (E 21047), has a documented provenance and illustrates the continued presence of games in a funerary context. It was discovered in 1901 by Albert Gayet at Antinoë in a female burial attributed to St Thaïs by Gayet who identified the pierced box as a prayer marker (Gayet 1902:47–8, 51). The burial was dated to the fourth century CE (May 1992:164) or sixth century CE (Drioton 1940:186) but is now considered to date to the second part of the seventh century CE after C14 analysis of textile and human remains (Calament & Durand 2013:330). The other step-type boards are attributed to the Byzantine period in general.
Boards for the game of _hounds and jackals_ , or _fifty-eight holes_ , have been found in elaborate and humble versions, in materials ranging from gold-inlaid ivory to wood and clay, suggesting that the game was played by members of all classes. The present study shows that problems of identification of the gaming equipment should be addressed not only archaeologically (fragments of boards with consistent arrangement of special holes, pegs versus pins, associated randomizing implements) but also visually and epigraphically, such as recent attempts at identifying the game in Sumerian proverbs.
##
## Roman Board Games Crossing the Borders of Egypt
The Ptolemaic period began with Ptolemy I Soter in 305 BCE, some years after the death of Alexander the Great in 323 BCE. It was followed by the Roman conquest of Egypt in 30 BCE when Egypt became a province of the Roman Empire. During the Ptolemaic period there was a strong Hellenistic influence throughout Egyptian material culture that may be reflected in board games of the times but other than dice with Greek lettering (Pedrizet 1931), there is no physical evidence for Greek gaming practices. The ensuing Roman conquest may have introduced games played by Roman soldiers who were based in the country and evidence for such games is still increasing. Some game practices are likely to have continued until at least the Arab conquest in the seventh century CE even though datable finds in Egypt have been limited to account for the period after the fourth century. Although one board game of Greek origin is discussed below, it is the gaming practices of the Romans that are found in the archaeological record thus far and that are discussed here in some detail.
The climate of Egypt played an important role in the preservation of game boards, playing pieces and dice; even if Egyptians played only a marginal role in the distribution of Roman board games, with the possible exception of providing a link to present-day Sudan, the examples from the Egyptian archaeological record are numerous. Preserved Roman game boards remain rare even in Egypt and Roman games have mostly been found carved in pavements and buildings. The presence of these latter so-called graffiti games in Egypt has confused archaeologists working on different time periods. Games found on monuments or next to other games have been frequently but erroneously associated with the period and the people associated with the building. This problem, which was touched upon in previous chapters as well, has also given several board games an Egyptian origin while the carvings may have occurred centuries after the building's erection.
Despite the presence of Roman games on several archaeological sites in Egypt, most of this material has remained unpublished. A study by Mulvin and Sidebotham (2003) as well as the work by Brun (2003, 2011) and Matelly (2003) are important exceptions to materials found in fortresses in the eastern desert of Egypt. The year 2003 also provided a publication of all the graffiti, including games, found on the roof of the temple of Khonsu in Karnak, Luxor, by Jacquet-Gordon. Most other publications refer to games found elsewhere in the Roman Empire or on sites in Sudan. They also include earlier studies of the Latin and Greek sources that provide the necessary background on game rules and game names.
With the help of the photo archive of the Gebel el-Silsila Survey Project in Egypt conducted by Maria Nilsson and John Ward, it was possible to confirm the presence of Roman game boards in a series of sites. The dimensions of these boards have not yet been recorded and their presence on each of these sites still needs further analysis. The study of game boards found at Silsila, Egypt and the games found at Sedeinga, Sudan, are part of current research projects by Alex de Voogt together with Maria Nilsson and Vincent Francigny, respectively. This is, therefore, a preliminary overview of Roman gaming practices in Egypt and serves as a reference for ongoing and future research on Roman games in this region.
### The sources
Both Greek and Roman game boards are occasionally found as grave goods but more frequently as carved outlines on rock faces, temple rooftops and pavements. This complicates the possible association of a game board with playing pieces and dice. Dice may be found in tombs or as surface finds unassociated with boards, and typically cannot be linked with game boards carved in public spaces. Casting devices including cubic dice, throwing sticks and astragali are associated with most board games of antiquity but may also have been part of dice games not requiring a board. Which Greek or Roman dice games were then popular in ancient Egypt remains unclear. Cubic and twenty-sided dice with Greek lettering have been attested in Egypt but their particular use is unknown. Instead the graffiti game boards can tell us only about which Roman board games were played in Egypt as Greek game boards are still absent from the archaeological record.
**Figure 6.1** Map of sites mentioned in Chapters 6 and .
From the literary and iconographic sources, four board games have been given most of the attention. For instance, Rieche (1986) mentions _ludus latrunculorum_ (henceforth _latrunculi_ ), _duodecim scripta_ as well as _three-men's-morris_ also known as the Roman _mill_ or _merels_ game (henceforth _merels_ ), for which the Latin name is unknown. Her work already suggests that the written and art historical sources do not necessarily describe all Roman board games found in the archaeological record. Lamer (1927) and, more importantly, Schädler (1998) added the Greek game of _five lines_ to this list. Schädler analyzed the Roman and possible Byzantine forms that are as ubiquitous as the other Roman games. It is not possible to claim that this list is complete, moreover, as Schädler (1998) already pointed out, some game boards may have been used for multiple games and some games may have been played on different configurations of boards. The discussion of the material is only a starting point rather than a definitive statement on these board games in Egypt.
Mulvin and Sidebotham (2003) attested _mancala_ , _merels_ (although with only one unclear example), several _duodecim scripta_ boards as well as multiple partial and complete _latrunculi_ boards. This provides a convincing context for Roman games, except that _mancala_ should probably be reinterpreted as the game of _five lines_. Jacquet-Gordon (2003:17, 18, 21) in her work on the temple roof in Karnak mentions a few unfinished _seeja_ boards (see Chapter 7) that in some cases can also be interpreted as unfinished examples of _five lines_. Since no other known Roman game boards were attested, it leaves these circular depressions open to explanations unrelated to Roman gaming practices. In the eastern desert, more specifically in the fortresses of Dawwi and Didymoi, fragments of _latrunculi_ and _duodecim scripta_ are attested mostly from datable strata going back to the second century CE (Brun 2003, 2011). From the Gebel el-Silsila Survey Project, the temple of Kom Ombo, built in the Ptolemaic period, shows multiple examples of _merels_ and _duodecim scripta_ as well as _latrunculi_ and a game of _five lines_. Again this provides a clear example of Romans playing multiple games in the province Aegyptus. An agglomeration of different games or games context (see fig. 6.2) facilitates the interpretation of the time period and the players if they are all acknowledged Roman games. Unfortunately, configurations that cannot yet be interpreted, as well as Arab or Ottoman games discussed in the next chapter, complicate such efforts.
In the Greco-Roman period the types of board games found in antiquity are beginning to multiply in the archaeological record. Games popular in Pharaonic Egypt are joined by Roman games and may have coexisted with them. Setting precise dates for Roman graffiti games is usually impossible (see Lamer 1927:2010 for a possible exception). Although only a relative date for these graffiti games can be ascertained, some excavated boards have been dated and, unlike the Pharaonic examples, the rules of play are better understood. The sources that hint at games' rules are found in literary texts in both Greek and Latin. The Latin sources mention books written by Suetonius on Greek children's games and by Emperor Claudius on board and dice games (Lamer 1927). While the works have not been preserved, references to them in later texts as well as poetic allusions and dictionary entries in both Greek and Latin allow some knowledge to be gleaned about the rules of play.
Apart from allusions to game rules, the written and iconographic sources also provide information about the presence of dice with a game board, the contexts of game play and their role in society. This information on Roman play culture may be identical in the Egyptian context. However, Roman games as well as Ottoman and Arab games, as discussed in the following chapter, are often associated with soldiers. This limits the players' group and may affect the common play contexts in Egypt. Descriptions of men and women playing board games together as found in Ovid's _Ars Amatoria_ (for a discussion, see Rieche 1986:42, 44) may not apply to Egypt. Despite the richer contextual information provided by the Roman sources, it is this societal part that is most likely different in Egypt. On the other hand, the game rules and board designs are not likely to be affected as studies of other board games in antiquity have already shown in detail (de Voogt et al. 2013).
Apart from the occasional game board or dice that can be dated as part of an excavation, the monuments on which Roman games were incised provide a _terminus post quem_ for the game and are not necessarily of the same time period. The ability to identify a pattern as Greek, Roman, Arab and/or Ottoman, as explained in this and the following chapter, is essential for identifying the correct time period to which these games belong rather than the other way around. Buildings rarely provide a useful date for the graffiti games found upon their stone.
**Figure 6.2** Example of a Roman games context at Palmyra, Syria: _merels_ , _five lines_ and the outline of _duodecim scripta_ (de Voogt 2010:1060).
### The game of _five lines_ or _πέντε γραμμαί_
Schädler (1998) was the first to link the different appearances of two rows of five playing fields in Greco-Roman contexts. These two rows may consist of lines, holes, squares or a combination thereof. When the configuration consists of rows of holes or sets of squares they can be classified as the Roman version of the Greek game of _five lines_ (Schädler 2008, 2013c:65). This shape has been attested in multiple contexts throughout the Roman Empire and often in the context of other Roman games. Despite Schädler's observation, this configuration also continues to be interpreted as _mancala_ , in most cases a problematic assumption that is discussed in more detail in the following chapter.
The game has been discussed comprehensively in several sources either using its Greek name _πέντε γραμμαί_ or the English equivalent _five lines_. Although it is originally a Greek game it seems closely connected with the Roman Empire as well and its introduction to Egypt can, at this stage, not be disentangled from the Roman games.
As Schädler (1998:16) mentions, the existing literary and archaeological evidence gives a good picture of the game. The game was probably part of Suetonius's book on Greek dice games, to which some references exist. The literary references have been discussed at length (e.g., Lamer 1927). It is said that both players of this game owned five lines and between these was the sacred line. The player who first managed to place his pieces on the sacred line was the winner. According to Schädler (1998, 2008, 2013c) two players played either on five or eleven lines or on separate groups of five lines. The earliest reference dates to the sixth century BCE and is attributed to the Greek poet Alkaios or Alcaeus who speaks about the rules (Schädler 2008:174). For that same period Schädler (2008:175) also mentions a terracotta model of a gaming table from Attica that approximates the game. The game rules and other artistic representations suggest the use of one cubic die, possibly two. The consistent design of two rows of exactly five cells in Roman Asia Minor made the possibility of _five lines_ more plausible than that of a very early form of _mancala_. The same situation exists in Egypt.
Mulvin and Sidebotham (2003:605–8) identified a number of configurations of two rows of five as _mancala_ games at Abu Sha'ar. The context alongside other Roman games strongly suggests that here also the game of _five lines_ was intended. Other than that a configuration of holes resembles a modern _mancala_ game, there is neither any evidence nor a necessity to identify them as representations of _mancala_ as it is only rows of five or fewer fields that have been attested in Abu Sha'ar.
Two rows of holes that have been scratched in rock surfaces and monuments are easily confused with so-called _mancala_ games. Both _five lines_ and _mancala_ games were found in Palmyra, for instance, where rows of five holes were used by Romans and rows that were longer than five holes were used by Arabs or, more likely, Ottomans (de Voogt 2010). The latter may even have expanded the Roman carvings to accommodate their own game. Schädler (1998) goes as far as to suggest that _mancala_ may have succeeded _five lines_ in late antiquity, possibly before the arrival of the Arabs in the seventh century. The evidence known from Egypt does not support this suggestion as the arrival of _mancala_ is so far connected only with the presence of the Ottomans and not the Christian and later Arab influences in the region.
**Figure 6.3** _Five lines_ at the Luxor temple (top) and at Qasr al Ghweita (bottom), which is located south of the Kharga Oasis, based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
The Gebel el-Silsila Survey Project photographed a few examples of two rows of five depressions, one at the Luxor temple and one at Qasr al Ghweita, which is located about 18 km south of the town of Kharga (fig. 6.3). Elsewhere in the Kharga Oasis, some designs suggest a similar game but, unfortunately, the site does not provide much of a context for Roman games apart from one _merels_ game design with two diagonal lines.
### _Duodecim scripta_ or _ludus duodecim scriptorum_
Despite the predominance of incised games, Egypt, or more precisely, the Egyptian border with Sudan, is home to one of the best-preserved wooden examples of the Roman game _duodecim scripta_. This particular board was found in a grave at Qustul dating to the period after the fourth century CE (Emery & Kirwan 1938:345). It shows three rows of squares arranged in the same manner as the many examples of this game recorded throughout the rest of the Roman Empire (fig. 6.4). The wooden example from the Qustul grave, found in near perfect condition in Sudan, illustrates that a tradition of wooden boards may have escaped the archaeological record in most of the Empire.
The game of _duodecim scripta_ or the _game of twelve signs_ is considered a precursor of the modern game of _backgammon_ with which, according to Austin (1934) and Schädler (1995), it shares a number of characteristics. First, it consists of rows of twelve fields, visibly separated in the middle making two sections of six, and is played by two players using two cubic dice. A later version, called _alea_ or _dice_ , was played with three instead of two dice, on a similar board using fifteen pieces for each player. Games tables from the fifth and sixth centuries CE from Aphrodisias and Ephesos, but also the example from Qustul, confirm the survival of this game into the centuries up until the Arab conquest.
The most common representation of _duodecim scripta_ in the archaeological record is a carving into a marble slab where the game is formed by rows of letters, each letter indicating a playing field. These ranges of letters look rather different and Lamer (1927:2010) and Merkelbach (1978), among others, did not believe this game to be _duodecim scripta_ , but a different thirty-six-fields game or a _latrunculi_ board. Since the work of Austin (1934, 1935) this is, or should no longer be, in contention. The letter combinations make so-called hexagrams, often witty comments categorized by Austin (1934:32, note 1) as "( _a_ ) maxims for players, ( _b_ ) jeers..., ( _c_ ) references to the circus, eating and other pleasures." There are examples using a Greek text (Lamer 1927:2010; Merkelbach 1978) but the remainder seems to be exclusively in Latin for which the hexagram was popular. The following example was found on a board in Trier, Germany (Austin 1934:31):
**Figure 6.4** Top: Drawing of the wooden _duodecim scripta_ board from Qustul, 77.5 × 37 cm (after Emery & Kirwan 1938:pl. 87). Bottom: Drawing of a partially preserved stone _duodecim scripta_ board from Dawwi for which no dimensions were recorded (after a photograph by J.-P. Brun 2003:184).
VIRTUS IMPERI
HOSTES VINCTI
LUDANT ROMANI
In English this hexameter could take the following form (tr. Alex de Voogt):
EMPIRE LAUDED
RIVALS BEATEN
ROMANS AT PLAY
Egyptian locations have so far not rendered any _duodecim scripta_ boards of this kind. The boards attested in Abu Sha'ar, Dawwi and Kom Ombo have small depressions, a feature that is sometimes found in addition to the lettering or instead of circles, squares, vertical bars, leaves, crosses and crescents (Murray 1951:30). The three boards at Abu Sha'ar date to the third and no later than the fourth century CE after which the fortress was abandoned. The table-size board at Dawwi can be dated to the second century CE (Brun 2003:134). It has been partly preserved with three rows of small black circles of which about eight are still visible in each row. The marks separating each set of six fields are also visible (see fig. 6.3). A small fragment of a possible _duodecim scripta_ board was found at Qusur al-Banat with only four small playing circles preserved as well as an outline of a semi-circle divider (Matelly 2003:594, 605), but this example is too small for a positive identification. The often rough carvings of graffiti games referring to _duodecim scripta_ in Egypt are in contrast with the decorative or more appealing examples found elsewhere but it is also possible that hexameters on stone in Egypt have not yet been recognized as referring to a board game.
Qustul (Emery & Kirwan 1938:345–6, pl. 87) provides one of few wooden examples of the game: a perfectly preserved inlaid board with three rows of twelve squares divided in the middle to create six sets of six fields. It is associated with fifteen black wood and fifteen white ivory pieces. In addition, a _pyrgus_ or dice tower was preserved as well as five ivory cubic dice. Due to the extraordinarily good preservation of objects in these graves, the practice of playing on wooden boards can be attested and may indicate that many boards once existed in Egypt in the private sphere that have not been preserved. The grave in Qustul dates to the Post-Meroitic era, i.e., after the fourth century CE.
Another example of a wooden board of which the ivory inlays are preserved together with some wood fragments is found across the border in Sedeinga, Nubia. This find is part of ongoing research on the game board discoveries at this location by Vincent Francigny and Alex de Voogt. Most of the material dates to the first centuries CE and was found in a rich elite cemetery of the Meroitic Kingdom. One grave had two associated cubic dice with fifteen black and thirteen white glass pieces preserved. Another had three dice associated with twenty-seven gaming pieces, some of black wood and others of white ivory as well as several game board fragments. It is evidence that the game was not just played in Roman Egypt but crossed the frontier of the Empire and found its way into the Meroitic Kingdom where at least two graves claimed this game among their grave goods.
The presence of _duodecim scripta_ outside the Roman Empire was already attested for Germania (in Vimose, Denmark) where wooden remnants of the board were also found. Krüger (1982:163) remarks that letters that make up the playing fields are understandably less common in this region as, in general, the players would not have understood Latin. In Germania Krüger (1982:162) attests two bifacial boards that feature _duodecim scripta_ on one side and _latrunculi_ on the other. It is again evidence that these games coexisted and were played by the same people.
In addition to the examples at fortresses in the eastern desert, the Gebel el-Silsila Survey Project identified four _duodecim scripta_ boards near the temple in Kom Ombo (fig. 6.5) showing that their presence is not limited to soldiers' quarters. The presence of _duodecim scripta_ boards carved in monuments or pavements is relatively limited compared to other Roman games. The preservation of this game in Qustul confirms, however, that the game's popularity was widespread and crossed into enemy territory.
Each of the game boards recorded here has three rows of play. A variation of _duodecim scripta_ with two rows is known as _alea_ and became the standard in Rome during the first century CE (Schädler 1995). This variation has not been attested in Egypt. The practice of playing _duodecim scripta_ continued after the fourth century CE as examples from Aphrodisias and Ephesos already indicated but which is also illustrated by the find at Qustul. Today's game of _backgammon_ was preceded by the Arab version known as _nard_ , which has been attested throughout the Islamic Empire. _Nard_ most likely replaced the Roman version in Egypt.
**Figure 6.5** _Duodecim scripta_ boards at Kom Ombo based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
### _Latrunculi_ or _ludus latrunculorum_
Falkener (1892:39) suggested that _ludus latrunculorum_ had its origin in Egypt, equating it with the Egyptian game of _ṯ w_ (see Chapter 4 for a discussion of this game). His comments are part of a long and problematic tradition in which board games from outside of Egypt are given an Egyptian origin. Falkener had only Latin descriptions of the rules for the Roman game and Egyptian depictions of the board for the Egyptian game, i.e., representations of people playing a board that was shown in profile. Only the pieces and not the configuration of the spaces on the board were visible to him when observing Egyptian iconography. Today it is clear that none of the Egyptian game boards have a configuration even comparable to _latrunculi_ ; that there is no reason to assume that _latrunculi_ was played by the ancient Egyptians; and that their connection can no longer be supported. Connections with later games such as _chess_ and _checkers_ are equally problematic. Austin (1934:25) correctly points out that there is no proof that _five lines_ or _latrunculi_ have any connection with _chess_ and finds such a suggestion "inaccurate and misleading."
The Roman game of _latrunculi_ or _ludus latrunculorum_ has been found at the far ends of the Roman Empire (Schädler 1994a). The configuration of this board varies but a field of 7 × 8 squares is quite common. Schädler (2013c) includes 7 × 7, 8 × 8, 9 × 9, 9 × 10, 10 × 11 and even 12 × 12 in the range of possibilities. Unlike _duodecim scripta_ this game does not require any dice. The game does not seem to have a modern counterpart despite its popularity in antiquity. The Greek game _πόλις_ , which is also played on a grid with pieces of two colors and without dice, is understood to be largely the same as the Roman game _latrunculi_ (Schädler 2013a:2844). This game is not discussed here as there is little evidence even outside of Egypt to assist the Egyptologist with identifying a board of this kind and to distinguish it from a _latrunculi_ board. Apart from graffiti boards and the occasional gaming table, _latrunculi_ is also attested in literature and art, and as a board game in progress in terracotta (see Schädler 1994a). An example of the latter was also found in the Fayum in Egypt and is now in the Petrie Museum (UC59258). It shows a board of 6 × 7 squares with seventeen "low domed" playing pieces, neatly placed on the individual squares (fig. 6.6). The placement of the pieces was evidence for Petrie (1927:55) that this was not a _checkers_ game. Instead its configuration fits the description of a _latrunculi_ board, although a small one.
**Figure 6.6** Drawing of a terracotta game board, 9 × 7.3 cm. Petrie Museum, UC59258 (after Petrie 1927:pl. XLVIII.177).
Rose (1996:160) speaks of a "chequerboard" in the Meroitic temple complex of Qasr Ibrim, which was a Roman-Egyptian outpost as well as a Meroitic one. The board on which she reports, consisting of 8 × 7 squares of approximately four by four centimeters each, has the appearance of a game. Modern gaming boards, such as those for _chess_ and _checkers_ , can have eight rows of eight squares and perhaps that is the reason why she mentions a possible eighth line in her notes. South of Qasr Ibrim there were images of 8 × 7 squares found in Mussawarat el-Sufra (location 526.524.02) as well as one with 7 × 14 squares (location 529.525s.26). At this site, the period in which the board was made is less clear since graffiti of many time periods has been attested along its walls and pavements and in large quantities. In line with other Roman games, it is likely that _latrunculi_ also reached the Meroitic Kingdom, which adopted its playing practice.
In the study of the fortress at Abu Sha'ar, Mulvin and Sidebotham (2003) found multiple examples carved in stone blocks, dating to the third and no later than the late fourth century CE. Brun (2011) reports several fragments of _latrunculi_ boards excavated at Didymoi that can be dated with more precision with the help of an archaeological context. The corner of a board with four rows of three squares still visible is dated to 96 CE or not long thereafter (Brun 2011:121, 143). A fragment with seven rows of at least four squares and another with three rows of three squares discernible comes from the period between 123 and 150 CE (Brun 2011:126, 153). The latter was found with fragments of boards featuring small circles and lines that may point to another game. At the fortress of Krokodilo, Matelly (2003:594, 605) reports on a board of which at least five rows of seven squares are visible. At this fortress two playing pieces made of quartz and one of stone as well as one cubic die made of stone were reported. These finds also date to the beginning of the second century CE.
At the stone quarry of Silsila, recent surveys have uncovered a number of board games from Roman and later periods, including at least two _latrunculi_ boards. In the associated Gebel el-Silsila Survey Project it became clear that _latrunculi_ may also be found in Kom Ombo (fig. 6.7). Although the game has been associated mainly with soldiers, these sites suggest that its appearance is not restricted to the fringes of the Roman Empire nor the outposts of its soldiers, but is found throughout Egypt and likely across its southern border in the heart of the Meroitic Kingdom as well. Its future after the seventh or even the fourth century CE is unclear as no comparable board games datable to that period have been attested so far.
**Figure 6.7** _Latrunculi_ board at Kom Ombo based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
### _Merels_ or _mill_ game
Although the _merels_ game (Murray 1951:37) played by the Romans seems insignificant in shape, it stands out as an easily recognized carved game in several excavations throughout the Empire, including Egypt. Mulvin and Sidebotham (2003) attested only one that was partially preserved and difficult to interpret, but the Gebel el-Silsila Survey Project revealed clear examples in the Kharga Oasis, Bachias in the Fayum Oasis, Dendera, Kom Ombo and Kom el-Dekka (Alexandria). The shape of these examples allows for nine intersecting points (fig. 6.8). The final shape, as recorded in Egypt, is a square intersected by two perpendicular lines. In Silsila there is also a version with additional diagonals, which can be considered just a different type (Schädler 2013a:2844).
**Figure 6.8** _Merels_ boards at Dendera (top left), Kom el-Dekka in Alexandria (top right), two boards together at Kom Ombo (second row), three single boards at Kom Ombo and a board at Silsila (bottom right) based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
Although there is no Latin name known for this game, the rules have been deduced from two poetic lines in Ovid's _Ars Amatoria_ (see for a discussion Rieche 1986). Väterlein (1976:59) remarks that it is the same as the _Kleine Mühle,_ a game still known in German-speaking Europe. It is indeed related to the _merels_ , _mill_ and _three-men's-morris_ game, and is perhaps the only board game dating to Roman times that survived the ages without any significant change. Each player tries to move his or her three pieces into a row. Although many variations of this game have been described and found around the world, today the basic principle is still the same.
Instead of a square, there have been frequent finds of circles intersected by three lines, making eight intersections with the circle and one in the center where all three lines overlap. Heimann (2014) has questioned the fact that this is the same game or even a game at all. An analysis of the rules has shown that a "loop takes effect also in everyday playing." Behling (2013) has argued that this is akin to a children's dexterity game described in ancient Greek literature in which nuts are thrown into the circle where they should not touch the lines. It suffices to state that circular patterns with intersecting lines are different from the _merels_ game.
According to Schädler (2013a), the game of _nine-men's-morris_ or _larger merels_ (Murray 1951:43), which is a more complicated version of this game, does not appear in the record until Byzantine times. In Egypt, the Gebel el-Silsila Survey Project photographed one such game next to one of the columns in the Ramesseum, across the river Nile from today's city of Luxor (fig. 6.9). Similar to the game of _three-men's-morris_ , this game and this game board have a wide distribution far beyond the Middle East. Acknowledging its presence in Egypt may assist in documenting this international presence and the people associated with carving them in monuments. The particular design of the one example spotted in the Ramesseum should not be considered a Roman game.
**Figure 6.9** Drawing of a _nine-men's-morris_ board as found on a column at the Ramesseum based on a photograph from the Gebel el-Silsila Survey Project.
### _Marbles_
The Greco-Roman period in Egypt resulted in the introduction of multiple games, including dexterity games and sports. Although only game boards and dice are given attention here, designs meant for dexterity games may sometimes interfere with the interpretation of the archaeological finds, such as the circle with intersecting lines described above.
Rieche (1986) explains that board and dice games were played by adults, both men and women, while children played dexterity games such as _marbles_. The game of _marbles_ is particularly relevant as it may use a marble lane carved in stone that confuses the identification of similarly carved board games.
Suetonius is credited with a _liber de lusibus puerorum_ or a book on children's games, which, similar to his book on Greek dice games, has not been preserved (Väterlein 1976:13). Rieche (1984:10–13) mentions games played with (wal)nuts and Väterlein (1976:37) even mentions the possibility of _marbles_ but neither go beyond the references made in the Latin sources. Schädler (1994b, 2013b) described and illustrated marble lanes as found on archaeological sites in Rome and pointed out their relevance (fig. 6.10).
In the basis, a marble lane consists of parallel or irregular rows of holes at one end and a line from which to start at the other end. The irregularly placed holes prevent the marble player from rolling into the holes at the very end, while the line indicates the starting position, possibly with a depression to store some marbles.
**Figure 6.10** Drawing of a marble lane (after Schädler 1994b:56).
The immediate relevance of this game is not found in the assumption that many children used marble lanes in Egypt, although this may be possible but cannot be ascertained at this time. Rather it encourages the archaeologist who encounters a group of holes in an irregular order to consider the possibility of a marble lane. This could consist of a starting line a few feet away from these holes, preferably with a largely flat surface, interspersed with an occasional hole, between them. Documentation of groups of holes rarely includes a survey of the vicinity that could point to or exclude such a possibility.
### Remaining configurations
There remain a number of configurations of squares and holes that are placed systematically and intentionally so that their presence does not appear random or coincidental. They may be variations of boards that are already known, incomplete, corrected or amended shapes, or they may be part of games not yet described. It is, of course, also possible that they are not game boards at all even if an alternative purpose is unknown. In any case, it is necessary to depict at least some of these examples for future comparison as these configurations commonly remain unpublished. Indeed, the suggestion of a new gaming practice is reinforced if the same configurations are found in different sites as well as in the presence of other game boards.
In the Beni Hasan quarries a clearly defined board design of two rows of eight square fields was photographed during the Gebel el-Silsila Survey Project. A similar design of two rows of seven squares is found in Silsila West (fig. 6.11). Such game configurations complicate an interpretation of _mancala_ as it is rather awkward to play on a square field without depressions needed to hold the multiple counters used in a _mancala_ game. These game configurations are not yet understood and it is appropriate to state that the games in Egypt are only part of a much larger puzzle.
One example from the Kharga Oasis illustrates how a complicated set of lines can be interpreted multiple ways. In this figure about six rows of ten squares can be distinguished (fig. 6.12). In the center of the configuration, there is a column of squares that has an unusual shape and the bottom row looks eroded. It is possible to distinguish a 5 × 5 _seeja_ board (see Chapter 7) as part of the right-hand side of this configuration. The carving at closer inspection suggests a number of attempts at creating a _seeja_ board while there is a possibility that a _tâb_ board was attempted as well. Since these are only suggestions, the presence of a game in this configuration requires a games context or other supportive evidence in this location to make a more educated guess.
**Figure 6.11** Two unidentified game boards at the Beni Hasan (left) and Silsila (right) quarries based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
Although a long list of possible game board designs could follow, most examples are better explained as eroded, partially completed or non-game designs. Without at least one comparable example elsewhere there is also no need to assume a "new" game board design. It remains essential that projects such as the Gebel el-Silsila Survey Project document such carvings in addition to the archaeologists active on a site.
### The borders of Egypt
The board games discussed in this chapter were illustrated with examples from Roman sites in Egypt and across its southern border. Towards the south is Qasr Ibrim, a temple complex and fortification that was conquered by Romans but with Meroites alternatingly having control of the site. Qustul, the location of one of the best-preserved _duodecim scripta_ boards, was never Roman and dates to the Post-Meroitic period, i.e., after the fourth century CE. It became part of the Christian kingdoms in Sudanese Nubia, outside of the Roman Empire. Sedeinga with its cubic dice and possible evidence of _duodecim scripta_ is squarely located in the Meroitic Kingdom and the discovery of games in that location coincides with the time of Roman Egypt. It was never conquered by the Romans and was an important center within the Meroitic Kingdom.
**Figure 6.12** Possible _seeja_ board at the Kharga Oasis based on a photograph from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
Abu Sha'ar is on the Red Sea Coast and its location can be securely placed within Roman Egypt. Its function as a military fortress follows the popularity of the games among the Roman military. Fortifications on the borders of Egypt (Brun 2003, 2011; Matelly 2003) and other parts of the Empire, such as at Hadrian's Wall (Austin 1934:26), show a similar pattern. So far only the work of Mulvin and Sidebotham (2003) is specifically dedicated to Roman games in Egypt. Most isolated examples have not yet been published or were part of studies that did not focus on the understanding of games. The various examples provided with the help of the Gebel el-Silsila Survey Project indicate that the presence of Roman games is much more prevalent than the published literature suggests. Although Roman soldiers may still be the main group of players, the games are not limited to fortresses on the border of the Roman Empire. The quarry site of Silsila and the Ptolemaic temple at Kom Ombo both feature multiple Roman games that complement the inventory of games found at the borders of the Roman sphere of influence.
This chapter provides the background for future studies of Roman games in Egypt. Game boards can be more readily recognized and interpreted in the context of the sites discussed and their interpretation is facilitated by comparisons with existing finds from Roman Egypt and the Roman Empire in general.
##
## Arab and Ottoman Invaders Scratching the Surface
Although the history of modern board games in Egypt cannot be covered in a single chapter and is not part of our study, both Arab and Ottoman game practices have sometimes confused the understanding of ancient games. These games are mainly found scratched on pavements and monuments, erroneously suggesting an ancient history by their association with Pharaonic architecture. They can also be found next to games that date to Roman or Pharaonic times, so that even the games context in which they are found gives rise to confusion. Only an understanding of these games in contexts outside of Egypt allows them to be differentiated. This chapter disentangles the graffiti games introduced after the Arab conquest up to modern times from the Roman and earlier game practices in the record by describing their main characteristics and by pointing at possible problems of identification.
### Graffiti games
Game boards that have been confused with those from antiquity have been uniquely and consistently discovered as graffiti games. If more recent games are scratched in existing buildings it is the date of the building or the presence of earlier games from antiquity that may lead to misconceptions. As emphasized in the previous chapter, the date of the building does provide a _terminus post quem_ for the date of the game, but does not imply the game was played at the same time the building was built, or even in use. Whenever possible, an understanding of the taphonomy of the building in which games were found may provide some understanding of the date of inscribed games.
The types of games that can be found as graffiti are limited. They can be a configuration of holes, usually small depressions, or a grid of lines creating a board with square fields. An instant recognition of these games is particularly helpful to identify the era in which the incisions were made, or at least to attribute them to the relevant population.
The time period in which Arab and Ottoman games were made is stretched between the seventh century CE and the present day, as some of these games are still being played in Egypt and neighboring countries. Even if this occurred in recent memory the local population may no longer be familiar with these games. Some games can only be attributed to Ottoman invaders, usually soldiers. This appears particularly true for _mancala_ games that are rarely found on rock faces or monuments outside the Ottoman sphere of influence. Research on this Ottoman tradition of carving games in stone is ongoing and the date and origin of such games is refined each time new finds are made (see de Voogt 2010, 2012; Charpentier et al. 2014).
There are three games that have been associated with the Arab and Ottoman Empires. The game of _seeja_ commonly consists of five rows of five fields, either small depressions or squares. The game of _tâb_ often has four rows of seven or more fields, frequently seen as small depressions as well. Finally, the game of _mancala_ usually has two rows of six or more cup-shaped holes, usually larger than those found for _seeja_ and _tâb_. The variation that is found with each of these games and the occasionally overlapping configurations between games has to be taken into account whenever a game board is identified on an archaeological site (see fig. 7.1).
Arab and Ottoman games can be found together with Roman games since they often share the location of play. More problematic for the archaeologist are those games that were reused by later Arab or Ottoman players. Examples include a reused _latrunculi_ board for making a _seeja_ board at Silsila—which is part of ongoing research on games in collaboration with Maria Nilsson and John Ward—, possible extensions of the game of _five lines_ to create _mancala_ boards at Palmyra (de Voogt 2010) and contexts such as the roof of the temple of Khonsu where games from Pharaonic times are joined by _seeja_ and possibly _mancala_ or _five lines_ games (Jacquet-Gordon 2003). Without an understanding of the latest of additions in such a games context, it is impossible to disentangle the different origins and time periods in which these games were made.
**Figure 7.1** Example of an Arab or Ottoman games context at Petra, Jordan (top) with a 7 × 7 _seeja_ board, _c._ 40 × 35 cm and a 4 × 12 _tâb_ board _, c._ 78 × 29 cm, and on Sai Island, Sudan (bottom) with mostly 2 × 6 _mancala_ boards _,_ a 5 × 5 _seeja_ board, _c._ 40 × 33 cm and a 4 × 12 _tâb_ board, _c._ 72 × 21 cm. Photographs by Alex de Voogt in 2014 and 2010, respectively.
### _Seeja_ or _siga_
In 1694 Thomas Hyde remarked on "an interesting game called _seejeh_ or _sutreng_ " as played by the Bedouin of Israel (translation Keats 1994:97). Without further explanation he states that the game is "mentioned in Egyptian papyri, and played in Ancient Persia, using glass pieces on carved stone boards." His description is in the context of his history of _chess_ , and although it is "quite unlike chess, and should not be confused with it," Hyde is probably one of the first Western games historians who became part of a long tradition of relating certain board games to ancient Egypt.
_Seeja_ —a transliteration of Arabic that has variations such as _sija_ , _siga_ and _seega_ —is a board game that does not resemble other games from antiquity. It commonly consists of five rows of five holes or squares, unlike most other games discussed here, which have either two, three or four rows of playing fields. It cannot be found in documents dating to Pharaonic Egypt nor is the placement of this game on ancient monuments evidence of its supposed long history. On the contrary, there is no evidence in Egypt or in neighboring countries that _seeja_ was played before the Arab invasion and in several cases before the Ottoman presence in the region.
Jacquet-Gordon (2003:12) in her study of the graffiti of the Khonsu temple roof states that she has no reason to suppose that the game, which she spells as _siga_ , is not ancient. She states that the dating of the graffiti on the temple remains obscure. However, on two occasions this game covers earlier graffiti, in one case "a tiny figure of Khonsu" and in the other a lion, suggesting a players' group that is not associated with the imagery of these figures. These two games consist of only two rows of five holes, which she interprets as an unfinished _seeja_ game (Jacquet-Gordon 2003:17–18) but, as has been discussed elsewhere, it is more likely to be an example of the game of _five lines_.
### _Seeja_ playing rules
The 5 × 5 configuration of _seeja_ is easily recognized and helps to date the games to the period after the Roman era. The grid can be found as a set of twenty-five squares or as a set of small depressions. The variation reminds one of the game of _five lines_ in that the appearance of the board is either a set of fields, small depressions or a combination of the two. In some cases the nature of the material may explain these preferences. Apart from the variation of squares and holes, there are multiple examples where the grid is slightly larger or smaller, usually not larger than seven rows of five or seven fields and commonly not smaller than four rows of four although the latter is often interpreted as an unfinished example.
_Seeja_ uses two distinct sets of playing pieces (Murray 1951:54–5). When played in the sand, differently colored sticks can be used that are then stuck into the board. On stone, any pebble, seed or small piece may do as long as the two opponents can be differentiated. There are no dice needed for this game and the number of players or sides is limited to two.
_Seeja_ is still played today and the board has different sets of rules. For instance, in Sudan a similar and popular game is called _dhala_ , which uses the same grid, often in the shape of depressions in the sand. Lane (1908:356–7) has described what he called _seegà_ for modern Egyptians with rules similar to those found elsewhere in the Middle East. He describes the board as commonly consisting of five rows of five holes, or seven rows of seven or even nine rows of nine. As in the game of _tâb_ and the game of _mancala_ , the size of the board varies but the rules are usually identical.
Lane (1908) calls a hole _eyn_ (eye) and a playing piece _kelb_ (dog). For a game played on a five-by-five board there are twelve pieces for each player. The two sets of twelve are differentiated by appearance. Lane has the first two pieces of each player placed as shown in fig. 7.2.
After this initial placement, players take turns putting two of their pieces on the board with each turn. They may be placed in any empty space except the central one. Once all the pieces have been placed, one player starts by moving one of his or her pieces from an adjacent field into the empty central space. If the opponent has no adjacent piece to place into the newly emptied space, the first player loses a piece as he or she is required to open up a place. If such a situation occurs again in the game, the same rule applies. Lane formulates the aim of the game as follows (Lane 1908:357):
The aim of each party is to place any one of his kelbs in such a situation that there shall be, between it and another of his, one of his adversary's kelbs. This, by doing so, he takes; and as long as he can immediately make another capture by such means, he does so, without allowing his adversary to move.
**Figure 7.2** Placement of first pieces on a _seeja_ board as discussed by Lane (1908).
Other sets of game rules may be present in Egypt but the rules above suffice to explain the nature of the game and the design of the playing board that comes with it. Lane (1908:357) adds that several boards "have been cut upon the stones on the summit of the Great Pyramid, by Arabs who have served as guides to travelers." This early observation by Lane should inform those who need to interpret the history of this game on top of ancient monuments in Egypt.
The Gebel el-Silsila Survey Project revealed examples of _seeja_ in el-Kab, Medamoud and several at Silsila (fig. 7.3). Also there are examples of four rows of five holes found at Edfu that may be related. In Silsila there are a few examples where more rows of depressions are found but where the five-by-five configuration stands out. Transformations of one game into another by adding or accentuating the desired configuration have been attested across time periods in Palmyra (de Voogt 2010) but can also be found in Silsila where a _latrunculi_ board was partially reused for _seeja_.
**Figure 7.3** Examples of _seeja_ boards at Silsila (top left), Medamoud (top right) and four at el-Kab, based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
### _Tâb_
The game of _tâb_ is not as well known today as the game of _seeja_ but it is also found in the Middle East, including Egypt and Sudan. Four rows of seven or more depressions are used while the players use a set of throwing sticks or stick dice as the preferred randomizer for the moves. The game is played by two opposing sides, each moving one set of colored pieces around the board. Each opposing side may have two players that work as a team. The team members alternate throwing the dice to determine the next move.
Unlike _seeja_ , which uses five rows and often has fields, the four rows of holes used for _tâb_ create a similarity with _mancala_ games. Four-row _mancala_ boards that are known for East Africa can be identical in configuration to a _tâb_ board. The size of the playing holes is likely to be different. In _mancala_ games, the holes should be able to contain multiple counters at a time while in _tâb_ , each depression usually holds just one or two pieces.
_Tâb_ , which is a transliteration of Arabic, has different spellings including _ṭáb_ as it is used by Lane (1908:353–6). It is attested for modern Egypt with a published complete description of the rules as they are used today. Similar descriptions are found in Niebuhr (1774:188–9) and the earliest attestation of this game in the Western literature is found with Thomas Hyde (see Depaulis 2001:56).
The game commonly requires four sticks, according to Lane often cut from a palm-branch. They are cut so that they have two sides; one is called white and the other black (Lane 1908:353):
Next, it is necessary to be provided with a "seegà." This is a board, divided into four rows of squares, called "beyts" or "dárs," each about two inches wide; or it consists of similar rows of holes made in the ground, or in a flat stone: the beyts are usually seven, nine, eleven, thirteen, or fifteen, in each row.
As Lane points out, the board varies in size and the size merely affects the duration of the game but not the playing rules. The value of the throws of the sticks are defined by the number of white (or flat) sides that are visible. Each white side counts for one. For example, three whites and one black scores three. The only exception is four black sides, which score six. A score of one—called _ṭáb_ or _weled_ (child)—, four or six requires the player to throw again and the player only stops throwing when a two or three is thrown in which case it is the turn of the opponent.
Each player owns two rows of _beyts_ , the outer rows are commonly filled with pieces, also called "dogs" according to Lane (1908:354). The word "dog" for gaming piece is also found in ancient Greece (Schädler 2002) and Pharaonic Egypt (see Chapter 3), but contrary to Falkener's suggestion (Falkener 1892:39) that does not link the game boards or game rules. The game of _hounds and jackals_ (see Chapter 5), also features dog-shaped pieces so that a continuous tradition of such pieces can be ascertained from ancient Egyptian to ancient Greek and Ottoman times. In each period, however, the games are known to be different and the majority did not originate in Egypt.
In order for a piece to move from the outer row the player has to throw a one. After this piece leaves the back row counterclockwise, it circulates in the inner two rows in a clockwise direction. If it lands on a field with an opponent's piece, then this piece is captured and removed from the board. If it joins a piece of its own, it may move as one, i.e., two or even three pieces move as if they are one piece. A three-piece combination is made by either moving a set of two to a field with one piece, or by moving one piece onto a field with already two. They can only be separated again by throwing a one. Once a player moves a piece into the outer row of the opponent, it can no longer be captured although it may capture pieces that have not been removed from that row by the opponent. The object of the game is to capture all the opponent's pieces. This technical information about the playing rules is relevant here since it determines the size of the playing fields as well as the playing pieces. Each field ideally accommodates three pieces, at least where the Egyptian variation is concerned (see also Murray 1951:95).
The game has been attested for Sudan, on the island of Sai (de Voogt 2014), where it was found carved next to a series of _mancala_ games as well as one _seeja_ game. In the context of two-row _mancala_ and _seeja_ , _tâb_ is shown to belong to a particular group of people active in a specific time period. So far, this time period has coincided with the presence of the Ottomans but the period between the arrival of the Arabs and the colonial conquests of the French and the British in the nineteenth century is such that perhaps other periods can be associated with these games for other sites. In the Gebel el-Silsila Survey Project, only the site of el-Kab showed possible examples for the game of _tâb_ (fig. 7.4).
**Figure 7.4** Examples of _tâb_ boards at el-Kab with one possibly combined with a _seeja_ board, based on photographs from the Gebel el-Silsila Survey Project. Courtesy Maria Nilsson and John Ward.
### _Mancala_
As with almost any board game discussed here, the origins of _mancala_ have at some point been placed in ancient Egypt. Parker (1909:589–91) enumerates several examples of rows of holes cut "in the roof-slabs of the Kūrna temple in Upper Egypt," "on the summit of the damaged portion of the great pylon built in Ptolemaic times at the entrance to the temple of Karnak, as well as on the tops of the walls there" and "at the Luxor temple." These were also mentioned by Murray (1951), Walker (1990:34–5) and mentioned but finally criticized by Schädler (1998). Parker (1909:Fig. 256) adds a photograph of the holes at the "Third Pyramid, Gizeh" (fig. 7.5), an image that is not much dissimilar of the rows of holes found on the remains of the temple of Tiye at Sedeinga or those found on top of the Soleb Temple, two New Kingdom Egyptian monuments both found in Sudan (fig. 7.6). Also, the Gebel el-Silsila Survey project revealed two such games at Dendera; one consisting of two rows of six holes and the other of two rows of five holes with two end-holes, i.e., an additional hole at each far end.
Johnston (1913:384) states that as early as 4000 BCE "there came all musical instruments superior to the musical bow and the drum, several types of games played with hollowed or divided boards and counters, and a good many Egyptian notions about religion" that "began to penetrate Negro Africa." Again this comment was picked up and repeated by Murray (1951:159) and Walker (1990). While Johnston placed the origin of all these matters in ancient Egypt, Schweinfurth (1874, 2:28) speaks about a game known by the Nubians as "mungala." He stated it to be "singular that this pastime be so familiar to the Mohammedan Nubians, who only within the last twenty years have had any intercourse at all with the negroes of the south; but in all likelihood they received it in the same way as the guitar, as a legacy from their original home in Central Africa." The accompanying illustration (Schweinfurth 1874, 2:26) shows a board of two rows of eight holes with two end-holes, a design and configuration unknown in the Middle East. Today's Nubians do not seem to play this game anymore and it does not seem to have traveled down the Nile. Although these early twentieth-century sources may be considered dated, and today it is difficult to find Nubians playing this game, the connection with Egypt is not a thought that has vanished.
**Figure 7.5** Drawing of a _mancala_ board at the "Third Pyramid, Gizeh," dimensions not recorded (after Parker 1909:Fig. 256).
**Figure 7.6** A 2 × 6 _mancala_ board carved in the temple of Tiye, Sedeinga, Sudan, _c._ 21 × 61 cm, and a partly eroded 2 × 6 _mancala_ board on top of the temple of Soleb, Sudan, _c._ 20 × 54 cm. Photographs by Alex de Voogt, 2011.
The esteemed games researcher Bell (1979:12), who based his view partly on rows of holes found on Egyptian ruins, suggested that it may have originated from "black Africa and had been taken to Egypt at a very early date." A previous publication edited by Grunfeld (1975:20) but with Bell acting as a consultant states: " _mancala_ games... have been played for thousands of years in Egypt, where boards have been found carved into the stone of the pyramid of Cheops and the temples at Luxor and Karnak." It is the presence of games on Egyptian monuments that has created a series of other connections as the temple of "Kurna" is mentioned to feature an unfinished " _alquerque_ diagram" (Grunfeld 1975:38) as well as a _nine-men's-morris_ game "probably carved by the workmen who built the temple around 1400 B.C." (Grunfeld 1975:59). Both the identification of the game and the associated date are problematic. It cannot be emphasized enough that the date of the erection of a building is not necessarily related to the period in which the game was scratched into its roof.
Graffiti games on Egyptian, Nubian, Roman or other monuments from antiquity should not be immediately interpreted as dating to the time of the monument itself. The latter can only be assumed if the games are revealed within an archaeological context that leaves no doubt as to its origin and age. The accessibility of the temple at Karnak, but also structures such as the Nubian pyramids (de Voogt 2012) and the many monuments dating to Roman times (Schädler 1995, 1998; Mulvin & Sidebotham 2003; de Voogt 2010, 2012) have shown that games from different periods and origins can be present. Contrary to the beliefs of Grunfeld, Bell and other authors on game history, the temples of Kūrna and Khonsu feature games played by the ancient Egyptians as well as later Roman or Arab and Ottoman visitors that came after them.
Jacquet-Gordon attested a "twelve-hole game board" (Jacquet-Gordon 2003:13) consisting of two rows of six holes on the roof of the temple of Khonsu but refrains from speculating that this is a _mancala_ game or that it was carved by ancient Egyptians. She mentions two games that were not assigned an entry number, one of which obliterated an older inscription. Only one "coherent graffito" is assigned a number, i.e., it received a separate graffito entry number, representing this board configuration. The presence of games covering inscriptions also makes an ancient Egyptian origin less likely.
Mulvin and Sidebotham (2003) identified two and three rows of holes of mostly five and sometimes four holes. They concluded that _mancala_ games were present and, in light of the games context and the presence of _mancala_ games in nearby geographical locations in today's world, this seemed a correct conclusion. But the places where _mancala_ is played on two rows of five or four are few. _Mancala_ games with three rows have been attested in modern times as well but only with six holes per row. More importantly, the game of _five lines_ is a more likely candidate and has already been attested in other Roman sites and fits with the limited number of holes per row as well as the presence of several other Roman games present on their particular site. The absence of any additional evidence that Romans played _mancala_ , such as imagery or literary references, further questions the conclusion of Mulvin and Sidebotham.
The Gebel el-Silsila Survey recorded two boards at Dendera that have a configuration and an appearance consistent with _mancala_. They are not found in a context that could identify their makers. More extensive examples were documented for Sudan, particularly on Sai Island (de Voogt 2014) where also an Ottoman games' context is present (see also fig. 7.1).
Modern Egyptians play a version of _mancala_ , but they are not known to use the rows of holes carved in rock and monuments even though that may not have been the case in Ottoman times. This is also true for its immediate neighbor to the south, Sudan. In South Sudan, i.e., south of Sudan, there is a lively tradition of _mancala_ that does not seem to have reached the Nile Valley since they prefer four rows of holes and do not prefer rock but rather sand to play their games. The same is true for Ethiopia and Eritrea where _mancala_ is popular, mostly found as wooden boards, and quite distinct from the _mancala_ versions in South Sudan and the rest of the Middle East. If anything, the Nile Valley has been an obstacle in the distribution of _mancala_ games with games from the Middle East not traveling further south, and games from sub-Saharan Africa not traveling further north.
The earliest description of _mancala_ rules, as opposed to possible game boards, in modern Egypt is probably by Carsten Niebuhr in 1774 who mentions Maronites playing in Cairo (Niebuhr 1774:185). The board has two rows of six holes and a starting configuration of six counters in each hole. A player picks up the contents of one of the holes on his or her side and spreads them one-by-one in counterclockwise direction. If the last counter ends up in a hole where the contents becomes either two, four or six, the player can capture its contents and also the contents of the holes directly preceding this hole if these holes happen to have two, four or six counters as well. The rules are similar to those still found in the Middle East. For instance, in Tartus, Syria, the board has two rows of seven holes but the rules of play are similar with captures of all those that have two or four in one hole (Alex de Voogt, field notes).
Ottomans are known to have played _mancala_ , as there is evidence found in illustrations and within the historical geographical confines of the Ottoman Empire. For instance, there is a banquet scene found in a sixteenth- or seventeenth-century Turkish album of miniatures and calligraphy that shows men drinking coffee, playing _backgammon_ and _mancala_ (Chester Beatty Library/Bridgeman Art Library CBL71659, Ms 439 f.9r). _Mancala_ games are still found in Turkey and Syria that share a more or less common appearance. There is also a frequently quoted reference to _mancala_ in Arabic (Murray 1951:165), a name of a game from which today's word _mancala_ has been derived. Yet, outside of the Ottoman Empire there has been little evidence that _mancala_ games were played on stone surfaces and, even on the illustration mentioned above, it is a wooden game board. Since the ancient history of _mancala_ is not sufficiently known, the attribution of stone or graffiti _mancala_ boards to ancient Egypt cannot be accepted without additional evidence.
The identification of _mancala_ games in rock is complicated by the fact that they may be confused with other games such as _tâb_ or the game of _five lines_ or it may not be a board game at all but a game of _marbles_ or part of an area where other activities using stone give the appearance of a set of playing holes. Identifying _mancala_ on a rock surface requires a careful outline of two rows of at least six holes, each hole with a size that accommodates a small hand, and a location that seems suitable for playing games. Even with these characteristics one cannot always be sure, but usually other games in the vicinity may point to the possible origin and may create a so-called games context that could confirm the presence of _mancala_.
There is no archaeological evidence that the ancient Egyptians played _mancala_ or _seeja_ , nor _alquerque_ and _nine-men's-morris_ for that matter. When _mancala_ boards appear in the archaeological record they are always found in locations where an Ottoman or late Arab presence can be assumed. When _mancala_ boards are found at archaeological sites in Syria, Sudan and Egypt, usually _seeja_ boards are present as well, suggesting that these games were played by the same people. Boards with two rows of five holes, particularly in the vicinity of Roman games, can be safely dated to Roman times and were most likely used for the game of _five lines_ (Schädler 1995, 2008). The material found in Egypt does not contradict this reading of the evidence.
##
## The Role of Board Games in Understanding Antiquity
The archaeological and historical records of Egypt have supplied about five thousand years of games history. The documentation of such continuity is a rare but crucial occurrence for understanding the roles board games played in the ancient world. The overview in this volume shows commonalities in the ways board games were harnessed for societal processes, as well as a changing array of social niches in which they were preferred. This kind of research into ancient games is only a beginning and the evidence that is currently available should advance rather than finalize the conversation about the role of games in culture.
### Spread of board games
The major board games in Egypt were either transmitted to neighboring regions or came to Egypt from foreign lands. Even in the third millennium BCE _mehen_ spread to Cyprus and possibly Mesopotamia. Shortly thereafter, _senet_ was brought to the Levant and Cyprus. Perhaps as early as the Middle Kingdom, the _game of twenty_ found its way from Mesopotamia and the Levant to Egypt, and became popular during the New Kingdom, only to disappear quite suddenly during the Late Period. _Hounds and jackals_ found its way rather quickly from Egypt to Anatolia and Mesopotamia, and maintained a long popularity outside of Egypt. _Five lines_ , _duodecim scripta_ and _latrunculi_ all came to Egypt once increasing Greek and Roman influence took hold in the years following the conquests of Alexander and Augustus. Finally, _tâb_ , _seeja and mancala_ appear to have arrived through Arab and Ottoman invasions and occupations.
This simplified narrative of game playing in Egypt masks multiple kinds of interactions that were relevant to the spread of all of these games. For _mehen_ and _senet_ , trade is likely the primary mechanism by which the games were introduced into new lands. For _senet_ in particular, the changing pattern of game board-producing sites appears to reflect what we know of shifting trade patterns. Trade is accompanied by migration as a mechanism by which games arrived in different regions, and evidence for the _game of twenty_ suggests that it initially came to Egypt through the migration of Levantine peoples during the Middle Kingdom. Migration may be seen in the appearance of _tâb_ and _seeja_ as well. The diffusion of _hounds and jackals_ outside of Egypt is less obvious, as it seems to have spread very quickly to distant regions that lack strong cultural or exchange relations (e.g., Anatolia), suggesting there is a lacuna in the evidence for the existence of this game in other portions of the Near East.
Greco-Roman games in Egypt can be explained by a complex combination of conquest, trade and emulation. Of course, the incorporation of Egypt into the Hellenistic world after the conquest of Alexander and the subsequent division of his empire into autonomous regions with Greek rulers led to the promulgation of Hellenic culture throughout the Near East, which encouraged the migration of Greeks and emulation of them by the indigenous population. Similar processes happened during the Roman Empire when legionnaires were stationed throughout the eastern part of the Empire. Complex interactions between Roman citizens living outside of Italy and local peoples led to processes of Romanization that were neither one-directional nor universal, and games can be seen as part of this process.
All of these processes—trade, migration, emulation—are ways in which people address a need to interact with one another across linguistic, political, socio-economic and religious boundaries. The concept of a "social lubricant," a practice that facilitates interaction across such boundaries, is relevant here. Research on social lubricants has traditionally focused on intoxicants and psychoactive substances, but games function similarly with regard to the facilitation of interaction. Board games are commonly played by two people. This requires people to interact before a game can take place, and thus the location of a game board may point to places where people were in contact with one another. The liminal nature of games, which allows for people to ignore socially-constructed boundaries (Turner 1982:27), functions similarly to ingested social lubricants in that they often take place in specific places and times, during which relationships can be strengthened, changed or created (Crist 2015; Crist et al. in press). Unfortunately, the archaeological evidence from Egypt is predominantly mortuary in nature such that it is difficult to identify places and spaces in which games were played. Instead, the broader geographic distribution of the games and comparison to contemporary social processes may provide insight on this function of Egypt's games.
Since the appearance of _mehen_ around 3000 BCE, board games have shown a regional popularity. When _senet_ increased in popularity, Egypt's main interactions with the outside world were to build relationships with local peoples with the purpose of creating economic ties in order to secure important goods that were absent from Egypt itself, such as copper and cedar, as discussed in Chapter 3. The appearance of _hounds and jackals_ in Anatolia and Mesopotamia could point to a similar economic impetus, although the specific mechanisms by which they would have traveled to these regions are not readily apparent.
Though migration was the process that initially brought the _game of twenty_ to Egypt, it was the popularity of the game among Late Bronze Age elites throughout the Near East that appears to have led to its adoption by the Egyptian nobility. The game was decorated in the distinctive International Style, using shared symbols of power among the elites. This form of emulation was the primary cause of the spread of this game, and not the Egyptian conquest of the Levant. Little evidence for the presence of _senet_ exists in the Late Bronze Age Levant even though it was at the height of its importance in Egypt. After the region-wide Late Bronze Age collapse, the _game of twenty_ disappeared from Egypt, never to reappear, as its functionality was eliminated along with the extinction of the system of elite exchanges.
More emulation can be seen in a potential re-introduction of _hounds and jackals_ during the Persian occupation of Egypt, as well as the arrival of Greco-Roman and later games. Whereas previously the clearest evidence for the adoption of a foreign game into Egypt was the _game of twenty_ , all of these later games demonstrate the changing political realities in Egypt, in which the Nile Valley's power was diminished but played a key role in the empires to which it was subject. It was through an influx of transplanted Greeks and Romans, such as at the fortress at Abu Sha'ar, that brought Greco-Roman games to the Nile Valley, but the state of the evidence at present makes it difficult to determine the degree to which these games were played by local Egyptians. Attitudes toward pagan practices likely brought about a changing landscape of games with the adoption of Christianity, and a Coptic game gained in popularity during Late Antiquity, as did _tâb, seeja and mancala_ during the Islamic periods.
This discussion of the changing landscape of games is again simplified, as the interactions that brought games to new places and the processes that affected their popularity across complex social networks are difficult to assess in the archaeological record. Ongoing archaeological research and new approaches to board games in Egypt will enrich, correct or confirm the above histories.
### Religiosity of board games
In researching cultural transmission of board games, de Voogt et al. (2013) found that games cross cultural boundaries with remarkable fidelity. They retain much of their morphology and, likely, many of the rules. That is not to say that people adopting a new game will assign it similar social connotations. One cannot assume that the religious connotations a game had in one culture were transferred to the adopting society.
The religious use of _senet_ is found prominently during the New Kingdom when texts, representations, and symbols on the boards themselves demonstrate its importance in this arena of life. What is less understood is whether the game _always_ had this connection. Its importance in mortuary ritual is made clear, as is its presence in tomb assemblages, but the nature of the archaeological record in Egypt does not provide much evidence for how it was played in domestic contexts. Graffiti games from various sites do suggest a non-cultic use, as does its presence in the Levant and Cyprus. In contrast, Levantine and Cypriot contexts do not support a religious or ceremonial function for _senet_ or for _mehen_ , which also had strong cultic symbolism in Egypt (Sebbane 2001; Crist 2015). The symbolism of these games was not translated into the cultures receiving these games; rather, they were adopted into existing cultural milieux.
The connection between divination and the _game of twenty_ outside of Egypt did not translate to its use in the New Kingdom. While it is possible the practice existed among the Levantine peoples settling in Egypt, there is no indication in the textual evidence that Egyptians used it for this purpose once it became popular among the nobility. Its divinatory use continued in Mesopotamia and the Levant alongside its symbolic use by elites, and lasted long after its demise in Egypt. The _game of twenty_ also does not appear to have been adopted into any form of Egyptian cosmology relating to the afterlife. Though it appears opposite _senet_ on game boxes, texts and scenes are silent on the game. This suggests a non-religious use for the _game of twenty_ , like many of the other games found in Egypt about which texts reveal no information.
Greek and Roman games, particularly those involving dice, are associated with gambling practices rather than divination and religious practice. _Senet_ did not become part of this tradition, and, with its eventual demise, the religious connotation of games disappeared from Egypt. At the rise of Christianity during Late Antiquity, pagan symbolism related to Egyptian games was long abandoned, and games from then on and continuing throughout the Islamic periods appear to have been strictly a secular practice.
### Site use
As has been stressed throughout this volume, attribution of graffiti games to certain time periods or practices requires care, as the structure on which they are found merely provides a _terminus post quem_ for the game pattern itself. The appearance of a Roman or Islamic game on a Pharaonic monument does not imply that the appearance of the game in Egypt should be ascribed to the date of that monument, rather it implies that the monument was in use or at the least unburied during later periods, and was utilized as a place in which to play a game, for whatever reason the players deemed fit.
Identifying the game itself is a crucial first step in dating the carved pattern. Some patterns, such as _merels_ , _duodecim scripta_ and the _game of twenty_ are sufficiently diagnostic that their presence on monuments allow for an easy identification. Others, such as _mancala_ , _five lines_ and _latrunculi_ are open to interpretation, and require further lines of evidence to positively identify them. Site taphonomy helps to ascribe graffiti games to certain time periods and aids archaeologists in understanding the chronology of the individual games' existence. Much of this information is lost, since major Egyptian monuments were excavated before scientific methods were adopted by archaeologists, but ongoing and future excavations identifying graffiti games already help in understanding how and when some of these monuments were buried and which populations visited the remains.
### Unidentified board games and new approaches
There are patterns that do not easily fit any of the currently known types. They are arranged as intersecting lines making fields of squares or rectangles as well as rows or other patterns of cupules. While it is likely that games existed in Egypt that we have yet to discover, the identification of patterns as games can only be made in conjunction with other kinds of evidence. These include patterns that appear multiple times, have associated gaming paraphernalia (playing pieces or casting devices), exist alongside other known games in games contexts (de Voogt 2010; Crist et al. in press), have a certain social context from the archaeological record (Crist 2015) or appear with corroborating textual or artistic evidence.
While efforts to identify new games are ongoing, there are undocumented examples of known types throughout Egypt as well. Newly found games at Palmyra in Syria, as well as at the pyramids of Meroe (de Voogt 2010, 2012), and those documented at various sites by Pusch (1979), Piccione (1990b) and Jacquet-Gordon (2003) suggest that there is a wealth of evidence for games that remains mis- or unidentified. For Egypt, this volume not only adds a range of Roman games found through the Gebel el-Silsila Survey Project but also contributes unpublished examples of graffiti _senet_ , misinterpreted _hounds and jackal_ boards, unpublished photographs of a series of Egyptian games and a scientific examination attempting to authenticate a long-disputed _mehen_ board as well as corrections of interpretations in earlier scholarly texts.
The discovery of games outside of Egypt has helped to refine our understanding of the ways in which Egyptian games were adopted and further elucidates the types of interactions Egyptians had with local peoples. In the past forty years, evidence for Egyptian games in Cyprus has ballooned significantly to the point where there are now more _senet_ boards that have been found outside of Egypt than within. Cyprus experienced this increase of awareness during the late 1970s and 1980s, and now most Bronze Age sites excavated since then have produced games. The Levant may be experiencing a similar expansion of awareness of this artifact type, as more games have been identified there over the past decade and a half than had been found previously.
Methods from anthropological archaeology (de Voogt et al. 2013; Crist 2015) examine existing material in the light of cultural and anthropological theory. This interest from other disciplines coupled with a better identification of cruder examples of games helps to reinvigorate the study of games in Ancient Egypt. Our work serves as a basis for Classical and anthropological archaeology as well as the field of Egyptology in general to provide a thorough understanding of existing evidence and an overview of the sources used for identifying and interpreting games in Egypt.
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## Author Index
Multiple author publications and editors are indexed under the name of the first author.
al-Ma'shani, A. _see_ Charpentier
al-Maqdissi, M. (ed.) _see_ Oates, J.
Albright, W.F. here, here, here
Allen, J. here, here
Amandry, P. here
Amélineau, E. here, here
Amiran, R. here
Anderson, J.R. (ed.) _see_ de Voogt
Arnold, D. here, here
Aruz, J. (ed.) here, here, here _see also_ Finkel, Tosi
Austin, R.G. here, here, here
Ayrton, E. here
Bagchi, T. _see_ Soni
Bagnall, R.S. (ed.) _see_ Schädler
Banks, E.J. here
Bardanis, M. here
Bardiès-Fronty, I. (ed.) xi _see also_ Dunn-Vaturi
Bauer, G. (ed.) _see_ Schädler
Becker, A. here
Behling, C.-M. here
Bell, R.C. here, here
Ben Zvi, E. (ed.) _see_ Guillaume
Bénédite, G. here, here, here
Ben-Tor, A. here
Benzel, K. (ed.) _see_ Finkel
Berger, J.-F. _see_ Charpentier
Bhattacharya, R.K. (ed.) _see_ Soni
Bieliński, P. here
Birch, S. here, here
Bisht, R.S. here
Blackman, A. here, here
Boaretto, E. _see_ Shai
Boraas, R. _see_ Richard
Borgi, F. _see_ Charpentier
Brandl, B. here
Breyer, C. here, here
Brink, E. _see_ van den Brink
Brodersen, K. (ed.) _see_ Schädler
Brown, W.N. here
Brun, J.-P. here, here, here, here, here
Brunner-Traut, E. here
Brunton, G. here, here, here, here
Bruyère, B. here, here
Bryan, B.M. (ed.) _see_ Allen, DuQuesne
Buchanan, B. _see_ Ellis
Buchholz, H.-G. here
Burke, A.A. here
Calament, F. (ed.) here
Capart, J. here, here
Carnarvon, Earl of here, here, here, here
Carpenter, E. _see_ Schuster
Carter, H., here, here _see_ Carnarvon
Caubet, A. here, here
Champion, C.B. (ed.) _see_ Schädler
Charpentier, V. here
Clarysse, W. (ed.) _see_ Tait
Claudius here
Cline, E.H. (ed.) _see_ Caubet, Guillaume
Clogg, P. _see_ Philip
Cochavi-Rainey, Z. here
Cohen, R. here
Collon, D. here
Crassard, R. _see_ Charpentier
Crist, W. here, here, here, here, here, here, here
Cuvigny, H. (ed.) _see_ Brun, Matelly
Dales, G.F. here
Dandoy, J.R. here
Daressy, G. here
Dasen, V. here
David, A.R. here
Davies, N. here, here
Davies, R. here
de Garis Davies, N. here
de Mecquenem _see_ Mecquenem
de Miroschedji, P. (ed.) _see_ Ilan, Schulman
de Morgan, J. here _see also_ Mecquenem
de Voogt, A.J. here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ Charpentier, Crist, Francigny, Rilly, Schädler
Decker, W. here, here, here, here, here, here, here, here
Depaulis, T. here, here
Deveria, Th. here
Dickmeyer, C.G. here
Donadoni Roveri, A.M. (ed.) here
Donadoni, S. (ed.) _see_ Kendall
Dorn, A. here
Dreyer, G. here _see also_ Kaiser
Drioton, E. here, here, here, here, here
Dunand, M. here, here
Dungworth, D. _see_ Philip
Dunham, D. here, here, here, here
Dunn Friedman, F. (ed.) _see_ Pierrat-Bonnefois
Dunn-Vaturi, A.-E. here, here, here, here, here, here, here, here _see also_ Crist, de Voogt
DuQuesne, T. here, here, here
Durand, M. (ed.) _see_ Calament
Edelman, D.V. (ed.) _see_ Guillaume
Eerkens, J.W. _see_ de Voogt
Effenterre, H. _see_ van Effenterre
Eliyahu-Behar, A. _see_ Shai
Ellis, R. here, here
Emery, W.B. here, here, here, here, here, here, here, here, here
Erdös, S. here, here
Erskine, A. (ed.) _see_ Schädler
Eustathius here
Evans, J. (ed.) _see_ Finkel
Eyckerman, M. here
Falkener, E. here, here, here, here
Faulkner, R.O. here, here
Finkel, I.L. here, here, here, here, here, here, here, here, here _see also_ Becker, Hoerth, Kendall, Piccione, Pusch, Soni, Tait
Finkelstein, I. (ed.) _see_ Guillaume
Firth, C.M. here, here
Fischer, H. here, here
Fiske, P.N. (ed.) _see_ Eyckerman
Förster, F. here
Fourrier, S. here
Francigny, V. here, here, here, here, here, here
Franco, I. here
Frankel, D. here, here, here _see also_ Webb
Frankfort, H. here
Freed, R. (ed.) _see_ Kendall
Frewer, E.F. (translator) _see_ Schweinfurth
Friedman, R.F. (ed.) _see_ Eyckerman
Fugmann, E. here
Gaballa, G. (ed.) _see_ David
Gachet-Bizollon, J. here
Gale, N. _see_ Webb
Gale, R. here
García Martínez, M.A. here, here
Gardiner, A. _see_ Davies, N.
