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Die Communauté de communes du Pays Gentiane ist ein französischer Gemeindeverband in der Rechtsform einer Communauté de communes im Département Cantal in der Region Auvergne-Rhône-Alpes. Der Verwaltungssitz ist im Ort Riom-ès-Montagnes.
Historische Entwicklung
Mit Wirkung vom 1. Januar 2017 schloss sich die Gemeinde Lugarde dem Gemeindeverband an, die bisher zur Communauté de communes du Cézallier gehörte.
Am 1. Januar 2019 verließen die Gemeinden Chanterelle, Condat, Montboudif und Saint-Bonnet-de-Condat die Communauté de communes Hautes Terres und schlossen sich dem Gemeindeverband an.
Mitgliedsgemeinden
Die 17 Mitgliedsgemeinden sind:
Einzelnachweise
Pays Gentiane
Gegründet 1993
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,213
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Lega Toscana () may refer to two political parties in Italy:
Lega Toscana, originally established as "Movimento per la Toscana" (), later known as "Alleanza Toscana" () and "Lega Nord Toscana" ()
Lega Toscana (2011), originally established as "Lega per la Toscana" (English: League for Tuscany)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,812
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Valperga is a comune (municipality) in the Metropolitan City of Turin in the Italian region Piedmont, located about north of Turin, in the Canavese historical region.
It is home to the Sacro Monte of Belmonte, a site of pilgrimage and worship close to it. The Sacro Monte was built in 1712 at the initiative of the Friar Minor Michelangelo da Montiglio. In 2003, the sanctuary was inserted by UNESCO in the World Heritage List.
References
External links
Official web site for European Sacred Mountains
Cities and towns in Piedmont
Canavese
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,711
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Will Copacabana Beach become the source of death in Brazil?
08/09/2020By travelweeklyPosted in News
The party never stops at Copacabana Beach in Rio de Janeiro. Even Coronavirus will not intimidate this beach most in the world see as the most beautiful land on the globe. Could it soon become the deadliest land as well?
Copacabana, the very name itself provokes images of beauty, sand, and ocean. The magnificent jungle-clad mountains rise from the ocean and seem to blend into the beautiful bend of Copacabana Beach, a world-renowned hotspot for tourists from around the world. The neighborhood lives up to its nickname, A Princesinha do Mar or Princess of the Sea. Copa (short for Copacabana) is a paradise with stunning beaches, lively streets, where the party never seems to stop. Apart from being Rio's egalitarian and eclectic neighborhood, romance and glamour are its obvious trademarks.
Brazil is entering the most dangerous phase of Coronavirus. With record infections, Brazilians just had enough and started to forget about social distancing. Recommendations by health experts to remain isolated are being challenged even by a nursing technician who worked in a field hospital for coronavirus patients.
'The coronavirus is being controlled a little more, that gave me the security to go out,' she said.
With more than 4,148,000 confirmed infections and 127,000 deaths from the virus, Brazil has the second-highest totals in behind only the United States. In recent weeks, Latin America's largest country has left a new case number plateau that had dragged on for almost three months and started seeing a reduction in the number of new confirmed cases.
But with an average of 820 deaths per day, its numbers are still considered high in Brazil.
Latest eTN Podcast
A pulmonologist at Brazil´s premier biomedical research and development lab, the Oswaldo Cruz Foundation, or Fiocruz, warned that if Brazilians are negligent the country could see a repeat of what happened in Europe, especially Spain, where second waves of new cases were seen.
People at Copacabana Beach in Rio are at the beach ignoring all rules of social distancing. The same is true in Sao Paulo, Brazil's worst-hit state with more than 855,000 confirmed infections and 31,000 deaths. Thousands of residents took advantage of the long weekend to travel to the coast.
Uganda Wildlife Authority opens all parks for tourism
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New metro line operated with Chinese-made automated driverless trains capable of speed of up to 74.5 miles (120 kilometers) per hour with daily capacity of 800,000 passengers now connects central Istanbul and […]
Don't leave home without it- Your Chase Card!
Don't leave home without your American Express card was a year long slogan. This may now be replaced by Chase. Don't leave home without it- Your Chase Card!
China allows 'organized group' travel to 'relevant countries'
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
\label{Introduction}
Dimerization is a common process in
physical, chemical and biological systems.
In this process, two identical units (monomers)
bind to each other and form a dimer
($A + A \rightarrow A_2$).
This is a special case of a more general
reaction process (hetero-dimerization) of the form
$A + B \rightarrow AB$.
Dimerization may appear
either as an isolated process
or
incorporated in
a more
complex reaction network.
The modeling of dimerization systems is commonly done using
rate equations, which incorporate the mean-field approximation.
These equations describe the time evolution
of the concentrations of the monomers and the dimers.
Assuming that the system is spatially homogeneous,
these concentrations can be expressed either in terms
of the copy numbers per unit volume or in terms of the
total copy number of each molecular species in the system.
The rate equations are reliable when the
copy numbers of the reacting
monomers in the system are sufficiently large for the
mean-field approximation to apply.
However, when the copy numbers of the reacting monomers
are low, the system becomes highly fluctuative,
and the rate equations are no longer suitable.
Therefore the analysis of dimerization processes
under conditions of low copy numbers
requires the use of stochastic methods
\cite{vanKampen1981,Gardiner2004}.
These methods include the direct integration of
the master equation
\cite{Biham2001,Green2001},
and
Monte Carlo (MC) simulations
\cite{Gillespie1976,Gillespie1977,Tielens1982,Newman1999,Charnley2001}.
The master equation
consists of coupled differential equations
for the probabilities of all the possible microscopic states
of the system.
These equations are typically solved by
numerical integration.
However,
in some simple cases, the steady state probabilities
can be obtained by analytical methods
\cite{Green2001,Biham2002}.
The difficulty with the master equation is that it consists
of a large number of equations,
particularly if the dimerization process is a part of
a larger network.
This severely limits its usability for the analysis of complex
reaction networks
\cite{Stantcheva2002,Stantcheva2003}.
Monte Carlo simulations provide a stochastic implementation
of the master equation, following the actual temporal
evolution of a single instance of the system.
The mean copy numbers of the reactants and the reaction
rates are obtained by averaging over an ensemble of such instances.
In both methods, there is no closed form expression for the
time dependence of the copy numbers and reaction rates.
Recently, a new method for the stochastic modeling of reaction networks
was developed, which is
based on moment equations
\cite{Lipshtat2003,Barzel2007a,Barzel2007}.
The moment equations
are much more efficient than the master equation.
They consist of only one equation for each reactive species,
one equation for each reaction rate,
and in certain cases
one equation for each product species.
Thus, the number of moment equations
that describe a given chemical network
is
comparable to the number of rate equations,
which consist of one equation for each species.
Moreover,
unlike the rate equations,
the moment equations
are linear equations.
In some cases, this feature enables to
obtain an analytical solution for the time dependent
concentrations.
In this paper,
we apply the moment equations to the analysis
of dimerization systems with fluctuations.
These equations are accurate even in the limit of
low copy numbers, where fluctuations are large
and the rate equations fail.
We show how to obtain an analytical solution for
the time dependent concentrations of the reactant
and product species
as well as for the reaction rate.
We identify and characterize the
different dynamical regimes of the system
as a function
of the parameters.
We examine the validity of our solution
by comparison to the results obtained from the master equation.
The analysis is performed for three
variants of the system: dimerization, dimerization
with dissociation and hetero-dimer formation.
The paper is organized as follows.
In Sec. \ref{sec2} we analyze the dimerization process
using the moment equations and provide an analytical solution
for the time-dependent concentrations.
In Sec. \ref{sec3} we extend the analysis to the
case in which dimers may dissociate.
In Sec. \ref{sec4} we generalize the analysis to
the formation of hetero-dimers.
The results are summarized
and potential applications are discussed
in Sec. \ref{sec5}.
\section{Dimerization systems}
\label{sec2}
Consider a system of molecules, denoted by
$A$, which diffuse and react on
a surface or in a liquid solution.
Molecules are produced or added to the system
at a rate
$g$ (s$^{-1}$),
and degrade at a rate
$d_1$ (s$^{-1}$).
When two molecules encounter each other
they may bind and form the dimer
$D$.
The rate constant for the dimerization process is denoted by
$a$ (s$^{-1}$).
For simplicity, we assume that
the product molecule,
$D$,
is non-reactive and undergoes degradation at a rate
$d_2$ (s$^{-1}$).
The chemical processes in this
system can be described by
\begin{eqnarray}
&& \varnothing \arrow{g} A \nonumber \\
&& A \arrow{d_1} \varnothing \nonumber \\
&& A + A \arrow{a} D \nonumber \\
&& D \arrow{d_2} \varnothing.
\label{eq:processes1}
\end{eqnarray}
\subsection{Rate Equations}
The dimerization system described above is characterized
by the average copy number of the monomers,
$\NA$,
and by the average copy number of the dimers,
$\ND$.
Denoting
the average dimerization rate
by
$\R$ (s$^{-1}$),
the rate equations for this system take the form
\begin{eqnarray}
\frac{d\NA}{dt} &=& g - d_1 \NA - 2\R \nonumber \\
\frac{d\ND}{dt} &=& - d_2 \ND + \R.
\label{eq:rate1}
\end{eqnarray}
\noindent
These equations include a term for each process
which appears in
Eq. (\ref{eq:processes1}).
The factor of
$2$
in the reaction term of the first equation
accounts for the fact that the dimerization process removes
two $A$ molecules,
producing one dimer.
The dimerization rate,
$\R$,
is proportional to
the number of pairs of $A$ molecules
in the system,
$\NA(\NA-1)/2$,
where the factor of $1/2$ is absorbed into the
rate constant $a$.
As long as the copy number of $A$ molecules is large,
it can be approximated by
\begin{equation}
\R = a{\NA}^2.
\label{eq:Rrate1}
\end{equation}
\noindent
Eqs. (\ref{eq:rate1}) form a closed set
of two non-linear differential equations.
Their steady state solution
is
\begin{eqnarray}
\NAss &=&
\frac{d_1}{4a}
\left(
-1 + \sqrt{1 + 8 \gamma}
\right)
\nonumber \\
\NDss &=&
\frac{d_1^2}{16 a d_2}
\left(
-1 + \sqrt{1 + 8 \gamma}
\right)^2,
\label{eq:ss_rate1}
\end{eqnarray}
\noindent
where
$\gamma$
is the reaction strength parameter given by
\begin{equation}
\gamma = \frac{ag}{d_1^2}.
\label{eq:gamma}
\end{equation}
\noindent
Two limits can be identified.
In the limit where
$\gamma \gg 1$,
the steady state dimerization rate satisfies
$\Rss \simeq g/2$,
and the steady state dimer population is
$\NDss \simeq g/2d_2$.
This means
that almost all the monomers that are generated
end up in dimers and
the monomer degradation process becomes irrelevant.
Therefore,
this limit is referred to
as the reaction-dominated regime.
The degradation-dominated limit is obtained when
$\gamma \ll 1$.
In this limit
$\NAss \simeq g/d_1$,
$\Rss \simeq a g^2 / {d_1}^2 = \gamma g$
and
$\NDss \simeq a g^2/({d_1}^2 d_2)$,
namely most of the monomers that are generated
undergo degradation and only a small fraction
end up in dimers.
The time-dependent solution for the population size of the
$A$
molecules can be obtained from the first equation in
Eqs. (\ref{eq:rate1}).
The result is
\begin{equation}
\NA = \NAss -
\frac{1}{2 a \tau}
\left(
1 + C e^{t/\tau}
\right)^{-1},
\label{eq:rate_solution1}
\end{equation}
\noindent
where
$\tau = 1/\sqrt{d_1^2 + 8ag}$
is the relaxation time
and the parameter
$C$
is determined by the initial conditions.
In the reaction-dominated regime, where
$a g \gg d_1^2$,
the relaxation time converges to
$\tau \simeq 1/\sqrt{8ag}$.
In the degradation-dominated regime,
it approaches
$\tau \simeq 1/d_1$.
The rate equation analysis is valid as long as
the copy numbers of the reactive
molecules are sufficiently large
\cite{Lederhendler2008}.
In the limit in which the copy number
$\NA$
of the monomers
is reduced to order unity or less,
the rate equations
[Eqs. (\ref{eq:rate1})]
become unsuitable.
This limit can be reached in two situations:
when the monomer concentration is very low or
when the volume of the system is very small.
In the limit of low copy number of the monomers,
the system becomes dominated by fluctuations
which are not accounted for by the rate equations.
A useful characterization of the system is
given by the system-size parameter
\begin{equation}
N_0 = \frac{g}{d_1},
\label{eq:N0}
\end{equation}
\noindent
which approximates the copy number
of the monomers in case that
the dimerization is suppressed.
The parameter
$N_0$
provides an upper limit for the monomer population size
under steady state conditions.
It can be used to characterize the dynamical regime
of the system.
In the limit where
$N_0 \gg 1$
the copy number of the monomers is typically
large and the rate equations are reliable.
However, when
$N_0 \lesssim 1$
the system may become dominated by fluctuations.
In this regime, the rate equations
fail to account for the population sizes and the dimerization rate.
In Fig. \ref{fig1} we present a schematic illustration of
the parameter space in terms of $\gamma$ and $N_0$,
identifying the four dynamical regimes.
\subsection{Moment Equations}
To obtain a more complete description of the dimerization process,
which takes the fluctuations into account,
we present the master equation approach.
The dynamical variables
of the master equation are the probabilities
$P(N_A,N_D)$
of having a population of
$N_A$
monomers
and
$N_D$
dimers
in the system.
The master equation
for the dimerization system
takes the form
\begin{eqnarray}
\label{eq:master1}
\frac{dP(N_A,N_D)}{dt} &=&
g [P(N_A - 1,N_D) - P(N_A,N_D)]
\nonumber \\ &+&
d_1 [(N_A + 1)P(N_A + 1,N_D) - N_A P(N_A,N_D)]
\\
&+&
d_2 [(N_D + 1)P(N_A,N_D + 1) - N_D P(N_A,N_D)]
\nonumber \\
&+&
a [(N_A + 2)(N_A + 1)P(N_A + 2,N_D - 1) - N_A(N_A - 1)P(N_A,N_D)].
\nonumber
\end{eqnarray}
\noindent
The first term on the right hand side
accounts for the addition or formation of
$A$
molecules.
The second and third terms account
for the degradation of
$A$
and
$D$
molecules,
respectively.
The last term describes the reaction process, in which two
$A$
molecules are annihilated and one
$D$ molecule is formed.
The dimerization rate is
proportional to the number of pairs of
$A$
molecules in the system, given by
$N_A (N_A - 1) / 2$.
Therefore, the dimerization rate can be expressed in terms
of the moments of
$P(N_A,N_D)$
as
\begin{equation}
\R = a
\left(
\NAs - \NA
\right),
\label{eq:Rmaster1}
\end{equation}
\noindent
where the moments are defined by
\begin{equation}
\langle N_A^n N_D^m \rangle =
\sum_{\substack{N_A = 0\\N_D = 0}}^{\infty}
{N_A^n N_D^m P(N_A, N_D)},
\label{eq:moments}
\end{equation}
\noindent
and $n$ and $m$ are integers.
Note that in the stochastic formulation, the
expression used for the dimerization rate,
$\R$,
is different than in the deterministic approach
[Eq. (\ref{eq:Rrate1})].
The two expressions are equal in case that
$P(N_A)$ is a Poisson distribution, for
which the mean and the variance are equal.
The master equation (\ref{eq:master1})
for the dimerization system can be analytically solved
to obtain the steady state probabilities
$P(N_A)$.
This solution can be found at refs.
\cite{Green2001,Biham2002}.
However, an
analytical solution for the time dependent case is currently
not available.
For dimerization systems in the degradation-dominated limit, the
probability distribution
$P(N_A)$
approaches the Poisson distribution.
However as
$\gamma$
increases, and the system enters the reaction-dominated regime,
$P(N_A)$
becomes different from Poisson.
In Fig. \ref{fig2}
we present the marginal probability distributions
$P(N_A)$ (circles)
as obtained from the master equation for four choices of the
parameters, each in one of
the four regimes shown in Fig. \ref{fig1}.
In the limit of
$N_0 \ll 1$
and
$\gamma \gg 1$ (a)
the system is in the reaction dominated regime
(quadrant I in Fig. \ref{fig1}),
and
the results obtained from the master equation deviate from
Poisson (solid lines).
Here the parameters are
$g = 0.5$,
$d_1 = 2$,
$a = 200$
and
$d_2 = 10$ (s$^{-1}$).
The distribution shown in (b),
where
$N_0 \gg 1$
and
$\gamma \gg 1$
(quadrant II in Fig. \ref{fig1}),
also deviates from the Poisson distribution.
Here the parameters are
$g = 100$,
$d_1 = 1$,
$a = 1$
and
$d_2 = 10$ (s$^{-1}$).
In the limit of
$N_0 \ll 1$
and
$\gamma \ll 1$ (c)
the system is in the degradation-dominated regime
(quadrant III in Fig. \ref{fig1}),
and correspondingly
$P(N_A)$
coincides with the
Poisson distribution.
Here the parameters are
$g = 0.5$,
$d_1 = 5$,
$a = 1$
and
$d_2 = 10$ (s$^{-1}$).
Finally, in (d)
$N_0 \gg 1$
and
$\gamma \ll 1$
(quadrant IV in Fig. \ref{fig1}),
the system is dominated by degradation and
as before,
the master equation results coincide with the Poisson distribution.
Here the parameters are
$g = 10$,
$d_1 = 1$,
$a = 5 \times 10^{-3}$
and
$d_2 = 10$ (s$^{-1}$).
In addition to the analytical solution mentioned above,
the master equation
[Eq. (\ref{eq:master1})]
can also be integrated numerically using standard
steppers such as
the Runge Kutta method
\cite{Acton1970,Press1992}.
In numerical simulations,
one has to truncate the master equation
in order to keep the number of equations finite.
This is achieved by setting upper cutoffs
$N_A^{\rm max}$
and
$N_D^{\rm max}$
on the numbers of $A$ and $D$ molecules,
respectively.
This truncation is valid if the probability for the number of molecules of
each type to exceed the cutoff is vanishingly small.
The population sizes of the
$A$
and
$D$
molecules and the dimerization rate are
expressed in terms of the first moments
of
$P(N_A, N_D)$
and one of its second moments,
$\NAs$.
Therefore, a closed set of equations for the time derivatives of
these first and second moments could directly provide all the information
needed in order to evaluate the population sizes and the dimerization rate
\cite{Lipshtat2003}.
Such equations can be obtained from the master equation using
the identity
\begin{equation}
\frac{d\langle N_A^n N_D^m \rangle}{dt} =
\sum_{\substack{N_A = 0\\N_D = 0}}^{\infty}
{N_A^n N_D^m \dot P(N_A, N_D)}.
\label{eq:moments_dot}
\end{equation}
\noindent
Inserting
the time-derivative
$\dot P(N_A, N_D)$
according to Eq. (\ref{eq:master1}),
one obtains the moment equations.
The equations for the average copy numbers are
\begin{eqnarray}
\frac{d\NA}{dt} & = & g - d_1\NA - 2\R
\nonumber \\
\frac{d\ND}{dt} & = & -d_2\ND + \R,
\label{eq:moment1_no_R}
\end{eqnarray}
\noindent
while the equation for
the dimerization rate is
\begin{equation}
\frac{d\R}{dt} = (2ag+4a^2) \NA + (10a - 2d_1)\R - 4a^2 \NAc.
\label{eq:Rmoment1_not_truncated}
\end{equation}
\noindent
Eqs. (\ref{eq:moment1_no_R})
have the same form as the
rate equations (\ref{eq:rate1}).
However the term for the dimerization rate,
$\R$,
as appears in the moment equations is different from the analogous
term in the rate equations
[Eq. (\ref{eq:Rrate1})].
Eqs. (\ref{eq:moment1_no_R}),
together with Eq. (\ref{eq:Rmoment1_not_truncated}),
are a set of coupled differential equations,
which are linear in terms of the moments.
Although we have written the equations only for the
relevant first and second moments,
the right hand side of
Eq. (\ref{eq:Rmoment1_not_truncated})
includes the third moment for which we have no equation.
In order to close the set of equations one must express this
third moment in terms of the first and second moments.
Different expressions have been proposed.
For example, in the context of birth-death processes
the relation
$\NAc = \NAs \NA$
was used
\cite{McQuarrie1967}.
This choice makes the moment equations
nonlinear,
which might affect their
stability.
Another common choice is to
assume that the third central moment is
zero (which is exact for symmetric distributions)
and use this relation to express
the third moment in terms of the
first and second moments
\cite{Gomez-Uribe2007}.
Here we use a different approach.
We set up the closure condition
by imposing a highly restrictive cutoff
on the master equation.
The cutoff is set at $N_{A}^{\rm max} = 2$.
This is the minimal cutoff
that still enables the dimerization process to take place.
Under this cutoff,
one obtains the following relation
between the first three moments
\cite{Lipshtat2003}
\begin{equation}
\NAc = 3\NAs -2 \NA.
\label{eq:third_moment}
\end{equation}
\noindent
Using this result, one can bring the moment equations
[Eqs. (\ref{eq:moment1_no_R}) - (\ref{eq:Rmoment1_not_truncated})]
into a closed form:
\begin{eqnarray}
\frac{d\NA}{dt} &=& g - d_1\NA - 2\R
\nonumber \\
\frac{d\ND}{dt} &=& -d_2\ND + \R
\nonumber \\
\frac{d\R}{dt} &=& 2ag\NA - 2(d_1 + a)\R.