Garstang, J. here
Gasson, P. _see_ Gale, R.
Gayet, A. here
Gempeler, A. _see_ Kaiser
Gilmour, G.H. here
Goedicke, H. here
Greenfield, H.J. _see_ Shai
Griffith, F.L. (ed.) _see_ Naville
Grossman, P. _see_ Kaiser
Grunfeld, F. here
Guichard, S. here
Guillaume, P. here, here, here
Hachmann, R. (ed.) _see_ Meyer
Haddad-Zubel, R. (ed.) _see_ Schädler
Haeny, P. _see_ Kaiser
Hall, H.R. here
Harcourt-Smith, W. _see_ Francigny
Haring, B.J.J. here
Harris-Cline, D. (ed.) _see_ Caubet
Hassan, S. here
Hawass, Z. here
Hayes, W.C. here, here, here, here, here, here, here, here, here, here
Heimann, F.U.M. here
Hendrickx, S. _see_ Eyckerman
Hepper, N. _see_ Gale, R.
Herb, M. _see_ Decker
Herodotus here
Herscher, E. here _see also_ Crist, Swiny
Hillbom, N. here, here
Hoerth, A.J. here, here, here, here, here, here
Hood, S. here
Hornung, E. _see_ Allen, DuQuesne
Huebner, S.R. (ed.) _see_ Schädler
Huizinga, J. here
Hyde, Th. here, here _see also_ Depaulis, Keats
Ilan, O. here
Iskander, J.M. here
Jacquet-Gordon, H. here, here, here, here, here, here, here, here _see also_ Piankoff
Jaritz, H. _see_ Kaiser
Jéquier, G. here, here _see also_ Mecquenem
Johnston, H.H. here
Junge, F. _see_ Kaiser
Junker, H. here, here
Kahn, J. _see_ Francigny
Kaiser, W. here
Kamrin, J. here, here
Kanawati, N. here, here
Kaplony, P. here
Karageorghis, V. here, here, here
Kassianidou, V. (ed.) _see_ Merrillees
Keats, V. here
Kendall, T. here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Keswani, P. here
Killebrew, A.E. (ed.) _see_ Burke, Sharon
Killen, G. _see_ Gale, R.
Kirwan, L.P. _see_ Emery
Kitchen, K. (ed.) _see_ David
Knapp, A.B. here, here
Kochavi, M. here
Konyar, E. (ed.) _see_ Michel
Krüger, T. here
Kurke, L. here
Lamberg-Karlovsky, C.C. _see_ Tosi
Lamer, H. here, here, here
Lane, E.W. here, here
Lansing, A. here, here
Lauer, J.-P. here
Lee, J. here
Lepsius, C.R. here
Levy, T. (ed.) _see_ Kaplony
Lieblein, J.D.C. here
Lilyquist, Ch. here, here
Loat, W.L.S. _see_ Ayrton
Loud, G. here
Macalister, R.A.S. here, here, here
Mackay, E.J.H. here
Mackenzie, C. (ed.) _see_ Finkel
Maeir, A.M. _see_ Shai
Maltby, M. (ed.) _see_ Dandoy
Marée, M. (ed.) _see_ Tiradritti
Mariette, A. here, here, here
Maspero, G. here _see also_ Deveria
Matelly, M.-A. here, here, here, here
May, R. (ed.) here, here, here, here, here
Maystre, C. here
McDonald, G.E. _see_ Dickmeyer
McDonald, H. _see_ Oates
Mecquenem, R. de here
Merkelbach, R. here
Merriam, A. here
Merrillees, R. here
Meyer, J.-W. here, here
Michaelides, D. (ed.) _see_ Merrillees
Michel, C. here _see also_ Collon
Miniaci, G. here, here, here
Montet, P. here, here
Mulvin, L. here, here, here, here, here, here
Murray, A. here
Murray, H.J.R. here, here, here, here, here, here, here, here, here, here
Muscarella, O.W. here
Nash, W.L. here, here
Naville, E. here
Needler, W. here, here
Newberry, P.E. here, here
Nicholson, P. (ed.) _see_ Gale
Niebuhr, C. here, here
Nilsson, M. here, here, here, here, here, here, here, here, here
Novacek, G.V. here
Oates, D. here
Oates, J. here, here
Oren, E.D. here
Ornan, T. here
Ovid here, here _see also_ Rieche
Pankhurst, R. here
Parker, H. here
Parlett, D. here, here, here
Parrot, A. here
Pauly, A.F. _see_ Lamer
Pedrizet, P. here, here
Pendlebury, J. _see_ Frankfort
Petrie, W.M.F. here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Philip, G. here
Piankoff, A. here
Piccione, P. here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Pieper, M. here
Pierrat-Bonnefois, G. here
Piperno, M. here
Plato here
Polak, F. (ed.) _see_ Guillaume
Popova, A. here
Porat, N. here
Prisse d'Avennes, É. here
Pritchard, J. here
Pusch, E.B. here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Quenet, P. (ed.) _see_ Oates, J.
Quibell, J.E. here, here, here, here, here, here _see also_ Petrie
Quirke, S. xii _see also_ Miniaci
Ranke, H. here, here, here, here
Rao, S.R. here
Rapp, G. _see_ Swiny
Rast, W. here, here
Redford, D.B. (ed.) _see_ Spanel
Reeves, N. here, here
Regev, J. _see_ Shai
Retschitzki, J. (ed.) _see_ Schädler
Richard, S. here
Rieche, A. here, here
Rilly, C. here
Röder, G. here
Rose, P. here
Rosellini, I. here
Rothöhler, B. here, here, here, here, here
Ruffle, J. (ed.) _see_ David
Rundle Clark, R.T. here
Saad, Z.Y. here
Salvador, Ch. _see_ Vandenbeusch
Salvatori, S. _see_ Piperno
Saqqara Expedition here
Sauvage, M. here
Schädler, U. here, here, here, here, here, here, here, here, here, here, here, here _see also_ Dunn-Vaturi
Scharff, A. here, here
Schaub, R.T. _see_ Rast
Schiaparelli, E. _see_ Hall
Schoors, A. (ed.) _see_ Tait
Schulman, A. here
Schuster, C. here
Schweinfurth, G.A. here, here
Schweitzer, U. here
Sebbane, M. here, here _see also_ Ilan
Seyffarth, G. here
Shai, I. here
Sharon, I. here
Shaw, I. xiii _see also_ Gale, R.
Shore, A.F. here, here
Sidebotham, S.E. _see_ Mulvin
Silva, J.N. (ed.) _see_ Schädler
Silverman, D. (ed.) _see_ Piccione
Simpson, W.K. here, here, here
Smith, A. _see_ Murray, A.
Smith, S.T. here
Soni, L.N. here
South, A. here
Sowada, K. here
Spanel, D.B. here
Spencer, A.J. here
Stager, L. here
Steiner, M.L. (ed.) _see_ Burke, Sharon
Stos, Z.A. _see_ Webb
Suetonius here, here, here
Sutton-Smith, B. here
Swiny, S. here, here, here, here, here, here, here, here, here, here, here _see also_ Crist
Tait, W.J. here, here, here, here, here, here
Taracha, P. _see_ Bieliński
Tarhan, T. (ed.) _see_ Michel
Teissier, B. here
Tibet, A. (ed.) _see_ Michel
Tiradritti, F. here, here
Tosi, M. here
Tournay, M.-M. here
Tournay, S. _see_ Tournay, M.-M.
Towry-Whyte, E. here
Tufnell, O. here
Turner, V. here
Ussishkin, D. (ed.) _see_ Guillaume, Sebbane
van den Brink, E. (ed.) _see_ Brandl, Kaplony, Porat
van Effenterre, H. here
Vandenbeusch, M. here
Vandier, J. here, here, here, here, here, here, here
Vandier d'Abbadie, J. here
Vassalli, L. _see_ Tiradritti
Väterlein, J. here
Vaturi, A.-E. _see_ Dunn-Vaturi
Vermaak, P.S. here, here
Vermeule, E. here
Voogt, A.J. de _see_ de Voogt
Voorhies, B. here
Walker, R.A. here
Walters, H. _see_ Murray, A.
Ward, J. here, here, here, here, here, here, here, here, here
Webb, J. here, here _see also_ Frankel
Welsby, D.A. (ed.) _see_ de Voogt
Wenig, S. (ed.) _see_ Kendall
Wiedemann, A. here, here
Wilkinson, J.G. here, here, here
Willems, H. (ed.) _see_ Tait
Williams, B.B. here, here
Winlock, H.E. here, here
Wissowa, G. _see_ Lamer
Wolff, S. (ed.) _see_ Stager
Wolsky, F. _see_ Vermeule
Woolley, C.L. here
Yadin, Y. here
## Subject Index
Names of games and transliterations of Egyptian are in italics. Names of gods, places and individuals are capitalized. Entries starting with "el-" are listed under the second part of the name.
_ nḫ w st_ here
_ nḫ_ here
A-Group here, here
Aba here, here
Abu Ballas here
Abu Rawash here, here, here
Abu Sha'ar here, here, here, here, here, here, here
Abydos here, here, here, here, here, here, here, here, here, here, here, here
Acemhoyük here, here
Aegean here, here, here
Aegyptus, province of here
Africa here, here
Central Africa here
East Africa here
Saharan Africa here
sub-Saharan Africa here
afterlife here, here, here, here, here, here, here, here, here, here
Ägyptisches Museum und Papyrussammlung, Berlin here, here, here, here
Ägyptisches Museum, Bonn here, here, here
Akhmim here, here
alabaster here
Alamgirpur here
_alea_ here, here
Alexander the Great here, here, here
Alexandria here, here, here, here
_alquerque_ here, here
Amara West here, here
Amarna, el-Amarna here
Amarna letter here
Amathus here, here
Amenemhat here, here
Amenhotep II here
Amenhotep III here, here
Amenhotep IV here
amethyst here
Amorite here
amulet, amuletic here, here, here, here
Anatolia here, here, here, here, here, here, here
animal _see_ playing pieces, zoomorphic
Ankhefensakhmet here, here
Ankhtifi here, here
Antef here
anthropoid _see_ board
anthropological archaeology here, here
Antinoë here
Anubis here, here
apotropaic here
Aphrodisias here, here, here
appendage _see also_ projection, protrusion
round here
trapezoidal here, here
turtle-head here, here, here, here
Aqhor _see_ Hornakht
Arab here, here, here, here, here, here, here, here
Arabic here, here, here, here
Aramatelqo here
Archaeological Museum of Heraklion here
Asasif, el-Asasif here, here, here, here, here _see also_ Thebes
Ashmolean Museum, Oxford here, here
Ashur here, here, here
Assyrian, Assyrians here, here, here
astragali _see_ casting devices
_b w_ here, here, here
_ba_ here, here, here
Bâb edh-Dhrâ' here, here, here, here, here
baboon here, here
Babylon here, here
Babylonian here, here, here, here, here
Bachias here _see also_ Fayum Oasis
_backgammon_ here, here, here
Ballas here, here, here
Baqet III here, here, here
Bedouin here
beer here
Bellapais _Vounous_ here
Beni Hasan here, here, here, here, here, here, here, here
Bes here, here
Beth Shean here, here, here
bifacial _see_ board shape
bird here, here, here, here, here _see also_ duck, goose, sparrow-hawk, waterfowl
black here, here, here, here, here, here
blue here, here
board shape
anthropoid here
axe-blade here
bifacial here, here _see also_ double-sided, reversible
box type _see_ box
double-sided here, here, here, here, here, here, here
ellipsoidal here, here _see also_ violin
hippopotamus here, here _see also_ zoomorphic
liver here
Palestinian _see_ violin
reversible here, here _see also_ bifacial, double-sided
shield here
turtle here, here
violin here, here, here, here, here
zoomorphic here, here
Bode Museum, Berlin here
Book of the Dead here, here, here, here, here, here
box, game box here, here, here, here, here, here, here, here, here, here, here, here, here
British Museum, London here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
brown here
Buhen here, here, here, here, here
burial here, here, here, here, here, here, here, here, here, here _see also_ funerary, mortuary, offerings
Byblos here, here, here, here, here
Byzantine here, here, here, here, here
Cairo Papyrus _see_ Great Game Text
calendar here
Canaan, Canaanites here, here, here
canine here, here, here _see also_ dog
caption here, here, here, here, here, here, here _see also_ playing scene
captives _see_ playing pieces, throwing sticks
casting devices here, here, here, here, here _see also_ randomization
astragalus, astragali here, here, here, here, here, here, here, here, here
cubic dice here, here, here, here, here, here, here
disks, two-sided here, here
knucklebones here, here, here _see also_ astragalus
oblong dice here, here
pyramidal _see_ teetotum
six-sided _see_ cubic dice
sticks, four-sided here
teetotums here, here, here, here
tetrahedrons here, here
throwing sticks here, here, here, here, here, here
twenty-sided dice, Greek lettering here, here
cat here
cedar here, here
celebration, celebratory here, here, here
Chalcolithic here
_checkers_ here _see also draughts_
checkered here, here, here, here
_chess_ here, here, here
child, children here, here, here, here, here
Christianity, Christian here, here, here
circle here, here, here, here, here, here, here _see also game of thirty-three circles_
semi-circle here, here, here
circus here
climate here, here, here, here, here _see also_ preservation
clockwise here, here, here _see also_ counterclockwise
coffee here
Coffin Text here, here, here
color, colored here, here, here, here, here, here, here, here, here
concentric circle _see_ spiral
configuration here, here, here, here, here, here, here, here, here, here, here
connecting lines _see_ linkage
conquest
Arab here, here, here, here
British, French here
Egyptian conquest of the Levant here
Egyptian conquest of Nubia here
Roman conquest here, here
copper here, here, here
Coptic
calendar here
game here, here
counterclockwise here, here, here, here
counters _see_ playing pieces
counting devices _see_ casting devices
Crete here, here _see also_ Archaeological Museum of Heraklion, Knossos
cross _see_ "X"
crystal _see_ rock crystal
cubic dice _see_ casting devices
cultural transmission here _see also_ anthropological archaeology
Cypro-Geometric here, here
Cyprus, Cypriot here, here, here, here, here, here, here, here, here, here, here, here, here
Dakhleh Oasis here, here
dance, dancers here, here, here, here, here, here
Darius III here
Dawwi here, here
Deir el-Bahari here, here, here _see also_ Thebes
Deir el-Bersha here
Deir el-Gabrawi here, here
Deir el-Medina here, here, here, here, here, here, here, here, here
Delta, Nile here
Demeter here
Demotic here
Den here, here
Dendera here, here, here, here, here, here
Deneia _Kafkalla_ here
depression here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ playing field
drilled/bored here, here, here
pecked here, here, here
dexterity here, here
Dholavira here
dice _see_ casting devices
dice tower _see_ pyrgus
Didymoi here, here
Dinkha Tepe here
disks _see_ casting devices
divination, divinatory here, here, here, here
Djet here
dog, hound here, here, here, here, here, here, here, here, here
domestic here, here, here, here, here, here _see also_ public life
double-axe _see_ board shape
Dra Abu el-Naga here, here, here, here, here, here
_draughts_ here _see also checkers_
draughtsmen _see_ playing pieces
drinking here, here, here _see also_ beer, coffee, social lubricant, wine
drinking cup here, here
_duat_ here, here, here
duck here
_duodecim scripta, ludus duodecim scriptorum_ here, here, here, here, here, here, here
ears here, here, here
Early Bronze Age here, here
Early Dynastic Period here, here, here
East Africa here
eastern desert of Egypt here, here, here
Eastern Mediterranean here, here, here, here, here
Edfu here, here
Egyptian Museum, Cairo here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Eighteenth Dynasty here, here, here, here, here, here, here _see also_ New Kingdom
Eleventh Dynasty here, here, here _see also_ Middle Kingdom
elite here, here, here, here, here, here, here, here
ellipsoidal _see_ board shape
elliptic here
emulation here
Enkomi here, here, here, here, here
Ephesos here, here
Eritrea here
eroded here, here
Ethiopia here
Eustathius here
excavation, excavated here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
exchange here, here, here, here, here _see also_ trade
eye here, here, here, here
faience here, here, here, here, here, here, here, here, here, here, here, here, here, here
Famagusta here
famous here, here, here, here _see also_ iconic
Fayum, Fayum Oasis here, here, here, here
feasting here, here
feet here, here, here, here, here, here, here
fennec here
festival here
Fifteenth Dynasty here, here _see also_ Second Intermediate Period
Fifth Dynasty here, here, here _see also_ Old Kingdom
finger, fingertip here, here
First Intermediate Period here, here, here, here, here, here
First Persian Period _see_ Twenty-Seventh Dynasty
Fitzwilliam Museum, Cambridge here, here
_five lines_ here, here, here, here, here, here, here, here see also rules of play
flute here, here
foot _see_ feet
forgery here
fortress here, here, here
Abu Sha'ar here, here, here, here _see also_ Abu Sha'ar
Buhen here, here, here _see also_ Buhen
Dawwi here _see also_ Dawwi
Didymoi here _see also_ Didymoi
Krokodilo here _see also_ Krokodilo
Tell Defenneh here
_forty-two and pool_ here
Fourth Dynasty here, here, here, here _see also_ Old Kingdom
fox here, here, here
frog here, here
funerary here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ burial, mortuary
gambling here, here
_game of fifty-eight holes see hounds and jackals_
_game of horseshoes_ here
_game of the robbers see game of tjau_
_game of thirty points_ here _see also hounds and jackals_
game of _thirty-one_ here
_game of thirty-three_ here, here, here, here
_game of tjau_ here, here, here
_game of twenty_ here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ rules of play
"head-and-tail" type here, here
liver-shape here
_royal game of Ur_ here
games context here, here, here, here, here
games of skill here
gaming pieces _see_ playing pieces
gaming/game table here, here, here, here, here, here, here, here, here, here, here
garlic here
Gath _see_ Tell es-Safi
Gaza, ancient Gaza here _see also_ Tell el-Ajjul
Gebel el-Silsila Survey Project here, here, here, here, here, here, here, here, here, here
gender here _see also_ women
Gerar _see_ Tell Jemmeh
Germania here
Gezer here, here, here, here
Giza here, here, here, here
glass here, here, here, here
goal of the game here, here, here, here, here
goose here, here
graffiti games here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Greek calendar _see_ calendar
Great Game Text here, here, here, here, here, here, here
Greco-Roman here, here, here, here, here, here, here, here, here _see also_ Hellenistic
Greco-Roman period here, here, here, here _see also_ Ptolemaic period, Roman period
Greek lettering _see_ twenty-side dice
green here
guessing, counting games here, here
Hadjiprodromou Collection, Famagusta here
hairpin here, here _see also_ peg, pin
Hama here, here
Hapy here, here
Har Yeroham here
Har-Behdety here
Harappa here, here
Hathor here, here, here, here
Hatshepsut here, here, here, here, here, here
Hawawish, el-Hawawish here, here
Hazor here, here, here, here
Pillared Building here
head _see also_ "head-and-tail"
animal here
anthropoid here
canine, dog, hound here, here, here
jackal here, here
leopard here
serpent, snake here, here, here, here
sparrow-hawk here
turtle here, here, here, here
waterfowl/goose/duck here
Heliopolis here, here
Hellenistic here, here, here _see also_ Greco-Roman
hepatoscopy here _see also_ divination
Hesy-Re here, here, here, here, here, here
hexagram, hexameter here _see also duodecim scripta_
hippopotamus here, here, here, here _see also_ board shape, game piece
Hor Aha here
Hornakht here, here, here
horse here, here
horseshoes _see game of horseshoes_
Horus here, here, here, here, here, here, here
hound _see_ dog, _hounds and jackals_
_hounds and jackals_ here, here, here, here, here, here, here, here, here _see also_ rules of play
hunting here
Hyena game here
Hyksos here
Ibi here
Iby here
iconic here _see also_ famous, McDonald, G.E.
iconography here, here, here, here, here, here, here
Idu here, here, here, here
Ilum-ishar here
Imenmes here, here, here
Indus Valley here, here
Inherkau here
Inshushinak here, here
Institut Français d'Archéologie Orientale, Cairo here, here, here
interaction here, here, here, here, here, here
Intermediate Bronze Age here
International Style here, here, here
intersecting lines here, here
intoxicant here _see also_ beer, psychoactive substance, wine
Iran here, here, here, here, here, here, here, here
Iraq _see_ Mesopotamia
Iron Age here, here, here, here
Isesi-merynetjer here, here, here
Isis here, here
Islamic here, here, here, here, here, here
Israel, Israelites here, here
Israel Museum, Jerusalem here, here
ivory here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
_ỉwỉw_ here
jackal here, here, here, here, here, here, here _see also hounds and jackals_
_jeu de dames see checkers_
Jiroft here, here
Jordan here
Kab, el-Kab here, here, here
Kababish here
Kahep here
Kaiemankh here, here, here, here
Kairere here
Kalavasos _Ayios Dhimitrios_ here, here
Kamid el-Loz here, here, here, here
Kanesh _see_ Kültepe
Karahoyük here, here
Karnak here, here, here, here, here _see also_ temple of Khonsu
Keramit here
Kerma here
Kerman here
Kharga Oasis here, here, here, here, here
Kheni here
Khety here, here, here
Khirbet Iskander here, here
Khnumhotep II here
Khonsu _see_ temple of Khonsu
Khonsumose here
Kition _Kathari_ here, here
Knossos here
knucklebones _see_ astragali
Kom el-Dekka here _see also_ Alexandria
Kom Ombo here, here, here, here
Kordofan here
Kotchati here
Kouklia _Skales_ here, here, here, here
Krokodilo here, here
Kubban here, here, here, here
Kültepe, ancient Kanesh here, here, here
Kūrna here, here
Kurru, el-Kurru here, here, here, here
Kushite here, here, here _see also_ Twenty-Fifth Dynasty
labyrinth here, here, here
Lachish here, here, here, here
Lahun here, here, here, here, here, here
lapis lazuli here
Late Bronze Age here, here, here, here, here, here
Late Period here, here, here, here, here, here, here, here, here, here _see also_ Twenty-Fifth, Twenty-Sixth, Thirtieth Dynasties
Latin here, here, here, here, here, here
_latrunculi_ here, here, here, here, here, here, here, here, here _see also_ rules of play
Lebanon here, here
leg, legs here, here, here, here
Lemba _Lakkous_ here, here
leopard here
letters here, here, here, here, here _see also_ Greek lettering, hexagram, twenty-side dice
Levant, Levantine here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
limestone here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
liminal here
lines _see five lines_ , linkage
linkage here, here, here, here, here
lion here, here, here, here, here, here, here
Lisht here, here, here, here, here
liver _see_ board shape
Lothal here, here
Louvre _see_ Musée du Louvre
_ludus duodecim scriptorum see duodecim scripta_
_ludus latrunculorum see latrunculi_
lunar, moon here
Luxor here, here, here, here, here
Ma'at here
magical here, here, here
Mahasna, el-Mahasna here
_mancala_ here, here, here, here, here, here, here, here, here _see also_ rules of play
marble, _marbles_ here, here, here, here, here, here, here, here
Mari here, here
Marki _Alonia_ here, here
Marki _Kappara_ here, here
Mashabei Sade here, here
mastaba here, here, here, here, here
Medamoud here, here
Medinet Habu here, here, here
Megiddo here, here, here, here, here, here, here
_mehen_ here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ rules of play
Memphis here, here
_men_ here, here, here, here, here, here
_merels_ here, here, here, here, here _see also mill game_ , _three-men's-morris_
Merenptah here, here, here
Mereruka here, here
Meretseger here
Meroe here
Meroitic here, here, here
Post-Meroitic here
Mery here
Merymaat here
Mesopotamia here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Metjetji here
Metropolitan Museum of Art, New York here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Middle Bronze Age here, here, here, here, here
Middle Kingdom here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ Eleventh, Twelfth, Thirteenth Dynasties
Middle Minoan period here
military here, here, here _see also_ soldiers
_mill game_ here, here _see also merels_ , rules of play
Mitanni here
Moalla here, here, here
Mohenjo-daro here, here
moon _see_ lunar
Morphou _Toumba tou Skourou_ here, here, here
mortuary here, here, here, here, here, here, here, here, here _see also_ burial, funerary
mouth here, here, here _see also_ snout
Muhattah Jaqub here
Musawwarat el-Sufra here
Musée du Louvre, Paris here, here, here, here, here, here, here, here, here
Musée Royal de Mariemont here
Musées Royaux d'Art et d'Histoire, Brussels here, here, here
Museo Egizio, Turin here, here, here
Museum of Fine Arts, Boston here, here, here, here
music, musicians here, here, here, here, here, here
Mut here
_mw_ here, here, here
Nahum here
naked here _see also_ nude
Naqada here, here, here, here, here, here
_nard_ here
Narodni Muzeum, Prague here
Naxos here, here
Nebenma'at here
Neferhotep here, here, here, here
Neferiretenef here
Neferukekashta here
Nefwa here
Neith here, here
Nekau here
Nephthys here
New Kingdom here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ Eighteenth, Nineteenth, Twentieth Dynasties
_nfr_ here, here, here, here, here, here, here, here, here, here
_nfrw_ here, here, here, here
_nine-men's-morris_ here, here, here, here
Nineferkaptah here
Nineteenth Dynasty here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ New Kingdom
Nippur here, here
nomads here
_nṯrw_ here, here
_nṯrwy_ here, here
Nubia, Nubians here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Nubia Museum, Aswan here
nude here, here _see also_ naked
object frieze here
offerings here, here, here, here, here, here, here, here
burial gifts/goods here, here
formulae here
offering table here, here
Old Kingdom here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ Third, Fourth, Fifth, Sixth Dynasties
Omphale here
Oriental Institute Museum, Chicago here, here, here, here, here
Osireion here, here, here
Osiris here, here, here
ostracon, ostraca here, here, here, here, here, here, here
Ottoman here, here, here, here, here, here
Oxyrhyncus Papyrus here
_pachisi_ here
paint here, here, here, here, here, here, here _see also_ pigment
painting here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ relief
Palestine Archaeological Museum, Jerusalem here
_palm tree game_ here _see also hounds and jackals_
Palmyra here, here, here, here, here
papyrus here, here, here, here, here, here _see also_ Cairo, Oxyrhynchus, Turin Papyrus
pavement here, here, here, here, here, here, here, here, here, here, here
pawns _see_ playing pieces
pegs here, here, here, here, here _see also_ pins, playing pieces
_pegs and holes_ here _see hounds and jackals_
_πέντε γραμμαί see five lines_
Pepi I, Pepi II here
Peribsen here, here
Persia, Persian here, here, here, here, here, here
Petosiris here
Petra here, here
Petrie Museum, London here, here, here, here, here, here, here, here
φόρωρ Ὥρου ὄικος, Φόρωρ here
Piankhy here
Piay here
pigment here, here, here _see also_ color, paint
pins here, here, here _see also_ pegs, hairpin
platter here
playing field _see_ depression, square
playing pieces here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ dog
animal, zoomorphic here, here
bird here, here
captive figurine here, here
conical here, here, here, here, here, here, here
couchant hare here
couchant lion here, here
couchant sphinx here
domed, low-domed here, here, here, here
draughtsman here
halma type here
hippopotamus here
pawn here, here, here, here
pyramidal here
reel-shaped here
spherical here, here
spool-shaped here, here
playing scene here, here, here, here, here, here, here, here, here, here, here, here, here, here
pleasure(s) here, here
poet, poetic lines, poetry here, here, here
_πόλις_ here
Post-Meroitic here
Predynastic here, here, here, here, here, here, here, here, here, here _see also_ Naqada
Prehistoric Bronze Age here
preservation here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ climate
private collection here, here, here
projection here, here, here, here _see also_ appendage, protrusion
Protohistoric Bronze Age here, here
protrusion here, here, here, here _see also_ appendage, projection
psychoactive substance here _see also_ beer, intoxicant, wine
Ptahmay here
Ptolemaic period here, here, here, here, here, here, here, here, here _see also_ Roman period
Ptolemy I Soter here
public life/space here, here, here, here
pyramid here, here, here, here, here, here, here, here
Pyramid Texts here, here, here,
pyramidal _see_ casting devices, playing pieces
pyrgus here
Qantir, el-Qantir here, here, here
Qar here
Qasr al Ghweita here
Qasr Ibrim here, here, here
Qau here, here, here, here
quartz here
Quft here, here
Qustul here, here, here, here, here, here, here
Qusur al-Banat here, here
Ra here, here, here, here
race game here, here, here, here, here
Rahotep here, here, here, here, here, here
ram here
Rameses II here
Rameses III here, here, here
Rameses V here
Rameses VI here
Ramesseum here, here
randomization here, here, here, here, here, here, here, here _see also_ casting device
Rashepses here, here, here
Re-Horakhty here
rebirth here, here
red here, here, here, here, here, here, here, here
Red Sea here
Rekhmire here, here
relief here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ painting
religion, religious, religiosity here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Reniseneb here
Rhampsinitus here
ritual here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
rock crystal here
rock face here, here, here, here
Roman period here, here, here _see also_ Greco-Roman period, Ptolemaic period
roof here, here, here, here, here, here, here, here
rosette here, here, here, here, here, here, here
_royal game of Ur_ here _see game of twenty_
rules of play here, here, here, here
_five lines_ here
_game of twenty_ here, here
_hounds and jackals_ here
_latrunculi_ here
_mancala_ here
_mehen_ here, here, here
_mill_ here
_seeja_ here
_tâb_ here
_s _ here
sacred line here
sacred symbols/text here, here, here _see_ symbols
Sai Island here, here, here, here
sailors here
Saite Period _see_ Late Period, Twenty-Sixth Dynasty
Saqqara here, here, here, here, here, here, here, here, here, here, here, here, here, here
Sacred Animal Necropolis here
scarab here
scorpion here, here
Second Intermediate Period here, here, here, here _see also_ Fifteenth, Sixteenth, Seventeenth Dynasty
Sedeinga here, here, here, here, here
Sedment here
_seeja_ here, here, here, here, here, here, here, here _see also_ rules of play
_senet_ here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here
Sennedjem here
Sennefer here, here, here
Senusret I here, here
Senusret II here
Senusret III here, here
serpent _see_ snake
Serra East here, here
Seshemnefer here
Setne here, here
Seventeenth Dynasty here, here, here, here, here, here, here _see also_ Second Intermediate Period
Shabaqo here
Shahr-i Sokhta here
Shatt el-Rigal here
_shield game_ here _see hounds and jackals_
_siga see seeja_
Silsila _see_ Gebel el-Silsila Survey Project
Silsila West here
simulacra here
Sinai here, here
singers here, here
Sixteenth Dynasty here, here _see also_ Second Intermediate Period
Sixth Dynasty here, here, here, here, here
skill _see_ games of skill
_skittles_ here
_šn_ here, here
snake, serpent here, here, here, here, here, here, here, here, here
_snakes and ladders_ here
snout here, here _see also_ mouth
social lubricant here
solar, sun here, here, here, here, here, here, here
soldiers _see also_ military
Egyptian here
Ottoman here
Roman here, here, here, here, here, here
Sotira _Kaminoudhia_ here, here
South Sudan here
sparrow-hawk here
special holes, positions, spaces, squares here, here, here, here, here, here, here, here, here, here, here
spiral, spiraling here, here, here, here, here, here, here
sports here, here, here
square here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ special holes
starting configuration, line, point, position here, here, here, here, here, here
State Hermitage Museum, Saint Petersburg here
status here, here, here
sticks _see_ throwing sticks
stone _see_ limestone
strategy game here
Sudan here, here, here, here, here, here, here, here, here, here
Sudan National Museum, Khartoum here
Sumer here
Sumerian proverb here, here
sun _see_ solar
surface find here, here, here, here
Susa here, here, here, here
Swiss Museum of Games, La Tour-de-Peilz here, here, here
symbolic, symbols, symbolism here, here, here, here, here, here, here, here, here, here, here, here, here, here
Syria here, here, here, here, here, here, here, here, here
_ṯ w see game of tjau_
_tâb_ here, here, here, here, here, here, here, here _see also_ rules of play
table _see_ gaming/game table
Taharqo here, here
tail here, here, here, here, here, here, here, here, here, here, here, here
Taya here
teetotums _see_ casting devices
Tel Arad here, here
Tell Abu Hatab here
Tell Ailun here
Tell el-Ajjul here, here, here, here _see also_ ancient Gaza
Tell Beit Mirsim here, here, here, here
Tell Brak here, here, here, here
Tell Defenneh here, here
Tell el-Farah south here
Tell Halaf here
Tell el-Hisn here, here
Tell Jemmeh here, here
Tell es-Safi here
Tell es-Saidiyeh here
Tell as-Sib here
Tell Sukas here
temple of/at:
Abydos here
Amun at Karnak here, here
Dendera here
Inshushinak at Susa here, here
Karnak here
Khonsu at Karnak here, here, here, here, here, here
Kom Ombo here, here, here
Kūrna here, here
Luxor here, here, here
Medinet Habu here, here
Qasr Ibrim here, here
Soleb here
Tiye at Sedeinga here
Userkaf here
Tepe Gawra here
Tepe Sialk here, here, here
Tepe Yahya here, here
terracotta here, here, here, here
Teti here
tetrahedrons _see_ casting devices
Thebes here, here, here, here, here, here, here, here, here, here, here, here, here _see also_ el-Asasif, Deir el-Bahari, Dra Abu el-Naga
Third Dynasty here, here, here, here, here _see also_ Old Kingdom
Third Intermediate Period here, here _see also_ Twenty-First–Twenty-Fourth Dynasties
Thirteenth Dynasty here, here _see also_ Middle Kingdom
Thirtieth Dynasty here, here, here
Thoth here, here
_three-men's-morris_ here, here _see also merels_
throwing cup here _see also_ pyrgus
throwing sticks _see_ casting device
Thutmose III here
Thutmose IV here, here
Tiye _see_ temple
_tjau see game of tjau_
track here, here, here, here, here, here, here, here, here _see also_ elliptic, spiraling
trade here, here, here, here, here _see also_ exchange
trapezoidal _see_ appendage
Turin Papyrus here, here, here, here, here, here
Turkey here _see also_ Anatolia, Ottoman
turquoise here
turtle _see_ appendage, board shape, head
Tushratta here
Tutankhamun here, here, here, here, here, here
Twelfth Dynasty here, here, here, here, here, here, here _see also_ Middle Kingdom
Twentieth Dynasty here, here, here, here, here, here, here _see also_ New Kingdom
Twenty-Fifth Dynasty here, here, here _see also_ Kushite, Late Period
Twenty-First–Twenty-Fourth Dynasties here, here, here, here _see also_ Third Intermediate Period
Twenty-Seventh Dynasty here, here
twenty-sided dice _see_ casting devices
Twenty-Sixth Dynasty here, here, here, here _see also_ Late Period
_Two rows of thirteen_ here
Tyre here, here
Ugarit here, here
underworld/netherworld here, here, here, here, here, here
unidentified games here, here
University of Pennsylvania Museum, Philadelphia here
Ur here, here, here
Valley of the Kings here, here, here, here
Vimose, Denmark here _see also_ Germania
violin _see_ board shape
votive here, here, here, here, here, here
_Vounous see_ Bellapais _Vounous_
Walters Art Museum, Baltimore here, here, here, here
wands here, here _see also_ casting devices
warship here
waterfowl here
_wḏ t_ here
white here, here, here, here, here
wine here _see also_ beer, psychoactive substances, social lubricant
women here, here, here _see also_ gender
wooden here
board here, here, here, here, here, here, here, here, here, here, here, here, here
casting device here, here, here
playing pieces here
World Museum, Liverpool here, here, here
"X" here, here, here, here, here, here, here, here, here, here
Yale Babylonian collection here
Yale University Art Gallery here
Zakros here, here
Zawiyet el-Aryan here, here, here, here, here
Zer-Ta here
Zimri-Lim here
zodiac here _see also_ divination
zoomorphic here, here, here, here _see also_ board shape, feet, playing pieces, table
**Bloomsbury Academic**
An imprint of Bloomsbury Publishing Plc
50 Bedford Square 1385 Broadway
London New York
WC1B 3DP NY 10018
UK USA
**www.bloomsbury.com**
**BLOOMSBURY and the Diana logo are trademarks of Bloomsbury Publishing Plc**
First published 2016
© Walter Crist, Anne-Elizabeth Dunn-Vaturi and Alex de Voogt, 2016
Walter Crist, Anne-Elizabeth Dunn-Vaturi and Alex de Voogt have asserted their rights under the Copyright, Designs and Patents Act, 1988, to be identified as Authors of this work.
All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers.
No responsibility for loss caused to any individual or organization acting on or refraining from action as a result of the material in this publication can be accepted by Bloomsbury or the authors.
**British Library Cataloguing-in-Publication Data**
A catalogue record for this book is available from the British Library.
ISBN: HB: 978-1-47422-118-4
PB: 978-1-47422-117-7
ePDF: 978-1-47422-120-7
ePub: 978-1-47422-119-1
**Library of Congress Cataloging-in-Publication Data**
Crist, Walter, author.
Ancient Egyptians at play : board games across borders / Walter Crist, Anne-Elizabeth Dunn-Vaturi, Alex de Voogt.
pages cm — (Bloomsbury Egyptology)
Includes bibliographical references and index.
ISBN 978-1-4742-2118-4 (hb : alk. paper) — ISBN 978-1-4742-2117-7 (pb : alk. paper) — ISBN 978-1-4742-2119-1 (epub : alk. paper) — ISBN 978-1-4742-2120-7 (epdf : alk. paper) 1. Board games—Egypt—History. 2. Egypt—Antiquities. 3. Egypt—Social life and customs. 4. Divination—Egypt—History. I. Dunn-Vaturi, Anne-Elizabeth, author. II. Voogt, Alexander J. de, author. III. Title. IV. Series: Bloomsbury Egyptology.
DT62.B5C75 2016
794'.0932—dc23
2015034318
Series: Bloomsbury Egyptology
Typeset by RefineCatch Limited, Bungay, Suffolk
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{"url":"https:\/\/www.zentralblatt-math.org\/matheduc\/en\/?q=an%3A2010b.00392","text":"History\n\nHelp on query formulation\nSums of two squares. (Summer av to kvadrat.) (Norwegian. English summary)\nNormat. 57, No. 3, 97-106 (2009).\nSummary: The characterization of numbers representable as the sum of two squares in terms of their divisors has been known since the 17th century (Girard, Fermat) but not proved until 18th century (Euler). A refinement involving the number $R(n)$ of representations of $n$ as sum of two squares was proved by Jacobi using theta functions. This article starts out by reproducing a very elementary proof due to Heath-Brown of Fermat\u2019s characterization of primes of type $4n+1$. Then it turns to discussing problems concerning asymptotic distribution, occurrence in prescribed intervals, infinite occurrences of patterns such as $n,n+h_1$, $n+h_2\\dots h_k$ for fixed $h_i$ (It is shown that $n,n+1$, $n+2$ occurs infinitely often, but of course $n,n+1,n+2,n+3$ may never all be sums of two squares.) Those are compared with the corresponding statements for primes. One may note that the asymptotic behaviour of $B(x)=\\sum_{n= a^2+b^2C\\log x$ for some suitable constant $C$, but so far it has only been shown for $h>Cx^{\\frac 14}$. But if we relax the condition to almost all such intervals, there is a complete solution.\nClassification: F65","date":"2019-05-21 21:07:21","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\": 0.8806787133216858, \"perplexity\": 367.03496462618835}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256571.66\/warc\/CC-MAIN-20190521202736-20190521224736-00550.warc.gz\"}"} | null | null |
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Q: Argument notation in Python documentation I read the Python documentation a lot and sometimes I am baffled by this notation:
os.path.join(path1[, path2[, ...]])
I somehow make that [, path[,...]] is a list but I would like to know if I am reading it correctly.
Bear with me, this is coming from a Java developer who is trying out Python. X)
A: The brackets indicate an optional parameter. The ellipses indicate a variable-length argument list.
A: That is for multiple arguments. You could call that method with 1 or more variables. That particular method could be called with:
*
*join(path1)
*join(path1, path2)
*join(path1, path2, <optional parameters>)
Option 3 can only be used when the path2 argument is present. If you have have use C, think printf("Number %d", number);
According to the python documentation, those optional parameters are for more paths. So you could call join(path1, path2, path3, path4) or with as many paths as you like.
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package org.qiaoer.photogallery;
import android.net.Uri;
import android.util.Log;
import org.json.JSONArray;
import org.json.JSONException;
import org.json.JSONObject;
import java.io.ByteArrayOutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.net.HttpURLConnection;
import java.net.URL;
import java.util.ArrayList;
import javax.net.ssl.HttpsURLConnection;
public class FlickrFetcher {
public static final String TAG = "FlickrFetcher";
public static final String PREF_SEARCH_QUERY = "searchQuery";
public static final String PREF_LAST_RESULT_ID = "lastResultId";
private static final String ENDPOINT = "https://pixabay.com/api/";
private static final String KEY = "2701371-6dd40f7d949e87e4267ed4e1b";
// private static final String QUERY_WORDS = "yellow+flowers";
private static final String EDITORS_CHOICE = "true";
private static final String IMAGE_TYPE = "photo";
private static final String PER_PAGE = "200";
private static final String PRETTY = "true";
public ArrayList<GalleryItem> downloadGalleryItems(String url) {
ArrayList<GalleryItem> items = new ArrayList<>();
try {
String jsonString = getUrl(url);
Log.i(TAG, "Received xml: " + jsonString);
parseItems(jsonString, items);
} catch (IOException e) {
Log.e(TAG, "Failed to fetch items", e);
}
return items;
}
public ArrayList<GalleryItem> fetchItems() {
String url = Uri.parse(ENDPOINT).buildUpon()
.appendQueryParameter("key", KEY)
.appendQueryParameter("editors_choice", EDITORS_CHOICE)
.appendQueryParameter("image_type", IMAGE_TYPE)
.appendQueryParameter("per_page", PER_PAGE)
.appendQueryParameter("pretty", PRETTY)
.build().toString();
return downloadGalleryItems(url);
}
public ArrayList<GalleryItem> search(String query) {
String url = Uri.parse(ENDPOINT).buildUpon()
.appendQueryParameter("key", KEY)
.appendQueryParameter("q", query)
.appendQueryParameter("image_type", IMAGE_TYPE)
.appendQueryParameter("per_page", PER_PAGE)
.appendQueryParameter("pretty", PRETTY)
.build().toString();
return downloadGalleryItems(url);
}
public String getUrl(String urlSpec) throws IOException {
return new String(getUrlBytes(urlSpec));
}
byte[] getUrlBytes(String urlSpec) throws IOException {
URL url = new URL(urlSpec);
HttpsURLConnection connection = (HttpsURLConnection) url.openConnection();
connection.setDoInput(true);
try {
ByteArrayOutputStream out = new ByteArrayOutputStream();
InputStream inputStream = connection.getInputStream();
if (connection.getResponseCode() != HttpURLConnection.HTTP_OK) {
return null;
}
int hasRead;
byte[] buffer = new byte[1024];
while ((hasRead = inputStream.read(buffer)) > 0) {
out.write(buffer, 0, hasRead);
}
out.close();
return out.toByteArray();
} finally {
connection.disconnect();
}
}
private void parseItems(String jsonString, ArrayList<GalleryItem> items) {
try {
JSONObject jsonObject = new JSONObject(jsonString);
JSONArray jsonArray = jsonObject.getJSONArray("hits");
for (int i = 0; i < jsonArray.length(); i++) {
JSONObject pictureItem = jsonArray.getJSONObject(i);
String caption = pictureItem.getString("tags");
String url = pictureItem.getString("previewURL");
String id = pictureItem.getString("id");
GalleryItem item = new GalleryItem();
item.setId(id);
item.setUrl(url);
item.setCaption(caption);
items.add(item);
}
} catch (JSONException e) {
e.printStackTrace();
}
}
/*byte[] getUrlBytes(String urlSpec) throws IOException {
ByteArrayOutputStream out = null;
InputStream inputStream = null;
URL url = new URL(urlSpec);
HttpURLConnection connection = (HttpURLConnection) url.openConnection();
try {
out = new ByteArrayOutputStream();
inputStream = connection.getInputStream();
if (connection.getResponseCode() != HttpURLConnection.HTTP_OK) {
return null;
}
int hasRead;
byte[] buffer = new byte[1024];
while ((hasRead = inputStream.read(buffer)) > 0) {
out.write(buffer, 0, hasRead);
}
out.close();
return out.toByteArray();
} finally {
connection.disconnect();
try {
if (out != null) {
out.close();
}
if (inputStream != null)
inputStream.close();
} catch (IOException e) {
e.printStackTrace();
}
}
}
public String getUrl(String urlSpec) throws IOException {
return new String(getUrlBytes(urlSpec));
}
public ArrayList<GalleryItem> fetchItems() {
ArrayList<GalleryItem> items = new ArrayList<>();
try {
String url = Uri.parse(ENDPOINT).buildUpon()
.appendQueryParameter("method", METHOD_GET_RECENT)
.appendQueryParameter("api_key", API_KEY)
.appendQueryParameter(PARAM_EXTRAS, EXTRA_SMALL_URL)
.build().toString();
String xmlString = getUrl(url);
Log.i(TAG, "Received xml: " + xmlString);
XmlPullParserFactory factory = XmlPullParserFactory.newInstance();
XmlPullParser parser = factory.newPullParser();
parser.setInput(new StringReader(xmlString));
parseItems(items, parser);
} catch (IOException e) {
Log.e(TAG, "Failed to fetch items", e);
} catch (XmlPullParserException e) {
e.printStackTrace();
Log.e(TAG, "Failed to parse items", e);
}
return items;
}
void parseItems(ArrayList<GalleryItem> items, XmlPullParser parser) throws IOException, XmlPullParserException {
int eventType = parser.next();
while (eventType != XmlPullParser.END_DOCUMENT) {
if (eventType == XmlPullParser.START_TAG
&& XML_PHOTO.equals(parser.getName())) {
String id = parser.getAttributeValue(null, "id");
String caption = parser.getAttributeValue(null, "title");
String smallUrl = parser.getAttributeValue(null, EXTRA_SMALL_URL);
GalleryItem item = new GalleryItem();
item.setId(id);
item.setCaption(caption);
item.setUrl(smallUrl);
items.add(item);
}
eventType = parser.next();
}
}*/
}
| {
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Q: Best practice of adding node(s) to elasticsearch cluster I'm using Zen Discovery module to connect all nodes together, but when I want to add a new node to the cluster, I have to change every node's discovery.zen.ping.unicast config to add the new node's ip, and then restart all nodes.
When the cluster's nodes grow up to a big number, it's impossible to do this manually, what should I do?
A: You do not have to change every node's configuration, especially not restart every node.
You only have to add one or more running nodes into the new node's discovery.zen.ping.unicast. That is enough for the new node to discover and join the cluster.
| {
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Chimney at Druid Mine, Ashburton
Mid 19th century chimney of stone rubble with a red-brick top.
Listed Building (II) 1206269: ENGINE HOUSE AND CHIMNEY AT DRUID COPPER MINE
CHIMNEY (XIX - 1801 AD to 1900 AD)
Google, 2012, Google Streetview (Website). SDV348799.
Chimney visible in woods from road.
English Heritage, 2013, National Heritage List for England (National Heritage List for England). SDV350785.
Engine House and Chimney at Druid Copper Mine. Ruined engine house and chimney. Probably mid C19. Stone rubble; chimney has a red-brick top. Engine-house roof removed. Engine-house consists of a single chamber, roughly square in plan. About 8.5m to E is the circular chimney. The mineshaft lies immediately W of the engine-house and to the N of this is a series of below-ground chambers. The engine-house is a tall building, probably 2-storeyed and with an underground chamber originally. Most of the walls survive, although some masonry has collapsed at the top. The E side has a tall round-arched doorway and there are similar, smaller openings on the S and W sides. The S side seems to have been decorated with pilaster-strips. On the N side is a tall narrow opening, presumably for a piece of machinery. The chimney is almost complete. At the top of the stone shaft is a cornice composed of projecting brick courses, and above that a short brick shaft with round-arched panels on its sides. The owner says the mine ceased working in 1888.
SDV348799 Website: Google. 2012. Google Streetview. http://maps.google.co.uk. Website.
SDV350785 National Heritage List for England: English Heritage. 2013. National Heritage List for England. Historic Houses Register. Digital.
MDV8015 Part of: Druid Mine, Ashburton (Monument) | {
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Q: UITextField - Keyboard not hiding when app comes from background In my iOS app, I've UITextfield at the bottom of the view. So, when user starts entering text, I'm sliding the view up so that users can see what they are typing.
While entering text, view moves upwards, then press home button and the app goes to background.
Now tap on the icon of the app and it brings the app to foreground
Now I noticed that view comes back to original position (X=0, Y=0) but keyboard is still visible.
How to hide the keyboard when the app comes to foreground.
I tried to hide the keyboard in viewWillAppear and viewWillDisappear. It didn't work.
A: -(void)applicationDidEnterBackground:(UIApplication *)application
{
[self.window endEditing:YES];
}
********** OR ************
you have to call Notification when you come from background. when you enter from background to foreground then calling this method of appdelegate.
- (void)applicationWillEnterForeground:(UIApplication *)application
fire the notification in it as below.
- (void)applicationWillEnterForeground:(UIApplication *)application
{
[[NSNotificationCenter defaultCenter] postNotificationName:@"HIDE" object:self];
}
Add notification in your viewcontroller as below
-(IBAction)HideKeyBard:(NSNotification *)noty
{
[txt resignFirstResponder];
}
- (void)viewDidLoad {
[super viewDidLoad];
[[NSNotificationCenter defaultCenter] addObserver:self selector:@selector(HideKeyBard:) name:@"HIDE" object:nil];
}
OutPut:
A: *
*Have a weak reference of UIView * in your App Delegate.
*Set it to whichever text field you show (from any view controller).
When the app comes to foreground, just call resignFirstResponder of the view and it will dismiss the keyboard. This solution does not care which view controller was displayed when the app went to background. All it needs to know is if there was a text view displayed and if yes, just resign it as first responder. If there was nothing displayed this variable will be nil and will be just a no-op.
Now, even if the same view controller is brought to front, the user has to just tap the text view to get the keyboard back which is perfectly acceptable.
A: The best way is to put window?.endEditing(true) in applicationWillResignActive on AppDelegate:
func applicationWillResignActive(_ application: UIApplication) {
window?.endEditing(true)}
hope this help you
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,683 |
Q: AttributeError: 'str' object has no attribute 'str' how to solve this When I apply f"" string on my text data it procduces the following error. AttributeError: 'str' object has no attribute 'str'.
The simple code is below, I am providing just one line so that it will save your time. I just want to apply f"" in this way. I know the problem but don't know how to figure it. Thanks
caption = f"{caption.str.lower().str.rstrip('.')}"
A: You may be familiar with the pandas library's .str accessor which make pandas ports of python string methods available to a Series. But if you're working with python builtin str types, you don't need the .str accessor. Simply call:
caption = f"{caption.lower().rstrip('.')}"
See the python builtin types docs on string methods for more info.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,025 |
Trupele de carabinieri este una din cele trei ramuri componente a Forțelor armate ale Republicii Moldova. Trupele au fost create în ianuarie 1992 în baza ordinului M.A.I. al Republicii Moldova, când se formează Regimentul mobil-operativ de carabinieri (U.M. 1001; comandant a fost numit Sergiu Ganea) și Regimentul de escortă (U.M. 1002; comandant fiind numit Valeriu Ceachir).
Trupele sunt destinate să asigure, împreună cu poliția sau independent, ordinea publică, apărarea drepturilor și libertăților fundamentale ale cetățenilor, avutului proprietarului, prevenirea faptelor de încălcare a legii.
În România, echivalentul acestei instituții se numește "Jandarmeria Română", înființată la 1850.
Istorie
La 13 martie 1992 subdiviziunea de carabinieri (u.m. 1002) pentru prima dată se îndreaptă în regiunea conflictului armat din Transnistria.
La 31 martie (1992) in timpul atacului în apropierea satului Coșnița, r-l Dubăsari, îndeplinind misiunile de serviciu au căzut în luptă plutonierul adjutant Victor Lavrentov și plutonierul Dumitru Roman.
La 30 mai (1992) subdiviziunea mixtă u.m. 1001 se îndreaptă în zona conflictului armat (s. Holercani).
La 29 iunie (1992), aflîndu-se la postul de pază al punctului de comandă al Marelui Stat Major al M.A.I. al Republicii Moldova în timpul bombardamentului concentrat de artilerie a fost rănit mortal ostașul Igor Mînăscurtă (u.m. 1001).
În baza ordinului M.A.I. din 24 august 1993 trupele de carabinieri încep să îndeplinească sarcinile de pază și apărare a obiectivelor diplomatice, pentru ce a fost formată u.m. 1026.
În februarie 1998 unitățile și subunitățile trupelor de carabinieri participă la lichidarea urmărilor calamităților naturale în s. Leușeni, Hîncești.
Structura
Comandamentul general al IGC
Direcția management operațional
Direcția management strategic
Direcția management resurse umane
Direcția securitate și investigații
Direcția cooperare și misiuni internaționale
Direcția aprovizionare și dotări
Secția financiară
Secția juridică și practică contravențională
Serviciul documentare
Serviciul tehnologii informaționale
Serviciul informare și comunicare cu mass-media
Subdiviziuni pe lângă IGC
Dispeceratul Operațional
Batalionul cu destinație specială "SCORPION"
Secția pregătire fizică și sport
Direcția medicală
Centrul de instrucție
Centrul de Asigurare
Direcția ceremonial militar
Subdiviziuni subordonate IGC
Direcția regională "CENTRU"
Direcția regională "NORD"
Direcția regională "SUD"
Batalionul cu destinație specială "SCORPION"
Batalionul cu destinație specială ,,SCORPION" este o subunitate de elită a Inspectoratului General de Carabinieri al Ministerului Afacerilor Interne cu atribuții în domeniul prevenirii și combaterii actelor de terorism, criminalității organizate și tulburărilor sociale în masă.
Batalionul cu destinație specială ,,SCORPION" are ca zonă de responsabilitate întreg teritoriul Republicii Moldova și este încadrat cu personal pregătit pentru îndeplinirea sarcinilor și atribuțiilor speciale, conform competenței.
Referințe
Legături externe
Pagina web a Trupelor de carabinieri
Legea nr. 806 din 12.12.1991 cu privire la trupele de carabinieri (trupele interne) ale Ministerului Afacerilor Interne
Forțele armate ale Republicii Moldova | {
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VERONICA MAGGIO
Sweden- the cold and sparsely populated country up north. Somehow Sweden has managed to maintain an amazing musical export over the years, from ABBA to Avicii and Max Martin to Swedish House Mafia. But there is a Swedish superstar you might not yet have heard of: Veronica Maggio. 2016 and Veronica Maggio is still undeniably Sweden's queen of pop since many years, and the biggest local artist. Despite lyrics sung in Swedish, a language spoken by no more than 10 million, Veronicas songs have managed to stream upwards of five hundred thousand plays on Spotify. Even when Veronica first showed up on the Swedish music scene in 2006 with her debut single "Dumpa Mig" (break up with me), one could sense that something unique was taking shape. Her style of lyric-writing was significantly different from the tradition of swedish pop lyrics up to date. A sort of "write about what you see"-perspective based on herself, her childhood, experiences and not so seldom about her shortcomings. Pop music with huge choruses, as contagious as the best, but revealing a darkness. Veronica sang about relations and parties, but also about restlessness, breaking up, hangovers and her lyrics often creating small film-scenes. Together with another Swedish artist, Håkan Hellström, who she later would collaborate with on the single "Hela Huset" (the Whole House), she managed to change the way swedes wrote lyrics, bring the Swedish language back up to the top of the charts and getting tons of followers in the process. Veronica lined up the hit singles - "Jag kommer" (the ambiguously titled "I'm coming"), "Välkommen in" (Welcome Inside), "Snälla bli min" (Please be Mine) and "Satan i Gatan" (an expression, much like "Goddamn!") to mention a few. Her profile rose for each album and she managed to balance huge commercial success, while remaining the critics favourite. At this point, Veronica has released four albums, won five Grammy's and received more than 25 gold and platinum awards. | {
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Asunción Bastida Pibernat (Barcelona, 6 de abril de 1902-8 de octubre de 1995) fue una diseñadora de moda española de alta costura cooperativa. Una diseñadora innovadora, que se considera fue la introductora en España del uso del algodón para la playa, la calle y los vestidos de fiesta, así como la apertura de boutiques de deportes.
Biografía
Contrajo matrimonio con Marcelino Mases i Cabeza en 1926 y abrió un negocio de prendas de punto en paseo de Gracia de Barcelona. Más tarde se trasladó a la Gran Vía, bajo el nombre de Modas Mases de Asunción Bastida, dedicada a la alta costura. En 1934, abrió una sucursal en Madrid. Con el estallido de la Guerra Civil española cerró sus dos establecimientos. En 1939, viajó a Italia, donde mantuvo contacto con las grandes casas de moda de Milán y Roma y reabrió su casa de Barcelona al final del año.
Empresaria
Fue miembro inicial de la Cooperativa de Alta Costura. Reabrió su establecimiento de Madrid y, también, realizó desfiles regulares en Sevilla. Perteneció al núcleo selecto de los llamados "cinco grandes" de la mencionada Cooperativa de Alta Costura. Bastida llegó a tener la autorización oficial de Christian Dior para reproducir sus modelos y poderlos firmar con la marca Dior. En el marco de las actividades internacionales de la Cooperativa, Bastida presentó colecciones en Estados Unidos durante la década de 1960, entre otras ocasiones, en 1963 en Miami y en 1965 en Nueva York. Trabajó para el cine español, como creadora de vestuario de los años cuarenta y cincuenta.
En la década de 1950, produjo en España la línea joven llamada a Jeunes hijas, del diseñador francés Jacques Heim. Asunción Bastida fue una de las primeras casas de alta costura en dedicar una sección de boutique y pret-à-porter en la década de 1950. De 1952 a 1968, fue directora técnica de la revista El boletín de la moda revista continuada por El boletín de la nueva moda hasta finales de los años 70. En 1970 cerró su casa de costura y continuó trabajando con el nombre Asunción Bastida S. A. hasta 1975. Está representada en la Colecció Textil Antoni de Montpalau y en la colección de moda del Museo del Diseño de Barcelona, donde se conserva también su archivo.
Referencias
Ferrando Miralles, 2014. «Boletín de la moda ". [Consultado en 2013].
Bibliografía
Ferrando Miralles, Josefina. "Assumpta Pibernat andamio". Diccionario biográfico de mujeres, 2014 [consultado: 30 de marzo de 2014].
Diseñadores de moda de Cataluña
Nacidos en Barcelona
Españolas del siglo XX | {
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Q: fancy header does not work in dissertation class overleaf As many before me, I have issues with fancyhdr. Something that should be easy to solve but is frustratingly non-cooperative.
I use a dissertation class project on Overleaf to write my thesis. This set-up is originally from Harvard, but I have altered it here and there.
Now, I would like to have a fancy header with chapter number and title at the top of every odd page (if not the first page of a chapter) using the following code:
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{Dissertate}[2014/01/24 v1.0 Dissertate Class]
\LoadClass[12pt,a4paper]{book}
\RequirePackage[a4paper, margin=2cm]{geometry}
\RequirePackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter}{}}
\fancyhf{}
\fancyhead[RO]{\thechapter}
Since it is a large group of code, I have created an open dummy project with my altered Harvard setup on Overleaf to access and try things with: https://www.overleaf.com/5483644888pkkryhrxzpsg
I did notice that somewhere in this project code the characteristics of a book class are lost/overwritten. For instance, the chapters do not open on the odd page at all time (very annoying), so something must be overriding these things. Unfortunately, I cannot seem to figure out what it is.
Could you help me figure this out?
Thanks!
A: The class file (Dissertate.cls) contains the following code:
\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter}{}}
\fancyhf{}
\fancyhead[RO]{\thechapter}
which means the the odd page header will only be the chapter number. And that is exactly what it does. Please note that the first chapter page (the one with the chapter title will not have a header as it uses page style plain (this is quite usual in LaTeX classes). So you only get a page header from the third page of the chapter. You must add more text to your example chapters to be able to see this.
To get the chapter title in the header you have to add the chapter title to the \chaptermark command, and to use \leftmark in the header, so change the above code to:
\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter. #1}{}}
\fancyhf{}
\fancyhead[RO]{\leftmark}
And if you have enough pages you will se that it works:
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/infinite-lines-of-charge.285510\/","text":"# Infinite Lines of Charge\n\n1. Jan 17, 2009\n\n### KillerZ\n\n1. The problem statement, all variables and given\/known data\n\nThe figure is a cross section of two infinite lines of charge that extend out of the page. Both have linear charge density $$\\lambda$$. Find an expression for the electric field strength E at the heigth y above the midpoint between the lines.\n\n2. Relevant equations\n\nWell E = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{r}$$ for an infinite line of charge.\n\n3. The attempt at a solution\n\nI am not quite sure what this question is asking. I think I have to integrate that formula to get an expression at y but I am not sure.\n\n2. Jan 17, 2009\n\n### Staff: Mentor\n\nNo need to integrate--use superposition. Use that formula to find the electric field from each line charge at the point in question. (What would \"r\" be? What's the direction of each field contribution?) Then just add the two vectors to find the total field at that point.\n\n3. Jan 17, 2009\n\n### KillerZ\n\nI placed the fields like this.\n\nI said:\n\nr = $$\\sqrt{y^{2} + (\\frac{d}{2})^{2}}$$\n\nE1 = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\nE2 = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\nENET = E1 + E2\n\n4. Jan 17, 2009\n\n### Staff: Mentor\n\nLooks good, but you're not done.\nActually find the resultant (in terms of the given variables).\n\n5. Jan 17, 2009\n\n### KillerZ\n\nLike this?\n\nENET = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$ + $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$ = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}} + \\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\right]$$\n\n6. Jan 17, 2009\n\n### Staff: Mentor\n\nNo. You must add them like vectors. (Find the x and y components.)\n\n7. Jan 17, 2009\n\n### KillerZ\n\nOk, I think I got it.\n\n(E1)x = (E1)cos$$\\theta$$\n(E2)x = (E2)cos$$\\theta$$\n\n(E1)y = (E1)sin$$\\theta$$\n(E2)y = (E2)sin$$\\theta$$\n\n(E1)x = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E2)x = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E1)y = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E2)y = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\nthen add the x components and the y components:\n\nENET = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}} + \\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\right]$$ i , $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}} + \\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\right]$$ j\n\n8. Jan 17, 2009\n\n### Staff: Mentor\n\nCareful with the signs of these components. (You should be able to look at the diagram and immediately have an idea of which way the total field will point.)\n\nCorrect the signs as needed and redo. Be sure to simplify your final answer as much as possible.\n\n9. Jan 17, 2009\n\n### KillerZ\n\nOk, I think I finally have it. I fixed the signs.\n\n(E1)x = -(E1)cos$$\\theta$$\n(E2)x = (E2)cos$$\\theta$$\n\n(E1)y = (E1)sin$$\\theta$$\n(E2)y = (E2)sin$$\\theta$$\n\n(E1)x = -$$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E2)x = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{\\frac{d}{2}}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E1)y = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\n(E2)y = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$$$\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}$$\n\nthen add the x components and the y components:\n\nthe x components cancel as they are equal but opposite.\n\nENET = $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}} + \\frac{2\\lambda}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\frac{y}{\\sqrt{y^{2} + (\\frac{d}{2})^{2}}}\\right]$$ j\n\n= $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{y2\\lambda}{y^{2} + (\\frac{d}{2})^{2}} + \\frac{y2\\lambda}{y^{2} + (\\frac{d}{2})^{2}}\\right]$$ j\n\n= $$\\frac{1}{4\\Pi\\epsilon_{0}}$$$$\\left[\\frac{y4\\lambda}{y^{2} + (\\frac{d}{2})^{2}}\\right]$$ j\n\n10. Jan 17, 2009\n\n### Staff: Mentor\n\nLooks good! (Cancel those 4s. )","date":"2017-01-16 10:53:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6251917481422424, \"perplexity\": 2186.0807157368095}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560279169.4\/warc\/CC-MAIN-20170116095119-00151-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
Issue 10.5: ALAC Reader "Soñadores" - California EE.UU. Los Angeles USA
Gelare Khoshgozaran, Eunsong Kim
Archiving for New Worlds
Monica Rodriguez's research as practice explores international and intergenerational struggles for Caribbean independence.
The Antilles for the Antilleans. This is how Puerto Rican nationalist Ramon Emeterio Betances calls for the independence of the Caribbean and the foundation of an Antillean Confederacy. Artist Monica Rodriguez describes her project Las Antillas para los Antillanos as a consideration of that call.
The project is an ongoing research collection that when exhibited takes the shape of a temporary "Caribbean research center." The collection includes a vast variety of objects, books, essays, drawings, and documents. It involves fifty-one collaborating artists (to date) living across the different islands in the Caribbean, including Beatriz Santiago Muñoz (Puerto Rico), Andy Roberts (Haiti and US) and Kelman Duran (Dominican Republic and Los Angeles). Connecting its remote participants, a digital version of the research project is consolidated in a Dropbox folder, to be elaborated and edited by its collaborators. Inspirations for the project include classical texts such as Roberto Fernández Retamar's Caliban essay, Eduard Glissant's concepts of "creolization" and "antillanité," and poems by Nicolas Guillen, among many other resources in English and Spanish. Among these texts also appears The Artist In The Caribbean, an open letter by CLR James originally delivered in 1959 at the University College of the West Indies Mona, in which James poses the following question:
"Is there any medium so native to the Caribbean, so rooted in the tight association which I have made between national surroundings, historical development, and artistic tradition, is there any such medium in the Caribbean from which the artist can draw that strength which makes him [sic] a supreme practitioner?" In a call to the importance of the nationalist artist, defined as a practitioner of a wide range of mediums, including literature, painting, architecture, etc., James compellingly states in his letter: "the universal artist is universal because he is above all national."
It is not uncommon among contemporary artists to incorporate extensive research, archiving and cataloging as part of their practice. Artists' research, an often uncompensated and unacknowledged labor, may exist as independently from the artwork, as a contextualizing backdrop or as a complementary element to the "final project." For Monica Rodriguez, the research is the process of the production. The work is the material revisited, collected and (re)created that fills the space of a gallery and turns it into a library or research center. More than displacing objects and items from the colonial archives—a curatorial process that borrows on loan—the artist is interested in the "building of an archive" that often takes a collaborative form.
Following her trajectory on archival transformation, in the Los Angeles iteration of Monica Rodriguez's Las Antillas para los Antillanos, as part of Open Air Prisons (September–November 2016), the project room at Los Angeles Contemporary Exhibitions was transformed into a "Caribbean research center." Books, publications, documents, and drawings were organized next to images of food, flags and listening stations. The mats and pillows on the floor, as well as the desks and chairs, implied the center as a space for study and contemplation, a space between a personal library and an artist's studio.
In Rodriguez's words, "The project uses Puerto Rican nationalist Ramon Emeterio Betances' call—Antilles for the Antilleans—as starting point to build connections between cultural producers among the Caribbean, a region where political and economic paradigms have prevented the development of a creative and fair flux between islands. This has historically been due to language barriers, cultural and ethnic differences, and because of the economic and political structures implements by colonial powers. It is, therefore, the archive's core intention, to prompt a much-needed dialogue about what it means to be Caribbean through an articulation of collective social exchange. In other words, only through collectivity can we (The Antilleans) work together towards a consideration of decolonization strategies through a creative supportive network."
The online collaborations include the use of "exquisite corpse" as a way to network artists across the Caribbean islands. Rodriguez's project investigates the history of Puerto Rico's colonization as beginning with the question of the Caribbean writ large. By examining the interconnections of Caribbean colonization, the project digs into the deep and complex histories of political struggle, complicating language and ethnic similarities. Rather, she emphasizes that the islands in the Caribbean have been separated because of such differences—and not by their political distances. Hence it is part of her project's ambitions to imagine the relationships and connections among them through a creative, critical and explicitly political archive building process. At the core of Rodriguez's practice is the political question of methodology: how is Caribbean identity investigated through creative practice, and how are the results of such investigations displayed.
Their display often takes the form of transference. Another work, Botin Ocupado (libros) (2014) features twelve wooden frames that contain Rodriguez's drawings of confiscated books, taken by the U.S. Department of Justice during or after the Puerto Rican Nationalist Revolt of 1950. The drawings of book covers depict the reading selections of anti-colonial protestors and their allies as well as the range of threats perceived and felt by the colonial forces. In a drawing on the left-hand side, Arte de la Guerra is drawn in black capital letters. Next is Edward Hallett Carr's La Revolucion Ruse de Lenin a Stalin 1917–1929 in a soft brown marron. The drawings set up the scope of contractions in confiscation. Blatantly political titles, such as La Lucha Por La Independencia de Puerto Rico by Juan Antonio Corretjer and El Fascismo by Juan Carreras appear in the same red and black palette. They are intermixed with Women in Kentucky Industries 1937 and the cookbook Scholar et Francois Recettes de Cuisine Practique, the letters drawn out in lapis blue with a pale pink depiction of a roasting pig. Through Botin Ocupado (libros) (2014) we learn that Popular Science and Reader's Digest were included in the confiscated materials. How are we to make sense of the threat that cookbooks, industrial histories, and contemporary science posed to colonial governments—in addition to the texts that could have only called for, not carried out, direct attack? Perhaps more importantly, the range of the drawings unfold the complex knowledge systems—life—of those associated with the Puerto Rican Nationalist Revolt of 1950.
Rodriguez's work sets up imaginative landscapes by re-situating artifacts from colonial archives through an auxiliary. Her drawings, sculptures, and databases work to divulge the detailed accounts incorporated into colonial archives yet still missing from our own understanding (what and how much was taken from the resisters and why, and what explanations do they offer today?). They show us the possibilities offered when the archive is rendered to narrate those it has taken from. In Botin Ocupado (libros) (2014) we learn not only what the US colonial state feared, what it marked as a potential threat, but also about the life of those working towards liberation. They read everything, about how to fight, the histories of fighting, and Reader's Digest. What could be harmful about practical cooking and contemporary news other than that, a better and thriving life for the anti-colonial, decolonial dreamer is in itself, the threat? Such are some of the questions leaping out of the drawings in Botin Ocupado (libros) (2014).
The project allows us to imagine a museum of confiscated items and gives flight to their potential variations: a museum of confiscated items across colonial regimes across the continents. While absurd and comical in its banality, it is also a profound entrance into the lives and communities of resisters. There is potential in understanding what power imagines will unravel its hold. We feel it through her beautiful, innocuous, DIY transformations of objects that induced so much fear in an all-powerful colonial government. In addition to the books in Botin Ocupado (libros) (2014), a series of molotov cocktails sits next to a box of matches, flags, and shovels. The presence of these items next to each other infers the potential for a fire. It has all the necessary elements but has yet to be ignited. What comes through is a rage that is well alive.
When we asked Rodriguez about the process of selecting archives, and her method of researching, she recounts that formal methods cannot easily be applied to government records of resistance. When she was in New York City interested in making work about the Puerto Rican Nationalist Revolt of 1950 she looked at the archives at Hunter College. She tells us that the records that became translated as drawings in Botin Ocupado (libros) (2014) were in boxes about Puerto Rican history but did not necessarily announce themselves as documents about the Revolt.
Rodriguez's bodies of work re-imagine classification systems and forms of knowledge for the historical to present day protester and new and old possibilities of political Caribbean affinities and accentuate the lived politics of the Master Archives. Her work is also an archive for new dreams: a world not yet entered but being imagined.
TAGS: dreamers Eunsong Kim Gelare Khoshgozaran Las Antillas Monica Rodriguez
Alma Ruiz
Art after LA/LA
Natalia Mendoza, Miguel Fernández de Castro
Unsettled: Limits and Domains
Issue 21: A Burning Song
Maya Juracán
A Future Without the Need to Name Ourselves
On a path that begins to outline a more diverse presence in Central American art, curator Maya Juracán talks with a few of the region's collectives about the possibility of visualizing futures where creative processes endowed with life expand and contract; healing and weaving genealogies to write another history of art.
Risseth Yangüez Singh, José Braithwaite, Milko Delgado
Exercises of Resistance: Recovering and Reconstructing Memory and Identity/ies
In an exercise in which criticism and imagination are inextricably linked, Panamanian artists Milko Delgado, Risseth Yangüez Singh, and José Braithwaite exchange opinions, expectations, and memories about the conflicted and problematic mestizo identity in Panama to consider the possibility of a world where Afro-descendant and Black memory of the land is not ignored.
Alex Santana, Lorena Cruz Santiago
Indigenous Visual Sovereignty: For All the Memories Yet To Be Recorded
Where does the sovereignty of images lie? Wandering critically through the fields of memory, the artists Lorena Cruz Santiago and Alex Santana share impressions of visual productions that, as an exercise of autonomy, move away from statism flowing in a stream of what original populations are and have been.
Elyla, Juan José Guillén (Purificación, Puri)
Loose Tongues in Mesoamerica
The artists Elyla and Purificación exchange dreams and critical notions regarding mestizo identities subjected to the colonial apparatus in Nicaragua and Guatemala, which, in tension with their mariconería, outline artistic practices that make imagining other worlds possible. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,128 |
\section{Introduction}
Researchers working in natural language processing (NLP) often treat hate speech as a binary, unified, concept that can be detected from language alone.
However, as a linguistic concept that relies heavily on social context, hate speech contains a variety of related phenomena~\citep{Brown2017part2}.
Hate speech is characterized by variation in linguistic features (e.g. implicit vs. explicit), context (e.g. platforms, prior conversations), and communities (social histories and hierarchies).
This paper focuses on a crucial aspect of this variation: how hate speech varies by the identity groups it targets.
To study this variation, we analyze hate speech datasets that include annotations for which identity group is targeted.
Drawing from multiple of these datasets, we sample new corpora that target the same identity group.
These identity groups vary according to several dimensions, including relevant demographic category (e.g. gender, religion) and relative social power (e.g. socially marginalized or dominant).
We empirically test which dimensions most clearly separate different forms of hate speech by
evaluating how well classifiers trained on one set of identities generalize to hate speech directed at different sets of identities.
We find that hate speech varies most prominently by the targeted demographic category and less so by the social power of the targeted identity group.
Theorists working in philosophy and sociolinguistics have drawn attention to how hate speech directed at marginalized groups differs from hate directed toward socially dominant groups \citep{Butler1997, Lakoff2000}.
However, we do not find that hate speech toward dominant groups is sufficiently different to consistently increase classification performance when removed from existing datasets.
Analyzing the most representative terms in hate speech directed toward different identities, we find that many words reflect identity-specific context such as histories of oppression or stereotypes.
These results have implications for NLP researchers building generalizable hate speech classifiers, as well as for a more general understanding of variation in hate speech.
\paragraph{Contributions}
\begin{enumerate}
\item An empirical analysis of variation in hate speech by target identity. Specifically, how well classifiers trained on hate speech directed toward specific identities generalize to hate speech directed at other identities.
\item An analysis of which dimensions of social difference (demographic category, power) among targeted identities reflect the most variation in hate speech.
\item A qualitative analysis of the hate speech terms most strongly associated with specific target identities.
\end{enumerate}
\section{Hate Speech}
Hate speech is an example of a ``thick concept'' with a set of related, but difficult to define meanings and understandings \citep{Pohjonen2017}.
Legal theorist Alexander Brown \citeyearpar{Brown2017part2} argues for a set of attributes that make an expression more or less likely to be considered hate speech, similar to Wittgenstein's ``family resemblances'' concept.
Key attributes include an incitement of emotion and violence, and a direction of that incitement toward a targeted identity group \citep{Sanguinetti2018, Poletto2021}.
Though others have studied the linguistic properties of this incitement~\citep{Marsters2019,wiegand-etal-2021-implicitly-abusive}, we focus on how variation in the identity group targeted by hate speech affects the linguistic characteristics of hate speech.
\subsection{Variation by identity}
Identities are central to hate speech.
Classifiers often learn to associate the presence of identity terms, especially derogatory ones, with hate speech and abusive language~\citep{Dixon2017,Uyheng2021}.
Computational studies of the targets of online hate speech have included measurement studies of its prevalence toward different targets.
\citet{Silva2016} and \citet{Mondal2017} searched for templates such as ``I hate \_\_\_'' to measure hate toward different identity groups.
We analyze datasets manually annotated with the targets of hate speech.
This captures a broader range of hate speech, including indirect hate speech and stereotypes.
\citet{elsherief2018hate,ElSherief2018} investigated differences between hate toward groups versus individual targets.
In contrast, we compare differences among identity targets.
\citet{rieger_assessing_2021} measured multiple types of variation, including by identity target, in hate speech from fringe platforms such as 4chan and 8chan.
We test if such differences affect the generalization of hate speech classifiers.
Many identities are involved in the production and recognition of hate speech, including the identities of those who produce hate speech and those who annotate hate speech datasets.
The post history and inferred gender of social media users have been found to be useful in predicting hate speech~\citep{Waseem_hovy2016,unsvag_effects_2018,qian_leveraging_2018}.
\citet{Waseem2016} find differences in hate speech annotations between crowdworkers and experts, while \citet{sap2022annotators} find differences by the political ideology of annotators.
We focus on identities presented in the hate speech itself.
\subsection{Generalizability}
In this paper, we evaluate the ability of hate speech classifiers to generalize across targeted identities.
\citet{Groendahl2018} find that hate speech models generally perform poorly on data that differs from their training data; we look at how shifts in the distribution of identity targets affects generalization.
\citet{Swamy2019} look at generalizability across subtasks of abusive language detection and find that a larger proportion of hateful instances aids generalization.
\citet{Pamungkas2020} and \citet{fortuna-etal-2020-toxic} find that hate speech models using variants of BERT \citep{devlin-etal-2019-bert} generalize better than other models.
We thus use a variant of BERT in our generalization experiments.
See \citet{Yin2021} for a more thorough survey on generalizability in hate speech detection.
\section{Data}
From surveys of hate speech datasets \citep{Vidgen2020,Poletto2021} and the Hate Speech Dataset Catalogue\footnote{\url{https://hatespeechdata.com/}}, we selected datasets with annotations for targeted identities.
We only selected datasets that do not restrict target identities in order to minimize differences in other properties (e.g, domain, year) when comparing across targeted identities.
This excludes hate speech datasets and shared tasks that focus on particular targeted identity groups, such as women or immigrants \citep{Kwok2013,Basile2019}.
We also did not consider hate speech datasets that label targeted demographic category, such as race or gender \citep{Waseem2016}, but do not specify the identity group targeted.
Demographic category is just one of the dimensions of similarities and differences among identity groups that we wish to compare for their affect on hate speech.
We included datasets from all domains, except those with synthetic data.
Since we only found one non-English dataset that contained unrestricted annotations for targeted identities \citep{ousidhoum_multilingual_2019}, we focus on hate speech in English in this work.
For generalization analyses, we sampled corpora specific to identity groups across datasets large enough to contain a minimum number of instances of hate speech against enough groups (described in Section \ref{sec:sampling}).
These are the first 4 datasets noted in \autoref{tab:dataset}.
All datasets are used in the analysis of removing dominant groups (Section \ref{sec:power_removal_comparison}).
\input{tables/dataset}
Datasets are resampled to a 30/70 ratio of hate to non-hate to eliminate a source of variance among hate speech datasets known to affect generalization~\citep{Swamy2019}.
Non-hate instances are upsampled or downsampled to meet this ratio, which was chosen as typical of hate speech datasets~\citep{Vidgen2020}.
If they do not already contain a binary hate speech label, dataset labels are binarized as described in Appendix \ref{sec:appendix}.
\subsection{Target identity label normalization}
\label{sec:norm}
Annotations for targeted identities vary considerably across datasets.
Some of these differences are variations in naming conventions for identity groups with significant similarity (`Caucasian' and `white people', for example).
Other identities are subsets of broader identities, such as `trans men' as a specific group within `LGBTQ+ people'.
To construct identity-based corpora across datasets, we normalized and grouped identities annotated in each dataset.
One of the authors, who has taken graduate-level courses on language and identity, manually normalized the most common identity labels in each dataset and assigned these normalized identity labels into broader identity groups (such as `LGBTQ+ people').
Intersectional identities, such as `Chinese women', were assigned to multiple groups (in this case `Asian people' and `women').
Hate speech was often directed at conflated, problematic groupings such as `Muslims and Arabs'.
Though we do not condone these groupings, we use them as the most accurate descriptors of identities targeted.
\section{Cross-Identity Generalization}
\label{sec:cross-identity}
We examine variation among hate speech targeting different identities in a bottom-up, empirical fashion.
In order to do this, we construct corpora of hate speech directed at the most commonly annotated target identities, grouped and normalized as described in Section \ref{sec:norm}.
We then trained hate speech classifiers on each target identity corpus and evaluated on corpora targeting other identities.
Along with practical implications for hate speech classification generalization, this analysis suggests which similarities and differences among identities are most relevant for differentiating hate speech.
\subsection{Data sampling}
\label{sec:sampling}
In order to have enough data targeting many identities and to generalize beyond the particularities of specific datasets, we assembled identity-specific corpora from multiple source datasets.
To mitigate dataset-specific effects, we uniformly sampled hate speech instances directed toward target identities from the first 4 datasets listed in \autoref{tab:dataset}.
We select these datasets since they contain enough data to train classifiers targeting a sufficient variety of identities.
The corpus for each target identity contains an equal amount of hate speech drawn from each of these datasets, though the total number of instances may differ among corpora.
Negative instances were also uniformly sampled across datasets, and were restricted to those which had no target identity annotation or an annotation that matched the target identity of the hate speech.
We selected target identities that contained a minimum of 900 instances labeled as hate across these four datasets after grouping and normalization.
We selected this threshold as a balance between including a sufficient number of identities and having enough examples of hate speech toward each identity to train classifiers.
In order to include a variety of identities in the analysis while maintaining uniform samples for each dataset, we upsample identity-specific hate speech from individual datasets up to 2 times if needed.
Corpora are split into a 60/40 train/test split.
Selected target identities and the size of each corpus can be found in \autoref{tab:corpora_stats}. These identity-specific corpora, which are samples of existing publicly available datasets, are available at \url{https://osf.io/53tfs/}.
\input{tables/dataset_stats}
\subsection{Cross-identity hate speech classification}
Due to the high performance of BERT-based models on hate speech classification~\citep{Mozafari2019,Samghabadi2020}, we trained and evaluated a DistilBERT model~\citep{Sanh2019}, which has been shown to perform very similarly to BERT on hate speech detection with fewer parameters~\citep{vidgen-etal-2021-introducing}.
Models were trained with early stopping after no improvement for 5 epochs on a development set of 10\% of the training set.
An Adam optimizer was used with an initial learning rate of $10^{-6}$.
Input data was lowercased and an uncased base DistilBERT model was fine-tuned using the Hugging Face Transformers package, Keras, and Tensorflow.
We removed URLs, hashtags and @mentions of users, but kept emoji in preprocessing.
To mitigate random variation, we trained separate DistilBERT models 5 times and report the average performances.
As a baseline, we also evaluated a logistic regression classifier with TF-IDF unigram features over the entire vocabulary.
This classifier used L2 regularization with a constant $C=1$.
Results from only the DistilBERT models are reported as they consistently outperformed the logistic regression model by 0.1 F1 or more.
Generalization performance trends across identities were similar for DistilBERT and logistic regression.
Code for these analyses are available at \url{https://github.com/michaelmilleryoder/hate_speech_identities}.
\subsection{Results}
\autoref{fig:identity_generalization} shows generalization performance, measured by F1-score on the positive class of hate speech, across identity splits.
We choose F1 on the `hate' class since that focuses on performance in detecting hate speech across different target identities, rather than the non-hate instances which may or may not target identities.
Generalization across target identities is poor, often dropping from over 70 F1-score when training and test sets match by targeted identity to less than 40 when they do not.
\input{tables/identity_generalization}
Following \citet{Uyheng2021}, we perform a PCA dimensionality reduction of this generalization performance to 2 factors in order to visualize which target identities exhibit similarities (\autoref{fig:identity_pca}).
\begin{figure}[tb]
\centering
\resizebox{\columnwidth}{!}{\includegraphics{img/identity_pca_legend_top.pdf}}
\caption{PCA of cross-identity hate speech classification performance. Hate speech classifiers trained on data targeting identities in the same demographic categories perform most similarly.}
\label{fig:identity_pca}
\end{figure}
Evident from this PCA is a clustering of identity targets by demographic category.
In particular, three clusters are evident: identities that reference religion are in a similar space, while identities that reference race and ethnicity are in a different space, as are terms that reference gender and sexuality.
We look specifically at the effect of these distinctions on hate speech in Section \ref{sec:category}.
Three identities included have relative social power in the European and North American English-speaking contexts from which our datasets were drawn: white people, Christians, and men.
These identities do not form a clear cluster in \autoref{fig:identity_pca}, though they contain factor loadings relatively close to 0 for both factors.
In Section \ref{sec:power}, we investigate how hate speech varies according to the relative social power of the identities targeted.
\section{Variation by Demographic Category}
\label{sec:category}
Poor generalization results across identity targets (\autoref{fig:identity_generalization}) suggest that hate speech varies significantly by the identities it targets.
Our results also suggest that this variation patterns largely by demographic categories such as race/ethnicity, gender/sexuality, and religion (\autoref{fig:identity_pca}).
We hypothesize that if demographic categories are particularly discriminative, hate speech classification performance will drop sharply when attempting to generalize across categories.
To test this, we manually assigned normalized and grouped identities to the categories referenced by the identity.
For example, the identity of `Asian' references race/ethnicity, while `Asian women' references both race/ethnicity and gender/sexuality.
In cases where target groups fit multiple categories (which is not common), we include instances in all corpora they reference.
Though targeted identities sometimes reference categories such as politics, interests, and age, the only categories that met a threshold of 900 hate speech instances uniformly sampled across datasets were race/ethnicity, religion, and gender/sexuality.
Details on corpora constructed by category can be found in \autoref{tab:corpora_stats}.
We then train DistilBERT hate speech classification models on each corpus and test on all others to measure generalization performance in the same way as for identity generalization.
Results can be found in~\autoref{tab:category_generalization}.
\input{tables/category_generalization}
Performance drops across identity categories, sometimes falling by almost half of the F1-score.
This suggests that for purposes of automatic classification, hate speech varies significantly by demographic category.
Classifiers generalize particularly poorly from race/ethnicity and religion to gender/sexuality, and less poorly between race/ethnicity and religion.
This may be because of the blurred lines in hate speech targets between racial and religious categories, for example, by conflating Muslims and Arabs or targeting Jews by both religious and racial characteristics.
\section{Variation by Power}
\label{sec:power}
Another significant dimension of variation among targeted identities is relative social power in the societies from which hate speech data has been drawn.
Work on hate speech detection in NLP is often motivated as an effort to fight sexism, racism, homophobia, and other oppressions of marginalized groups, and improve participation of these groups online~\cite{mathew2021hatexplain,Jurgens2019}.
However, this work often frames hate speech as a property of language without considering social context.
Abstracting away from the particulars of targeted identities, datasets often include hate speech directed at any identity group, regardless of the social context of power or marginalization.
Such datasets thus include hate speech directed toward groups with relative social power, such as white people or men in English-speaking European and American contexts.
Calls are growing to consider the role of power and historical oppression in NLP work~\citep{blodgett-etal-2020-language, field-etal-2021-survey}.
Moreover, some theorists of social meaning in language argue that hate speech is fundamentally different when directed at social groups with power \citep{Butler1997,Lakoff2000}.
They note that such speech does not reference the same historical threat of possible violence and recurring oppression as does hate directed toward marginalized groups.
From a lens of social dominance theory~\citep{Sidanius1999}, hate speech serves either to perpetuate or challenge group hierarchies depending on its target.
Activists have called for social media platforms to incorporate this social context by treating hate speech toward marginalized groups as more serious than hate directed toward groups with relative social power \citep{Nurik2019, Dwoskin2020}.
For these theoretical and practical reasons, we consider empirical differences in hate speech based on the social power of targeted identity groups.
Similar to previous experiments, we test the generalization of classifiers across identities with different levels of social power.
We also test for effects on classification performance when removing hate directed toward socially dominant identity groups from hate speech datasets.
If this type of hate is sufficiently different, including it could ``muddy'' the concept we are after and reduce the effectiveness of classifiers in identifying hate speech.
Removing it would more closely match commonly stated motivations of NLP work on hate speech.
\subsection{Generalization}
Just as with demographic categories, we construct separate corpora of hate speech directed at identities with relative social power and identities with relative social marginalization.
We manually label normalized, grouped identity terms with a coarse-grained label as either \textit{dominant}, \textit{marginalized}, or \textit{other}.
This labeling was done by one of the authors familiar with the North American and European English-speaking contexts from which hate speech datasets were drawn.
Identity groups certainly have different social power depending on the setting.
For example, though LGBTQ+ people are generally marginalized, gay men in LGBTQ+ spaces can have higher social power relative to people with more marginalized genders and sexualities~\citep{stulberg2018lgbtq}.
Our goal in annotation was to label identity groups for which there would be broad agreement of enduring dominance or marginalization in North American and European English-speaking societies.
All other cases were marked \textit{other}.
This included political identities such as `Republican' or `liberal', since political power is generally transient in these societies.
Some targeted identities were intersectional, that is, contained multiple identity groups, such as ``white women'' or ``transgender men''.
These cases were taken case-by-case, considering the marginalization of each identity component and marking \textit{other} for many tough cases.
A full list of identities labeled as dominant and marginalized is available in \autoref{tab:marginalized_dominant_identities} in Appendix \ref{sec:appendix}.
Any identities not in these lists were marked \textit{other} by default.
Some datasets all annotators to mark multiple targeted identities.
We marked these instances as directed to \textit{marginalized} groups if there was only \textit{marginalized} or \textit{other} identities targeted.
Instances with both \textit{marginalized} and \textit{dominant} identities targeted were marked as \textit{other}.
Details on corpora constructed by power are in \autoref{tab:corpora_stats}.
As with identities and demographic categories, we evaluated the ability of DistilBERT hate speech classification models to generalize across marginalized and dominant identity targets (\autoref{tab:power_generalization}).
\input{tables/power_generalization}
Generalization does not suffer as much across target identities with differences in social power, particularly when trained on the corpus of hate directed at marginalized identities.
This suggests that which target identities have power does not structure variation in hate speech as much as differences in demographic category.
\subsection{Removing hate speech toward socially dominant groups}
\label{sec:power_removal_comparison}
We further evaluate the effect of removing hate speech toward socially dominant groups on classification performance.
We hypothesize that if it is sufficiently different, as some theorists argue, then it may act as noise.
For this experiment, we resample all 7 hate speech datasets listed in \autoref{tab:dataset} separately instead of combining across datasets as in generalization experiments.
This allows us to see trends across even more datasets than we could examine if uniformly sampling from just those with enough to reach a certain threshold.
We resample each dataset to exclude or include hate toward dominant social groups.
All instances are the same between these samples except for instances of hate speech toward dominant social groups and those instances replaced by them.
This allows a comparison across samples of equal size and hate speech ratio.
Removing hate speech toward any set of target identities could improve performance since the remaining instances are more likely to be similar to each other.
For this reason we compare removing hate speech toward dominant groups with removing hate speech toward a set of non-dominant identities.
We select these ``control'' identities to be similar in frequency across datasets to identities labeled as dominant.
Specifically, we match each identity labeled as dominant with the non-dominant identity that has the closest log frequency distribution across datasets (by Euclidean distance).
We perform 5x2-fold cross-validation with a DistilBERT model to estimate performance with and without dominant or control identities.
Parameters are the same as were used with the models built to test generalization, and 10\% of training sets are used as development sets for early stopping.
Two out of the 7 datasets, \citet{elsherief2021latent} and HateXplain, show significant improvement after removing hate speech toward dominant social identities.
However, when removing the control identities, 2 out of the 7 datasets, Civil Comments and HateXplain, also show significant improvements, while the Social Bias Inference Corpus shows a significant decrease in performance.
This does not show convincing evidence that hate speech toward dominant groups is sufficiently different to act as noise for hate speech classification.
\section{Lexical Variation Across Target Identities}
To explore how hate speech varies by target identity, we examine the words most strongly associated with each target identity and grouping of identities.
We use the Sparse Additive Generative Model~\citep[SAGE;][]{Eisenstein2011} to find words that are most representative of each hate speech corpus.
SAGE finds representative words by learning a generative model that contrasts terms in documents in a section of a corpus with a background frequency distribution over the whole corpus.
We run SAGE over 3 separate corpora: one where each section is an identity-specific split, another with category splits, and another with splits by relative social power.
We run SAGE with a vocabulary size of the most frequent 3000 words and a smoothing rate of 50.
Larger vocabulary sizes and lower smoothing included less informative, specialty words that did not occur frequently in the corpus.
The 10 most representative terms for each of these splits are shown in \autoref{tab:top_terms}.
\input{tables/top_terms}
Identity terms, many of them derogatory, form the bulk of these representative words.
This provides more evidence for the centrality of identities to hate speech~\citep{Uyheng2021}.
Some representative words relate to identity-specific histories of oppression.
For example, `oven' and `gas' are representative terms of antisemitic hate speech.
Identity-specific stereotypes are also visible: `terrorist' and `bomb' are top terms in hate speech against Muslims and Arabs.
Current culture wars issues are also relevant.
For example, transphobic attitudes around bathrooms are reflected in the top terms in hate speech targeting LGBTQ+ people.
`BLM', for the Black Lives Matter movement, is a top term associated with anti-Black hate speech.
The difficulty in a binary distinction of dominance and marginalization can be seen through the most representative words in hate directed toward groups with high relative social power.
As a marker of Christianity, `Catholic', for example, could be seen as dominant in European and American contexts where Christianity has historically been a religion with relative social and cultural prominence.
However, some white nationalist groups such as the Ku Klux Klan have targeted Catholics as outside idealized Christian Protestantism~\cite{Burris2000,Berlet2006}.
`Redneck' and `trash' are top terms in hate targeting white people, and `virgin', a top term in hate targeting dominant groups, is used in jokes stereotyping incest.
Such terms target poor white people based mainly on class.
Also in the top terms against white people is `mudshark', a derogatory term targeting white women who have relationships with Black men.
These terms target groups that are marginalized within broadly dominant groups: white women, poor white people, and Catholics.
Such examples show how social power is relative, complex, and intersectional.
They also evidence a tendency for hate speech to target marginalized groups, even within groups that have higher relative social power.
\section{Discussion}
Our results demonstrate that hate speech varies considerably according to which identities are targeted.
We show evidence that classifiers trained on hate toward one target identity generalize poorly to other target identities, especially across demographic categories such as race/ethnicity, religion and gender/sexuality.
These results suggest that the designers of hate speech classifiers pay attention to the distribution of targeted identities in training data.
Many commonly used hate speech datasets do not specify this information.
If the distribution skews toward a particular identity group (such as anti-Black racism), then using such a classifier on data that has a different distribution (e.g., mostly antisemitic) would likely give poor performance.
More generally, these results suggest a value in treating hate speech as a social and linguistic category with lots of internal variance.
This variance depends in part on the social context around targeted identities.
Classifiers trained on hate speech toward dominant or marginalized groups suffered somewhat when tested on the opposite group.
However, we did not find evidence that removing hate speech toward dominant groups clarifies the hate speech signal enough to consistently increase performance beyond what might be expected by removing a random set of targeted identities.
This suggests that differences based on the social context of power do not affect the language of hate speech enough to be easily detectable by machine learning classifiers.
Differences in severity between hate speech targeting socially marginalized or powerful groups is more likely a matter of interpretation by those with social knowledge of power in a particular society.
\section{Conclusion}
We present a meta-analysis of hate speech datasets annotated for identity group targets.
This analysis shows that hate speech differs significantly by target identity, as classifiers trained on hate speech toward one identity do not generalize well to other identities.
We then examine what factors of social context structure this variation by target identity.
We find evidence for hate speech varying substantially by demographic category, and less so by the relative social power of targeted identities.
These results reinforce the importance of variation by social context within hate speech and suggest that researchers pay attention to variation by target identity.
Future work may address improving generalization across target identities by strategically sampling training data or incorporating multiple identity-specific classifiers.
Similar analyses may also be conducted on multilingual hate speech datasets in future work.
\section{Limitations and Ethics}
As a meta-analysis of existing datasets, this study is limited by the availability of hate speech data labeled with target identity.
Performance estimates with and without hate speech toward dominant groups would be more reliable with more labeled hate speech toward socially dominant groups.
The scarcity of hate speech against socially dominant groups is not coincidental: this speech is less prototypically considered hate speech than that against marginalized groups.
This can be seen in the dataset from \citet{Kennedy2020}, for example, where annotators rate the average severity of hate against dominant groups as less than the average severity of hate against marginalized groups.
Another limitation is that datasets each have their own definitions of hate speech and associated annotation criteria, which may vary considerably.
We attempted to mitigate the effects of any one dataset's definition with uniform sampling (see Section \ref{sec:sampling}).
Since we take these annotations as representative of hate speech, it is necessary to be mindful that we are not capturing any true sense of ``hate speech'', but simply what annotators have identified as hate speech.
However, we wished to investigate the role of target identity in existing hate speech classification approaches, for which existing datasets and their associated definitions are most relevant.
These datasets are only in English and largely reflect European and American societies.
Our findings are specific to this context.
Experiments on multilingual datasets may reveal other trends and reflect different social associations around identity terms, which are culturally specific.
When sampling identity-based corpora from datasets, we attempted to control for the idiosyncrasies of any particular dataset.
However, the sizes of the resulting identity-specific corpora vary depending on how much hate speech directed toward them occurs across datasets.
This could influence our generalization experiments.
Classifiers trained on identities with small corpora still perform well on test sets of identities with the same demographic category, the general trend we report.
As seen in \autoref{fig:identity_pca}, identities with lots of data sometimes exhibit behavior similar to identities with not as much data.
These factors lead us to doubt that corpus size has a large impact on generalization results.
Care must always be taken to specify that differences based on identity, in this case hate speech directed toward identities, are due to social, not biological, factors \citep{Hanna2020,Lu2022}.
We attempt to be clear that these differences are the result of social context.
\section*{Acknowledgements}
This work was supported in part by the Collaboratory Against Hate: Research and Action Center at Carnegie Mellon University and the University of Pittsburgh.
The Center for Informed Democracy and Social Cybersecurity at Carnegie Mellon University also provided support.
We thank the researchers who made their annotated hate speech data publicly available, which enabled this meta-analysis.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,588 |
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export default mongoose.model('Input', inputSchema); | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,067 |
\section{Introduction}\label{sec:intro}
The macroscopic properties of neutron stars, like masses, radii, and tidal deformabilities, are highly sensitive to the nuclear microphysics of the stellar interior. Nonetheless, relations between pairs of these observables are often remarkably \emph{insensitive} to internal structure: while a neutron star's properties depend individually on the extreme-matter equation of state, certain combinations of them effectively do not.
Several nearly equation-of-state~independent relations among neutron-star~observables have been studied under the designation of \emph{approximate universal relations} (see \citet{YagiILQreview} for a review).
These include I-Love-Q relations between the moment of inertia $I$, the tidal deformability (or Love number) $\Lambda$ and the rotational quadrupole moment $Q$ \citep{YagiILQscience,YagiILQ}; effective no-hair relations among the lowest few multipole moments \citep{Pappas,Stein,Yagi_nohair}; and binary Love relations linking the tidal deformabilities $\Lambda_{1,2}$ of the members of a binary system \citep{Yagi_BiLoveLett,Yagi_BiLove}.
Neutron-star universality has been proposed as a tool for constraining observationally inaccessible properties \citep{YagiILQreview}, enhancing gravitational-wave parameter estimation \citep{Yagi_BiLove,Chatziioannou}, reducing uncertainty in electromagnetic radius measurements \citep{Psaltis,Baubock}, and testing general relativity (see \citet{Doneva} for a review), among other applications.
The equation-of-state~insensitivity of the relations connecting a single neutron star's various properties is thought to arise from an emergent symmetry in strongly gravitating stars \citep{Yagi_why}. This kind of universality can be used to translate a measurement of e.g.~a neutron star's tidal deformability into a determination of the same star's moment of inertia with percent-level error \citep{YagiILQscience}.
In conjunction with the assumption that all {neutron star}s share a common equation of state---a consequence of fundamental nuclear many-body physics---one can moreover establish approximate universal relations between the properties of \emph{different} {neutron star}s, like the binary Love relations.
The relations need not be restricted to members of a binary system; measurements of one neutron star~have implications for the properties of all other cold, $\beta$-equilibrated {neutron star}s in the universe.
Indeed, identical universal relations with comparably small dispersion hold whether the {neutron star}s are composed of hadronic, quark \citep{YagiILQscience} or two-phase hybrid hadron-quark \citep{Paschalidis,Wei} matter.
We caution, however, that neutron-star universality is violated by nonbarotropic equations of state, such as those describing young, hot {neutron star}s \citep{Martinon,Marques}, and by the presence of strong stellar magnetic fields, like those associated with magnetars \citep{Haskell}; inferences derived from universal relations are therefore valid for weakly magnetized isolated {neutron star}s long after formation and binary {neutron star}s long before merger.
Furthermore, universality appears to be broken by certain non-standard equations of state~with strong phase transitions \citep{Bandyopadhyay,Han,Lau,Annala17}.
Disagreement between universal-relation based predictions and direct measurements of astrophysical {neutron star}s could thus be a signature of such equations of state.
The universal I-Love relation and a specially adapted binary Love relation were combined by \citet{Landry_pulsar} to infer the moment of inertia of PSR J0737-3039A, the primary component of the double pulsar, with $\approx 30\%$ accuracy based on tidal deformability constraints from GW170817 \citep{LVC_GW170817,LVC_eos}.
We extend this technique to make general inferences about the properties of {neutron star}s, placing bounds on tidal deformability, moment of inertia and radius $R$ as a function of stellar mass $M$ via universal binary Love, I-Love and I-compactness relations.\footnote{A different universal relation has been used elsewhere in conjunction with GW170817 to constrain the maximum mass of nonrotating {neutron star}s \citep{Rezzolla}.}
We take the constraint $\Lambda_{1.4} = 190^{+390}_{-120}$ (median and symmetric 90$\%$ confidence interval) on the canonical deformability of a $1.4\,M_{\odot}$ neutron star~established by \citet{LVC_eos} as our primary observational input.
Their study assumed a common equation of state~\citep{Chatziioannou} and reprised the initial GW170817 parameter estimation of \citet{LVC_GW170817}, which found $\Lambda_{1.4} \leq 800$ at 90$\%$ confidence, assuming small neutron-star~spins, by performing a Bayesian analysis of the gravitational-wave strain data recorded by Advanced LIGO \citep{LIGO} and Virgo \citep{Virgo}.
The original parameter estimation results were also combined with priors on the equation of state~from parametric piecewise-polytrope \citep{Annala18} and perturbative QCD \citep{Most2018} models to obtain the constraints $\Lambda_{1.4} \in [120,1504]$ (allowing for first-order phase transitions) and $\Lambda_{1.4} > 375$ (95$\%$ confidence, assuming purely hadronic composition), respectively.
Similarly, \citet{LandryEssick} used updated parameter estimation results from \citet{LVC_properties} with a broad non-parametric equation-of-state~prior to find $\Lambda_{1.4} = 160^{+448}_{-113}$ (maximum \textit{a posteriori} and highest-posterior-density 90$\%$ confidence interval).\footnote{In the remainder of the paper, quoted error bars refer to symmetric 90$\%$ confidence intervals about the median unless otherwise specified.}
We present general tidal deformability, moment of inertia, and radius bounds associated with these constraints for comparison with those derived from \citet{LVC_eos}.
Refs.~\cite{De,LVC_properties} also measured neutron-star~tidal deformability with GW170817, but they reported the chirp deformability $\tilde{\Lambda}$ rather than the canonical deformability $\Lambda_{1.4}$.
The former is a mass-weighted average of the tidal deformabilities of the {neutron star}s involved in the coalescence, and is therefore specific to the event GW170817; the latter is a generic constraint on the equation of state~that is more easily incorporated in our universal relations.
Likewise, multimessenger parameter estimation studies of GW170817 and its electromagnetic counterpart, combining gravitational-wave and kilonova observations, yielded intriguing constraints on $\tilde{\Lambda}$ \citep{RadiceDai,Radice,Coughlin2018}, in addition to other macroscopic observables \citep{Bauswein}.
The conclusions of Refs.~\cite{De,LVC_properties} are similar to those of \citet{LVC_eos}, favoring a relatively soft equation of state, while the multimessenger analyses indicate a preference for a somewhat larger tidal deformability, corresponding to a slightly stiffer equation of state~consistent with the findings of \citet{Annala18}.
Besides gravitational-wave measurement of the tidal deformability, masses and radii have been measured for a variety of pulsars and binary {neutron star}s via radio and X-ray astronomy.
However, only a handful of \emph{simultaneous} mass and radius measurements exist \citep{OzelFreire}.
Even the most precise of these, obtained by fitting spectra for accretion-powered thermonuclear bursts on the surface of {neutron star}s in low-mass X-ray binaries, may be affected by substantial systematic errors \citep{MillerLamb}.
Nonetheless, we extract radius estimates for six bursters studied by \citet{Ozel} from our general constraints, and find that they are consistent with the corresponding electromagnetic measurements.
In the cases we consider, the universal-relation based constraints on $R$ turn out to be more precise than the direct radius measurements themselves, after accounting for the uncertainty in the burster masses.
Additionally, we estimate moments of inertia for a few short-period double {neutron star}s whose relativistic periastron advance may be measurable with next-generation radio observatories, like the Square Kilometre Array \citep{SKA}.
Future direct measurements of $I$ can be compared to these gravitational-wave predictions to test the universality of the equation of state~\citep{Landry_pulsar}.
We perform a similar moment-of-inertia calculation for millisecond pulsars of known mass.
Using their measured angular frequencies $\Omega$, we compute their dimensionless spins $\chi := cI\Omega/GM^2$ to be $O(0.1)$.
For the fastest spinning pulsars in double neutron star~systems, we find instead $\chi \sim 0.01$, in keeping with conventional expectations \citep{Damour,Hannam}.
In anticipation of more accurate neutron-star~radius measurements from the NICER observatory \citep{NICER}, we demonstrate how the binary Love, I-Love and I-compactness relations can be combined into an effective $R(M,\Lambda_{1.4})$ relation that is insensitive to the equation of state.
This derived relation can be employed to place multimessenger constraints on tidal deformability using gravitational waves from binary neutron star~mergers in conjunction with radius measurements from X-ray binaries.
Taking simultaneous mass and radius measurements for the six thermonuclear bursters as our input, we tighten the GW170817-derived bounds on canonical deformability to $\Lambda_{1.4} = 196^{+92}_{-63}$, assuming all the observations are equally reliable.
The constraint's improved precision, relative to previous results, reinforces existing observational support for a particularly soft equation of state.
Obtaining this type of multimessenger constraint from universal relations is simpler than performing a joint Bayesian analysis and does not require modeling the equation of state.\footnote{Note added: A Bayesian analysis of this kind---but focused on the stellar radius, rather than the tidal deformability---is presented in \citet{Fasano}, which appeared shortly after completion of this paper.}
We describe our universal-relation based inference of neutron star~properties below. The equations of state~used to compute the relations are presented in Sec.~\ref{sec:uni}, with the piecewise polytrope representation we adopt detailed in Appendix~\ref{sec:pwp}. The binary Love, I-Love and I-compactness fits are introduced in Secs.~\ref{sec:bilove}-\ref{sec:ic}. Sec.~\ref{sec:scheme} explains our inference method. The results of the inference for general neutron-star~observables, as well as for specific systems, are presented in Secs.~\ref{sec:gen} and \ref{sec:inf}, respectively. Multimessenger constraints on neutron star~tidal deformability are calculated in Sec.~\ref{sec:joint}. Lastly, we discuss our findings in Sec.~\ref{sec:disc}.
\section{Universal relations} \label{sec:uni}
To infer the tidal deformabilities, moments of inertia, and radii of astrophysical neutron stars from gravitational-wave observations, we require universal relations linking each of these properties to the canonical deformability deduced through Bayesian parameter estimation \citep{LVC_GW170817,LVC_eos}.
The desired I-Love, binary Love, and I-compactness relations have been computed elsewhere, but for consistency in modeling the error in the fits we recompute the latter two here with the fiducial set of equations of state~used in \citet{Landry_pulsar}.
We also specialize the binary Love relation of \citet{Yagi_BiLove,Yagi_BiLoveLett} to our purposes.
We therefore briefly recapitulate our choice of equations of state, and our calculation of sequences of neutron-star observables, before presenting the specific fits employed for the universal relations.
\citet{Landry_pulsar} computed the I-Love relation and a binary Love relation between $\Lambda_{1.4}$ and PSR J0737-3039A's tidal deformability using a collection of 53 unified equations of state~from relativistic mean-field (RMF) theory and Skyrme-Hartree-Fock (SHF) theory.
The equations of state, plotted in Fig.~\ref{fig:eos}, are consistent with studies of the bulk properties of finite nuclei and infinite nuclear matter near nuclear saturation density, as well as the observational lower bound on the neutron-star~maximum mass \citep{Antoniadis}, which we conservatively take as $1.93\,M_{\odot}$.
The set includes RMF models with hyperonic $npe\mu Y$ matter, in addition to hadronic RMF and SHF $npe\mu$-matter models, and spans a wide range in stiffness and phenomenological behavior.
Although none of these equations of state~include quark matter, as per the Introduction, we expect the universal relations for purely hadronic stars to hold to the same level of accuracy for quark stars and two-phase hadron-quark hybrids.
A complete listing of the equations of state~is given in Sec.~2 of \citet{Landry_pulsar}, and we adopt the same set for our calculations here.
\begin{figure
\centering
\includegraphics[width=0.66\columnwidth]{eos.pdf}
\caption{Pressure $p$ as function of rest-mass energy density $\rho$ for the RMF and SHF equations of state. The equations of state~are colored by type and composition, with a few labeled explicitly for reference. The dividing densities of the three-segment piecewise polytrope representation we adopt for calculations with the equations of state~(see Appendix~\ref{sec:pwp}) are indicated with vertical lines, and the piecewise polytrope parameters are shown schematically.}
\label{fig:eos}
\end{figure}
The neutron-star~observables are determined by integrating the equations of stellar structure for a choice of equation of state~and central density.
Specifically, the Tolman-Oppenheimer-Volkoff equations \citep{Tolman,Oppenheimer} fix the stellar mass and radius, Hartle's slow-rotation equation \citep{Hartle} sets the moment of inertia, and the field equation for the quadrupolar tidal perturbation governs the tidal deformability \citep{Hinderer08}.
A stable sequence of {neutron star}s is obtained from successive choices of central density such that the resulting masses span from 1 to $1.93\,M_{\odot}$.
(For consistency, we truncate every sequence at $1.93 \, M_{\odot}$, even if the equation of state~can support a more massive star.)
For the purpose of these integrations, we adopt a piecewise polytrope representation of the equation of state~\citep{Read}.
This phenomenological parameterization is commonly used in astrophysics and gravitational-wave astronomy because it accurately reproduces with four parameters the neutron star~properties one would calculate from a tabulated version of the equation of state.
We determine the accuracy of the piecewise polytrope fits to the RMF and SHF equations of state~in Appendix~\ref{sec:pwp}, and list the best-fit parameter values in Table~\ref{tb:params1}.
\subsection{Binary Love relation} \label{sec:bilove}
We calculate a binary Love relation between the tidal deformability of a $1.4\,M_{\odot}$ star and that of a star of mass $M$ by performing a three-dimensional fit to $(M, \Lambda_{1.4}, \Lambda)$ data for a stable sequence of neutron stars with each of the 53 equations of state described above.
Expanding in canonical deformability and stellar mass, we posit a functional form
\begin{equation} \label{bilove}
\log_{10} \Lambda = \sum_{m=0}^{4} \sum_{n=0}^{1} a_{mn} M^m (\log_{10} \Lambda_{1.4})^n
\end{equation}
for the relation and perform a least-squares fit for $a_{mn}$.
The resulting coefficients are listed in Table~\ref{tb:coeff}.
Projections of the fit surface into the $M$-$\Lambda$ plane are superimposed on the underlying neutron-star~data in Fig.~\ref{fig:bilove}, which also shows the fit residuals
\begin{equation} \label{biloveres}
\Delta \Lambda = \frac{|\Lambda-\Lambda_{\text{fit}}|}{\Lambda_{\text{fit}}} .
\end{equation}
The residuals are calculated in the full three-dimensional space, but are projected into the $M$-$\Lambda$ plane in the plot.
The maximum residuals as a function of mass can be approximated by
\begin{equation} \label{resfit}
\Delta \Lambda (M) = b_0 + b_1 M + b_2 M^2 ,
\end{equation}
with the coefficients $b_n$ given in Table~\ref{tb:coeff}; for simplicity, we suppress the $\Lambda_{1.4}$-dependence of the residuals in our representation of the dispersion.
The function $\Delta \Lambda(M)$ is used to model the errors in the fit \eqref{bilove}.
Specifically, denoting the best-fit tidal deformability relation from Eq.~\eqref{bilove} as $\Lambda_{\text{fit}}$, we take $\Delta \Lambda \, \Lambda_{\text{fit}}$ to be half the width of the symmetric, two-sided 90$\%$ confidence interval of a Gaussian distribution
\begin{equation} \label{biloveprob}
P(\Lambda|M,\Lambda_{1.4}) = \frac{1}{\sqrt{2\pi{\sigma_{\Lambda}}^2}} \exp\left[-(\Lambda-\Lambda_{\text{fit}})^2/2{\sigma_{\Lambda}}^2 \right]
\end{equation}
centered on $\Lambda_{\text{fit}}$ that characterizes the uncertainty in the relation due to its \emph{approximately} universal nature.
Here, $\sigma_{\Lambda}(M) = \Delta \Lambda(M) \, \Lambda_{\text{fit}}/1.645$ is the standard deviation derived from the fractional errors $\Delta \Lambda(M)$.
The fractional errors are $O(1\%)$ near $1.4\,M_{\odot}$ and rise to $\approx 50\%$ at the high-mass edge of the relation.
\begin{figure}
\centering
\includegraphics[width=0.66\columnwidth]{BinaryLove.pdf}
\caption{Binary Love relation calculated with our set of 53 equations of state. The black dashed lines in the upper panel are selected $\Lambda_{1.4}=\text{constant}$ slices of the three-dimensional fit \eqref{bilove} to the sequences of $(M,\Lambda_{1.4},\Lambda)$ data. The fit and data are projected into the $M$-$\Lambda$ plane for display purposes only. The fit residuals are plotted in the lower panel, where the purple dashed curve approximates the maximum residuals in accordance with Eq.~\eqref{biloveres}.}
\label{fig:bilove}
\end{figure}
\setlength{\tabcolsep}{6pt}
\begin{table}
\centering
\caption{Coefficients of the fits \eqref{bilove}, \eqref{resfit}, \eqref{ilove} and \eqref{ic} for the binary Love, I-Love and I-compactness relations.} \label{tb:coeff}
\begin{tabular}{lcccc}
\hline \hline
\multicolumn{2}{c}{$\Lambda(M,\Lambda_{1.4})$} & $\Delta \Lambda(M)$ & $\bar{I}(\Lambda)$ & $C(\bar{I})$ \\ \hline
$a_{00} = -9.4469 \phantom{\times 10^{-1}}$ & $a_{01} = \phantom{-}4.6152 \phantom{\times 10^{-1}} $ & $b_0 = \phantom{-}3.7152$ & $c_0 = \phantom{-}6.5022 \times 10^{-1}$ & $d_0 = \phantom{-}4.8780 \times 10^{-2}$ \\
$a_{10} = \phantom{-}3.9702 \times 10^{1}$ & $a_{11} = -1.2226 \times 10^{1}$ & $b_1 = -5.2874$ & $c_1 = \phantom{-}5.8594 \times 10^{-2}$ & $d_1 = -4.2829 \times 10^{-1}$ \\
$a_{20} = -4.9173 \times 10^{1}$ & $a_{21} = \phantom{-}1.4214 \times 10^{1}$ & $b_2 = \phantom{-}1.8876$ & $c_2 = \phantom{-}5.1749 \times 10^{-2}$ & $d_2 = \phantom{-}1.2468 \phantom{\times 10^{-1}} $ \\
$a_{30} = \phantom{-}2.4937 \times 10^{1}$ & $a_{31} = -7.1134 \phantom{\times 10^{-1}}$ & - & $c_3 = -3.6321 \times 10^{-3}$ & $d_3 = -9.0716 \times 10^{-1}$ \\
$a_{40} = -4.7288 \phantom{\times 10^{-1}}$ & $a_{41} = \phantom{-}1.3416 \phantom{\times 10^{-1}}$ & - & $c_4 = \phantom{-}8.5909 \times 10^{-5}$ & $d_4 = \phantom{-}2.3302 \times 10^{-1}$ \\
\hline \hline
\end{tabular}
\end{table}
The binary Love relation \eqref{bilove} is similar to the one introduced by Refs.~\cite{Yagi_BiLove,Yagi_BiLoveLett}, but is specially adapted to our purpose.
While the original binary Love relation effectively links $\Lambda_1(M_1)$ and $\Lambda_2(M_2)$ via the mass ratio $M_2/M_1$, assuming a common equation of state, ours essentially sets $\Lambda_1 = \Lambda_{1.4}$ by fixing $M_1 = 1.4\,M_{\odot}$, and accordingly we use $M_2$ itself in place of the mass ratio $M_2/(1.4 \, M_{\odot})$.
Moreover, Refs.~\cite{Yagi_BiLove,Yagi_BiLoveLett} use the combinations $\Lambda_s = (\Lambda_1+\Lambda_2)/2$ and $\Lambda_a = (\Lambda_1-\Lambda_2)/2$ in place of the individual tidal deformabilities to improve the universality of the fit.
Doing the same would unnecessarily complicate our inference, as a closed-form expression for $\Lambda_2$ cannot be obtained from a log-log polynomial fit for $(\Lambda_s,\Lambda_a)$.
In any case, the dispersion in our modified binary Love relation is nearly as small as in the original formulation.
\subsection{I-Love relation} \label{sec:ilove}
We adopt the I-Love relation from Eq.~(7) of \citet{Landry_pulsar} directly, as it was computed with the same set of equations of state considered here. The coefficients of the log-log polynomial fit
\begin{equation}\label{ilove}
\log_{10}{\bar{I}} = \sum_{n=0}^{4} c_n \left( \log_{10}{\Lambda} \right)^n
\end{equation}
for the dimensionless moment of inertia $\bar{I} := c^4 I/G^2 M^3$ as a function of tidal deformability are given in Table~\ref{tb:coeff}.
The fit, the ($\Lambda$,$\bar{I}$) data, and the residuals
\begin{equation} \label{iloveres}
\Delta \bar{I} = \frac{|\bar{I}-\bar{I}_{\text{fit}}|}{\bar{I}_{\text{fit}}}
\end{equation}
can be seen in Fig.~4 of \citet{Landry_pulsar}.
The maximum residuals are approximately constant over the relevant range of $\Lambda$, amounting to no more than 0.6$\%$ error.
We therefore take this value to define the half-width of the 90$\%$ confidence interval of the Gaussian uncertainty in the fit, modeled in the same manner as above, such that
\begin{equation} \label{iloveprob}
P(\bar{I}|\Lambda) = \frac{1}{\sqrt{2\pi{\sigma_{\bar{I}}}^2}} \exp\left[-(\bar{I}-\bar{I}_{\text{fit}})^2/2{\sigma_{\bar{I}}}^2 \right] ,
\end{equation}
with $\sigma_{\bar{I}} = \Delta \bar{I} \, \bar{I}_{\text{fit}}/1.645$.
\subsection{I-compactness relation} \label{sec:ic}
Our universal relation between the dimensionless moment of inertia and the stellar compactness $C := GM/c^2 R$ is based on a similar one from \citet{Breu}.
Quasi-universal I-compactness relations predate the I-Love-Q relations in the literature \citep{Ravenhall,Bejger,LattimerMoI,Baubock}, but generally exhibit a lesser degree of equation-of-state~independence \citep{Chan,YagiILQreview}.
\citet{Breu} discovered that the relation's universality could be enhanced by using the normalization $\bar{I} = c^4 I/G^2 M^3$ for the dimensionless moment of inertia, as in Refs.~\cite{YagiILQscience,YagiILQ}, rather than the conventional definition $I/M R^2$.
Hence, theirs is the version of the I-compactness relation we calculate here; however, we fit for the inverse relation, namely $C(\bar{I})$.
Taking ($\bar{I}$, $C$) data for our stable sequences of {neutron star}s, we perform a least-squares fit to the model
\begin{equation}\label{ic}
C = \sum_{n=0}^{4} d_n \left( \log_{10}{\bar{I}} \right)^{-n} ,
\end{equation}
displaying the resulting coefficients in Table~\ref{tb:coeff}.
The fit and the residuals
\begin{equation} \label{icres}
\Delta C = \frac{|C-C_{\text{fit}}|}{C_{\text{fit}}}
\end{equation}
are shown alongside the underlying neutron star~data in Fig.~\ref{fig:ci}.
The maximum residuals are roughly constant as a function of $\bar{I}$, so we take the maximum value of $3\%$ to define the half-width of the 90$\%$ confidence interval about the mean of the Gaussian distribution describing the error in the relation,
\begin{equation} \label{icprob}
P(C|\bar{I}) = \frac{1}{\sqrt{2\pi{\sigma_{C}}^2}} \exp\left[-(C-C_{\text{fit}})^2/2{\sigma_{C}}^2 \right]
\end{equation}
with $ \sigma_C = \Delta C \, C/1.645$.
\begin{figure}
\centering
\includegraphics[width=0.66\columnwidth,]{CI.pdf}
\caption{I-compactness relation calculated with our set of 53 equations of state. The fit \eqref{ic} is shown as a black dashed line, and the fit residuals are displayed in the lower panel.}
\label{fig:ci}
\end{figure}
\section{Inference scheme} \label{sec:scheme}
Equipped with the distributions \eqref{biloveprob}, \eqref{iloveprob} and \eqref{icprob} describing the probabilistic mappings defined by the universal relations, we can translate a gravitational-wave measurement of $\Lambda_{1.4}$ into constraints on {neutron star}s' tidal deformabilities, moments of inertia, spins and radii. Our inference of these observables is described in general terms here; we apply it to the observational input from GW170817 in the following section.
We suppose that $P(\Lambda_{1.4} |\,\text{GW})$, the posterior probability distribution for the canonical deformability given a gravitational-wave observation $\text{GW}$, is known.
The target of our inference is taken to be a neutron star~for which a mass posterior $P(M|\,\text{EM})$ is available from electromagnetic observations $\text{EM}$.
(The case of general constraints on neutron-star~properties, absent a specific target system, is treated separately below.)
The posterior distributions from the independent gravitational-wave and electromagnetic measurements serve as priors for our inference of the target's properties.
Firstly, the posterior distribution $P_{\Lambda}(\Lambda|\,\text{EM},\text{GW})$ for the target's tidal deformability, conditioned on the gravitational-wave and electromagnetic observations, is computed by marginalizing the binary Love relation over the two priors:
\begin{equation} \label{plambda}
P_{\Lambda}(\Lambda|\,\text{EM},\text{GW}) = \int P(\Lambda |\,M,\Lambda_{1.4}) P(M |\,\text{EM}) P(\Lambda_{1.4} |\,\text{GW}) \, dM \, d\Lambda_{1.4} .
\end{equation}
The posterior distribution for the target's dimensionless moment of inertia can then be calculated via the I-Love relation as
\begin{equation} \label{pibar}
P_{\bar{I}}(\bar{I}|\,\text{EM},\text{GW}) = \int P(\bar{I}|\,\Lambda) P_{\Lambda}(\Lambda|\,\text{EM},\,\text{GW}) \, d\Lambda
\end{equation}
by marginalizing over the tidal deformability.
The posteriors for the moment of inertia $I = G^2 \bar{I} M^3/c^4$ and the dimensionless spin $\chi = G \bar{I} M \Omega/c^3$ follow by a change of variables and a marginalization over mass:
\begin{align} \label{pi}
P_I(I |\,\text{EM},\text{GW}) &= \frac{c^4}{G^2}\int \frac{P_{\bar{I}}(c^4 I/G^2 M^3 |\,\text{EM},\text{GW}) P(M |\,\text{EM})}{M^{3}} \, dM , \\
P_{\chi}(\chi |\,\text{EM},\text{GW}) &= \frac{c^3}{G}\int \frac{P_{\bar{I}}(c^3 \chi/G M \Omega |\,\text{EM},\text{GW}) P(M |\,\text{EM})}{M \Omega} \, dM . \label{pchi}
\end{align}
When inferring $\chi$, we assume that the {neutron star}'s rotational frequency $\Omega$ is known exactly.
From Eq.~\eqref{pibar}, we can also infer the posterior distribution for the target's compactness $C = GM/c^2 R$ through
\begin{equation} \label{pc}
P_C(C |\,\text{EM},\text{GW}) = \int P(C|\,\bar{I}) P_{\bar{I}}(\bar{I}|\,\text{EM},\text{GW}) \, d\bar{I} ,
\end{equation}
which makes use of the I-compactness relation and leads immediately to an inference of the target's radius via
\begin{equation} \label{pr}
P_R(R |\,\text{EM},\text{GW}) = \frac{G}{c^2} \int \frac{P_C(GM/c^2R |\,\text{EM},\text{GW}) P(M |\,\text{EM})}{R^2} M \, dM .
\end{equation}
In the event that the target's mass is known exactly, $P(M |\,\text{EM})$ reduces to a Dirac delta function in $M$ and the mass marginalizations are trivial.
Given the posterior distributions \eqref{plambda}, \eqref{pi}, \eqref{pchi} and \eqref{pr}, we can compute the median value of $\Lambda$, $I$, $\chi$ and $R$ for the target star and extract symmetric confidence intervals for each observable.
General constraints on neutron-star~properties, rather than inferences for a specific target system, can also be calculated by dispensing with the mass marginalizations altogether and computing the posterior distributions as a function of mass based on the gravitational-wave observation alone, i.e.~
\begin{align} \label{plambdam}
P_{\Lambda_M}(\Lambda|\,M;\text{GW}) = \int P(\Lambda |\,M,\Lambda_{1.4}) P(\Lambda_{1.4} |\,\text{GW}) \, d\Lambda_{1.4} , \\
P_{\bar{I}_M}(\bar{I}|\,M;\text{GW}) = \int P(\bar{I}|\,\Lambda) P_{\Lambda_M}(\Lambda|\,M;\text{GW}) \, d\Lambda , \label{pim} \\
P_{R_M}(R |\,M;\text{GW}) = P_{C_M}(GM/c^2R |\,M;\text{GW}) \, GM/c^2R^2 . \label{prm}
\end{align}
Here, we have defined $P_{C_M}(C |\,M;\text{GW}) := \int P(C|\,\bar{I}) P_{\bar{I}_M}(\bar{I}|\,M;\text{GW})\, d\bar{I}$.
By calculating confidence intervals about the median \textit{a posteriori} for each value of mass in the domain, we can place constraints on the $M$-$\Lambda$, $M$-$\bar{I}$, and $M$-$R$ relations that govern all old, cold {neutron star}s in the universe.
\section{Implications of GW170817 for neutron star properties} \label{sec:results}
We apply the inference described above to astrophysical {neutron star}s, using GW170817---and, specifically, the measurement $\Lambda_{1.4} = 190^{+390}_{-120}$ from \citet{LVC_eos}---as our observational gravitational-wave input.
However, since only the median and symmetric $90\%$ confidence interval for $\Lambda_{1.4}$ were reported in \citet{LVC_eos}, we must model the full posterior distribution $P(\Lambda_{1.4}|\,\text{GW})$. In order to preserve the asymmetry evident in the confidence interval, we choose to represent it as a generalized beta prime distribution
\begin{equation} \label{model}
P(\Lambda_{1.4}|\,\text{GW}) = \frac{p \,\Gamma(\alpha+\beta)}{q\,\Gamma(\alpha)\Gamma(\beta)} \left(\frac{\Lambda_{1.4}}{q}\right)^{\alpha p-1}\left[1-\left(\frac{\Lambda_{1.4}}{q}\right)^p\right]^{-\alpha-\beta}
\end{equation}
with parameters $p=1$, $q=0.934$, $\alpha = 2.856$ and $\beta = 191.509$, where $\Gamma(z)$ is the gamma function.
With these parameter selections, the distribution has the same symmetric $90\%$ confidence interval as implied by the gravitational-wave measurement, and its median of 198 is only shifted mildly relative to the actual value.
Thus, our model for $P(\Lambda_{1.4}|\,\text{GW})$ closely reproduces the features of the measurement reported in \citet{LVC_eos}.
Using this posterior probability distribution, we first infer general constraints on the $M$-$\Lambda$, $M$-$\bar{I}$, and $M$-$R$ relations, and then extract specific bounds for individual {neutron star}s of interest.
Because our universal relations' fits and errors are based on data for $M \in [1, 1.93] \, M_{\odot}$, we focus on {neutron star}s with (median) $M \leq 1.93 \, M_{\odot}$ in this paper to avoid extrapolation insofar as possible.
\subsection{General constraints} \label{sec:gen}
Following Eqs.~\eqref{plambdam}-\eqref{prm}, we calculate, as a function of mass, symmetric $90\%$ confidence intervals about the median for each of the neutron-star properties attainable from the GW170817 $\Lambda_{1.4}$ measurement by way of the universal relations.
The resulting constraints on $\Lambda(M)$, $\bar{I}(M)$ and $R(M)$ are plotted in Figs.~\ref{fig:mlambda}-\ref{fig:mr}.
The canonical deformability measurement maps to the colored band with decreasing slope in the $M$-$\Lambda$ plane seen in Fig.~\ref{fig:mlambda}, reflecting the fact that $\Lambda$ is a monotonically decreasing function of mass.
We observe that several of the stiffer reference equations of state, e.g.~NL3, DDME2, and IOPB-I, lie outside the 90$\%$ confidence region, in keeping with the preference found by other studies \citep{LVC_GW170817,De,LVC_eos,LVC_properties} for a relatively soft equation of state.
Similarly, the $\Lambda(M)$ constraint transforms to the colored band in the $M$-$\bar{I}$ plane shown in Fig.~\ref{fig:mi}.
Its decreasing slope reflects the monotonicity of $\bar{I}(M)$ for realistic equations of state, and the same stiff models are disfavored.
The corresponding $R(M)$ constraint is depicted in Fig.~\ref{fig:mr}.
The median $M$-$R$ relation reveals that neutron-star~universality imposes near-constancy of the radius over the mass range of interest.
The colored region of the plot excludes radii larger than 13.0 km and less than 8.7 km at 90$\%$ confidence for stars with $M \in [1,1.93]\,M_{\odot}$.
Evaluating the constraint at $M = 1.4 \, M_{\odot}$, GW170817 implies $R_{1.4} = 10.9^{+1.9}_{-1.5}$ km for the canonical radius.
This value is compatible with upper bounds of $\approx 13$-14 km computed via equation-of-state~modeling \citep{Annala18,Nandi,Fattoyev_GW170817,Most2018} or a universal chirp-deformability--radius relation \citep{Raithel2018}.
It also overlaps with the result $R_{1.4} = 12.2^{+1.0}_{-0.8}$ km obtained by \citet{RadiceDai}'s joint gravitational-wave and electromagnetic parameter estimation for GW170817.
To illustrate how the inferred bounds on the properties as a function of mass depend on the choice of priors and assumptions made in the initial parameter estimation for $\Lambda_{1.4}$, in Figs.~\ref{fig:mlambda}-\ref{fig:mr} we also show the general constraints stemming from \citet{LVC_GW170817} ($\Lambda_{1.4} \leq 800$, without the common equation of state~assumption), \citet{Annala18} ($120 \leq \Lambda_{1.4} \leq 1504$, modeling the equation of state~as a piecewise polytrope), \citet{Most2018} ($\Lambda_{1.4} > 375$, modeling the equation of state~via perturbative QCD calculations), and \citet{LandryEssick} ($\Lambda_{1.4} = 160^{+448}_{-113}$, modeling the equation of state~with a Gaussian process).
Since these results are only shown for comparative purposes, we do not perform the full inference described in Sec.~\ref{sec:scheme}.
Rather, we simply map each $\Lambda_{1.4}$ constraint through the best-fit universal relations, accounting for uncertainty by inflating upper and lower bounds by a factor of the fractional error in the fit.
In this way, we obtain conservative estimates of the alternative constraints' implications for neutron-star~properties.
As can be seen, \citet{Annala18}'s upper bound is stiff enough to allow all the reference equations of state. Meanwhile, depending on the analysis, the constraint's lower bound excludes a varying fraction of the region compatible with neutron-star~universality.
We note that the maximum \textit{a posteriori} from \citet{LandryEssick} is omitted in the plots, as it is similar to the median from \citet{LVC_eos}.
\begin{figure}
\centering
\includegraphics[width=0.66\columnwidth]{LambdaMass.pdf}
\caption{Constraints on the mass--tidal-deformability relation $\Lambda(M)$ from GW170817 and the universal relations. The green line and shaded region show the median and symmetric $90\%$ confidence interval derived from \citet{LVC_eos}'s $\Lambda_{1.4}$ measurement. Upper (respectively lower) bounds stemming from alternative constraints are indicated by solid (dashed) colored lines; the input $\Lambda_{1.4}$ constraints can be read off from the $\Lambda(M)$ curves at $M = 1.4\,M_{\odot}$ (dotted vertical line). $\Lambda(M)$ relations for a few reference equations of state~are shown in black. The tidal deformability inferred for the $1.338\,M_{\odot}$ PSR J0737-3039A by \citet{Landry_pulsar} on the basis of \citet{LVC_eos}'s (\citet{LVC_GW170817}'s) canonical deformability measurement is indicated with the pink error bars (orange point).}
\label{fig:mlambda}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.66\columnwidth]{MoIMass.pdf}
\caption{Constraints on the mass--moment-of-inertia relation $\bar{I}(M)$ from GW170817 and the universal relations. $\bar{I}(M)$ relations for a few reference equations of state, as well as the dimensionless moment of inertia inferred for the double pulsar by \citet{Landry_pulsar}, are also shown.}
\label{fig:mi}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.66\columnwidth]{RadiusMass.pdf}
\caption{Constraints on the mass-radius relation $R(M)$ from GW170817 and the universal relations. $R(M)$ relations for a few reference equations of state~are also shown. The posterior 90$\%$-credible contour from the simultaneous mass and radius measurement of EXO 1745-248 is overplotted in brown, demonstrating the consistency of the universal-relation based inference with electromagnetic observations of {neutron star}s.}
\label{fig:mr}
\end{figure}
\subsection{Individual neutron stars} \label{sec:inf}
Next, we extract constraints on the properties of specific {neutron star}s of interest---primarily pulsars with well-measured masses from electromagnetic observations---following Eqs.~\eqref{plambda}-\eqref{pr}.
In the literature, universal relations have been proposed as a tool for improving the precision of a measurement of neutron-star~radius, spin, or moment of inertia \citep{Maselli,YagiILQreview,Bhat}; however, until recently \citep{Landry_pulsar}, the application has always involved translation of one observable (e.g.~$\Lambda$) to another (e.g.~$I$) for the same system.
Here we use GW170817 to infer the properties of individual {neutron star}s in other systems.
We compute their tidal deformabilities, moments of inertia and radii using the $\Lambda_{1.4}$ constraint from \citet{LVC_eos}.
\subsubsection{Double neutron stars} \label{sec:dns}
We begin by inferring the tidal deformability, moment of inertia, and radius for the pulsar component of several short-period double neutron-star~systems.
Double neutrons stars in tight binaries are good candidates for an electromagnetic measurement of the stellar moment of inertia if radio pulses from one member of the system are detectable, as they can be used to determine the post-Keplerian parameters of the orbit with great precision \citep{KramerWex}.
In particular, a sufficiently precise measurement of the system's relativistic periastron advance can distinguish the part due to spin-orbit coupling, which is proportional to the spin of the pulsar; knowledge of its angular frequency can then be used to extract the moment of inertia, which depends sensitively on the equation of state.
No electromagnetic neutron-star~$I$ measurements exist at present, but they may be feasible with the Square Kilometre Array and other next-generation radio observatories \citep{Kehl}.
The best-studied candidate for a future moment of inertia measurement is the double pulsar, PSR J0737-3039 \citep{Burgay03,Lyne04}.
Its 1.338$\, M_{\odot}$ primary component's moment of inertia was estimated by \citet{Landry_pulsar} as $1.15^{+0.38}_{-0.24} \times 10^{45}$ g cm$^2$ based on GW170817 and universal relations.
We revisit the calculation here, ignoring the uncertainty of less than one part in $10^{-3}$ in the pulsar's mass---i.e.~taking $P(M|\,\text{EM})=\delta(M-1.338\,M_{\odot})$---when using Eq.~\eqref{pi}.
Despite using a slightly different binary Love relation than \citet{Landry_pulsar}, we find a nearly identical moment of inertia constraint, $I = 1.16^{+0.33}_{-0.25}\times 10^{45}$ g cm$^2$.
Furthermore, we infer the pulsar's tidal deformability to be $269^{+439}_{-170}$, and its radius as $11.0^{+1.9}_{-1.5}$ km.