\label{eq:moment1}
\end{eqnarray}
\noindent
Numerical integration of these equations provides all the required moments,
from which the population sizes and the dimerization rate are obtained.
\subsubsection{Steady State Analysis}
The steady-state solution of the moment
equations
takes the form
\begin{eqnarray}
\NAss &=&
\frac{g(a + d_1)}{2ag + d_1 a + d_1^2}
\nonumber \\
\NDss &=&
\frac{ag^2}{d_2(2ag + d_1 a + d_1^2)}
\nonumber \\
\Rss &=&
\frac{ag^2}{2ag + d_1 a + d_1^2}.
\label{eq:ss_moment1}
\end{eqnarray}
\noindent
In the limit of very small copy numbers
the approximation appearing in
Eq. (\ref{eq:third_moment})
is valid.
Thus, in this limit the
moment equations
provide accurate results, both for the population sizes (first moments)
and for the dimerization rate (involving a second moment).
To evaluate the validity of the moment equations in the limit of
large copy numbers, we compare
Eqs. (\ref{eq:ss_moment1})
with the solution of the rate equations
(\ref{eq:ss_rate1}),
which is valid in this limit.
Consider the large-system limit, where
$N_0 \gg 1$
[Eq. (\ref{eq:N0})].
In this limit,
the common term in
the denominators in
Eqs. (\ref{eq:ss_moment1}) approaches
${d_1}^2 (2 \gamma + 1)$,
where
$\gamma$
is the reaction strength parameter,
given by Eq. (\ref{eq:gamma}).
Thus, in the degradation-dominated limit,
where
$\gamma \ll 1$,
the steady state solution of the moment equations approaches
\begin{eqnarray}
\NAss &=&
\frac{g}{d_1}
\nonumber \\
\NDss &=&
a \frac{g^2}{d_2 {d_1}^2}
\nonumber \\
\Rss &=&
a \frac{g^2}{{d_1}^2}.
\label{eq:ss_moment1_gamma_small}
\end{eqnarray}
\noindent
Here we use the fact that in order to
satisfy both the large-system limit
($N_0 \gg 1$),
and the degradation-dominated limit
($\gamma \ll 1$),
one must also require
$d_1 \gg a$.
The results appearing in
Eqs. (\ref{eq:ss_moment1_gamma_small}) are consistent
with the results of the rate equations in this limit.
We conclude that the moment equations are also reliable for large
populations under the condition that the
system is in the degradation-dominated regime.
To test the results of the moment equations
for large systems in
the reaction-dominated regime
we examine the case of
$\gamma \gg 1$.
Here Eqs. (\ref{eq:ss_moment1}) are reduced to
\begin{eqnarray}
\NAss &=&
\frac{a + d_1}{2a}
\nonumber \\
\NDss &=&
\frac{g}{2 d_2}
\nonumber \\
\Rss &=&
\frac{g}{2}.
\label{eq:ss_moment1_gamma_large}
\end{eqnarray}
\noindent
In this limit,
the monomer copy number
$\NAss$,
obtained from the moment equations,
does not match the
rate equation result.
Nevertheless, the results for the dimer population size,
$\NDss$,
and for the dimerization rate
$\Rss$,
do converge to the results obtained from the rate equations.
We thus conclude that the accuracy
of the moment equations is maintained
well beyond the small system limit.
The equations provide accurate results for the dimer copy number,
$\NDss$,
and for the dimerization rate,
$\Rss$,
for both small and large systems.
As for the monomer copy number,
the moment equations provide an accurate
description in all limits,
except for the limit where both
$N_0 \gg 1$
and
$\gamma \gg 1$
(quadrant II in Fig. \ref{fig1}).
In Table \ref{tab1}
we present a characterization of the different
dynamical regimes and the applicability of the
moment equations for the evaluation of
the copy numbers and the dimerization rate
in each regime.
In Fig. \ref{fig3}
we present
the monomer copy number
$\NAss$ (circles),
the dimer copy number
$\NDss$ (squares)
and the dimerization rate
$\Rss$ (triangles),
as obtained from the moment equations,
versus
the reaction strength parameter,
$\gamma$.
The rate constants are
$g = 0.01$,
$d_1 = 1$
and
$d_2 = 5$
(s$^{-1}$).
The reaction rate,
$a$,
is varied.
These parameters satisfy the small-system limit
$N_0 \ll 1$.
The moment equation results are in excellent agreement
with those obtained from the master equation (solid lines).
However, since the populations are small,
the results of the rate equations
show deviations (dashed lines).
In Fig. \ref{fig4} we present
$\NAss$ (circles),
$\NDss$ (squares)
and
$\Rss$ (triangles),
as obtained from the moment equations,
versus
the reaction strength parameter,
$\gamma$.
Here the rate constants are
$g = 10^{3}$,
$d_1 = 0.1$
and
$d_2 = 0.1$
(s$^{-1}$).
As before,
the reaction rate,
$a$,
is varied.
These parameters satisfy the large-system limit
$N_0 \gg 1$,
and thus the rate equation results (dashed lines) are accurate.
Although the populations are large for the entire parameter range
displayed,
the results of the moment equations are in excellent agreement
with those obtained from the rate equations.
The only deviation appears in the results for the monomer population
in the limit
$\gamma \gg 1$.
For the parameters used in this simulation it was impractical to simulate
the master equation.
In any chemical reaction it is important to
characterize the extent to which
fluctuations are significant.
From the
master equation, one can evaluate the fluctuation level in the
monomer copy number,
given by the variance
\begin{equation}
\sigma^2 = \NAs - \NA^2,
\label{eq:sigma}
\end{equation}
\noindent
where $\sigma$ is the standard deviation.
The problem is that the expression for
$\sigma$ includes
the first moment
$\NA$,
which
is not always accurately accounted for by the moment equations.
However, when the copy number is sufficiently large,
the rate equations
apply, and thus one can extract the value of
$\NA^2$
in this limit from the rate equations.
On the other hand, the moment equations account correctly for
the second moment
$\NAs$
by
$\NAs = \R/a + \NA$.
Using this relation,
the result at steady state is
\begin{eqnarray}
\sigmass^2 &=&
\left\{
\begin{array}{ccccr}
\frac
{g
\left[
d_1^3 + a^2(d_1+g) +a (2d_1^2 + d_1 g + 2 g^2)
\right]}
{\left(
2 a g + a d_1 + d_1^2
\right)^2}
& & {\rm for} & & N_0 \le 1
\\ \\
\frac
{g (g + d_1 + a)}
{2 a g + a d_1 + d_1^2} -
\frac{d_1^2}{16 a}
\left(
-1 + \sqrt{1 + 8 \gamma}
\right)^2
& & {\rm for} & & N_0 > 1.
\end{array}
\right.
\label{eq:moment_rate_sigma}
\end{eqnarray}
\noindent
In Fig. \ref{fig5} we present
the coefficient of variation
$\sigmass / \NAss$ (circles),
as obtained from
Eq. (\ref{eq:moment_rate_sigma})
versus the system size parameter
$N_0$.
The parameters are
$d_1=1$,
$a=1$,
$d_2=5$ (s$^{-1}$)
and $g$
is varied.
Here
$\NAss$
was extracted from the moment equations for
$N_0 \le 1$,
and from the rate equations for
$N_0 > 1$.
In the small-system limit
($N_0 \ll 1$),
the average fluctuation becomes much larger than
$\NAss$.
The system is thus dominated by fluctuations.
As the system size increases,
$\sigmass$
becomes small with respect to
$\NAss$,
implying that the system enters the deterministic regime.
In order to validate our results,
we compare them with results obtained
from the master equation (solid line).
For
$N_0 < 1$
the agreement is perfect, as in this limit the moment equations are
expected to be accurate.
A slight deviation appears for
$N_0 > 1$,
where
$\sigmass$
is constructed by
combining results obtained from
the moment
equations and from the rate equations.
In both limits,
Eq. (\ref{eq:moment_rate_sigma})
is found to provide a good approximation
for the fluctuation level
of the system.
\subsubsection{Time-Dependent Solution}
The time dependent solution for
$\NA$
can be obtained by solving the two coupled equations for
$\NA$
and for
$\R$
in Eqs. (\ref{eq:moment1}).
The equation for
$\ND$
receives input from these two equations.
However, the dimers are the final products of this
network and
$\ND$
has no effect on
$\NA$
and
$\R$.
Thus the equations for $\NA$ and $\R$ can be decoupled
from the equation for $\ND$.
One obtains a set of two
coupled linear differential equations
of the form
\begin{equation}
\dot{ \vec{ N}} = {\bf M} \vec N + \vec b,
\label{eq:moment1_matrixform}
\end{equation}
\noindent
where
$\vec N = (\NA, \R)$,
$\vec b = (g, 0)$
and the matrix
${\bf M}$
is
\begin{eqnarray}
{\bf M} =
\left(
\begin{array}{cc}
- d_1 & -2
\\
2 a g & -2(d_1 + a)
\end{array}
\right).
\label{eq:M}
\end{eqnarray}
\noindent
The two eigenvectors of the matrix
${\bf M}$
are given by
\begin{eqnarray}
\vec v_1 =
\left(
\begin{array}{c}
\frac
{2a + d_1 - \omega}
{4 a g}
\\
1
\end{array}
\right);
&&
\vec v_2 =
\left(
\begin{array}{c}
\frac
{2a + d_1 + \omega}
{4 a g}
\\
1
\end{array}
\right),
\label{eq:eigenvectors}
\end{eqnarray}
\noindent
where
$\omega = \sqrt{4 a^2 + d_1^2 + 4 a d_1 - 16 a g}$.
The corresponding eigenvalues are
\begin{eqnarray}
-\frac{1}{\tau_1} =
\frac{1}{2}
(-2 a - 3 d_1 - \omega);
&&
-\frac{1}{\tau_2} =
\frac{1}{2}
(-2 a - 3 d_1 + \omega).
\label{eq:eigenvalues}
\end{eqnarray}
\noindent
Using
the matrix
${\bf Q} = (\vec v_1, \vec v_2)$,
one can write
Eq. (\ref{eq:moment1_matrixform})
as
\begin{equation}
{\bf Q}^{-1} \dot{ \vec{ N}} =
{\bf Q}^{-1} {\bf M} {\bf Q}
{\bf Q}^{-1} \vec N +
{\bf Q}^{-1} \vec b.
\label{eq:moment1_matrixform_Qued}
\end{equation}
\noindent
The result is a set of two un-coupled differential
equations of the form
\begin{eqnarray}
\dot{ \vec{ f}} &=&
\left(
\begin{array}{cc}
- \frac{1}{\tau_1} & 0
\\
0 & - \frac{1}{\tau_2}
\end{array}
\right)
\vec f + \vec k,
\label{eq:moment1_uncoupled}
\end{eqnarray}
\noindent
where
$\vec f = {\bf Q}^{-1} \vec N$
and
$\vec k = {\bf Q}^{-1} \vec b$.
The solution of Eq. (\ref{eq:moment1_uncoupled}) is
\begin{eqnarray}
\vec f(t) &=&
\left(
\begin{array}{c}
k_1 \tau_1 + C_1 e^{-{t}/{\tau_1}}
\\
k_2 \tau_1 + C_2 e^{-{t}/{\tau_2}}
\end{array}
\right),
\label{eq:moment1_uncoupled_solution}
\end{eqnarray}
\noindent
where
$C_1$ and $C_2$
are arbitrary constants.
Multiplying
Eq. (\ref{eq:moment1_uncoupled_solution})
from the left hand side by the matrix
${\bf Q}$
one obtains the time dependent solution of
Eq. (\ref{eq:moment1_matrixform}),
which is
\begin{eqnarray}
\NA &=&
\frac{g(a + d_1)}
{2 a g + a d_1 + d_1^2}
+
{\bf Q}_{1,1} C_1 e^{-{t}/{\tau_1}} +
{\bf Q}_{1,2} C_2 e^{-{t}/{\tau_2}}
\nonumber \\
\R &=&
\frac{a g^2}
{2 a g + a d_1 + d_1^2}
+
{\bf Q}_{2,1} C_1 e^{-{t}/{\tau_1}} +
{\bf Q}_{2,2} C_2 e^{-{t}/{\tau_2}}.
\label{eq:moment1_time_solution}
\end{eqnarray}
\noindent
The first terms on the right hand side of
Eqs. (\ref{eq:moment1_time_solution})
are the steady state solutions
$\NAss$
and
$\Rss$
as they appear in Eqs. (\ref{eq:ss_moment1}).
The second and third terms represent the time-dependent parts of
$\NA$
and
$\R$.
These terms exhibit
an exponential decay with two
characteristic relaxation times,
$\tau_1$
and
$\tau_2$.
Practically, since
$\tau_1 < \tau_2$,
the effective relaxation time for the monomers is
$\tau_A = \tau_2$.
In the limit of small copy numbers, where
$N_0 \ll 1$,
Eqs. (\ref{eq:moment1_time_solution})
account correctly for the copy numbers and for
the reaction rates.
In this limit
(where $g \ll d_1$),
one obtains
$\tau_A \simeq 1/d_1$.
In the limit of large copy numbers, where
$N_0 \gg 1$,
one has to distinguish between degradation-dominated and
reaction-dominated systems.
Consider a degradation-dominated system, where
$N_0 \gg 1$
and
$\gamma \ll 1$.
These two conditions require that
$d_1 \gg a$.
As before,
the effective relaxation time is approximated by
$\tau_A \simeq 1/d_1$.
This result is consistent with the
results of the rate equations in this regime
[Eq. (\ref{eq:rate_solution1})].
A peculiar result arises in the reaction-dominated
regime, in the limit of
large populations.
In this limit
$\omega \simeq \sqrt{4 a^2 - 16 a g}$.
For
$a < 4 g$
the parameter
$\omega$
becomes a purely imaginary number.
This result leads to spurious oscillations in the solution
presented in
Eqs. (\ref{eq:moment1_time_solution}).
As shown for the steady state solution
[Eq. (\ref{eq:ss_moment1})],
the moment equations consistently fail in the limit of reaction-dominated
systems with large copy numbers.
To obtain the relaxation time in this limit, one can rely on the
results obtained from the rate equations, which give
$\tau_A \simeq 1/\sqrt{8 a g}$.
The relaxation times in all the different limits are summarized in
Table \ref{tab2}.
Finally, we refer to the time evolution of the dimer population
$\ND$.
The equation for
$\ND$
is the second equation in
Eqs. (\ref{eq:moment1}), where
$\R$
is to be taken from
Eqs. (\ref{eq:moment1_time_solution}).
The solution of this equation takes the form
\begin{equation}
\ND = \NDss +
\tilde C_1 e^{-{t}/{\tau_1}} +
\tilde C_2 e^{-{t}/{\tau_2}} +
C_3 e^{-{t}/{\tau_3}},
\label{eq:ND_time_solution}
\end{equation}
\noindent
where
$\NDss$
is taken from
Eqs. (\ref{eq:ss_moment1}),
$\tilde C_i = {\bf Q}_{2,i} C_i / [d_2 - (1/\tau_i)]$,
$\tau_3 = 1/d_2$
and
$C_3$
is an arbitrary constant.
The effective relaxation time for the copy number
of the dimer product depends on the value of
$\tau_3$.
If
$\tau_3 < \tau_A$,
the copy number of the dimers relaxes rapidly.
Thus,
the time required for
the dimers to reach steady state is determined by
the monomer relaxation time, namely
$\tau_D \simeq \tau_A$.
In the opposite case,
where
$\tau_3 > \tau_A$,
the monomer population reaches steady state quickly,
and the production rate of
the dimers acts in effect as a constant generation rate.
Correspondingly, the relaxation time for
$\ND$
in this limit is approximated by
$\tau_D \simeq 1/d_2$
(Table \ref{tab2}).
In Fig. \ref{fig6}(a)
we present the time evolution of
$\NA$ (circles)
$\ND$ (squares)
and
$\R$ (triangles),
as obtained from the moment equations.
The parameters are
$g = 2 \times 10^{-3}$,
$d_1 = 0.05$,
$a = 100$
and
$d_2 = 5$
(s$^{-1}$).
These parameters correspond to the small system limit and to the
reaction-dominated regime (quadrant I in Fig. \ref{fig1}).
The moment equations (symbols) are in perfect agreement with the
master equation (solid lines).
The rate equations (dashed lines) deviate from the stochastic results
both in evaluating the steady state values of
$\NA$, $\ND$ and $\R$,
and in predicting the relaxation times of
$\NA$ and $\R$.
According to the rate equations,
this relaxation time should be
$\tau_A \simeq 1/\sqrt{8 a g} \simeq 0.8$ (s),
while according to the stochastic description
$\tau_A \simeq 1/d_1 \simeq 20$ (s).
In
Fig. \ref{fig6}(b)
we present results for a system in the large population
limit and in the regime of reaction-dominated kinetics
(quadrant II in Fig. \ref{fig1}).
Here the parameters are
$g = 10$,
$d_1 = 0.5$,
$a = 1$
and
$d_2 = 10$
(s$^{-1}$).
Under these conditions the moment equations fail to produce the correct
time transient, and give rise to an oscillatory solution (symbols).
In this regime the results obtained from the rate equations
(dashed lines) are accurate
and coincide with
the master equation
results (solid lines).
Note that even in this case the
moment equations provide the correct values
for the dimer production rate,
$\R$,
and for the dimer population,
$\ND$,
under steady state conditions.
In Fig. \ref{fig6}(c)
the parameters are
$g = 0.01$,
$d_1 = 2$,
$a = 10$
and
$d_2 = 10$
(s$^{-1}$).
These parameters satisfy the small system
limit and are in the kinetic regime
dominated by degradation
(quadrant III in Fig. \ref{fig1}).
The results are in perfect agreement with those obtained from the
master equation (solid lines).
However, the rate equations (dashed lines),
although displaying similar relaxation times,
show significant deviations in the steady state values of
$\ND$ and $\R$.
In
Fig. \ref{fig6}(d)
we present results for the case of a
large system, where
the parameters are
$g = 10$,
$d_1 = 0.5$,
$a = 10^{-3}$
and
$d_2 = 0.05$
(s$^{-1}$).
These parameters correspond to a system in the degradation-dominated regime
(quadrant IV in Fig. \ref{fig1}).
Although in this system the copy numbers are large, the
results obtained from the moment equations (symbols)
are in perfect agreement with those obtained
from the master equation (solid lines) and from the rate equations
(dashed lines).
Here the relaxation time for the monomer population is
$\tau_A \simeq 1/d_1$
and for the dimer population it is
$\tau_D \simeq 1/d_2$.
\section{Dimerization-Dissociation Systems}
\label{sec3}
To generalize the discussion of the previous Section we now consider
the case where the dimer product
$D$
may undergo dissociation
into two monomers,
at a rate
$u$ (s$^{-1}$).
The chemical processes in this system are thus
\begin{eqnarray}
&& \varnothing \arrow{g} A \nonumber \\
&& A \arrow{d_1} \varnothing \nonumber \\
&& A + A \arrow{a} D \nonumber \\
&& D \arrow{d_2} \varnothing \nonumber \\
&& D \arrow{u} A + A.
\label{eq:processes2}
\end{eqnarray}
\subsection{Rate Equations}
The rate equations for this reaction take the form
\begin{eqnarray}
\frac{d\NA}{dt} &=& g - d_1 \NA - 2\R + 2 u \ND \nonumber \\
\frac{d\ND}{dt} &=& - (d_2 + u) \ND + \R,
\label{eq:rate2}
\end{eqnarray}
\noindent
where $\R = a \NA^2$.
These equations are similar to
Eqs. (\ref{eq:rate1}),
except for the
$u$ terms
which account for the dissociation.
We define the effective reaction rate constant as
$\aeff = a[d_2/(u+d_2)]$
such that
the effective dimerization rate is
$\Reff = \aeff \NA^2$.
Under steady state conditions,
Eqs. (\ref{eq:rate2})
can be written as
\begin{eqnarray}
g - d_1 \NA - 2\Reff &=& 0
\nonumber \\
\Reff - d_2 \ND &=& 0.
\label{eq:rate2_eff}
\end{eqnarray}
\noindent
They take
the same form as
Eqs. (\ref{eq:rate1}), for dimerization
without dissociation,
under steady state conditions.
The steady state solution for these equations is
\begin{eqnarray}
\NAss &=&
\frac{d_1}{4 \aeff}
\left(
-1 + \sqrt{1 + 8 \gammaeff}
\right)
\nonumber \\
\NDss &=&
\frac{d_1^2}{16 \aeff d_2}
\left(
-1 + \sqrt{1 + 8 \gammaeff}
\right)^2,
\label{eq:ss_rate2}
\end{eqnarray}
\noindent
where
\begin{equation}
\gammaeff = \frac{g \aeff}{d_1^2}
\label{eq:gammaeff}
\end{equation}
\noindent
is the effective reaction strength parameter.
In the limit where
$d_2 \gg u$,
most dimers undergo degradation.
The dissociation process is suppressed,
and the effective reaction rate constant is
$\aeff \simeq a$,
namely the solution approaches that of
dimerization without dissociation.
In the limit where
$d_2 \ll u$,
most of the produced dimers end
up dissociating into monomers,
and correspondingly
$\aeff \rightarrow 0$.
In this limit, the dimerization and dissociation processes
reach a balance.