This radius value is no different, within uncertainty, than that of a canonical 1.4$\, M_{\odot}$ star.
PSR J1946+2052 is another especially promising candidate for a moment of inertia measurement, since it resides in the tightest double neutron-star~system discovered to date \citep{Stovall}; it is also the fastest-spinning pulsar with a neutron-star~companion that will merge within a Hubble time.
However, given its relatively recent discovery, the pulsar's mass has not yet been determined with precision---only an upper bound of 1.31$\, M_{\odot}$ exists.
Nonetheless, if we model $P(M|\,\text{EM})$ as flat for $M \in [1.0,1.3] \, M_{\odot}$, we are able to estimate its moment of inertia as $0.96^{+0.37}_{-0.26}\times 10^{45}$ g cm$^2$ by marginalizing over the mass uncertainty.
In Table~\ref{tb:dns}, we report the inferred properties of these and several other pulsars in double neutron-star~systems, such as PSR B1913+16, the Hulse-Taylor pulsar \citep{HulseTaylor}.
As with PSR J0737-3039A, we take their masses to be known exactly for the purpose of the calculation, except for the aforementioned case of PSR J1946+2052. The errors reported in the table therefore account for the approximate nature of the universal relations and the uncertainty in the $\Lambda_{1.4}$ measurement from GW170817.
Although the pulsar masses are clustered in the small range of $\approx 1.3$-$1.6 \, M_{\odot}$, the inferred 90$\%$ confidence intervals for the tidal deformabilities are distributed over an order of magnitude.
The $\Lambda$ uncertainties are typically lopsided, with larger error bars on the upper side, because the monotonically decreasing function $\Lambda(M)$ behaves roughly like $1/M$, tending to a constant value at large $M$.
For the moments of inertia, we find that they are typically constrained by GW170817 to $\approx 30\%$ accuracy, with median values of $\sim 1 \times 10^{45}$ g cm$^2$.
Given the weak mass dependence of the radius for $M \in [1,1.93] \, M_{\odot}$, the median radius for nearly all the pulsars in Table~\ref{tb:dns} is $11.0$ km.
\begin{table}
\centering
\caption{Inferred properties of pulsars in double neutron star systems. The tidal deformability, moment of inertia, radius, and dimensionless spin are calculated via universal relations from the $\Lambda_{1.4}$ constraint of \citet{LVC_eos}. Orbital periods, masses and rotational frequencies are drawn from the listed references. The measurement uncertainty of no more than $\pm 1$ in the last digit of $M$ is ignored for the purpose of the inference, with the exception of PSR J1946+2052, for which we assume a flat probability distribution for $M \in [1,1.3] \,M_{\odot}$.}
\label{tb:dns}
\begin{tabular}{lclcccccc} \hline \hline
Pulsar & $P_{\text{orb}}$ [d] & $M$ $[M_{\odot}]$ & $\Omega$ [rad s$^{-1}$] & Reference & $\Lambda$ & $I$ $[10^{45}~\text{g cm}^2]$ & $R$ [km] & $\chi$ \\ \hline
B1534+12 & 0.421 & \phantom{$<\;$}1.333 & 165.76 & \cite{Fonseca} & $276^{+449}_{-174}$ & $1.15^{+0.33}_{-0.24}$ & $11.0^{+1.9}_{-1.5}$ & $0.012^{+0.004}_{-0.003}$ \\
B1913+16 & 0.323 & \phantom{$<\;$}1.438 & 106.44 & \cite{Weisberg} & $163^{+286}_{-106}$ & $1.27^{+0.37}_{-0.27}$ & $10.9^{+1.9}_{-1.5}$ & $0.007^{+0.002}_{-0.002}$ \\
B2127+11C & 0.335 & \phantom{$<\;$}1.36\phantom{0} & 205.81 & \cite{Jacoby} & $241^{+399}_{-153}$ & $1.19^{+0.34}_{-0.25}$ & $11.0^{+1.9}_{-1.5}$ & $0.015^{+0.004}_{-0.003}$ \\
J0453+1559 & 4.072 & \phantom{$<\;$}1.56\phantom{0} & 137.24 & \cite{Martinez} & $\phantom{0}89^{+172}_{-60}$ & $1.41^{+0.41}_{-0.29}$ & $10.9^{+1.9}_{-1.5}$ & $0.009^{+0.003}_{-0.002}$ \\
J0737-3039A & 0.102 & \phantom{$<\;$}1.338 & 276.80 & \cite{Kramer} & $269^{+439}_{-170}$ & $1.16^{+0.33}_{-0.25}$ & $11.0^{+1.9}_{-1.5}$ & $0.020^{+0.006}_{-0.004}$ \\
J1756-2251 & 0.320 & \phantom{$<\;$}1.34\phantom{0} & 220.76 & \cite{Ferdman} & $267^{+435}_{-168}$ & $1.16^{+0.34}_{-0.25}$ & $11.0^{+1.9}_{-1.5}$ & $0.016^{+0.005}_{-0.003}$ \\
J1906+0746 & 0.166 & \phantom{$<\;$}1.29\phantom{0} & \phantom{0}43.61 & \cite{Leeuwen} & $344^{+542}_{-215}$ & $1.11^{+0.32}_{-0.24}$ & $11.0^{+1.9}_{-1.5}$ & $0.003^{+0.001}_{-0.001}$ \\
J1946+2052 & 0.078 & $<\;$1.31\phantom{0} & 370.47 & \cite{Stovall} & $710^{+1516}_{-490}$ & $0.96^{+0.37}_{-0.26}$ & $11.0^{+1.9}_{-1.6}$ & $0.031^{+0.009}_{-0.007}$ \\
\hline \hline
\end{tabular}
\end{table}
\subsubsection{Millisecond pulsars} \label{sec:psr}
Precise mass and angular frequency measurements exist for a number of millisecond pulsars thanks to detailed studies of their regular radio pulses.
Here we calculate their moments of inertia as a way to infer their dimensionless spins.
We focus on a subset of the millisecond pulsars considered in \citet{OzelFreire}, and list their masses, angular frequencies and inferred properties in Table~\ref{tb:psr}.
The subset includes PSR J0437-4715, the closest and brightest pulsar detected to date \citep{Reardon}, and PSR J1713+0747, one of the most precisely timed pulsars \citep{Zhu}.
We model the uncertainty in the pulsars' masses as Gaussian, converting the standard deviations reported in the original references listed in the table to $90\%$ confidence intervals.
With this model for $P(M|\,\text{EM})$, we follow the prescription of Sec.~\ref{sec:scheme} for computing confidence intervals about the median moment of inertia.
Overall, we find that the errors bars on $I$ are slightly larger than for the double {neutron star}s in Table~\ref{tb:dns} on account of the broader mass uncertainties for the millisecond pulsars.
Incorporating the pulsars' known angular frequencies, we then infer the stars' dimensionless spins.
We find that the universal relations permit $\chi$ to be inferred from GW170817 with $\approx 30\%$ accuracy in an approximately equation-of-state~independent way.
The fastest-spinning pulsar we consider, PSR J1909-3744, is found to have $\chi = 0.147^{+0.043}_{-0.031}$.
The astrophysical spin distribution for millisecond pulsars is known to extend up to at least $\chi \sim 0.4$ \citep{Hessels}, while binary {neutron star}s that merge within a Hubble time are expected to have much smaller spins $\chi \lesssim 0.05$ \citep{Damour,Hannam,Landry_pulsar}.
Hence, for comparison, we also infer the dimensionless spin for the pulsar components of the double neutron star~systems listed in Table~\ref{tb:dns}.
We find that the pulsars of this kind have dimensionless spin $\chi \lesssim 0.04$ at 90$\%$ confidence, while the millisecond pulsars in Table~\ref{tb:psr} have dimensionless spins below $\chi \approx 0.20$.
One could systematize this dimensionless spin inference for all known pulsars to establish a virtually equation-of-state~independent upper bound on the spin distribution, whose precision would improve as more gravitational-wave events are detected.
Because the universal relations used here were developed in the context of slowly rotating stellar models, one might suppose that they do not apply to rapidly rotating millisecond pulsars.
However, Refs.~\cite{Pappas,Chakrabarti} showed that they also hold for stars in rapid uniform rotation,\footnote{Because we evaluate the stability of our neutron-star~sequences in the absence of rotation, we are excluding supramassive (i.e.~rotation-stabilized) {neutron star}s, for which the universal relations deteriorate at high compactness \citep{Lenka}.} despite earlier claims to the contrary \citep{Doneva_rot}.
In any case, for stars with moderate rotation ($\chi \sim 0.1$), spin corrections to the moment of inertia are negligible, as they enter at $O(\chi^2) \sim 10^{-2}$.
In addition, we note that our spin analysis depends implicitly on the assumption that the progenitors of GW170817 rotated slowly, with $\chi \leq 0.05$, through the priors adopted in \citet{LVC_eos}'s parameter estimation.
The low-spin assumption is consistent with dimensionless spin estimates for the fastest-spinning pulsars in double neutron-star~systems \citep{Damour,Hannam,Landry_pulsar}.
However, for a spin inference that is independent of this assumption, one could repeat the calculation with the upper bound $\Lambda \leq 1400$ from \citet{LVC_GW170817}, which instead requires only $\chi \leq 0.89$ \textit{a priori}.
Indeed, this was done for the double pulsar in Sec.~5 of \citet{Landry_pulsar}.
\begin{table}
\centering
\caption{Inferred properties of millisecond pulsars. The tidal deformability, moment of inertia, radius and dimensionless spin are calculated via universal relations from the $\Lambda_{1.4}$ constraint of \citet{LVC_eos}. Masses and rotational frequencies are drawn from the listed references. The Gaussian errors in $M$ have been converted to the $90\%$ confidence level.}
\label{tb:psr}
\begin{tabular}{lccccccc} \hline \hline
Pulsar & $M$ $[M_{\odot}]$ & $\Omega$ [rad s$^{-1}$] & Reference & $\Lambda$ & $I$ $[10^{45}~\text{g cm}^2]$ & $R$ [km] & $\chi$ \\ \hline
J0437-4715 & $1.44 \pm 0.12$ & 1091.31 & \cite{Reardon} & $163^{+344}_{-116}$ & $1.28^{+0.40}_{-0.29}$ & $10.9^{+1.9}_{-1.5}$ & $0.076^{+0.022}_{-0.016}$ \\
J0751+1807 & $1.64 \pm 0.25$ & 1795.20 & \cite{Desvignes} & $\phantom{0}59^{+227}_{-51}$ & $1.50^{+0.51}_{-0.39}$ & $10.7^{+1.9}_{-1.6}$ & $0.114^{+0.036}_{-0.026}$ \\
J1713+0747 & $1.31 \pm 0.18$ & 1374.84 & \cite{Zhu} & $310^{+710}_{-232}$ & $1.13^{+0.40}_{-0.30}$ & $11.0^{+1.8}_{-1.5}$ & $0.103^{+0.029}_{-0.023}$ \\
J1802-2124 & $1.24 \pm 0.18$ & \phantom{0}496.79 & \cite{Ferdman10} & $439^{+939}_{-326}$ & $1.05^{+0.38}_{-0.28}$ & $11.0^{+1.8}_{-1.5}$ & $0.038^{+0.011}_{-0.009}$ \\
J1807-2500B & $1.3655 \pm 0.0034$ & 1500.93 & \cite{Lynch12} & $234^{+391}_{-149}$ & $1.19^{+0.35}_{-0.25}$ & $11.0^{+1.9}_{-1.5}$ & $0.109^{+0.032}_{-0.023}$ \\
J1909-3744 & $1.47 \pm 0.05$ & 2131.98 & \cite{Reardon} & $139^{+261}_{-94}$ & $1.31^{+0.38}_{-0.28}$ & $10.9^{+1.9}_{-1.5}$ & $0.147^{+0.043}_{-0.031}$ \\
J2222-0137 & $1.20 \pm 0.23$ & 191.46 & \cite{Kaplan14} & $509^{+1062}_{-397}$ & $1.02^{+0.40}_{-0.29}$ & $10.9^{+1.7}_{-1.6}$ & $0.015^{+0.004}_{-0.003}$ \\
\hline \hline
\end{tabular}
\end{table}
\subsubsection{Low-mass X-ray binaries} \label{sec:xray}
Neutron stars in X-ray binaries are the best candidates for electromagnetic radius measurements. Radius estimates for a few systems already exist, although their accuracy is a matter of some debate \citep{MillerLamb}.
The most precise measurements involve thermonuclear bursters in low-mass X-ray binaries; by fitting for the spectrum of the thermal emission, which is related to the burst luminosity by a factor of the surface area, one can determine the radius from the observed flux \citep{OzelFreire}.
Observations from the NICER mission are expected to place even tighter and more accurate constraints on neutron star~radii via pulse profile modeling \citep{OzelNICER}.
For the time being, we focus on six bursters in low-mass X-ray binaries for which simultaneous mass and radius measurements exist \citep{Ozel}.
In Table~\ref{tb:xray}, we list the median and symmetric $90\%$ confidence intervals for the neutron-star~masses and radii extracted from the $M$-$R$ posteriors associated with the electromagnetic observations.\footnote{The mass-radius posteriors are available in tabulated form at \url{http://xtreme.as.arizona.edu/NeutronStars/}.}
(Note that the masses and radii reported in Refs.~\cite{OzelFreire,Ozel} are given instead as maxima \textit{a posteriori} with symmetric uncertainties at the $68\%$ confidence level.)
The confidence intervals are calculated from the marginal distributions $P(M|\,\text{EM}) = \int P(M,R|\,\text{EM}) \,dR$ and $P(R|\,\text{EM}) = \int P(M,R|\,\text{EM}) \,dM$, respectively, with $P(M,R|\,\text{EM})$ constructed from the available posterior samples.
Taking the calculated $P(M|\,\text{EM})$ as our mass prior in Eq.~\eqref{pr}, we obtain a GW170817-based radius estimate for the {neutron star}s through the universal relations.
The inferred radii are consistent with the $R_{\text{EM}}$ values obtained from the direct measurements via $P(R|\,\text{EM})$.
This can also be seen in Fig.~\ref{fig:mr}, where---as an example---we overlay the 90$\%$ confidence contour of $P(M,R|\,\text{EM})$ for EXO 1745-248 on our $R(M)$ constraints.
In Table~\ref{tb:xray}, besides the inferred radius, we also show the tidal deformability and moment of inertia calculated for each burster.
We note that, for the thermonuclear bursters considered here, the universal relations and GW170817 actually provide a more precise radius determination at the 90$\%$ confidence level than the direct observations, after marginalizing over the mass posterior $P(M|\,\text{EM})$.
\begin{table}
\centering
\caption{Inferred properties of {neutron star}s in low-mass X-ray binaries for which simultaneous mass and radius measurements exist. The tidal deformability, moment of inertia and radius are calculated via universal relations from the $\Lambda_{1.4}$ constraint of \citet{LVC_eos}. Masses and direct radius measurements $R_{\text{EM}}$ are obtained from the $M$-$R$ posteriors associated with \cite{Ozel}, as described in the text.}
\label{tb:xray}
\begin{tabular}{lcccccc} \hline \hline
Neutron star & $M$ $[M_{\odot}]$ & $R_{\text{EM}}$ [km] & $\Lambda$ & $I$ $[10^{45}~\text{g cm}^2]$ & $R$ [km] \\ \hline
4U 1608-52 & $1.59^{+0.54}_{-0.47}$ & $10.2^{+3.7}_{-2.7}$ & $74^{+532}_{-72}$ & $1.45^{+0.61}_{-0.53}$ & $10.7^{+1.9}_{-1.7}$ \\
4U 1724-207 & $1.81^{+0.36}_{-0.48}$ & $11.5^{+2.5}_{-2.5}$ & $24^{+291}_{-23}$ & $1.64^{+0.54}_{-0.54}$ & $10.4^{+2.0}_{-1.6}$ \\
4U 1820-30 & $1.76^{+0.44}_{-0.43}$ & $11.2^{+3.2}_{-2.6}$ & $32^{+297}_{-31}$ & $1.60^{+0.56}_{-0.52}$ & $10.5^{+2.0}_{-1.6}$ \\
EXO 1745-248 & $1.60^{+0.36}_{-0.42}$ & $10.3^{+2.7}_{-2.4}$ & $72^{+477}_{-67}$ & $1.45^{+0.56}_{-0.50}$ & $10.7^{+1.9}_{-1.6}$ \\
KS 1731-260 & $1.59^{+0.61}_{-0.62}$ & $10.4^{+3.8}_{-3.4}$ & $67^{+587}_{-65}$ & $1.47^{+0.63}_{-0.57}$ & $10.6^{+1.9}_{-1.7}$ \\
SAX J1748.9-2021 & $1.73^{+0.43}_{-0.56}$ & $11.3^{+2.9}_{-2.9}$ & $37^{+450}_{-36}$ & $1.57^{+0.57}_{-0.59}$ & $10.5^{+1.9}_{-1.7}$ \\
\hline \hline
\end{tabular}
\end{table}
\section{Multimessenger constraints on tidal deformability} \label{sec:joint}
Typical multimessenger probes of the neutron-star~equation of state~involve gravitational-wave and electromagnetic measurements of the same system.
However, the universal relations provide a means to translate observations of low-mass X-ray binaries into quantities, like tidal deformabilities, that are normally measured via gravitational waves from binary neutron star~mergers.
The independent gravitational-wave and electromagnetic measurements can then be combined to tighten the constraints on the tidal deformability as a proxy for the equation of state.
We use the simultaneous mass and radius measurements for the aforementioned bursters in conjunction with GW170817 to improve knowledge of the canonical deformability, starting with EXO 1745-248 as an example.
The symmetric 90$\%$ confidence intervals for its mass and radius, calculated from the $M$-$R$ posterior samples associated with the electromagnetic observations, are given in Table~\ref{tb:xray}.
The uncertainty of $\approx 25\%$ in its radius at 90$\%$ confidence is characteristic of the best current measurements; radius measurements with a better level of precision ($\approx 15\%$ at 90$\%$ confidence) are expected from pulse profile modeling with NICER \citep{OzelNICER}.
To infer the canonical deformability implied by EXO 1745-248's measured mass and radius, we link $R$ and $\Lambda_{1.4}$ through the universal relations by combining the probability distributions \eqref{biloveprob}, \eqref{iloveprob} and \eqref{icprob}, such that
\begin{equation} \label{Rinf}
P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM}) = \frac{G}{c^2} \int \frac{P(M,R|\,\text{EM}) P(GM/c^2 R|\bar{I}) P(\bar{I}|\,\Lambda) P(\Lambda|\,M,\Lambda_{1.4})}{R^2} \, M \, d\Lambda \, d\bar{I} \, dR \, dM .
\end{equation}
This amounts to using the fits \eqref{bilove}, \eqref{ilove} and \eqref{ic} successively to produce a function
\begin{equation} \label{Rmap}
R(M,\Lambda_{1.4}) =\frac{c^2}{GM} \sum_{k=0}^{4} d_k \left[ \sum_{l=0}^{4} c_l \left( \sum_{m=0}^{4} \sum_{n=0}^{1} a_{mn} M^m (\log_{10} \Lambda_{1.4})^n \right)^l \right]^{-k} ,
\end{equation}
while also accounting for the uncertainty in each universal relation.
Equation~\eqref{Rinf} allows us to convert the probability distribution $P(M,R|\,\text{EM})$ constructed from EXO 1745-248's $M$-$R$ posterior samples to a posterior distribution for the canonical deformability, $P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM})$.
This posterior distribution is plotted in Fig.~\ref{fig:joint}.
Calculating its median and symmetric 90$\%$ confidence interval, we find $\Lambda_{1.4} = 139^{+284}_{-82}$.
In other words, the constraint $R_{\text{EM}} = 10.7^{+1.9}_{-1.6}$ stemming from X-ray observations of EXO 1745-248 translates to these bounds on canonical deformability, as the universal relations map the mass-radius posterior $P(M,R|\,\text{EM})$ to the distribution $P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM})$ shown in the figure.
\begin{figure}
\centering
\includegraphics[width=0.55\columnwidth]{JointPosteriors.pdf}
\caption{Posterior distributions for $\Lambda_{1.4}$. Our model \eqref{model} of the posterior for \citet{LVC_eos}'s $\Lambda_{1.4}$ measurement (green) and the posterior distribution inferred from \citet{Ozel}'s electromagnetic observations of EXO 1745-258 (orange) are shown. The $\Lambda_{1.4}$ posteriors derived from several other observations of thermonuclear bursters are plotted in gray. The combined distribution resulting from the set of electromagnetic observations, plus GW170817, is shown in blue. The median and symmetric 90$\%$ confidence interval of the combined distribution are indicated with the dashed and dotted vertical lines, respectively.}
\label{fig:joint}
\end{figure}
We subsequently repeat the EXO 1745-248 analysis for the other {neutron star}s listed in Table~\ref{tb:xray}, obtaining posterior distributions $P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM}_i)$ for bursters $i=1,...,6$.
We then combine these indirect constraints on $\Lambda_{1.4}$ with the direct measurement from GW170817, $\Lambda_{1.4} = 190^{+390}_{-120}$, to get joint electromagnetic and gravitational-wave constraints that are tighter than the individual measurements.
The combined posterior distribution is computed as
\begin{equation} \label{joint}
P(\Lambda_{1.4}|\,\text{EM,\,GW}) = P(\Lambda_{1.4})\, P(\text{GW}|\,\Lambda_{1.4}) \prod_i P_{\Lambda_{1.4}}(\text{EM}_i|\,\Lambda_{1.4})
\end{equation}
by multiplying the likelihoods $P(\text{GW}|\,\Lambda_{1.4})$ and $P_{\Lambda_{1.4}}(\text{EM}_i|\,\Lambda_{1.4})$ with a chosen prior $P(\Lambda_{1.4})$, lending equal weight to each observation.
The likelihoods are related to the posteriors by Bayes' theorem:
\begin{equation} \label{lhoods}
P(\text{GW}|\,\Lambda_{1.4}) = \frac{P(\Lambda_{1.4}|\,\text{GW})}{P(\Lambda_{1.4})} , \qquad P_{\Lambda_{1.4}}(\text{EM}_i|\,\Lambda_{1.4}) = \frac{P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM}_i)}{P_{\Lambda_{1.4}}(\Lambda_{1.4})}
\end{equation}
up to normalizations.
The common prior $P_{\Lambda_{1.4}}(\Lambda_{1.4})$ for the electromagnetic observations is calculated from Eq.~\eqref{Rinf} assuming a uniform distribution in $M$ and $R$, i.e.~replacing $P(M,R|\,\text{EM})$ with $P(M,R) = \text{constant}$.
The mapping \eqref{Rmap} is such that small values of canonical deformability are more likely \textit{a priori}, despite the uninformative mass-radius prior.
The prior $P(\Lambda_{1.4})$ in Eq.~\eqref{joint} is chosen to be identical to the one appearing in Eq.~\eqref{lhoods} for the gravitational-wave observation.
Then, Eq.~\eqref{joint} reduces to
\begin{equation}
P(\Lambda_{1.4}|\,\text{EM,\,GW}) = P(\Lambda_{1.4}|\,\text{GW}) \prod_i \frac{P_{\Lambda_{1.4}}(\Lambda_{1.4}|\,\text{EM}_i)}{P_{\Lambda_{1.4}}(\Lambda_{1.4})} ,
\end{equation}
which yields a median and symmetric 90$\%$ confidence interval of $\Lambda_{1.4} = 196^{+92}_{-63}$.
This joint posterior is plotted in Fig.~\ref{fig:joint}.
As can be seen, the collective impact of the burster measurements is to substantially reduce the size of the error bars on $\Lambda_{1.4}$ relative to the gravitational-wave observation alone; meanwhile, the median is hardly changed.
This is because most of the electromagnetic mass-radius measurements imply a smaller canonical tidal deformability than GW170817 \textit{a posteriori}, thereby cutting off the long tail of $P(\Lambda_{1.4}|\,\text{GW})$ that extends to large values of $\Lambda_{1.4}$; simultaneously, the bulk of the observations provide minimal support for $\Lambda_{1.4} \lesssim 60$.
Hence, the incorporation of electromagnetic observations of {neutron star}s in low-mass X-ray binaries appears to disfavor some of the stiffer candidate equations of state~that remained compatible with GW170817, while corroborating a canonical deformability of $\approx 200$.
However, we remark that the combined constraint is only as reliable as the simultaneous mass and radius measurements themselves.
Fig.~\ref{fig:joint} shows that the $\Lambda_{1.4}$ posteriors for 4U 1608-52 and KS 1731-260 are outliers relative to both the GW170817 posterior and the other burster posteriors.
Since $\Lambda_{1.4}$ is a unique property of the equation of state, which is common to all {neutron star}s, the discrepancy among maxima \textit{a posteriori} for the electromagnetic measurements indicates that the observations are not, in fact, equally accurate.
As we have not accounted for possible systematic errors in the X-ray observations, it will be interesting to see whether this inference of $\Lambda_{1.4}$ is corroborated by future data from NICER.
\section{Discussion}\label{sec:disc}
In this paper, we used universal relations and constraints on canonical deformability from GW170817 to bound the mass--tidal-deformability, mass--moment-of-inertia and mass-radius relations satisfied by all cold neutron stars.
We found that the neutron star~radius is constrained to be roughly constant for $M \in [1,1.93]\,M_{\odot}$, with radii larger than 13.0 km ruled out at 90$\%$ confidence.
The mass-radius relations that are compatible with GW170817 are also consistent with existing simultaneous mass and radius measurements for six thermonuclear bursters.
Moreover, we inferred tidal deformabilities, moments of inertia, dimensionless spins and radii for individual {neutron star}s of interest.
The moments of inertia of a few double {neutron star}s were constrained to $\approx 30\%$ accuracy at 90$\%$ confidence by GW170817 and the universal relations, while the canonical neutron-star~radius was inferred as $R_{1.4} = 10.9^{+1.9}_{-1.5}$ km.
The dimensionless spins for a set of millisecond pulsars with well-measured masses were calculated to be $\lesssim 0.20$, and those for a set of pulsars in double neutron star~systems were found to be $\lesssim 0.04$.
The spin inferences presented here could be extended to the full population of pulsars with measured masses and rotational frequencies to obtain a spin distribution that is less dependent on equation-of-state~modeling.
The current $\approx 30\%$ level of precision in the inferred spins will improve as more binary neutron star~mergers are detected.
The gravitational-wave based predictions for the properties of specific {neutron star}s can be compared to direct electromagnetic measurements to test the universality of the {neutron star}~equation of state.
Recently, a number of candidate equations of state~that generically violate the universal relations because of multiple first-order phase transitions or non-standard phases of matter have been proposed \citep{Bandyopadhyay,Han,Lau,Annala17}.
Systematic disagreements between the moments of inertia or radii inferred here and those measured directly via radio or X-ray observations could be interpreted as evidence for such features in the equation of state.
Alternatively, because the universal relations are different in some modified theories of gravity \citep{Doneva}, a discrepancy could instead indicate support for a modification to general relativity.
Finally, we investigated how the universal relations can be used to tighten the constraints on $\Lambda_{1.4}$ by combining a gravitational-wave measurement of tidal deformability with electromagnetic observations of {neutron star}s in low-mass X-ray binaries.
Successively employing the binary Love, I-Love and I-compactness relations to create an equation-of-state~insensitive $R(M,\Lambda_{1.4})$ relation, we mapped simultaneous mass and radius measurements into posterior probability distributions over $\Lambda_{1.4}$, which were then combined with the corresponding posterior from GW170817.
Based on the resulting joint distribution, we refined \citet{LVC_eos}'s canonical deformability constraint to $\Lambda_{1.4} = 196^{+92}_{-63}$ at 90$\%$ confidence.
This inference of $\Lambda_{1.4}$---the most precise to date---is consistent with many (e.g.~\cite{LVC_GW170817,LVC_eos,Annala18}), but not all (e.g.~\cite{Most2018}), previous GW170817-based estimates, and favors a decidedly soft equation of state.
As part of the calculation, we found that the most probable $\Lambda_{1.4}$ values derived from observations of different {neutron star}s are not mutually consistent, nor are they all consistent with the canonical deformability implied by GW170817.
Indeed, the maxima \textit{a posteriori} inferred from observations of 4U 1608-52 and KS 1731-260 are considerably lower than the most probable value indicated by the gravitational-wave event.
Since the derived $R(M,\Lambda_{1.4})$ relation enables us to map disparate radius measurements to a common quantity, $\Lambda_{1.4}$, regardless of the equation of state, and since that quantity can be measured independently using gravitational waves, the joint inference technique presented here may be useful in redressing systematic errors affecting current probes of neutron-star~radii.
In any case, additional gravitational-wave observations of binary neutron star~mergers and more accurate radius measurements, like those expected from NICER, will permit the universal-relation based bounds on canonical deformability to be further refined.
\acknowledgments
The authors thank Reed Essick and Luciano Rezzolla for helpful discussions about this work, and acknowledge Katerina Chatziioannou for pointing out a mistake in an earlier version of Sec.~\ref{sec:joint}. P.~L.~was supported in part by the Natural Sciences and Engineering Research Council of Canada, and by NSF grants PHY 15-05124 and PHY 17-08081 to the University of Chicago. B.~K.~thanks the Navajbai Ratan Tata Trust, which also provided partial support for this work.
| {
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Adobe After Effects CC 2019 Mac Crack V16 is the latest Full Version of the most advanced Video compositing and editing software For Mac OS X which is now available with direct download link at gsmboxcrack. With After Effects CC 2019 For Mac + Crack you can create cinematic movie titles, intros, and transitions. It enables yo to add new life to your animation with ease.
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Štitasto ptičje mlijeko (laščet, lapčet, pupa. lat. Ornithogalum umbellatum), lukovičasta trajnica iz roda ptičjeg mlijeka, porodica Asparagaceae. Rasprostranjena je po čitavoj Europi, osim Skandinavije, gdje je naknadno uvezena, a odakle se raširila i u Sjevernu Ameriku, Australiju i Novi Zeland.
Naraste do 25cm. Raste samoniklo po livadama i voćnjacima, a uzgaja se i kao ukrasna biljka. Lukovice su otrovne, ali i jestive nakon tremičke obrade, pa se nakon nje mogu i sušiti i mljeti u brašno.
Za vrstu postoje brojni sinonimi. Jedan od naziva za nju je i betlehemska zvijezda
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Hyacinthus umbellatus (L.) E.H.L.Krause
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Ptičje mlijeko | {
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Prince of Persia Remake Now Set for 2022 Release (Maybe)
Last year Ubisoft confirmed that Prince of Persia The Sands of Time (originally released in 2003) was getting a remake. Given that I rather enjoyed the original game (except the absolute garbage ending provided in the DLC), this was definitely something I was keeping track of on my gaming radar. Ever since the official announcement, however, it has been pretty clear that work on the game has been more challenging than Ubisoft expected.
Firstly it was supposed to release in January 2021, then it was delayed into February/March, and following an official update last February (when the game did not materialise), Ubisoft then said that the Prince of Persia remake was now officially in a state of being 'indefinitely postponed'.
Well, following a report via DSOGaming, while we still don't have any firm news regarding when this game is going to come out, Ubisoft has provided an update with a provisional timeframe. Albeit, it's not exactly encouraging.
Following an update to the game's official website, while Ubisoft does ramble on a bit about thanking its fans and the hard work they're putting into this remake, etc., etc., in truth, the whole thing pretty much boils down to one sentence. They are hopeful that this will be ready and completed for release sometime in 2022. – So, even with this not entirely optimistic update, they are pretty much confirming that despite being originally set for release around November last year, it's going to possibly end up coming out nearly 2 years later than planned!
Why Such a Long Delay?
Although opinions may differ, and I'm pretty sure Ubisoft wouldn't confirm this themselves, it does seem that the delays all initially stemmed from the original announcement trailer (embedded above). Put simply, the reaction to it from both fans and the media was exceptionally negative, and I think rightly so. I mean, does it look that much better from the original really?…
An opinion shared by many is that Ubisoft attempted to pass off a very loose remaster as a remake and got caught out by the gaming community. As such, delays have now been necessary for them to actually create a remake rather than just a crap graphical gloss over.
Admittedly though, and as noted above, I am genuinely hopeful that when this does release, it will be a decent enough game to give me lots of nostalgia replaying it again. I just largely hope for two things; Firstly, that it actually does look like a true next-gen remake, and secondly, please Ubisoft, fix the originals DLC ending!
What do you think? Are you looking forward to the Prince of Persia remake? – Let us know in the comments!
Mike Sanders
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{"url":"https:\/\/plainmath.net\/16632\/a-set-of-ordered-pairs-is-called-a","text":"A set of ordered pairs is called a _______.\n\nTrent Carpenter 2021-06-13 Answered\nA set of ordered pairs is called a _______.\nYou can still ask an expert for help\n\n\u2022 Live experts 24\/7\n\u2022 Questions are typically answered in as fast as 30 minutes\n\u2022 Personalized clear answers\n\nSolve your problem for the price of one coffee\n\n\u2022 Math expert for every subject\n\u2022 Pay only if we can solve it\n\nClelioo\nA set of ordered pairs is called a relation.\nIf the x-coordinates of each ordered pair are unique, that is, each input of the set corresponds to only one output, then the set can be more specifically called a function.","date":"2022-08-08 23:09:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 18, \"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\": 0.7577729821205139, \"perplexity\": 1148.7839929524237}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570879.1\/warc\/CC-MAIN-20220808213349-20220809003349-00094.warc.gz\"}"} | null | null |
A Estrada Nacional 99 (em sueco: Riksväg 99) é uma estrada nacional sueca com uma extensão de 367 km, que atravessa as províncias históricas de Norrbotten e da Lapónia, ligando as cidades de Haparanda e Karesuando. Passa pelas localidades de Övertorneå e Pajala.
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{"url":"http:\/\/mymathforum.com\/physics\/347324-why-wouldn-t-horizontal-component-force-make-yo-yo-move.html","text":"My Math Forum Why wouldn't the horizontal component of the force make yo-yo to move ?\n\n Physics Physics Forum\n\nOctober 27th, 2019, 06:14 AM \u00a0 #1\nNewbie\n\nJoined: Aug 2019\nFrom: India\n\nPosts: 23\nThanks: 1\n\nWhy wouldn't the horizontal component of the force make yo-yo to move ?\n\nThe problem statement is:\nA yo-yo rests on a table (see the attachment) and the free end of its string is gently pulled at an angle $\\displaystyle \\theta$ to the horizontal as shown. What is the critical value of $\\displaystyle \\theta$ such that the yo-yo remains stationary, even though it is free to roll? (This problem may be solved geometrically if you consider the torques about P, the point of contact with the table).\n\nNow, the thing is that this problem can be solved ( I mean answer can be made to come right) if we assume a few things and those assumptions are my problem.\n\nThe proposed solution is: (see the second attachment for reference) The angle between $\\displaystyle r$ and $\\displaystyle R$ is $\\displaystyle \\theta$ because of the principle the angle between two lines is the angle between their normals and $\\displaystyle r$ is perpendicular to the line of force because it is tangent to that inner circle and $\\displaystyle R$ is perpendicular to horizontal line.\nNow just equate the torques produced by these forces:\n$\\displaystyle F~\\sin\\theta ~r \\sin\\theta = F~\\cos\\theta~(R-r\\cos\\theta)$\n$\\displaystyle r \\sin^2\\theta = R~\\cos\\theta - r~\\cos^2 \\theta$\n$\\displaystyle r= R~\\cos\\theta$\n$\\displaystyle \\theta = \\cos^{-1} (r\/R)$\n\n(I have included the solution with all possible intricacies only because to save your precious time, I know these things are so easy for you and my intention was not to exhibit anything it was purely for clarity and time saving.)\n\nNow, how can we ever ignore the translational effect of $\\displaystyle F~cos\\theta$? I mean it would cause the the yo-yo to translate in horizontal direction if there were no friction. And if there were to have a friction then for critical angle our equation would be very simple\n$\\displaystyle F~\\cos\\theta + Friction =0$\n$\\displaystyle \\theta = \\cos^{-1} (-Friction\/F)$.\n\nSo why we are not considering it? I mean why are we just assuming that $\\displaystyle \\Sigma F=0$ and $\\displaystyle \\Sigma \\tau=0$ (I must clarify that some people in US call torque as moment and denote it by M, here moment has been phrased as torque and is represented by tau \\tau). What are the reasons for taking those two assumptions?\n\nThank you. Any help will be much appreciated.\nAttached Images\n Physics.jpg (11.2 KB, 1 views) solution 1.jpg (9.0 KB, 1 views) Solution2.jpg (13.5 KB, 1 views)\n\nLast edited by skipjack; October 27th, 2019 at 09:15 AM.\n\n October 27th, 2019, 08:12 AM #2 Math Team \u00a0 \u00a0 Joined: Jul 2011 From: Texas Posts: 3,101 Thanks: 1677 Reference the diagram ... As shown above, all forces acting on the yoyo (string tension F , weight mg, normal force N, and static friction f) act through point P, the point of contact with the ground, hence the net torque on the yoyo is zero. As you stated, the angle $\\theta$ where the yoyo remains in equilibrium is $\\theta = \\arccos\\left(\\dfrac{r}{R}\\right)$ Reviewing the equilibrium of the translational forces ... $F\\cos{\\theta} = f$ $N + F\\sin{\\theta} = mg$ ... and the rotational forces about point C (the center) $F \\cdot r - f \\cdot R = 0$ $F \\cdot r - F\\cos{\\theta} \\cdot R = 0$ $F(r - \\cos{\\theta} \\cdot R) = 0$ $F \\ne 0 \\implies r = R\\cos{\\theta} \\implies \\cos{\\theta} = \\dfrac{r}{R}$ Thanks from topsquark, romsek and Adesh Mishra\n October 27th, 2019, 08:38 AM #3 Newbie \u00a0 Joined: Aug 2019 From: India Posts: 23 Thanks: 1 I have one doubt : the tension force is applied at the inner circle so we should find the effective lever arm from P to that point. But you have made that as if string force is acting on P. I mean that tension force is in contact with the inner circle so I think it would cause a net torque at P. I request you to please clear my doubt\nOctober 27th, 2019, 08:53 AM \u00a0 #4\nMath Team\n\nJoined: Jul 2011\nFrom: Texas\n\nPosts: 3,101\nThanks: 1677\n\nQuote:\n Originally Posted by Adesh Mishra I have one doubt : the tension force is applied at the inner circle so we should find the effective lever arm from P to that point. But you have made that as if string force is acting on P. I mean that tension force is in contact with the inner circle so I think it would cause a net torque at P. I request you to please clear my doubt\nConsider a door hinged at point H with a string exerting a force outward from the door edge as shown in the \"view from above\" attached ... would it exert a torque on the door?\n\nIs not the lever arm relative to the hinge zero?\nAttached Images\n door_force.jpg (13.8 KB, 1 views)\n\nLast edited by skeeter; October 27th, 2019 at 09:43 AM.\n\n October 27th, 2019, 09:56 AM #5 Newbie \u00a0 Joined: Aug 2019 From: India Posts: 23 Thanks: 1 Thank you so much for being so clear.\n\n Tags component, equilibrium, force, horizontal, make, move, yoyo\n\n Thread Tools Display Modes Linear Mode\n\n Similar Threads Thread Thread Starter Forum Replies Last Post leo255 Calculus 1 December 16th, 2014 08:48 PM limes5 Real Analysis 0 June 27th, 2013 05:18 AM mr calculus Applied Math 1 February 21st, 2013 03:39 PM subwayheaven Algebra 1 January 25th, 2013 10:02 AM\n\n Contact - Home - Forums - Cryptocurrency Forum - Top","date":"2019-11-19 20:09:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7242016792297363, \"perplexity\": 1141.5777023071687}, \"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-2019-47\/segments\/1573496670255.18\/warc\/CC-MAIN-20191119195450-20191119223450-00514.warc.gz\"}"} | null | null |
Q: Raid controller can't erase RAID config I bought a Dell PowerEdge T320 that came with 2 1TB HDD and I just ordered 6 more used 3TB HDD. When setting up an array I'm able to see the original 2 HDD that came with the server but the 6 used drives are showing up as "foreign". I'm trying to erase the RAID configuration from those HDD's but I'm getting the following message
One or more of your disks were secured using a different controller. This controller does not support security features. You need a controller with security to unlock a secured disks.
Would updating the BIOS add the "security features" needed to resolve this problem?
This is a server I bought for myself to learn and host some applications.
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{"url":"https:\/\/no.overleaf.com\/latex\/templates\/syllabus\/bdqfmfnncffs","text":"# Syllabus\n\nAuthor\nJames Russell\nAbstractThis is a sample LaTeX syllabus for the Structuring Content Class as part of the Teaching and Communication Certificate at NC State.","date":"2021-01-18 07:41:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 1, \"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\": 0.19287040829658508, \"perplexity\": 9374.799217404205}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703514423.60\/warc\/CC-MAIN-20210118061434-20210118091434-00394.warc.gz\"}"} | null | null |
{"url":"http:\/\/cinf201.artifice.cc\/notes\/import-export.html","text":"# Import\/export\n\nDatabase systems typically support data import and export. Data import is useful when the data already exist in some external system or files, but needs to be imported into a different database. Export is handy for backups and transferring data to another database.\n\n## Import\/export in SQL format\n\nThe simplest format to import\/export data for MySQL is using the database\u2019s own SQL language. Supposing you have a SQL file full of table creation and data insert commands, \u201cimporting\u201d the data is as simple as executing the SQL code:\n\nmysql < myfile.sql\n\n\nIn order to create such a SQL file from an existing database, you can use the mysqldump command from the Linux command line (not from the mysql prompt):\n\nmysqldump cinf201_jeckroth > myfile.sql\n\n\nIf you only want the tables but no data in the tables, use the -d option:\n\nmysqldump -d cinf201_jeckroth > myfile-only-tables.sql\n\n\n## Import from CSV\n\nWhen we need to import data from a non-SQL source, such as a spreadsheet, we can use the CSV (comma-separated values) format. A CSV file looks like this:\n\ncolname1,colname,colname3\n35.2,0.0,foo\n6.1,2.2,\"bar, baz\"\n\n\nIn other words, the first row contains column names, and the following rows contain values for each column. Columns are separated by commas. If a value has commas, like \u201cbar, baz\u201d, the value will be surrounded in quotes. Sometimes the first row, containing column names, is missing. The data types (e.g., INT vs. DOUBLE vs. DATETIME, etc.) must be inferred or known ahead of time; the CSV file does not tell you anything about data types.\n\nIn order to import a CSV file, you must first create a table. Then, use the LOAD DATA command:\n\nLOAD DATA INFILE '\/tmp\/myfile.csv'\n[REPLACE | IGNORE]\nINTO TABLE tbl_name\n[{FIELDS | COLUMNS}\n[TERMINATED BY 'string']\n[[OPTIONALLY] ENCLOSED BY 'char']\n]\n[IGNORE number LINES]\n[(col_name,...)]\n[SET col_name = expr,...]\n\n\nThe LOAD DATA command supports different kinds of files, not just CSV. But if you wish to import CSV data, indicate FIELDS TERMINATED BY ',' and, if you expect any fields to have commas (like a \u201cLastname, Firstname\u201d column), be sure to quote the field in the CSV file and add OPTIONALLY ENCLOSED BY '\"', e.g.,\n\nLOAD DATA INFILE '\/tmp\/myfile.csv' INTO TABLE mytable FIELDS TERMINATED BY ',' OPTIONALLY ENCLOSED BY '\"';\n\n\nAlso note that the MySQL server must be able to read your file. On londo, normal file permissions are insufficient for these purposes. Thus, we\u2019ll use the following strategies:\n\n\u2022 Copy your CSV file to the \/tmp directory; give your file a funny name so it does not conflict with anyone else\u2019s. E.g., use this copy command: cp myfile.csv \/tmp\/jeckroth-myfile.csv.\n\u2022 Next, fix the permissions on the file so MySQL can read it: chmod a+r \/tmp\/jeckroth-myfile.csv. (FYI: a+r means \u201call can read\u201d).\n\nThe additional elements of the LOAD DATA command can be used for the following purposes:\n\n\u2022 REPLACE:\n\u2022 IGNORE number LINES: skip a header line in the CSV file with this feature.\n\u2022 (col_name, \u2026): indicate which columns should be filled from the CSV data; if not specified, all columns will be filled in the order specified when the table was created.\n\u2022 If you need to ignore some columns in the CSV file, you have can use a variable (say, @dummy) in that position: LOAD DATA \u2026 (col1, col2, @dummy, col3, @dummy, col4, col5) for example.\n\u2022 SET col_name = expr, \u2026: for each row, set some other columns; the expr value may refer to other columns on the same row.\n\n## Export to CSV\n\nSELECT select_statement...\nINTO OUTFILE '\/tmp\/myfile.csv'\nFIELDS TERMINATED BY ',' OPTIONALLY ENCLOSED BY '\"'\n\n\nDue to the same file permission issues on londo, be sure to save the outfile in \/tmp. Then copy it to your own directory.","date":"2017-07-26 00:34:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2212081104516983, \"perplexity\": 5541.111335228832}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-30\/segments\/1500549425737.60\/warc\/CC-MAIN-20170726002333-20170726022333-00609.warc.gz\"}"} | null | null |
DON'T TELL ME WHAT TO DO
Copyright © 2017 by Dina Del Bucchia
All rights reserved. No part of this book may be reproduced in any part by any means—graphic, electronic, or mechanical—without the prior written permission of the publisher, except by a reviewer, who may use brief excerpts in a review, or in the case of photocopying in Canada, a license from Access Copyright.
ARSENAL PULP PRESS
Suite 202 – 211 East Georgia St.
Vancouver, BC V6A 1Z6
Canada
_arsenalpulp.com_
The publisher gratefully acknowledges the support of the Canada Council for the Arts and the British Columbia Arts Council for its publishing program, and the Government of Canada, and the Government of British Columbia (through the Book Publishing Tax Credit Program), for its publishing activities.
This is a work of fiction. Any resemblance of characters to persons either living or deceased is purely coincidental.
Cover and text design by Oliver McPartlin
Edited by Robyn So
Library and Archives Canada Cataloguing in Publication:
Del Bucchia, Dina, 1979-
[Short stories. Selections]
Don't tell me what to do / Dina Del Bucchia.
Short stories.
Issued in print and electronic formats.
ISBN 978-1-55152-702-4 (ebook).
I. Title. II. Title: Do not tell me what to do.
PS8607.E482538A6 2017 | C813'.6 | C2017-904059-6
---|---|---
| | C2017-904060-X
Contents
Keeping Things Alive Is Too Much Work
Don't Tell Me What to Do
Miss Supreme
Under the "I"
Cold Cuts
Particleboard Man
A Beautiful Feeling
Hamsters
Haul
Nest
Sometimes We Can Be That Way
Sleep Talk
Instructions for Having an Affair
Full Price
The Gospel of Kittany
Keeping Things Alive Is Too Much Work
"We call them blades, but they can't cut nothing," Ron says, chuckling at what he thinks is a joke.
Val rakes a swath of yellow grass into another pile; the scratch of the rake blocks out her neighbour's voice. Mounds of shorn grass dot her wide yard, rustle in the wind. They almost pulse as if under their scruff lies a mulchy beating heart.
"Okay, Ron. Whatever you say." She wipes a gardening glove across her sweaty forehead. It's too early for Ron's nonsense and for it to be so hot, but the right time of day to get work done.
Ron has been watching her since she came out this morning. Not sure what was so interesting about her pushing the mower around in neat lines, about the way she rakes, slower than she used to. And he doesn't seem to be going anywhere as she admires her grassy hills before loading them into the wheelbarrow and carting them to the alley. And he's still there to see her uncoil the hose and spray down the dusty flowerbeds, turn on the sprinkler.
"Not looking too good there. Everything looks near dead if you ask me."
Val didn't ask him and what does she care what he or anyone else thinks. She goes inside, slams the sliding glass door. He better not set up camp outside her front window and watch her watch people root around in other people's storage lockers on television.
She makes herself more coffee, puts bread in the toaster. A splash of water hits the kitchen window. The placement of the sprinkler is slightly off. The house doesn't need watering.
Her granddaughter, Rae, is always talking about the environment, how we're wasting everything away.
"Resources," she says. "Water. Every time you turn on the faucet the world dies a little. It doesn't just come flowing out of the tap from nowhere."
Val knows water flowing out of a tap isn't magic. Just because she didn't go to college, dropped out of high school to get married to someone with a decent wage, doesn't mean she's confused about the world. She dumped her husband fifteen years ago. Doesn't need a GED to know it's the smartest move she ever made. Her days are her own. Her life her own because someone doesn't think they own her.
Rae comes by for bottle drives, bikes over with energy-efficient light bulbs, a special box to collect batteries, checks up on people's recycling, and sets up robot-like composters around the neighbourhood. Each one decorated by local artists, painted with vegetable-based paints. At night the composters look like intruders, something from space ready to abduct humans, crumple their bodies into manure. She spends weekends raising money for clean water in far-away countries by selling organic, gluten-free pastries.
Val's grandson, Justin, is always talking about snowboards. So he's into water too. Frozen water crystals all packed up on a mountain.
"It's best in the powder, Gramma. The board moves so sick."
"I'd be sick if I was swooshing around on a board down a steep hill."
"Are you going to tell me you had to walk to school up a mountain in your day?"
"Nope. I know things are different."
The environment and snowboarding. Both endeavours seem equally taxing to Val. This weekend when they come for lunch, she'll be prepared to talk about it all. She can look up anything on her tablet.
"Hi, Mom. Your yard looks so sad." Her son, Ben, hugs her. Insults always pair best with hugs.
"Yard doesn't have feelings."
"Nice one, Gramma," says her grandson.
"But—" Ben tries to get a word in.
"But, it's smack dab in the middle of summer, and I do my best without a tree for shade out there. Can only water on odd days. Yard's just balding, like you."
He stands at the back window, children on either side of him. The three of them looking out at the dusty brown scraps of grass, shrivelled weeds, bare patches that sprout more evenly than the lawn itself.
"Do you like that? Is it better than television? Staring at an old woman's failure. Just paint a picture and put it up in your own damn living room. Then you can look at it every day, like some kind of art. Okay, that's enough."
She ushers them into the kitchen, huffs until they're all seated at the table, and pours juice from a plastic jug into their glasses.
Rae says, "I told you to get rid of your plastics that aren't BPA free. I don't want you to get cancer." She examines the pitcher, rotates it in front of her. "There must be some way to re-use this."
Crisp salad sits in the middle of the table in a metal bowl.
"I've got green things for you right here."
Val points. She brings in chips and a pyramid of triangular sandwiches. Everyone takes a sandwich. No one mentions how unhealthy chips are as they heap handfuls on their plates until there isn't much room left for salad.
Ten years ago, when the teenagers were kids, Val's son took her with them on a family vacation to the Grand Coulee Dam. It seemed like everything had to be educational, even a weekend getaway. Val packed a cooler with triangular sandwiches and chips in Ziploc bags.
They all stood on the viewing deck. The cascades of water looked so beautiful. Photos were taken of all of them standing surrounded by the rich blue water, the crisp blue sky, and the slate grey of the massive structure.
Inside they walked past photographs of construction, the before and after. Rae asked why things were less green, then more green? Justin stared at the rushing water through the window, an almost dangerous level of interest. Val kept coming back to the massive images of the dry dam. When the valves were shut it was so smooth. Concrete slopes looked simple and pristine. Flat lines, cold contours. It was more beautiful that way. Clean. Constructed to harness nature.
When the water burst through, everything became animated, chaotic. Even though it was controlled, the gushing sound was overwhelming. There was a laser light show too. Educational and electric. It was everything the river was not: flashy and colourful, an attempt to take the flow into man's hands.
Val had wished they'd been photographed in front of the grey, empty, curved flumes. So peaceful.
"Time for dessert. It's pie and ice cream. And, yes, I got organic blueberries, dammit."
Val half-heartedly drags the sprinkler a few feet from where it's spraying up against the house on one side and the shed on the other. Water tings and spatters against glass and metal. Droplets dribble down the outside of the window. She likes the sound. She likes the look of it. Streaks blur her view of the inside of her home.
"You're a drip, Val."
Ron's sipping on a Mike's Hard Lemonade through a straw. He's in his same spot, camped out with a beverage on his weathered lawn chair. If it wasn't empty on occasion, it could be said that Ron's moulded to the fabric seat. If he didn't move his hand to his mouth, his arm might be glued to the rest. Beside him is what looks like a new cooler. Blue plastic, white lid, not too big. Easily fits a six-pack, and maybe a box of popsicles. He's always in the chair or not. Just appears and disappears.
"Yeah, you're working real hard over there. You're gonna break your wrist if you're not careful."
"You know I'm only teasing." Ron holds out his Mike's Hard to Val, winks at her.
"Do I?"
There's barely a need to cut the grass anymore. The only yard work is this half-assed watering, a vain attempt to keep some of the lawn alive. It's not working. Everything's brown. Even the clover that didn't succumb to the sugar and water, the corn gluten meal, the chemical poisons. She'd fed that clover perfectly good sugar and perfectly good poison. She'd wanted to kill that clover, and now it was dead. Because everything was dead. And now everyone was on her case about how terrible it looked. They'd been on her case about how terrible the clover had looked too. Ron told her that clover was a sign of low nitrogen or something. Her son gave her lawn a solemn look, gave her the downturned eyebrows of pity.
"It's real hot out, Val." Ron drains the yellow sugary alcohol like a kid drinking a pop.
"Oh, really?" Val's hands are still gripped tight to the hose. She might as well just shut the whole thing down. Give up. Go inside. Or head to the store and stock up on her own supply of cold alcoholic beverages. Nothing so sweet though. Just regular old beer.
"C'mon, now. Won't hurt you to wet your throat. Or any other parts."
"I'm not in the mood for that talk. Ever."
"Be fun. Come on and have a drink." Ron opens his cooler, jammed as tight as can be with pastel pink and yellow bottled drinks.
Val yanks the hose toward herself, stands in the spray. Behind her Ron's caught in the shower too. He drops one drink, gets up, and runs inside. The first time she's seen him move in years.
Each night it gets worse. Val can't sleep. Her house heats up like a brick oven, and even with three fans pointed at her she can't cool down. The blades chop the air. Val imagines each cutting through this thick heat, this stench. She wills each swipe to help her ease into sleep, tries to count the rotations like sheep to help herself stay in control. She lolls for a few minutes at a time, falling into strange dreams. None of it feels like sleep.
She's standing on a knife-edge, and it's dangerous but cool. Her ex-husband is chastising her from their old Winnebago, that she's ruined their house, let it go to rot, but he won't just drive away. She wakes up in a wheelbarrow full of crispy grass, but really wakes up in bed.
In some gardening magazine her son gave her a subscription to, a subscription she didn't want or care about, she remembers reading that lawns were originally for rich people. Poor people were throwing their shit out their windows into the street, onto other people. They didn't have a patch of green to try not to kill. Rich people had servants to fluff up their shrubs.
Everyone wants to pretend to be rich. That's where lawns came from. Everyone thinking they're better than everyone else. Everyone wanting to pretend they're the aristocracy and not peasants working just to make their life not a terrible, shitty mess.
How the hell did her son get the idea that she wanted a gardening magazine? What did she do that made anyone think she cared about gardening? Just get old? Did she look useless? It's supposed to be contemplative, he'd said. Val was thinking about shit all the damn time. She had plenty of time to think, being a retired woman with a paid-off mortgage.
The clock says four a.m. It's the coolest part of the night. She might sleep until six-thirty a.m. Could plan her day around a nap that'll barely register.
What would happen if people looked into her yard and didn't see green grass? What would happen if she just tilled the whole thing up? If she decided to let it die? What else could live out there so simply? Who decided to put grass out there anyway? Her house isn't Versailles. She Googles it all. Makes a list of supplies she'll need to transform her lawn from the living dead into something else.
"It's dead anyway," she says into the sticky air, the whir of fan blades, herself.
"It's dead anyway," she says to her son at their next Sunday family lunch.
He'd suggested they eat outside. Val reminded him that there's no table, no chairs, no umbrella, no fans to stave off heat. He said that's okay. They can spread a blanket, have a picnic. Rae thought it a nice idea. Val said a flat out no. End of discussion. Justin was out with friends, skateboarding for a birthday or something.
Val opens the curtains so they can look outside, see the backyard in all its faded glory. "I miss the voles."
"Grandma, they were so cute. Me, too, I hated when Grandpa tried to poison them."
The voles were cute, like the small mice found at pet stores for snake feed.
"I can't believe you remember that, Rae. You were barely two years old."
Those voles used to annoy the shit out of her husband. Oh, he tried to murder them, tried to flush them out with the hose once. He tried everything, but nothing worked. Those voles outsmarted and outlasted him. And yet, somehow, once he left they never came back, moved on to greener lawns, she supposed. Wanting the voles back to ruin her lawn is funny, since she's already ruined it herself. A sad lawn isn't a happy home for any self-respecting vole.
She baked Ben's favourite cookies in the middle of the night, and a lemon pie. Insomnia baking. It made her feel useful. Hopes it's useful now. Slicing pie, she gives him the fattest slice.
"Can I borrow your truck?"
"Sure. For what?"
Explaining to her son why she needs to borrow his truck is more annoying that she'd realized it would be. She doesn't need a lawn. She's got a plan. To get rid of the lawn. To tear it up, to smooth it out. Fill the space with perfect unmowable concrete, concrete that won't burn in August heat. Concrete that will shine in the daylight. That she can sweep away dirt from. That she can admire in any season.
"That's not very environmentally sound, Grandma."
"Wasting water isn't good for the environment. And that's all I'm doing out there, Rae."
"That's a big job, Mom."
"Don't need it. Who goes back there anyway? Nobody sees it except for that creep, Ron."
No one does go back there. She was never one for hosting parties, hadn't invited anyone into her yard in years and years. It's all work and no play.
"We could have barbecues. You could start a book club back there."
"Can pretend to talk about books and drink wine inside just fine. Just let me know what days I can have the truck. I won't scratch it. It's a very nice truck. This is what I'm doing. Keeping things alive is too much work."
She packs the cookies into a plastic container for him to take home.
"You kept me alive."
"Yeah, and your sister. And she barely talks to me. Two people is enough to keep alive for one lifetime."
It's so hot that she dreams her whole yard is on fire. Blades of grass are tiny tinder that explode into flames. There are dark storm clouds above but no rain. She watches the grey and orange clouds and smoke and fire from inside her house, which is impermeable to the blaze. She wakes up with a rash on her wrist. Scratches it as she continues her research on her tablet, makes a full list of supplies for the next day.
Her number one live reality show fan, Ron, sits in a new lawn chair with a 7up in one hand and a Molson in the other. He observes her moves, rolling the rototiller down the ramp from the back of the truck to the edge of the yard. It sputters before it starts and at first feels like too much for her to handle. But she steadies her hands, braces her arms, and rolls it across her brown and yellow lawn. The richer earth becomes the top layer. Ron yells things at her as she propels the machine, but she can't hear him, just the roar of domestic destruction. She smiles as she eliminates each crispy blade. It takes all morning just to do one corner. But time doesn't matter. She's satisfied.
Once the grass is gone, and it's just soil, there'll be one more raking session. Val is sure Ron will watch that too. Tear up the grass, smooth the soil, move grass and excess soil with a wheelbarrow to the garbage bin. A little bit each day, she chips away.
In the evenings, she comes inside through the basement door after working and makes sure the curtains are shut. She takes off her striped work gloves, her heavy socks and boots, too hot for this weather, her souvenir Mexico T-shirt, stained with rust, the shorts her husband left behind, too big for her but full of pockets. Her wilted bra, her underwear from a drawer of identical underwear, everything striped. Naked, she gathers the mess and puts it in the washing machine, doesn't turn it on. In the windowless basement bathroom, she stands in the shower stall for too long, just cold water washing over her. She forces herself to scrub head and body with a bar of soap. Head wrapped in a towel, she puts on her robe, starts the wash cycle, eats dinner, watches television, and sleeps on the couch.
Preparing the subbase was no big deal, laying the fine grade stone and compacting it. Even building the form and mesh didn't trouble her. Throughout the early stages, it was work, but she didn't notice. Ron was the only nuisance. Getting her yard torn up wasn't easy, but making the cement is hard. She struggles with the bags of cement, with the old mixer. Her arms feel like saggy grocery bags. It takes three days for her to figure out how to angle the mixer. The consistency wasn't right at all, and that took her longer than she wanted it to.
When she finally pours, she wants to make sure she gets it right. Wants it to look as nice as can be. She smooths it out until it's even. That first section is small but exactly as she pictured it. A test patch.
At night, she celebrates with a couple of beers in the quiet of her living room, television on, muted. The sun goes down, and in the slightly cooler air, she sits in her favourite chair until she's sucked back half a six-pack. She cracks a fourth, one more to help her get some rest. She hears kids running through the alley, the clang of her back gate opening and slapping shut.
In the morning she sees the damage done. Names, initials, hearts, and profanity scrawled. They even left their sticks sitting in the now dry cement. She cracks one off, the nub sticking out, and throws the rest into the alley.
She stretches, puts on her gloves, and fires up the mixer. Ron shouts over the din of the cement mixer, "Can't trust anyone's offspring these days." She can barely hear him. He's a low buzz. She stares at bags of concrete piled near the shed, checks her watch. The hardware store won't be open for two more hours. Time to whip up another batch before that. She'll call Ben to bring the truck and the alarm company to come on Monday morning.
Usually she hits up the smaller hardware store closest to her. They know her and give her what she's looking for, and they don't ask questions and don't play any music. But they didn't carry anything to break up a solid square of concrete.
In this big chain they have everything, including some terrible radio station. She stalks the aisles trying to find the rental desk.
"Do you need help, ma'am?" A young man with a ponytail asks.
"Rentals."
"Come with me."
Val huffs and follows him. He points to a desk at the far end of the construction section but continues to walk with her until the two of them get there, like she needs a police escort.
"This lady is looking to rent something, Jim."
"Thanks, Cary."
"Glad you two explained everything to each other."
She drops her elbows onto the desk, and Cary and his ponytail leave to escort other customers around the store, like a misguided gentleman.
"I need a saw or something to help me break up a slab of concrete in my lawn."
He stares at Val.
"I'd like it today."
"Okay. So, is this from a patio or some other permanent structure that you're getting rid of?"
"No. It's a piece of my lawn. But some kids vandalized it, and now I need to take out this whole piece and do it again."
He stares at Val.
"I'd like to get back to work as soon as possible, so something like a chainsaw."
"Okay. Do you know how to use this equipment?"
"I don't know what equipment I'll be renting yet, so I don't know if I know."
"Well, some of these tools are more heavy duty than others."
"That's what I want. Something that'll get the job done."
"Will you be using the tools yourself?"
Val raises an arm. He doesn't move. She grabs for a catalogue of items that's attached to the desk and starts flipping through it. She should have just ordered it up on her tablet if she knew she was coming to this place. Then she wouldn't have to talk to any people. Or stare into this man's face while he decides whether or not he's got more useless questions for her.
"Show me something I can use!" Val shouts.
"Mom? Mom?"
She hears Ben's voice. He rounds the corner with Cary and his trusty ponytail.
"Ben, did you bring the truck?"
"Sorry. Yes. Sorry."
"Your mother is looking for something to break up concrete."
"Yes. She needs to undo some damage done. What do you recommend?"
Useless Jim goes back into a storage area and returns with two saws.
"One of these two will do the job well and are a snap to use. It's like slicing off icing on a cake. Sometimes a sledgehammer is good to break up the bigger pieces. And, of course, we also have safety gear too."
"I have—" Val starts to say.
"She has all the safety equipment. I've made sure of it," Ben interrupts.
Val slams shut the open catalogue and stomps up the aisle.
"And a sledgehammer."
"Mom, where are you going?"
"Need more cement. Always need more cement."
Val hauls as many bags as will fit into her trunk to the till. As she loads them in, Ben wheels up a dolly with the saw.
"Thought it would all be a lot more. Bigger. Don't need the truck," she says.
"I can take all of this for you, Mom."
She takes the saw from his hands and straps it into the back seat, clicks the seatbelt to keep it in place.
"Thanks for carrying it out to the car."
Val takes the rest of the day off. Drives her rented saw around town. They pass Ben's elementary school, the playground there already half concrete, for tetherball and square ball next to the basketball court. Up the hill, they cruise by the church where she was married, small and in the middle of a large parking lot. The rink nestled between the river and a parking lot. She drives through neighbourhoods just to feel the pavement under the tires, to watch the shades of grey stretch out in front of her at every turn.
"Excuse me. Is this your house?"
A woman in perfectly creased capri pants mimes knocking on an invisible door next to Val's shrubs.
"Do I look like I live here?"
"Sorry. You can never be sure who's a worker bee and who's not now, can you?"
Val stabs her shovel into the bag of cement. Uses a stick to check the consistency of what's in the mixer. The woman steps to the edge of the area where Val has freshly poured, looks down into the grey wetness.
"Well, I am a representative of the Neighbourhood Enhancement Committee. And we have had a lot of talk about what you're doing here. And we discovered that though it doesn't violate any bylaws, we would like to know, first, what you're doing, and secondly, if you will stop doing whatever it is you're doing."
Val shovels a mound of concrete in a pile in front of the woman. A dollop jumps onto the cuff of her capris. Val shovels another dollop. Ruining the smooth finish of her day's work.
"We would really appreciate if you could come to our next meeting and talk to us about what this is." The woman clutches her purse to her chest, leans back on one leg, and points a circle around all Val's hard work. After her finger travels all the way around, she places her hand on her hip and giggles.
"No."
"Well, if you decline our offer, again, though it's not illegal, we can petition you to restore your lawn to grass, like all of the others in the neighbourhood."
"What's that crazy bitch doing over there, Val?" Ron shouts over from his yard, holds up a Smirnoff Ice, and points with his ice cream sandwich at the woman in the capris.
Val slops more concrete near the woman's sandaled feet. Then she fires up the mixer before telling her, "Get the fuck off of my property."
Once the woman's scoffed and skedaddled, Val arms herself with her steel trowel, fixing her angry mess. Her arms are defined now, and she catches herself watching her arm, marvels at the bulge of her now very visible biceps almost as often as she marvels at the way the grainy mixture turns into an even line. Everything around her is finally starting to look the way it should, and no one seems to appreciate it.
She stands in the alley past the gate, surveys the whole lot. Every space now filled, some still slick and wet, some dry. Something looks wrong. The edge of the lawn along the street is too green, too much life. The shrub along the front street looks absurd. Exhausted, Val leans on the robot-like compost bin. It won't be right unless she does it. She puts her gloves back on and yanks out every plant by every root.
"Val, you're like a machine," Ron shouts, this time an empty bottle his only prop.
Val waves at him. Almost a thank you.
The skies are getting dark, and the man on the weather channel said a forty percent chance of showers. Val drags out thick sheets of plastic, bought just in case of this emergency, and drapes them over the yard. It looks like she's growing something underneath, after all.
She sleeps well. Dreams of clear skies. When she wakes up in the middle of the night, she hears soft rain. The weatherman was right.
It's like Christmas morning. She gets up early, forces herself to stay in bed just a little longer. To hold out. The rain has stopped. The air smells fresh.
Every inch of former grass an unmoving slab. The wall's not quite tall enough but it'll do. Ron won't be blocked out entirely. But something about that feels okay. Ron's a fixture too. Steady and solid, even though he's not doing anything special, only being annoying. But his voice has been her perpetual companion. Even when she's not looking at him, she knows he's there.
She removes the plastic sheeting like she's unwrapping a gift. Underneath beautiful greys, hard and clean. Let them bring their petitions. Let her son come over any time with his kids, and they can have a picnic here. Let him walk up right now and see her in her cement glory and judge her after it's all done. No one can jackhammer a fulfilled dream.
The August sun beats down. She presses her cheek to the wall. So cool. Rolls her body on the concrete lawn. She lifts up her shirt, the front of her pressed to smooth cement. Skin and mortar, clay. Her heartbeat slows for the first time in years. Beats in a slow rhythm. She hears the rush of water as it cascades down a flume, breathes her to life.
Don't Tell Me What to Do
I probably can't drink another shot. But I do. I drink two more. I put my leg up on the side of the pool table and stretch it, even though there's a sign screwed into the side that says "do not sit on pool table!" with a little clip art old man wagging his cane at me. If Gus comes out from behind that bar, I'll just say that I'm not _sitting_ ; this is stretching, and it's for my better health. Break the rules just enough to annoy.
Gus looks over and makes an angry face. He's old enough to be my dad, but he acts like we were at prom together, close forever. Just because I like music from the eighties doesn't mean we're buddies. I like him, though. He's always making sure the women's washroom is kept clean, and he throws guys out the back door if they're being assholes to me or the other girls who hang out here.
"Alex. We should go home." Robert grabs my hand, but I pull it away.
"Don't tell me what to do."
He knows I hate that more than anything. Being treated like a child, being told what I should do. Like the time he tried to convince me not to bring my half-coffee–half-Bailey's into the movie theatre. No one in this town gives a shit if I want to get a little tipsy while I watch _Spider-Man_.
Robert is old, too. Like, he actually did hang out with Gus at prom. They've been friends for thirty years. They used to live in a basement together before they each got married, then divorced. Robert's been divorced twice. Three times if you count common-law.
I don't care that Robert and Gus both look like they're going to ground me. I run up on stage and grab the mic from the Rocktown Hillbillies frontman. He just laughs, keeps playing "Back in Black." I belt out the lyrics. A group of girls in the back cheer, and I can see them clink their bottles of Canadian together, beer bobbling out the tops. They come up to the front and start singing along, dancing in their short denim skirts and V-neck T-shirts. They're shouting wildly, blocking the small dance floor. Robert is trapped in the corner by the dartboard. I'll come down when I'm ready to drink with these girls, escape into the bathroom to see if anyone has any drugs, any stories, anything fun at all.
"Why did you have to do that?" Gus says. I come out of the bathroom, and he's standing there with a dolly and a keg.
"Why did I have to have a fun time? I don't know. This is a bar." I point to the required poster on the wall telling us to drink responsibly and get a safe ride home. "Where's Robert?"
"He's not a safe ride home." Gus points at Robert, slumped over in a booster seat in the hallway.
I bite at my nails, an old habit I wish I could kick. "I'll have a quiet drink, and we can wait for you to drive us home."
"Can you do anything quietly?"
"Oh yeah." I lean in and whisper in his ear, lips close enough to brush the skin.
Gus drops us off at three a.m. Even though I kept drinking until Gus told me he had to clean the glass I was drinking from, I feel fine. The two of us carry sleepy, boozy Robert from the car, up the steep wooden stairs that connect the alley parking to our back door. I can't get my key to work while trying to keep Robert upright, so we prop him up against the door until I finally get the old lock to click. Robert tumbles to the floor, half inside the kitchen, half outside on the back deck. A rough snort out of one nostril, but he doesn't wake up. Gus gets down and picks him up over his shoulder, walks him into the living room, and sets him on the old sectional. He's stronger than he looks.
Gus eyes up the plywood in place of a countertop. Our kitchen isn't finished. Robert can never pull his shit together, and everything in there is torn apart. We've been cooking with a microwave, a toaster oven, and a barbecue. It's not bad, but the weather is starting to get cold, and Robert hates standing outside in the breeze with his BBQ tongs, and I am constantly charring meat to a blackened crisp when I attempt to grill. I'm not exactly a gourmet cook, but I could put things in the oven if we had an oven. I could make a good tomato sauce if there was somewhere to put a decent-sized pot.
"Do you like what I've done with the place?" I ask Gus.
"Oh, sure. You didn't do this."
"Well, I did put that kettle over there on the floor next to the outlet. How else would I get instant coffee in the morning?"
"I could help Bob with this place. I've told him he can check out my carpentry shed."
I offer him another beer. We might not have much food around, but our fridge is full of beverages. He declines, moves toward the still open door. The fall chill bristles the hairs on my arms.
"Thanks, though," he says.
"Okay then," I say.
Maybe I should have offered him chips and pretzels, my snack-meal staples stashed away at the back of the Lazy Susan.
He's just standing on the deck, staring out at the blue-black night. There are so few lights on, other houses all asleep at this hour. The littlest birds won't be up for another hour or so. No twittering. The crickets aren't even making their quiet racket.
"I don't know what you eat around here," Gus says, "but tomorrow you should come eat at mine. You and Robert. I'll cook. I've got some tricks up my sleeve."
It's gonna rain, and I know he won't want to barbecue in a soaking-wet sweatshirt. "All right, we will. I'll get Robert to pick up some beers."
"No need, Al."
He's the only one who calls me Al. His acting like we're buddies from the glory days, reminiscing about girls we used to pretend to fool around with.
"Okay then. Good night, Gus. See you tomorrow." I'll still force Robert to hit up the liquor store. No sense in being rude.
I reach out for his hand, just want to touch it for a moment. He grips it and gives me a firm handshake. Cups his other hand around too, like this is a serious agreement we've got, an important dinner, where maybe there will be too much beer.
After Gus lets go of my hand, leaves to go across the yard to his own house, I head inside. Robert's yelling for more pillows. I grab two from the spare bedroom and toss them on top of him, place a bucket beside the couch in case he's too lazy to barf somewhere appropriate. He'll pass out again in a minute. I grab the chips, the pretzels, two bottles of beer, and a roll of paper towel. Upstairs in the bedroom, with the curtains open wide, I spread it all out on the bed and eat and drink and stare at the night sky. I wait until the sun crests the mountains, tints the room in orange light, and I can see into the distance.
"Why did you have to make plans without asking me?"
"Gus is your best friend. Fuck."
Robert's been grumpy and on my case since he woke up this afternoon with a hangover, and I told him we needed to get beer for dinner at Gus's tonight. I went on a long walk up and down the streets until I figured he'd be awake and I could use the vacuum. I cleaned the whole place and swept up the sawdust from today's attempt to finish part of the kitchen counter. Even tended to a bloody fingernail he broke while getting angry at a saw. At this rate we'll have a dismantled kitchen until I'm his age.
"Why can't you ever just listen to me?" he says.
"All I do is listen to your mouth sounds all day and night," I say. "You couldn't even carry this beer, because carrying stuff and whining at the same time is too much multitasking."
I knock on Gus's door with a flat palm. He doesn't answer quickly enough, so I burst in, shout his name, " _Gus!_ " and run upstairs, yelling his name. I find the kitchen and put the beer into the fridge. Robert slowly clomps up the stairs far behind me. I've never been in Gus's place. For a single guy, it's nice. New fridge, nice dining room table, antique china stacked in a glass cabinet. He's even got photographs on the walls. Never had kids, but his old dog is framed in the living room, along with beautiful bucks, wolves in the meadow, and a large print of the waterfall on the other side of the mountain.
Robert's kids don't talk to him much. I've never even seen them, except once, accidentally. I was driving by the mall along the highway, and Robert was there waving his hands around, trying to wrangle blurry shapes into a car.
Gus comes upstairs with a bottle of wine and pie in a box. He's wearing a button-up shirt that looks ironed, not one of the free T-shirts that comes in a beer box that he usually wears to work.
"Hi, Bob, Alex. Sorry, had to grab a few supplies."
Robert ignores us both and heads to the fridge for beer, grunts something to himself, and gets comfortable on the couch. He probably wants to sit in front of the TV, watch that reality show about hand-fishing and not say another word for the rest of the night.
I turn away from Robert, and Gus motions for us to walk out the sliding glass door. The table is set on the back patio, blue plates, cutlery, wineglasses. Gus has made himself a gorgeous patio set, an umbrella rests in a stand, a firepit of cinder blocks in the shape of a perfect circle. I take the wine from Gus and screw off the top, pour each of us a glass.
"Don't you want a beer?" Gus asks. "I just decided to try this out. Since we're eating deer steaks for supper."
"No, I'll drink this. Save the beer for Robert."
Gus must think I'm like Robert, drinking the same old thing day after day, year after year. Never getting bored of anything when, really, I'm bored half the time I'm not at the bar or working at the Esso. Even at the Esso, I'm bored half that time too. Little halves of my life spent staring into space, trying to figure out how not to be bored.
I sniff the wine like people do to get a sense of it. It smells like the wooden boxes of berries we used to get from our neighbour every summer. I pull the sleeves of my hoodie up around my fingers, grip the wine glass with a cotton-bundled fist. I sip it. Try to take it slow.
"I know it's a bit cold, but I can put on a fire." Gus steps to the firepit, arranges kindling.
I set the glass on the table. Shake it to test how sturdy it is. Solid. I think about what it's like to do things for yourself like that. To know how to make things happen, how to hammer and nail, make something out of little parts, scraps, and pieces. I gulp my wine. The flavours are sweet and tart, and I'm liking it. Maybe I should start making my own wine. I'd be good at selling alcohol.
"I'm starving!" Robert shouts from the living room. He never did manage to put the TV on, just sitting alone with his can of Blue like it's his best friend.
Gus heads inside and by some miracle comes back out ten minutes later with Robert holding a fresh salad. Gus has a bowl of peas and rice and a platter of something juicy and delicious. Robert sits, and Gus takes each of our plates and sets the steak and rice in pretty portions. The food looks even more tempting on the blue plates. I wonder if he thought of that when he was cooking. How to make food pretty.
"Tasty," I say, forkful of rice in my mouth.
"Yeah. Gus, you can cook. No shitting you. Like good." Robert ignores his vegetables like a child, swipes bread into the juice.
We eat quietly. Because it's good food, and because sometimes the three of us together don't have that much to say. Behind Gus's house there are other small houses and their yards. Kids throw a basketball against a garage door. A man collects his children's mess of toys. An old woman snips at her vegetable garden with nail scissors. Some yards empty, just tools and chairs.
"Any excitement with you guys?" Gus asks.
"Besides the renovation, not much. Though that's keeping me pretty busy." Robert says as he fingers the meat, then drops a slice back on his plate.
"It's so boring around here lately," I say, staring at the woman clipping leaves and weeds.
"It's a slow town." Gus fills my glass from the rapidly dwindling bottle of wine.
"There's plenty to do," Robert says, opening another beer.
"Oh yeah?" I say. "There's nowhere fun to go. It would be nice if there was something big around here."
"You gonna talk about that mall again? How can a mall be interesting?" Robert says.
When I was a kid, my mom took us to West Edmonton Mall while we waited for my dad to come back from working up north. This was before he got his job at the pulp mill and lived with us full-time for two years, before he moved one town over and started another family, before my mom gave up on trying to enjoy her life. Before she stopped caring about living.
For three days, my sister and I ate fried foods in the food court and went on waterslides and rides. And, of course, we got cute new things. Discmans, CDs, cool outfits. My mom let us do whatever we wanted, because my dad was coming home with money and we didn't have to worry. We stayed in the Fantasyland Hotel. The room looked and smelled like a tropical island, like a far-away vacation, like someplace beautiful people go to feel even more free.
"That's a nice memory," Gus says.
"It's a stupid mall. Girl dreaming of a mall trip," Robert says, sloshing beer onto his jeans.
"Yeah, that's me. I'm a stupid girl."
"I used to love taking road trips as a kid," Gus says. "The only time we ever got to go anywhere or do anything." He smiles and hands me a slice of bread.
After dinner Robert demands we leave before we even get to the pie, and when I refuse he takes off into the bathroom with a woodworking magazine. Hopefully he'll learn something useful while taking a shit.
"Does wine go with pie?" Gus asks.
I'm a bit tipsy and say, "Wine can go with anything."
"I have one more bottle downstairs in the laundry room."
"I'll go get it," I say.
Downstairs is even cleaner than upstairs. The laundry room doesn't have a thing out of place except for the wine bottle sitting on a box of Tide. Before I head upstairs I snoop in the bathroom, nothing interesting. The main room in the basement is nearly empty. An old stereo on the far wall, speakers in each corner. I walk into the room on what has to be shiny, new wood floors. I slip around on the wood in my socks. Do a skating spin I learned in that one year of lessons. I check to see if things are plugged in, what kind of music Gus puts on in his private den. I drop the bottle of wine, and thankfully it doesn't break. I bend down to pick it up. Kneel a moment, a little light-headed. Perfectly embedded in the floor are circular pieces of metal. I trace my finger around one of the metal circles.
"Are you okay?" Gus is standing in the doorway.
"Yeah. Fine. I dropped the wine." I hold up the still intact red. "It's so neat and clean down here."
He comes over and tries to help me up, but I just stay sitting on the floor, touching the metal, fascinated by the design.
"Come on. Let's go upstairs." Gus seems anxious.
"You don't want me in your precious room? What are these?" I sprawl my body on the floor and reach around me, grab at all the metal rings.
Gus inhales deeply and stands over me. For the first time since I've known him he seems sort of powerful, like I should listen to him.
"They're handles."
"For what? Are these trapdoors?"
I scramble to a sitting position and try to pry one of them open. My long nails make it hard for me to get a decent grip. Gus sits down beside me and pulls on one of the rings. A piece of the floor the size of a cupboard door swings up on an invisible hinge. The wooden floor-doors are so perfectly lined up they look decorative and not functional. Under each is a perfect square hole, also made of wood. Each one houses treasures. We go through every organized section. Neatly organized boxes of photographs. Hockey cards. School medals. Letters from his ex-wife. Metal cash boxes.
"This is incredible. Like, I can't believe this," I say to Gus. He's behind one of the open doors, and I can only see his eyes over the top of the shiny wood. We probably need a third bottle of wine.
"The house is small and I hate clutter. Just wanted somewhere to keep all my things. I got new furniture for down here that'll be delivered in a few weeks. Then this will just look like any other den."
I peek around the side of the door. "What's in those cash boxes?"
He doesn't answer me, examines his impeccable craftsmanship. I crack open the bottle of wine and take a long drink, offer it to Gus. He takes the bottle but doesn't drink. We're sitting on the floor like it's kindergarten show and tell, passing around an adult juice box.
He starts to close the doors one by one. When he gets to the cash boxes I hold it open with all my strength. He tries to move my hand, but I won't budge. I squirm over as close as I can to him without losing my grip. Our bodies are close, and I lean in and bite his ear. He lets go. We kiss, my hands still holding the door open, his hands on either side of him, no other part of us touching. He pulls back and one by one opens the metal boxes. Inside are rolls and rolls of toonies. Hundreds of tight little paper coin rolls bundled together like shiny silver in sleeping bags having the biggest sleepover.
"Holy shit," I say.
I take another swig of wine. Gus does the same.
"Okay. But why so many toonies? A bank doesn't even have so many toonies."
From upstairs, Robert is yelling. I drink as much wine as I can in one gulp.
Gus carefully closes every door and walks over the smooth finish.
"Let's go up, Alex." Gus grabs my hand and doesn't let go until the last possible second before Robert will see our fingers entwined.
At home Robert's on the floor this time. He stumbled in the door before me, and I'm full of wine. Without someone to help me move him I just fold a blanket over him, prop his head up on a stack of magazines, and then get my chips and pretzels, two bottles of beer. It's started to rain. The sunrise won't be visible, but the beats of the rain will hopefully lull me to sleep before then.
I lie. All week. As often as I can, I lie to Robert and go to Gus's on my lunch break, or on my days off. Robert's on shiftwork, and he doesn't pay that much attention to me anyway.
Starting on Monday, I just pop over and he's not even there. I walk into his backyard and sit by the firepit. Stare at the even blocks, the black ash from the fire a few nights ago. I check the basement door. Locked. He must be the only person in town who locks their door. I walk up and down the streets until I get to the bar, join the girls drinking beer late on a work night. We clink our bottles in cheers. Gus isn't behind the bar.
The next day he's home, and I invite myself in. He makes us coffee, and we lay on the living room floor together mostly not talking, barely touching the outside of our hands. His are the slightest bit clammy.
The next day, I walk over at eleven in the evening. Gus is sleeping, but he lets me in and we watch a talk show, and I slink into his lap. After he falls asleep I grab his phone and take a few selfies, only two with my shirt pulled up, my lips puckered in perfect duck face.
Then on Thursday I bring over a bottle of wine, and as soon as I get in the door I take off my shoes and my shirt and walk him straight to the bedroom.
Gus holds me around the waist. It almost seems like he's going to cry, that feeling of his chest quivering a little against mine.
"Bob treats you like—"
"You don't have to say things."
I take his rough hands and put them on my cotton bra cups. He doesn't curl his fingers around them, palms flat, fingers stiff and straight. He looks up into the old ladyish light fixture, a dim bulb.
"He's been my best friend since grade three, and he biked home to get my dad when I fell thirty feet out of our tree fort. I was best man at both of his weddings. And I know he's trying. That last divorce, last set of kids, it wore him out. You must wear him out."
He finally stops talking. I grab the back of his shirt and yank it off. As I fling the T-shirt off, his hands come apart from my boobs for a second, but he places them back right away, like he doesn't want to offend me.
"Move in with me. You can take the spare room even. He's my friend, but I don't want to see you in that house with him. He's confused and just can't be responsible for another woman right now. Do it. Listen to me."
We make out with no tops on, like two teenagers scared to take off any more clothes because someone's parents should be getting home any minute. He's the one who stops, tells me not to take my jeans off. He takes one hand and reaches into his jeans pocket and folds two newly cut keys into my palm.
Things move fast with these middle-aged guys. They're used to being married, even if they've been divorced for ten years, like Gus. I fell asleep in my bra and jeans, and now it's well beyond sunrise and I've been out all night. The keys to Gus's place rattle around in my almost empty purse. It's my day off, so I head home to sleep all day. Robert's already at work when I get in. There's a full pack of empties on the deck.
I wake up at three in the afternoon and look into my open closet. The rolling suitcase I moved in with is still sitting there, underneath a pile of T-shirts and shorts from when it was warm enough to let my legs breathe. Four months ago, I'd thought this wouldn't be the great romance of my life and that it wouldn't last. And I was right about both of those things, but I had my own bed to crawl into, and the house wasn't far from work. When I moved in there was a kitchen countertop, and Robert smelled of the fruity woman's body wash his wife had left behind six months earlier. He was raw but sweet. That must have been what both of his wives fell in love with. I didn't fall in love with anything. I liked that he was always up to meet me for a drink, didn't question why I never went back to the house with my bitchy roommates, didn't say a word when I rolled my shit into the bedroom upstairs and didn't leave.
When Robert storms into Gus's wood shop that evening and finds his arms around my waist, his hands on my hands pushing a slab through the planer, I almost let my hand go through too. Gus steps back, and I pick up another piece of wood, tap it against a pile of sawdust, send a little cloud into the air. I know just what to say.
"I thought Gus could teach me a thing or two, so I could help you with the renovation."
The buzz of the wood getting smaller is probably what my voice sounds like to Robert's ear right now. His face tenses, one eye shrinks to the size of a dime. That's how angry he is. He's going to lose an eye to rage.
"I don't need help. You insulting me?"
Gus jumps in. "I'm not insulting anything, Bob. Alex just wants to learn."
"I'm not blaming you, Gus. This one, she is always trying to make me look stupid."
"The kitchen has been in ruins for three months," I say. "Maybe you need an expert. Like Gus."
Robert kicks the sawdust pile. Gus turns off the machine. The only sound is everyone breathing heavy.
"Maybe I could help you," I say again. "You don't let me do anything."
Robert moves to grab my arm, but I slam the two-by-four against the floor. I slam it again, and again, then drop it. I look at Gus first. He's staring at me, looks tense, but he doesn't move. Robert stares at the unmoving band saw in the corner. I walk over to Robert and grab his arm, lead him out into the backyard. I look behind me, mouth "one hour."
I convince Robert I'll meet him at the bar. That I need to shower all the dust and wood and sweat off my body. I strip in the hallway and promise I won't go behind his back again, that I won't get in his way with construction. That he's got this. I turn on the shower and wait for the clunk of the door shutting. I turn off the water and run upstairs, put on my favourite jeans, T-shirt, and hoodie, and grab my suitcase waiting in the closet.
Robert's car is parked up the street because his construction supplies take up all of the carport and part of the alley. He doesn't lock the doors. The spare car keys were still in the fish-bowl in the unfinished basement bathroom.
I park a street up from Gus. He's on a steep hill. When I get inside he looks relieved, like maybe I wasn't going to show up. His arms wrap around me, like he's trying to protect me from something. He just holds me there for a long time. I push my open eyes into his chest, look at the plaid fabric of his shirt up real close. Then he's grappling with my belt, and I guess he was protecting me from his raging boner, which I feel through his jeans and which probably got harder because he was holding me so close. He was too awkward in the bedroom, so I wrestle him to the floor of the living room. Our naked bodies bristle on the carpet. Gus is so gentle. His hands roll over my skin so lightly that he might as well be across the room. He moves like I'm made of glass. I imagine Robert judging us both for being total sex wusses, for not rattling the house until it falls down. His delicate thrusts are too respectful.
"Tell me about the toonies," I say, biting his ear.
"What?"
"Tell me all about your fat rolls of toonies," I whisper. Grind against him.
"I have so many toonies," he says. Flips me over onto my back and takes control.
"So hot," I say.
"How much money? Tell me about the money." I roll him back over so I'm on top.
He just groans, and I pull back, slow down. "Don't make me stop. How much, Gus?"
"Over thirty-five grand." He groans again, and I speed up.
"Holy fucking shit." I rock him to completion and collapse.
"We're like two loonies that make a toonie," he mumbles.
"Thanks, Gus. That means a lot coming from you."
I go to bed before the sunrise for once, Gus's arm draped over me and his clean striped quilt. I try to wait to see slivers of light from behind the curtains, but my eyelids dip and I sink into sleep. His bedroom window faces north anyway.
In the morning, Gus has to go in to wait for deliveries. I'm pretending to be asleep. Before he gets out of bed he strokes my cheek, rubs each finger over the apples, and doesn't say anything. It's nice, and I keep my eyes closed, stay so still. He kisses my fingers and quietly puts on his Labatt's T-shirt and jeans and slips out the front door.
The second he's out the door I toss the covers off myself, dress, and head downstairs with my suitcase. The main room in the basement looks even prettier in the morning light, the wood grain shining. Gus must have just polished the floor. In the front pocket of the suitcase, I've stashed Robert's screwdriver. The one he threw into the yard when he couldn't quite get a light switch cover to cooperate. I thought I'd remember exactly which hidey-hole had the toonies, but I pop open two with the screwdriver before I see the matte grey cash boxes.
Each roll feels heavier than a can of beer. I reach into the first case and fill each hand, let the money sit there, perfectly neat and packaged. They look like cute little gifts. I can't take them all, but I fill the case so I can still actually carry it without putting my back out. No time to count, but I probably have eight grand. I close every cash box, stack them back in order, close the wooden lid, and rub clean every surface I've touched with the sleeve of my hoodie.
I totter up the hill with the heavy case dragging behind me. It feels like it barely has wheels at all as I scrape the bottom along the dusty concrete, keep having to set it upright as I struggle to get it to the car. I grab seven rolls and drop them in my purse. Knees bent, I inhale deeply into my lungs, shove my arms underneath the bottom of the case and get it halfway into the open trunk. Sweat drips into my eyes. I put all my weight on the case to keep it from falling. A nice man walking his dog offers to help me as I heave the case into the trunk. I decline, say fuck a few times, and topple my whole body onto the suitcase, which thankfully ends up thudding into the trunk. Then I pat his pit bull and climb into the driver's seat.
I try not to speed, even though all I want is to get somewhere fast. If Robert reported his car stolen, then I'm in big trouble sooner. But he hasn't been driving lately, since he's been drinking so much, and he walks up the hill to work. He's supposed to pick up his kids on the weekend, so hopefully I can get three days of driving in before he gets super suspicious. He might bail on those kids anyway. He's reliable that way.
First stop is the mall in Cranbrook. New outfits. I need some. Why bring clothes when your goal is to steal a shit ton of toonies that you can spend, spend, spend. New jeans, not on sale, that cost over a hundred dollars. Slim T-shirts that fit perfectly. A black leather jacket and a wool jacket the colour of fresh blood. I get two dresses, decide I will treat myself to a sit-down dinner as soon as I get to West Edmonton Mall. One pair of heels, one pair of sneakers, and a bigger black purse made of real leather are the last things I buy before taking a quiet minute in the women's washroom. I come out wearing my new black jeans, grey T-shirt, and leather jacket. It looks like a sexy criminal's uniform from a movie. Underneath I'm wearing new underwear, black lace everything.
I buy myself an iPhone, fling my shattered android into the trash can along with my old clothes, before I head to the food court. I toss two coins to the kid at Taco Time for toonie Tuesday and spend another at Dairy Queen for an Oreo Blizzard. I stop by the liquor store on my way out and buy a 26 of rye. I keep my favourite hoodie, put it on the seat next to me as I drive, an old friend. We're on the lam together.
At the motel, I turn on the television. I didn't want to start getting too fancy until I got to the real destination, but the place had a big sign flashing free cable, and I'd almost driven the car into a semi two hours ago. I want to be fresh when I get to the mall, and a few hours of sleep would help. But when I crawl into my glamorous Roman-themed bed, I won't want to get out for days.
The pop machine outside the lobby is too old to take toonies, but luckily I've got other change from my purchases. I load up on ginger ale and ice. Drop another buck for Hawkins Cheezies. I fill the sink with ice, unwrap a plastic cup, and fill it with rye, spritz the top with ginger ale. Then I put on _So You Think You Can Dance_ and strip my new clothes off to the theme music. On the edge of the nightstand, I crack open two paper rolls like eggs and spray the silver and gold coins onto the bed. In my new La Senza underwear I jump on the bed, let them ting and clink around me. I roll around, let the coins stick to my skin, hold up handfuls and let them slap down on my body. A muscular, young black guy and a tiny white girl dance to Drake. "Started from the Bottom (Now We're Here)." I laugh and crack open another roll, let the coins cascade onto my naked belly.
I know I should get to a bank, change this money into paper. I wonder if Gus has these rolls marked somehow. Even though he's as organized as hell, I don't think he ever imagined anyone would steal his toonies. All night, the TV flickers without sound. I dream of a bright, open sky, purple- and pink-like flowers suspended in air.
The Fantasyland Hotel won't take cash. I need a credit card. A woman with makeup that looks like a sprayed beauty queen mask tells me this, and I try not to punch her fake eyelashes off her face. My new purchases and the money roll behind me in my old suitcase. Now that I've been denied Fantasyland, the little plastic tires feel like they're leaving a trail of black tar behind me. I won't be going upstairs into a Roman bath, with a glass of booze and a fluffy robe waiting on the arm of a marble statue.
I walk by the huge replica of Columbus's _Santa Maria_. Nearby there are submarines indoors, and that makes me still think that anything is possible. I sit in the food court to think. I order a Coke and a giant tub of fries smothered in everything, and I eat it all, even the slimy green onions. It's exactly what I wanted: creamy, salty, spicy, crunchy, gooey. As many flavours and textures that could fit into a paper bowl, be scooped with a plastic fork, washed down with bubbly sugar.
I'd heard stories about people living in the mall, that there's an entire task force of security monitoring and trying to remove people from here. A guy told me that a girl lived for two months in a condemned storage room at the Brick. He also said she ate Cinnabon every day, but that seems unlikely. It's easy to get tired of Cinnabon. That's not what I want to do with myself, but I could see a few days of crashing in here until I get things sorted out.
I find Bourbon Street, with its neon lights and bright colours, and head straight to the Tony Roma's. Rolling up with this suitcase not only looks more ridiculous the longer I do it, but it's also starting to take its toll on my arm. The throb pulses down to the wrist. Every ten minutes I'm switching hands so my biceps won't be uneven. I don't want one bodybuilder's arm. That's how stupid people get caught. And by spending dozens of toonies like it's a normal thing to do.
"Table for one please, ma'am." I smile at the hostess as though I'm selling something.
"Okay, I could do for six p.m. or eight p.m." This girl in her tight black dress isn't smiling, just looking cautiously over the long sheet of names.
"Six. I'm meeting someone later. My husband. That's why I've got this suitcase. I forgot my credit card, so I guess he needs to check us in."
"Mm hm. Okay, six will work great. Now, did you know that there's a place to check your bag? You could always leave it in one of the storage lockers."
"Oh, it's full of very precious cargo. Personal items that it would be impossible to replace. It'd be a real tragedy." I thrust my huge purse onto the desk, and two rolls of toonies jump out and bang into a stemmed glass of water. The girl grabs the glass and lets the rolls of coins slam to the floor. I bend to pick them up, toss them into my dangling purse.
"Mm hm. Well, that's fine then. We'll see you and that suitcase at six p.m." She smiles and tilts her head like a dog I should give a treat to.
Instead I just smile back like I'm the fanciest fuck in the world and hoist my heavy coin-filled purse onto my shoulder. I have more purchases to make. Accessories, makeup, a box of Band-Aids in case my feet blister in my new shoes. And I have to figure out where I can get a credit card, how I can sort this money into paper and plastic before it gets me in trouble.
I check the screen on my brand new iPhone. Almost four p.m. I need to find a bank. I walk a huge section of the mall, first just trying to find one of those ugly directory maps. I pass the ice rink, keep walking, almost stop in another food court even though I'm holding out for my Tony Roma's ribs. Dolphins, fountains, huge mall sculptures. The amusement park looks too shiny, and if I didn't have to take a suitcase full of toonies on with me I'd ride the Mindbender, shake loose some of this stress. When I do find a directory, I'm overwhelmed with how huge the map is, how far away a bank seems to be from where I am, how the mall seems so much larger than my whole town. How can there not be more banks, when some of the same stores are listed on the map three times?
I root around in my new purse to make sure my wallet is still in there, then race toward the nearest bank. My arms and legs are starting to feel tight and hot. Teen girls in groups block my path, and I slip past them. My story is that I have an eccentric aunt, and she gave me all of these toonies. A family of six with two double strollers takes up all the room between kiosks, but I dodge them. I whip through a group of old people getting their mall walk exercise. My story is that my eccentric aunt died in an eccentric way. All her ceramic figurines crashed on her in her sleep and suffocated her, and I inherited a huge chest from her home, and it was filled with toonies.
I can see the warm green of a bank logo in the distance. There will be people in blouses and button-up shirts in there whose jobs are probably to ask a lot of questions. The only bank account I've ever had was opened for me by my mom at the local credit union when I was fourteen. My story is ridiculous. I _wish_ I had a rich, eccentric aunt.
I backtrack to a money exchange I passed earlier and get them to change six rolls into six fifties. An amount that doesn't seem too suspicious. Like, maybe I just emptied a piggy bank I've had all my life. My ankle is blossoming into a blister, so I sit and watch people go up and down escalators. Then I remember I have a phone that works, and I add more apps. By the time I make it back to the bank, people in blouses and dress shirts are pulling closed the metal gate, locking it up for the night.
With an hour until my fancy dinner I shuffle through the mall, buy red lipstick, liquid eyeliner, and a rhinestone necklace that looks like it could choke me. I take everything into the washroom with me, get ready while women have a last-minute piss before they go to their waiting homes or hotel rooms.
I tuck a tissue into my shoe to deal with the pain and, with no real plan, walk to the hotel. Maybe I can leave a cash deposit, pretend my husband is trapped in a tiny airport with no way of getting out for the night. Then I could get everything figured out in the morning.
When I walk into the lobby, there's a new woman at the desk, hair long and shiny. And she's talking to a man who's got the exact same tiny half-moon bald spot as Gus. My face tightens. He could have flown here this morning. Or afternoon. Two flights a day into Edmonton from that shitty airport. I turn and almost take out a toddler with the suitcase as I rush away as fast as I can.
In my new heels and the sexier of my new dresses, black and backless, I sit in the lounge and wait for my table to be ready. My suitcase is stored underneath the bar, my legs clipped underneath, crossed at the ankles, always in contact with the full bag. If Gus is in this mall, he won't know where to find me. The hotel was his only clue.
The guy at the M A C counter had given me a whole new face, and I paid for every tube and tub, brush and bronzer he used. He was so pretty, and he made me look like I oozed glamour. The lipstick was called Pure Fantasy, and the orange hue made my lips like a summer sky. I puckered, took a selfie with my new phone. I wish I could Facebook it, but that would be too obvious. I'm no genius, but I know I can't go doing that.
As I sip my cocktail, the happy hour special with a shot on the side, two mall cops stroll by, their little radios buzzing away with words and codes that don't mean anything to me. I should have spent more time with Robert, listening to the police scanner. They ignore me. I kick the suitcase, but it's already as far in the dark under the bar that it can be.
I hope those two assholes didn't conspire together. If Gus showed up here alone, that's fine. I could take Gus ratting me out. He deserves to. And I bet he'd still forgive me. That loveable bastard. But Robert. He owes me. A car is a small price for that juicy slice of my youth, his bitchery, the sounds of his unskilled carpentry.
The same girl with the black dress and mm hms from earlier comes over and tells me that my table will take a little longer. Asks if I want an appetizer. I do. I dip calamari in creamy sauce. Deep fried seafood is a perfect food group.
Now the mall cops are on the other side of the bar. Everything in here is so visible, so easily identifiable. Even though it's still early, I feel as if I could fall asleep already if someone picked me up and tucked me into bed.
When I was a kid, I remember going into a women's clothing store here. It was the first time I wasn't in a store for kids or a department store. The clothes were so adult. I could picture women's bodies filling them out, their skin peeking from the top and the bottom, hair cascading down backs to meet zippers. This dress I'm wearing, these shiny green shoes, they're what I pictured all women wore. I was allowed to buy one outfit, and it was too big, but that didn't matter. The blouse was cute and professional, the little skirt short and flirty. I wore it to school, even though everyone else was in crewnecks and jeans that fit them properly. I loved how I looked, that I'd picked it out myself. When a girl made fun of me after gym class, called me a lady loser, I punched her in the leg. She screamed that ladies don't punch, and so I punched her other leg to prove that they did.
The mall cops are talking to the Mm Hm Girl at the front; she moves her arm to point. I'm staring straight at her. She motions for them to go behind a partition. I take my shot. I drop calamari down my throat like I need it to live. I order another drink to calm my nerves. The girl goes back to her podium to greet people. I don't see mall security.
Robert's car isn't down in the mall parkade. I was too worried to keep it close to me for another day. Parked it by the side of the road and walked in my leather jacket, even though I was freezing. Looking cool feels good, warms the soul. Good riddance anyway. Don't need anything of Robert's anymore. Sweet Gus's toonies are with me. Shiny starts to a new life. The mall is so anonymous. No one should know me here, but if I don't get these toonies sorted out now my life won't ever change. If I get caught though, this time I won't be able to snake my way out of it.
Mall cops walk out the front door, and I'm glad I got another drink; I take a triumphant sip. The Mm Hm Girl is talking to someone else now and looking down at a phone. Out from behind the partition steps a man and his bald spot. My chest squeezes my heart, my lungs. It might be easier to be alive if you knew you could force your own ribs to puncture your organs at will. Is it Gus? It has to be. Even from a distance I can see he's concerned. I can tell that look in his eyes, his look of wanting to help.
I slink along the short wall separating the booths, ask the bartender where the bathroom is. He points into the main part of the restaurant, near the kitchen. Ignoring the tall wooden door with a sign that says "ladies," I scoot into the kitchen. I run in my heels, pulling the suitcase, don't look or stop as people shout that I'm not supposed to be there. I click through the kitchen until I'm out the back door, in a long hallway that leads to the back rooms of other stores. I wait quietly behind a cart of folded T-shirts until someone comes along and pulls it through a door, then I wedge my toe to keep the door open and slip inside.
I bump into a perfectly put-together woman in chinos, with a walkie-talkie.
"You can't be back here," she snaps. "This is our stockroom."
"I'm so sorry," I say. "I got lost on the way from the bathroom."
"It's strange how often this happens. Come out this way, please."
She leads me through a room of bags and belts and out into the store, and I thank her. The suitcase is wearing me down. If I can get enough money into my purse I can ditch it. Earlier I should have exchanged more rolls for paper cash.
Everything is brightly lit, so clean and fresh. I probably need a new outfit and to get out of these heels and this dress. If I need to run again, I should throw my hair into a ponytail. Grabbing as many slim pants and denim and cotton things as I can, I head into the fitting room and lock the door. I decide I'll spend as long as I can trying everything on, looking at my new appearance. I struggle to get out of the dress because the zipper's stuck.
"Fuck, fuck, fuck, fuck, fuck, fuck."
The fucks don't release the teeth of the zipper from the lining of the dress. These pants would look so chic and slim; in these shirts I could look like anyone else, someone on vacation somewhere hot.
I start to peel off the black dress, pull hard to get it over my head, but it tears as I bring the waist over my shoulders. I hang it on the nice wooden hanger, frayed fabric and loose threads burst from the rip along the middle. I didn't look at myself enough times before I ruined it, and now I'll never get to wear it again. Maybe Gus had seen me from the back as I slipped through the swinging kitchen door. Admired me as I escaped. My ass in a tight black dress, waving goodbye. Maybe it hadn't been him at all.
Except he probably won't just let me leave like that. Came all the way here looking for me. Fuck him for actually listening to my dreams. At least Robert had the common decency to ignore me.
The lightest-coloured khakis are the first thing I grab. I pull them on along with a tight, white V-neck tee. It's so clean and fresh. In the mirror, I see a different girl. Definitely someone who's not me. A person who orders coffee with very specific instructions about temperature and caffeine and foam. Someone who owns a forty-dollar water bottle. Someone who knows how to put on all of this makeup perfectly herself, every single morning.
My eyes feel fuzzy. "Fuuuuuuuck," I intone quietly.
An employee knocks lightly on the door.
"Hi there. Are you doing okay with everything? Do you need any other sizes?"
"No thanks." I try to sound like a chipper shopper.
She shuffles away to help other customers. I push all the clothes I haven't tried on into a pile on the floor. I roll myself into the pile of pale shirts and soft denim. This is a brief rest. I wish I'd taken a few more bites, one more shot at Tony Roma's. I could have savoured the flavours while I waited out that very nice man I robbed. The one I don't want to be with, no matter how kind he is, how much he cares about me. He looked at me like it was hard to look at me, like he felt too much. The fuzziness in my eyes feels like when you're in a dream and you can't cry, but you think if you could then you'd be okay after all.
I stretch out my legs, knock the suitcase over on top of my legs. It hurts. I kick it off and sit up. Blisters on my feet, and now there will be bruises on my legs. In the mess of clothes, I unzip the suitcase, drop rolls of coins into the purse. To make sure I'll still be able to walk without crashing over to one side, I stand up and walk around in the little square of the fitting room, do a jumping jack. I add a sweater and a dark denim jacket to my outfit. Only my scraggly hair still looks like me.
On the walk to the cash register, I push the suitcase behind a rack of men's plaid shirts. It physically hurts to let it go. I buy a V-neck in every colour, a pair of jeans, plain canvas shoes. I explain that my dress ripped and that I'll need to wear these clothes out. I pay for them with toonies, my last big toonie hurrah purchase, and I ask if I can go back into the change room to undress and cut off the tags. She loans me a pair of scissors.
After I've taken off the clothes, cut the price tags, and handed them over the door, I look at myself in the mirror. I whip my hair into a ponytail, gathering it all in one hand, and hack at it with the scissors. I stuff the hair into my purse, heavy with toonies, and zip it up. I layer myself in new clothes, two full bags of things that I would never normally wear sitting on the change room bench beside me. I grab them and my purse and head out the door. My shoulder throbs with the weight of it all, but in the fake light of the mall I feel light, fresh, and ready to run.
Miss Supreme
At the hair salon, Carol Winter looks at Sashay as if she were a dollar store Barbie. She smirks at the little girl and struts out the door, fingers flipping her fresh bob before Linda has a chance to tear out a chunk of Carol's newly tinted hair. Sashay did her own styling today—swiped mascara across her lashes, dabbed sky blue along the lids to spark her grey eyes. A smear of Tulip Bliss lipstick dots one of her teeth, small, crooked nubs with gaps between. The whole look is charming. Carol sucks, judging a child with her eyeballs like that.
"Can't you make her hair any bigger?" Linda asks.
Rhonda, their long-time hairdresser, sighs, brushes an errant tendril behind Sashay's ear. Bulbs of curl sprawl across Sashay's head. Butter blonde strings hold a can of hairspray, but still, her scalp is visible, hair too thin to tease to high heaven. Linda knows this, that Sashay didn't inherit her full locks, got her father's wispy hair. Sashay flicks her tongue at her reflection.
"With extensions. Or a full wig," says Rhonda.
"Sashay do you want a wig or extensions?"
Sashay grabs Linda's hand and pets it, gives it a wet kiss.
"I want big hair like Mama. I want to win. Are we almost done? This chair is making my butt itch."
Rhonda's committed to Sashay's hair and prepping her for the Miss Supreme Pageant. They've only ever participated in small-time local pageants before. This one has some prestige. It's in a hotel, not the legion hall, in a big city, with 150 girls in mascara and expensive gowns. But Sashay is loveable and spunky. She just has a hair problem.
Rhonda did Linda's hair for a pageant in high school, a French twist, which looked simple and clean but wasn't bold enough. It paled next to the other girls' bouffants, not to mention their rhinestone-encrusted dresses and bleached teeth. Linda placed third runner-up in the Miss Jewel City pageant. No money, a small tiara, the smallest throne at the back of the float. She'd lived in the same town ever since, with the third-best lawn on the block.
The last of Linda's vacation money was sunk into the dental flipper, a set of perfect teeth to fit over Sashay's missing baby and new grown-up ones. It's something they can't do without. Glitz pagaents mean no gaps. Kevin's health benefits are pretty good, but the Chiclet-white fake teeth are considered a nonessential. The insurance provider didn't think her argument that they were for Sashay's career counted as a medical reason. Instead of a trip to their family dentist, she drove Sashay in secret to a specialist while on a dress-shopping trip. Sashay can't be expected to parade around on stage with her real teeth. She might as well get on stage, pee in a sandbox, and smear melted crayons on her clothes.
At the till, Linda hands over a bottle of homemade Chardonnay and some blondies, the only appropriate baked good payment for a hairdresser. Rhonda hugs Linda and the bottle. Linda has time to figure out what barter to bring next week for her own bold highlights and cut. Thin as it is, Sashay's hair is the perfect flaxen shade, and Linda dyes her own to match. Still in the hairdressing gown, Sashay jumps out of the chair and runs into Linda's arms, scattering snips of hair everywhere.
Kevin shakes her in the middle of the night.
"What is this bill?"
The flipper. Linda pretends to sleep. Holds the blankets to her face tight, almost suffocating. She imagines his pride, seeing Sashay hold her bouquet, crown sparkling on her head, the $500 prize money fanned out in front of her perfect little face. A winner.
"I thought you said the dress was $300?"
"Did I?" It's not the flipper bill.
"This says $350. The pageant budget is out of control."
She jumps out of the covers, skitters to the edge of the bed. "I know. I'm sorry. She also needs extensions."
"No. She can use her real hair."
"But—"
"No, Linda. No."
Hands clenched to his boxers, she tugs them down, works her tongue and mouth until he says yes. The next morning she gets up to find a note. _Sorry to be tough on you. No more money on this pageant_. He's left thirty dollars for groceries.
Sashay was an angel at the spray tan salon. The ladies gave her a special rate and a treat. She munches the Snickers Fun Size in the doorway of the post office. Linda pops open the PO box to find a small package. Sashay's eyes get big when she sees the box.
"This is for you, Sassy."
"Is that earrings, Mama?"
Since getting her ears pierced for her fifth birthday, Sashay asks for earrings every time they enter a store. At the supermarket, at the bakery, at the hardware store. Her eyes pop like a lizard when she sees shiny things and tidy boxes. The dresses she wants always have the most sparkle.
Linda opens the box. Nestled in padding and a sealed and sterilized plastic bag is Sashay's flipper.
"It's not jewels. It's your flipper, princess."
"Ooooh! I wanna try it on."
Wide mouthed, Sashay waits, tapping her toes in rhythm with her talent routine dance number. With a steady hand, Linda tries to lock it in place. It fits. Cheeks stretched in a near smile, Sashay grimaces.
"Your face will get used to it. I promise."
"Hi, Mrs. Shue," says Sashay to their mailbox neighbour, slurring, spitting, and practising blowing kisses.
"Hello, there."
The woman looks down at the fake teeth and up to meet Linda's eyes. "Your daughter does those pageants."
"Yes."
"These are my new teeth!" Sashay hisses.
"I see that, Sashay. Like that reality show. So offensive. No offense. My niece from Fernie drove all the way to Calgary for one a month ago. Cost my sister a fortune, but they have money to waste on dressing like cupcakes and strippers."
Linda plucks the flipper from Sashay's mouth, wipes saliva on her jean shorts, and carefully puts the flipper back in the bag and then into the padded box.
"Sashay is building character. And learning so much."
"Of course. My daughter's going to be in the provincial spelling bee."
"Does she have to be on stage for that? For all eyes to see?"
Linda plunges her fist into her coat pocket, the box in her firm grip. Taking Sashay's hand, she stomps out to their parked car. Sashay wipes her spitty palm on Linda's bare leg.
Linda drops Sashay off at Rhonda's for pageant practice: poise training, finger pointing, pouting and smiling, walking and turning, the subtle art of dancing with your hips without gyrating too much.
She takes side roads nearest the mountains, engulfed in trees. Two towns over is still close enough for people to know who she is, but there will be more strangers, a higher percentage of people who don't know she sells insurance part-time in a strip mall.
Linda ransacks the dollar store, fills her plastic basket with phony ponytails, clip-on strips of plastic hair. The final price is less than twenty dollars. Enough left over to make a payment on the flipper before Kevin sees the bill, hidden in her underwear drawer.
Linda strokes the hairpieces, waxy and obviously fake. This might be the last time Sashay will be like this, Linda's little buddy, Mama's girl. Soon she'll be at school, and they won't be together as much. She'll find new things to love, new friends, new excitement. Her spunk will give her so many ideas. Linda will still be exactly where she is now.
They sit together at the computer. Kevin is mad about the cheap hair too. He doesn't understand that otherwise she won't be competitive.
"You've never spent this much before."
"Those pageants were low stakes."
"Do you even know what you're doing?"
Linda plays montage after montage. Girls in their enormous glowing hair, winning trophies as big as their bodies. The judges, their words cutting, their praise only for participants with the most polished looks. _Frizzy hair, unkempt, raised in a barn, disgraceful, sad_.
Kevin looks like he has the worst case of heartburn on record.
"We've already spent the money. We need her to be her best. We can't humiliate our own daughter. Do you think she's not good enough? Do you think she's not worth it? Don't you want her love?"
He doesn't answer, turns off the computer monitor.
"We've just got to go all in and be there. You have to be there."
He runs a hand through her unwashed hair.
Kevin and Sashay watch cartoons in the living room. It's close to dinnertime, but Linda is on a deadline. In the laundry room, she shoves an embroidery needle through the segments of bleachy hair until the edges form a semi-circle, or something slightly larger than a semi-circle. Then she takes more long curls and adds them to the base, over and over again. She keeps piercing her fingers. Seamstress wasn't on her resumé, but she has to keep going, has to fit in this craft project while making sure they have clean clothes for the day trip.
"Honey, I'm starving over here."
"Me too, Mama."
Knuckles stiff, she takes the last track of hair and threads, makes sure it's as tight as possible. Careful not to smear blood on the light hair, she sets it down on the foam head and washes her hands in the wide laundry tub, wraps her hands in two clean towels, and carries her creation into the living room.
She presents it to Kevin and Sashay. Sashay strokes the curls, pulls them like a pig's tail, and lets them snap back.
"Wow," Kevin says.
"Is that for me?"
"Of course."
"I love you, Mama," Sashay says as she burrows into Linda.
They're running late, and the car seat won't clip, broken clasp. Another thing she forgot to fix, can't replace just yet. Instead of trying to fix it, Linda plunks Sashay down in the back seat, straps the seat belt on. The neighbour's dog shit in their carport, and Linda stepped in it. Rhonda cancelled at the last minute because some kid at the salon gave her strep. The fake hair wig concoction sits on a Styrofoam wig stand in a Styrofoam-filled box in the trunk, and they don't have a hair expert to fix it. Rhonda was against the homemade wig from the start.
The drive is long and hot, and she's paranoid that the costumes will be wrinkled, that there won't be a steamer, that Sashay will vomit at some point in the next eighteen hours. Kevin is driving them so that she can coach Sashay with focus. On the way back they'll nap together in the back seat, after a fast food stop for burgers in crinkly paper, a pee break.
Kevin drops them off to park the car. In the hotel lobby, Sashay runs to a little girl decked out in a purple tracksuit. Large gems spell out "Jewel" on the back. She's wearing sunglasses. Her parents are nowhere in sight.
"Hi, Jewel!"
"Your mom smells like dog farts."
Sashay looks confused, hurt, then softens to Jewel's giggle, relents to the hilarity of dog farts. Linda wants to be angry at Jewel's rudeness but is more annoyed at the accuracy of her nose.
Linda ushers Sashay into the ballroom to register. Politeness and decorum, signing papers, and obtaining nametags. Her daughter eyes the tourists rolling suitcases in and out of the elevators.
"Are we going to sleep in a hotel room?"
She's told Sashay many times they are not going to sleep in the hotel. They'll drive back tonight. They'll survive the day on snacks in her purse.
"We have to hurry. There isn't much time."
Since they don't have a hotel room, they're stuck with the washroom. There's a dressing room for prep, but when they walked by, it was already cluttered with hot rollers, screaming kids, and racks and racks of dresses. She can't focus on perfection when there is so much going on. Linda works fast, and Sashay sits frozen on the toilet seat. Makeup goes on with only a little fuss. The hair stayed perfect, the Styrofoam peanuts a cloud of cushion. As she affixes the pins, a strand of artificial hair breaks off. A minor glitch. She holds the strand, dotted with glue, to Sashay's head until it seems secure.
A girl in frothy buttercup tulle runs through the washroom and into a stall.
"Mom! My butt is so angry!" she shouts to no one visible. Frat-a-tat explosions rumble from her tiny body, a stream of splashes quickly follow. "My butt!" she shouts again.
A moment later, a woman in high heels and yoga pants clicks in on the tile floor and looks around with worry. Linda points to the stall, and the woman slides in with her child. The two of them whisper to each other over the other, more audible, noises.
Linda scoops up Sashay's dress, and instantly her daughter's arms go up over her head. With precision, she shimmies it onto her body, zips up the back, and fastens the hook and eye; then Sashay turns around to face her. Sashay's dress looks beautiful, shimmery sky blue with silver stars and rhinestones on the bodice, the layers of her skirt as bouncy as her hair, finished off with a silver belt of gems. Her fake teeth bright as sunshine. Even in the bathroom lights, Sashay looks lovely and perfect. Linda wants to snap a good photo, but Kevin has the camera bag with the fancy camera with him. It would have been another thing for her to carry. She snaps a quick photo with her phone and stops before uploading it to social media. It feels like bad luck.
"Are you ready?"
Without a word, Sashay struts out to the competitors' entrance, turns at the last moment, and waves proudly. No hug, no need for encouragement. Linda takes the shedded fake hairs and forms a small ball in her hands.
In the audience the mothers whisper, tell jokes. Linda read online that the most famous pageant girl retired at the age of six. On the day she announced her retirement, her Facebook fan page filled with death threats and calls for the girl to commit suicide. Linda admired the girl's mother, the way she went on _Good Morning America_ and told people to stop hating on a little girl. Her daughter will make lots of money and be famous and loved. More people love her than send anonymous hate posts on the internet. Linda knows this must be true. A blonde doll of a girl is born to be admired.
Kevin stands behind their seats in the conference room, proud in his "Sashay Sashays" T-shirt. After just two tries, the iron-on printer paper worked for them, Linda and Kevin together in the basement last night, their daughter asleep upstairs, unaware of this surprise that might not work out.
"I can't get her face to not look like a pug's," Linda said, picking at an image of her child with one eye and a peeling nose stretched across a yellow shirt.
"Give me the iron," he says.
He made the picture smooth and even across the cotton. They looked like a family and a team.
The first part is beauty, which seems easy but is actually more stressful than talent. With talent you can rely on costumes or props or something else entirely. With beauty, it's all on these little girls to appear poised, to have personalities.
Jewel hits all her marks, but her smile seems put on, too toothy, if that's possible. Pink seems to have fallen out of favour, and gowns are in blues, greens, aqua tones. Everyone looks like they're floating under water.
Sashay comes out in her shimmery sky blue and nails every move. Even though it seemed like she didn't pay attention during Rhonda's training sessions, she picked up all the important things. Turns, positions, head tilt. As she waves, the crowd gives loud applause and cheers. The judges smile and nod at each other.
"She could win. She could win big."
"Sashay is a winner. Our winner. That's a lot of money fanned around that trophy, Linda."
"It's not about the money."
"Well, it's nice, honey. It'd be real nice."
They hold hands and sit in their seats to watch the rest of the girls parade across the stage. Before Linda goes to help Sashay get into her talent costume and prep her, Kevin holds her tightly, strokes her back and hair.
"Next time we should stay in a hotel," Kevin says.
During her talent portion, Sashay juts her hip out when the music starts and doesn't stop giving her all. Foot taps, booty swirl, kicks, shimmy, shimmy, shimmy, body roll, shake. She blows kisses that could knock out a tall building. But then she goes too big with her windmills and jumps, and her hair starts to come undone. Blonde curls hurtle from her head onto the stage, an errant tendril hits the judge's table, her wig flying off, pieces of her stripped, broken in front of everyone.
The murmurs from the crowd turn into muffled laughs and tsks. Some stop looking because it's too humiliating. Others fire up their cameras. The judges, so taken with her earlier, are scratching at their scorecards. Linda can't see their faces but knows they're not smiling now. Kevin squeezes Linda's hand, a different pressure from earlier, a solid, heart-thudding urgency. Sashay performed perfectly, but it doesn't matter. All the girls have big hair, sparkles, eyes wide open like dolls. Her daughter does and doesn't look like the others. Underneath her spray tan is her real skin, peachy with mosquito bite marks. Linda's drugstore mascara is supposed to be waterproof, but charcoal streaks through layers of foundation.
Sashay doesn't stop dancing, stomps the fallen hair as she goes into her grand finishing move, a spin and half flip. She doesn't realize the damage is done, that her hair looks like it's been attacked by a feral cat, what's left unadorned with sparkle. Hands thrown in the air, she smiles so big that the flipper pops out for a second before she reins it in with her tongue.
Under the "I"
Outside the front doors of the Elks Hall, women jostle to light up. The clink of brooches and pins heard as coats collide, as they wrestle to get under the awning, the best spot to smoke and complain. They hold Craven As like daggers, ready to fight. In the dim neon of the "Bingo" sign, the tops of their heads glow, a glisten in their perms. Wind whips falling snow into their cluster and they start back inside, ready to savour refills of burnt coffee and fountain pop. Shirley butts out, watches her younger sister, Mary Beth, rush for the doors, her lavender polyester pants crinkling against her Depends.
Shirley's station is decorated with fresh cards, the used ones folded into squares and stacked to her left. Her cards all touch each other in an even layout. No mismatched edges, all colour coded, numbered, in order. Unlike Mary Beth's area, hers has nothing extraneous, no charms, no insipid dolls, no heart-framed family. Shirley rounds the shoulders of her coat over the back of a folding chair. Coffee steams in Styrofoam to her right.
"Your cards are on my side," Mary Beth says.
"They're perfectly organized," says Shirley.
Mary Beth sticks out her elbow to show the cards' infringement on her space. Grinding her teeth, Shirley sets up again, moves everything to please her sister, stamps each card down with just enough force to shake the table a little. Mary Beth furrows her brow.
"Our table needs a win," Shirley says.
"I need a win," Mary Beth says.
"Pish," says Shirley.
As children, they would fight like this. No pinches under the dinner table or swats on the way to church. They would bicker, unlike sisters, more like an old married couple in a struggle for power. Their relationship a familiar blend of connection and conflict. Mary Beth played volleyball and Shirley played basketball, games different enough to make competition more interesting when they compared points and team esteem and parental attendance, especially if they both played the same night. Debates over physical prowess were more complicated.
As they got older, they never fought over boyfriends, only over whose was better, more attractive, more likely to become a manager at the smelter instead of a lowly labourer. Mary Beth hated for Shirley to hold back information, preferred a kiss and tell, and she always knew when she wasn't getting enough gossip by the nervous way Shirley would rap her fingers on the back of her other hand. Their late night teenaged conversations across their bedroom would culminate in smoking, their new shared pastime. They would puff and shush each other, huddled together at their open bedroom window.
Now, in their seventies, they squabble over pot roasts and pound cake precision, the merits of supermarket mystery novels and bingo, the one activity that levels the playing field. Someone could win at bingo, a win measured in dollars. It's all up to chance.
Last year, Mary Beth's Arnie died. Shirley's Frank is still holding on, a millwright who retired early. He watches television crime dramas while she's out and falls asleep in the armchair before the killer is revealed. When she wakes him up, he spurts and shouts that he knows who did what. Sometimes he accuses her, then zonks right back out, and Shirley leaves him there. On those nights, the bed is hers. She stretches in all directions, can dream. Frank in bed next to her isn't uncomfortable, but he gets in the way.
The bulky snack cart lurches around the room, two perky uniformed volleyball players needed to propel it from one table to the next. Chips jiggle on metal clips, individually wrapped Rice Krispies treats stacked high below them; the coffee urn gleams between their doughy girl faces. Shirley watches the young athletes struggle to turn the cart around a corner, wonders if their thin arms could even spike a ball. Shirley hands a loonie to the girl wearing her warm-up jacket around her waist, pours her own coffee.
All the women at their table have their own rituals. Mary Beth has her full display of junk, like that stuffed gorilla the size of her head. Annie mutters to herself with each jittery spat of her dauber. Janine alternates as she dabs, pink then green then purple then pink again. Pauline likes to be precise, but her eyes are bad, and she refuses to wear her bifocals. Her card is always a Rorschach mess. Shirley takes another sip of dark coffee, scalding caffeine her ritual, the heat already seeping out of it.
The caller repeats himself, and everyone giggles. He's so young, not quite fifty, wears jeans. Everyone says he reminds them of someone else. Son, nephew, that boy from down the lane with the cowlick. He calls the letter in a low voice, then bellows the number. Mary Beth touches one of her photos. In the corner of her eye Shirley sees this, knows her sister is looking for a stroke of luck.
Forty years ago, the two of them had worked the afternoon shift together in the hospital kitchen, would commiserate over their husbands' complaints of late dinners and children unable to complete their homework. Mary Beth got Shirley the job when there was an open spot to fill after a girl got knocked up by a surgeon. To feed and clean for money felt special and irritating in an important way, new emotions that waved over Shirley like fragrant soap.
They worked well together, had a system. Rigged the deep fryer to hold more frozen fries, baked the cheese sandwiches instead of frying to leave time for smoke and gossip breaks. They shared inside jokes in the walk-in freezer.
After three years together at the sinks, at the cutting boards, Shirley had to quit to stay home with her son, who got drunk and walked into a moving car. Her son spent most of his time reading comic books and ignoring her when she asked him what he wanted for lunch. Figuring that without a job she had free time, Mary Beth asked Shirley to pop over and feed her family too.
"You've got time. The girls are big enough now to set the table. And with you gone, I've got to pick up the slack at work," Mary Beth had said.
And Shirley did it, day after day, cooked for two families, wasn't thanked twice as much. Every square inch of her son was bandaged for months, and once he was mostly healed and she had forgiven most of him for being an idiot, she decided, with much encouragement from Frank, not to go back to work. She continued to cook for Mary Beth, still up at the hospital perfecting skins on puddings. She fed Mary Beth's husband, a manager at the smelter, and their young daughters, then ran home to her own family. Sometimes she burnt the roast in her own oven, but she never set off the smoke alarm at her sister's.
"Bingo," hollers Annie.
It's a small pot, only forty-two dollars. They all sigh. Shirley recaps her green dauber. Mary Beth looks at her, whines with her eyes. The two of them haven't had a bingo in weeks, although Mary Beth shouldn't complain. A month ago she got two bingos in one night, and one was on a double pay card, so she had enough money to treat everyone for breakfast the next day at the Zellers restaurant, which they sometimes do, but she didn't. Spent it on a fancy As-Seen-On-TV Ultra-Mop, which she talks about as much as possible.
Those afternoons at Mary Beth's house had been easy. The girls had piano or dance lessons until at least five o'clock. Arnie walked in the door at a quarter past four and was never a nuisance in the kitchen. By the time he'd put his coat in the closet, hung his hat on the hook, dropped his shoes on the mat, and unbuttoned his cuffs, dinner would be in the oven. Arnie would ask about her day and tell her that her housedress was flattering. He smiled with his eyes and had one dimple. She'd put the kettle on. Standing at her side, he would reach up high to get the good tea from the top shelf, curl an arm around her waist to sneak past her in the narrow space between the stove and table. One touch, one motion swayed into another. Their bodies came together in the hallway or alcove. Arnie was firm but generous, and his breath tasted of wedding almonds.
The table set with dinnerware and a meal, Shirley would be satisfied, dressed, out of the house before the girls were back. Back at home, Frank never noticed her flushed cheeks as she whisked through the kitchen, still didn't ask how her day was, and at night he grunted the same old way in their bedroom twice a week. He did comment on her new habit of whistling as she prepared supper. He liked it, and that made Shirley happy.
Someone at their table needs to win big. They could use a little fun. Shirley would love some fun, a morning out with the ladies, sniping and cackling at each other. The caller repeats himself, and Mary Beth shuffles. She examines her cards, covers them with her arms as though it's a test she wants everyone else to fail.
"I think this card's lucky," Mary Beth says.
"Maybe," Shirley says.
After the girls left for college, Mary Beth quit the hospital, and Shirley had to quit her afternoon visits to her sister's house. The first childfree weekend, Shirley invited her sister and Arnie to dinner. Over beef and potatoes, the two couples smoked their way through talk of who had the more capable children, whose relationship yielded the more ridiculous arguments, whose wedding photos held up to the test of time. Mary Beth hated the bridesmaid's dress she'd had to wear to Shirley's wedding and used it as a stone to throw at the whole ceremony. Shirley always felt the dress was flattering, that in fact Mary Beth had looked better, more radiant than she did in her ivory gown.
Their husbands escaped to watch hockey in the next room. A relief for Shirley. She hadn't looked anyone in the eye all evening, kept her attention on the meal. Had carefully planned multiple courses, excuses to get up from the table. Shirley and Mary Beth picked at the leftover tomatoes from the salad. In the living room, Frank threw an ashtray at the wall. Mary Beth flinched. The wrong person must have scored.
"Well, I know you agree. I got the better husband," Mary Beth said.
Shirley poured two cups of coffee, and they sat and smoked and sipped. They stared at each other, then Mary Beth's eyes changed focus, cast down on Shirley's fingers as they slapped at the skin of her other hand.
"It's not a contest," said Shirley.
"You prefer Arnie. Of course you do. Frank's crass. No offence," Mary Beth said, then blew a stream of smoke into her sister's face.
"I'm not offended."
"I know."
Mary Beth added more sugar, more milk until her coffee looked beige. Shirley listened to the clink of the spoon in the cup, the men's conversation in the next room, her own throat swallowing saliva. Arnie broke off talk of a penalty to go to the bathroom. Shirley finished her coffee and walked out of the kitchen.
She found Arnie in the bathroom, knew he'd been waiting for her, waiting for a moment. She'd liked the way he'd combed his hair, swept to the side. He closed the door, pressed her back against the glass shower door. They kissed in a fury, sucked at the pulse of each other before she moved away. Hands clasped with Arnie's, she sat on the toilet seat, wet with a dribble of Frank's pee. Frank, too lazy to lift the seat. Frank yelled for Arnie to check out the shit those asshole referees think they're pulling. She opened the bathroom door and pushed Arnie away. Their skin slowly lost contact, until only the tips of their fingers touched for a few final seconds of electricity.
After that, instead of afternoons with Shirley, Arnie spent them with rye on the rocks. For the rest of his life. Shirley took up badminton, and Mary Beth joined her even though she suffered from tendonitis, strain from hauling steamer trays. They played three times a week. Shirley let Mary Beth win some of the time. The loser had to buy the winner coffee and a muffin after the game. They lingered at the coffee shop and never compared husbands again.
Across the room, the Botanical Society table celebrates another win. Shirley huffs, scouts the room for those volleyball players and their charcoal coffee. When she gets home she can stay up late, long after Frank's gone to sleep, sip rye and play solitaire, compete against herself, her own personal cycle of win–lose.
A year ago, Arnie fell all the way to the bottom of the hill from the pub to the lip of the river. Died taking a leak, pants splayed, Y fronts open, wet and pickled. On her walk home from her book club, Shirley found him. In the spring snow, she bent to him, brushed soggy flakes from his face, took a last fond look at him before she zipped him up, walked over to the police station, and called his wife. That night, Mary Beth couldn't find her glasses, so Shirley drove over, picked her up, sat in silence while Mary Beth complained that she had to change out of her pyjamas just to look at Arnie's dead body.
"No luck. I was wrong," Mary Beth says.
"You deserve a win. Maybe Wednesday," Shirley says.
Annie's meagre bingo is their only win of the night, and they let her off the hook. No breakfast tomorrow. Mary Beth takes her time collecting her charms; her cramped, sore hands make it difficult to pack her duffle bag neatly. Shirley helps put away everything except the photographs. One of her sister's grandchildren, one of Arnie. After she's stowed everything away, Shirley takes Mary Beth's hand. They sit like this, ready to go but not ready yet, hands together.
At the end of the night everyone is at their slowest as they reclaim belongings, fasten coats, knot scarves. Outside they stand together, smoke one last one before heading home. Shirley and Mary Beth stand side by side. They exhale old breath, then smoke, then breath, and more smoke until the subtleties of grey subside, and they stamp out their cigarettes, their wordless goodbye.
Cold Cuts
"Pass me the obituaries," I say to Mark, holding out my hand. His curly head is buried in the paper. It gets delivered in the afternoon, but he likes to read it the next morning while he waits for his on-call call. He's always going over yesterday's news while he waits for work to let him know he can work. I haven't had a job since I got fired from the printing place for photocopying my boobs. I'd worked there only six months, and paper was obviously not my niche.
"There is no separate section for obituaries."
I keep my hand extended until he relents and slaps the paper into my hand. I'm looking for funerals to attend. Good ones, with lots of extra food to bring home. It's good to have a purpose, one that isn't sitting around here pretending I'm good at domestic shit.
"The Rossis are having one on Saturday," I yell, because I like to be loud in the morning to wake myself up, and it keeps Mark on his toes.
"Which Rossis?"
Mark's hunched over the kitchen table eating a green-wrapped mandarin orange. As he chews a segment, he presses the crinkles out of the paper with his index fingers. When he gets up for a glass of water, I crumple it into a ball and throw it at him.
"The ones up the street. Four blocks away. By the park. It's the closest one." There are so many old people in this small town, and they're dropping like flies, so I've decided to go about this the easy route and pick the funeral within walking distance. Mark's car is a piece of shit. We need to keep ourselves fed, and I'm a go-getter, but no one around seems to want to acknowledge that.
"Nat, are you really going to do this?" Mark picks up the ball of green paper and smooths it out on the counter.
"No, _we_ are going to. It's a good idea. My idea. I'm writing it on your calendar. Don't be such a pussy."
The first time I crashed here, I made fun of Mark's calendar. His mom gave it to him. She'd gotten it free from some insurance company, and each month has a photograph of a different old man and his old car. I'm only supposed to have seen three months' worth of polished steel and commemorative car show hats because Mark says it's bad luck to flip ahead. He pins the bottom of the calendar down. The morning after that first night, I wrote "Mark sucks!" in one square of each month. The calendar hangs right above the shoe rack by the door, an unfriendly greeting from unattractive, old losers. Thank fuck, Mark hasn't posed with his crappy car for some small town insurance promo.
"I went to school with his granddaughter, so it's not like we don't know the Rossis. And that old guy used to yell at us to get off of his lawn, or he'd knock our 'bony asses into the river, goddamn it!' Oh, memories. And now he's gone."
Mark pulls the curtains open, and the kitchen fills with light, revealing sticky rings on the counter. He pulls out a fresh sponge and wipes the surface clean.
"We used to play on the other side of the river," he says.
Mark throws the sponge into the sink and comes back to finish his orange. I grab the last piece before he gets to it.
"Which is the wrong side of the river?" I ask.
Mark frowns at me and puts his hands in his pockets. "Whatever side you're on, Nat."
I hold the orange slice up to his mouth, and he hesitates before opening. I pop it in gently, and he bites down, satisfied. I can only fuck with him so much, but he needs me around to shake things up.
I thought Mark was a creep because he never tried anything with me. No gawking when Amy and I made out at a bush party. Never felt his hand climb up under my skirt. He didn't pin me to the side of his car, his dick ready-aim-fire in his pants. Mark was the guy at the party holding an always half-empty beer, slotted in the corner while everyone else did the talking and the dancing and the living. He was just there. I didn't understand why people invited him everywhere. He just took up space.
Four months ago he held back my hair while I puked my guts out. A chunk of sausage landed on his pant leg, but he didn't care, just told me to stop apologizing. Amy left me in a hallway after she convinced me we could each shotgun a two-litre of cider. When I asked someone where the bathroom was, Mark opened the door, and I crawled in. He didn't wipe the strings of saliva and booze from my mouth either. I liked that.
After the puke incident I looked closer. He was still in the corner, but he laughed. He spilled his beer. I even saw him adjust his package once. That made it all right to talk to him a few weeks later.
"Hi. Mark, right?"
"Yeah. Hi."
He was deep into some NHL conversation with Jer, some idiot who wanted to take naked pictures of me last year. When he asked, I had my period, so I told him and he didn't think that was very sexy. Later I found out that Jer took naked pictures of himself instead. He's a douchebag.
"Hey, Nat, I've got an itchy sack. Could you shave it for me? I've got..." Jer rooted around in his pocket as if any of this was funny, "about a dollar thirty and some fifty–fifty tickets from last week's game. You in?"
Mark started to walk away, but I grabbed his arm, then kicked Jer in the shin before he headed into the living room.