The effective dimerization rate vanishes
and $\NAss \simeq g/d_1$.
\subsection{Moment Equations}
In order to conduct a stochastic analysis we present the master equation
for the dimerization-dissociation system, which takes the form
\begin{eqnarray}
\frac{dP(N_A,N_D)}{dt} &=&
g [P(N_A - 1,N_D) - P(N_A,N_D)]
\nonumber \\ &+&
d_1 [(N_A + 1)P(N_A + 1,N_D) - N_A P(N_A,N_D)]
\nonumber \\ &+&
d_2 [(N_D + 1)P(N_A,N_D + 1) - N_D P(N_A,N_D)]
\nonumber \\ &+&
a [(N_A + 2)(N_A + 1)P(N_A + 2,N_D - 1) - N_A(N_A - 1)P(N_A,N_D)]
\nonumber \\ &+&
u [(N_D + 1)P(N_A - 2,N_D + 1) - N_D P(N_A,N_D)].
\label{eq:master2}
\end{eqnarray}
\noindent
This equation resembles
Eq. (\ref{eq:master1}),
except for the last
term which accounts
for the dissociation process.
The master equation can be solved
numerically by imposing suitable
cutoffs,
$N_A^{\rm max}$
and
$N_D^{\rm max}$.
However an analytical solution is currently unavailable.
To obtain a much simpler stochastic
description of this system we refer to the
moment equations.
Following the same steps as
in the previous Section,
we impose the minimal cutoffs on the master
equation, that enable all the required processes to
take place.
More specifically, we choose
$N_A^{\rm max} = 2$
in order to enable the dimerization.
We do not limit the copy number of the dimer, $N_D$.
However, we do not allow $N_A \ne 0$ and
$N_D \ne 0$ simultaneously, because $A$ and $D$
molecules do not react with each other.
These cutoffs reproduce the closure condition of
Eq. (\ref{eq:third_moment}).
They also gives rise to another closure condition,
which is needed here, namely
$\langle N_A N_D \rangle = 0$.
The closed set of moment equations
takes the form
\begin{eqnarray}
\frac{d\NA}{dt} &=& g - d_1\NA - 2\R + 2u\ND
\nonumber \\
\frac{d\ND}{dt} &=& - (u + d_2)\ND + \R
\nonumber \\
\frac{d\R}{dt} &=& 2ag\NA - 2(d_1 + a)\R + 2au\ND.
\label{eq:moment2}
\end{eqnarray}
\noindent
The steady state solution of these equations is
\begin{eqnarray}
\NA &=&
\frac{g(\aeff + d_1)}{2g\aeff + d_1 \aeff + d_1^2}
\nonumber \\
\ND &=&
\frac{\aeff g^2}{d_2(2g\aeff + d_1 \aeff + d_1^2)}
\nonumber \\
\R &=&
\frac{ag^2}{2g\aeff + d_1 \aeff + d_1^2}.
\label{eq:ss_moment2}
\end{eqnarray}
\noindent
Note that this solution resembles the steady state solution shown in
Eqs. (\ref{eq:ss_moment1}), except for the replacement of
$a$
by
$\aeff$.
As before, the validity of the moment
equations can be characterized by the
system size parameter,
$N_0$,
and by the effective reaction strength parameter
$\gammaeff$.
For small systems, where
$N_0 \ll 1$,
the approximation underlying the moment equations is valid,
and thus the moment equations provide accurate results for
$\NA$,
$\ND$
and
$\R$.
In the limit of large systems, where
$N_0 \gg 1$,
the validity of the moment equations can be evaluated by comparison with
the rate equations.
Two limits are observed.
In the degradation-dominated limit, where
$\gammaeff \ll 1$,
the solution obtained from the moment equations
(\ref{eq:ss_moment2})
converges to the solution obtained from the rate equations
(\ref{eq:ss_rate2}).
The moment equations are thus valid in this limit for
the monomer copy number,
$\NA$,
as well as for the dimer copy number,
$\ND$,
and its production rate,
$\R$.
However, for large systems in the reaction-dominated limit, where
$\gammaeff \gg 1$,
the moment equations converge to the rate equations only for
$\ND$
and
$\R$.
In this limit the monomer population size,
$\NA$,
is not correctly accounted for by the moment equations.
In conclusion,
the validity of the moment equations
is the same as in the case of dimerization without dissociation
(Table \ref{tab1})
under the substitution
$\gamma \rightarrow \gammaeff$.
In Fig. \ref{fig7}
we present
$\NAss$ (circles),
$\NDss$ (squares)
and
$\Rss$ (triangles),
as obtained from the moment equations for
the dimerization-dissociation system
versus the effective reaction strength,
$\gammaeff$.
Here the parameters are
$g = 0.02$,
$d_1 = 1$,
$a = 2500$
and
$d_2 = 1$
(s$^{-1}$).
The variation of $\gammaeff$
along the horizontal axis was achieved by
varying the
dissociation rate constant,
$u$.
For these parameters the system is in the
small population limit, namely
$N_0 \ll 1$.
The moment equation results are found to be in
perfect agreement with the results
obtained from the master equations (solid lines).
However, the rate equations (dashed lines) show significant deviations
for a wide range of parameters.
These deviations are largest when the dimerization process is dominant
($\gammaeff > 1$)
as the effects of stochasticity become important.
In Fig. \ref{fig8}
we present
$\NAss$ (circles),
$\NDss$ (squares)
and
$\Rss$ (triangles),
as obtained from the moment equations,
versus the effective reaction strength,
$\gammaeff$.
Here the parameters are
$g = 1000$,
$d_1 = 1$,
$a = 1$
and
$d_2 = 1$
(s$^{-1}$).
The dissociation rate constant,
$u$,
was varied.
For these parameters the system is
in the large population limit, namely
$N_0 \gg 1$.
Although the population sizes of the
monomer and of the dimer are large, the
moment equations are in perfect
agreement with the rate equations (dashed lines)
in the limit of
$\gammaeff \ll 1$.
For
$\gammaeff \gg 1$
this agreement is maintained for the dimer population size
and for its production rate.
In this limit the monomer population
size is not accounted for by the moment equations.
Slight deviations in $\ND$ and $\R$
appear within a narrow range around
$\gammaeff \simeq 1$.
In this narrow range
the effective reaction strength
parameter is far away from either of its limiting values.
In any case, these deviations are insignificantly small.
In the case of the dimerization-dissociation process the
moment equations (\ref{eq:moment2})
are a set of three linear coupled differential equations.
As opposed to the case of
dimerization without dissociation, here the equation for
$\ND$
does not only receive input from the other two equations, but
also generates an output into those equations.
This does not enable one to solve the first and third equations independently
to obtain a time dependent solution as shown in the previous Section.
Here the time dependent solution will include three characteristic time scales
for the relaxation times of both
$\NA$,
$\ND$
and
$\R$.
To obtain these time scales we first write
Eqs. (\ref{eq:moment2}) in
matrix form as
\begin{equation}
\dot{ \vec{ N}} = {\bf M} \vec N + \vec b,
\label{eq:moment2_matrixform}
\end{equation}
\noindent
where
$\vec N = (\NA, \ND, \R)$,
$\vec b = (g, 0, 0)$
and
\begin{eqnarray}
{\bf M} =
\left(
\begin{array}{ccc}
- d_1 & 2 u & -2
\\
0 & -(u + d_2) & 1
\\
2 a g & 2 a u & -2(d_1 + a)
\end{array}
\right).
\label{eq:M2}
\end{eqnarray}
\noindent
The time dependent solution of the moment equations is given by
\begin{equation}
N_i = N_i^{\rm ss} +
\sum_{j=1}^{3}
{{\bf C}_{ij} e^{-{t}/{\tau_j}}},
\label{eq:moment2_time_solution}
\end{equation}
\noindent
where
$i,j = 1,2,3$.
Here,
$\vec N^{\rm ss} = (\NAss, \NDss, \Rss)$,
and the matrix elements
${\bf C}_{ij}$
are determined by the initial conditions of the system.
The relaxation times
$\tau_j$
are
\begin{equation}
\tau_j = - \frac{1}{\lambda_j},
\label{eq:tau_i}
\end{equation}
\noindent
where
$\lambda_j$,
$j=1,2,3$,
are the eigenvalues of the matrix
${\bf M}$
[Eq. (\ref{eq:M2})].
The time dependent solution obtained from the
moment equations applies in the limits
where
$N_0 \ll 1$,
or in the limits where
$N_0 \gg 1$
and
$\gammaeff \ll 1$.
In the limit where
$N_0 \gg 1$
and
$\gammaeff \gg 1$,
the time dependent solution should be
obtained from the rate equations
[Eqs. (\ref{eq:rate2})].
\section{Hetero-dimer Production}
\label{sec4}
Consider the case where the reacting monomers
are from two different types of molecules,
$A$
and
$B$.
Each of these molecules is generated at a rate
$g_A$ ($g_B$)
and degraded at a rate
$d_A$ ($d_B$).
The two molecules react to form the dimer
$D = AB$
at a rate
$a$ (s$^{-1}$).
The dimer product undergoes degradation at a rate
$d_D$ (s$^{-1}$).
For simplicity, here we assume that the
process of the dimer dissociation is suppressed.
The chemical processes in this system are thus
\begin{eqnarray}
&& \varnothing \arrow{g_A} A \nonumber \\
&& \varnothing \arrow{g_B} B \nonumber \\
&& A \arrow{d_A} \varnothing \nonumber \\
&& B \arrow{d_B} \varnothing \nonumber \\
&& A + B \arrow{a} D \nonumber \\
&& D \arrow{d_D} \varnothing.
\label{eq:processes3}
\end{eqnarray}
\noindent
The average copy numbers of the reactive monomers and of the dimer product
are described by the following set of rate equations
\begin{eqnarray}
\frac{d\NA}{dt} &=& g_A - d_A \NA - \R \nonumber \\
\frac{d\NB}{dt} &=& g_B - d_B \NA - \R \nonumber \\
\frac{d\ND}{dt} &=& -d_D \ND + \R,
\label{eq:rate3}
\end{eqnarray}
\noindent
where
$\R$,
the dimer production rate, is given by
\begin{equation}
\R = a \NA \NB.
\label{eq:Rrate3}
\end{equation}
The master equation for this system describes the time evolution
of the probabilities
$P(N_A,N_B,N_D)$
for a population
$N_A$
molecules of type
$A$,
$N_B$
molecules of type
$B$
and
$N_D$
dimers
$D$
in the system.
It takes the form
\begin{eqnarray}
\label{eq:master3}
\frac{dP(N_A,N_B,N_D)}{dt} &=&
g_A [P(N_A - 1,N_B,N_D) - P(N_A,N_B,N_D)]
\nonumber \\ &+&
g_B [P(N_A,N_B - 1,N_D) - P(N_A,N_B,N_D)]
\\ &+&
d_A [(N_A + 1)P(N_A + 1,N_B,N_D) - N_A P(N_A,N_B,N_D)]
\nonumber \\ &+&
d_B [(N_B + 1)P(N_A,N_B + 1,N_D) - N_B P(N_A,N_B,N_D)]
\nonumber \\ &+&
d_D [(N_D + 1)P(N_A,N_B,N_D + 1) - N_D P(N_A,N_B,N_D)]
\nonumber \\ &+&
a [(N_A + 1)(N_B + 1)P(N_A + 1,N_B + 1,N_D - 1) - N_A N_B P(N_A,N_B,N_D)]
\nonumber
\end{eqnarray}
\noindent
In the stochastic description the production rate of the dimer
$D$
is proportional to the number of pairs of
$A$ and $B$
molecules in the system, namely
\begin{equation}
\R = a \NANB
\label{eq:Rmoment3}.
\end{equation}
\noindent
A more compact stochastic description can be
obtained from the moment equations.
Here one must include equations for the first moments
$\NA$, $\NB$ and $\ND$,
and for the production rate, which involves the second moment
$\NANB$.
The results for the first moments can be obtained by tracing over the
master equation as shown in
Sec. \ref{sec2}.
However, when deriving the equation for
$\R$
one obtains
\begin{eqnarray}
\frac{d\R}{dt}
&=& ag_B\NA + ag_A\NB - (d_A + d_B)\R
\nonumber \\
&-& a^2(\NAsNB + \NANBs - \NANB),
\label{eq:Rmoment3_not_truncated}
\end{eqnarray}
\noindent
which includes third moments for which we have no equations.
To obtain the closure condition
we follow the procedure presented in Sec. \ref{sec2}
and impose highly restrictive cutoffs on the master equation.
Here the cutoffs are chosen as
$N_A^{\rm max} = N_B^{\rm max} = N_D^{\rm max} = 1$.
These are the minimal cutoffs that enable the
dimerization process to take place.
Under these cutoffs,
the third order moments appearing in
Eq. (\ref{eq:Rmoment3_not_truncated})
can be expressed by
\cite{Barzel2007}
\begin{equation}
\NAsNB = \NANBs = \NANB.
\label{eq:mixed_moments}
\end{equation}
\noindent
One then obtains a closed set of moment equations
\begin{eqnarray}
\frac{d\NA}{dt} &=& g_A - d_A\NA - \R
\nonumber \\
\frac{d\NB}{dt} &=& g_B - d_B\NB - \R
\nonumber \\
\frac{d\ND}{dt} &=& -d_D \ND + \R
\nonumber \\
\frac{d\R}{dt} &=& ag_B\NA + ag_A\NB - (d_A + d_B + a)\R.
\label{eq:moment3}
\end{eqnarray}
\noindent
As in the case of the homo-molecular dimerization presented above,
the validity of the moment equations
extends well beyond the cutoff restriction.
It can be characterized by four parameters.
The first two are
$N_0^A = g_A/d_A$
and
$N_0^B = g_B/d_B$,
which provide the upper limits
on the monomer population sizes
$\NAss$
and
$\NBss$,
respectively.
The second two parameters are the reaction strength parameters,
which in the case of hetero-dimer production are
$\gamma_A = a g_A /(d_A d_B)$
and
$\gamma_B = a g_B /(d_A d_B)$.
In the limit where the populations are
small, the moment equations provide accurate
results for all the moments appearing in
Eqs. (\ref{eq:moment3}).
When the populations are large,
the moment equations provide accurate results
for the dimerization rate,
$\R$,
and for the dimer population
$\ND$.
However, if the reaction strength parameters are also large,
the moment equations will not correctly account
for the monomer population sizes,
$\NA$
and
$\NB$.
In Fig. \ref{fig9} we present
$\NAss$ (circles),
$\NBss$ (squares),
$\NDss$ (triangles)
and
$\Rss$ ($\times$),
versus the reaction strength parameters
$\gamma_A = \gamma_B$,
as obtained from the moment equations.
Here the parameters are
$g_A = 10^{-2}$,
$g_B = 10^{-2}$,
$d_A = 1$,
$d_B = 10$,
$d_D = 0.2$
(s$^{-1}$),
and the parameter
$a$
is varied.
These parameters are within
the limit of small populations.
The results are in perfect agreement
with those obtained from the master
equation (solid lines).
The rate equations (dashed lines)
show strong deviations, which are mainly
expressed in the reaction-dominated regime.
In Fig. \ref{fig10} we present
$\NAss$ (circles),
$\NBss$ (squares),
$\NDss$ (triangles)
and
$\Rss$ ($\times$),
versus the reaction strength parameters
$\gamma_A = \gamma_B$,
as obtained from the moment equations.
Here the parameters are
$g_A = 10^{3}$,
$g_B = 10^{3}$,
$d_A = 1$,
$d_B = 10$,
$d_D = 0.2$
(s$^{-1}$),
and the parameter
$a$
is varied.
These parameters are within the limit of large populations.
Nevertheless the results obtained from the moment equations
for
$\NDss$
and for
$\Rss$
are in good agreement
with those obtained from the rate equations (dashed lines)
in both the reaction-dominated limit
and in the degradation-dominated limit.
For the monomer population sizes,
$\NAss$
and
$\NBss$,
the moment equations apply only in the limit where
$\gamma_{A} < 1$
and
$\gamma_{B} < 1$.
\section{Summary and Discussion}
\label{sec5}
We have addressed the problem
of dimerization reactions
under conditions in which fluctuations are important.
We focused on two types of reactions,
homo-molecular dimerization ($A+A \rightarrow A_2$)
and hetero-dimer
production
($A+B \rightarrow AB$).
Common approaches for the stochastic simulation of such
reaction systems include the direct integration of the
master equation and Monte Carlo simulations.
The master equation involves a large number of coupled equations,
for which there is no analytical solution in the time-dependent case.
Monte Carlo simulations are often computationally intensive and
require averaging over large sets of data.
As a result,
the relaxation times
and the steady state populations
for given values of the rate constants
can only be obtained by numerical calculations.
Here we have utilized the recently proposed moment equations method,
in order to obtain an analytical solution for
the relaxation times and for the steady state populations.
The moment equations provide an accurate description of
dimerization processes in the stochastic limit, at the cost
of no more than three or four coupled linear differential
equations.
Another useful feature of these
equations is that in certain cases they also apply
in the deterministic limit.
Using the moment equations we
obtained a complete time dependent solution for
the monomer population
$\NA$,
the dimer population
$\ND$
and the dimerization rate
$\R$,
in the case of homo-molecular dimerization.
Expressions for the relaxation times
and the steady state populations were
found in terms of the rate constants of the
different processes.
In the case of hetero-dimer production the moment
equations include four coupled linear
equations.
These equations can be easily solved by direct numerical
integration.
However, a general algebraic expression
for this solution is tedious
and was not pursued in this paper.
Stochastic dimerization processes appear in many natural systems.
Below we discuss several examples.
One of the most fundamental chemical
reactions taking place in the interstellar medium is
hydrogen recombination, namely
${\rm H} + {\rm H} \rightarrow {\rm H}_2$
\cite{Gould1963,Hollenbach1970,Hollenbach1971a,Hollenbach1971b}.
This reaction occurs on the surfaces of microscopic dust grains
in interstellar clouds
\cite{Spitzer1978,Hartquist1995,Herbst1995}.
The resulting
${\rm H}_2$
molecules
participate in further reactions
in the gas phase,
giving rise to more complex molecules
\cite{Tielens2005}.
They also play an important role in
cooling processes during gravitational collapse and star formation.
In recent years there has been much activity in the computational
modeling of interstellar chemistry.
While the gas phase chemistry can be simulated by rate equations
\cite{Pickles1977,Hasegawa1992},
the reactions taking place on the dust grain
surfaces often require stochastic methods
\cite{Biham2001,Green2001,Charnley2001}.
This is because under the extreme interstellar
conditions of low gas density,
the population sizes of the reacting H atoms on
the surfaces of these microscopic grains
are small and highly fluctuative
\cite{Tielens1982,Charnley1997,Caselli1998,Shalabiea1998}.
The processes taking place on the grains are
the accretion of H atoms onto the surface,
the desorption of H atoms from the surface,
and the diffusion of atoms between adsorption
sites on the surface.
These processes can be described by the
dimerization system discussed in Sec. \ref{sec2}.
In recent years, experimental work was
carried out in an effort to obtain the relevant
rate constants
and for certain grain
compositions these constants were found
\cite{Pirronello1997a,Katz1999,Hornekaer2003,Perets2005,Perets2007}.
The solution of the moment equations,
as appears in Sec. \ref{sec2},
provides the production rate of molecular hydrogen
on interstellar dust grains,
in the limit of small grains and low fluxes,
where fluctuations are important.
In the biological context,
regulation processes in cells can be described by
networks of interacting genes
\cite{Alon2006,Palsson2006}.
The interactions between genes
include transcriptional regulation processes as well as protein-protein
interactions
\cite{Yeger-Lotem2004}.
Due to the small size of the cells,
some of these proteins my appear
in low copy numbers, with large fluctuations
\cite{McAdams1997,Paulsson2000,Paulsson2004,Friedman2006}.
Deterministic methods are thus not suitable for the
modeling of these systems.
Dimerization of proteins is a common process
in living cells.
In particular, many of the transcriptional regulator
proteins bind to their specific promoter sites on the DNA
in the form of dimers.
It turns out that such dimerization, taking place before
binding to the DNA, provides an effective mechanism for
the reduction of fluctuations in the monomer copy numbers
\cite{Bundschuh2003}.
In a broader perspective, complex reaction
networks appear in a variety of physical contexts.
The building blocks of these networks are intra-species
interactions and inter-species interactions.
Thus, the analysis presented in this paper of
homo-molecular and hetero-molecular dimerization
processes, lays the foundations for the analysis of more complex networks.
Complex stochastic networks are difficult to simulate
using standard methods,
because they require exceedingly long simulation times.
The moment equations,
applied here to dimerization systems,
provide a highly efficient method for the
simulation of complex chemical networks.
This work was supported by the US-Israel Binational
Science Foundation and by the France-Israel High
Council for Science and Technology Research.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,575
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\section{Introduction}
\label{sec:Intro}
In the $\Lambda$CDM standard scenario of structure formation, clusters of galaxies form hierarchically by the accretion and subsequent merging of smaller units in a bottom-up manner \citep[e.g., the reviews of][]{Planelles,KraBor}. As the merging and virialization of subcluster units is a slow process and new groups of galaxies are arriving and falling into a cluster from the environment even at the present cosmic epoch \citep[][and references therein]{TulShay, A05, Co12}, this process has left its traces by the simple presence of substructure in most clusters.