"I'm only on my first drink of the night," I said, gripping Mark's arm.
"Mark, the game starts in ten," the douchebag yelled.
"Okay. I'll be there."
I cut to the chase so Mark could watch hockey and I could go smoke. "Good job with the hair holding. That's usually a girl's job, but you knew what you were doing." I patted him on the back as if he'd scored the first goal of the game.
"Don't worry about it. The puke on my pants was thanks enough." He tapped me on the shoulder, a beer firmly in his hand. It was all over my sweater before I could pull away.
"I got beer on you. Sorry."
I took off my sweater and threw it on a chair. My shoulder glistened with drops of beer. Mark pulled his sleeve over his fist and wiped them away.
" _Mark. Game's on_."
Jer never shut up.
"Thanks again. See you later? Second floor bathroom? Eleven-thirty? I need to finish this up early tonight. Better to puke before midnight." I stroked my hand along his sweater and gripped his fingers. My nails dug in just enough. He looked me in the eye before I let go and headed out to the porch.
From the kitchen I saw him watching me during the third period. We met in the bathroom at eleven-thirty. We kissed for a while, his tongue weak, fighting a losing battle against mine. He remembered my sweater smelling of beer and offered to drive me home in his ugly car. I pushed him out onto the street while he tried to unlock the passenger side door. Then he dropped me off at my parents' house. All night he watched me from the other room, he stalked me, and he kissed me. When I tried to undo his pants on the way home, he grabbed hold of my hand and set it underneath his on the stick shift, our fingers curled together.
A tray stacked with deli meats is set precariously close to the edge of the counter. Some careless mourner must have unloaded it too quickly, telling a nostalgic story, and crying. I'll keep my fond "kids being thrown in the river" memory to myself.
Mark is impressive. He was made for funerals, so quiet, solemn, respectful. A real suckhole. He sits in a high-backed chair and listens to ladies talk about card night.
Mark doesn't know it yet, but I could be an entrepreneur. I'm working the room. I am making a difference. I am being productive. I hug old women; I deposit their used tissues in the garbage; I help carry platters of cheese to the table and bowls of olive pits into the kitchen. This is a lot more work than I did at my last job. Consoling people in shifts, doling out looks of deep understanding, touching wet lipstick- and antipasto-stained napkins. Only once at my old job did a woman sob in my presence and that was because three boxes of legal paper fell on her foot.
"Old Mr. Rossi. All us kids in the neighbourhood knew him. What a joker." I repeat this line as I make the rounds.
No one eats the buns and cold cuts and cheese. They're all separate, so everyone has to make their own sandwich. When my grandmother died a few months ago, there was way more food. Way more options. She always liked to take care of us all and set aside money to make sure her funeral was full-on catered with hot items, things you could make a meal out of. This spread is fine but not spectacular. Better for carting home, I guess. I have no problem slapping some pre-cut items together, being resourceful.
Mrs. Rossi is working harder than anyone. When someone gets up off the couch, she plumps each brown floral cushion until it looks as though no one had been sitting there at all. That's how it will look tomorrow when she wakes up alone.
"I know Stephanie. We went to high school together," I tell a man in a short-sleeved shirt who doesn't speak much English.
Mark doesn't remember what Stephanie looked like, even though she was always in front of a microphone at assemblies talking about charity phone book deliveries and cake walks and animal rescues.
I almost dump a full bowl of mixed olives onto a girl in black velvet as she steps out of the bathroom. Stephanie. In the seven months since graduation, I'd forgotten how large her hips were. Made for babies and backdoor lovin' is what we used to say.
"Stephanie?"
"Yes."
She seems distracted. When my grandmother died, I grumbled and acted uncivilized. At least that's what my mother said. I didn't even eat any of that food, just stared at it from the bathroom door. Foolish. The heels of Stephanie's shoes dig holes into the carpet. Mourners walk hard.
"It's Natalie."
"I know."
"Sorry. Hope you're holding up."
"Thanks. I am. I'm okay."
I try to wave Mark over. The woman next to him can't let him go. He's opening her childproof pill bottles and handing her each dose. He is supposed to be here for me, be my assistant. Stephanie smiles at the room, her eyes unfocused. She's not looking at anyone.
"I heard your grandmother died recently. Sorry to you, too. I was away at school then, I didn't..."
"Yeah. There is so much food here."
I hold up my bowl of olives.
"Thanks for helping."
She doesn't need to say it. She thinks I'm here because I know how hard it is to lose a grandparent. No sense telling her she's wrong. Her lip quivers a bit, and she looks deeply into the olive bowl. I put the olives on a side table, reach out, and pull her in close for a hug. Stephanie falls against me, and I feel her shake. It only lasts a moment, and then she's composed again, her face screwed up as though she's remembering hard.
"Mark!" I say.
I yank Stephanie along as I make my way to the dining room. It's time to pack up. Mark stands beside the table of food, a highball glass filled with wine in his hands. The pill-bottle woman has left and most people are embracing Mrs. Rossi and getting the hell out of there. The sun is almost down, just a thin layer of red visible in the sky. It'll be snowing by dinnertime.
"Mark, remember Stephanie? Stephanie, Mark."
Mark puts his arm over her shoulders and looks down at her with sincerity. Stephanie melts into his chest and heaves her shoulders. Mark walks her over to the couch. He's unbelievable.
Empty, cheap plastic containers bulge in my purse. While Stephanie breaks down on Mark, I pull out the containers. I walk around the table loading up while everyone else dabs their eyes and thinks about the next time they'll all have to sit around like this, in someone else's house, some other old dead person buried and gone and mourned with a plate of salami, roasted red pepper stains on their clothes. On the way out, I shoot the guns at Stephanie and mouth that I'll call her.
"That was harder than manual labour." My purse smells like a day-old sandwich.
Flakes as big as my fist fall around us. Mark takes my hand and walks me down the steep embankment.
"You made out well," he says and taps my bag. I dig my heels into the snow and make him pull me. As soon as we're on level ground, I let go and run ahead, kicking fresh powder at him. He doesn't try to catch up.
I thought Mark was a bit off because he said it didn't matter that I didn't have a job. He thought I'd figure this all out; being young and from a broken home, I'd need extra time to find out where I should be. I punched him in the arm three times for judging me. So, I have a dad who skipped town and a bitchy mom who hated him and parts of me for being parts of him. I couldn't even believe Mark would use those stupid words right in from of me. Broken. When I thought about it later, he seemed pretty fucking wise for an on-call carpenter's assistant. I wasn't sure I could trust him. When my mother told me to get another job or get out, I pounded the pavement for three days with my resumé printed on stolen paper. This garbage town didn't get me. If I couldn't get job interviews, I'd figure things out myself. So I called Mark to come haul my stuff over in his shitty car on the fourth day.
Mark works for two weeks straight and makes enough for two months' rent. I'm getting ready for my funeral, which he says isn't necessary.
"It's important to me, Mark. Don't be such a selfish dick."
I need him to soothe their pain, be my soft-spoken working man, the boring, normal distraction. The dead guy was pretty rich, and the food will be heaped on the table, but lately Mark's been buying groceries and thinks we should use them. I told him I don't want to cook food I haven't earned. He refuses to put on the khakis I've ironed, even though this is the first time I've ironed his clothes.
"I want to stay home today. Just watch the game."
There's a six-pack of beer in the fridge but no cider, which is what I've been drinking lately. I drank the rest of it last night. He only bought what _he_ wanted this time. I haven't figured out how to convince old ladies and their overbearing children to fork over their leftover booze.
There's a lot more laughing at these gatherings than I thought there'd be; families remember the most ridiculous parts of a life. Last week, at Jerry DePaolo's celebration of life, someone had brought up the school musical from 1963, when Jerry had tumbled through the plywood backdrop, but still managed to sing the song with a concussion. The entire house had broken into the "Modern Major-General" song from _The Pirates of Penzance_. Dancing and pretending they had head injuries, the group swept me up, and I bobbed my head as if I knew the song, the words, the whole story. Some middle-aged sad sack told me about the song and the musical for thirty minutes after that.
We haven't been to a party in over a month, but we've been to six funerals. Mark didn't like when I convinced him to go to two in one day, but I say, "Get while the getting is good." He held the widow's hand at both funerals while I organized cold meatloaf into Ziploc freezer bags.
"Nat."
"What?"
"We have a fridge full of food here. You don't have to go funeral foraging."
"I want to!"
"Why? The food I bought will go bad if we don't eat it. Why do you need to go?"
The house where they're gathering is behind the nearby school playground. I used to play in the sandbox there, when I was a kid. You could see right into the front window while digging to nowhere with a plastic shovel. The family there was a lot older than mine, but they always had parties that spilled onto the front lawn. Everyone having a great time, saying cheers, and getting into sports arguments that didn't end in fist fights or smashed bottles.
Mark strokes my crossed arms. Those widows are vulnerable, but not me. Swipes of his rough palms on my skin aren't going to make me any less angry. He's left his half-empty beer on the table, and I try to kick it over. I miss.
"Nat. Please? There is no reason to go."
"I ironed your pants."
"And they look great. I can still wear them."
"Just sit around the house and wear khakis instead of sweatpants? What are you?"
"I'm not going."
His hands are still on me, and he's too good at maintaining eye contact, so I look away first. I kick at the table one more time, get the corner, and the bottle wobbles.
"Just one day off? I worked hard all week. And you know I hate fighting. Please. When do I say no? Why are you pushing this?"
Mark kisses my cheek and rests his head beside mine. He's freshly shaven and smells shower clean, and I think of formaldehyde and mothballs.
"Quit interrogating me. I'm going. I can do shit on my own."
I break away and head over to the bedroom. I strip out of my sweatshirt and stand in my bra and panties and press the steaming iron over my only black pants, my Saturday mourning pants that make my ass look good even when it shouldn't.
Jer told me that Mark had never had a steady girlfriend before me.
"He's probably a fucking virgin. Why would he want a slut like you? You're more my style."
Jer was wrong. About two things. Mark had a steady girlfriend, Jenny Parker, in grade six. And he'd fucked a girl on a hockey trip and another one that he'd dated for a month before she dumped him because he drove a truck with a canopy. She was embarrassed to have sex with someone with a canopy. He sold the truck to some hunter and bought the crappy car. I wouldn't have minded the canopy. I don't really mind the crappy car. Don't even mind fucking in it. Mark likes to do it in the bedroom, in a bed.
It's snowing hard, but I put on spike-heeled boots. Mark told me to take his car, but I don't want to drive in the snow. Or owe him anything. I've never driven his car before. He's enjoying himself, sitting in front of the television as the afternoon turns into darkness too quickly. I hear goodbye when I storm out the door. Probably happy I didn't make a mess of his beer. Tiny flakes make a veil in front of me as I walk. I slow down, watch my pin-heeled feet to make sure they don't slip.
I thought Mark was abnormal because he didn't know how to yell. His voice is soothing, low, and sometimes I don't quite catch what he says. I yell out that I can't hear him, that he should speak up. This makes him laugh. He doesn't take my anger seriously. Which makes me seriously furious.
I thought he was abnormal because he refused to make obscene comments about me to his friends. Because he wasn't like Jer and their bro gang. I told them about our sex life, bragged about it for him. I thought Mark wasn't ready for me because he sometimes stuttered when I came into the room. I thought Mark was horny just because he'd kiss me goodnight.
A car swerves at me, and I jump into the snowbank. I didn't see anything coming at me, only the wild flakes and the tips of my boots as I watched myself shuffle down the middle of the street. No snowplows have come through yet, and it's been coming down heavy for hours. Snow up past my shins, I stand in the cold, realize my purse has fallen somewhere in this whiteout.
The car's going fast, a shiny black rush in the snow. The hill is long, and whoever is driving pumps the brakes and manages to slow down before the park, drive smoothly for a few metres. Then the car catches ice. The driver does everything right. Tries to turn into the skid, but the car hits something in the bank, and the whole thing flips, then flips two more times, and rests near the snow-blanketed teeter-totters.
The crash is quiet, snow muffling the sound of the impact. I stare at the car's tires, the bottom of the vehicle facing up, collecting snow now too. I pull myself out of the bank and find my purse on the other side of the road. I hadn't zipped it up, and things are missing. I salvage my wallet and some lip gloss, but my keys are gone.
I run to the car, slipping and sliding down the hill. It looks cradled in a white sheet. My heels stick into the snow, and it feels like I've been walking for hours when I get to the car. I reach for the door, stumble. I take a step back to steady myself and lose one of my boots in the snow, pulling out a socked foot. I trip trying to find the boot and land against the driver's side window. When I look in, I can just see her white hair through the window. I brush snow away with my glove and see a veiny head hang limp from her jowls, not even a neck anymore. She doesn't move or whimper, but I see a strand of smoky breath filter out of her mouth.
Sometimes old people still drive. They make their way to friends' houses or even go visit other old people clogging up hospital rooms. The hospital is in a good location. They get a nice view in those last moments, snow coming down onto the mountains, landing gently on the roofs of houses in the valley. My view is the wall of the house next door and the alley full of Christmas trees.
I try the door, pull on the handle, but it's locked. The driver's side window is shattered but still intact. I push on the glass, puncture a hole just big enough to push a finger in. If I try to smash through the brittle mess, I'll cut her hand, which dangles right there. Might even slash her face.
She'd turned from the top of the hill on her way somewhere, and then, there I was walking right in the middle of the goddamn road thinking about whose pants I iron. I want to blame Mark for this, but he wasn't in the road. He's at home, warm and nice and drinking at a responsible pace. Mark had washed the sheets this morning, because after drinking all that cider I'd been too lazy to get out of bed to puke. Mark always walked on the sidewalk, even if it was covered in snow. She'd cranked the wheel to miss me. And now I'm not mowed down, bloodying that snowbank. I only lost my keys.
I watch for more breath. My fingers still slotted through the glass, I grab her hand. It's freezing cold. I crouch in the snow and hold her fingers until I see another weak stream of air come out of her mouth.
I haul myself out of the bank, pull my boot over my snow-clotted sock. The funeral is around the corner. I run through the drifts, boot full of snow crunching and melting through my sock. At the end of the block, I turn back to see the car, already turned almost white by snow, her gloveless hand and jowls invisible to me. I take a turn, climb up another hill.
At the hospital, I tell some lady in pink teddy bear scrubs about the accident. She tells me to calm down, and I tell her I am calm and that she should calm down but also that this is serious. Hospital workers scuffle around behind a glass partition. A nurse says that someone is already on their way down and that I can wait.
I leave and walk down the hill on the other sidewalk, which somehow has been sanded. When I get to the bottom, I watch paramedics pull the driver from the car, pile blankets around her on the stretcher. A few neighbours stand around too. Some look out from their front windows. I don't know if she's dead or not.
After the ambulance is gone, I walk a few blocks, sit on the steps under the covered green staircase that leads home. I wait until I think the funeral will have ended and someone else will have taken the canapés into the kitchen. I wait a little while after that, just to be sure.
When I walk in the door, Mark's wearing his khaki pants. I can see them on his legs, propped up on his old ottoman. He's got a half-full beer in his hand, the game on quietly, as if he's mourning something. The click of the door startles him, and he jerks around to see what's happening as I shake snow from my coat.
"Sorry," I say as I strip off my wet clothes, drop my boots on the rubber mat underneath the calendar.
"How was it?"
I go sit on the couch beside him, reach over and take a sip of his beer.
"It was shit. There'll be better ones," I say, my voice barely raised.
Particleboard Man
_Hey, you've reached Ryan and Cindy. Can't take your call, but don't worry, we'll get back to you real soon_.
There's no real good reason to keep the message, but she can't think of a good reason to erase it either. Cindy tells herself she'll wait until the tape snaps, and it'll be gone for good. Strong as steel that tape, over ten years on record, eleven and a half months since he's been gone. She doesn't get a lot of calls, just from her mother, her friend Heather, who's almost always on night shift, the dentist. Mostly she listens to Ryan in the early evening. When she feels like muting _Wheel of Fortune_ , clicking play to hear the purr of "real soon" or the curt, hard "can't take your call." She remembers the way he'd clicked the record button, his wrist at a ninety degree angle, his elbow resting on the wicker placemat under the phone, the bowl of keys beside the answering machine.
Cindy clamps her hand around her whisky fizz and thinks about all the things she's neglected to do in her rush to make it to the bar: wash her work clothes, let the cat out, set up her voice mail, throw out that piece of shit answering machine. The old, heavy machine somehow still works after twenty years of loyal service. Ryan's voice coos through the message whenever anyone calls and she can't get to the phone in time.
Two nights ago, Heather came with wine and Chinese food. They didn't even talk about Ryan. Heather was worried about her dad's surgery the next morning and her son's new girlfriend. Cindy told a story about a man she'd seen driving a car with two pigs strapped in seatbelts in the back seat. They snorted and laughed, made pigs' faces. But after a bottle and a half, they moved from the kitchen into the living room. Cindy stared hard at the answering machine, and Heather ran over and forced Cindy's hand over the delete button.
"No!" Cindy shouted and grabbed at Heather's wrist. The two of them wrestled above the particleboard table, the cordless phone and answering machine rattling as their knees knocked against the thin faux wood legs.
"You've got to let go! This is old!"
Heather pinches Cindy's neck. Cindy yelps and whips her arms, tangled with Heather's behind her. They break free, and Cindy clutches the answering machine in her arms. Heather is slumped on the arm of the couch. Both breathe heavily.
"I'm not ready."
"Okay," Heather said, "but I'm ready to help you rip off that Band-Aid when it's time. Should we drink some more wine?"
"Of course," Cindy said.
Ryan and Cindy. The message lies to the people who call. At thirty-eight, she's only been with one man, and now he's just a voice, two sentences held in outdated technology. His voice used to mean that she was useful. Asking where his underwear has gotten to, what time he has to be at the doctor's office, which drawer has the bottle opener. Cindy, wordless, would find the underwear, or hand him the doctor's reminder card, or open his beer for him. At night she could get him to brush his teeth with a smile, could steer him to bed with one hand on his shoulder. Even sex was seamless, though not so quiet. The one time she would let her vocal cords loose. At least, for the first few years that's how it was. Then her silence became as annoying as a jackhammer is to someone not wearing protective ear-gear. At least, that's pretty much what he said to her.
The clink of beer bottles in the bar comforts her. People are around her. Lots of people. Young men. The sort of people who might talk to her. Maybe not tonight. But one day. But hopefully tonight. After they'd invited her over to a quiet corner table, she could tell them about the busted-up dirt bike in her garage and the whirlpool bath she'd had installed. As long as it wasn't Travis Smith, from Smith and Son's Sunshine Pool & Spa, because he already knew about the bath and didn't seem very interested in the dirt bike as he wheeled the dolly past Cindy in her neon and orange vest and steel toes. These people will be interested in her, want to know what she's about. She's been in every Thursday, just once a week for three weeks. No need to seem like a drunk.
Construction workers shoot pool in the corner, and a group of guys sit at a round table with two pitchers of beer, talking and sometimes pointing at the TV. Cindy watches the activity in the room in the reflection of Budweiser and Pilsner and Kokanee mirrors hanging behind the bar. In her peripheral vision, she sees three paint-splattered guys trying to build a beeramid with their empty bottles. Adorable. But it doesn't matter how cute Cindy thinks they are; it's what they think of her that matters. Brown glass clatters and bottles roll. The paint-splattered guy with the long hair and the one with the shaved head pour their beer dregs onto their buddy who knocked over their hard work. In retaliation he pulls the ponytail of the long-haired guy. It all looks like so much fun. Vibrant, like what her co-workers must do after a shift together.
She'd been the younger woman with Ryan, but by just enough. That was then. It was all about then. When there was a then, there had been so much time, a future. Time, in the future, when she could make decisions and wait. But Ryan did not like waiting. Ryan said he wanted to be in the now. And that is why he had to leave. But to Cindy, that was their sameness. They both wanted to be in the now. She didn't want to think about what would happen in the future. She just wanted it to be there, somewhere away from her and Ryan and their wicker mat and velvety sofa and their Napoli pizza delivered to the door on Friday nights. Ryan, with his Canucks jersey and his 501s, and his hairy wrists protruding from the microwave popcorn while they watched every city's worth of _CSI_ on tape. He meant he wanted other things to happen in the now. That is why. They had a different idea of what should happen in the now they were both living in.
Though she didn't mark it on a calendar, Cindy knows the exact last moment Ryan looked at her with interest. It was the last time they had sex. He came into the kitchen, and she had spilled red wine on her shirt and had taken it off to toss into the washing machine. She was wearing just a bra and jeans, and he had come at her shoulder with his teeth like it was prime rib. While they were down on the kitchen floor, he looked into her eyes like it was the first thrill, and she was so grateful that she grabbed for a tea towel and wrapped it around his head to hold it in place and keep their eyes locked on each other forever. But it only lasted two minutes, and then it was over. That was six years ago.
His new girlfriend is sweet and a teller at the credit union, with a teal green sedan and a well-defined chin. Cindy's chin is on its way out. The new girlfriend was very helpful when Cindy needed to assess her RRSPs, that's for sure. And Cindy has quite a few years on this new girlfriend. Being five years younger than your man isn't much when he can easily get himself a nice fertile one who's fourteen years younger than you are and clearly better at math because she works in a bank and is essentially a financial professional. Maybe she thinks that the future is far away too. Maybe he's recorded his hard consonants on her answering machine. She probably has her own personal voice mail. Of course she does.
Adding up the age differences and subtracting how many years apart isn't the kind of number crunching Cindy is used to. On a regular day she counts cars instead of money, but mostly she stands in the hot sun or rain, reflective vest over her T-shirt and the jeans she wishes she could just throw away. Seated at the bar in a pink skirt and white sandals feels right, more who she is on the inside. Although Cindy shifts constantly, tugs at her skirt even more constantly. The jeans she should just throw away are a protective stone-washed shell and cover her problem areas in a way that a silk camisole tucked into this skirt never could.
The bald guy stands next to her at the bar. He smiles at her, not politely. Cindy thinks this might be a seductive pose, the face of a man who knows he needs a more experienced woman. She also knows that she will have to try harder to appear like the more experienced woman.
"You gotta get back out there. He can't be having all the fun. You deserve to jump on some young guy, really take a ride."
Heather doesn't know how long it's been. Cindy hasn't been able to confide in her about that part of it. She knows it's been a long time, that Cindy hasn't flexed certain muscles in a long time. She doesn't know the date, doesn't know that Cindy stays up at night and tries to imagine other men's naked bodies lying next to her in bed, but can't really picture their arms around her, the warmth of their skin, that they have genitals at all. When she closes her eyes they're young and faceless, wearing T-shirts and plaid pyjama bottoms. Just like Ryan. He's even in her terribly uninteresting fantasies.
They sat on the couch with more wine, and Heather was quiet for a while.
"Sorry about the answering machine. I wouldn't just delete it, you know."
Heather hugged Cindy, a big hug. Their full glasses touched, and some of Cindy's wine splashed onto a tan throw pillow. She raced to get salt and OxiClean.
"I know. But you're right. I have to do something. I just can't tell if I'm ready. How can anyone tell? Especially after so long."
Cindy sprinkled salt over the surface of the setting red stain, pressed granules into the fabric. Cars drove by outside. In the dark, Cindy could see their lights zipping along the pavement, bright blips. People going home to loved ones, people finding their place.
"What about if we go out? You can look at guys, have some drinks, forget about that barfbag Ryan."
This isn't the first time Heather's suggested a night out, that they actually consume alcohol in public like they did when they were younger, when she met Ryan at a Little League fundraiser, and they held hands in the Lions Hall, and she felt her whole life warm up to the right temperature. Cindy has always turned her down. She already has to be in public for work all day long. Holding up a stop sign, waving people through. At night she wants to take a break from strangers, people. Heather is the only one she likes visiting with, at her place or here at home.
Heather pours more wine into Cindy's glass, empties the bottle into the wide bowl with a stem. They cackle about the man across from Heather who can't seem to get his house paint to match. When the wine is gone, Cindy looks down at her fluffy socks. The socks of someone who doesn't need to impress anyone. What would she wear?
"Okay. I'll go out. Tomorrow. I have the next day off."
"Really? You'll go out? Do you need me to be your wing woman?"
"Does that mean you serve me wings while I try and pick up men?"
"That would be a better use for the term. Do you want me there?"
Cindy turned Heather down. Her dad's surgery is in the morning. He'll be in recovery, and Heather needs to be there for her family. Her lazy brothers certainly won't be bringing food to her mom or ferrying her back and forth to the hospital. On her own is what Cindy needs if she's going to try and spend a night with another person. She's had enough practice lately spending nights alone in bed. That's why she's so out of practice touching another person.
"Babe, can I buy you a drink?" But this is not to her. The bald guy is talking to the bartender manning the taps in her black apron and grommet belt. The cotton of her Nickelback T-shirt so shoddy it will stretch in places and shrink in other places the first time it's thrown in the washing machine and dryer. But for now it's perfect, free of pills and snags, the plastic iron-on transfer still firmly rooted to the chintzy fibres. A woman should not think this way when she wants to get what the girl in the tight and shoddy concert tee could get. Maybe she should mention to the girl that she should wash in cold water and lay flat to dry.
The bartender huffs and fills a pint glass with foam, then grabs another, slides it under the spigot to collect the good beer. When she gives the full glass to the guy, Cindy realizes that they must already be a couple. She knows what he wants. He ignores her and walks back to the pool table, doesn't compliment her on the tightness of her jeans or ask when she gets off shift. He knows. He knows and doesn't care. She'll come home to him with her booze-scented clothes and overloaded key chain and wake him up in the middle of a dream about _Maxim_ 's hottest female fighter pilots, and he'll grumble that she needs to make it up to him, and then she'll get her chance to ignore him, throw her shirt and jeans into the hamper to mingle with his sweaty clothes, and they'll go to sleep, backs facing each other.
"You've sucked that dry," the bartender says as she wipes the bar with a grey towel and grabs Cindy's empty glass from her hand. She's been slurping and clinking the ice cubes.
"Are you ready for another drink?"
The bartender holds up the glass, and Cindy bobs her head up and down. _Yes, another drink_. She could buy two for herself, just to get in the mood. Walk around the bar showing them off. In case anyone missed what she was drinking, now they would know, now they would see her taking her little sips and sitting quietly contemplating life and the problems with trying to live it. Someone would come up and set one in front of her when this was done and say, "This one's on me."
Her old melty ice cubes tumble out into the sink with a crash, and then the bartender is tossing new ones, cold and fresh, into a new glass and squirting the soda gun with indignation. The whisky comes down fast into the shot glass, and then Cindy is presented with a perfect new cocktail.
Cindy opens her mouth to thank the girl but is interrupted by the loud clang of a telephone. She retreats and clams up. The bartender reaches under the bar and pulls the receiver to her ear.
"Hello, Brat's Pub."
Cindy leans over and drinks through the straw, imagines that someone has to be watching her. Some man in this bar sees a woman alone drinking whisky. Cold, sugary booziness squirts into her mouth. She pulls back to admire the rosy lipstick ring left behind on the blue plastic. When she finishes this drink, she'll need to replenish a coat of Revlon Super Lustrous 440 Cherries in the Snow, but then, maybe the man who'd been watching her would think she was leaving and give up. His shyness would win and she would lose. She wonders which man is watching her. It's uncouth to turn around and search for him. She occasionally glances into the beer mirrors to check out if she is being checked out. But now that she's really thinking about it, she can feel his eyes on her. It is only a matter of time.
"Uh-huh. Five-fifty." The bartender covers the mouthpiece of the phone with her hand while she asks Cindy for payment. Cindy pushes three toonies across the bar.
"Keep the change." She gives the bartender her widest smile, eyebrows raised.
"Uh-huh. Yeah. Thanks." The bartender hangs up the phone, and Cindy isn't sure if she's the one being thanked or yeahed or uh-huhed.
Her bedside table is cluttered with magazines and biographies. Cindy had studied all the right older women: Goldie Hawn, Demi Moore. And locally too, she'd been paying attention: the woman who actually had a pool boy and had made love to him, the baker from across town who'd married her daughter's boyfriend. Her only concern was that so many of them seemed to have a rule about going without a bra. This was not something she had ever thought her breasts could handle. She disregarded this little amendment to her wardrobe, while taking the rest of her cues from them. She had also carefully thought to put a pair of slinky underwear into a Ziploc bag in her purse. When he took her home, she could excuse herself to freshen up, run her hand seductively along the hallway while making eye contact, and in the bathroom quickly change out of her Spanx and into the lacy ones before he'd change his mind about her.
The beeramid comes crashing down again. An empty Budweiser rolls across the floor and stops directly under her stool. The long-haired guy jogs over, beer stains mixed with the paint on his white coveralls. Cindy grips her whisky fizz and freezes. He bends over to retrieve the bottle but stumbles and grabs her thigh for support. Cindy clenches every muscle in her body and sucks at her drink. Using her leg to brace himself, he stands up, proudly holding the loose bottle. Cindy focuses on her almost empty drink and tries not to stare at him.
"Sorry." He brushes her leg as though he's dirtied it. Cindy smiles. He didn't call her lady or ma'am.
"Don't worry about it." Cindy looks over at him. It seems appropriate that after someone has groped and caressed a part of her body, even if it was for his own safety, she make eye contact. His eyes are green and dopey like a kitten's.
"Oh no. I got some on you there." He points at her lap. A few drops of whisky fizz dot her pink skirt. Cindy takes the napkin from under her drink and covers up the marks. Her face feels hot, and home seems far away. She finishes the last drops of liquor in her glass and clinks the ice cubes with the straw. The bald one and the other guy throw coasters at their friend, and he deflects them so they go flying behind the bar.
"What did you do, Jamie?" The bartender raises her voice and throws her arms across her Nickelback-clad chest.
"Nothing. Shut up."
Cindy reaches for her purse. She could call home and listen to the answering machine one last time before she gets with a new man. Her phone is almost dead.
"Are you okay?" The bartender grabs Cindy's newly empty glass and dumps the ice cubes.
"She's fine. I already said sorry." Jamie is teetering around like the top of the beeramid. He puts his arm around Cindy. Silent and still, Cindy tries to assess the meaning of the gesture.
"Doesn't she look fine? She looks fine." Every time he says "fine," Jamie's fingers press down on her shoulder. She thinks she likes this heavy hand. Cindy looks at the bartender, who looks back at Cindy as if she isn't fine. Maybe she should have given her a better tip.
"God, Jamie. Leave her alone. And no more fucking beeramids!"
"I was just going to buy her a drink. And maybe I don't want to leave her alone." Jamie puts the empty bottle on the bar and indicates that he wants another.
"Whatever." She stalks away to serve an old man in a ball cap.
Another coaster comes flying and hits Jamie in the back. His arm is still firmly around Cindy's shoulders.
"Mike doesn't want me talking to his woman," Jamie whispers into Cindy's ear. His breath is tart and warm. Not unpleasant for someone who's consumed enough beer to construct a beeramid and splash alcohol onto her. "She was my high school girlfriend. And she all loved me and shit. I was only in it 'cause her dad sold pot. Don't let Mike know. It's fun to piss him off."
"Oh." Cindy hopes they are all out of coasters.
Mike's woman brings over their drinks. Jamie raises his beer up for Cindy to admire.
"These are on special." Jamie's beer is still raised.
"Well, that is special." Cindy raises her whisky fizz, and Jamie knocks his bottle into it. A splash jumps out over the side and onto her lap.
"Shit. Sorry again," Jamie removes his arm, and Cindy feels empty. He wipes at her skirt with a painter's rag from his pocket.
"Why don't you go sit back down with your friends?" She could leave now. Two-and-a-half drinks seems like a perfectly respectable night. And a man has touched her shoulder with force. Jamie's arm is gone and probably also her chance to show him the sexual power of a woman with age as an asset. Even though she's a woman who hasn't exerted any sexual power in years, who actually could use a step-by-step instruction manual to jog her memory.
Jamie ignores her and sits on the stool beside her. "You look really familiar. Where have I seen you?"
"Nowhere."
She wasn't a friend's mother who had baked him cookies or the nurse who'd taken his blood. She didn't participate in community theatre or volunteer to raise money selling raffle tickets.
Jamie puts his fingers under her chin and with his painty hands, delicately turns her head toward him. He brushes the curling ironed curls, which are falling out anyway, from her face with his enormous thumbs and pulls all of her hair up and holds it in a loose bun on top of her head. A stray curl falls and lands in front of her eye. Despite his inebriation he sweeps it gracefully into his hand. His eyes look into hers. He touches her lip with the tip of his finger.
"On the roadside. Fuck, I knew I'd seen you somewhere. The side of the road near where they were blasting." He keeps his fist full of hair firmly rooted to her head, his other hand pointing at her while he speaks.
"You never look pissed off like those other road safety people, or whatever you're called. And we drive by, like, four times a day."
Cindy doesn't know what to say to Jamie. In this moment of contact she is unprepared. She hasn't even known how to leave a message on her own answering machine, what to say. Her job is a particular kind of communication. Silent. Hold up a sign, wave people along, hold up a hand, smile and nod. A call and response without her having to raise her voice. She has to be quiet and use her hands to guide drivers who so often are blasting music. No one wants to listen to what she has to say. But she has to tell them what to do. Cindy doesn't even mouth the words any more, just rolls her wrists, like a bigger, more meaningful royal wave.
"I usually drive that old paint truck. But I own a GMC. Very roomy. Got it lifted. I take it out when I'm on my own time." Jamie lets go of her hair, and Cindy hopes that her curls will bounce around her face in a pleasing manner. She tries to look at herself in the beer mirrors, but can't tell. Then Jamie leans in and he seems happy, so it must look attractive.
Cindy thinks he might be trying to impress her. Talking about his shiny truck and the spacious cab. She has driven herself tonight. He pulls back from her, taking her hand to pull her off the stool, but he stumbles and their fingers disconnect. Another coaster hits Jamie in the eye. Mike's woman comes out from behind the bar and kicks the bottles on the floor. She slaps Mike's arm and shoves him into the table. Jamie tumbles back into Cindy's arms.
"We're getting kicked out." He grabs his beer and chugs it.
"Okay," Cindy says and drops her head to her chest.
"C'mon. Let's get out of here before those assholes have to pay their tab. Drink and dash on them." Jamie pushes her off the stool, throws a hand around her waist, and ushers her out the door.
Outside the air is the same as in Brat's Pub. Warm and sticky and a little bit stale. The sun is going down over the mountains pink and lovely, like the way she feels in her skirt. This moment is a bit more romantic than when she'd watch the sunset at the end of her shift, and hope to share it with someone, and be dressed in something more flattering than a reflective vest. Jamie tips himself onto her. He needs her to keep himself upright. The number of bottles left on the barroom floor indicates that there are likely a lot of drinks making their way through his bloodstream. The weight of him fills her up again, and she puts her own arm around his waist to keep balance. His face is buried deep in her neck. Cindy thinks he must be sniffing her shampoo and perfume, which mix quite well, and not the remnants of the strong sunscreen she wears at work, which shouldn't mix with anything.
Cindy gets to her car and stops. There is still the possibility that Jamie just wants to go home and be alone. That he has a girlfriend who waits for him with the barbecue on and steaks marinating in the fridge beside the cold beer he shouldn't drink more of, but she'll let him do it anyway out of love. Maybe he is short of money and hopes that if Mike and the other guy were to catch up to them and demand money he wouldn't get into a fight if he's with a woman like Cindy.
"Why are we stopped here?" Jamie pulls her closer, even though Cindy thought they were already as close as could be.
"This is my car."
"You mean this one?" Jamie pushes Cindy up against the door and falls on top of her. He grinds into her leg. An old couple walks their dog along the sidewalk, and Cindy smiles and waves at them politely. Their dog barks, and they huff at her public display and walk off.
"Pretty nice ride. What's your name?"
"Cindy."
"Cindy, you have a pretty nice ride." Jamie's hand is on the curve of her buttocks. Cindy hopes the Spanx are doing their job. His hand inches under her skirt. Spanx are far too complicated to peel off in the middle of the street. They require a woman to brace herself.
"Thank you, Jamie. Would you like a ride home?" Cindy peels herself off and unlocks the passenger door. Jamie falls into the seat and tries to play with the radio.
"There's no music, Cindy."
Cindy thinks about buckling him up so he doesn't rattle around. She closes his door and moves to the driver's side. Before she gets in, she calls Heather.
"Hey, lady. How's your pick-up night going?"
"Good. I've got a drunk young man sitting in the passenger seat of my car. And I've had two-and-a-half drinks. I shouldn't even be driving. And I don't know what to do about anything. He rubbed his hard-on right into my thigh. He is cute and has long hair! I might just drop him off and go home and finish that bottle of Merlot. Also, how is your dad? Was his surgery successful? Are you okay?"
"Cindy. You listen to me. You got this. You take him home, you walk him to the door, and then you just put your body right up to his. If he doesn't want anything, you'll know, but I bet he does since he already had his hard dick up against your leg."
"But what if I'm terrible? Like, the whole thing is sealed shut with mortar?"
" _Cindy!!!_ Does this car have music?" Jamie is yelling and pawing at the driver's side window.
"Heather, I've gotta go."
"You don't have to try hard. He's going to like it. And the surgery was fine, and I'm fine, and you're damn fine. And I love you, and text me in the morning, dammit."
Before she has a chance to respond, Heather's hung up. She opens the door and plunks herself down on top of Jamie's hand. He giggles and worms back into his seat. When she starts the car, talk radio buzzes in the background.
"Can I change the station?"
"Sure."
Jamie traces circles on Cindy's thigh while he gives her directions to his little house. She notices that the lights are off when they pull into the driveway. Jamie walks up to and then in the front door, and without prompting, Cindy follows him. He sits on the floor and removes his shoes. Cindy closes and locks the door. Jamie looks up at her standing in his hallway and doesn't tell her to leave. He grabs her calf to help himself up again, and it's just like back at the pub. Familiar and close.
"Nice house."
"Yeah. I know."
Cindy steps out of her white shoes and onto the gritty welcome mat. She holds out her hand and makes what she thinks is a seductive face but is more of a regular smile. Jamie grabs her hand, and they fall together again, the way it's meant to be. Cindy kisses the underside of his chin. It tastes like primer. She licks it a little to see if she can find a spot that's not too painty. Jamie is hard against her thigh. He walks her into the living room and throws her down on the couch. Jamie climbs on top of her, still in his paint clothes. He untucks her shirt from her skirt and then kisses her. Cindy reaches up and pulls out the elastic holding his hair in a ponytail. His hair smells like freshly cut apples. Maybe they use the same shampoo.
Jamie is having trouble with his zipper. He sits up, and Cindy bolts up beside him. He yanks at it, but it won't budge. She gets off the couch, stands him up, and easily unzips him.
This younger man is like a starter younger man. Shaggy hair and not a lot of style, what she had recently learned about style anyway. He is not Ashton Kutcher. He is not an actor with agents and managers telling him to dress better, to behave more adult while still remaining cute. Under his coveralls he wears the uniform of a regular young man: boxer briefs and a faded T-shirt that remind her of something she's lost.
Cindy can't close her eyes while they kiss. She's distracted by the mismatched couches, DVDs, flyers on the coffee table, giant speakers, empty bottles, posters that don't make sense to her. Ryan didn't have things like this when she was with him. He had an entertainment system and a leather couch and loveeat. Cindy feels lighter here though, like she's in a new world, and everything is strange and exciting.
Jamie reaches under her shirt and rests his hand somewhere near her bra clasp. If they weren't clutched together, Cindy would wave him on through with a flick of her wrist, but she doesn't know how to get him to take direction. She has to wait in the now. He yanks on it and pulls the elastic tight. She is tired. Tired of waiting. Tired of feeling like she can't have control. He can't undo a bra, unzip his own pants.
She pushes him back onto the couch, reaches behind herself, and unclasps the bra, unrolls the Spanx. Jamie stands, drops his boxer briefs, and totters into her with his hard-on swaying until she grabs and directs it. She feels rattled with his penis inside her, and then she feels like she thinks she's supposed to. Wet and excited. But worried about how fast he's thrusting, and if he's noticed the flaws in her body she tries not to examine in the fog of the bathroom mirror. Her breasts rub against his chest hair, and he jackrabbits into her pelvis.
Cindy feels him falling and straightens herself to keep him balanced. She flips back, and her hip bashes into the flimsy leg of a table. The weight of his body against her, she fights hard not to lean against this thing that won't hold the two of them up. She brushes his long hair out of her face and looks down to see that he, too, has a particleboard table with a bowl full of keys.
Jamie collapses on her, and the particleboard snaps. Keys and coins jingle to the floor. Cindy falls hard, wishing she still had a layer of Spanx to cushion her. Jamie whimpers and puts his hand to his forehead. He must have bashed it on the wall.
"Fuck. Shit. Fuck," he says.
Cindy pushes the broken table bits away from the now shaking Jamie.
"I'm sorry," he says.
Cindy reaches out to touch his back.
"Just give me a minute," he says, pressing his head into her chest.
He's just another man, Cindy thinks, as she cradles his drunken head in her arms. "It's okay. Maybe another time," Cindy says.
"Just give me a minute."
Maybe another time.
Cindy hopes that Jamie won't be sick. He's had a lot to drink. If he does need to throw up, she will lead him to the bathroom and hold back his hair, like an experienced woman. Then she will put him to bed on his side. And she'll wish that she could have finished something she started. But at least she is a woman living in the now. She started something.
But he doesn't throw up. He stands up, staggers to the kitchen, and opens the fridge. She hears the crack of a beer opening. She's still in a sweaty heap, braless and Spanxless. Through the sound of his chugging, she makes for the door, grabs her shoes in one hand, her bra in the other, and kicks the Spanx into the middle of the living room, smack dab in the middle of his mess. She won't need them when she gets home, opens the door, drops her keys on her own little table, unplugs the answering machine, and tosses it from her bedroom window.
A Beautiful Feeling
"Happy birthday!"
Joanna hands Pamela a card. Pamela delicately takes the card between two fingers and says nothing. She's holding a ribbon-wrapped succulent in a decorative pot. The card still between two fingers, Pamela holds the beautiful potted succulent out to Joanna.
"I made this pot just for you! In pottery class."
Joanna stares at the plant. A single subtle swirl of orange sorbet blooms from between rubbery leaves. It's not any occasion.
"Because you've got a dental appointment next week. Min mentioned it."
Min and Joanna are total work BFFs, rarely not together. Last time Joanna went to the dentist, her novocaine had lasted for six hours, and Joanna hadn't been able to eat anything for an entire day. Pamela took note of this.
"It's just a cleaning."
"Yes. But still. It's always better to have a gift from someone before any kind of trauma. Especially dental. When you get to your desk, you'll have this bright plant to greet you."
"Well, happy birthday, Pamela. And thank you so much for this. You're always so thoughtful."
"Oh, thank you for the card." Pamela tucks the card under her arm and walks across the office.
Pamela's sedan trunk brims with pastel-wrapped boxes. The weekend spent perusing the craft fair had inspired her. Tables full of gorgeous glass beads dangling from delicate chains, illustrated and hand-stamped cards for every occasion, small kits with special tools for drawing personalized designs on balloons. It's a never-ending supply of great giving ideas, small tokens of gratitude, ways of ensuring the people in your life know that you are there and that you care.
"Give and you always get a beautiful feeling" is a cross-stitch that hangs above the door of her craft room. She also made prints of it to hang in the common lounge at the office. And postcards in case anyone has need to send one to someone else in need of knowing they are important.
Joanna's pot wasn't exactly as she'd envisioned it. It didn't capture Joanna's quiet strength or strong work ethic in a way that felt authentic. But Pamela ran out of time. With so many other gifts to create, prepare, wrap, and distribute, sometimes a thing falls a little off the mark. It wasn't perfect, but neither is any of us. The plant itself was perfect, though. The exact plant for the angle of Joanna's desk and the temperature in the office. The plant will thrive. It will bloom on Joanna's desk for years.
Their office used to be such a dreary place. And, of course, it's still a place of business, but with a few small touches, some effort over time, it's transformed so much in the twenty years she's worked there. And why shouldn't a workspace be lovely? According to the Time Use Institute, most people spend, on average, fifty-six percent of their waking hours at work, and then for thirty-three percent of the day they're asleep. It's just reasonable and good for the soul to care for your workplace people.
Pamela starts to unload her trunk, makes a mental note to box up the slow cooker she bought last week. With no children and no husband, and no desire for either, the size of this crockpot seems a bit ridiculous, even if it was on clearance. It will serve a much better purpose at work, filling the bellies of her co-workers with healthy, homemade meals. The gift of nourishment.
Min gets to the office early to set up for Asha's baby shower. A lunch break baby shower isn't very exciting, but at least it can look like a party even though no one's allowed to slip Asha third-trimester martinis.
Laden with crepe paper and a shiny silver "it's a girl!" balloon, she unlocks the main door and goes to punch in the security passcode, but it's already been deactivated. She left last night after everyone else, because it was a slow workday with a hangover. Fuck her if she forgot to set it, and now some dirtbag is inside fucking around with their very important, but mostly uninteresting, documents.
There's a rattling noise, like someone trying to open a locked door. Then a crash like metal instruments clattering to the floor.
Min attaches the balloon to an office chair. Sets her backpack on the seat.
"Hey, fucker?! Who's in here?"
No response. She can only hear more rustling, the sound of items being stacked or packed. She flicks on the lights. Then hears a bang.
"Helloooooo!? I have been in two bar fights and one scrape on a volleyball trip."
Again, no response, but now the rush of the air conditioner has kicked in, masking noises from elsewhere in the building.
Min pushes the office chair outfitted with the balloon in front of her as a shield. At the corner, she grabs the fire extinguisher and peeks around. No one. Another bang. She screams, shoves the balloon chair down the middle of the aisle, and stomps down the hallway.
"Get out of this office!" Min shouts, barrels through the cubicles with the fire extinguisher poised over her head. She can hear someone in the supply closet. In the corner, there are canvas bags loaded up with something she can't quite see. A large and dark object blocks the door to the break room. Slowly, Min makes her way to the door of the supply closet, and from the side swings the extinguisher into the frame. Pam comes out from the supply closet, removes her earbuds and eyes the fire extinguisher in Min's hands.
"What are you doing with that fire extinguisher? I could have waltzed right into it."
Min pants and drops to the floor with the fire extinguisher. Over in the corner, she can see that the canvas bags are filled with party supplies, that the dark shape in the break room door is a mass of black-and-white paper flowers in the shape of an arch. They make her single balloon look pathetic.
"How could you not hear me yelling at you at the top of my lungs?"
"I was listening to a crafting podcast, like I do every morning."
Pamela offers to help Min get upright, but Min ignores her outstretched hand, rolls around on the carpet muttering swears. The break room looks fancy. Cloth tablecloths and jars of black-and-white jellybeans, black-and-white patterned scarves, and a checkerboard rug brought in especially for the day.
"Shit, Pamela, this is decked out. But fuck. I've been binging on horror movies."
"Oh, Min. You'll be all right. Let me make you some tea. Do you want a cookie?"
Her first boss had been stern, made everyone nervous because he valued traditional office culture, yelling when things didn't seem to go his way. There was hardly any socializing. On the walls hung cheaply framed inspirational signs left over from the eighties. Men in business suits decorated with pink and aqua neon zigzags proclaiming power, an odd angle of skyscrapers pushing success, a close-up of the front of a black Bentley to show the results of hard work.
Pamela tried to understand his way, because she wanted to do a good job. She'd gotten good grades, had secured this good job, and wanted to continue the theme of good. But after her first mistake, deleting an appendix while proofing her first-ever report, he'd exploded at her in front of everyone at the meeting.
Whenever her father had been angry, her mother would bake him something: a blueberry pie, almond cookies, a batch of muffins. So Pamela had spent her weekend baking. Her boss didn't seem to notice, but her co-workers did. Dozens of iced cookies consumed by mid-morning. They happily chomped as she happily watched their chomping. In that moment, to her the entire office had evolved into a better place.
The skateboard comes in its own case. Neon-coloured, with a custom-painted deck. The image is Min's childhood dog, Pork, done in vibrant geometric shapes by her favourite local artist. Min opens the case, looks in, closes it. Opens it and looks in again. Then she shuts the case, fastens the clasp, and places the whole thing on the lunchroom counter.
"I got a gift last week," Min says.
"Well, Min, you deserve one every week," Pamela says.
"Do I?"
Min can't take her eyes off the sturdy case. The polished wood handle, the coppery hinges.
"Yes, of course. And, honestly, this is an 'I'm sorry' gift. I want to apologize for scaring you the other day. Usually no one comes in early. And no one helps with party set-up at that hour."
The gift is the exact skateboard style that Min had posted about on Facebook two months ago, along with the idea of paying tribute to her dog with a portrait. An expensive thing that she couldn't justify buying for herself, but hoped the hints would take hold of her girlfriend.
"I saw Elle had posted about it online. That you wanted it. And it was a post hidden from you, and she wondered if anyone had seen it in town. Well, I just happened to notice this exact thing when I was browsing on the weekend."
"It's a custom board."
"I mean, I noticed where I could order one."
Min walks over to the coffee pot and pours herself a cup, the last little bit of caffeine left in there. She rummages through the top cupboard where they usually keep Baileys stocked at Christmastime, hidden behind a giant can of coffee whitener. Someone else had gotten to it first. She shakes her head, then shakes her booze-free coffee cup, and hot splashes jump onto the counter and the floor. She tosses the last few drops into the sink.
"Are you two Facebook friends? Are you friends with my girlfriend?"
"Oh, no, no, no, no, no. But she has a public profile, so I just went on and took a look to see what might be a good idea to get for you. It comes with a case!"
"I know it does, Pam. I'm touching it right now. I can't believe it comes with a friggin' case."
Pamela is already preparing a new pot of coffee, knows they are used to her doing it. Without coffee, their days would be less enjoyable, and she enjoys making days more enjoyable, not less.
Min grabs the handle and swings the case a little aggressively. "This is nicer than what my mom got me for Christmas. Shit!"
Pamela forgot to wash her tea mug in the morning, and before she leaves for the weekend she has to make sure it's clean, just in case it attracts mice or insects or produces a strange smell. That would not be a very nice gift for someone to find on a Monday morning. A rodent as an officemate.
"She is fucking out of control. She bought me a fucking expensive skateboard!" Min shouts.
"Min, she is not out of control. She was trying to be nice," Joanna says.
Pamela hears her co-workers talking, stops and collects herself outside the lunchroom window, sets her mouth into a smile. She sees Min throw the bottom half of a muffin against the lunchroom wall.
Pamela walks in to wash her mug, hopefully safe from flying crumbs. The water trickles over her hands, but she adds too much soap. The mug overflows with froth, and it bubbles up her wrists.
"Hello, girls. I hope you're both having a wonderful afternoon."
"Hi. How's your day?" Joanna says. Min doesn't say anything.
Pamela rinses the mug, sets it in the drying rack next to scratched Tupperware containers and barely rinsed communal plates. She pumps out a long strand of paper towel and wipes her hands on the scratchy surface.
Pamela reaches into the zippered pocket of her purse and holds out a fresh pack of gum to each of them. Their hands don't move from their afternoon coffee mugs.
"Does anyone want some? I bought a huge package, and I can only chew so much gum." Pamela whips her hands around, waves the gum in front of their coffee-clutching fingers.
"Sure. Thanks! This is actually my favourite flavour." Joanna reaches out for the gum, and Pamela hands it to her, then flicks her eyes onto Min. Pamela smiles at her, waits for a response, but Min keeps her eyes on the contents of her cup, as if she's searching in there for something she lost.
To commemorate the company's twenty-third anniversary, Pamela fills every office, cubicle, and common area with balloons. The company colours are teal and yellow and are represented evenly in each space. She had to come in at three a.m. to ensure no one would be there before her. And also to make sure she didn't scare anyone else. Min's reaction to the skateboard was not ideal. It was a bit rude. Pamela doesn't understand how someone could treat another person with indignity after they'd received something so special. Min hasn't even said a word to her since the gum incident, not even a thank you email.
In every twenty-third balloon, Pamela has left a special message and a Starbucks gift card for twenty-three dollars.
"You're a special cog in the wheel!"
"Never give up on giving!"
"A gift is from the heart!"
The new intern ruins everything. Kristen has a latex allergy, and as soon as she walked into the office everything went wrong. Joanna, as the head of the health and safety committee, calls for an ambulance.
"She can't breathe! Her skin. There are red welts everywhere," Joanna says into her cell phone.
Pamela rushes to collect some of the balloons from the intern's area, get them out of her space, and also so she can regift them.
"Get out of the way!" Joanna screams into Pamela's face. She stomps on several balloons in her kitten heels, and Pamela pops a few more as she stumbles to make room for her office-mates who are scrambling to help.
The girl's mouth is locked in a neutral position. Never did Pamela conceive that something like this could happen, that someone could react like this to something as whimsical and lovely as balloons.
"Oh dear," Pamela says, as she stands in the corner and watches as more balloons are popped and discarded as the paramedics rush in with the stretcher. Min kicks more balloons down the hall as they pick Kristen up from the carpet.
"Everyone, go back to your desks," their boss says. "We will make sure Kristen is taken care of, and we'll let you know how she's doing as soon as we know."
"I'll call her parents," Min says, and Joanna nods.
They both walk right past Pamela. She rushes back to her desk and tries to find out more about Kristen. How could she know so little about someone in the face of a tragedy that clearly needs to be dealt with?
She searches through emails, tries to find Kristen on Twitter, on Instagram, a website. A Google search turns up nothing conclusive. Why does her name have to be Kristen Smith? Is she even a proper millennial, with no social media presence? Who is this girl?
Pamela sits at her desk rearranging her file folders until everyone else has gone home. She creeps into the shared cubicle space, settles onto Kristen's desk chair, and begins a thorough rifling of every single nook and cranny, drawer and folder. Her desk is free of photos. Only folders full of work in her filing cabinet. The only indication that Kristen likes anything outside of work is the small pink eraser shaped like a bunny that hops over the end of an HB pencil.
Another weekend filled with essential trips to more out-of-the-way locales. Pamela is on a quest, and it means no stopping until the goal has been reached. _Would Indiana have stopped his search for the Ark of the Covenant?_ she often thinks to herself after several hours of luckless rummaging.
This time she's successful. Pamela lines the craft table with vintage fabric buttons. Each she carefully selected and has now colour-coded.
Pamela keeps an old ledger on the bookshelf underneath her hot glue guns and an arsenal of glue sticks. She re-covered the dull navy blue plastic with a child's wallpaper from a home decor shop that was closing. Small bears in business suits frozen against delicate yellow stripes. There's something cute and professional about it. Inside she tracks her spending. Each year a co-worker gets a larger gift budget. Frequenting remote second-hand shops and odd little stores is how she saves money. Then when she really needs to splurge on more elaborate special gifts she has that money set aside. Just like she did for Min. Even though it wasn't enough. She'll have to do something more. Something personal.
She fires up the glue gun and clears the table of everything that isn't going to make this her best gift yet.
"Name please?"
As the woman at the patient information desk scrolls on her computer for an interminable amount of time, Pamela adjusts her bags. Her purse keeps slipping off her shoulder as she tries to delicately balance the tote on the opposite side. It currently houses timothy grass and a small white rabbit.
A bunny is easy to come by just after Easter. So many rejected gifts. Such a tragedy to see a loved one's hard work thoughtlessly rejected by their family. Craigslist turned up a dozen right away. This one seemed best. Unnamed yet, poor soul, and a dwarf. He looks just like the rabbit eraser on Kristen's pencil. The only downside is that she isn't able to bring the house hutch she'd worked so hard on. Too large to present to someone in a hospital. But a snuggle with a new friend is enough. That is healing.
"I don't see your name here, ma'am."
"Excuse me?"
"No. This patient had very specific instructions. Family only."
Tiny claws hook into the canvas, paw at her ribcage. She lowers the bag slowly, pulls it into her arms.
"Well, that's absurd. I am one of her co-workers."
"Are you family?"
"A workplace is like a family."
The bunny rustles around, presses her feet into Pamela's stomach, ready to bound. She rolls the top edge of the bag over itself, holds it closer to her chest.
"I'm sorry. Family. Only. You can leave her a card or something here, and we'll get it to her."
Pamela shakes her head. She turns away and drops her hand into the bag, pets the soft head and ears. Gets into a smooth rhythm of stroking the fur as she walks a long way toward a door that says NO ENTRY. When she turns back around, everyone is working or staring at their phones. She pushes through the door and slips in. The next room is really a hallway, full of people in beds. She ignores them, the nurses rushing around, looks for another door. The next one requires a key card, but beside it is an elevator. She pushes the button and waits. Small powerful legs start to pummel her chest. The elevator is empty, and Pamela remembers the floor her cousin was on last year while recovering from falling out of a tree. It's all she has to go on.
On the fifth floor, she looks around to find another locked door and a nurse.
"How did you get in here?"
"Oh, I must have taken a wrong turn. I was told to take an elevator to this floor to visit a friend."
"You can't be in here."
The nurse swipes her card and points at the hallway. Pamela trips on the edge of the door and squeezes the canvas bag. A loud grunt escapes the bag, accompanied by even more intense attacks from little rabbit feet.
"Shhhhhhhhh."
"What is in that bag?"
The rabbit continues to make noises, her kicks violent, desperate. Pamela goes down the list of everything she briefly read about rabbit care. She bought the right hay, fed her pieces of apple and carrot, didn't touch her adorable feet. The sounds coming out of her are louder, more aggressive. She didn't read about this. Assumed bunnies are docile and silent: the ideal pet to smuggle into a hospital.
"Is there a live animal in that bag?"
"Excuse me?"
Pamela tucks the bag under her arm. A shrill screech responds from her armpit, and the rabbit's teeth pierce the fabric, make contact with Pamela's skin through her blouse. She screams in unison with the stowaway bunny, grabs her bitten arm, and drops everything. Now there's no screaming or grunting. The bag is still. Fragrant hay has spilled out onto the floor around her.
"I'm calling security."
"Please! I am just here to see if someone is all right! I'm very concerned for her."
The nurse is already talking to someone on a wall phone while glaring at Pamela. The words "unstable," "confused," "prohibited area" are what she can make out. She can't adjust her eyes, stares at the silent, motionless bag. Deep in her stomach a queasiness, in her chest she feels the warmth and motion of the small animal against her heart. She should have looked up more about rabbits, their fragility, their distinct communication methods. She should have been less impulsive, spent the weekend with the bunny first, taken detailed notes on her behaviour, kept a photo journal.
Down the hall, a security guard walks toward her. He looks calm and kind, not angry or annoyed. Probably the type of man who is very grateful for every gift he receives, who doesn't take generous people for granted. When he tells her that she has to leave, he reaches down to get her purse, hands it to her. This gesture secures her feelings about this man. She takes note of his name tag.
"Okay. I'm going to escort you out now."
"I just want people to feel special. I am not one to leave a mess."
He starts walking her away from the nurse, the bag on the floor. She runs from him and grabs for the bag. She swoops it up easily, because it's empty. No nameless bunny corpse. She's alive somewhere in the hospital, already abandoned twice.
"What we experienced last week was the very definition of thoughtlessness." Lisa has gathered them in the conference room to discuss the balloon incident.
Pamela can hear people whispering Kristen's name, can't quite make out what they are saying about her, if she's in the hospital, out of the hospital. Pamela had taken a few days off after her trip to the hospital. She rarely took vacations, often stayed late or arrived early, and had more days owed to her than anyone else on staff. It's her first day back since, and she hasn't checked her emails, heard any word about how Kristen is doing. Her now useless gift for Kristen, nailed and glue-gunned to perfection, is still sitting on Pamela's craft table at home.
"The health and safety committee will be creating a new handbook. There are many small violations and safety concerns that they will be taking seriously as they implement changes and create a better work environment. As well, they are looking for new members. The original team was only two members, and one of them was Kristen. Min has recently joined and has taken on the task of recruiting." Lisa points at Min, who flashes a goofy smile and waves her hand like a beauty queen.
Min and Joanna sit in front of the flower wall on the opposite side of the room from Pamela. Since the installation, the conference room has smelled of dead flowers, and she can see how many plants need to be picked or pruned. As if they can read her thoughts, Min and Joanna start plucking flowers from the wall. Each of them grabbing the deadheads. But also they're plucking fresh blooms, grabbing at the new buds and tossing them to the floor. Min pulls a mauve pansy and crushes it in her fist. Actively destroying Pamela's gift to the entire office.
"There is a card for Kristen at the end of the table. I know Kristen was our newest team member, but please sign it with your most thoughtful sentiment. As well, there's a fund to send her a gift basket, one that adheres to her strict diet and won't cause another horrific allergic reaction. Thank you all. See you at next week's meeting."
Pamela rushes to the card, wants to make sure her words are clear: "We learn from our mistakes, the best is always ahead of us. Warmth and health, Pamela."
She steps aside and sees Min's hands, dusted with pollen. Min picks up the card, gets pollen on the pristine white back. She rolls her eyes and elbows Joanna in the side. Joanna throws her a look and hands Min the card-signing pen.
"Hi, Joanna. Min, I would like to say something." Pamela steps close to them to speak.
"Is it sorry for almost killing our co-worker? Because that would be nice to hear."
"I would like to join the safety committee."
Min laughs, drops the card to the table, and doubles over. "Um. No. What about that?"
Pamela is shaking. "Excuse me?"
"No. You are not joining any committee I'm a part of."
"Min, come on, sign the card and let's go. She didn't know," Joanna says.
"And you brought in a balloon a few weeks ago, Min. It could have happened to any of us."
"It didn't happen to you. And I took your advice from your sign in the lounge. I gave away everything you've given me to the Salvation Army. Giving is great!"
"Well, I—"
"Pamela! Come into my office, please." Lisa motions for Pamela to follow her.
As she walks down the hallway, behind her she can still hear Min and Joanna talking, but can't make out a coherent syllable. Everyone muttering. She sits on a chair in front of Lisa's desk and watches her mouth move. All she can envision is the stripped blossoms, petals on the carpet being trod on.
"Did you hear me? You'll need to tone everything down, all right?
"Excuse me?"
"I think it's nice on special occasions, but right now it all needs to stop."
Her head is flooded with images: gifts still wrapped, glue guns at ease, deep frowns, her alone at home with her thoughts, dead flowers, dead dreams. No one hates a gift. Not if it's given by someone so good at giving.
After her talk with Lisa, she leaves work early. She says she has a doctor's appointment, but that's just not true. Everyone knows Pamela would schedule an appointment only before or after work, would start hours earlier to get her work done, would not leave everyone in the lurch. Her heart is beating too fast, her head unfocused. She goes straight to the craft store, pays to park her car for an entire day.
She drags a rolling cart through the aisles, examines the scrapbook papers in detail, holds paint pens up to her eye to really see the nuance in each colour. Pamela marvels at the variety of glue sticks available now, pats the perfect squares of felt, and pushes her hand into the rows of yarn, her skin immersed in textures and tones. Her heart beats less rapidly.
She turns the corner and sees a woman and her child in the ribbon aisle. It's like a painting she would hang in her living room, everything colour-coordinated to perfection. It's a favourite because of the level of organization. There are no missteps in the ribbon aisle.
"Mama, what kind of ribbon do you like best for birthdays?"
Pamela's ears perk up. She pulls her still empty basket behind her and looks at the silver ribbons.
The woman whispers to her small child that she doesn't need a gift for her birthday. "Honey, you're the only gift I need."
"Don't say that to your child," Pamela shouts. Her eyes sting with tears. The woman speaks to her, but she can't hear what she's saying, her head foggy again, her heart galloping. Tears are streaming down Pamela's face. A guttural whine escapes her body. She's staring down at the little girl, the mother's voice getting louder, and the child's small eyes also welling up with tears. She's small and scared. Pamela feels this too. A red-vested employee rushes over, is also talking, but Pamela still can't hear anything. The woman picks up her sobbing daughter and holds her face to her sweatshirt.
The staff member reaches for Pamela's arm and starts escorting her to the escalator. Pamela looks at this nice woman just doing her job, and she slaps away her arm and runs. Hand still clutching the empty basket, she winds her way through aisles, knows the way to get somewhere without anyone noticing. She's got a craft map stored in her brain. At the back of the store, she skips under the red rope and silently cries in the dark, empty craft classroom until the store closes.
The next day, Pamela goes in to check on the flower wall, make sure some of the plants are still going to survive. Something she cultivated in this office won't disappear. She can't hear the dull noise of the generator and of water being sent to their little roots. She checks the control panel. Everything is fine. She listens again. Nothing. Above the wall, she can see slashed pipes. Someone has cut the water supply to the flower wall.
Min and Joanna walk past the conference room windows, sipping their coffees as if everything were normal. Pamela tries to make a mental note, envisions herself in her craft room. She imagines swatches and scissors and catalogues and online sales, but when she opens her eyes there is nothing bright. The wilting flowers smell like boiled compost, and her nostrils stiffen. She's unsure where to begin giving again.
"I am the most thoughtful woman here!" Pamela shouts into the empty conference room.
Hamsters
Danilo lives in the white house with the green trim. The paint is chipped, but the flakes of trim are still green, and the brown of the wood shows through. Most of the house still looks white. It was already in disrepair when his parents bought it after they arrived in Canada from Manila fifteen years ago.
Inside, Danilo's mother puts a meal together for him and his younger sisters, five-year-old Imee and Maggie, "five months until seven," as she likes to remind him. His father has to work out of town for now, somewhere up north. There were layoffs, and he took the next job that came along. Once he makes enough money, he'll open a corner store, where Danilo imagines he will have to stack crates full of pop every afternoon.
After his mother reheats the pork and steams the rice, she will grab her purse and come outside in her scrubs to say goodbye for the night, even though it's still bright and scorching. Out front, Danilo leans on an old rake, comatose in the sun. His sisters play bad guy versus good guy, Maggie brandishing a water pistol, and Imee waving a rubbery fly swatter caked with fly guts. Danilo can't decide who is bad and who is good. Wiping sweat from his head, he thinks about which chore he can put off today and where he left his sunglasses. Rake held like a guitar, he taps out "Welcome to the Jungle" on the handle.
"I got you with my gun." Maggie is gleeful that she made a hit.
"So. That water is warm already." Imee never admits defeat.
They chase each other, run around him over and over again, as though he isn't their older brother but a tree that managed to grow out of the dead grass. The edge of the lawn is the colour of dust, but nearer the house, in the shade, the blades spring green and long. Danilo will have to mow along the perimeter of the house with the heavy electric mower.
"What's wrong with your eyes, Dani?" Maggie shouts as she passes him by.
"Nothing, Maggie. His eyes are fine." Imee starts to slow down. The shot of water has taken its toll.
"It's called squinting, twerps."
"Maggie broke your sunglasses."
"Did not."
Danilo stays still a moment, times it right, then drops the rake, swings his arms down, and scoops up the girls, one under each arm. Imee and Maggie shriek with joy.
Next door, Mr. Anducci waters his flowers and tries to ignore the shrieks. Danilo hates the garish pink and purple petals and the fact that the old man waters them every day, even though they're on water restrictions and only allowed every second day. Their mother doesn't even let them fill up their plastic dolphin pool every day. The Anduccis have had fifty years to perfect their house and yard, and the old man has a fit over any imperfection. He almost slapped Imee for picking one of his precious flowers.
"Please put us down," Maggie pleads, but he flips her over and lets her dangle upside-down at his side. Her tiny toenails glint with sparkly polish.
A newer sedan pulls up to the curb and Mr. Anducci turns off the water. Danilo puts his sisters down and checks out the car. He isn't that impressed, though he knows it has a sporty engine. When he can drive in a few years, he dreams of a less family-friendly car, something that looks cool on the outside. Mr. Anducci's granddaughter, Chelsea, opens the passenger door and leaps out into the sun, slams the door without a goodbye or thank you, and the car drives away. She swings a canvas bag over her shoulder and storms through the gate, her flimsy skirt swaying. Mr. Anducci sets his hose on the concrete steps, and she runs to his arms. When they've hugged enough, he pulls a red bill out of his pocket and presses it into hers. Danilo turns away.
"Maggie! Chelsea is here."
"Oh yeah, I see her. She's wearing a new tube top. Chelsea!"
Chelsea waves at the girls and laughs. Mr. Anducci waves at them too. No one waves at Danilo.
"Chelsea, can we play with you?"
"Please? You have the best stuff."
Even though she's thirteen, Chelsea sometimes humours Imee and Maggie. Danilo dreads dragging them home after they've been playing dolls or doing makeup at the Anducci's house. He is never invited, so he stays home and does his chores with music cranked, and sometimes he hangs out with Shane and his brothers from down the street, and they jam, or he'll stay home and practice his guitar alone. No one talks about a rock star's younger sisters.
"I'm very busy, girls," Chelsea says as she arranges the hem of her skirt.
Danilo can't stand this bitchy game girls play.
"Please," Maggie whimpers in her sweetest voice.
"We got new hamsters." Imee contributes what she thinks will be the clincher.
"Don't beg her. Let's go inside." Danilo reaches out to hold their hands, but they protest and slap him away. He dreams of new and better sunglasses and squints back up at the sky. Chelsea stands silently, checking her nails. Mr. Anducci glowers.