The degree of substructure present, along with the mixture of galaxy types, is therefore taken as a measure of the evolutionary stage of a cluster. The less substructure present the more evolved and relaxed a cluster of galaxies is. A very rough morphological classification proposed by \citet{Ab75}, following \citet{Zw61} and \citet{Mo61}, distinguishes between `regular', mostly rich clusters showing little substructure but a high degree of concentration and symmetry with few spiral galaxies, and `irregular', mostly poor clusters with a high degree of substructure and asymmetry containing many spiral galaxies (see also the classification schemes of \citet{BauMo71}, and \citet{RoSa67} which capture more morphological details but are based on the same basic distinction). The prototypes of a regular and a irregular cluster are the nearby, best-studied Coma cluster and Virgo cluster, respectively.
However, even the rich, regular \& relaxed Coma cluster (A1656), which is in the focus of the present paper, shows significant substructure (\citeauthor{Biviano96}, \citeyear{Biviano96}; \citeauthor{BivianoRev}, \citeyear{BivianoRev};
\citeauthor{CD96}, \citeyear{CD96}, hereafter CD96;
\citeauthor{N03}, \citeyear{N03};
\citeauthor{A05}, \citeyear{A05}, hereafter A05). The basic substructure of Coma is its binarity induced by the two dominant giant galaxies NGC 4889 and 4874. However, as much as 17 subgroups were identified in Coma by A05 using the hierarchical method of \cite{SernaGerbal}.
There is a host of sophisticated statistical methods for the analysis of substructure in clusters of galaxies \citep[e.g.,][and references in both]{WB13,Y15}. An alternative, rather simple approach to substructure, though applicable only to relatively nearby, well cataloged clusters, is to look directly for bound companions around massive cluster galaxies.
In the plausible infall scenario of cluster formation (see above), the infalling groups would consist of a small number of giant galaxies that are surrounded by swarms of bound dwarf galaxy satellites — a phenomenon best known from the Local Group. After the infall, most of these satellites would get stripped off (`liberated') and henceforth move freely in the cluster potential. But a small fraction of the dwarfs, presumably depending (among other things) on the mass and type of the mother galaxy, is expected to survive as companions.
An elegant statistical method to look for bound companions in clusters was developed and applied to the Virgo cluster by \citet[hereafter F92]{F92}. It is the purpose of the present paper to apply \citeauthor{F92}'s (\citeyear{F92}) method to the Coma cluster, the second-best cataloged rich cluster which is clearly of a different type than Virgo. In a nutshell \citeauthor{F92}'s method works like this: First the cluster sample is divided into a small sample of `primaries' (galaxies brighter than $M$ = -19) and a larger sample of `secondaries' (galaxies fainter than $M$ = -19). Then the number of secondaries is counted within a step-wise growing distance from each primary. These numbers have to be compared with the statistically expected numbers of apparent companions produced by projection effects. This is achieved by repeating the counting process for a large number of Monte Carlo clusters where the positions of the primaries are randomly changed along circles around the cluster center. Any excess of the observed numbers of secondaries around primaries over the corresponding mean numbers from the pseudo-clusters would then indicate the presence of gravitationally bound companions around the primaries. This procedure can be performed as a function of (primary and secondary) luminosity, morphological type and velocity. There are further refinements possible, for example by the introduction of a `interaction parameter' (see F92, also below). \citeauthor{F92}'s procedure bears some resemblance to the two-point cross correlation function but has the advantage that the information about the primaries' location in the cluster is not lost. Compared to the \cite{SernaGerbal} method it is also possible to give a confidence level for the presence of bound satellites. The density excesses due to bound companions found by \citet{F92} are small but significant, \citeauthor{F92} estimated a minimum of 7\% of cluster members to be bound companions in the Virgo cluster.
\citeauthor{F92}'s (\citeyear{F92}) work found surprisingly little follow-up. \citet{CWG95} used his procedure to look for pairs in \citeauthor{D80A}'s (\citeyear{D80A}) clusters, but there is so far no \citeauthor{F92}-type analysis of another morphologically resolved nearby cluster. Other very nearby clusters like the Ursa Major and Fornax clusters at roughly Virgo distance lack the richness warranted for a \citeauthor{F92}-type study. So the next target for such a study is clearly Coma. There is in fact some previous work on dwarf companions in the Coma cluster due to \citet{SHP97}. These authors studied the distribution of faint dwarfs around a number of early-type giants in a restricted area covering the cluster core and found on average 4$\pm$1 objects per giant in excess of the local background. However, there is meanwhile good coverage of the whole cluster to a decent depth to allow a full-fledged \citeauthor{F92} analysis of Coma to be persued.
The results for Coma are expected — and indeed are shown here — to differ significantly from those for Virgo due to the different population and dynamical state of the two clusters. The unrelaxed Virgo cluster exhibits a higher fraction of late-type members than the more virialized, denser Coma cluster \citep[e.g., the review of][]{Boselli}, which is part of the well-known morphology-density relation of galaxies \citep{D80A,D80B}. Infalling groups seem to be preferentially centered on luminous spirals. Thus, one may expect to find bound satellites in clusters preferentially around spirals or in subgroups and not around individual early-type giants. Spirals generally exhibit a higher number of companions than E's and S0's, as shown for example by \cite{BothunSull}. Even more important concerning a systematic difference between the clusters, we expect that individual primaries, in particular the old members of of early type, have lost their satellites in the dense environment of Coma due to interactions with the cluster potential as a whole or with other individual galaxies (by tidal interaction, harassment etc., see \citeauthor{Boselli}, \citeyear{Boselli}). In principle, the bound companions, instead of being survivors from infall, not having been ripped off their mother galaxy, could also have been captured later on in the cluster, especially by very massive cluster galaxies. However, numerical simulations indicate that this capture process is at best secondary \citep{BMP99}.
There are many processes in a cluster environment that tend to transform galaxies from late-type to early-type morphology (giants and companions alike) and systematically wipe out subclustering, including the presence of bound dwarf companions. Our simple statistical analysis, while providing a lot of morphological details with the observed subclustering, is not able to distinguish these processes.
The purpose of the present paper is primarily to carry out a \citeauthor{F92}-type analysis for the Coma cluster in as much detail as possible, to have a comparison with \citeauthor{F92}'s (\citeyear{F92}) original work on the Virgo cluster.
The rest of the paper is organized as follows: In Sect.~\ref{sec:samples} the catalogs used are presented. Sect.~\ref{sec:Method} deals with the applied MC methods, the computation of the projected secondary density around primaries $\Sigma(r)$ and the interaction parameter. The results are presented in Sect.~\ref{sec:RESULTS} and discussed in Sect.~\ref{sec:discussion}. A short conclusion and prospect are given in Sect.~\ref{sec:concl}.
\section{Samples}
\label{sec:samples}
While the Coma cluster has been studied extensively for the last few decades, there does not yet exist such a complete catalog as the VCC by \cite{VCC} for the Virgo cluster. Nonetheless, plenty of catalogs of Coma cluster galaxies are available. The most frequently cited work is the catalog of \citet[GMP83]{GMP} listing magnitudes and colors, complete to $m_{b_{26.5}} = 20$, for $\sim$ 6700 galaxies centered on the Coma cluster core. However, it does not contain any redshift data or membership assignments. This information was later on added by \citet[MA08]{MA08} for a subsample of the GMP83 catalog, which is the principal data base of the present paper (s. Sect.~\ref{sec:MA08_sample}). Another important compilation is \citet[CD96]{CD96} who give 465 members with measured radial velocities, however being restricted, like most other catalogs, to relatively bright magnitudes \citep[e.g.,][]{Bej,Mob01,Lobo,Edw02}. Other catalogs go down to fainter magnitudes but are limited to a relatively small area \citep[e.g.,][]{Bernst,Trent}. There are also specific catalogs of dwarf galaxies in Coma \citep[e.g.,][]{SeckHar}.
For our purposes, the catalog of choice has to (1) cover a sufficiently large area, (2) contain redshifts, (3) provide morphological information, (4) include faint galaxy populations. This is best met by the work of MA08 who give the morphology of all members and provide redshifts for more than $70\%$ of them, with a completeness limit of $M_B = -15$. Still, it
does not cover the cluster outskirts, which is why we use the catalog of \citet[WLG11]{WL11} as a complementary data base (missing morphological information, though).
\subsection{The MA08 catalog}
\label{sec:MA08_sample}
MA08 used the deep and wide field $B$ and $V$ imaging of the Coma cluster by \cite{Ad06} obtained with the CFH12K camera at the Canada-France-Hawaii Telescope (CFHT) to examine 1155 objects from the GMP83 catalog. Of these, 473 galaxies were assigned Coma cluster membership due to morphological appearance, apparent size, surface brightness, and redshift (for the detailed criteria see MA08). The data is complete down to $M_B = -15$ and extends to $M_B = -14.25$.\footnote{The MA08 catalog is free for download at\\ http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/490/923 \citep{MA08}.}.
In the following sections a division between (bright) primary and (faint) secondary galaxies is applied (limit at $M_{B}=-19$). Of the 74 primaries a substantial fraction (29) is excluded due to their proximity to the cD galaxies and a general crowding in the central region. We formally exclude primaries in the innermost cluster region defined by an ellipse with minor axis 250 kpc (s. Sect.~\ref{sec:MA08_results}). The remaining set of 45 giants is further split into several subsamples of different morphological type. To keep the analysis statistically meaningful, the morphological classification given in MA08 had to be simplified. The classification bins used here, for primaries as well as secondaries, are given in Table~\ref{tab:P_morph}. The columns are: (1) The abbreviation used for the set (E/S0: early types, S: late types, dE: dwarfs, Irr: irregulars); (2) types included with the MA08 morphology assignments; (3) the number of primaries in each set; (4) the number of secondaries in each set (in parentheses: the number of secondaries with measured velocities). The numbers refer only to the region outside the central region indicated by the inner ellipse in Fig.\,1.
\begin{table*}
\caption[Morphology grouping of primary galaxies (brighter than $M_{B}=-19$)]{Morphology grouping of primary galaxies (brighter than $M_{B}=-19$) and secondary galaxies (fainter than $M_{B}=-19$)}
\begin{tabular}{ccc>{\centering\arraybackslash}m{0.25\textwidth}}
\hline\hline
Morphology ID & MA08 Types included & Number of Primaries & Number of Secondaries (with measured $v$) \\
(1) & (2) & (3) & (4)\\
\hline
\\
E/S0 & cD, E/S0, S.0, E0-E5, Ep, SA0, SA0/a, SB0, SB0/a & 28 & 102 (99) \\
\\
S & S.a, S.b, SAa-SBc & 17 & 66 (63) \\
\\
dE & dE0-dE4, dS0, Ec & - & 184 (100)\\
\\
Irr & Im, Sdm, Sm, Sm/dS0 & -& 47 (17)\\
\hline
\end{tabular}%
\label{tab:P_morph}%
\end{table*}
The redshifts in the publicly available sources of the MA08 catalog are given only with three significant digits, for example as $z$ = 0.0231. This poses a problem for our velocity analysis in Sect.~\ref{sec:250kpc}, as the spectrum of velocity differences $\delta v$ becomes discrete instead of continuous. To get more accurate velocity data, we made a query in the NASA/IPAC Extragalactic Database (hereafter NED\footnote{http://ned.ipac.caltech.edu/}) and found velocities for 353 galaxies. Only these NED velocities are used here. Where more than one value was given, we chose the one with the smallest error (which is not necessarily the most reliable one). A Kolmogorov-Smirnov-Test and an Anderson-Darling-Test were performed to check that the velocity distribution is left unchanged, statistically, by the substitution of MA08 velocities with NED velocities.
The present work is based mainly on the sample described above, composed of the 473 members of the MA08 catalog, velocities taken from the NED, and the morphological classifications from Table~\ref{tab:P_morph}.
As the exact boundary coordinates of the CFHT field are not given in MA08, they are estimated here. The coordinates $x_{min}$, $x_{max}$, $y_{min}$, $y_{max}$ are approximated by the coordinates of the outermost galaxies covered (including non-members) plus a margin of $0.01^{\circ}$, such that $x_{min} = \min\limits_i(\alpha_i)-\alpha_c-0.01$ or $y_{max} = \max\limits_i(\delta_i)-\delta_c+0.01$ ($\alpha$ and $\delta$ for right ascension and declination in J2000, "c" for central). So the resulting field has a coverage of $\sim 0.73 \times \SI{0.84}{\deg}^2$. In Fig.~\ref{fig:Karte_Loc} the positions of all galaxies in the MA08 catalog are shown. Primaries are divided morphologically, and the principal regions distinguished are indicated by the gray ellipses.
\begin{figure*}[htbp]
\centering
\resizebox{15cm}{!}{\includegraphics[width=0.9\linewidth]{Karte_MA08.pdf}}
\caption[Inner and outer region for Coma primaries in the \citet{MA08} sample with morphological distinction]{Map of Coma cluster galaxies covered by MA08. Black crosses indicate galaxies fainter than $M_{B} = -19$ (the secondaries); circled dots: early-type primaries; dots in triangles: late-type primaries; bold circled crosses: NGC 4889 (left) and NGC 4874 (right). The inner (minor axis $b = \SI{250}{\kilo pc}$) and outer ($ b = \SI{500}{\kilo pc}$) gray ellipses separate the central, inner and outer regions considered in the analysis. Primaries within the innermost ellipse ($b = \SI{250}{\kilo pc}$) are excluded from the analysis. The rectangle indicates the area covered by the MA08 catalog which is relevant for density computations (s. Sect.~\ref{sec:MA08_sample}).}
\label{fig:Karte_Loc}
\end{figure*}
\subsection{The WLG11 catalog}
\label{sec:WL11_sample}
\cite{WL11} studied the galaxy populations of several clusters including Coma and compared them to a semi-analytic model. To enlarge the sample of Coma members provided by MA08, these authors composed their own catalog of Coma cluster galaxies based on SDSS data, kindly made available to us by Dr. Thorsten Lisker, co-author of WL11.
The WL11 catalog is based on the SDSS DR7 \citep[$M_R \leq -16.7$]{SDSS} and covers a very large area (out to $\sim 2.5^\circ$ from the cluster center).
Cluster membership for 923 galaxies was determined spectroscopically. Another 383 galaxies were judged statistical members and 835 are statistical background (small crosses for both in Fig.~\ref{fig:Karte_WL11}). But only the 923 galaxies with measured redshifts are used here ($M_R \lesssim -17.2$). The sample contains 197 primaries and 695 secondaries (limit at $M_R = -20$) shown as red and black points in Fig.~\ref{fig:Karte_WL11}.
The WL11 catalog does not contain morphological information. But as it covers a much larger area (out to $\sim \SI{4.2}{\mega pc}$) than the MA08 catalog it is used here to study the outskirts of the cluster. Generally, it is difficult to determine a boundary of the cluster. \cite{LokM} for instance give a virial radius of $\sim \SI{2.8}{\mega pc}$. For practical reasons we defined a boundary simply by excluding all primaries with major axis distance from center $a_k$ larger than \SI{3.9}{\mega pc} (referring to the position shuffling method, s. Eq.~\ref{a-b-ratio2}). This limit is shown as outermost ellipse in Fig.~\ref{fig:Karte_WL11}. This guarantees that the density of secondaries within \SI{300}{\kilo pc} around each primary can be computed without applying any boundary corrections. For the MA08 data this is not the case, there boundary effects have to be corrected as described in Sect.~\ref{sec:density_comp}. Again the primaries close to the cD galaxies are not included (s. Sect.~\ref{sec:WL11_results}).
\begin{figure*}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics[width=1\textwidth]{Karte_WL11_neu.pdf}}
\caption[Map of Coma galaxies in the WLG11 sample]{Map of Coma cluster galaxies listed in WL11. Circled red dots: Primaries (brighter than $M_{R} = -20$); black dots: secondaries; small crosses: galaxies without redshifts which are not considered in this analysis. The subsamples, i.e. annular regions considered in Sect.~\ref{sec:WL11_Diff} are indicated by gray ellipses and Roman numbers (steps of $\sim \SI{0.5}{\mega pc}$). The black rectangle indicates the field covered by MA08 shown in Fig.~\ref{fig:Karte_Loc}.}
\label{fig:Karte_WL11}
\end{figure*}
\section{Method}
\label{sec:Method}
\cite{F92} split his sample of Virgo cluster galaxies \citep{VCC} into 83 primaries and 1157 secondaries (brighter or fainter than $M_B = -19$, respectively). He determined the surface density of secondaries around primaries by counting the number of secondaries within a step-wise growing distance from each primary divided by the area searched. To correct for projection effects Ferguson created pseudo-clusters by changing the positions of the primaries azimuthally around the cluster center (holding the radial distances fixed). This randomization was performed 40 times. Then he compared the mean surface densities ($\Sigma$) in the real data with the corresponding mean value from the pseudo-clusters ($\hat{\Sigma}$). Any excesses of the observed over the expected numbers (surpassing one standard deviation) were finally interpreted as indication for the presence of bound companions; the statistical significance of the excesses was assessed by a Kolmogorov-Smirnov-Test. This kind of procedure was repeated with a subsample of galaxies that have measured redshifts (79 primaries and 290 secondaries). Small velocity differences ($\lesssim \SI{250}{\kilo m.s^{-1}}$) between primaries and secondaries were taken as additional evidence for their gravitational binding. A further refinement of the analysis was done with respect to primary position in the cluster, primary morphology and luminosity. F92 also introduced an interaction parameter to compare a galaxy's strongest local interaction to its interaction with the rest of the cluster.
This is the basic procedure adopted for the present Coma cluster analysis, with only slight deviations which are discussed in the following. In Sect.~\ref{sec:Method_Density}-\ref{sec:MC_prop} the generation of the pseudo-clusters and the computation of $\Sigma$ and $\hat{\Sigma}$ are explained. The interaction parameter method is described in Sect.~\ref{sec:IP_Shuffling}.
\subsection{Monte Carlo methods and companion density $\Sigma(r)$}
\label{sec:Method_Density}
\subsubsection{Generation of uniformly distributed primary positions}
\label{sec:position_method}
The generation of pseudo-clusters is done by randomly assigning new positions to the primaries uniformly along an ellipse around the center of the Coma cluster ($\alpha_c = \SI{194.9668}{\degree},\delta_c = +\SI{27.9680}{\degree}$, WL11). For this purpose a coordinate system $(x,y)$ is introduced with the cluster center as its origin.
F92 shuffled the Virgo primaries azimuthally, simply in want of a clear symmetric form of the cluster. However, the Coma cluster is more regular and exhibits a fairly well defined elliptical shape \citep[e.g.,][]{BiEps05,CaMet}. It is therefore more adequate for Coma to randomize the primary positions uniformly along ellipses instead of circles, based on the following equation.
\begin{equation}
\label{a-b-ratio2}
\frac{x_k^2}{a_k^2}+\frac{y_k^2}{b_k^2} = 1 \quad
\Leftrightarrow \quad \sqrt{x_k^2 (1-e)^2 + y_k^2} = |b_k|\,\,\,,
\end{equation}
with $(x_k,y_k)$ as position of the $k$-th primary. Substituting $b_k$ back into the left formula leads to $a_k$.
The ellipticity $e = 1 - \frac{b}{a}$
($a,b$ are the semi-major and semi-minor axis lengths) is estimated by different ways (e.g., from the standard deviation in x- and y-coordinates of the galaxies). For Coma we find $e \approx 0.13$.
The elliptical shuffling method is non-trivial. A simple drawing of a random azimuthal angle $\phi_{ran}$ (uniformly distributed) in polar coordinates would lead to a non-uniform distribution of random positions on an ellipse.
The same holds true for points uniformly distributed along a circle which are linearly transformed to an ellipse.
Thus, an acceptance-rejection algorithm is performed using two probability density functions (pdfs) $g(t),f(t)$ such that ${C \cdot g(t) \geq f(t)} ~ \forall t$ and for some $C \geq 1$. We construct $f(t)$ from the two-dimensional line element ${ds= a\sqrt{1 - \epsilon^2\cos^2(t)}}$
and the perimeter $\mathcal{L}$ of the ellipse:
\begin{equation}
\label{eq:pdf}
f(t) = \frac{ds(t)}{\mathcal{L}} = \frac{\sqrt{1 - \epsilon^2\cos^2(t)}}{\int_0^{2\pi} \sqrt{1 - \epsilon^2\cos^2(t)} dt}\,\,\,,
\end{equation}
where $\epsilon$ is the eccentircity; $\epsilon^2 = 1-b^2/a^2 = 1-(1-e)^2$.
We use ${g(t) = \mathcal{U}[0,2\pi]}$ and ${C = \frac{2\pi a}{\mathcal{L}}}$. In this way the algorithm produces $\phi_{ran} \sim f(t)$. As a last restriction the positions drawn at random must fall into the CFHT frame of the MA08 catalog. This is implemented directly into the acceptance-rejection algorithm. For the WL11 primaries no problems occur since all major axes $a_k$ are smaller than \SI{3.9}{\mega pc}.
\subsubsection{Velocity shuffling}
\label{sec:V-Shuffle}
Another possibility to create pseudo-clusters is the shuffling of primary velocities. To keep the same overall velocity distribution the new radial velocities are not just randomly generated numbers. Instead all the primary velocities of the sample are redistributed randomly to a new primary.
This is achieved by implementing the Fisher-Yates shuffle \citep[also known as Knuth shuffle,][p.542]{TrendsNet} which produces permutations ${\pmb{v_{\tau}} = (v_{\tau_1}, v_{\tau_2},...,v_{\tau_N})}$ of the primary velocity vector $\pmb{v}$ with equal probabilities for each permutation $\tau$. With this algorithm also primaries which are placed closer to the cluster center than \SI{250}{\kilo pc} can be included. Positions stay unchanged in these pseudo-clusters.