"Okay, tomorrow after I get home from tanning at the pool," Chelsea says. She flicks her chestnut hair, exposes bare shoulders streaked with tan lines. Danilo can't help but notice that her canvas bag digs into her shoulder, that it makes her slouch on one side. Stacks of plastic bangles weigh down her arms. They hang like uncooked strands of noodles.
"Dani! Girls!" His mom stands at their own gate, her hair smoothed into a tight bun, purse clamped in her right hand.
Imee and Maggie don't budge, eyes fixed on Chelsea, still luring them with her older girl coolness. If she was fat and wearing a T-shirt and no lip gloss, no jangly bracelets, the girls might not mention her at bedtime as if she were their own private goddess.
"Goodbye, Maggie. Goodbye, Imee. See you tomorrow," Chelsea coos and, surprisingly, waves at Danilo too before heading into the house. Mr. Anducci stays outside.
"She remembered my name."
"She remembered my name too, Maggie."
"I'm leaving, girls." Their mother tries again to get their attention.
"Your lawn is a disgrace!" Mr. Anducci is back to watering and yelling. Danilo's mother looks at the plastic toys dotting the dried-out grass.
"Sorry, Mr. Anducci. We'll clean it up. Right, Dani?"
"You don't have to apologize, Mom," Danilo says. The girls cower behind his legs.
"You _should_ be sorry. So brown. It makes the neighbourhood look bad." Mr. Anducci turns his hose onto his own lawn.
Their mom pulls her keys out, and Imee and Maggie kiss and hug her goodbye. Danilo hugs her too. After she's taken off in the hatchback, he goes to confront Mr. Anducci, but despite his large belly and bad leg, he's already inside with his beautiful granddaughter and obedient wife.
At dinner, the girls gobble up leftover meat and make sculptures with their rice. They all do the dishes together, Imee standing on the counter to dry and Maggie on a stool that makes her tall enough to reach the sink to rinse. Danilo washes and puts the dishes in the cupboard. Across their battered lawn is a direct view into the Anducci's kitchen window. Chelsea and Mrs. Anducci clear plates and talk rapidly. Danilo thinks his sisters must be disappointed that her delicate fingers, more suited to giving them flashy manicures, are being used for menial labour. A plate slips from her grasp and crashes to the floor, and Chelsea picks up the broken pieces, holds them out, and apologizes. Danilo smiles and looks to his sisters, but they haven't noticed a thing, Imee too busy telling Maggie that she's rinsing dishes the wrong way.
The tape player blares Guns N' Roses. When his sisters get ready for bed they want to hear "Girls Just Want to Have Fun." Danilo cranks up "Appetite for Destruction" and pretends he doesn't know what they're talking about. They point at themselves and yell.
"Girls! Girls!"
Danilo doesn't see the value of Cyndi Lauper, but they're still young and haven't refined their taste in music yet. Rooting through the Barbie lunchbox full of cassettes, he examines each one and then tosses it back in.
"I see everyone else in here. Here it is, Madonna."
They shake their heads and yell and squeal.
"No! No! Girls! Girls!"
"Don't you like Madonna?" Danilo thinks she's a bit better looking than Cyndi, which must be why girls like girl music. To admire a pretty girl singing and want to be just like her.
"She's okay. We don't love her," Imee says, speaking for the two of them. Maggie diligently brushes her teeth in the doorway to show Danilo that she's taking bedtime seriously. After a few more fake-outs, he slides the right cassette into the player and turns the volume up. Imee jumps out of bed and rushes to get her toothbrush too. They dance in their nightgowns and sing through the Crest foam, then spit and dance and sing all over again.
He brings in their new pets, and the girls kiss the hamsters goodnight.
"Goodnight, Casey." Maggie nuzzles his brown face.
"Goodnight, Cyril Sneer," Imee whispers into his peachy fur.
"Danilo?" Maggie's voice is higher than is comfortable.
"You ask every night. You can't sleep with the hamster, Maggie."
He tucks her in tight.
"You don't know everything," Maggie pouts.
"You'll crush him."
He puts Casey and Cyril Sneer back in their home, a tall white bucket filled with cedar shavings.
"No, I won't."
"No, _I_ won't because I'm so much smaller than Maggie," Imee says with a sinister giggle.
"Are you saying I'm fat?" Sensitive, Maggie pushes her whole body under the covers and whimpers. Danilo pats the ball of her covered in blanket.
"No one is fat, and no one sleeps with hamsters. And no one will get to see Casey and Cyril if you don't stop fighting and act cool."
Maggie's head pops out of her blanket. She gives him a nonchalant look and props her head up on her elbow. "Whatever you say, boss."
Wrangling the bucket in one hand, he clicks off the light and closes the door. The critters rustle in their fresh shavings. He puts their bucket in his bedroom and heads downstairs.
"I'm cool. Right, Maggie?" Imee's voice reaches him in the living room just before he switches on the TV to watch music videos.
Mr. Anducci is staring at him again. Danilo pushes the old mower over the healthy grass, careful not to run it over the cord. He's finally gotten the girls to pick up their toys and put them in the toy box, and now he's clipping the grass. Brown grass can't turn green overnight, but at least he can make all of it a similar length. Mr. Anducci leans on his hoe, looks in Danilo's direction. It makes him uncomfortable. Maybe Anducci will freak out and yell at his mom again. Maybe he's gloating about his immaculate lawn, perfect garden, and vines bursting with grapes along his metal fence. Sturdy and clean, no chipped paint, no missing posts. He should take care of their yard, share some home and garden tips instead of being a jerk.
Inside, his sisters wait for their date with Chelsea. They're fighting over who will wear the blue dress because they've decided that must be her favourite colour.
The temperature is higher than yesterday, and Danilo wishes the girls were playing outside today. Then he'd have an excuse to fill up the pool and climb in with them. If his mother wasn't so nervous about deep water, he could have taken them to the public pool where they could study Chelsea in her bikini, observe her attracting boys and brushing them off. Instead, Imee has spent all morning making a pile of clothes on the floor as Maggie clings tightly to the coveted dress. He hears their shouts through the intermittent lulls when the motor stops working and he has to kick it to start it back up. It takes pushes and kicks to use the heavy machine, the process so slow he's not sure when he'll finish. Slash and Axl don't mow the lawn. They don't have brown grass either.
Mr. Anducci hobbles out of his yard, and Danilo kicks the lawn mower for the millionth time. Happy not to have the old man stare at him while he tries to deal with this crappy chore and crappy machine, he lifts his shirt up to swipe his face. Sweat soaks through the cotton, creates a salty, black licorice smell that stings his nose, reminds him of his father. When he drops his shirt, he sees the old man limp toward the extension cord on the mower and unplug it. Danilo kicks the base hard.
"What are you doing?"
He stands tall, but the older man towers over him, even though he's shrunken over the years.
"This is not how you do it."
Mr. Anducci takes the mower from Danilo and flips it over onto its side. The blades are coated in flecks of green and may be a bit rusted. Danilo wrestles with the weight of the metal and the man to right the mower again. All he wants to do is finish the job and not have to deal with his neighbour's anger problems, and strum a few songs before reheating dinner. "Get away from my lawn mower."
Mr. Anducci ignores him, drops to the ground to fiddle with the blade. "See here, boy. This is your problem."
"What problem?"
He wavers on his bad leg, tightens something with a small screwdriver from his pocket. "Now you won't have to kick it like an angry animal."
Danilo is not the one who's angry all the time, the one getting into everybody's business. Mr. Anducci plugs the mower back in and motions for Danilo to push it. It runs smoothly, making a neat row of cut grass along the side of the house. He turns it around and pushes the other way, and there is no lull. He can't hear two little girls arguing about who's got the cuter outfit. He turns it off.
"And you cut it too short. That's why it burns."
Again, Mr. Anducci's eyes are fixed on him, even though he's already fixed the mower, gotten him to mow in the first place. The old man points to Danilo's arms. "Do you lift weights?"
A disgusted look on his face, he shakes his head.
"You are strong though."
Though he's lean, his arms are getting big. While he practices his chords, the curve of his bicep is visible, and he glances at it to check out how it looks. This summer it looks good.
"I can't move as well as I used to. I have some old pipes that I need to get rid of. I need more room for my garden. They're behind my greenhouse. You can move them out to the alley for me. Next summer I can show you how to make that better."
Mr. Anducci points his hairy hand toward Danilo's backyard, a concrete pathway beside the carport, and a small plot of dirt dominated by a single row of vegetables, weeds, and lawn chairs. His mother only had time to properly tend the garden for the first few months of his life. Before he even ate solid food.
"I have to watch my sisters."
"It will only take a few hours. Tomorrow."
The conversation is broken by several loud thuds coming from the upstairs window, followed by crying. Danilo nods his head. He could use a break from caring for the hopeless lawn and sisters.
"I'll pay you something."
Danilo thinks of Chelsea's spindly arms, just enough muscle to hold up a fifty-dollar bill.
Dinner is Imee talking over Maggie talking over Imee. Chelsea braided their hair, and when they unravel them in the morning they will have almost-perms. That is why neither of them must get their hair wet and so can't take baths, to make sure they look their best. Nail polish remover stung her scraped thumb, but Imee pretended that it didn't because she squirmed while Chelsea drew the gold star, and it was her fault they had to start all over. Maggie made up a dance routine and sang, but not too loud, and Chelsea told her that she had a beautiful voice. Chelsea is going to give them some of her old clothes to wear. Chelsea doesn't have a favourite colour, but she does like blue, but also red and purple and definitely pink. During the school year, Chelsea goes to catechism on the same day as they do, but across the bridge. Chelsea shaves her legs. While they talk about Chelsea, they make heart shapes with their rice.
Before bed, Casey and Cyril Sneer make a brief appearance to be kissed and brushed. Danilo puts on "Material Girl" while his sisters sing into their toothbrushes. Chelsea likes Madonna more than Cyndi Lauper.
Danilo is surprised that the Anduccis have garbage too, but not surprised that it's hidden from view. The pipes smell strange, like there's been a long-dead animal or vegetable rotting inside them while they've been stacked against the fence and pinned to the greenhouse. He's careful not to crack the plastic greenhouse wall as he hauls them out into the alley, one by one. On his hands he wears a large old pair of Mr. Anducci's work gloves. They make his wrists look like copper wire.
Cars line the gravel lane of the alley, most of them second vehicles bought second-hand for teenagers to cruise in, to and from each other's houses. Lines drip and sway with laundry in backyards. Barbecues sit idle until fathers come home to light them and throw on slabs of meat. Every evening, Danilo can smell steaks and hamburgers being grilled.
A pack of kids runs from yard to yard hurling water balloons. A stray skids to a stop behind Danilo and splats him in the back of the knee. Pipe in his hands, he turns, careful not to knock the boy out. It would have been nice if the kid had got him in the head or chest. He's boiling. Kicking out a cloud of dust, the boy jerks through a fence and out of sight. Danilo hacks dirt into his lungs. Between coughs, he hears Imee bawl, a high shriek followed by blubbering. He ignores it, drops the pipe, goes back for another. Hopefully, his mother will sleep through the noise.
" _Dani!_ Help me." It sounds as though she's been in a serious car crash.
Danilo moves to avoid hitting a parked car and sees her, red-faced, in the corner of the yard. He walks over.
"Dani," Imee sobs into her closed hand.
Danilo puts down the rusted pipe and pulls Imee over the fence. His loaner gloves leave a smear of red metal dust along her pink T-shirt. She scowls at the dirt.
"Dani! Now I'm dirty too," Imee scolds him and wriggles to the ground.
"Sorry, Imee."
Danilo takes off his gloves and throws them down. One hand still in a tight fist, she examines her shirt. He knows she will make him pay for this later, for ruining her matching shirt and skirt with the printed kittens and her carefully selected matching rubber flip-flops from her basket of coloured flip-flops. It will cost him at least one dollar to ply her with a new pair from Kresge's.
"We'll clean you up."
"Fine. I look ugly now."
"You look the same."
"No."
"Why are you crying?"
Danilo watches her face change as she remembers her distress. She pulls open her tiny hand and points to her index finger. Chipped green wood protrudes from her brown skin, ringed with red.
"I got a sliver."
"Were you climbing on the fence?"
Maggie is at the fence looking on in fear. She must have encouraged whatever game caused the injury.
"We were only playing adventure models, Dani," Maggie yells from their yard.
He yanks the wood out, and Imee yelps. A stubborn piece remains under a thin layer of flesh. She looks down at it and wails.
"It... will... never... come... out," she screams between sobs.
He shushes her with a finger to her lips. She spits. "Dirty hands."
Mr. Anducci stumbles out of the house, limps out to the garden. Chelsea is right behind him, her tanned legs fully visible beneath her neatly rolled up denim shorts. She frowns at Danilo.
"What's the matter, cutie?" Chelsea trills this out, a bird's voice coming from her stiff body. She bends at the waist and puts her hand on Imee's hot black hair. The presence of the queen of girlishness calms Imee down enough to stop shaking.
"You? What did you do?" Mr. Anducci's accusations make Danilo want to run the rusted pipe through the fence.
"She got a sliver."
"It'll be okay." Chelsea pulls Imee in for a hug.
"And Dani got me dirty with his gross gloves!" Imee pushes herself further into Chelsea's body. Clinging to the fence, Maggie sings Cyndi Lauper to get everyone's attention. Each verse, her voice gets louder. Danilo thinks about what Axl Rose must be doing right now.
"If that fence wasn't such a piece of rotten shit, then this never would have happened. Are you okay, little one? Of course you're not. That place is a hazardous zone." Mr. Anducci reaches down to comfort Imee, but she jerks away from him.
Without thinking, Danilo kicks his gloves. Mr. Anducci glares at him and huffs. Imee hides from the old man and cries hard again. Somewhere Axl Rose is in a hot tub with champagne, writing epic rock and roll.
"Gianni, you need to finish eating and take your medicine. And her little hand could get infected." His wife says this from the door, fanning herself with a romance novel.
Chelsea takes Imee's uninjured hand. Maggie's singing changes to a wild rendition of "Like a Virgin." Mr. Anducci glares harder at Danilo.
"Gianni! Get in here and take your pills now," Mrs. Anducci says and slaps her romance novel against the screen door. Her husband takes a quick glance at the greenhouse before he limps back inside.
"I'll take care of you, Imee," Chelsea says.
"Thank you, Chelsea. You're so nice and so pretty."
Chelsea laughs at Imee's precociousness. Danilo watches the two girls walk to the house: Chelsea's preened hair a few shades lighter from the sun, Imee's as black and thick as ever, kinked with an almost-perm. At the door, Chelsea scoots Imee in ahead of her and waves for Maggie to come over too. Maggie climbs over the fence. One less chance for sibling rivalry.
There are still stacks of pipe piled behind the greenhouse, but Mr. Anducci gives Danilo five dollars and tells him he can finish tomorrow. He goes home and practices his finger-plucking, no girls to disturb him.
After the wound was treated, they performed one of their dance routines for Mr. Anducci, and he told them they were very talented. Mrs. Anducci packed up food for the girls to take home. They helped her make ravioli, and Imee wants to keep them in her room to show off. Danilo convinces her they should eat them for dinner instead. Maggie is proud because hers look much neater than her sister's and have more filling.
Over their handmade ravioli swimming in red sauce, the girls tell Danilo that they told Chelsea he can sort of play the guitar. Then they had an air guitar contest, and nobody won because they were all so great. The girls too hyper to be helpful, he sends them to play and does the dishes himself. There is nothing to view in the Anduccis' window but an empty dining table.
At bedtime, Imee flashes a neon Band-Aid over her sliver wound. Maggie flashes a matching plastic bangle on her wrist. Imee whines that she left her new bracelet in Mrs. Anducci's living room. She took it off because she didn't want to get Italian food on it.
The hamsters are forgotten, and he thinks about how ridiculous it would be to bring ravioli in to be kissed goodnight. He goes down the hall to grab his guitar. His bedroom smells rotten. Hamster droppings. Cyril Sneer is sweet like always, but when he reaches his hand in to fish out Casey he gets nipped. With a quick snap he lifts him by the scruff and underneath his white belly is a mound of pink. The babies are hideous, and even though they squirm and squeak, their eyes are closed, and they look dead. Remembering the ravenous mother who ate her babies and the angry father who almost killed her in the kindergarten classroom, Danilo drops Casey back into the bucket. He makes Cyril Sneer a new, smaller home out of an ice cream pail, an ashtray for a food dish. There is no extra water bottle, so he uses an empty jar lid. Cyril will just have to make do.
Downstairs, Danilo sits in the quiet heat of the evening. This summer is almost over, but maybe next summer he could get a real job, and he won't have to work for his father counting out change for popsicles. His guitar playing will be killer by then too, and maybe he can start a band with Shane and his brothers. An all G N' R cover band until they write some of their own songs. He'll have to get an electric guitar.
He hears a knock at the door. Shane and his brothers are at the family cabin, and no one else would come by so late. He opens the door, and Chelsea breaks the night in a breezy sundress. A nervous grin on her face, she holds out a lime green plastic bangle to him.
"Imee forgot this. I gave it to her."
Danilo takes the bracelet from her, heavier than he imagined it to be. She looks over Danilo's shoulder and into his house.
"Thanks."
"Is that your guitar?" Chelsea pushes past him and heads for his guitar sitting on the couch. She sits beside it and puts her arm around it, rests her head on it and looks at it lovingly.
"It's like your date."
"Thanks for bringing this over."
He tosses the bangle onto a chair but stays by the open door. Chelsea moves away from his guitar and sits up straight. She folds her hands into her lap. He plays with the doorknob and watches Chelsea make herself comfortable on his couch, in his house. With her own bangled hand, she pats the seat beside her and smiles. Her teeth are the slightest bit crooked. He shuts the door.
"Come and play something."
"You've never even talked to me before. Ever." He picks up his guitar and sits on the other end of the couch, rests his instrument on his knees.
"I thought you probably think that I'm immature or something."
"Why?" Danilo looks at her smoothly shaven legs. They are very similar to his own, thin with hints of muscle running down them. That sucks.
"You're just such a quiet guy. And I play with your little sisters all the time." She looks down at her folded hands.
"So?"
"I like them. They're sweet." Chelsea looks over at him, his guitar resting uselessly on his legs. "I like guitar music." Her voice is bright.
"You like pop music."
Danilo strums a few bars of "Material Girl." Chelsea laughs, and Danilo stops playing. She grabs the neck of his guitar.
"No, I like it. Keep going."
"No. I don't know it very well. I don't practice it all the time."
"What do you know well?"
"You won't like it."
"Just play it." She kicks his leg and bumps the guitar on his lap. It twangs and tips, and he grabs it before it hits the floor. Careful with each chord change, he plays a very slowed down version of "Welcome to the Jungle" and mouths the words. Chelsea sways to the music. He screws up and starts over. Then he screws up again. He finally gets through the first verse and puts down his guitar.
"I like it better that way." Chelsea lays down on the couch, her feet crossed in the air.
"I'm just learning, so I can't play it like it sounds on the record. I'm not Slash."
"That's okay." Chelsea pats him on the leg.
"The album only came out about a month ago."
He sets his guitar on the floor. She stretches her arms out in front of him, and he tickles her the way he does when his sisters get tired and yawn and stretch. Chelsea laughs and tries to grab his arms away, but he's stronger than she is. Her body twirls and twists, and when Danilo stops tickling she lands with her head on the arm of the couch, her body lying across his lap. She slaps him lightly on the cheek. She looks relaxed, and Danilo feels relaxed too.
"Can you play 'Papa Don't Preach'?"
"I don't know that one."
"It's on the newer album."
"My sisters don't have the newer album."
"I'll make them a copy."
"Please don't."
They both laugh. She won't stop making eye contact with him. He can't stop smiling at her crooked smile. He wants to reach down and touch her smooth legs, but instead leans in and kisses her. The arms he thought were so weak a few days ago wrap around his neck and hold on tightly. They stop for breath and look at each other. Danilo looks to see if Chelsea is going to hesitate, decide that he is unworthy, run away. She doesn't. He wishes he'd put on a tape, kisses Chelsea's warm lips again. She continues to grip his neck.
The sound of a running hose breaks through their kiss. Hard streams of water against the wall of the house. Dani chances to open his eyes and through the slit between the curtains sees Mr. Anducci stationed across the fence. In his own yard, he's got his hose pointed at Danilo's lawn. He drowns in another kiss, but the sound of water beating against his home distracts him. The guitar drops to the floor, and he gets up.
"What's wrong?"
"Nothing. I think I hear my sister crying."
"I don't hear anything." She moves toward him, puts her hand on the crook of his thigh. He blocks it.
"She has a soft cry."
"Which one? They both wail."
She laughs. He feels the pressure of her hand on the cotton of his shorts, but it's like she's touching underneath the fabric, underneath his skin, underneath the place where songs come from.
"I've got to go check on them."
Dani shifts, pulls up the neck of his guitar between his legs, separates the bare flesh of their summer skin.
"I can wait," she says, lounging with an arm above her head, cradling a bouquet of her hair.
"No. They take forever to get back to sleep."
"Fine."
She gets up and adjusts her strap back onto her shoulder, walks across the room stretching her arms high, exposing another few inches of thigh. Dani sinks into the couch, crushes his palm against his leg.
"Shit."
"What?"
"My nonno is outside. He can't see me in here." She plops back down beside him on the couch. "He wants me to marry a nice Italian boy."
"We're not getting married."
"Not that there really are any."
She slides her body down the couch so she's almost vertical. An errant leg kicks; her toes strum the strings, a long, messy note. Dani stops the sound with his hand.
"They're all so selfish. With their cars and their biceps. And their moms. Nothing closer than a mom to her son. Momma's boys. All of them. Trying to please her while they wait for her to serve them dinner and stuff. At the table waiting for food in their stinking clothes. It's not worth it. But that's what they all do."
With a wriggle of her toes, she strums the guitar again. "Everyone expects me to be a mom in five years. I can feel it."
Danilo's mother was seventeen when he was born, close to eighteen, but not quite. His parents got on a plane, got away from their old lives, before he needed luggage.
"Shit. I can't go out there."
"Well, I'm going up to check on them."
"Fine. Shit."
"He won't care."
The look she gives him says that he will care. A lot.
"Wait outside the back door."
"Boring," she says but picks herself up. At the door she tries to kiss him, but their heads move away at the same time. Neither one apologizes. Or says goodbye.
Through the door, he can hear her quiet breathing, the waiting. He creeps upstairs. Out the hall window, he can see Mr. Anducci's changed direction, trying to nourish the vegetable garden, the stream jumping the fence. Danilo's property drowning. Chelsea hasn't emerged from his yard yet. He assumes she'll wait in the darkness until her grandfather packs it in for the night. Instead she pulls around the far corner of the yard and comes through the back gate. Mr. Anducci turns off the hose, and brings her into his arms. She stands in his embrace, and Dani can tell that's she lying about something. They go inside. In the kitchen window, he watches Mr. Anducci gesture to Chelsea and his wife, and they hustle to the fridge. He looks away, walks down the hall to the girls' room.
He presses his ear to his sisters' door to check that they aren't awake. Still asleep, no cross-bedroom chatter, no bedspring squeaks from restlessness. They are sound asleep, worn out after a long, stimulating day. He heads to his own bedroom, sets his instrument under his bed, and throws himself on top of it.
Danilo wakes up in his clothes to squeaks from the bucket. Instead of two hamsters there are a dozen, Cyril and Casey not boys after all. He must have touched some of the babies earlier when he fed them, because Casey has eaten two of them. The others wriggle in the blood of their departed siblings. Danilo finds them so strange. They look like baby moles. They could be baby anythings. They look barely alive, but someone will have to take care of them too. In the ice cream bucket, Cyril Sneer is scratching to get up the side, sliding down, trying again. At night, they have the most energy, rustling and playing until morning. He takes Cyril's ashtray, dumps out the food, and uses it to scoop up the remaining newborns.
Through the kitchen window, Danilo sees that all of the lights are off in the Anduccis' house. He goes out the kitchen door and into the night. He doesn't know what time it is. It's finally cooled off. With a steady hand, he throws the babies one by one over the fence. They land between the neat rows of the garden. Danilo can't see them once they've landed, their thin bodies hidden by lettuce and zucchini leaves.
Sometimes Imee and Maggie don't remember they have two hamsters. They won't get to meet the newborns, with their skin pink and quivering like Chelsea's lips. He'll have to break the news to them that Casey is a she and not a he. Danilo is sure they'll be thrilled to have another girl in the house.
Haul
_Persona: Test Video_
Camera is small. Frame. Small. Close. Not too close. Small frame. A small girl is framed. Short. Propped up on nautical striped pillows. Her sensible decor not visible through the lens of her webcam. Her face, though. Yes. Cheeks ripe as genetically engineered peaches and of a similar colour. Orange and red and pink and purplish tones along the ridges of her round cheeks. Lips lined a little sloppily, lipstick steadier. Revlon Plum Velour, a drugstore staple. Gloss is L'Oréal Colour Riche Le Gloss, in Golden Splash. Cream shadow is Sunset, M A C. $19. Discontinued.
She smiles and pulls back, forces her cheeks and lips not to strain against her fake presentation. The face of someone who always excels at things but is trying something for the first time. A tightness in the jaw. Unpolished fingers fidget with something just out of frame. She holds up a skirt with bronze shimmer-thread woven through it, the pattern like fireworks. She wants to explode.
_Haul Index_
Three-button blouse. Red. Polyester. $12.99. On sale. Forever 21. Will match navy, black, stripes, polka dots, some florals. Would look good for school presentation days.
Jeans. Indigo. Stretch. $39. Not on sale. American Eagle. Can wear on weekends to visit family but not for special occasions, birthdays, holidays, etc.
Jeggings. Light wash. V. stretchy. Denim/cotton/spandex. $19. On sale. Forever 21. Casual wear good for winter months with long sweaters, or as transitional pieces for fall.
Necklace. Chunky. Gold-coloured. $6.95. Not on sale. Forever 21. Will match all colours.
Skirt. Mullet. Diamond pattern. Multicoloured. $8. On sale. Forever 21. Will match a variety of pastels and primary colours. Can wear to school or on special occasions.
_Note_
According to _Seventeen_ , mustard and gold can be worn together. August photo spread "Fall into Colour," page 58.
_Reading and Research_
_Vogue_ , August and September issues, current year. What to research: boots and coats, pattern mixing. Take notes on how to discuss different types of stripes.
_Teen Vogue_ , May–August. Current year. What to research: denim and leather, lace and feathers. Take notes on names for varying lengths of pants and skirts.
_Nylon_. All issues, current year. What to research: edgy style, personal style, not following trends. Take notes on what "edgy" means.
_Seventeen_ , September issue. What to research: Fall trends, all. Take notes on which trends will potentially play out best through video presentation.
Complete twenty #ootd studies and analysis reports per week. Watch shopping haul videos by _Candy Girl_ , _Becky123_ , _SureStar99_ , and _GorgeousGirlStar_. Take notes on presentation and clothing selections. Do not pay so much attention to hair.
Update: Research of blogs revealed that most popular videos are not perfect. Use bad lighting. Well, not as good. A bit grainy too. But don't want face to look shiny or greasy.
_Day Planner_
September 2—Shopping trip. Oak Valley Mall, Central Plaza, 8th Avenue.
September 4—First day of high school. Do not want to try too hard. Will wear old jeans and favourite tank top. Maybe new red flats, as old black flats have a hole. Though black flats are also favourite. Will make new first impression.
September 18—Group science proposal due. French test.
September 23—Movie club. Social studies assignment due.
September 29—English essay. Update Haul Index. Review Notes.
_Notes_
Budget looks good. Birthday stockpile still has some left over, since Grandma's card was late, and she felt generous. Will ask for money to go to the movies. Research plots of current teen movies. Become a haul keener. Become a style keener. Become friends with two girls.
_Catalogue_
Dim light. After homework, after dinner, after teeth are freshest, hair brushed down. She showers at night, keeps mornings free for homemade mochas and closet organization. She films herself doing the part she likes most. Eyes trained on a rainbow-coded spreadsheet. Cells filled with important information. The most important. Documents filled with notes. Goals typed in bold red textboxes: stay organized, make good impressions, try hard, be seen. Hands stampeding over her keyboard, keeping track of school work and this new work in equal measure. Key clacks are more interesting than piano lessons, than Taylor Swift's "Trouble." It's music, composition.
_Presentation 1_
Drape garments over arm. Discuss fabric, colour, design elements.
No.
_Presentation 2_
Reach into bag and present each garment with both hands.
Twirl garments, from front to back. Hold garments in front of body.
No.
_Presentation 3_
Use arms for manual zoom. Bring garments close to the cam and then back again. Make sure all of the pieces are shown in full. Hold garment in front of face.
No.
_Notes_
Revisit presentation folder for further presentation ideas. Watch clips of previous presentations. Take further notes. Put notes in "Notes" folder.
Revisit "Don't Leaf This Alone! Cardigans the Colour of Fallen Foliage!"
_Words to Use_
1. Cute
2. Fun
3. Flirty
4. Stylish
5. Cool
_Haul Video_ , September 15
"Hey, everyone. I'm new to posting, but I'm a longtime fan of so many of you, and you make some awesome videos. It inspired me. Thank you so, so, so, so, so much. First, I'd like to show some tops things I got at the Gap. This one is fuchsia with polka dots and buttons all the way up. This one is yellow and printed with tiny foxes. So cute. Also buttons all the way up. Both are cotton blends. I also bought two T-shirts, one in green, one plain black, but both cute and, like, soft. Nice. One pair of leggings in Navajo green and red and yellow. Same trip, I got these jeans with zippers on the ankles, indigo wash. Copper zippers. Oh, no. They're from Old Navy. Oh, no. Ok. The jeans are Old Navy. Can't believe I got that wrong. Thanks for watching and stay stylish!"
She uses the first presentation method mixed with the second. Reaches in with both hands. Discusses fabric and colour. Drapes garment.
She stutters a little. Voice a slight quaver, her face is stiff. Stiffer than she wants it to be. Eyebrows look like tight crochet pasted above her eyes, wide and wanting. Her arm juts into the frame and clicks off.
This one, she will broadcast the next morning. No edits. One clean cut. Keep it simple. Straightforward. Just like studying for an exam.
_Notes_
After watching video, there are three things. Be yourself. Be yourself. Be yourself. Also, try not to do that thing with the eyebrows. Turn the overhead light down a little more. Use bedside lamp.
No comments on video. Retweet every second day.
_Haul Index_
Shoes. Brogues. Patent. Pleather. Silky laces. Two-toned purple and teal. $28. Forever 21.
Boots. Dark brown. Leather. Flat with side zippers. $80. Aldo Ballet flats. Black with zebra print stripes. Faux suede. $14. On sale.
Heels. Purple. Faux-suede. One-inch height. Detachable gold bow. $25. On sale. Urban Outfitters.
Toms. Grey. $60. Foot Locker.
Slippers. Fluffy. Hello Kitty-shaped. Cotton and fleece. $18. On sale. Claire's.
_Notes_
Overbudget. Had to borrow money from sister. Parents will not give more money. Will do extra chores for two weeks. Most things bought on sale. Will try and be better about sales.
_Haul Index_
Satin, chiffon, boning. Cocktail dress. $120. On sale. Club Monaco. So fancy. Will look great on camera. Must have. Did not even try on.
_Day Planner_
October 1—Math test. Science dissection. Review monthly Haul update.
October 10—School dance. Not going. No one asked if going. Despite recent purchase of satin, chiffon cocktail dress. See: Haul Index. Create resumé.
October 16—Shopping trip. Apply for jobs at stores you do not purchase clothes at. Skip social studies group project meeting. Project due in one month.
October 24—English essay.
October 31—Too old to trick or treat. Hand out candy.
_Catalogue_
The sun is still down when she wakes up. By computer glow, she makes a new column in the spreadsheet. Date worn. Most of them read the same. Never worn, never worn, never worn. She checks for new followers. Four on Twitter. Her hand reaches out to tap the screen, though it's not a touchscreen. A finger brushes lightly over each of the new avatars. In her oldest hoodie, she grabs her backpack filled with worksheets: some created by her, some her teachers. Downstairs, the smell of hot cereal and her older sister's complaints about how high school has made her sister even weirder.
_Haul Video_
"Fall is here and this is the all-shoe haul. Feet are important. Keep yours looking and feeling great in this fantastic footwear."
She details every element of the shoes. Colour, texture. She has them fly around her head like sartorial spaceships. She is enthusiastic. She memorized this script. Verbatim. She read it verbatim.
_Notes_
Must remember, what happened can't happen again. Parents almost saw. The closet door needs to be able to close. Fully close. Tight. Must remember to sell items on Craigslist. Save up for necessary leather jacket. All the other girls are showing their leathers.
Comments are positive. People like her enthusiasm. She is still not herself, but people like her.
_Persona_
Jittery. She tries to coax the clothes into the plastic tubs, into the segmented cloth baskets, into organization. Shirts spill from the top shelf of the closet, pant legs peep out from plastic shopping bags kept under the bed.
Unworn dresses in the closet hide behind coats that rudely bulge. She layers clothes on hangers, fits the fresh tags into necks and sleeves. Bulky bodies. They are her and not her. These clothes still smell like packing peanuts, saran wrap slips, and cardboard. But she went in and bought them.
_Notes_
View count is up by two hundred. This is good news.
Comments are half complimentary, half creepy.
Ten new Twitter followers. Only two are bots this time.
_Day Planner_
November 3—Book report. Read something short. A novella.
November 13—Social studies presentation day.
November 20—Math test. Shopping day. Pick one mall only. After school.
November 24—English assignment due. French quiz.
November 26—Window shopping. No purchase. Get smoothie for energy.
_Catalogue_
She did not show up for her Social Studies presentation. Spent day at the mall doing research. Watching girls in their twenties walk in and out of stores. Their confidence, their legs and arms in motion, laughs and nods. Observing the motion of skirts, listening for the rustle of wool coats and scarves.
She takes photos with her phone. Click, click, click. Scrolls through. She wants a new skirt, a new sweater, a collar necklace in silver and gold. She wants an Orange Julius.
Her parents went into her closet last night to hide Christmas gifts for her sister, and everything became clear. Plastic bags full of clothes with tags, unworn sweaters, shoes in boxes. They said her neatness, organizational skills, intelligence wasted on this nonsense. She is not getting anything for Christmas. She might still pass Social Studies.
_Haul Video_ , November 25
"It's one month to Christmas, so I treated myself to cute and flirty sweaters. If sweaters can be flirty. This has hearts on it, so it's much flirtier than this one here."
She tosses the heart sweater onto her bed like it's nothing.
"This striped one I picked up on sale at H&M. So cute, right? Yeah. It's—"
A sigh. Silence.
She keeps filming, nods her head, keeps it close to her chest for too long. An awkward silence in a shiny visual medium.
She stands, head still down, walks with her back to the camera, opens the closet door. She pushes the clothes back with both hands as though opening the doors to a saloon. Inside the closet, she burrows into the fabric, the colour-coordinated shopping bags, the hung dresses, boxed shoes. Her body shifts and folds to fit. Her head slips between a cardigan and a sundress. Cotton, polyester, and wool blends crowd her. She backs in further, the vibrant material a curtain. Against the back of the closet she presses her shoulder blades into the wall, makes herself as thin as a flimsy skirt. Invisible to the lens, she can still see the green glow of the camera through the fabric until her body is fully woven into the tangle of textiles.
Nest
Sara rounds fine sandpaper over the curve of a domed wooden roof. The sides of the structure are curved as well. The house designed to look round and soft, not sharp. A request made by her client. This is just the model, what she brings to clients for inspection. First comes the initial consultation, then a scale drawing for approval, then proper blueprints, and then the scale model for approval. If all goes well, Sara will make just under $20,000 for creating one doghouse. Since her business, Barkitects Pet Designs, opened over a year ago, she's made a dozen pet homes and now has contracts for a dozen more in the next six months.
Today's meeting is with a client and her Pomeranian. Even though the dog is small, the paycheque is big. She often finds pet size is disproportionate to the amount of money an owner is willing to pay for a designer pet home. They consider their toy pets more in need of protection. Mrs. Doty specifically asked if there was any way of making the house in the shape of Posh's favourite toy. She gets out of the car and brings the model to the front door, rings the bell. Hopefully, it'll look enough like a hot dog in a bun to please Mrs. Doty and chipper Posh. From behind the heavy oak door, she can hear both of them scuffling to the door, the lilting pant of their breath, and the yips they use to communicate. A few sighs and oohs and aahs is all Sara needs to return home elated to celebrate another design success with her wife, Kate.
That night over dinner at the Izakaya place, she tells Kate about the contract, that they're almost stable enough. Almost enough to really pursue this baby, seriously talk to their doctor, seriously consider Kate's dream. Kate wants to be pregnant, carry Sara's egg and donor sperm inside her, feel the beat of a little Sara stir. Sara's doctor says her eggs are ready when she is. The cost has always been a concern. The cost of high-end sperm and in vitro is as expensive as several pet mansions.
"Almost? You're raking it in, babe. Raking in money."
"I have student loan debt."
"So does every asshole with a graduate degree," says Kate.
"You're right."
"I know. I'll start making the appointments."
In a city of never-ending construction, a never-ending real estate bubble, a never-ending supply of nouveau riche, Sara didn't think studying architecture would be a mistake. Her wife agreed, believed in the creative potential of Sara. When they started dating, Sara had made a tiny replica of Kate's childhood tree house, the one she'd lost in a brushfire. From a grainy photograph, she'd reproduced the tilted floor, the slope of the roof, the frayed rope that swung below the entrance, the two-by-four steps nailed to the tree. Kate was in awe of how perfect it was. It felt like she could crawl right into it and read Archie comics until sundown.
Sara dreamed of creative housing, interesting and unique. Recycled materials, stability, shape, colour. She saw images in her head that would enhance the glass and concrete landscape. She graduated and looked for a job and found there was very little out there, and what was there wasn't top shelf work. No one wanted to hear her ideas on brightening the urban image of the city. But more than anything, she needed to make money, needed to find a way to make Kate not regret all her encouragement. Feisty Kate, who'd worked and waited years to live her own dream of raising little Katelets and Saralets. She'd promised Kate that after graduation they could start a family. On graduation day, Kate high-fived her and kissed her, whispered in her ear that they were going to be the coolest moms.
Months later, she caught up with her adviser, Jed. Talking to him, they'd tried to think of answers to this problem. Over eight-dollar lagers they discussed all the same old things, the way to edge ahead in a competitive market. Where were the new ideas? Where were the new ways of making a name? Could there be a new Ando, Erickson, Gehry? Sara joked that she might as well design for dogs. Jed ordered another round to celebrate.
"Celebrate what?" Sara said, tipsy and distressed.
"Your new career. Architect to the dogs."
A little research revealed that the pet industry is an area of growth. Pet supplies, pet care, pet bakeries, pet masseuse, pet estheticians. No one had ventured into high-end, one-of-a-kind, designer pet domiciles. It was an untapped market. With financial help from Kate, they developed their business plan. They pulled it together so quickly that there was no time for Sara to question her decision. The name was a post-coital masterpiece. The two of them naked and sweaty on the living room couch in the summer heatwave, when Sara made a loud, throaty orgasmic exclamation. Kate looked up from between Sara's legs and started barking. Knew they had the branding figured out. Kate pushed her in the right directions. The website was cute but not twee, interactive but not annoying.
"We're good at working together," said Kate. "This is how we'll be as parents. Look how we collaborate on this shit? We killed it."
Sara spends the next morning organizing her day planner, checks that she has all her appointments in both her paper and digital planners. Kate has left a Post-it note on top of Sara's day planner:
_Remember. Sperm meeting, Monday, 3:20. I took the day off. xo_
She had forgotten. Kate probably saw it wasn't marked on the calendar but a client was. She texts Kate: _I have a mtg Mon aft_.
Kate responds: _I know. But really? Important?_
Sara stares at the question marks, their judging little swoops and dots. She'll cut her meeting short. She has to.
_I'll be there_ , she texts back, _xo_
Sara spent two years applying to schools and failing to get in, then four years of grad school, six months' worth of unemployment, another eight of unsteady income, all told more years and months of waiting than Sara thought Kate would put up with. Now that Sara is finished pursuing her dream, it's time for Kate to fulfill hers.
Sara knew they should have talked about it more. That during all the nights she spent locked away in her studio across the yard in their garage, all the mornings she crawled home to sleep while Kate assessed insurance claims, there was another person waiting at the end of the timeline. A little person that they would bring into their home and feed and take care of.
The more Sara thought about it, the more she considered this person a stranger, an interloper. They worked well together; they made things happen. Without Kate's commitment, there would be no Barkitect firm. This person might be cute, but there would be interference.
"We nailed it," Kate says, clutches Sara's calloused fingers in her manicured ones.
The appointment with the fertility clinic had been very encouraging.
"Gonna get us some spermatozoa, gonna implant that in my lady bits," Kate says.
"Don't get cocky."
"I'm not cocky, I'm confident. Your doghouses are blowing up our bank accounts, and now I'm going to be one of those glowing ladies who gets to be offended and pleased when other people want to touch my bulging belly."
"You're not pregnant yet."
"No shit, honey."
Kate dances around the car while Sara scrambles in her bag for her keys. Kate humps the front of the car and makes gestures toward her flat stomach. She's beautiful, healthy. She's funny, even in these moments when Sara wants her to stop squirming and sit quietly in the passenger seat, listen to talk radio while Sara drives. There are times while Sara renders drawings that she stops to think about how sexy Kate is, how her hip juts out while she tosses sweaters from the closet, the way her lower lip droops with concern. Her auburn hair in an explosive braid each morning makes it hard for Sara to leave the heat of their bed. In a few months, she will swell and stretch; hormones will tread all over her body, and Sara won't know who this person is. Two strangers. One inside Kate, the other someone she used to know.
"The woman seemed sketchy to me," says Sara.
"She seemed sketchy? You talk to women who dress their dogs like Scarlett O'Hara and offer you money to build canine-sized Taras and milk bone fountains. And the highly educated doctor who's going to squirt some baby juice our way is sketchy?"
"She had a weird haircut."
"She had a bob."
"It was threatening."
Sara rushes across town for her next appointment, has to park blocks away and run; her short, sideswiped bleached hair flips into her eyes as she tries to run, not run. Her undercut keeps her head from building up with sweat. The client, May, wants to meet in a small park downtown. She told Sara she'd be standing near a tree on the north side. She's asked Sara to construct a home for a bird. On the drive over, she belts out classic rock, lets the radio blare, gets pumped. She's never worked with metal before, but is excited for the chance to try, add another mode to her arsenal. It must be an ornate cage, something grand. She already has a line on some scrap metal she can upcycle into something Victorian. The woman mentioned in her email that she was interested in something classic and traditional.
A small woman with straight black hair wearing a cream coat stands next to a tree. This must be her. Sara walks over to her and holds out her hand. May points up at the tree.
"Hi, I'm Sara. Thank you so much for considering my services. I've never designed anything for a bird before. Mostly dogs and cats, some rabbits, a chinchilla once."
"Shhhhh," the woman says.
"Sorry. Would you like to sit on the bench over there and discuss..."
"Please. Just be quiet for a moment."
Sara stays silent and waits, tries not to stare. The air is warm and smells of bark and cotton. She notices that there are two clouds in the sky, each one bulbous, like cartoon clouds. Sara looks back to May, who twists her mouth and a strange trill comes out from between her lips. A small, dowdy bird emerges from the tree and lands at May's feet. The little brown thing hops about near the woman's pristine velvet flats.
"She doesn't live in my house, but we're close. I wouldn't consider her my pet, but we have a relationship."
"Excuse me?"
"I want you to create a home for her. I want her to come live with me. It's time. Something inside that will make her feel at home. Classic. Natural. Only materials a real bird would use. No substitutes. Authentic method of construction. I've been studying bird homes."
Sara feels pricks of heat in her cheeks. Is this woman mocking her? Is this Barkitecture backlash? Even Sara's surprised it's taken this long for someone to have a go at her. As ridiculous as she knows her business is, she always assumed the few online comments about her business would be the extent of negativity she'd experience.
"Why are you wasting my time?"
"Excuse me?"
"You'll still have to pay for this consultation."
"Of course. And then another payment after you've submitted the drawings. And the model, and then the final product. I think the best way to proceed is to develop a solid schedule and payment plan, don't you?"
May moves to the base of the tree, and the bird cocks her head up. Kneeling, May reaches out a hand to stroke the bird's tiny tail feather. It doesn't look at ease, but it doesn't fly away. This woman is serious. Sara falls into the bench, rubs her eyes to stop her head from spinning.
"I'll know if you're somehow faking it," May says.
Faking it. What does it mean to authentically create a bird's nest? Bold as she can be, as innovative as she is, Sara can't imagine what this project will be like. There are levels of questioning herself that she doesn't know if she can overcome.
"I'm not sure I can do that."
"Well, there's $100,000 in it for you."
Sara watches May watch her bird. On May's arm is a designer handbag. The bird's brown feathers match the tones in the Coach logo. They look coordinated, like women and their lapdogs. Sara wonders what a bird would look like in a Juicy Couture jacket.
"I have the money. If that's what you're worried about."
"It's not the money," Sara says, even though part of it is.
"I've seen your homes. There is something special in them. I want something special. I want to know that we are going to be happy together, that someone has taken care to make something that doesn't _feel_ like home—that _is_ home."
Sara isn't sure how anyone could know that this small brown bird is the same small brown bird, when all brown birds look the same. May stands, and the bird ruffles and takes off. May's eyes, dewy with love, watch her land on a branch to preen. Then a second later, the bird leaps into flight. May's whole body turns as she watches the small brown spot disappear in the sky.
"Okay," Sara says, "I want to do it."
Sara tells her about the bird, the woman, the cashmere coat, the paycheque as she scrubs Kate's back in the shower. Kate is thrilled.
"What if I fail?" Sara asks Kate, backed into the shower stream.
"Are you kidding? If anyone can make an authentic bird-house for an eccentric woman with more money than brains, you can."
Soaped and slick they reach for each other, Kate panting with excitement as they kiss, as Sara reaches between her legs, a celebration of fresh ideas and clean skin.
Every day, Sara traipses through the park, studies the bird. She's energetic, sometimes frantic, in the way she moves from grass to tree. Larger birds seem unsure about venturing. For hours, she watches the finch hop along the grass or the busted stone fountain. Sara hasn't asked May if she's anthropomorphized this bird, but it doesn't seem like her way. Sara can't help it, though.
She makes notes, sketches twigs and grass, imagines the way they'll fit together in a nest. She searches for empty nests while she and Kate walk to get groceries or gelato or new socks. And just like in grad school, she spends her nights in her studio, drawing, rendering, thinking, rethinking. She questions every move she makes. The library becomes an oasis. Structure, shelving, books, kiosks.
On the way to their doctor's appointment, Sara stops several times on the street, catches glimpse of twigs in trees. Examines them.
"See. These are not finch nests. Too sparse. See the way this looks like it could break apart?"
"I'll break you apart if you don't move faster. Someone else will snatch up this appointment if we don't get there on time. I am getting the fullest physical and fertility assessment that money can buy."
"Sorry, I didn't know we were late-late."
"What if it starts with you getting distracted and we miss this appointment, and then we miss all the important ones, and in the end someone takes my baby-making cocktail?"
Sara stops walking. After a few steps, Kate realizes she's huffing along on her own and turns. Sara's look is supposed to say, "Are you serious?" but, more likely, Kate thinks Sara's calling her an idiot. Kate's look tells her that she is serious, that they need to hustle. That she doesn't give a shit about finches.
"I thought you might be interested in this. It's an intricate project," says Sara.
"If we're not ten minutes early, we're late."
The curves of the nest are at odd degrees. There's no symmetry. Everything is angled to confuse. And each nest so different. Each finch as finicky as a West Vancouver housewife asking her to incorporate bunk beds for Persian kittens.
Immediately Sara has gone into research mode. Unsurprisingly, the females, of which May's bird friend is one, are doing the hard work of building the nest. Nice bird women in their unflattering brown house-feathers forming a comfortable cup-shaped home for their families to enjoy. To keep it all together, sticks and grass, twigs and leaves. For comfort, feathers, and, like in any good home, a bit of collected debris. There are thousands of pages of information on finches, their ways, the people who love to watch and court them with seeds and berries. Mostly vegetarians. No way to woo a finch with a pulled pork sandwich.
May approved the drawings, sent the first $25,000 the day after Sara hung them from the park bench and walked May through the construction procedure.
"I feel good about this," she'd said to Sara.
"Thank you," was her response, but what she wanted to do was an impression of Kate, running down the length of the park, arms thrown in the air, a joyful flailing of limbs.
The last appointment hadn't been ideal. They have to go back to find out the results. Kate said that the doctor said they should still do a few more tests. The next slot isn't available for three weeks, at a time when Sara has to rush to a mountainside supply shop that specializes in natural building materials, but tiny ones. She learned about it after befriending niche dollhouse enthusiasts online. It's very specialized and somewhat secretive and is open only six hours a month. Kate is angry about going alone but Sara promises she will spend some of her bird nest money on a gift for her.
Thinking of names has become a new game. They can play over breakfast, before bed, while driving, walking. Anytime. It's a good distraction.
"Cora?" asks Sara.
"Is she going to be born eighty?"
"No. Sorry."
"I thought we had it narrowed down to our top five for each sex."
"Oh, I didn't know Alana was still in the mix."
"Well, it is. Although our baby is going to be so cool it won't matter if she's called Assface."
"I think it will."
"Oh, Sar, you're so practical. I mostly love that about you."
"I love you too."
Sara meets with May on the park bench near the tree. Their designated client meeting spot.
"I've been studying their nests. There was even an old one near my place. The architecture of them is fascinating," Sara says.
"Remember, the individual bird is important to study, too. This is not merely technical."
The bird tends to her empty nest or, at least, that's what it looks like. Sara is still not sure about the intricacies of finch behaviour, let alone this particular finch, who seems to have an intimate relationship with a very wealthy woman. She makes a mental note to read more about behaviour. When they bob their heads they could be saying no. They could be acting out some important drama. She's never worked so hard on something, never been more intrigued, confounded, pushed to understand the creature that will live in her invention.
"Sara!"
Eyes trained on a finch video on her iPad, she looks up at Kate, her narrow shoulders drenched in spring rain. A striped straw protrudes from her lips. A jumbo milkshake. A bad omen. In the corner of her lip, a bit of liquidy chocolate ice cream pools and sputters. Then Kate is on the kitchen tiles, her body limp and shapeless.
The doctor wasn't even going to try, she told Sara. There was no point. Her uterus was being an asshole, she said. Basically, her body hated babies, and they were not allowed to live in there.
"I shouted that I wanted to burn my body down! Then they gave me an Ativan."
None of her other doctors had ever mentioned that she had an inhospitable body cavity. Why had she assumed? Why had she let this dream go on without being prepared?
"I'm sorry I wasn't there."
"You were working. You can't be everywhere."
"I probably could have taken an hour."
"It doesn't matter where you are. My uterus is an asshole."
For the next week Sara keeps Kate comfortable. Camped out on the armchair, her body curled away from the television, the window, trained on the corner of the room. Sara knows she imagined feeding the baby there, little fruit of her loins at her breast. Everything she can do to stop her from weeping, she does. Trips to the store result in bags of candy and chips, prescription meds, and Victoria Gin. She wraps Kate in blankets, threads pillows around her slouched form.
When Sara had asked about adoption, Kate had accused her of being unsympathetic. Had shouted and thrown her gin and ginger to the ground.
"You need to go to the doctor," Kate said. "You need to have your innards explored. You need to make a little Sara for me to love."
Every day after work, Sara meets May in the park. These meetings to sit and observe together. The first few tries, the nests are terrible. Sara knows this, and yet after her sixth she wants to present something to May, show her that she's been working hard on this project, that she's paid attention. In a small cake box from the bakery she's packed the nest in gold-flecked tissue paper, thinking it makes the structure and tones of the nest stand out. May's face remains stern as ever. She merely hands the box back.
"Please try again. I do appreciate the work you're doing."
They sit in silence, side by side on a bench, the same one they sat upon on the day they first met. May's breathing is calm and steady. Everything about her is so focused. The bird appears like it is willed by May. The only time her face softens is when the bird comes close to her, lands right on the knee of her designer jeans.
When the bird is near May, then Sara can take a closer look at them both. May's posture is less perfect than she originally thought. May's eyeliner is always thin and smooth. She favours neutral tones, except for her lip colours, which vary and are always vibrant. She is stern, but while the bird hops on her leg, looks up at May, her mouth cracks open slightly, feathers of a smile along the edges of her lips. For what seems like an hour, the bird happily enjoys May's leg as a perch. Then the bird trills a small song, as if she's asking May a question. In response, May reaches a small hand down, almost in slow motion she's so careful, and strokes the top of the bird's head. Usually so energetic, the bird remains still, petted like a content cat on a lap.
Sara fights her impulse to reach out and touch the bird herself, slip her hand flush with May's thigh.
Loud jazzy notes burst from Sara's pocket, her brassy ringtone for Kate, and the bird flies off instantly. May looks away in the direction of the bird.
"I'm sorry," Sara says, but May's gaze stays focused on the trees, doesn't acknowledge the apology.
Sara answers, hears Kate's agitated voice asking for more Reese's Peanut Butter Cups and gin, and muffles the call as she hurries over to a different bench to talk.
"I can't be at your beck and call. I'm working."
"Oh shit. Sorry. I thought I was your wife."
The light never goes out in her tiny garage studio. The light is always out in the house. They mostly communicate by phone, if they communicate at all. A text about toilet paper. Sara can see Kate sometimes in the light of the television, which Kate leaves on for company, though she doesn't pay attention to the exploits of Real Housewives. Sara has built a human nest out of sweaters and blankets, out of new pillows she bought herself. She never has to leave the studio, can work and sleep and dream all in one place.
Sara sneaks into the kitchen in the middle of the night for a snack. Kate has been sitting in the dark living room and shouts into the dark kitchen.
"God, this woman must be fucking special. You spend so much time with her."
Sara doesn't respond, spoons soup into her mouth, and stares at the curves of her bowl. A tight nest looks indestructible. They're designed to keep the eggs protected, encased. Everything reminds her of how this project seems to be failing.
She can't deny that she is intrigued by May. This woman who seemed so strange and unfathomable has elements to her, depth that Sara can't resist experiencing, even for brief moments in the wind-whipped park. Underneath the tree, May becomes more elegant than a yuppie in a gilded home.
"Are you ever going to come home again?"
"You're the one who told me to take the project!"
"I told you to do it for us. And this fucking baby that doesn't exist."
Sara doesn't know what to do. She plugs in her iPod, blasts noise, and walks back to her studio.
When she finally presents the finished nest to May, there is no fancy paper, though Sara did have to build a small stick structure to transport it safely. They meet in the park, as always. This time, May's face changes immediately. Not quite a smile but a flush of approval. She briefly holds the nest in her hands, then passes it back to Sara, gets a tissue from her purse, and dabs at her eyes.
"If this doesn't work, nothing will," May says.
"I really hope it does."
She's hired some men who work with movie actor animals to capture her bird, bring it to her gorgeously appointed home filled with expensive furniture, art, all the neutral clothes a closet can hold.
"I'm nervous," May says.
The bird is in her usual spot, neck jittering from side to side, so unaware of what is about to change. May watches her, clamps a hand on Sara's shoulder, a tight grip on her now tense body.
"Do you want to see where I'll be installing it?" she asks Sara.
Sara watches the bird take off and land softly on their usual bench.
"I'm too nervous."
Sara gets the call from May as she hauls a hamster mansion to a townhouse in Coal Harbour. At first, the bird refused to leave her perch near the skylight, onyx pin eyes trained on the sky as it changed from day to night. After a few days of sorrowful singing, May trying her best to be accommodating like a tree, like a shrub, like a cloud, a puff of air, the bird soars in the loft, rounds the space, and finally lands in her nest. May watched to see if she would react. All through her trials, Sara had imagined the bird would reject it, tear each slim piece apart until the binder failed, and the whole structure would disintegrate, would disappear like dust fading into sunlight. Instead the bird rested.
"I am so grateful," May says.
In the background, Sara thinks she hears the short trill the bird makes when she's near May. No eggs to protect, no outside air, no tall trees or dirt. No manufactured bench, all of them sitting in silence, waiting for a meaningful moment. Sara isn't sure how to feel. But she's made her client happy. May is overjoyed.
The house is dark and quiet, no sign that anyone lives there, that two people used to share small touches as they passed each other on the staircase. Sara stares at the front door, considers knocking. Inside, it's confirmed. Kate is gone. Dishes in cupboards, no crumbs on the counter, throw pillows fluffed and perfect in the elbows of the couch. Nothing comfortable about this emptiness. Sara leaves through the back door, skirts the space in the drive where Kate's car should be. In her studio, she doesn't turn on the light. No work to be done. She coils herself into the blankets and sweaters. Her failed nests surround her as she hums a song she imagines a different kind of bird might sing.
Sometimes We Can Be That Way
Before the party, Phil waits for Karen, watches a puma tear into a deer in high-definition. She walks in front of the TV wearing a dress in the royal blue colour he once said made her eyes look sky clear.
"How do I look?" she asks, swirls the skirt to expose her thighs.
"Fine," he says and swishes his arm to remove her from his view. She backs up, and his eyes refocus on blood and fur.
She's not sure if he even registered that she's wearing clothes at all, or if it would matter if she isn't. In the closet, she holds hangers to her chest, drapes dresses and skirts over her frame. All the clothes on her side face toward her; his wardrobe hangs the other way. Backs of shirts ignore her. Stiff seams, closed buttons. She changes into a black skirt and top just before they leave. If she tries to use colour they'll be late.
After they moved in together, she changed the sheets once a week and cancelled her subscription to _Cosmo_. Weekends felt less desperate. Her desire for a small dog vanished. When someone called to tell her she'd been promoted, he gripped her shoulders and shook her with joy. He filled the other half of her closet with his buttoned shirts and tailored blazers. Their clothes looked beautiful together.
Her company gathers at a mid-range hotel to celebrate an unexpected victory. Girls in unfashionable vests glide past her with palm-sized napkins and finger-sized morsels, legs pumping to get back to the kitchen once their trays are emptied. Drinks are ordered from the velvet and oak bar across the room, where Phil holds the hand of a stranger, grips it and pumps, his best handshake. This is how he shows off, proves he is the manliest, most dedicated husband. Her co-workers love him at these events because he never intimidates as he mines the room for attention and gives a good amount in return. At home, his attention is always somewhere else.
Someone hands Karen a neat Scotch as she stands in a circle with her boss and his favourites. All she wants is to feel a light buzz. They clink glasses, down their thimblefuls of booze. It's not enough to make her easy. Celebrations feel like work. Eye contact that doesn't look forced, head nods slotted in without appearing like she's lost, conversations about TV personalities she's never heard of that don't result in awkward pauses. Often when she lingers, stares, she wonders if she looks as though she needs to have a quick nap.
At the bar she gets a double. The straw stabs her in the nose. She's distracted by watching Phil across the room, the way he talks to other people, pays attention, seems invested in their words and lives. Hands flail as he talks to her assistant, rapt by some story he'll claim later to be too tired to tell her. She stalks along the wall toward them, hides behind a leafy plant, listens and sips. Her all black clothes a fine choice for subtle spying, gathering intel on Party Phil. Get an idea of why he finds other people so fascinating.
He tells her assistant about how he works from home, his rules for self-motivation: make schedules and stick to them, prime yourself with a daily pre-dawn run, visualize end results with vivid details, make up a theme song about your self-worth and sing it whenever you're deterred. Laughing at this, she chokes on her drink, sputters. She's never heard him sing a personal theme song.
They must hear her. Phil pushes the leaves aside, reveals Karen crouched behind the dirty pot, a straw dangling from her lips, tears in her eyes. She stumbles out from behind the green fronds.
Karen can't stop laughing, has to sit on the edge of the planter, wipe her hands across her eyes and cheeks to sop up the tears. When her assistant asks what's so funny, she just keeps laughing because she doesn't know if she's laughing at him or herself.
"He does have a regimented schedule. It's called _TV Guide_ ," she says.
"I just like to learn, Karen. It's not like I'm screwing around all day. I don't even subscribe anymore. It's all online."
They must still have a subscription. He might as well highlight the pages, put reminders in his phone, weekly pings to rouse him to the couch for another episode of _Deadliest Science Fair Projects_, or whatever it is he watches. Half his day is spent watching wildlife, space shuttles, guys with goggles proving each other wrong. His office is cluttered with stacks of papers fanned with sticky notes and an entire year of unread magazines. And yet he couldn't answer Karen two weeks ago as to when was the last time the two of them ventured out of the house for more than a trip to the grocery store.
"Karen?"
"Whatever. It's your thing. It's fine," she says.
Fine. It's fine, she thinks, bites her straw. Fine. Fine is not a fine word, and no one should use it to refer to another person's appearance. She slurps up cool fizz and alcohol. Karen's assistant looks at her as if she's been slashed with the shards of a broken bottle.
"What?" Karen asks.
"Your nose, there's blood."
With the back of her hand she swabs at her nostril, and a red smear appears on her skin. Eyes wide and crossed, her assistant excuses herself, says she has to make a phone call. Phil watches Karen struggle in her purse for a tissue. There isn't one. She tries to catch his eye, but he's staring at her bloody nose. He motions that he's heading to the bar. Her gnarled straw protrudes from the cubes of melting ice mingling with crimson drops. One of the vest-girls spins past, and Karen chases after her to abandon her glass on the empty tray.
In the bathroom mirror, she examines herself, her wounded nose, her simple black clothes. She notices rosy pops on her lips, must have drank some of her own blood. She dabs at the colour with balled-up toilet paper, then swipes hard until she can't tell it was ever there.
On her way back to the fray, she nearly barrels into Phil. He eats a breaded shrimp, turns to her. Mouth full, he doesn't speak, holds up another breaded shrimp. When she takes it from his hand, their skin doesn't touch. The deep-fried keeps them apart. She chews the warm hors d'oeuvre and thinks of the late dinner of snacks they ate so many years ago. When they both wanted to impress each other. After her car broke down, unfixable, on their first date. They got deep fried things to go, crispy golden food in little cardboard buckets that they ate in his bed. Each morsel tasted tender and sweet and full of old school sleepover comfort. With his hand, he'd fed her a piping hot onion ring, wiped hot sauce from her chin with his thumb, licked off her mess. After she swallowed the shrimp, she wanted to take his hand, put pressure on it with her fingers. She didn't.
Before she met him, life was sloppy because she liked herself but not really anyone else. Being a pair righted her, and she didn't apologize for it. Now they are like frayed socks that don't ball up neatly together, elastic shredded.
At home, Phil's forced to stare at Karen's ass while he fixes her broken zipper. Once she's unzipped and in bed, lights out, she sees him in silhouette as he folds his clothes and sets them on the dresser.
"Good night," he says to their dark room.
"Thank you," she says, but he's closed the door.
She hears the television, and sometimes there are cheers, and sometimes people are laughing. This is how she falls asleep, listening to the real joy and fake amusement through the speakers they picked out together. When he comes to bed he's groggy. He pulls at the covers and builds a wall with half the pillows, as though protecting himself against her.
She wakes in the middle of the night and pushes the rest of the pillows toward him, piles blankets on, makes him look comfortable enough to sleep on. Ear pressed to the cotton wall, she can't get shrimp out of her head, can hear him crunch the breaded coating, see him grin as if he'd caught the shrimp himself.
Before they were together, creaks in the night woke her. Wind in the folds of the curtain made her juvenile. After they met, she wanted to be a grown-up. He would put her coat on at the end of a meal and hold the restaurant door open. She would whisper in his ear and wrap her arm around him in the cab. She would massage every inch of his body, and he would thank her over and over until they were both asleep. But small thrills are gone, deep-fried holds no power. Memories are tricky and rude. Being a grown-up is full of paperwork and saying fine and cars that work properly almost all the time. On the other side of the pillow wall, he is gentle as a deer, sleeping vulnerable beside her.
In the grim grit of too early morning, she pushes herself into this puffy pile of linens, puts the whole weight of herself against it as though she must tear through and get to the other side. Karen's body pushes and won't stop pushing. She braces her legs on the floor and keeps pushing, until she's shoved the mass of everything on the bed onto the floor, including Phil. Karen's breathing as if she's been running hard, in pants and gulps. When she hears the thump of his body hitting the low pile carpet, she feels satisfied. Phil groans.
She crawls onto the bed, props her hands up under her chin, and smiles down at the heap of Phil and their bedding and giggles at the possibility of what will happen next.
Sleep Talk
" _Uhnnn_."
I pull the sheets up high and try not to listen.
" _Again!_ "
Sunbeams bump up against my eyelids.
"Mommy's got a marigold."
I vow to rip out the flowers along the patio.
" _Uhnnn_."
His morning breath burns my earlobe.
"Marigolds! _Ha, ha, ha!_ "
I haul myself out of bed and wince at the coldness of the floor. The sun is up, but it's still chilly out and too early to be climbing around the house trying to function. Some people love spring mornings, but I hate them. Everything is just coming awake and too little and too fresh and too unlike me in the morning, except for my husband, who is sleeping and is not fresh or little but very loud in his sleep and regular size.
Back when we first got married, Mitch didn't even mumble in his sleep. Little snorts here and there, maybe a sigh, but no words, no phrases, no sentences. Even a year ago, it was just a loud exhale if anything at all, nothing like this. Nothing loud enough to wake me up several times in the course of a night. His midlife crisis must be occurring in his sleep.
I force my eyes to stay open wide as I shower. I could easily fall asleep standing up like a withered old nag while the conditioner burns a hole in my cornea. I press my ear to the bathtub wall and hear him laughing about the little orange petals.
I peek in on him before heading downstairs. Our bedroom is quiet now, and his hand is groping for the empty space where I'm supposed to be sleeping. Marigolds were the first thing we planted when we bought this house. It's funny to think that he even remembers. I blow him a kiss.
Somewhat weary, I head out early to run errands. I picture Mitch and Ween having father-daughter time over sugary cereal and healthy omelettes. They never complain that I can't cook. Keeping their lives organized and clean seems to suit them just fine. The note I've taped to the kitchen table isn't nagging. _Hey guys, I fixed Ween's backpack—it's on the desk. Don't forget to lock the back door again! Mitch, doctor at 3. Dinner at 6 (I'll get Chinese?) because Dad's curling (2nd night this week!). No one eat the candy I bought for Grandma!!_ I always punctuate with an xoxoxo.
When I get to work the copy machine is broken, and everyone is crowded around it. I join in because I'm too tired to think about mortgages. We all stare too long and too hard, and I feel my eyelids quiver. Someone notices.
"Tara, coffee?"
I nod my head yes, not even knowing who I'm speaking to but grateful that someone is going to give me a boost this morning. I take the cup without acknowledging who my caffeine saviour is and then pour back some scalding gulps. This day will turn out to be okay.
"Again?"
Mitch is wearing oven mitts. He questions me about his nighttime behaviour as though I have answers. Strands of Ween's long wet hair lace my fingers as I pull them into a braid, while she gets our advice on science fair projects. The kitchen table is full of books I checked out of the library.
"Yes. It was pretty loud. The loudest you've ever been in your sleep," I say.
"I didn't hear anything. What about crystals?"
"Crystals? Get real, kid. What about something with explosions?"
The braid is too loose, so I take it apart and start again.
"Something about flowers."
Ween jerks her head around in shock. "Mom. Flowers are lame."
I turn her head back to face her homework and point to a book about rockets. I twist her hair into a bun and wrap a checkered hair tie around it. "I was talking to Dad, Janine. Flowers. You were very into marigolds."
"That's weird. I'll try to keep it down. Somehow," Mitch says.
"Dad's right. Explosions are the way to go."
The timer goes off, and Mitch turns to the oven. The aroma of confetti cupcakes fully distracts Ween from explosions and flowers. When she jumps up to take one from the tins, her loose bun opens, and wet hair slaps her back. "I wanted a braid, Mom."
We decorate the warm cupcakes at the table, careful not to drip icing on the library books. Ween forces Mitch to take the ones she did to his curling team. I walk him to the car, his arm around my waist.
"Honey, draft beer and cupcakes?" I ask.
"Once they've had enough beer, they won't be thinking about food and beverage pairings."
I give him the guilt look, but he kisses me over and over, on the mouth, on the cheek, on the neck, one hand resting on the open car door, the other on the small of my back.
"Please?" he says.
I take the cupcakes back inside and hide them in the coat closet until Ween goes to bed.
Bursting out from under the duvet, I struggle to find the alarm clock. Blue spots dangle in front of my face—I must have bolted up too fast. When they fade, I realize I have only been asleep for one hour. Mitch has snuck into bed sometime during that time.
"Whisky sour and a Strongbow."
His mouth makes different shapes when he talks in his sleep. The vowels are too open, and the rest is a slur. I've tried shaking and kicking and biting, but Mitch is comatose when he's chatty. I pinch his nose tight with my fingers and cup his mouth, a technique my brothers liked to use on our dog when he would snore. Just like that dog, Mitch snorts snot onto my fingertips, nips my palm, and turns over. I wipe my hand on my nightgown.