\subsection{Computation of $\Sigma(r)$}
\label{sec:density_comp}
To compute the secondary density around primaries (hereafter simply referred to as density) a circle with radius $r$ is chosen around a particular primary galaxy and all galaxies within this area are counted and divided by the area. This is done for a series of growing radius, that is the separation $r$ is grouped into bins $[0,r_i]$. Thus the area searched is always a circle and not an annulus, and the densities distributions plotted for the binned radii $r_i$ are cumulative, not differential. For differential distributions the numbers of galaxies are too small for statistical purposes. Computations in Sect.~\ref{sec:RESULTS} are done with
\begin{equation}
\label{eq:dens0}
\begin{aligned}
\Sigma_i &= \Sigma(r_i)= N_i/A_i\,\,,\\
A_i &= r_i^2\pi\,\,.
\end{aligned}
\end{equation}
$N_i$ is the number of secondary galaxies within $A_i$ (area of the $i$-th circle). As Coma does not subtend a large solid angle, all galaxies are assumed to lie on the same plane tangential to the sky and euclidean distances $r$ are calculated.
For primaries closer than \SI{300}{\kilo pc} to a boundary, the density calculated by Eq.~\ref{eq:dens0} has to be corrected by taking into account the effective search area falling inside the boundaries $(A_{eff})$:
\begin{equation}
\Sigma_{i,corr} = \frac{\Sigma_i \cdot A_i}{A_{eff}}\,\,\,.
\end{equation}
\subsection{Properties of MC results}
\label{sec:MC_prop}
To compare the densities $\Sigma(r_i)$ thus computed for the real cluster averaged over all primaries
with the expected densities from pseudo-clusters, we had to calculate the corresponding mean density $\Sigma^j(r_i)$ for each single Monte Carlo run ($j = 1,2,...,n$) and finally
the mean density $\hat{\Sigma}(r_i)$ and standard deviation $\hat{\sigma}_i$ for the whole set of $n$ pseudo-clusters.
The Monte Carlo (MC) results were tested for stability in different ways. Numerous experiments with various $n \geq 50$ of MC runs were executed and the results did not show any significant changes. Thus, we can assume to have a stable situation and reliable results by choosing $n = 1000$ for the position and velocity randomizations. Different random number generators (RNG) were tested, again without noticeable differences. Our MC runs were done with the standard and very rapid `Mersenne Twister' RNG algorithm which provides extremely uniformly distributed random numbers. Using different seeds, that is with the current time stamp, with zero or prime numbers, yielded stable results. The large period of $2^{19937}-1$ ($\approx 4 \cdot 10^{6001}$) guarantees the avoidance of correlated random numbers.
\subsection{Interaction parameter and direct position shuffling method}
\label{sec:IP_Shuffling}
\subsubsection{The interaction parameter method}
\label{sec:IP_theory}
F92 used a second approach to look for bound companions which avoids the need to distinguish between primaries and secondaries. The basic idea is to define a nearest neighbor of each galaxy not in terms of spatial proximity but of gravitational influence. F92 introduced an interaction parameter (IP) to identify each galaxy's neighbor with the largest gravitational pull $\mathcal{M}/r^2$ ($r$ describing the projected separation between the galaxies). The mass $\mathcal{M}$ of a galaxy is notoriously difficult to know but can be sufficiently well represented by its luminosity $L$. We apply this method here as well. For the MA08 sample we employ $L$ in units of blue solar luminosity $L_{\sun,B}$.
For galaxies of the WL11 catalog $L_R$ and $M_{R}$ are used instead.
The principal IP used here is the same as in F92 but defined directly by $L$ instead of $M$:
\begin{equation}
I_{Ferg,l} = \frac{\underset{k}{max} \left( L_{k}/r^2_{lk} \right)}{\sum\limits_{k \neq k_{max}} \left( L_{k}/r^2_{lk} \right) }\,\,\,.
\label{eq:IP_Ferg_theory}
\end{equation}
For the $l$-th galaxy this IP compares the interaction with its nearest neighbor (indexed as $k_{max}$) to that with the rest of the cluster. In Sect.~\ref{sec:IP_2_05} galaxies which have $I_{Ferg} > 2$ (strongest interaction with one galaxy twice as high than with all others) will be considered as bound (F92). This is only valid in a statistical way and may be wrong for a particular galaxy. \citeauthor{F92}'s IP clearly does not take into account the cancellation of gravitational forces by different neighbors nor the three dimensional structure of the cluster. Also, the analysis of $\Sigma$ is always performed up to separations of \SI{300}{\kilo pc}, while according to the definition of $I_{Ferg}$ in Eq.~\eqref{eq:IP_Ferg_theory} the 'nearest neighbor' of a galaxy might have a larger separation.
We explore alternative IPs that take more than one influential neighbor into account. First, a substructure IP which compares the gravitational interaction for a galaxy with all its neighboring galaxies within \SI{300}{\kilo pc} ($k'$) to the interaction with the rest of the cluster ($k''$, separations $r_{lk''} > \SI{300}{\kilo pc}$):
\begin{equation}
\label{eq:IP_substructure_theory}
I_{sub,l} = \frac{\sum\limits_{k'} \left( L_{k'}/r^2_{lk'} \right)}{\sum\limits_{k''} \left( L_{k''}/r^2_{lk''} \right) }\,\,\,.
\end{equation}
This IP should show a significantly higher fraction of galaxies with $I_{sub} > 2$ than \citeauthor{F92}'s IP.
As a second variant we define a group IP by comparing the three most strongly interacting galaxies (if those galaxies are closer than \SI{300}{\kilo pc}, $k'$) with the interaction by the rest of the cluster:
\begin{equation}
\label{eq:IP_group_theory}
I_{group,l} = \frac{\underset{k'}{max^*} \left( L_{k'}/r^2_{lk'} \right) + \underset{k' \neq k_{max}}{max^*} \left( L_{k'}/r^2_{lk'} \right)+\underset{\substack{k' \neq k_{max}\\ k' \neq k_{max,2}}}{max^*} \left( L_{k'}/r^2_{lk'} \right)}{\sum\limits_{\substack{k \neq k_{max}\\k \neq k_{max,2}\\k \neq k_{max,3}}}\left( L_{k}/r^2_{lk} \right) }\,\,\,,
\end{equation}
where "$*$" denotes that those maxima are only computed if the corresponding galaxies are located closer than \SI{300}{\kilo pc} to the $l$-th galaxy.
This IP should be sensitive to the presence of subgroups within the cluster. We assume that if such subgroups exist, $I_{group}$ should show approximately the same values for the galaxies contained in a group. Therefore the number of galaxies with $I_{group} > 2$ should range somewhere between the number of $I_{Ferg}$ and $I_{sub}$.
As hitherto only gravitational interaction is considered, a last IP is introduced which is sensitive to tidal interaction. It is almost the same IP as the one in \citet[hereafter Ka05]{Ka05}:
\begin{equation}
\label{eq:IP_Ka05_theory}
I_{Ka05,l} = \underset{k}{max}\left[ log(L_{B,k}/r^3_{lk}) \right]\,\,\,.
\end{equation}
The only difference is that in Ka05 a constant $C$ was added which allows to identify isolated galaxies ($I_{Ka05,l} < 0$).
\subsubsection{Direct position shuffling method}
\label{sec:direct_pos_Meth}
The pseudo-clusters to calculate IPs which then can be compared with the real IPs based on the MA08 and WL11 catalogs are created by a direct position shuffling. In this method the positions of two randomly chosen galaxies $k$ and $l$ are swapped directly.
As in F92 this exchange is performed $\mathcal{N}^2$ times where $\mathcal{N}$ is the total number of galaxies in the sample. In this way it can happen that a galaxy changes its position several times in one run, while another may stay in place.
\section{Results}
\label{sec:RESULTS}
\subsection{Results for the $\Sigma(r)$-analyses of the MA08 catalog}
\label{sec:MA08_results}
The settings for the MC simulations analyzed in this section are given below. They are valid throughout this paper, unless mentioned otherwise.
\begin{itemize}
\item Primary-secondary discrimination at $M_{B}=-19$ (74 giants and 399 possible companions at the outset)
\item Primaries have to lie outside an ellipse with minor axis 250 kpc centered on the cluster center (45 primaries left)
\item Distance to Coma: \SI{100}{\mega pc}, $H_0=\SI{70}{\km.s^{-1}.\mega pc^{-1}}$, $\Omega_{\Lambda}=0.7$, $\Omega_m=0.3$, Distance modulus $\mu=35$ and therefore a scale of $\SI{0.46}{\kilo pc.\arcsec^{-1}}$ \citep{A09}
\item To calculate the absolute magnitudes, a small correction of $0.25$ is added to $\mu$ leading to a value of $\mu'=35.25$ (MA08).
\item The cluster center is defined as in WL11 at $\alpha_c = \SI{194.9668}{\degree}$ and $\delta_c = +\SI{27.9680}{\degree}$.
\item $\Sigma(r)$ is computed up to \SI{300}{\kilo pc} for bins $[0,r_i]$ with $r_{i+1} = r_i + \SI{20}{\kilo pc}$ and $r_1 = \SI{20}{\kilo pc}$.
\item Random number generator (RNG): Mersenne Twister seeded with the current time stamp
\item $n=1000$ Monte Carlo runs
\end{itemize}
In order to avoid any influence on the companion density by primaries lying in the crowded central region around the two cD galaxies, the primary set underlies a restriction: As the position shuffling is performed elliptically, the computed minor axis $b_k$ (s. Eq.~\eqref{a-b-ratio2}) for the randomization of any primary used has to be larger than \SI{250}{\kilo pc}. This prevents the random placement of a primary within an ellipse with $b = \SI{250}{\kilo pc}$ around the cluster center, which is located midway between NGC 4874 and NGC 4889. We note that the point of this exclusion is not only to avoid the crowded region as such; the problem is that the azimuthal or elliptical randomization that close to the center produces overlapping sets of secondaries. F92 chose a radial distance of \SI{200}{\kilo pc} for the center of the Virgo cluster. This distance is slightly extended here due to the presence of substructure around the cD galaxies \citep[e.g.,][]{N01}. $\hat{\Sigma}(r)$ of the remaining 45 primaries is stable even if the ellipticity is slightly changed, which also supports the appropriateness of the chosen value of \SI{250}{\kilo pc}.
In this section, the results of the MC simulations with the settings above are presented. There are mainly two kinds of outcomes: Firstly, the surface density of companions around primaries and secondly, the distribution of separations between primaries and secondaries. Both sets are further split into several subsets in Sect.~\ref{sec:250kpc_PrimMorph}.
As an additional criterion to search for physically bound companions, the velocity differences $\delta v$ between primaries and secondaries can be used. Results found in this way have a stronger weight, especially because the observed field is not very large and the areas searched for satellites are quite often overlapping. As mentioned in Sect.~\ref{sec:MA08_sample} we take all velocities from the NED. Measured velocities are available for all primaries but only for part of the secondaries (see Table~\ref{tab:P_morph}).
F92 grouped $\Sigma$ into four bins of $\delta v$ from \SI{125}{\kilo m.s^{-1}} up to \SI{500}{\kilo m.s^{-1}}. Here we use two bins of velocity differences only: $\SI{0}{\kilo m.s^{-1}} \leq \delta v \leq \SI{250}{\kilo m.s^{-1}}$ and $\SI{250}{\kilo m.s^{-1}} < \delta v \leq \SI{500}{\kilo m.s^{-1}}$. Over-densities for velocity differences in the first bin are a strong indication of physical interaction, that is gravitational binding of primary and secondary galaxies. When $\delta v$ lies in the second bin, we expect a massive primary or a whole subgroup able to hold its satellites over a larger separation. In `calmer' outer regions we generally expect a higher fraction of bound satellites.
Two methods can be employed for galaxies with redshifts. The first method is the usual elliptical position randomization of the 45 primaries, where $\Sigma(r)$ and $\hat{\Sigma}(r)$ are determined only for the 279 secondaries with a velocity. In the second method the primary velocities are shuffled to create a pseudo-cluster (s. Sect.~\ref{sec:V-Shuffle}, also referred to as $v$-shuffling). Evidence is only taken as acceptable if both methods produce similar results. For the central primaries (minor axis smaller than \SI{250}{\kilo pc}) $\Sigma(r)$ can be analyzed only with the second randomization method. This set contains additional 24 early-type (including both cD) and 4 late-type galaxies.
\subsubsection{Complete set of primaries and secondaries}
\label{sec:250kpc}
In Fig.~\ref{fig:8Var}a, top left, we show the surface density of all secondaries around all 45 primaries. The open circles with error bars indicate the mean density $\hat{\Sigma}_i$ and standard deviation $\hat{\sigma}_i$ of 1000 MC runs. The black dots give the mean densities $\Sigma_i$ of the data from MA08. This type of figure is the basic tool of our analysis.
The effects sought for should fulfil one of the following conditions:
1) The mean MA08 density $\Sigma_i$ lies outside the error bars of the pseudo-clusters at least for three bins in a row (on the same side, significance at $1\hat{\sigma}$ for $r_i, r_{i+1}, r_{i+2}$), or 2) the observed density deviates from the model at the $2\hat{\sigma}$-level or more for at least one bin.
As can be seen, there is no such effect with respect to either specification. Instead of the expected over-densities there are rather under-densities. Toward the largest separation the observed and model densities match, as required, but for smaller separations the observed densities are systematically too low, which we ascribe to a combination of different biases stemming from local deviations from the assumed cluster ellipticity etc. For further analysis, the sets are differentiated in the following subsection.
The same is found by setting an upper limit for the velocity separation, $\delta v \leq \SI{250}{\kilo m.s^{-1}}$, as shown in Fig.~\ref{fig:8Var}g, upper right. The effect is even slightly accentuated by the apparently missing companions for small primary-secondary separations at $r_i = 40 - \SI{80}{\kilo pc}$.
\begin{figure*}[hbt]
\centering
\resizebox{15cm}{!}{\includegraphics[width=0.85\textwidth]{Fig3.pdf}}
\caption{a)-c): Mean density of secondaries around primaries depending on their separation r. It should be noted that the $r_i$ do not indicate annular bins. $\Sigma(r_i)$ represents the number of secondaries within a circle of radius $r_i$ (s. Sect.~\ref{sec:density_comp}). Black dots indicate the values for the MA08 set, the open circles and error bars (in red) show the mean and standard deviation of 1000 pseudo-clusters — a): whole sample, b): early-type primaries only, c): late-type primaries only.
\newline
d)-f): Cumulative distribution of primary-secondary separations $r$ for the MA08 data (black) and 1000 pseudo-clusters (dashed, red) — d): whole sample, e): early-type primaries only, f): late-type primaries only.
\newline
g): The same as a) but restricted to galaxies with velocity difference smaller than 250 km $s^{-1}$.}
\label{fig:8Var}
\end{figure*}
Another way to compare real-cluster with pseudo-cluster data is to calculate the `cumulative distribution functions' (CDF) for the separation of primaries and secondaries for both and to analyze the difference by a Two-Sample-Kolmogorov-Smirnov-Test (KS-Test).
There is one fundamental difference to the analysis with densities $\Sigma(r)$, namely one must assume that the borders of the field do not have any influence on the distribution (F92), as it is no longer possible to simply divide the numbers by the area searched.
The null hypothesis stating that the CDF of the primary-secondary separations of the real cluster and the pseudo clusters are drawn from the same unknown background distribution is rejected if the p-value of the KS test is smaller than a significance level of $5\%$. As the KS test is rather conservative \citep{EngCou}, for p-values close to the $5\%$-significance level a Anderson-Darling-Test (AD test) is performed in addition. While the KS-Test is sensitive with respect to global differences, the AD test is more powerful to discover differences near $0$ and $\SI{300}{\kilo pc}$ where the CDF converges to 0 and 1 respectively \citep[e.g.,][]{EngCou}. Furthermore, the AD-Test needs less data to reach sufficient statistical power. However, for very small numbers (e.g.,less than $\sim 10$ primaries) this test is performed neither.
The CDF for the whole sample is shown in Fig.~\ref{fig:8Var}d, top middle panel. The KS test does not reject the null hypothesis but the additionally performed AD test does point to a difference in the distributions (p-values are $p_{KS} = 0.080$ and $p_{AD} = 0.046$, respectively, see Fig.~\ref{fig:8Var}d). However, if only galaxies with measured velocities are used for the tests, there is no evidence for different background distributions left.
\subsubsection{Dependence on type, $\delta v$, location in cluster, and luminosity}
\label{sec:250kpc_PrimMorph}
The 45 primaries encompass 28 early-type (E/S0) galaxies and 17 late-type (spiral) galaxies (see Table~\ref{tab:P_morph}). The secondary densities for these two morphological subgroups are shown in Fig.~\ref{fig:8Var}b,c. The under-densities that appear for the whole sample in Fig.~\ref{fig:8Var}a are clearly present also for the early-type primary sample (Fig.~\ref{fig:8Var}b) but not for spiral primaries (Fig.~\ref{fig:8Var}c), that is the effect is due to early-type primaries. When the velocities are restricted as for the whole sample in Fig.~\ref{fig:8Var}g, there is no such effect seen in either subset (not shown here), casting doubt on the reality or relevance of the under-densities.
More confusing still are the CDF's for the two morphological subgroups, shown in Fig.~\ref{fig:8Var}e,f. Countering the expectation from the density results, here it is the E/S0 sample that is indifferent ($p_{KS} = 0.208$), while the spiral primary sample exhibits significant under-densities in the separation range $r \sim 150 - \SI{260}{\kilo pc}$, with a strong signal ($p_{KS} = 0.005$ and $p_{AD} = 0.004$, — and if restricted to secondaries with velocities $p_{KS} = 0.031$ and $p_{AD} = 0.020$). This contradiction calls for further differentiation of the sample.
First we look for any dependence on the location of the primaries in the cluster by splitting the primary sample into an inner set of 33 primaries located within an ellipse of minor axis $b$ restricted to $\SI{250}{\kilo pc} < b < \SI{500}{\kilo pc}$ and an outer set of 12 primaries with $b \geq \SI{500}{\kilo pc}$ (this division is indicated in Fig.~\ref{fig:Karte_Loc}; remember that the innermost region $b < \SI{250}{\kilo pc}$ is avoided because of confusion). However, this splitting alone does not reveal any new results. So the positional criterion is combined with morphology, yielding four subsamples encompassing
21 inner early-types, 12 inner late-types, 7 outer early-types, and 5 outer late-types. By restricting the sets additionally to secondaries with velocity data, some effects do appear, but almost all of them concern the outer samples where statistical testing is impossible due to the small numbers involved. We find only a significant signal for the inner late-type set ($p_{KS} = 0.023$), again for under-densities, that is apparently missing companions.
Next, we try a differentiation with respect to primary luminosity by dividing the whole into 17 high-luminosity ($M_{B} < -20$) and 28 low-luminosity primaries ($-19 \geq M_{B} \geq -20$). The only significant result is a $2\hat{\sigma}$-excess for $r_i = \SI{20}{\kilo pc}$ and large $\delta v$ in the high-luminosity sample. More revealing is the combination of luminosity and morphology, while nothing significant is found for the combination of luminosity and location. Adding the $\delta v$ criterion finally gives the following effects, see Fig.~\ref{fig:Lum_E_S}:
\begin{itemize}
\item Excess of possible companions around high-luminosity late-types (s. Fig.~\ref{fig:Lum_E_S}c)
\item A lack of secondaries around high-luminosity early-types and low-luminosity late-types (s. Fig.~\ref{fig:Lum_E_S}a and d)
\item A small excess around low-luminosity E/S0 giants (s. Fig.~\ref{fig:Lum_E_S}b)
\end{itemize}
For the low-luminosity late-type set we find statistical confirmation of under-densities by the KS test ($p_{KS} = 0.040$).
\begin{figure}[htbp]
\resizebox{\hsize}{!}{\includegraphics[width=1\linewidth]{Lum_E_S.pdf}}
\caption{Mean density of secondaries around different subsets of primaries. The set used, along with the $\delta v$ restriction, is indicated in each panel. Full black dots represent the data; colored open circles and error bars the mean and standard deviation of 1000 pseudo-clusters (created by shuffling primary positions).}
\label{fig:Lum_E_S}
\end{figure}
In the following the morphology of the 399 secondaries is considered. The secondaries are divided into four groups, as defined in Table~\ref{tab:P_morph}: 102 early-type (E/S0) galaxies, 66 late-type (S) galaxies, 184 early-type dwarfs (dE) and 47 irregulars (Irr). The density of each subsample of secondaries is first computed around all primaries and then also around the primary subsamples (inner/outer and early/late). The main results are shown in Figs.~\ref{fig:Sec_Morph} and \ref{fig:Sec_nurE}.
Overall, secondaries of any morphological type tend to be depleted out to $\sim \SI{200}{\kilo pc}$. This is exemplified for dwarf ellipticals and spirals in Fig.~\ref{fig:Sec_Morph} (however, we note the excess for the very first separation bin of spiral secondaries). This holds true even when the velocity difference is restricted to $\delta v \leq \SI{250}{\kilo m.s^{-1}}$, where companions are expected most. In contrast, and surprisingly, when larger velocity difference are considered
($250 \leq \delta v \leq \SI{500}{\kilo m.s^{-1}}$, that is when stronger binding forces are required, there is a strong hint for the presence of E/S0 companions at small separations, as seen in Fig.~\ref{fig:Sec_nurE}. Unfortunately, this remains unconfirmed by KS or AD testing.
\begin{figure}[htbp]
\resizebox{\hsize}{!}{\includegraphics[width=1\linewidth]{Sec_Morph_v250.pdf}}
\caption{Mean density of dwarf elliptical (dE, left panels) and spiral (S, right panels) secondaries around all primaries. In the lower panels the same is shown for primary-secondary pairs with a small velocity difference $\delta v < 250$ km s$^{-1}$ only. Full black dots represent the data; colored open circles and error bars the mean and standard deviation of 1000 pseudo-clusters.}
\label{fig:Sec_Morph}
\end{figure}
\begin{figure}[htbp]
\centering
{\includegraphics[width=0.6\linewidth]{Sec_nurE.pdf}}
\caption{Mean density of E/S0 secondaries around all primaries with large velocity differences (\mbox{$250 \leq \delta v \leq \SI{500}{\kilo m.s^{-1}}$}). Full black dots represent the data; colored open circles and error bars the mean and standard deviation of 1000 pseudo-clusters.}
\label{fig:Sec_nurE}
\end{figure}
As a last breakdown of the sample also the secondary luminosity is regarded. The secondaries are divided into five groups: 62 galaxies with $-19 < M_{B} \leq -18$; 74 (only 72 with measured $v$) with $-18 < M_{B} \leq -17$; 86 (74) with $-17 < M_{B} \leq -16$; 98 (47) with $-16 < M_{B} \leq -15$; and 79 (24) with $M_{B} > -15$. In the last bin probably more galaxies exist, but the MA08 catalog is only complete down to the limit of $-15$.
In Fig.~\ref{fig:Lum_E_S} we found hints for bound or missing companions depending on the luminosity of primaries. Now we examine the secondary density around these primary subsets in dependence of secondary luminosity and velocity difference. The results, restricted to interesting parameter combinations, are shown in Fig.~\ref{fig:Lum_sec}. The following features can be noted:
\begin{itemize}
\item Secondaries with ${M_{B} > -16}$ around high-luminosity primaries show a small excess (over-density) for $r_i \approx 60 - \SI{200}{\kilo pc}$ (Fig.~\ref{fig:Lum_sec}a,b).
\item For secondaries in the next brighter magnitude intervals, ${-17 < M_{B} \leq -16}$ and
${-18 < M_{B} \leq -17}$, there a hint of an excess at large separations, $r_i > \SI{150}{\kilo pc}$ (Fig.~\ref{fig:Lum_sec}c-f).
\item Secondaries in the brightest magnitude bin, ${-19 < M_{B} \leq -18}$ (Fig.~\ref{fig:Lum_sec}g-i) show the clearest sign for companions at small separations, $r_i <$ 100 kpc, but only for the high velocity difference bin, $\delta v \leq \SI{250}{\km.s^{-1}}$, very significantly around high-luminosty primaries (Fig.~\ref{fig:Lum_sec}h).
\end{itemize}
This last-mentioned Fig.~\ref{fig:Lum_sec}h is essentially reproducing the strong effect seen in Fig.~\ref{fig:Sec_nurE}. Hence the best evidence for bound companions so far is for bright early-type (E/S0) secondaries around bright giant galaxies, in particular luminous spirals, if Fig.~\ref{fig:Lum_E_S} is taken into account.
\begin{figure*}[htb]
\centering
\resizebox{15cm}{!}{\includegraphics[width=0.85\textwidth]{Lum_SEC.pdf}}
\caption{Mean density of different sets of secondaries around high- and low-luminosity primaries depending on separation $r$. Secondaries are distinguished by luminosity (bin of absolute magnitude) and velocity separation, as indicated in the individual panels. Black dots indicate the values for the MA08 set, the open circles and error bars show the mean and standard deviation of 1000 pseudo-clusters.}
\label{fig:Lum_sec}
\end{figure*}
\subsection{Interaction parameters for the MA08 catalog}
\label{sec:IP}
In Sect.~\ref{sec:IP_Shuffling} we introduced a set of different 'interaction parameters' (IP's) as an additional tool to look for bound companions. Following F92, $I_{Ferg}$ is used to look for bound pairs of galaxies. $I_{group}$ and $I_{sub}$ were constructed to have additional IP's that are more sensitive to subgroups around three influential neighbors and to general substructure up to a size of \SI{300}{\kilo pc}, respectively. Finally, $I_{Ka05}$ which is based on tidal interaction is briefly considered as well.
\subsubsection{CDF's of the interaction parameters}
\label{sec:IP_cdf}
It should be remembered that the IP method does not distinguish between primaries and secondaries. The pseudo-clusters are therefore built by direct swapping the positions of randomly chosen galaxies (see Sect.~\ref{sec:direct_pos_Meth}). In Fig.~\ref{fig:IP} we show the cumulative distributions (CDF's) for all four IP's. The p-values of the two statistical (KS and AD) tests applied to the CDF's are also indicated. Obviously, for $I_{Ka05}$ and $I_{group}$ there is no significant difference between real and model cluster (panels b and d), whereas \citeauthor{F92}'s IP is just on the border of statistical significance (panel a). However, $I_{sub}$ is very clearly different for MA08 data and the pseudo-clusters, with a high statistical significance.
A more detailed discussion is given in the following section using Table~\ref{tab:IP_frac}. First, we perform a test which is only feasible with \citeauthor{F92}'s IP.
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics[width=0.9\textwidth]{IP_KS.pdf}}
\caption[Cumulative distributions of the interaction parameters]{Cumulative distributions for the different interaction parameters discussed in the text. They are indicated along the abscissa. The black curves give the corresponding CDF's of Coma cluster data from MA08, the red curves show the randomized data for comparison (from galaxy position swapping). The p-values for the statistical tests are indicated in each panel.}
\label{fig:IP}
\end{figure}
\subsubsection{Bound companions ($I > 2$) and free members ($I < 0.5$)}
\label{sec:IP_2_05}
In the following, galaxies with $I > 2$ are assumed to be gravitationally bound whereas such with $I < 0.5$ are treated as free-floating members of the cluster. Galaxies with $0.5 \leq I \leq 2$ are not statistically evaluated. We focus here on the IP of F92 which is sensitive for pairs. By definition the chosen limits imply that a galaxy is twice or half as strongly bound to another one than to the cluster as a whole. If a very large fraction of galaxies with high IPs are physically bound, then the dispersion of $\delta v$ is expected to be around \SI{250}{\kilo m.s^{-1}} or less according to F92. On the other hand, with no bound companions at all the dispersion should be $\sim \sqrt{2}\sigma_v$ ($\approx \SI{1530}{\kilo m.s^{-1}}$), where $\sigma_v$ is the mean velocity dispersion of Coma ($\sigma_v = \SI{1082}{\kilo m.s^{-1}}$, CD96). In other words, in this case most of the high IP values are caused simply by projection.
Fig.~\ref{fig:delta_v} shows the distribution of $\delta v$ (up to \SI{5000}{\kilo m.s^{-1}}) between galaxies with $I_{Ferg} > 2$ and their nearest neighbors for the 1000 pseudo-clusters (panel a) and the 29 galaxies with these properties in MA08 (panel b) for which $\delta v$ could be determined. Best-fitting Gauss curves are also shown in the figure. Reassuringly, the pseudo-cluster best-fitting $\sigma_1$ is nearly the same as $\sqrt{2}\sigma_v$ (1540 versus 1530 km $s^{-1}$).
For the MA08 data the number of galaxies with large values of $I_{Ferg}$ is of course quite small, but to draw a comparison to \citeauthor{F92}'s work (only 45 galaxies, too) his approach is taken here as well. The best-fitting dispersion $\sigma_* = \SI{1310}{\kilo m.s^{-1}}$ of these galaxies can be modeled by a (normalized) superposition of two Gaussians in the following way: Assuming that part of the galaxies are indeed bound in pairs, they should be drawn from a narrow Gaussian with width $\sigma_2 = \SI{250}{\kilo m.s^{-1}}$, while those not bound should follow $\sigma_1 = \SI{1530}{\kilo m.s^{-1}}$ as described above. More formally, the total distribution function $\sigma_*$ as a combination of the two Gaussians is (with $\beta\in[0,1]$):
\begin{equation}
\label{eq:sumGauss}
\sigma_*^2 = \beta \sigma_1^2 + (1-\beta) \sigma_2^2\,\,\,.
\end{equation}
Solving Eq.~\eqref{eq:sumGauss} for $\beta$ leads to a fraction of roughly 0.75, implying that $\sim 25\%$ of those 29 galaxies with $I_{Ferg} > 2$ are indeed physically bound ($\sim 2\%$ of all 353 studied galaxies). The corresponding best-fitting superposition is shown as dashed blue curve in Fig.~\ref{fig:delta_v} is shown for the MA08 data. This result matches the expectation that only few bound satellites in Coma exist. However, as the galaxy census of Coma is certainly far from being complete, in particular for faint dwarf members which are most prone to be bound companions, this fraction of companions is bound to raise in the future, and $2\%$ must be regarded as a lower limit.
A third (dotted green) curve is shown in the MA08 part of Fig.~\ref{fig:delta_v} modeling a combination where the bound pairs have a high velocity difference of 500 km s$^{-1}$. This is indeed in accord with our previous findings that over-densities are found preferably for velocity differences in the range between 250 and \SI{500}{\kilo m.s^{-1}} (e.g., Fig.~\ref{fig:Sec_nurE}). In this case, the fraction $\beta$ in Eq.~\eqref{eq:sumGauss} becomes $\sim 0.3$ for galaxies with $I_{Ferg} > 2$ drawn from a Gaussian with $\sigma_2 = \SI{500}{\kilo m.s^{-1}}$, which is not very different from the case before. As can be seen in Fig.~\ref{fig:delta_v} all three curves may describe the histogram due to the small number of galaxies. Even a Gaussian curve with $\sim \sqrt{2}\sigma_v \approx \SI{1530}{\kilo m.s^{-1}}$ would fit into the picture. It is therefore clearly impossible to give an exact fraction of bound companions. However, a much larger fraction of bound satellites (e.g., as large as in the Virgo cluster with $7\%$, F92) can almost certainly be excluded.
\begin{figure}[htbp]
\centering
{\includegraphics[width=8cm]{deltav.pdf}}
\caption[Distribution of $\delta v$ between galaxies with $I_{Ferg} > 2$ and their nearest neighbors]{Histogram of $\delta v$ for the 1000 pseudo-clusters (panel a) and for the $I_{Ferg} > 2$ pairs in the MA08 data (panel b). Both histograms are overlaid with a best-fitting Gauss curve (red). For the MA08 data two combinations of Gauss distributions representing a population of bound pairs and one of projected pairs are drawn as well (dashed and dotted curves in panel b). The area under each curve is always normalized to the total number of galaxies involved.}
\label{fig:delta_v}
\end{figure}
$I_{sub}$ and $I_{group}$ aim at the gravitational interaction between more than just two neighboring galaxies as $I_{Ferg}$. The above analysis based on velocity differences between a galaxy and its nearest neighbor is therefore not possible for these IPs. The same is true for $I_{Ka05}$, as no limit for bound or free galaxies can be determined easily (s. Sect.~\ref{sec:IP_theory}). Nevertheless, the distribution of the interaction parameters $I_{sub}$ and $I_{group}$ and their dependence on morphology can be studied and compared to the one in 1000 pseudo-clusters in a different way, as follows.
In Table \ref{tab:IP_frac} we list the fractions (percentages) of bound pairs ($I > 2$) and free-floating cluster members ($I < 0.5$) for $I_{Ferg}$, $I_{sub}$, and $I_{group}$, comparing the real cluster data from MA08 with the random clusters for each morphological type. The numbers referred to in the following discussion are emphasized. The columns give: (1) Morphological type; (2) number and percentage of galaxies with $I > 2$ and $I < 0.5$ for \citeauthor{F92}'s IP in the data and for the randomized clusters; (3) the same for substructure IP and (4) for the group IP. The percentages for the pseudo-clusters are mean values. The standard deviations $\hat{\sigma}$ are similar for all types and amount to $\hat{\sigma} \approx 0.8-0.9\%$ for $I_{Ferg} > 2$, and to $\hat{\sigma} \approx 1.2-1.4\%$ for $I_{sub} > 2$ and $I_{group} > 2$. For the free members the values are $\hat{\sigma} \approx 1.4-1.5\%$ for the IP of F92, $\hat{\sigma} \approx 0.4-0.5\%$ for the substructure IP and $\hat{\sigma} \approx 1.3\%$ for the group IP.
\begin{table*}[htbp]
\small
\caption{Fractions of bound companions and free members according to the IP-analysis.}
\begin{tabular}{l>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}>{\raggedright\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{4em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{4em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}>{\raggedleft\arraybackslash}m{2em}}
\hline\hline
Type & \multicolumn{4}{c}{$I_{Ferg}$} & \multicolumn{4}{c}{$I_{sub}$} & \multicolumn{4}{c}{$I_{group}$} \\
\multicolumn{1}{c}{(1)} & \multicolumn{4}{c}{(2)} & \multicolumn{4}{c}{(3)} & \multicolumn{4}{c}{(4)}\\
\hlin
& MA08 & & 1000 pseudo-clusters & & MA08 & & 1000 pseudo-clusters & & MA08 & & 1000 pseudo-clusters & \\
& N & \% & N & \% & N & \% & N & \% & N & \% & N & \% \\
\hlin
\multicolumn{13}{c}{$I > 2$} \\
\hlin
E/S0 & 14 & 9.1 & 13377 & 8.7 & 109 & \textbf{70.8} & 116291 & \textbf{75.5} & 32 & 20.8 & 33147 & 21.5 \\
S & 8 & 9.1 & 7651 & 8.7 & 54 & \textbf{61.4} & 66291 & \textbf{75.3} & 15 & \textbf{17.0} & 19162 & \textbf{21.8} \\
dE & 16 & 8.7 & 15711 & 8.5 & 129 & \textbf{70.1} & 138516 & \textbf{75.3} & 34 & 18.5 & 39402 & 21.4 \\
Irr & 1 & \textbf{2.1} & 4092 & \textbf{8.7} & 35 & 74.5 & 35540 & 75.6 & 12 & \textbf{25.5} & 10132 & \textbf{21.6} \\
\hlin
\multicolumn{13}{c}{$I < 0.5$} \\
\hlin
E/S0 & 105 & 68.2 & 103762 & 67.4 & 3 & 1.9 & 4311 & 2.8 & 37 & 24.0 & 36793 & 23.9 \\
S & 66 & \textbf{75.0} & 59135 & \textbf{67.2} & 4 & \textbf{4.5} & 2518 & \textbf{2.9} & 30 & \textbf{34.1} & 20939 & \textbf{23.8} \\
dE & 127 & 69.0 & 124286 & 67.5 & 3 & 1.6 & 5106 & 2.8 & 45 & 24.5 & 44045 & 23.9 \\
Irr & 36 & \textbf{76.6} & 31630 & \textbf{67.3} & 2 & 4.3 & 1345 & 2.9 & 9 & \textbf{19.1} & 11239 & \textbf{23.9} \\
\hline
\end{tabular}%
\label{tab:IP_frac}%
\end{table*}%
As can be seen in Table~\ref{tab:IP_frac} (col. 2), the fraction of galaxies with $I_{Ferg} > 2$ in MA08 is marginally higher than in the randomized clusters for all types except the irregulars. In the pseudo-clusters the fraction of each type is approximately the same, for all IPs and for free members as well, which is expected from the direct position shuffling method. The low fraction of bound irregulars may be a small number effect, if true it would contradict our previous findings. The numbers for $I_{Ferg} < 0.5$ are more revealing. Here late-type galaxies, both spirals and irregulars, are significantly overabundant among free-floating members when compared to the pseudo-clusters (at $5\hat{\sigma}$-level).
$I_{sub}$ compares the gravitational pull between a galaxy and its neighbors within \SI{300}{\kilo pc} to the pull it is exposed to by the rest of the cluster. Hence as expected, the percentage of bound (non-bound) galaxies of every type is strikingly larger (smaller) than for $I_{Ferg}$. A systematic deviations between the distributions of $I_{sub}$ in Coma and the pseudo-clusters was already noted in Sect.~\ref{sec:IP_cdf}, Fig.~\ref{fig:IP}c, in the sense that there are more galaxies in the real cluster with small $I_{sub}$ and less with large values compared to the pseudo-clusters. This is confirmed here. Moreover we see that the effect is essentially owed to spiral galaxies. There are considerably fewer late types included in substructure than in the randomized clusters (at $10\hat{\sigma}$-level). Spirals are apparently underabundant in the known substructures within the MA08 area. But even luminous early-types and dwarfs follow this pattern, albeit less strongly (still at $3-4\hat{\sigma}$). Overall, these results taken together are in accord with the view that Coma is a fairly relaxed cluster.
The group IP is a weaker variant of the substructure IP, describing the three strongest gravitational interactions of galaxies within \SI{300}{\kilo pc}. The values differ notably only for the spirals (again) for $I_{group} > 2$ ($7\hat{\sigma}$) and $I_{group} < 0.5$ ($3\hat{\sigma}$). Another point is that the irregulars experience statistically more interaction with three neighbors than with their nearest neighbor. This seems to indicate that irregulars are rather bound to subgroups (infalling groups) rather than single primaries.
\subsection{Results for the $\Sigma(r)$-analyses of the WLG11 catalog}
\label{sec:WL11_results}
The settings for the MC simulations analyzed in this section are almost the same as in Sect.~\ref{sec:MA08_results}, except for the following changes: 1) the primary-secondary discrimination is at $M_{R}=-20$ (228 giants and 695 possible companions), and 2) absolute magnitudes can be taken directly from the catalog of WL11.
Again primaries with minor axis (referring to the position shuffling, s. Eq.~\eqref{a-b-ratio2}) smaller than \SI{250}{\kilo pc} are excluded. A second cut is applied to avoid density correction due to boundary effects (s. Sect.~\ref{sec:WL11_sample}), leaving 197 primaries in the sample. A primary of this set is not necessarily a primary in the MA08 analysis and vice versa. This time all secondaries have velocity data (thanks to SDSS). The velocity bins of primary-secondary $\delta v$ will be the same as before.
Position shuffling is found to produce constantly lower random-densities than the $v$-shuffling method. To be conservative, we therefore rely on, and show here only, the results obtained with the velocity randomization method. The prize, however, is that no statistical tests can be executed, as the galaxy separations remain the same with this method.
\subsubsection{Complete set with $\delta v$ distinction}
\label{sec:WL11_All}
Fig.~\ref{fig:WL11} provides a first look at the complete WLG11 sample. The only differentiation made is by primary-secondary $\delta v$, divided into the usual two bins. There is a clear excess of possible bound companions apparent, in particular (again), at the $2\hat{\sigma}$-level, for high velocity differences ($250 < \delta v \leq \SI{500}{\km.s^{-1}}$).
In the following we elaborate this result with respect to primary position, and primary and secondary luminosity. A distinction by type cannot be done, as WGL11 lacks morphological information.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{WL11.pdf}
\caption[Mean density of secondaries around primaries for the WLG11 data]{Mean density of secondaries around primaries for the complete WL11 set, separated into two bins of velocity difference as indicated in the panels. Black dots represent the data; open circles and error bars in red give the mean and standard deviation of 1000 pseudo-clusters created by shuffling primary velocities.}
\label{fig:WL11}
\end{figure}
\subsubsection{Dependence on location in cluster and luminosity}
\label{sec:WL11_Diff}
As a first step the primary sample is divided into a number of elliptical annuli around the cluster center.