"Oh, Julie."
My feet hit the floor as Mitch exhales this name. Feet stuck to the carpet fibres like flypaper, I hold my breath. Staying still, I wait for him to confirm the name, or say another one, throw some other characters into the mix, have a dream conversation with a Ted or a Dr. Heller. I don't think I know any Julie. Or maybe he said something else, and I'm too worn to pick up on it. Without regular sleep, words don't resemble real communication.
" _No!_ "
He pushes his pillow out of the bed, and just like that old dog, Mitch twitches his dream out. I pad down the hall in my robe, tripping on the belt.
"Honey? Weenie? Mama's going to crash with you tonight, okay?"
Ween doesn't budge. I don't have the power to wake anyone up. I shift her over onto the other side of the bed. If only we'd bought her that bunk bed she wanted so badly, then I would have an alternative. The warmth from her body is lovely. A light rain taps gently on her window, and when I look outside there is the smooth dark blue of night. Ween's room is a sleep refuge.
"Oh, yes! _Yes! Fuck yes!_ "
I turn over and hope this is all in my own dreams.
"You... are... so... hot!" Mitch pants in spurts.
Before I run down the hall, I make sure that Ween is still sound asleep. I need to get there to shut him up, but trying not to wake her makes me slow.
"Dirty, barnburning _fuck!_ "
I stand in the hallway waiting for more, but he's done for now, so I crawl back in beside Ween, carefully pull her hair into even segments and braid it tightly.
Polly likes to talk about everything and anyone. She is my favourite and least favourite gossip. Lunches are never boring, although I'm concerned that my current state of mild confusion might make this conversation like an acid flashback. Too bad I didn't drop any acid back when I was less uptight. I probably wouldn't know much about anything trivial or scandalous if it wasn't for Polly. Today there is no pleasant banter; we're straight into the deep stuff before I have time to fully peel the plastic wrap off the sandwich Mitch made me.
"Mike's dick is partially inside his stomach. And it's been like that forever, but his parents thought it would just drop on its own because they were Jehovah's Witness or something and didn't think that it would be a problem. But now it is, and he has this new girlfriend, who is young and very stupid, and basically he has to either be a bachelor forever or have some crazy operation."
"Mike from upstairs, or Mike the janitor?" I brush the crust of my sandwich against my lips and feel like I might be sick.
"And the company won't pay for the operation because they say that it's cosmetic, so he might just have to have a halfer forever."
"How did you get this information? About Mike..."
"From upstairs. Mike from upstairs. Tara! I'm wasting this story on you, Tara, I swear up and down his teeny cock."
There is a dollop of mayo oozing out onto my thumb, and the thinly sliced capicolla is rolled so tidily between the brown bread it couldn't look less appetizing. And the colours are all wrong. This penis talk is making me think too much. I wonder if I could just walk away and find a quiet place to nap.
"Anyway, the information came from Roberta, who gave him a down-the-pants jobbie at the Christmas party two years ago, and it was ridiculously small and weird, and so she asked, and he told her the whole story. Cried in her cleavage."
The uneaten sandwich looks like I imagine a fart to look if farts weren't made of gas, and I dry heave big and bold.
"That's disgusting."
"Sorry, Polly."
I settle down, glad that nothing actually came up. Polly is furiously eating her beet salad, staining her mouth a charming shade.
"Were they both drunk at the time?" I am trying hard to remember if I was at that party.
"God, fucking mercy, Tara, of course they were drunk. That's how I know it's all true. Mike had the elixir of truth in his system, and Roberta got the goods right out of him."
"How was your day at work, babe?"
Mitch works in an office. There are fluorescent lights and other people that work around him. He tells blue collar people what to do, and sometimes he fires them. We usually don't talk about work at home. That's for what we call "Living Time." It creeps into our lives when we leave the house during the day, but no need to know too much about work. Work is how we make money to buy handmade wood furniture and special gifts for each other, just because. Work is for making new acquaintances and talking about work-related things and work-related stress. Home is for keeping house and relaxing and delicious meals and love. Or that is what it's supposed to be.
Instead of lying to Mitch or confronting him again in the broad evening light, I just grab him and press my face into his chest and inhale, not knowing whether or not someone else has pressed her face into this shirt. He still smells like our laundry detergent, the pine scent strengthened by the needles that sometimes blow on to the fresh laundry when I set the basket near our open bedroom window. He still smells like someone who lives here and wears clothes that are washed and folded by his wife.
"I missed you. You weren't in bed last night. Was I talking in my sleep again?"
"Yes."
"What did I say?"
Hiding the fear in my eyes is as necessary as looking directly into his. "Nothing interesting."
Four nights in a row, Mitch belts out Julie's name and different sexual iterations of pleasure.
"Bandolier banging!"
"Julie!"
"Gonna bust that box!"
"My Julie."
"Sweet, fragrant, marigold muff!"
"Julie! Julie! Julie!"
The flower metaphor. That's when it stung. She must be a redhead. Julie. My whole body bloomed into anger, like those prickly purple flowers atop those even pricklier weeds.
Turned up as loud as it goes, the speakers are still terrible, but I don't think that's the problem. I had rigged my old Dictaphone from college to record Mitch's murmurings, thinking I'd be better at analyzing them when it isn't two a.m., and I'm not jarred awake. I slept on the couch, let the sound of Ron Popeil lull me to sleep, and woke up fairly rested. The sounds it recorded are vague and wispy. I thought to put it under the mattress on my side of the bed so he wouldn't see it, but all I can hear is the rustle of sheets and the creaky moans of the box spring.
Mitch and I just let things go. As long as my note is obeyed, I don't care if he wants to curl three nights a week or go for a drink after work. There is a schedule, and it works, and I write it on a piece of graph paper every morning. And right now there is something that is not on the graph paper, and I don't know what it is.
Mitch loves to curl. A dreary game made drearier by the bad lighting and poor quality vodka in their Black Russians. He has three games a week and a bonspiel. He likes to think he's one of the guys. I hate it and stay home to watch bad TV with Ween. We've only been twice. But have we ever been invited? When Polly finally notices that I'm out of sorts, she stops talking about Jen Vanelli's addiction to prescription inhalers.
"This is good shit, and you're acting like we're talking world events. What are you thinking about?"
"Curling."
"Curling?"
"I'm not sleeping much."
Polly convinced me to eat outside today, even though the wind keeps whipping our paper napkins off the bench. Her salad of choice today is covered in creamy dressing and bacon bits.
"So you're not sleeping much and thinking about curling."
The bonspiel runs the whole weekend. Three days of drinking and rocks and fleece jackets and hurry hard. And I'm going.
"That husband of yours curls?"
I nod. Polly offers me some bacon bits, but I don't respond.
"All right. Why aren't you sleeping?"
"Mitch yells in his sleep."
"Really?"
Polly is fascinated. Her eyes pop, and she clutches her salad container to her chest. The wheels in her head are turning, and I can only hope they'll also help me. I don't even care that this will become Polly's lunchtime conversation with someone else on a different bench tomorrow.
"About another woman."
"Well, this is something to think about."
Ween and I are packed into the plastic seats with the other wives. Some of them don't even know who we are, two strangers watching the final bonspiel of the year. Mitch doesn't know we're here. Ween is thrilled that we are and that it is a surprise for her dad. I feel like an undercover agent.
I don't see Mitch anywhere, but my view is limited. There are curlers drinking and laughing and spilling coffee in every corner. His rink's not supposed to be on the ice for another hour, but he's been out of the house for two. I thought I'd find him bonding with his buddies at the bar, throwing down a few mid-morning drafts.
"I should have made a sign for Dad."
Most of the women are seated with us. Slouching and supportive, aging, bleached-out former beauties wearing tank tops, teen daughters checking their nails, curling rink grannies with curling rink sweatshirts. A cross section of women I can't imagine Mitch having anything to do with. I scour the crowd to see if there is anyone worth my time.
"Why didn't we make a sign? Or a shirt like that lady's?"
There are a few younger women on a couch by the fireplace, sipping coffee and lacing up their curling shoes. A group of younger men is standing around them, pouring from a flask into their mugs.
"Mom!"
Ween whacks me in the arm.
"Look at that lady's shirt. God."
Ween points at the woman sitting next to us. It's one of the grannies, sporting the name of Mitch's rink on her chest. She's done it in gold and red glitter paint, cut out photos of the team ironed on the front and back. I can't help that it makes me uncomfortable. Her energy is huge, and she jumps up out of her chair. The excitement of my plan wanes when I see Mitch slide out to his team. His reaction is disheartening. The smile on his face is enormous as he taps a teammate and points up at us. His happy waving proves nothing. If he'd been surprised to see her, had stumbled on the ice, that would be a sign that something was up.
"That's my grandson out there. Who's yours?"
"That one. The one smiling and waving."
"That's nice. Haven't seen you here before."
I dread having to stay for the entire game, the entire weekend.
"Look at Dad! The surprise worked, Mom."
Another terrible night. It doesn't matter where I sleep; these thoughts, his shouts, they find me on the couch, in Ween's room, in the basement. The sounds have transitioned. Mitch's nightly content is more like his waking conversation. It's domestic and devoted.
"I love you."
"Muffin, you can do this."
"Sure, your mom's place on Sunday."
"I love roast!"
I get up at night and Google. Too many Julies in the world. I try Facebook. No Julies connected to Mitch or me. She could have a hidden profile. She could be lying about her boring name. I try thinking about every woman I've ever met in town. I can't think. I can't sleep. I don't even know if my brain is operating normally, or if at this point I'm making up a whole mess of stories in my head because of the not sleeping and not thinking.
For a minute it's quiet, me sitting at the kitchen table in the morning, and then happy feet hit the stairs.
"Good morning, Tara." He ruffles his own hair while reaching out to me from the kitchen doorway. I've poured him a fresh cup of coffee, which is sitting patiently on the counter.
"Thanks for coming to the bonspiel."
"Beer's in the fridge."
He opens the door and pulls out the milk.
"It's seven a.m."
"Milk, eggs, juice, fruits are in the fridge."
Mitch rests his coffee mug on my shoulder. He doesn't remove it, so I look up at him and see that he's smiling down on me. My face is too tired to say anything. I dip my head forward, and the towel wrapped around my hair falls into my oatmeal. Mitch grabs a dirty dishcloth and swipes at the goo. I take another spoonful before throwing my head back, and the oatmealy towel splats to the floor.
Thankfully, I don't offer Ween a beer when she marches down the stairs. Mitch has made her snowman-shaped pancakes by dropping three little ones side by side on the pan. At nine she still falls for it. My own child is a total idiot, and I know she got her gullibility from me.
"Ween! Snowmen are up. Here's a bacon scarf." Mitch throws a jaunty piece of bacon on the pancakes and three chocolate chips for buttons.
"This snowman looks stupid."
Ween and Mitch stare at me, baffled by my comment. I kick my slippers out of the kitchen ahead of me and try to imagine how they can be so fucking chipper. Before I climb the stairs, I remember all the things I was supposed to remind Ween of in my regular daily note. Instead of going back to write it down, I just yell out a reminder for each stair.
"Lunch is in the fridge, add a banana. A page of your science homework is on the couch. Piano lessons at three-thirty. Come straight home. Dinner will not be takeout, so don't ask. Your father's putting stew in the slow cooker. Badminton practice is at seven."
By the top stair, my voice is shrill, and I take a moment to think about whether or not I sound like my mother. She never left notes.
I nap in my desk chair on my lunch break. The window is half open, and I dream that a bird flies into my sleeping mouth, and I choke. When I wake up, there are two dirty wrens looking at me from the telephone pole. They look evil, so I shut the window and try to get another ten minutes of REM.
"Tara!"
Squawks from the birds and Polly's voice combine in a force that begs not to be ignored. But I do anyway.
"Tara!"
I feel Polly's head hovering over me.
"Tara. You need to deal with this problem. Now. Look at you."
I know how I look. Rough. Tired. Beaten. And Polly is supposed to help me, not insult me. I squint my eyes tighter.
"Leave me alone." I shove my limp hand in the direction of Polly's face.
"Over the weekend, I did some thinking," she says.
"Polly. I don't want to hear about someone getting something edible stuck in one of their orifices and having to go to the emergency room and the doctor being some handsome guy she used to date in high school."
Polly pulls one of my eyes open with her fingers.
"Your problem. I was thinking about your problem. And by the way that story is one hundred percent accurate."
I don't want to think about my problem. I want to sleep in my office, keep my eyes shut, and imagine that I don't have a problem, that there is no mystery to be solved, because I don't have the power to solve it.
"Why don't you just talk to him?" she says.
"This is what you think of?"
Tears burn hot in the corners of my eyes. I crumple a stack of credit checks into my face.
"Well, you need to figure this out, or you're going to be an even bigger mess."
Polly puts her hand on my head, and I feel like I could explode into raging sobs. I don't want him to know that I know, that I suspect.
"Talk back. Get the truth out of him."
I've shipped Ween off to a sleepover on a school night and look like a very cool parent. I've set the table, and the wine is decanting. And if Mitch doesn't want the truth to come out with wine, he can have beer or Scotch, or a fine brandy or a wine cooler. It's all the same to me.
"Tara?"
My earlier outburst is my excuse for this meal, for Ween's sleepover. I take Mitch's coat at the door and lead him to the dining room.
"You didn't—"
"No, I didn't cook this."
I laugh at this brief display of naiveté, and we settle in for gourmet food and a few bottles of pinot noir.
I rub Mitch's feet after the meal, sitting on the couch together with no TV, no music, just us. I've poured him a neat Scotch and slipped some brandy into his decaf espresso as well, both of which are sitting next to him on the coffee table. He's already drunk, and as I lull him to sleep with my thumbs on his instep I panic that maybe he only talks in bed, that he'll have a peaceful evening nap if he's in the living room.
"Honey, thank you." He slurs when he says this, just like in his sleep talk.
"Mitch. Shhh. Just relax."
I point to the Scotch, and he obeys, takes a hearty sip. I try to keep a rhythm with my massage, try to maintain my veneer of calm. His eyes close, and I wait. I smooth my hands over his feet and up his ankles and calves, and I wait. I wait until I hear the first mumbles of sleep, the loud exhale or satisfaction of being in his dream world. I'm not sure if I continue to wait for him to say something or jump right in.
"Fuck."
I'm not ready. It's like phone sex or dirty talk, things I've never done or am no good at.
"Yes... Mitch."
His lips look swollen, like pepperoni, and are slick with saliva. During dinner, he sloshed wine onto his shirt, and now he's sweating through it. No need to be nervous around this man.
"Oh yes."
"Do you like that?"
I pluck at each toe to steady my voice.
"Oh, yes."
"Say my name, say I'm sexy."
He must recognize my voice. Even in his sleep, he must. But he doesn't. He slurs out this woman's name.
"You're sexy, Julie. So sexy."
He sounds like a child, words coming out of his mouth loose and sloppy.
"What about your wife?"
"What?"
"Your wife?"
"You're my wife, baby marigold."
I grab his big toes and squish them tight in my fists. Then I slide out from under his sleeping legs and up to bed. From there, I hear one last blast before I fall asleep.
"Now, suck 'em, Julie. Suck my toes."
It's the best I've slept in months. Ever since the first disruptive wake-up as Mitch shouted expletives. Coffee in my hand, I'm awake and I see. Mitch is the same as always. Jokes with Ween, ruffles his hair, is slyly sweet with me because he knows I'm creeped out by a more sickly affection.
But maybe Julie loves it. At night, Julie comes to him, his dream wife. And there she is, so loving, and she can do it. Marigold pubes, Julie there to dote and careen toward him in public spaces. Quiet in the stands, she watches him curl three nights a week. Maybe has her own homemade glitter shirt, a fleece that Mitch nuzzles into.
He never seems tired after his escapades. The courtship of hot sex, the wedding that somehow I missed, the dinner at his mother-in-law's house, all the roast he seems to eat, the toe-sucking fetish that hopefully is only a sleep turn on. Maybe Julie will have a child. A little girl like Ween, but more Julie-like. Whatever that means.
I watch Mitch pack our lunches. Perfect baguettes full of carefully selected meat and cheese in our matching lunch bags. Ween grips his arm to try and sneak extra cookies into hers. This level of happiness is almost sickening. But it's good. And Julie is good. And we're all in this together.
Instructions for Having an Affair
**Read this first:**
Don't think it's as easy as going to some bar. It never is. That kind of thinking is what will cause disappointment.
**Get Organized:**
Details. That's what it's best to excel at. Like in an Excel spreadsheet. Keep track of things in Kodachrome code. Colours that correspond to fantasies, set to make reality.
Organize everything on the computer in secret documents. Filenames must be boring:
"Dessert Recipes"
"June Meeting"
"Raw Images"
And yet, they allude to something dirty or dangerous for you. Only you need to know the details. Only you need to know how to make yourself a wet mess. Even if it's just while looking at a few typed sentences on a black-and-white grid in a document on a desktop computer. The world might want us to be more elaborate than we want to be. Alleviate pressure by knowing your limit, by playing within it, by laying yourself out on your desk in the middle of the night and using an old joystick in ways that make you glad no one uses joysticks for playing video games anymore.
**Communication:**
"Honey." Still call your husband "Honey." If that's his nickname, that's the name you must use. Call him "Barracuda," or "Babe," or "Al," if that's his name, and you keep it serious, you keep it in the realm of the real without that added element of twee nuance. Not everyone needs it. But needs, that is what we're talking about. And yours are not being met in some way.
Figure this out. Answer questions. Do you hate the phrase oral pleasure, but your partner insists on using it? Are you sleepier than usual? Ask yourself: what is missing? Attention? Lust? The deep pull of emotion that used to rule your life and bring you joy? Find your own questions. Work with a deep integrity to answer them. Put them on the spreadsheet. Save the document with full knowledge that you are setting out to make your dreams come true.
**Locations for Meetups:**
The grocery store aisle with the most potential is the baking section. You can go homemade and shop to impress. Boxed cakes are surprisingly an aphrodisiac. There is something in their preservatives. The powder itself can function like mild cocaine. Draw lines on your hand and snort in the aisle. You will laugh and then feel like you can't take it anymore. This will cause hands to come together, mouths to breathe into each other.
Never waste your time in a coffee shop. Coffee culture is decidedly unsexy. No one is thinking about their bereft heart in a coffee shop. No one is thinking about anyone else's well-being in a coffee shop. If you want pastries and caffeine as a supplement for love, that's fine. Even encouraged. Do not misunderstand what it is. Because it is merely something that is unrelated to affairs.
Libraries are quiet, and people are often lonely. Leave your name in the search bar of the communal computer and your location in the stacks.
A park bench is a starting place. An art gallery too committal. A casino is a wonder, so loud and bright and full of potential.
**Locations for Sex:**
There are only "do nots" in the section. Do not think about your own home. If they are single, you may think about theirs but it must not be the only place. You will lose yourself. Not in a fun, wild, sexual way. In the way that causes you to forget to pick up children, to turn off your phone reminders, causes a small rash that develops into a small dragon that haunts you every time you try to have a shower. Fire-breathing, pocket-sized, the taunting and harassment only ceases when you're heartbroken.
Hotel and motel rooms are fine as anything. Alleys are fine too, if you're not squeamish. Hand sanitizer and Wet Ones are a great investment. Have them in your car or purse or backpack. Like a Boy Scout, you must be prepared even after you've left your spreadsheet hidden under domestic chores.
Off-season dugouts are good. Bring blankets. You are an adult. You own some things that will help make your life easier.
Practice levitating, and you can do it almost anywhere. Above the crowds. Bodies pressed in bliss.
Hollow out a large animal. This is not ideal, but once you try it, you'll be surprised how easy it is to desire it again and again.
This list could go on and on.
**Locations for Emotional Outpourings:**
This is best done in parked cars. There is no better place. Hidden on a side street. In a rooftop Best Buy parking lot. Parking garages are not ideal. The idea that someone could see you crying, yelling, confessing some other indiscretion to your already secret lover is essential. Fog up a window with feelings.
**Anecdote:**
I am only telling you this to make your experience more honest. The way it's gone for me has been like this.
I make a decision. To have an affair. I create worksheets for myself. I fill them out. I shred them. I fill them out again. I create spreadsheets. I fill them in.
My fantasies are big and bold. I want to teleport into my lover's bedroom, into their car, shrink down to the size of a cell and enter their body. I want to explore the inside of someone I love, and touch the tender vessels that make them breathe and walk, and stand in awe of the beauty of another person. I want flowers. I know they die. I still want them. I want someone to hold my hand and drag me to a dark corner in a public place.
Dark corners are not to be feared in an affair. This is where skirts are lifted, pants are unzipped, bodies crumpling together in pleasure and sorrow and minute moments of happiness.
None of this is revolutionary. Remember this.
I want to be loved and fucked. I want my partner to feel an unnamed shame. I want this person I committed to for life to question their own choices, to try to understand me like they used to, but to fail because we are more apart than ever.
I want so much to not feel like a monster. But I will. The tiny dragon is just like me. Judgmental and rude. I can't levitate or teleport. I can only make coffee for myself in the morning, no time to go to some café full of young people with time on their hands, with hope for the future, even for just the weekend. I can only ball up my fists and punch the laundry bag. I can only drive to work and cry as I sing the loudest to any song that comes on the radio, even if the lyrics are wrong. I can only remember the way I used to feel when my love first took my body and just held it, like I was a magical creature, something undiscovered and unlike anything else.
I write this down. I make another spreadsheet. I plan for the future.
Unplanned, I snapped to action in that moment I kissed the mail carrier. We snuck into the mail truck and fucked like teens, sloppy and without finesse on other people's lives, letters, and bills. Then I walked. Forgetting everything else. Just for that day. Then I only remembered those minutes in the truck. Flashbacks. Flashes. Not magical.
**Conclusion:**
But don't mistake the level of dream fulfillment. It might not work out. You might come crawling back home with a fistful of greasy coins you didn't put into the gambling machine slots.
Full Price
Girls armed with plastic forks crowd around a chocolate pastry in the lunchroom. The air is filled with white sugar, cheap cocoa, and four different kinds of perfume. They spear cakey bits, swing them up to their lips and snap their teeth down around white forks smeared with saliva and doughiness. The brown paper bag with the round green logo, so familiar to Sam, is torn like a downed mouse after the hunt. The gooey insides gone, thin wrapper left dotted with brown flecks of cake.
Sam walks past them to the manager's office at the end of the narrow room. These girls have names that frequently appear on _Nancy Grace_ , bold letters under her face, names that match up with new ticker tragedy. When she knocks on the open door, they toss their forks to the table. One of them looks up at her, still chewing her tiny morsel. One of them says her name and another echoes the syllable, fear in her voice. One of them balls the garbage up in her hand.
"Hi, girls," Sam says, waves less like a greeting than as a way of shooing away their liquid-bright eyes.
"How are you?"
"You look great."
"Sorry."
"Yeah, me too."
Still talking, they file out onto the sales floor. The store opens in fifteen minutes.
Sam wants to apologize for being late. She'd been able to wake up early every day before today, her first day back to work. When she'd had nothing to do but be a widow and count out how many Lean Cuisine dinners were left in the freezer, it had been easy to get out of bed. Last night had been her first bad night since the week of the accident. Each night for a week after the midnight call telling her Fred had tried to save the bank deposit, had lost, and had been pushed in front of a speeding SUV, she'd stayed up until the birds chirped, telling her she'd missed out on sleep. In their bed, she felt drowned by the sheets.
Jill sits in front of a frozen computer screen, her hand furiously moving the mouse over the surface of the desk. Sam knocks again. Jill slams the mouse against the arm of the chair.
"What?"
"Sorry, Jill. I'm late."
Jill looks over at Sam and smashes her head against the keyboard.
"God. I am so sorry. Sam. You're back." Her voice is muffled, her chin and lips clicking against the keys as she talks.
"It's okay."
"This place is a hellhole."
Jill gets up and moves to where Sam stands with her hands clasped against the zipper of her mid-rise jeans, name tag already pinned to her striped pin-tucked blouse. Without warning, Jill crushes Sam against her chest. This is not like the hugs at the funeral, the hugs at the hospital, the hugs from her family so afraid to break her after she's already been broken. Sam hugs her back, and Jill pulls away. Looking down to avoid eye contact, Sam notices Jill's not wearing any shoes with her perfectly new, perfectly Gapified outfit. Medicated foot cream odor fills Sam's nose.
"We've missed you, babe. We've all been thinking about you."
Sam nods like a distracted child. She makes an effort to breathe through her mouth.
"Are you ready to get out there?"
"Yeah."
"There was supposed to be something here for you, but I thought you were back tomorrow. I'm a jerk. Did you get the flowers and basket we sent?"
"Yeah. Thanks."
"I know it probably didn't mean much, but we had to send something."
"Anything I should know about before I start my shift?"
"Sorry. It's terrible. So frigging terrible. We've got a promotion on polos."
The rest of the staff is huddled around the cash desk, listening to another manager, a new guy with bright eyes and slim trousers and an accusing look. He tells everyone what their goals are for the day and pushes the polo promotion, then sends staff off in all directions. They walk past Sam and smile, or pat her arm, or look away. Half of the cake girls will be stationed in the fitting room. Sam will be at the men's denim wall folding jeans into tight blue shapes.
"You're late," the new manager says.
"She's fine!" Jill yells from across the store.
"Sorry. I'm Sam. You're new."
"Not that new, honey," he says, bolts across the floor with his clipboard.
Sam pulls pairs of boot cuts out of their cubbyhole homes and presses them back together with the edges in rigid symmetry. On her right is a cube laden with pastel one-pocket T-shirts. She leans over and breathes in their fresh Styrofoam chip scent. To her left is the khaki wall, a totally different fold, shape, wear. A completely different kind of man wears khakis. Fred wore khakis. Even before they were married, even before they'd kissed under the snowy fibreglass ceiling of the mall parkade on a rainy winter morning, he'd worn khakis. It was part of his job, but he also liked the way they fit, the neutrality of them.
He managed a Starbucks on street level, and she worked on the lower level of the mall, underneath that Starbucks. He was the same height as her and had a crop of wheaten curls on his head, a supple chocolate leather belt around his waist, and a polite stutter. Before her shift, she'd order a Tall, to go. He knew her order, and by the time she was at the front of the line it would be ready to go, cardboard sleeve nestled around the cup. She would notice when his curls were taking over his head and usually a day or two later would be complimenting him on a new haircut.
One day he gifted her a travel mug, leftover and green with snowflakes from the Christmas line. He admitted when he handed it to her, full of steaming Gold Coast blend, that it was on sale, no big deal, but thought she looked like the kind of person who would use a travel mug if she had one around. His honesty was charming, the way he put himself out there, but not too hard, not by spending full price. She imagines the cake girls would laugh at someone buying them a gift for less than the original retail price, tossing it to the ground in a dramatic attempt to break it and the giver's heart.
After that, the baristas had looked at her funny when she walked in and set her shiny travel mug on the counter to be filled. She couldn't tell if they thought she was weird or if they were jealous. They all seemed protective of him, and yet, none of them had snatched him up, taken him home after a late shift, coaxed him onto their futon couches, and made him a husband. Sam wasn't scared to take him home, to wait for him in the parkade that night, and let him know she wasn't headed to her own car. Sitting shotgun, she tapped her foot impatiently. His car smelled of takeout containers and cherry air freshener and him. Dark roast and hair gel and Right Guard and ocean. Permanent on his skin, a salty, tingly aroma.
From where she's stationed, Sam can see two of the girls admiring themselves in the fitting room mirrors. They poke at their abdomens with willowy fingers, can almost wrap their hands around their neck-thin waists. Sam can't believe that the four of them are satisfied with sharing a measly pastry. She used to eat a pastry every day. When she worked the morning shift, she went before work. If she worked a mid, then after the lunch rush, when she got to take her break. On the late shift, she'd wait until after closing, get one to go. At home in her pyjamas, with the TV far too loud, she'd sink her teeth into an oat fudge bar, or a coffee cake, or a plastic-wrapped slice of banana bread. She always hoped he'd be at the bar. And most of the time he was. Fred never gave her free pastries. He was a stickler for the rules.
Being a young widow is a strange burden, carries nostalgia that she isn't sure she wants. She's a working widow, punished by cotton socks and perfect folded jeans. There had been no time to stay at home. The bills were piling up, the modern-day retail romance over. The mall dream had died, and so had Fred, and Sam had to go back to telling older women to go up a size without also telling them their legs aren't sausages.
Sam wanders the men's section, runs her hands over the chilled cotton of the shirts and shorts, the brightness of everything. She helps a man pick out cargo shorts and a woman find a buttoned shirt in the right plaid for her husband. From the fitting room, the cake girls come and tell her to switch out. They go the front of the store, stand by the window, and watch for cute guys coming in.
When she gets to her new post, there are women waiting for different sizes, a man trying to roll a cuff on a pair of jeans that are too long for him, another fighting with the lock on his door. There are rumpled clothes on the floor that she picks up and piles on a shelf. Sam calls the new manager for sizes in trousers and a blouse. He shows up a minute later, shoves the hangers into her chest, snorts, and throws his hands across his clipboard. Sam knows he's watching, making sure she does all the sales steps. After she compliments the women and curls a perfect roll on the bottom of the man's jeans, Sam looks up and the new manager is gone. The guy with the lock problem emerges and explains why each piece isn't right for him as he hands them to her. He is so apologetic it makes Sam's chest burn. In the corner of the fitting room she takes deep whiffs of the shirts as she folds them, inhales the scent of every man who's tried on a new look. One of them must have a flake of comfort or pain or remembrance.
After two hours of folding, she's called to the cash desk. Someone is taking lunch, and she needs to be refreshed on cash procedures.
"Since I hear that was where you worked before," says the new manager. "Don't forget. Polos."
In the enclosed square of the cash desk, she tidies up stray tags, moves hangers into their correct bins. She sells a family of jeans to a woman with a torn purse. The cake girls beg Jill to switch their breaks so that they can take their lunches two by two. She slams open the lunchroom door, and two of them skip through it together. Sam would hate to have that job. There have been positions open, but she can't imagine dealing with employees like they're her children. She is the kind of person who likes having someone else in control. Looking pained, Jill splits up the other two girls, and they slouch off in different directions. She walks over to Sam, grips the desk, and leans back for a stretch.
"Sam, did you see these new scents? Some of them don't even smell that bad. Remember Midnight Tango?"
"Sadly, I do."
Jill holds the round bottles under Sam's nose for her to test. One peachy, another more citrus, one spicy, one too woody, and it makes her back away. Another stings her nostril, a hint of alcohol, but mostly a breezy, saline ocean smell.
"Nice, huh?"
Sam nods and puts a hand up to her eyes. Through the glisten at the start of tears, she sees a woman with a stroller roll up to the desk. Jill swoops up the bottles and arranges them back into a display. Sam moves without thinking, passes barcodes over the scanner. The scent sinks into her, embeds itself in her head, waves through her skin. In two weeks they would have been married for six months. In six weeks, Sam will be twenty-three.
When she looks up from the till, the woman and baby are gone. The new manager glares at her over top of his clipboard. Another woman piles camisoles on the desk.
"Did you know that all polos are eighteen-fifty? All colours, men's and women's."
Sam chokes through the spiel, points at the tables laden with vibrant shirts. The bottles of eau de toilette are two feet away. Smells sell. During a staff meeting, that's what someone had said, that people are more likely to buy from a store they like the smell of. The scents the company had tested at the time all had colour names. Green Hope, Pink Joy, Blue Kiss, Yellow Wisdom. Sam had thought of how hard it must be to name things, and even harder to distill the essence of a colour.
"Are you okay?" The customer asks her.
"You would look great in a coral polo," Sam says.
Two cake girls come back into the store with salad in a plastic bowl. They stare as though Sam is the wrong kind of person. Her hands shake; her skin breathes in fake ocean; her taste buds crave dark roast.
Jill hands Sam a tissue, steps in to finish the transaction. Sam is relegated to reorganizing the stockroom for the rest of her shift. Sizes organized smallest to largest, boxes of lip gloss and scents moved from one shelf to another. Sam dabs at her wet cheeks with the sleeve of a cardigan and vows to buy it when she gets her next paycheque.
At the end of her shift, Sam waits at the door. Jill stands next to the new manager while he checks the bags of the cake girls, examines their insides with a suspicious eye. Sam holds her leather satchel tight under her arm. Giggling, the cake girls sweep out the door together as one mass of human. The new manager grabs for Sam's bag.
"That woman in the slim fits has been waiting five minutes at the till," Sam says.
With an elbow, Jill nudges him and huffs. He flits away.
"You'll get used to him."
Jill moves to Sam and again embraces her. Jill smells like sweat from being in that stuffy office working out a schedule all day, hopefully finding enough shifts for Sam to make do. Sam walks out of the store. On the escalator, she reaches into her purse and feels the cold glass bottle, pumps a spray onto her hand. At the top, she cups her hand to her face, covers her eyes and nose and mouth as she passes Starbucks, inhales the stolen ocean.
The Gospel of Kittany
A slender boy with a neat bun emerges from underneath a geometric-printed linen skirt. He kneels at Kittany's feet, keeps his hands to himself now. She sighs and brings her thighs together.
"You did very well today." She wipes his mouth with a soft blue washcloth, holds his face in her hand for a moment, shares a half smile, then points to the door. He exits the gazebo, splits the filmy white gauze with his perfect young body. Wind doesn't whip through this wide swath of land; there's only ever a slight breeze. Mountains hold the valley in a tight circle. The parade of young men in and out of this gazebo, her bedroom, the rose garden, and the milking shed produces a steadier whip of air.
Kittany taps a message on her phone, then takes a selfie and sends it to her Instagram followers. She shares when she's satisfied. In two minutes, she will have hundreds of little hearts beating underneath her image. She uses hashtags sparingly. She doesn't wear makeup. But she gets her followers' attention. She's filled with love and affection.
Kittany. That was her nickname as a child. Her given name, Brittany, just never conveyed how cute she was. Her small feline lips and almond eyes gave her the look of a kitten at the height of its precociousness. Her father would spin her in the front yard, and she would mew and giggle. Her mother would pet her long soft hair. Kittany, Kittany, Kittany.
After she'd started _The Light_ online, it took a few months to build steam, to crack through to the people who would take her seriously in the way she wanted to be taken seriously. The site gained traction through her commitment to social media. She'd already had a decent following from her modelling career. She wasn't huge, but some of her most interesting shoots circulated on Tumblr in the right circles. Posing as zookeeper to taxidermy fails, standing in a hall of mirrors for a nude editorial, wearing a ballgown and tiara with a kaleidoscope of butterflies in a cactus field. Modelling had given her a platform, and now she was expanding it. She couldn't sell products anymore; she had to promote something bigger.
Her advisor, Tim, had been a small-time talent agent before she came along. It was easy to wrangle him into helping her with this movement. She was twenty and alluring and didn't take no for an answer. Her voice high but charming. He had always been supportive of her, had taken her modelling career seriously when others had doubted. She'd made money because of their work together.
When she'd appeared at her first casting call in a light denim skirt and a souvenir aquarium T-shirt, Tim barely batted an eye at her. Someone at her school had mentioned there was an open call for teen models. But when they asked her to pull her hair back, put on the plain black crewneck to do test shots, something must have clicked. He approached her father, standing behind a folding chair, compulsively checking his watch, and shook his hand.
"Your daughter has something," he said but didn't elaborate. That's not how these things work. It must be kept vague, up to interpretation.
One of her parents' friends had said that she was tall enough and even attractive enough to be a model, and so she looked into the logistics: age range of models, how much money they made, how quickly they could retire. Realizing her age was optimal, she begged her parents to take her to the model call. Her mother wasn't thrilled, had dropped a jar of salsa on the kitchen floor in shock. This wasn't a serious pursuit, something long-term. They'd negotiated. If she managed to get work, all money saved would be put away for school. It could be a learning experience, a chance to find out more about herself. Her father would drive her, his little girl next to him in the car as they made their way anywhere she needed to go.
That was eight years ago. Since then, Tim and Kittany have travelled the world, become business partners, founded this beautiful movement together. She couldn't tell what he thought of fourteen-year-old Kittany, so committed to doing everything by the book, constantly studying fashion magazines and Jane Austen. But by the time they'd moved into the warehouse space, started organizing _The Light_ , he seemed to understand that she meant business.
Early on he'd questioned her name. They were sitting on folding chairs in a meeting circle. It looked like a group therapy session, except they were passing around a fat growler of homemade alcohol, everyone swigging if they had a good idea, something approved by Kittany.
"Brittany, or even Brit, are not what I'm about anymore. I need to leave those names behind. My parents gave me this nickname, and it's going to be who I am. Cats, even kittens, are independent. They have a sexual look."
"What if you use a man's name?" he asked.
She threw the bottle of grain alcohol on the tile floor. Brown shards of glass exploded all around her. Alcohol pooled at her feet, soaking through her slim designer flats.
"What if you shut up like a man?" she'd said and stood up on her chair.
Everyone stopped talking and watched the pools of booze curl around on the concrete. Tim spent that evening mopping and cleansing with lemons and sleeping in a cold grey corner. She couldn't get rid of him. He was smart. And she liked that he wanted to question authority, that he acknowledged her as authority.
Kittany has the Light. This is what she makes clear to her followers. Though she doesn't call them followers. She calls them her Devotees. Because she believes in devotion, because devotion is love. Love comes from lust. Sex is the way to connect with light, hers being the brightest.
Boys line up to please her. They believe they will be touched by something intangible that will make them better. That by pleasuring her they are holding fast to hope. They believe. Kittany is careful not to break the law. So many who came before her made huge mistakes because they believed they were owed everything. Kittany wants to earn her Devotees. To have them follow her, please her, help her create a new world order from scratch. There are codes to follow in society that aren't laws or strict rules but ways of being. Don't break the trust of those who are devoted is her main one.
They've been hidden away for almost a year. Amassing Devotees wasn't difficult. At twenty-two, she's the youngest leader of a new religion in decades. She tweets her devotionals. She's been featured in _Buzzfeed_ , _Vice_ , _New York Magazine_ , and many small online publications. She had a six-page feature in _Vanity Fair_ ; Annie Lebovitz photographed her. Annie, wearing a satin sleep mask, was transported from the airport in the passenger seat of a Smart car driven by the largest of Kittany's guardians. She was delivered back to the airport the same way, with the radio playing classic rock and her eyes again covered in the sleep mask. She didn't see the large gate that separated the Devotional from the one-lane highway and lush hills.
A large print of the photograph hangs in the attic loft of Kittany's home. She's wearing a long, low-cut black dress, her long honey hair in a messy bun on her head, copper and silver and gold necklaces dangling at different lengths, some punctuated by rough crystals and stones. She'd been listening to a lot of Stevie Nicks, combed through every photo of her ever taken. The backdrop of the photo is the gazebo, the cornerstone of the Devotional. The gauzy curtains are wide open and behind her, the earth tones in the mountains seem to mimic her hazel eyes and not the other way around. That's the magic of Lebovitz in post-production. Kittany gives out small versions of this image, encased in vintage lockets, to the young girls who flock to be a part of what she's doing. They believe. That they can achieve high levels of devotion, carry light in their bodies, too. They dress like her, wear their hair how she's wearing hers, sometimes even mimic her tone of voice.
Tim waves his hand through the curtains and then stomps into the gazebo. He's supposed to make his presence known, not interrupt her, be respectful. No one is supposed to interrupt anyone. Kittany doesn't tolerate general rudeness.
"Thanks for knocking," she says, rubs homemade lavender balm into her dry elbows.
"How is anyone supposed to knock on a fucking curtain?"
"You know what I mean."
He grabs the patchwork pouf from under Kittany's feet and sits down in front of her.
"Well, I've been trying to take things literally," he says, eyes the moisturizer.
Kittany digs deep into the squat Mason jar and pulls out more balm, massages it into her cuticles. Elbows resting on his knees, head resting on his hands balled together, he looks like he's ready to question her about something she doesn't want to be questioned about. She presses her newly moisturized feet through the tent of his arms and rests them on his lap.
"Can I ask you to tell me why you're here right now?" she asks.
"I want to talk to you about your presentation. The way you're presenting your presentations, really. I still feel like something is missing. An elevation of ideas. Or the way you act, as if they're not elevated ideas. You're not giving yourself a high enough power."
She rubs her feet together, presses them against each other, heels close to his crotch.
"Is that the way you talked to the guy you managed from those used car commercials? Wondering if he could present himself better? You were a talent agent. Your job was to trust the talent, convince other people of that talent. Not to completely disregard everything they say and do. There is no higher power, Tim."
Atheism and agnosticism are popular. Growing in numbers. She is a part of this. There is no higher power. There is The Light. There are orgasmic good times. There is love in many forms: some more superficial, some deeper, some platonic, some romantic, and some frivolous but pure, like the love of artisanal salted caramel ice cream. Making a person lovable, herself included, wasn't as easy. She's still working on it. How she appears, every selfie, every Snapchat, every statement she makes. They can't be statements. That's for politicians. She has to remain genuine.
Getting in touch with people who used to casually worship God, or yoga, or home renovation television seemed difficult at first. But, in fact, that was so easy.
When they were an urban organization, it was always open calls for gatherings. Post events on Facebook. Leave artfully designed handbills at Whole Foods. Construct a beautiful setting with paper crafts in a warehouse space. Supply fancy canapés and artisanal drinks in mason jars. People need to believe they'll live lavishly. Trade in pleasure. Then see the importance of frugality too. Get them crafting with recycled materials. Old clothes into a quilt. Restructure a Coach purse into a compost bin. Guilt. That's the foundation of belief.
Sermons are for old men who want to tear people down. She deals in presentations. When she was a child, her mother was the one who convinced their family to stop going to church. They could be good people without it. It was the first time she'd heard the word "patriarchy," though coming out of her mother's mouth she wasn't sure how it related to sitting in pews and group hymn singing. Kittany was disappointed. At church, there were so many people; she could socialize with kids and have adults touch her face and tell her how beautiful she was. Her father always let her get ice cream and pie after church. Her mother said they could still get pie on Sundays. But they never did.
Today she is due to deliver an online presentation. The video accompaniment is ready and after she's done talking, live-streaming everyone the way they live so peacefully, flicking from slides of young men politely handing her slippers and robes in the morning to clips of mountain sheep locking horns to get access to a ewe, gain possession of a woman because they are animals, not humans, the whole thing will be available for $2.99 on iTunes.
"Kittany, I know there's no higher power."
"We're still building this, and I have the attention of a lot of people. Time. That's what we need. How many Twitter followers will it take to get you to stop harassing me?"
"Kardashian numbers, Kit. That's ideal."
The heel of her foot bumps his package with intent to harm. He whimpers and covers himself. Kittany ignores it, composes a tweet:
@GospelofKittany Sometimes we just need to send love. Let's devote ourselves to love. We are all in this together. #loveotional
Tim's phone buzzes in his pants pocket. He gets updates whenever Kittany tweets, posts on Facebook. Whenever she does anything. An instant Google alert to remind him that she's the one making things happen.
He even gets the bad news, the negative reviews, as he calls them. Even though she doesn't believe what they do can be reviewed like a blockbuster. It's not the same type of output, creation. This is something organic.
Young people have been positive, loving her posts, sending her private messages, crying over how she's transformed their lives, smiling and confessing how much they love her, making homemade shirts with her face on them, holding up signs that quote her messages of love. But older people have tried to take her down with words. A few articles crop up every so often, and she does her best to listen, read, absorb but not let it affect her. The work, her love, is what matters, what must be cultivated. If a columnist for the _Globe and Mail_ thinks she's vapid, then they aren't looking hard enough. At the work she's really doing. At the response. At themselves.
She doesn't think she has all the answers. Sometimes it's okay to be criticized. "Consider critics. How else will we know that we're doing something wrong, how else do we consider how to be better?" She'd tweeted with a #betterlives. It got 10K retweets and responses from across North America, some from across the world. People who felt changed, who told stories in 140 characters of becoming more aware of the world around them, cooking for the homeless, volunteering to build houses for strangers in another country, recycling their beer cans for the first time in their lives.
Not every tweet is equal. A few months ago, she posted a selfie posed in the rushing creek with her hair cascading over one eye, the rushing water further brightening her luminous skin. The caption read "It's okay to feel free."
This triggered an onslaught of reactions.
"Kittany is coy."
"Kittany is too beautiful."
"Kittany is sexist."
"Kittany acts like it's so easy to just do whatever you feel like."
"Kittany is smug."
"Kittany doesn't understand what life truly means."
"Kittany is blasphemous."
"Kittany only worships Kittany."
Tim and her team had analyzed the responses, yet couldn't quite figure out what had caused the reaction. The image, the words, her wet skin, her hair, a squirrel on the grassy bank in the corner of the frame, a combination of everything? She keeps these negative quotes about herself. She saves them, as the tweets and comments scroll on a screen above her kneeling desk. Apparently, Beyoncé does something similar. Kittany admires her, wishes she was better at dancing. It brings people together.
**Presentation 22: Seeds of Change**
Open on a sweeping shot of the Devotional. Mountains and trees, vibrant and glowing. Young men, shirtless, pick fruit from trees with shimmering leaves.
Voice-over: _Mountains and trees are the perfect shade of green. We don't try to improve on nature's spectrum. But we are imperfect. And yet, that is what makes us perfect. That is what makes us able to understand our need for more, our need to change, our need to grow. And, yes, sometimes our failure. But we are together_.
Kittany: Hello, I'm Kittany, founder of _The Light_ , here at the Devotional. Every time we make a move in our lives, things change. Even when we think things stay the same, that our routines are locked in, that we've figured out the best way to live our lives, there are moments of adjustment. A broken water heater that prevents us from cleaning the dishes, which prevents us from cleaning our home, which prevents us from feeling clean.
Change happens, even when we try to ward it off in every way. We become anxious. And why wouldn't we? The world does not want us to be as one. The world does not want us to indulge in simple pleasures. It wants to drive us apart.
Cut to the Boy, pacing in front of the gazebo. He's barefoot, and his curly hair makes him look like a poodle puppy. There's worry in his eyes, as if someone's forgotten him at a gas station.
Voice-over: _Even the young and beautiful experience anxiety. How can we make a change for those in our lives, those who seem to have things we wish we had? How can we put love ahead of jealousy? Really see other people with our eyes and the eyes that burst from within us, from somewhere we can't accurately define: the irises of the soul. We can never know the burdens of others, unless we communicate and show love_.
Cut to Kittany approaching the Boy with artisanal toast and fresh jam. She hands him the toile-printed plate, then holds out her arms. He sets down the plate and bombards her. In her arms he's burrowed deep.
Close up: The Boy inhaling Kittany's chest, deep breaths in. She strokes his bare arm.
Kittany: Small things can become big gestures. Each small step to affection might seem insignificant. But that's because of our perception. See each little moment, each slice of toast, smear of jam. We make those things with our hands, and then we use our hands to embrace each other. Touch someone today. Observe the change. And write to us here. We love hearing your stories. We see you. Sending Love and Devotion.
A text graphic starts small, zooms forward until it fills the screen: Try Change xoxo
"Thank you. You did so well today. I know you haven't been involved in many presentations." Kittany pets the Boy's curls.
"Thank you, ma'am." He stands over her, taller by at least half a foot.
"Don't call me ma'am. I'm not old enough for that." He's only eighteen, and she doesn't feel like a ma'am, though she's careful to never call herself a girl.
"Sorry."
"Don't apologize."
"Okay."
They stand in front of the gazebo as the camerawomen and crew move the cameras, booms, and mics. The women all smile at her, and she gives them reassuring looks. They'll need to edit everything together for the iTunes release. Kittany's proud of them. Efficient and skilled, they always make sure the music, the graphics, every detail is on message. The artistry of these videos can't tread into the realm of cheese.
"Great one today, Kit," Marian says, pats her on the back and struts with her camera bag on her shoulder.
"I think we should do some more long shots next time," Greta says.
Kittany nods her head. Thinks about visual impact.
"Thank you for continuing to do such good voice-over work, Emma. I know they'll re-record you, but it really helps for me to hear the words as we move and interact."
While she's talking to Emma, the Boy turns to her as if he has something to say. Before the Boy can talk, Tim barges between them, grips Kittany's shoulder. "Kit. Did you get that toast recipe from _goop_ last week?"
She's talked to too many men today, even though she's surrounded herself with women. Somehow there are still always so many men talking. Kittany ignores Tim, walks hand in hand with the Boy to the wellness centre. They need to recharge with massages and triangles of avocado. The Boy will be silent, in awe of her. He won't obstruct her devotion.
After reviewing articles and books and news reports, Kittany knew that total control wasn't her game. All those men writing their manifestos and enforcing them with an iron fist only made them more vulnerable to people wanting to take them down. L. Ron Hubbard, David Koresh, Jim Jones. They were wanted men, not because of their bad ideas but because of what they did. Because of their abuse, how they didn't respect people. Every one of them used their people like stepstools. While trying to convince their followers of their undeniable brilliance, were they not also thinking of them as underlings at the same time? If people don't want to be invested in what she's creating, they can leave. Devotion can't come from control. Love doesn't live in abuse.
Before the Devotional was a real place, she'd gone to visit her mother. She was proud of Kittany for giving up modelling, proud, too, of how she'd managed herself in that world—she'd made money; she wasn't holed up in a hotel room with rock stars and opiates. In the kitchen, Kittany and her mother were canning all the excess vegetables from the garden. Her father had gone on the first of many trips to his parents, who were both suffering with bad backs. Without Kittany and her father around to eat, the fresh vegetables were going to go rotten.
"I see you're still wearing those crystals. We used to love crystals in the seventies, too. I should get some of my old ones out for you."
"It's not about fashion, Mom."
"Oh, I know, Kit."
"It's about a lot more."
"Well, I still don't understand what it is you're doing now. Are you a party planner?"
Her mother is chopping the stems off beets while Kittany stands at the sink rinsing green beans. Right after Kittany started modelling, her mother had gone through a brief weight-loss kick. The dark wooden cupboards filled with pre-packaged bars and shakes. Unlike the way their kitchen was usually stocked, a mix of fresh vegetables, hearty breads, and cookies and candies were stashed in an old tin in the pantry. It didn't last long. Her mother couldn't handle the way the other women talked about themselves at meetings, how expensive and tasteless the boxed food was.
"Mom, this project I'm working on. It's something really big."
Kittany abandoned her green bean post and rushed to the hallway, came back with a stack of papers. Her mother held up her purpled hands as Kittany showed her the design for the Devotional, the first draft of her first presentation, photos of her standing in the empty field where it will all happen. She recited a message of love. Her mother's face dropped.
"Is this what you want to do with your life? How you'll spend all that money you just made? It's absurd."
Kittany grabs the papers, holds them to her chest. She stands and opens her mouth but doesn't speak. As she turns to leave, her mother reaches out, tries to catch her arm. Instead, she wipes a beet stain all over Kittany's $195 T-shirt, leaves a smear on the papers in her arms. Their last communication.
Her mother didn't want to understand her, or her goals. She didn't even want to try. This is why Kittany hasn't spoken to her in two years. Her father had long been too embarrassed about how much she'd bared as a model. The reason her parents are now separated is because of her career. Her father couldn't handle what he'd helped Kittany to achieve, and her mother couldn't invest in his absurd moaning about it. Her father's moved back to his small hometown across the country, to be near his aging parents. That day, Kittany left her mother alone in the only place she'd considered home. She didn't have to explain why she needed to make a new place for herself.
The people at the Devotional have jobs, work to do. But every minute of their lives isn't scheduled. It's Canada in the twenty-first century, and they have cable and craft beer and whatever else they need to unwind. People are filming crimes on their iPhones and documenting cats shoving a variety of objects off tables. It's so easy to get in trouble if you demand every ounce of every person. It's so easy to get into trouble. Images can cause problems.
At night there are organized events: beer tastings, knitting circles, movie nights, games nights. The first few weeks there were so few of them, and every night they gathered around a campfire, drank spirits, told stories, listened to her personal playlist on an iPhone in a tin cup. As more joined, it seemed necessary to organize. But it's not a home for the elderly. No one enforces participation. You don't have to make a turkey out of a tracing of your hand.
She remembers modelling. Every hour taken up by something she didn't feel devoted to. Everyone whose salary she paid cared about how she looked, what she ate, the kind of salads she posted pics of online. When she was alone, she slept or stayed awake thinking about how everyone in her life was thinking about her. Being alone didn't mean relaxing or enjoying herself. Some nights she'd lay awake, images of term papers and science experiments floating behind her closed eyes. Missing schoolwork and basic kid drudgery. That seemed like recreation.
As soon as she'd gotten a little bit of online attention, even fifteen hundred Instagram followers, her presence felt huge to her. Bigger than she'd dreamed. Because at that time, she hadn't dreamed at all. She was still a high school student, shuffling around to casting calls and photo shoots, and there they were, double tapping little hearts under photos of her in underwear sewn by children just a few years younger than she was.
In bed, Kittany tweets and retweets. So many young people are sending her their stories, their feelings, are revealing themselves in words and photos and emojis. A young woman in Alberta always believed she could never come out to her family. A boy from Nova Scotia who'd suffered horrible burns shared photos of his body, even though he was so ashamed of his skin that he hadn't looked at himself in the mirror in over two years.
And now, older people her parents' age, they too are sharing. There are a few out there who are able to understand devotion. Reclaiming their lives, understanding the value of things they'd never considered: women breaking free of loveless marriages, retired couples taking selfies together, the transformative power of memes. Lately these are her bedtime stories. She hates to admit to herself that usually these warm her heart more than the stories from people closer to her age.
One woman has been emailing her once a week. She caught the eye of the team filtering her emails because she is from Kittany's hometown.
The tone was sharp and sweet. The first had been a simple message of praise.
"This is good work you're doing."
This woman is anonymous. Not concerned with being seen. No avatar. Sometimes she gives criticism.
"When you say that love comes from lust, are you really thinking about what love means? It feels callous."
Sometimes concern.
"You might want to consider taking a break. Just a day for yourself."
It is hard not to see her mother in this woman. It is hard not to think about her mother sometimes. It is hard not to reach out.
"Can I talk to you?"
Emma's booming voice breaks through the silence of the night. Kittany pats the left side of the bed, and Emma sits beside her.
"I know you don't like gossip, but this is something I need you to hear. I didn't mean to." Emma stops and nervously gropes at the dangling tassels on the cotton drapes.
Kittany gets up from her bed and walks to the bar in the far corner of the tent. She knows Emma wouldn't interrupt if it wasn't something actually important or something that would burden her or others to keep to herself. She's also talks to Kittany like she's a real person, not some magical entity.
"I heard him talking. I was walking by, and I didn't mean to, but he was so loud. Yelling. And then it sounded like he was, I don't know, pushing someone, and then something fell over, and then I came here."
"Can you tell me what he said?"
Kittany hands Emma a glass, and she nods to pour the whisky. They both take a drink.
"It was hard to hear for sure, but it was something like 'I want to make sure we've got her while she's still viable. Women, they don't age like men do. We have time. There's time. But how much time?' Something like that. Shit, I'm so sorry."
She'd heard him say things like this before. Back in her modelling days. "Most men remain virile and sexually alluring for years. Well into their sixties." "Women need to put in more effort." And "Bodies have a clock all right, on when their ability to make money expires." As a teenager, she'd just put on her headphones and become absorbed in sounds that weren't his voice. Not because of the content but because he was always talking so much, and like any teenager, she got tired of adults shouting on high about anything and everything they felt. Maybe they didn't know better.
Her skin, her face, her body, it's all going to fall, change. The value system Tim trades in will leave her behind. Kittany radiates with rage. This is rudeness. She's given him so much of her. Allowed him to represent her ideas, her ideals.
"Can you stay for a while? Can we talk about something else?"
"Sure," Emma says, and they sit together, pass the bottle back and forth.
"Did you read _Eat Pray Love_?"
"Part of it," Emma says. "I got half way through pray and then I fell asleep on a Greyhound bus and left it on the seat when I got off. I didn't get to the love part."
"It was over-rated. Not sexy enough. I listened to the audiobook, though."
"Nothing is ever sexy enough for you."
Kittany chokes on her swig of whisky.
"Do you really think that's true?"
Emma yawns. Kittany knows she was up early today on an errand to a farm, sourcing some goats for the Devotional. Baby goats will look adorable in photographs, and everyone is already excited to make and share cute baby goat videos.
"Real answer? Sort of. You like to push things. Sometimes further than others would. Not in an offensive way, though."
Emma yawns again, body fading. Her wrist tips to the side, and the last bit of her drink almost drains onto the bed. Kittany takes the glass and sets it on the nightstand. She cradles Emma and strokes her hair. The night is so quiet. Sometimes it startles Kittany still, no street noise, vehicles, people shouting to themselves. A coyote howl doesn't count as night noise. Emma slumps into sleep, her dark hair spread across Kittany's lap and onto the creamy pillows.
**Presentation 23: Physical Devotion**
Kittany sits in the Wellness Centre lobby eating fresh, fat strawberries.
Voice-over: _Love. This is what we believe in. Each other. The flowing wave of feeling that overcomes us when we are able to connect as one. Sometimes we can love a small act. The simple joy of tasting the sweet flesh of a strawberry. But sometimes we need a more intimate connection_.
_Through these acts of sexual touching, we come to love. First we engage with consent; we look at each other; we nod our heads or say we want to proceed. Proceed to experience Devotional transformation. And in the end, we change through sex, through fucking or making love_.
Cut to the Boy. Kittany removes her custom-embroidered silk robe. Underneath she's wearing a slim floral cotton bandeau. The Boy rushes to her side. He pulls the bandeau up over her breasts. He kisses them. She nods. He presses his face between them, brings his hands to each, and touches them until she tells him to remove the bandeau.
The camera is too far away to see her tongue, but it can focus in on her face, head thrown back, not looking to see who is judging, who is worried, who cares what she looks like while she is trying to make her body feel better, to make the world a better place. Behind them, the sun is cresting over the mountain, a rosy slip of sky.
Voice-over: _The body has secrets, but the body is also a storyteller. Those secrets can be revealed to another person through body language, through gestures, and through our sensual connections_.
"You know why this is bad?" Tim is furious. He stalks onto the simple set like a caricature of an angry Hollywood mogul.
Kittany wanted the scene filmed in the morning. Not just for the morning light, those golden hues that create a warm glow, but also because she wasn't in the mood to be combative. Tim is a night owl and usually sleeps in. Kittany high-fives Greta and gives Emma a hug, a long hug.
"I know why you think it's bad, but it's on brand. Not everything is about what you think."
"On brand? A slutty, immature brand! You're fucked," Tim shouts through his grogginess.
She calls out to her number one performer and stalks away, Tim stumbling after her with his morning coffee in a metal camp mug. The Boy runs ahead to catch Kittany, her hand outstretched to him.
#sensualconnections starts trending. Kittany enlists a group at the Devotional to Instagram themselves in whatever poses feel beautiful to them. She selects only those who are engaged and excited about baring their bodies, about entwining their limbs. They make eleven-second make-out videos. Encourage their devoted to do the same. Young people everywhere slipping tongues into each other's mouths, showing skin, baring parts of their bodies in slips and flashes and filters. Their stories fill 140 characters, detail sensual experiences, fumbling first encounters and masturbation fails, their never-ending virginity. Young women sexualized by older men. Boys unsure of how not to offend girls. It's like this for a while. Honest and loving. Kittany, alone at night, taking time to look and read. She catches the glimmers of fear in their eyes when they pose, the vulnerable curls of their lips as they kiss, the delicate trails of saliva, the stretch marks and appendix scars, the acne, the makeup applied without the help of professionals. They'd all look gorgeous in a magazine. A special publication for the devoted. Her people seeing her people in glossy print.
Free Wi-Fi was to allow those living in the Devotional to express themselves, to communicate easily. The IT team was great at fixing any issues, and not once did Kittany think to snoop, to set up surveillance on phones and laptops.
But right under her nose, Tim pulled out his phone and contacted everyone he could think of who would be looking for a scoop, a hot tip. He didn't need to hide. In the middle of the night, while she lay in bed, propped on pillows, scrolling, dabbing feeds with emojis, he got in touch with all his old contacts. He told them that she was almost always naked and made everyone compliment her constantly, that she had to look in a mirror for an hour every day, that she forced people to touch her sexually, that she forced herself on people, that she was abusive. The rumours didn't need to be true. The spin cut her, everyone waiting to go negative.
Major news outlets reported that she was a sex addict, that the boy was underaged, that she was flying too close to the sun. Entertainment television reported that she was paying those living at the Devotional for sex. Religious groups protested her hold on youth, alleged that teen pregnancy was spiking, that chlamydia was making a comeback with eighteen to twenty-five-year-olds. And all of it was her fault. She was ruining society, a poison to the youth of today.
"Would anyone care if she wasn't young and photogenic?" asks one talking head.
"There's no doubt she thinks she can get away with anything. She's used to getting away with everything because she's beautiful," responds another.
"Exactly. Kids listen to what comes out of her mouth because she might as well be a pop star, mindlessly singing the word 'baby' over and over again." The talking head pounds the desk for emphasis.
"I give her religious movement another six months. If she doesn't Heaven's Gate them all—coerce them into a glamorous mass suicide. Gorgeous bodies, dead in designer sneakers."
On the night of her nineteenth birthday, Tim had taken her phone right out of her hand and shot so many photos of her in an exclusive booth at a cocktail lounge. Sipping from expensive drinks, as she has on so many other nights, she's surrounded in the dim light by other models. Together they crowd around the booth, all dressed to make the world jealous of their ease at being alive.
"Put these online," he'd instructed her.
Tim knew it was the right moment. She was transitioning from a child model that people wanted to know, be near to, touch, to an adult model that people wanted to know, be near to, touch. The next morning, as she looked at the confetti stickers and hearts and stars decorating her posts, she realized how calculating Tim was, how he knew his business. She could legally drink, a new stage in her career that had been so carefully constructed. It was serious business.
And this was no different. To him, spreading lies was the right way to go. A counterpoint to her presentation. Balance in the universe. A challenge for her to overcome. He did it all from a hundred yards away while she wrote notes to herself, while she thought about the lives of these young people growing up, hating themselves. And then he stole an ATV and slipped away, the phone she'd paid for zipped safely into his waterproof jacket pocket.
Kittany pulls up her old photo shoots, compares her poses then and now. Her arm draped over her head in the underwear ad is identical to her arm in a recent post about letting go of anger. A coy look on her face in a black and white shot from her first year as a model pairs perfectly with last year's "Love Yourself" meme series. This is no different. How many times have her devotees described her as beautiful, commented on her presentation outfits, screamed in Snapchat their jealousy of her glossy hair. Her face still the reason she coerces people.
Kittany emails the anonymous woman. She wants to check in with her; it's her first instinct. But she hasn't received a response. And Kittany is constantly checking to make sure.
In seclusion, the news playing on her iPhone, she scrolls and scrolls. For the first time since all of this happened, she cried. She covered her body in thin blankets, even though it was the heat of summer. Like when she was first modelling on those long days, she couldn't stand her body anymore. It was part of her but also not, like she could rip it from herself, and then lightly slip out of it and float away.
After a night of ignoring anyone who came to her bedroom, a night alone shouting, sobbing, running every mistake and failure in her mind, she posts a video. A close-up of her face. Smears of mascara. Red-streaked eyes. A sniffle of snot threatening to emerge from a nostril. In the allotted seconds, she expresses the importance of intimacy, of consent. How every expression of ourselves we might normally be afraid of is actually beautiful.
"Everyone. All ages. Women, especially, if you think you've aged out of beauty: ignore. You're sexy and magical. Share with me. I'm opening my direct messages. I want to see you. I am not the only face."
Then she uses her finger to scribble over her own face, like black Sharpie straight over the screen.
And they do send her photos. Flooded with grainy selfies, black-and-white butts, wrinkled cleavage. These women are her, and she will be them. Their revealing captions, their stories, their images. People need to see these. Does it matter if she loses followers? They'll come back when they look in the mirror in forty years, when they realize how their eyes and hearts have betrayed them.
One woman sends nudes, six-second sex tapes, a story revealed in body parts. She's sixty-eight. Her son bought her a selfie stick for a trip to Belgium, but instead she's at home in the same bedroom she's had since the day she got married. Above her bare shoulder, a dresser with a wooden jewellery box and family photos. In a shot of her upper body while lying on her bed, Kittany can see a cheap painting of a sunset. The video reveals two older bodies moving together, a pair of department store underwear rolled down a thigh.
Overwhelmed, Kittany can't stop staring at each photo, watches the tiny videos over and over. If only these women were here. They should be here.
Kittany thinks of her mother. Across the country, sitting in her kitchen nook in the early morning dark, drinking coffee. Before she pats on powder, lines around her mouth rising and falling as she chews granola, as she glows in the light of the local morning show. Her mother's body changed from when Kittany first remembers her, from a figure eight to a Weight Watchers casualty twig to a soft strength. She changed her mother's body just by living inside it. Genetics.
Her anonymous woman doesn't send a photo, remains visually inaccessible. But she does send a message. "How are you feeling?"
Kittany imagines Tim holed up in a hotel room with cheap scotch and Bugles, ready to sell a new lie to the press. Reporting her new "obsession" with older women and an unhealthy fixation on aging. Photoshopped images of her with horribly faked, horrible plastic surgery. She never claimed to be psychic, but if Tim doesn't go this route she'll be surprised and a little disappointed. Kittany knows him so well.
Her phone buzzes again, but it's not a notification. It's the Boy. She lets him in. They lay on their stomachs on top of the sheets, scrolling through photos.
"Look at these women," Kittany says.
"Okay."
"Really look."
The Boy scrunches up his aquiline nose.
"See them," she says.
_You're so beautiful_ , everyone always said. _You should be a model_. Kittany knows she should do a lot of things. She should keep trying. Admit her failure. Admire. She should cut off Tim's phone, his paycheque. She should ask these women how she can make them a part of her movement. She hands the phone to the boy and lets him explore, on his own, the bodies.
Kittany lies on her back and looks up at the tent ceiling. By winter, the yurts will be ready. Solid structures erected to protect them from cold and snow. In the lantern light, the creases in the canvas look like a crooked smile. She closes her eyes. Touches the smooth sides of her mouth. Kittany scrolls through an imaginary timeline of her body, her face. By the time she's her mother's age, she'll have documented most of her life. It will be beautiful.
**Acknowledgments**
Thanks to Kathryn Mockler at _Joyland_ and the Writers' Trust of Canada for giving time and space to stories from this collection.
I want to send heartfelt thanks (and refreshments) to Kellee Ngan, Shay Wilson, and Meghan Waitt for looking at earlier drafts of these stories and who are always there to give the best notes and hugs.
To my creative companion for life, Daniel Zomparelli, I give all the thanks. You pushed me, commiserated with me, and drank with me. Without you I couldn't have completed this book.
Thanks to the Lying Bastards, who have seen the roughest of my work over the past twelve years and helped me see these stories through. Sally Breen, Keri Korteling, Nancy Lee, Judy McFarlane, Denise Ryan, Carol Shaben, and John Vigna are all gems as writers and humans.
So much gratitude to Jen Sookfong Lee, Nancy Lee, Anakana Schofield, and Jenny Slate for your generous words about my words. Thanks to my editor Robyn So, who understood what I was trying to do. And huge thanks to the delights at Arsenal Pulp Press: Brian, Robert, Susan, Oliver, and of course, Cynara, who is a pal and a confidante, and a totally stellar publicist.
Love and gratitude to Roxan Marucot and Jason Bay who've lived with (and near) me and fed me and have been in my life as long as the oldest stories in this book. And special love to Jag Dost who always checks in on me at the right time.
Thanks as always to my family for their love and support, especially my mom and dad who always thought short stories were cool.
To my nieces, far in the future, I hope you're both as fierce and tender as the women in this book.
PHOTO: SHAY WILSON
**Dina Del Bucchia** is the author of three collections of poetry: _Coping with Emotions and Otters_ (Talonbooks, 2013), _Blind Items_ (Insomniac Press, 2014), and _Rom Com_ (Talonbooks, 2015), the latter written with her _Can't Lit_ podcast co-host Daniel Zomparelli. Her short story, "Under the 'I'," was a finalist for the Writers' Trust RBC Bronwen Wallace Award in 2012. Her work has also appeared in such places as _Event_ , _Matrix_ , _The Fiddlehead_ , _SAD Mag_ and _Joyland_ , and as art at Old Friends' exhibition Funny Business (Gallery Atsui) and at That One Thing You Said: an exhibit of visual poetry at Verses Festival of Words. She is an editor of _Poetry is Dead_ magazine and is the Artistic Director of the Real Vancouver Writers' Series. She lives in Vancouver. **_dinadelbucchia.com_**
Table of Contents
1. Cover
2. Title Page
3. Copyright
4. Contents
5. Keeping Things Alive Is Too Much Work
6. Don't Tell Me What to Do
7. Miss Supreme
8. Under the "I"
9. Cold Cuts
10. Particleboard Man
11. A Beautiful Feeling
12. Hamsters
13. Haul
14. Nest
15. Sometimes We Can Be That Way
16. Sleep Talk
17. Instructions for Having an Affair
18. Full Price
19. The Gospel of Kittany
20. Acknowledgments
21. About the Author
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