For a better comparison with the results for the MA08 catalog, the first separation is again fixed at minor axis length $b$=\SI{250}{\kilo pc} and named inner and outer as in Sect.~\ref{sec:250kpc_PrimMorph}. The outer region is further subdivided into two distance regions of width \SI{0.25}{\mega pc} out to 1\,Mpc, and 5 regions of width \SI{0.5}{\mega pc} out to 3.5\,Mpc. The sets are given in Table~\ref{tab:WL11_loc}. The columns list the following: (1) name of the set, (2) range for the primary minor axis $b$, (3) number of galaxies in each subsample. The whole set of elliptical annuli is also shown in the Coma cluster map of WGL11 data, Fig.~\ref{fig:Karte_WL11}.
\begin{table*}
\centering
\caption[Local grouping of WLG11 primaries (brighter than $M_{R}=-20$)]{Distance grouping of primary galaxies (brighter than $M_{R}=-20$)}
\begin{tabular}{lp{16em}l}
\hline\hlin
Set ID & Minor axis range [\SI{}{\mega pc}] & Number of galaxies \\
(1) & (2) & (3) \\
\hlin
inner & ]0.25,0.5] & 31 \\
outer & ]0.5,0.75] & 24 \\
outer II & ]0.75,1] & 17 \\
outer III & ]1,1.5] & 31 \\
outer IV & ]1.5,2] & 34 \\
outer V & ]2,2.5] & 20 \\
outer VI & ]2.5,3] & 26 \\
outer VII & ]3,3.5] with restriction of $a < \SI{3.9}{\mega pc}$ & 14\\
\hline
\end{tabular}%
\label{tab:WL11_loc}%
\end{table*}
The results are shown here in form of a table instead of the usual density distribution: see Table~\ref{tab:WL_Loc}. The columns list: (1) primary subsample according to Table~\ref{tab:WL11_loc}, (2) velocity difference bin, (3) net secondary density around primaries (WL11 data minus random values). For each bin $[0,r_i]$ we give either a "0" or a series of "+" or "-" signs in the table. A "0" indicates that the WL11 mean density $\Sigma(r)$ lies within one standard deviation of the pseudo-cluster mean $\hat{\Sigma}(r)$. One, two or three "+"-signs are indicating a $1\hat{\sigma}$, $2\hat{\sigma}$ or $3\hat{\sigma}$ over-density. The latter also stands for over-densities at more than the $3\hat{\sigma}$ significance level. Similarly, under-densities are indicated by "-"-signs.
\begin{table*}[htbp]
\tiny
\caption[Net densities of secondaries around primaries depending on their location for the WLG11 data]{Net densities $\hat{\Sigma}_{net}(r)$ of secondaries around primaries depending on their location according to Table~\ref{tab:WL11_loc}. Symbols are explained in the text.}
\begin{tabular}{p{5em}p{3.5em}rrrrrrrrrrrrrrr}
\hline\hlin
Primary \newline Location & $\delta v$-bin \newline [\SI{}{\kilo m.s^{-1}}] & \multicolumn{15}{c}{$\hat{\Sigma}_{net}(r)$} \\
(1) & (2) & \multicolumn{15}{c}{(3)} \\ \hlin
\multicolumn{2}{c}{Separation r [\SI{}{\kilo pc}]} & 20 & 40 & 60 & 80 & 100 & 120 & 140 & 160 & 180 & 200 & 220 & 240 & 260 & 280 & 300 \\
\hlin
Outer II & 0-250 & 0 & 0 & ++ & 0 & + & + & 0 & + & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
& 250-500 & 0 & ++ & + & + & + & 0 & 0 & 0 & 0 & 0 & + & + & 0 & 0 & 0 \\
\hlin
Outer III & 0-250 & 0 &++ & + & 0 & 0 & + & 0 & 0 & 0 & + & + & + & + & + & 0 \\
& 250-500 & 0 & 0 & + & 0 & + & ++ & + & + & + & + & + & ++ & ++ & ++ & ++ \\
\hlin
Outer IV & 0-250 & 0 & + & 0 & 0 & 0 & 0 & + & 0 & + & + & + & + & + & + & + \\
& 250-500 & 0 & 0 & 0 & + & + & ++ & ++ & ++ & + & ++ & + & + & + & + & + \\
\hlin
Outer V & 250-500 & 0 & + + & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hlin
Outer VI & 0-250 & + & + & 0 & 0 & 0 & 0 & 0 & + & 0 & 0 & 0 & + & + & + & + \\
& 250-500 & 0 & 0 & 0 & - & - & 0 & - & - & - & 0 & 0 & 0 & - & - & 0 \\
\hlin
Outer VII & 0-250 & 0 & 0 & 0 & 0 & + & ++ & + & + & + & + & + & + & + & 0 & 0 \\
& 250-500 & +++ & +++ & + & + & + & 0 & + & + & + & + & 0 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}%
\label{tab:WL_Loc}%
\end{table*}%
Not given are the (null) results for the inner and next outer regions where no clear over-densities for the complete sample were found for the WL11 sample either. But Table~\ref{tab:WL_Loc} confirms that over-densities (bound companions) abound in the outskirts of the cluster, and again more so in the high-$\delta v$ bin. To cover the outskirts of the cluster was of course the motivation to work with WGL11 data in addition. We cannot aim at interpreting the results for the individual distance regions, however. They are lumped together again in the following.
The primary sample is split into a 63 high- ($M_R < -21$) and 130 low- ($-21 \geq M_R \geq -20$) luminosity galaxies. For the high-luminosity primaries we find $1\hat{\sigma}$-over-densities of secondaries in the innermost \SI{60}{\kilo pc} separation range. For low-luminosity primaries
there are $1-2\hat{\sigma}$-level over-densities for separations larger than $\sim \SI{100}{\kilo pc}$ for $\delta v \leq \SI{250}{\kilo m.s^{-1}}$, and over-densities at the same significance level over almost the whole range of $r_i$ for the higher $\delta v$-bin. We show these results in combination with a distinction by secondary luminosity in the following.
The secondaries are divided into three luminosity groups: 213 galaxies with $-20 < M_{R} \leq -19$, 257 with $-19 < M_{R} \leq -18$, and 225 with $-18 < M_{R} \leq -17$. The resulting secondary densities, restricted to cases where a noteworthy signal is present, are shown in Fig.~\ref{fig:WL11_Lum_Sec_Low} for secondaries around low-luminosity primaries, and in Fig.~\ref{fig:WL11_Lum_Sec_High} for secondaries around high-luminosity primaries.
We note that secondaries of all luminosities show a mild over-density over almost all separations, where the luminosity class $-18 < M_{R} \leq -17$ pops up with very significant over-densities at separations below ca. 100 kpc for high velocity differences (Fig.~\ref{fig:WL11_Lum_Sec_Low}b). The over-densities, again at small separations, around high-luminosity primaries mentioned above are clearly owed to the most luminous secondaries, in the low- and high-$\delta v$ bin alike (Fig.~\ref{fig:WL11_Lum_Sec_High}).
\begin{figure}[htbp]
\centering
\includegraphics[width=1\linewidth]{WL11_Lum_Low.pdf}
\caption[Mean density of different secondary subsets around low luminous primaries]{Mean density of secondaries around low-luminosity primaries. Secondary luminosity and velocity difference bin are indicated in each panel. Black dots: WL11 data; open circles and error bars in red: mean and standard deviation of 1000 pseudo-clusters created by primary velocity shuffling.}
\label{fig:WL11_Lum_Sec_Low}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\linewidth]{WL11_Lum_high.pdf}
\caption[Mean density of secondaries with $-20 < M_{R} \leq -19$ around high luminous primaries]{Mean density of the brightest secondaries around high-luminosity primaries. Otherwise like Fig.~\ref{fig:WL11_Lum_Sec_Low}.}
\label{fig:WL11_Lum_Sec_High}
\end{figure}
\subsubsection{Interaction parameters for the WLG11 catalog}
\label{sec:WL11_IP}
Repeating the IP analysis from Sect.~\ref{sec:IP} for the WL11 sample, we first find a notable difference in that the CDF for $I_{sub}$ (compare Fig.~\ref{fig:IP}) is no longer significantly different between data and random model ($p_{KS} = 0.262$).
In addition, the best Gaussian fit to the $\delta v$-distribution for pseudo-clusters leads to a smaller value for the cluster dispersion $\sigma_v = \frac{\SI{1260}{\kilo m.s^{-1}}}{\sqrt{2}} \approx \SI{890}{\kilo m.s^{-1}}$ than before (see Fig.~\ref{fig:WL11_delta_v}a). For comparison, the dotted line in Fig.~\ref{fig:WL11_delta_v}a shows a Gaussian based on $\sigma_v = \SI{1080}{\kilo m.s^{-1}}$ as used in Sect.~\ref{sec:IP_2_05} \citep{CD96}. All commonly used dispersion values for Coma are in the range $\sigma_v \approx 1000 - \SI{1100}{\kilo m.s^{-1}}$. \cite{Edw02} found a lower value of $\sigma_v \approx \SI{980}{\kilo m.s^{-1}}$ but only for the giant population. The velocity dispersion of the dwarf population is even higher than the overall mean (e.g., \SI{1096}{\kilo m.s^{-1}} by \cite{Edw02} or \SI{1213}{\kilo m.s^{-1}} by \cite{Chi10}). However, the explanation for this apparent discrepancy probably lies in the observation that the low-density outskirts of a cluster are dynamically cooler and hence exhibit smaller velocity dispersion than the central part.
Studying infalling substructures, A05 found a velocity dispersion of $\sigma_v \approx \SI{973}{\kilo m.s^{-1}}$ for the whole cluster, still being on the high side but going in the right direction.
The IP analysis can be performed in any case. Fitting a Gaussian to the distribution of $\delta v$ for the 63 $I_{Ferg} > 2$ pairs in the WL11 data leads to $\sigma_* \approx \SI{900}{\kilo m.s^{-1}}$ (see Fig.~\ref{fig:WL11_delta_v}b). Now we can again decompose this distribution into a component of projected pairs with $\sigma_1 \approx \SI{1260}{\kilo m.s^{-1}}$ (obtained from the pseudo-clusters) and a component of bound companions with either $\sigma_2 = \SI{250}{\kilo m.s^{-1}}$ or $\sigma_2 = \SI{500}{\kilo m.s^{-1}}$. Using Eq.~\eqref{eq:sumGauss} one finds $\beta \approx 50\%$ for the former case and $\beta \approx 40\%$ for the latter. These two solutions for combined Gaussians are overlaid in Fig.~\ref{fig:WL11_delta_v}b. The combination with the narrow $\delta v$ component clearly does not provide a good fit and one should favor the second solution with $\sigma_2 = \SI{500}{\kilo m.s^{-1}}$. In this case Fig.~\ref{fig:WL11_delta_v}b suggests that $\sim 60\%$ of pairs with $I_{Ferg} > 2$ are bound companions. Thus around $4\%$ of the 883 galaxies in the WL11 set may be considered bound companions, not necessarily as satellites of an individual primary but more likely of a subgroup.
\begin{figure}[htbp]
\centering
{\includegraphics[width=8cm]{WL11_deltav.pdf}}
\caption[Distribution of $\delta v$ between galaxies with $I_{Ferg} > 2$ and their nearest neighbors]{Histogram of $\delta v$ for the 1000 pseudo-clusters (panel a) and for $I_{Ferg} > 2$ pairs in the WL11 data (panel b). Both histograms are overlaid with a best-fitting Gauss curve (red). For the pseudo-clusters a Gaussian with the velocity dispersion given by CD96 is shown for comparison (dotted line in panel a). The observed WL11 histogram is modeled by two different combinations of Gauss curves as explained in the text (dashed and dotted curves in panel b). The area under each curve is always normalized to the number of galaxies.}
\label{fig:WL11_delta_v}
\end{figure}
\section{Discussion}
\label{sec:discussion}
Building on the work of \cite{F92}, several statistical methods have been used to search for bound companions in the Coma cluster. The results are briefly summarized and discussed in this section. Some aspects were discussed already before along with the presentation of the results, especially for the IP analyses.
\subsection{Comparison of mean secondary density around primaries}
\subsubsection{MA08 catalog}
\label{sec:dis_MA08}
We first determined the mean density of secondary galaxies ($M_B > -19$) around primary galaxies ($M_B < -19$) for the MA08 catalog and compared it to the mean of 1000 pseudo-clusters created by shuffling the velocities or positions of the primaries. We analyzed the sample with and without using redshift data (bins for the primary-secondary velocity difference $\delta v$ are: $[0,250]$ and $]250,500]\SI{}{\kilo m.s^{-1}}$).
A general outcome of this statistical treatment was the frequent or even dominant occurrence of under-densities (Figs.~\ref{fig:8Var}-\ref{fig:Lum_sec}), which are difficult to interpret in terms of physical effects. Boundary effects cannot be blamed for this, as the the Monte Carlo clusters are run within the same boundaries as the real cluster. More likely, these under-densities are an artefact from the simple random shuffling which assumes that the galaxies are distributed randomly in a smooth, single-component cluster potential, on top of which we would have those companions around single primary galaxies. But we know of course from substructure studies of Coma \citep[e.g.,][]{Biviano96} that the galaxies are clustered on all scales in a self-similar way, as is expected from hierarchical clustering. The resulting under-densities are then also expected to be particularly strong in the dense, inner part of the cluster covered by the MA08 catalog, even though the very innermost core region was avoided here for possible confusion. Indeed, in the outer, low-density parts covered by the WLG11 no such under-densities appear (cf. Figs.~\ref{fig:WL11}-\ref{fig:WL11_Lum_Sec_High}). In any case, under-densities, if they are not highly significant, will in general not be taken at face value here.
Although the 'companion signals' are generally only of low significance (at $1\hat{\sigma}$ for at least three bins in a row) and can be statistically backed with a KS or AD test only in a few cases, some of the findings are in accord with previous observations and are strong enough to be at least indicative of various evolutionary effects that are expected to play a role in a cluster environment. The most noteworthy observations are the following.
\begin{enumerate}
\item Over-densities around high-luminosity late-type (spiral) galaxies are detected for small and medium separations (up to $\sim \SI{160}{\kilo pc}$) at a significance level of $1\hat{\sigma} - 3\hat{\sigma}$ for five bins (cf.\,Fig.~\ref{fig:Lum_E_S}c).
This is where we expected to find satellite galaxies. \cite{And96} describes a morphology-velocity segregation, with the late-type galaxy population of the cluster having a higher velocity dispersion ($> 700-\SI{750}{\kilo m.s^{-1}}$) than the early-type one ($\sim 700-\SI{750}{\kilo m.s^{-1}}$). According to CD96 this likely means that late-type cluster galaxies are still in a stage of infall into the cluster core. Our finding that some bound companions still exist around the spirals clearly supports this scenario.
\item Slight over-densities are found around low-luminosity early-type galaxies at medium separations and high velocity differences (see Fig.~\ref{fig:Lum_E_S}b). In the scenario just mentioned, where early types would be the oldest and therefore most virtualized cluster members, dwarfs bound to individual early-type giants are not expected to exist, with the exception of extremely close companions (as discussed below). More conceivably they are bound to small subgroups. Several groups in the Coma region with a velocity dispersion of $\sim \SI{300}{\kilo m.s^{-1}}$ were identified by A05, but only a few of them are located within the MA08 area. As these authors give no detailed description of the group members, we cannot judge whether the excesses we found is due to those groups.
We note that points 1 and 2 generally agree with the old finding by \cite{BothunSull} that companions are more numerous around spirals than E/S0's, although that sample encompassed preferentially isolated galaxies.
At this point we have to mention the Coma cluster study of \citet{SHP97} who found an excess of low-surface brightness dwarfs (dEs) around high-luminosity early-type giants, claiming that an E/S0 giant has on average 4$\pm$1 dwarf companions. This study cannot be directly compared with our's, however, for a number of reasons. First, \citeauthor{SHP97} confined their dwarf search to the dense core region which we cut out to avoid confusion. Second, their (numerous!) low SB dwarfs are much fainter than our secondaries. Third, bound companions were searched for only in a very small circle ($r <$ 20\,kpc) around the giants (this was their way to avoid confusion). The method employed the modeling of the light distribution of a giant and then subtracting the model from the image. The number of companions was statistically assessed by number background subtraction. In our study we are essentially blind for such extremely close companions that are plausibly bound to an individual giant even in the very dense core region. However, the following two observations are well in accord the \citet{SHP97} finding.
\item An excess of E/S0 secondaries is revealed for higher velocity differences (see Fig.~\ref{fig:Sec_nurE}). This is expected from the morphology-density relation of galaxies in general \citep[e.g.,][]{D80A,D80B} and for the Coma cluster in particular \citep{And96}. Strangely, the over-density is not seen for small $\delta v$ (Fig.~\ref{fig:Sec_Morph})
\item An excess of faint ($M_B > -16$) companions is found around high-luminosity giants for velocity differences between primary and secondary that are indicative of gravitational binding for massive giants (cf.\,Fig.~\ref{fig:Lum_sec}a,b). Again, this is expected, as faint dwarfs dominate among satellites simply due to the luminosity function of galaxies. The excess is in accord with the kinematically blind over-densities found around luminous spirals under point 1, and with the very close companions around E/S0 giants found by \citet{SHP97}, as mentioned above. Unfortunately, a morphological splitting of our small kinematic sample was not feasible.
\item Finally, for brighter secondaries we observe strong under-densities around high-luminosity primaries at small $\delta v$ (Fig.~\ref{fig:Lum_sec}d,g), but not so at larger $\delta v$ where we find significant over-densities (Fig.~\ref{fig:Lum_sec} h). Notwithstanding the general problem with under-densities (see above), it is tempting here to interpret this effect in terms of galactic cannibalism: According to Chandrasekhar's formula for dynamical friction \citep[$F_{dyn.fric.} \sim \mathcal{M}^2 v_M^{-2}$,][p. 644]{BinneyTremaine}, more massive, slowly moving galaxies tend to be more affected by cannibalism than less massive, faster moving ones. However, this explanation might be too simple and other interactions among the galaxies and the cluster more important.
\end{enumerate}
\subsubsection{WLG11 catalog}
\label{sec:WL11disc}
As the Coma cluster clearly exceeds the area covered by MA08, the catalog of \citep[][without morphological information, however]{WL11} was analyzed as well.
Interestingly, no faint companions at small velocity differences are detected around
high-luminosity giants in this catalog. There is a slight over-density of secondaries at small separations for high velocity differences (Fig.~\ref{fig:WL11}), probably due to the very significant over-density of modestly faint secondaries around low-luminosity primaries (Fig.~\ref{fig:WL11_Lum_Sec_Low}b). On the other hand, there is also an over-density of luminous secondaries around the most luminous primaries, for small and high $\delta v$ (Fig.~\ref{fig:WL11_Lum_Sec_High}). Without further morphological information it is difficult to make sense out of this. But given the large mean distance of the WL11 galaxies from the cluster core, the presence of companions around low-luminosity, less massive primaries might indicate the expected absence of strong tidal forces out there.
Splitting the outer cluster region into a number of elliptical annuli (Fig.~\ref{fig:Karte_WL11} and Table~\ref{tab:WL11_loc}), we find significant excesses of secondaries at different separations $r$ and mostly in both $\delta v$-bins for regions II-IV and the outermost region VII (Table~\ref{tab:WL_Loc}). The excess in the outer II region (especially for large values of $\delta v$) might be caused by the presence of the groups around spirals NGC 4921 and NGC 4911 \citep{N03}, again supporting the scenario of infalling groups centered on late-type giants mentioned under point 1 above.
The large substructure around the cD galaxy NGC 4839, on the other hand, might explain the over-densities at high velocity differences in the outer III/IV regions. Even though this primary lies within the outer III area, the group exceeds the boundary to the next outer area. It can be seen for instance in \cite{N01} that the X-Ray area is stretched in the SW direction. Some primaries might fall from the outer IV region into this group having still some bound companions. Moreover, A05 have located some more groups in this region as can be seen in their \mbox{Fig.~\ref{fig:Lum_E_S}}.
\subsection{Interaction parameter method}
Aside from \citeauthor{F92}'s (\citeyear{F92}) interaction parameter, $I_{Ferg}$, which is a measure of the binding of a galaxy to its nearest neighbor in relation to the gravitational influence from the rest of the cluster, we introduced also a substructure and a group IP, $I_{sub}$ and $I_{group}$, which are more sensitive to the binding of a galaxy to a larger mass aggregate (of up to a scale of 300 kpc in the case of $I_{sub}$). The cumulative distributions for these IP's are essentially the same for the real MA08 sample and a randomized pseudo-cluster, with the exception of $I_{sub}$ where a marginally significant difference is found (Fig.~\ref{fig:IP}). This meets the expectation that the Coma cluster is primarily a smooth, relaxed cluster with only moderate (though still evident) substructure. It should be remembered that the infalling groups around NGC 4839 \citep{N01} and NGC 4921 and 4911 \citep{N03} are not or only partly covered by the MA08 field. Defining galaxies with $I > 2$ as bound (to a neighbor galaxy or a subgroup) and $I < 0.5$ as free (bound only to the whole cluster), and differentiating according to morphological type, we find that more late-type (spiral) galaxies than the average type are free ($I < 0.5$), or fewer are bound, when compared to a mean randomized pseudo-cluster (cf. Table~\ref{tab:IP_frac}). However, this is probably simply mirroring the general morphology-density relation and, in lack of a luminosity differentiation, is not in conflict with our previous finding that late-type giants (luminous spirals), surmised as being the centers of infalling groups, tend to have companions.
The most important use of the interaction parameter method introduced by F92 is the assessment of the statistical abundance of bound companions in a cluster. By analysing the distribution of velocity differences between galaxies with $I_{Ferg} > 2$ and their nearest neighbors we could estimate that roughly $25\%$ or $30\%$ of these (29 cases) are genuinely bound companions, assuming a typical $\delta v$ of 250 or 500 km\,s$^{-1}$, respectively; the remaining pairs would be projection cases. This amounts to $\approx$ 2$\pm 1\%$ of the total cluster population. Given the brightness limitation of the MA08 catalog ($M_B \lessapprox -15$), this percentage may be underestimated, as their could be more bound satellites among the fainter dwarf cluster members not enclosed in the catalog. Additionally, as explained above, \citet{SHP97} reported evidence for a mean number of very close dwarf satellites around E/S0 giants of 4$\pm$1 per giant. We note, however, that this would add at best another $\lessapprox 2\%$ bound companions (10 giants with 4 satellites among 2250 dwarfs).
Going through the same procedure with the WL11 sample which covers a larger area (at a somewhat fainter magnitude limit), the estimated number of bound companions among cluster members is $\approx 4 \pm 1\%$. Again this might be slightly underestimated for the reasons given above. However, overall we regard a relative abundance of $2-4\%$ bound companions in the Coma cluster, down to a limiting magnitude of $M_B \approx -15$ and disregarding extremely close companions hidden in the high-surface brightnesss central part of the primaries, as a realistic, robust estimate. This must be directly compared to \citeauthor{F92}'s (\citeyear{F92}) estimate of $7\%$ for the Virgo cluster. It should be noted that the limit of absolute magnitude of the redshift sample used for the IP analysis is roughly the same for the Virgo cluster in spite of its proximity. Hence the abundance of bound companions in Coma is about half of that in Virgo. This is in accord with the expectation we have for a regular, more relaxed cluster such as Coma versus a irregular, less relaxed cluster such as Virgo.
We regard this as the principal outcome of our study.
\section{Conclusion}
\label{sec:concl}
\citet{F92} created pseudo-clusters and compared them to the Virgo cluster in order to search for bound companions around giants. In this study we have applied his methods to the Coma cluster. The catalog of \cite{MA08} was used for a detailed analysis of a $\sim 0.73 \times \SI{0.84}{\deg}^2$ field centered around NGC 4889 and 4874. As expected, we find fewer companions in Coma than in Virgo due to the different evolutionary states of the two clusters, Coma being more regular and relaxed than Virgo. Introducing an interaction parameter, we estimate 2 - 4$\%$ of all Coma members to be bound satellites, as compared to $7\%$ found in Virgo found by \citet{F92} for an equivalent luminosity and separation range.
The mean surface density $\Sigma(r)$ of galaxies fainter than $M_B = -19$ around primaries (brighter than $M_B = -19$), corrected for the expected mean from pseudo-clusters, was analyzed for various parameter constraints with respect to luminosity and morphology of primaries and secondaries, as well as velocity difference. Density excesses, interpreted as possible companions, are mainly found around very luminous late-type galaxies (spirals), preferentially at large cluster-centric distances. This is in accord with the infall scenario of cluster formation. Moreover, we find that only the faintest dwarfs ($-16 < M_B$) seem to be satellites of individual primaries. Brighter secondaries might have been accreted to the substructures around the central dominant galaxies, or simply added to the general cluster potential. There are also hints of galactic cannibalism in the cluster.
The same statistical tools are used to search for bound companions in the sample of \cite{WL11} which covers a larger region (circled area out to $\sim \SI{4.2}{\mega pc}$) but has no morphological information. Clear excesses of secondaries (fainter than $M_R = -20$) are visible for almost all regions except for the innermost $\sim \SI{750}{\kilo pc}$.
Summarizing, also in Coma some bound companions exist but the fraction is clearly lower than in Virgo. The methods used are suitable for a search for companions in Coma and could be applied also to other galaxy clusters. An important requirement for such a task is the existence of a cluster catalog covering a large area and a sufficiently large range of galaxy magnitudes. Also morphological assignments and measured radial velocities for most of the member galaxies should be available.
The results obtained are often difficult to interpret and do not provide much new physical insight without a more detailed cluster modeling. It might be rewarding to apply \citeauthor{F92}'s method also to morphologically and kinematically detailed $N$-body clusters in various evolutionary stages drawn from cosmological simulations and compare the outcomes with the results found by \citep{F92} for Virgo and by the present study for Coma.
\begin{acknowledgements}
We thank Thorsten Lisker for making the WLG11 data available to us. We thank an anonymous referee for helpful comments. BB is grateful to the Swiss National Science Foundation for financial support.
\end{acknowledgements}
\bibliographystyle{aa}
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\section{Introduction}
\emph{Algebras of chiral differential operators},
or \emph{algebras of CDOs},
are certain vertex algebras closely related to
the geometry of loop spaces and
the quantum theory of $2$-dimensional $\sigma$-models.
Locally, the algebra of CDOs on e.g.~an affine space
$\mathbb{A}^d$ is an elementary conformal vertex algebra
(a $\beta\gamma$-system) and it has some appropriate
holomorphic and smooth versions
(see \S\ref{algCDO}--\S\ref{sec.CDO.Rd}).
The global construction of CDOs,
first given by Gorbounov, Malikov and Schechtman,
shares various features with string geometry and $\sigma$-models.
In particular, the obstructions for a complex manifold $M$
to admit a sheaf of holomorphic CDOs $\mc{D}^{\mathrm{ch}}_M$ are certain
refinements of $c_1(M)$ and $c_2(M)$;
and if $M$ is closed, then the genus-$1$ partition function
of the conformal vertex superalgebra $H^*(M;\mc{D}^{\mathrm{ch}}_M)$ is
essentially the Witten genus of $M$.
\cite{GMS1,myCDO}
In the realm of algebraic geometry, Kapranov and Vasserot
have provided an interpretation of CDOs in terms of the notion of
\emph{formal loop spaces} they introduced.
\cite{KV1,KV}
Intuitively, the ``formal loop space of $X$'' is
the formal neighborhood of the subspace of constant loops
(i.e.~$X$ itself) inside the space of all loops,
but this has yet to be made precise for, say, smooth manifolds.
Meanwhile, physicists have explained how holomorphic CDOs
describe the so-called large volume limit of the half-twisted
$\sigma$-model.
\cite{Kapustin,Witten.CDO,Tan}
More recently, using his own mathematical formulation of quantum
field theory, Costello has given a new construction
of the Witten genus for complex manifolds, referring to
it as an ``analytic avatar'' of holomorphic CDOs.
\cite{Costello1,Costello2}
This paper investigates two related topics about
CDOs on \emph{smooth} manifolds:
(i) their interactions with smooth Lie group actions and
(ii) construction of geometrically meaningful modules.
First, anticipating an interpretation of CDOs in terms of
``formal loops'' (\`a la Kapranov-Vasserot), we study how
a Lie group action on a manifold can be lifted to
a ``formal loop group action'' on an algebra of CDOs.
This turns out to be a condition on the equivariant
first Pontrjagin class.
The case of a principal bundle receives particular attention
and gives rise to a vertex algebra that plays an important
role in the rest of the paper
(perhaps in the overall theory as well).
Using the abovementioned ``formal loop group actions'' and
semi-infinite cohomology, we then introduce a construction
of modules over CDOs.
Intuitively, each of these modules should be
the space of sections of some vector bundle over
a ``formal loop space'', as a module over
``differential operators''.
The first example we study leads to a new and more conceptual
construction of an arbitrary algebra of CDOs.
The other example, called the spinor module, may be useful
for formalizing the physical meaning of the Witten genus for
smooth string manifolds.
The following is a more detailed overview of the paper.
\begin{subsec}
{\bf Overview of the paper.}
\S\ref{sec.review} recalls the definition of
an algebra of CDOs on a smooth manifold
(Definition \ref{defn.CDO})
as well as the general construction and classification of
these vertex algebras
(Theorems \ref{thm.globalCDO} and \ref{thm.globalCDO.iso}).
The explicit generators-and-relations construction described
here should be compared to the more conceptual one
in \S\ref{CDOP.CDOM}.
Let $G$ be a compact connected Lie group,
$\mathfrak{g}$ its Lie algebra and
$\lambda$ an invariant symmetric bilinear form on $\mathfrak{g}$.
Notice that $\lambda$ determines a centrally extended
loop algebra $\hat{\mathfrak{g}}_\lambda$ (\S\ref{sec.Gmfld})
and also a vertex algebra $V_\lambda(\mathfrak{g})$
(Example \ref{VAoid.Lie}).
If $\mathfrak{g}$ is simple, we will use the more common notation
$V_k(\mathfrak{g})$ in place of $V_{k\lambda_0}(\mathfrak{g})$,
where $k\in\mathbb{C}$ and $\lambda_0$ is the normalized Killing
form.
Let us introduce the following terminology:
a \emph{formal loop group action} of level $\lambda$ on
a vector space $W$ is a $\hat{\mathfrak{g}}_\lambda$-action on $W$
whose restriction to $\mathfrak{g}\subset\hat{\mathfrak{g}}_\lambda$
integrates into a $G$-action;
and if $W$ is a vertex algebra then we also require that
$\mathfrak{g}$ acts by inner derivations.
(In the main text this is called
an \emph{inner $(\hat{\mathfrak{g}}_\lambda,G)$-action};
see Definition \ref{formalLG.action}.)
Notice that in the latter case the action is
induced by a map of vertex algebras
$V_\lambda(\mathfrak{g})\rightarrow W$.
Suppose $P$ is a smooth manifold with a smooth $G$-action
and $\mc{D}^{\mathrm{ch}}(P)$ is an algebra of CDOs on $P$.
Conjecturally, there should be a description of $\mc{D}^{\mathrm{ch}}(P)$
in terms of the ``formal loop space of $P$''.
This motivates the main result in \S\ref{sec.LGaction}:
\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.formalLG.action}.}
{\it
The $G$-action on $P$ lifts to a formal loop group action on
$\mc{D}^{\mathrm{ch}}(P)$ of level $\lambda$ if and only if
\begin{align*}
8\pi^2p_1(P)_G=\lambda(P)
\end{align*}
where $p_1(P)_G$ is the equivariant first Pontrjagin class
and $\lambda(P)$ is the image of $\lambda$ under the
characteristic map
$(\mathrm{Sym}^2\mathfrak{g}^\vee)^G\cong H^4(BG)\rightarrow H^4_G(M)$.
Moreover, the said action is primary with respect to
a suitably chosen conformal vector of $\mc{D}^{\mathrm{ch}}(P)$.
}
\vspace{0.05in}
\noindent
The key ideas behind this result are:
the use of the Cartan model for equivariant de Rham cohomology
(recalled in \S\ref{sec.Cartan});
and the observation that the $G$-action on $P$ and the vertex
algebra structure of $\mc{D}^{\mathrm{ch}}(P)$ together determine a Cartan
cocycle for $p_1(P)_G$ (Lemma \ref{lemma.Cartan.closed}).
In fact, there is a more refined statement detailing
a bijection between the formal loop group actions in question
and certain Cartan cochains.
From now, $P$ is the total space of a principal $G$-bundle
$\pi:P\rightarrow M$ and the algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)$ is
equipped with a formal loop group action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$.
\S\ref{sec.CDOP} is a more detailed study of the
vertex algebra $\mc{D}^{\mathrm{ch}}(P)$ in this situation.
First we consider the centralizer subalgebra
\begin{align*}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}=C\big(\mc{D}^{\mathrm{ch}}(P),V_\lambda(\mathfrak{g})\big)
\end{align*}
i.e.~the subalgebra invariant under the formal loop group action.
While the structure of $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ is not fully
understood, a notable part of it is equivalent to
the Atiyah algebroid $(C^\infty(M),\mc{T}(P)^G)$ (\S\ref{sec.invCDO}).
For this reason the vertex algebra $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$
has more interesting modules compared to an algebra of CDOs
on $M$.
This will become more apparent later.
In the example $P=G$ with $G$ simple, the two $G$-actions on itself
by left and right multiplications can be lifted to two
\emph{commuting} formal loop group actions
$V_k(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(G)$ and
$V_{-k-2h^\vee}(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(G)$, where $k\in\mathbb{C}$
is arbitrary and $h^\vee$ is the dual Coxeter number
(\S\ref{CDOG}).
Hence there is an inclusion of, say, $V_k(\mathfrak{g})$ into
$C\big(\mc{D}^{\mathrm{ch}}(G),V_{-k-2h^\vee}(\mathfrak{g})\big)$.
If $k\neq-h^\vee$, then $V_k(\mathfrak{g})$ has a Sugawara vector
and it turns out to be also conformal for the above centralizer
(\S\ref{invCDOG}).
This seems to suggest that the said inclusion may in fact be
surjective, but the author has yet to find either a proof or
a counterexample.
In the case where $\pi$ is a principal frame bundle of $TM$,
we find a better description of $\mc{D}^{\mathrm{ch}}(P)$ such that
(among other things) the formal loop group action becomes
manifest.
This occupies the second half of \S\ref{sec.CDOP} and
leads to the following result, which is central to
the rest of the paper:
\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.CDOP}.}
{\it
Suppose we have
a principal $G$-bundle $\pi:P\rightarrow M$,
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta$ on $\pi$ that induces
the Levi-Civita connection on $TM$;
an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$;
and a basic $3$-form $H$ on $P$ with
$dH=(\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho)(\Omega\wedge\Omega)$,
where $\Omega$ is the curvature of $\Theta$
(see also \S\ref{conventions}).
These data determine a conformal vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, which is generated by primary fields of
weights $0$ and $1$, and equipped with a primary formal loop
group action of level $\lambda$.
For its detailed definition see the main text.
}
\vspace{0.05in}
\noindent
Even though this vertex algebra arises as a particular
algebra of CDOs, it should really be regarded as a more
fundamental object for both conceptual and aesthetic reasons.
In fact, from vertex algebras of the form $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$,
there is a natural way to recover \emph{all} algebras of CDOs
and also construct many other interesting objects (see below).
Moreover, the definition of $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ is
arguably more appealing than that of an arbitrary algebra
of CDOs.
\S\ref{sec.ass} introduces the following construction:
given a vector space $W$ with a formal loop group action
of level $-\lambda-\lambda_{\t{\sf ad}}$, we can define a module over
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ as follows
(Definition \ref{defn.ass} \& Lemma \ref{lemma.ass})
\begin{align} \label{ass.module}
\Gamma^{\mathrm{ch}}(\pi,W)
:=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},\,
\mc{D}^{\mathrm{ch}}(P)\otimes W\big).
\end{align}
The notation $H^{\frac{\infty}{2}+*}$ refers to
\emph{semi-infinite (a.k.a.~BRST) cohomology},
which can be defined using the Feigin complex
(recalled in \S\ref{sec.Feigin}--\S\ref{sec.semiinf}).
This is analogous to the construction of associated vector
bundles of $\pi:P\rightarrow M$.
In fact, $\Gamma^{\mathrm{ch}}(\pi,W)$ should be the space of sections of
a vector bundle over the ``formal loops of $M$'', as a module
over ``differential operators''.
Notice that if $W$ is a vertex algebra, then so is $\Gamma^{\mathrm{ch}}(\pi,W)$
and there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,W)$.
In our first example of (\ref{ass.module}), $\pi$ is a principal
frame bundle of $TM$ and $W=\mathbb{C}$ (hence $\lambda=-\lambda_{\t{\sf ad}}$).
By Theorem \ref{thm.CDOP}, the definition of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
depends on the choice of some $H\in\pi^*\Omega^3(M)$ that satisfies
$dH=\mathrm{Tr}\,\rho(\Omega)\wedge\rho(\Omega)$.
The main result in \S\ref{CDOP.CDOM} is a new description
of CDOs as alluded to earlier:
\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.CDO.semiinf}.}
{\it
The vertex algebra
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})=H^{\frac{\infty}{2}+0}(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P))$ is an algebra of CDOs on $M$.
Up to isomorphism every algebra of CDOs on $M$ is of this form.
}
\vspace{0.05in}
\noindent
The proof consists of two parts:
first we identify the two lowest weights of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
(\S\ref{sec.ass.C.0} \& \S\ref{sec.ass.C.1})
and work out their structure
(Proposition \ref{prop.ass.C.VAoid});
then we use a certain property of its conformal vector
to deduce that $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is completely determined
by its two lowest weights
(Proposition \ref{prop.ass.C.conformal} \&
Corollary \ref{cor.ass.C.freeVA}).
There is also an extension of Theorem \ref{thm.CDO.semiinf}
for supermanifolds, including a special case that recovers
the chiral de Rham complex (\S\ref{CDO.cs}).
In our second and last example of (\ref{ass.module}),
$G$ is a spin group $\textrm{Spin}_{2d'}$,
$\pi$ is a spin structure on $M$ and
$W=S$ is the spinor representation of $\widehat{\mathfrak{so}}_{2d'}$.
It follows from Theorem \ref{thm.CDOP} that the definition of
the \emph{spinor module} $\Gamma^{\mathrm{ch}}(\pi,S)$ depends on the choice of
some $H\in\pi^*\Omega^3(M)$ that satisfies
$dH=\frac{1}{2}\mathrm{Tr}\,(\Omega\wedge\Omega)$ (\S\ref{sec.ass.S}).
Geometrically, $\Gamma^{\mathrm{ch}}(\pi,S)$ should be the space of sections
of the ``spinor bundle on the formal loops of $M$''.
In \S\ref{sec.FL.spinor} we carry out an analysis of $\Gamma^{\mathrm{ch}}(\pi,S)$
that parallels the one in \S\ref{CDOP.CDOM} and culminates
in a more explicit description of it in terms of
generating data (i.e.~a subspace and three types of fields)
and relations.
This is summarized in Theorem \ref{thm.ass.S}.
Notice that our work in \S\ref{sec.FL.spinor} should be viewed
mainly as preparation for further study.
The author \emph{speculates} that the spinor module $\Gamma^{\mathrm{ch}}(\pi,S)$
always admits an action of the $(0,1)$ superconformal algebra and
thereby provides at least a partial mathematical account
of the physical interpretation of the Witten genus
(\S\ref{ass.S.sconformal}).
The appendix reviews the notion of vertex algebroids
and their relations with vertex algebras.
Although vertex algebroids have a rather complicated definition,
they serve as a convenient tool for dealing with the vertex
algebras in this paper.
\end{subsec}
\begin{subsec} \label{conventions}
{\bf Notations and conventions.}
In this paper, every vertex algebra $V$ is graded by
nonnegative integers which we call \emph{weights};
its component of weight $k$ is denoted by $V_k$ and
its weight operator by $L_0$, i.e. $L_0|_{V_k}=k$.
For any $u\in V$, we always write the Fourier modes of
its vertex operator as $u_k$, $k\in\mathbb{Z}$, such that
$u_k$ has weight $-k$.
For any conformal vector $\nu\in V$ that we consider,
$\nu_0=L_0$.
Given a smooth manifold $M$, we write
$C^\infty(M)$, $\mc{T}(M)$, $\Omega^n(M)$ for its spaces of
smooth $\mathbb{C}$-valued functions, vector fields and $n$-forms.
Also, all ordinary cohomology groups and their equivariant
versions have complex coefficients.
Given a Lie algebra $\mathfrak{g}$ and a finite-dimensional
representation $\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$, we write
$\lambda_\rho$ for the invariant symmetric bilinear form on
$\mathfrak{g}$ given by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, $\lambda_{\t{\sf ad}}$ is the Killing form.
Square brackets $[\;]$ are used for supercommutators between
operators of any parities, while curly brackets $\{\;\}$
are reserved for a different use (see \S\ref{VAoid}).
Repeated indices are always implicitly summed over all possible
values, unless a specific range is indicated.
\end{subsec}
\begin{subsec}
{\bf Acknowledgements.}
The author is currently supported by the ESPRC grant
EP/H040692/1.
He would like to thank Matthew Ando for a suggestion
that largely motivated Theorem \ref{thm.formalLG.action},
and Dennis Gaitsgory for answering some questions about
semi-infinite cohomology.
\end{subsec}
\newpage
\section{CDOs on Smooth Manifolds}
\label{sec.review}
A sheaf of CDOs is a sheaf of vertex algebras locally
modelled on a very basic vertex algebra (known as the
$\beta\gamma$-system), and provides a mathematical
approximation to a quantum field theory of much
interest.
\cite{MSV,GMS1,GMS2,KV,Kapustin,Witten.CDO,Tan}
This section reviews the construction and
classification of sheaves of CDOs on a smooth
manifold, following the formulation in \cite{myCDO}.
\begin{subsec} \label{algCDO}
{\bf The algebra of CDOs on $\mathbb{A}^d$.}
Let $d$ be a positive integer.
Define a unital associative algebra $\mathcal{U}$ with
the following generators and relations
\begin{eqnarray} \label{Weyl}
b^i_n,\;a_{i,n},\;n\in\mathbb{Z},\;i=1,\ldots,d,\qquad
[a_{i,n},b^j_m]=\delta^j_i\delta_{n,-m},\qquad
[b^i_n,b^j_m]=0=[a_{i,n},a_{j,m}]\,.
\end{eqnarray}
(This is an example of a Weyl algebra.)
The commutative subalgebra $\mathcal{U}_+$ generated
by $\{b^i_n\}_{n>0}$ and $\{a_{i,n}\}_{n\geq 0}$
has a trivial representation $\mathbb{C}$.
The induced $\mathcal{U}$-module
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}(\mathbb{A}^d):=\mathcal{U}\otimes_{\mathcal{U}_+}\mathbb{C}
\end{eqnarray*}
has the structure of a vertex algebra.
The vacuum is $\mathbf 1=1\otimes 1$.
The infinitesimal translation operator $T$ and weight
operator $L_0$ are determined by
\begin{eqnarray*}
\begin{array}{lll}
T\mathbf 1=0, &
[T,b^i_n]=(1-n)b^i_{n-1}, \phantom{aa} &
[T,a_{i,n}]=-na_{i,n-1} \vspace{0.03in} \\
L_0\mathbf 1=0, \phantom{aa} &
[L_0,b^i_n]=-nb^i_n, &
[L_0,a_{i,n}]=-na_{i,n}
\end{array}
\end{eqnarray*}
The fields (or vertex operators) of $b^i_0\mathbf 1\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0$
and $a_{i,-1}\mathbf 1\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1$ are respectively
\begin{eqnarray*}
\begin{array}{l}
\sum_n b^i_n z^{-n},\qquad \sum_n a_{i,n}z^{-n-1}
\end{array}
\end{eqnarray*}
which determine the fields of other elements by
the Reconstruction Theorem \cite{FB-Z}.
This vertex algebra has a family of conformal elements
of central charge $2d$, namely
\footnote{
This vertex algebra is the tensor product of $d$ copies
of the $\beta\gamma$-system.
It admits other conformal elements that define different
conformal weights.
\cite{Kac}
}
\begin{eqnarray*}
\qquad\qquad\qquad
a_{i,-1}b^i_{-1}\mathbf 1+T^2 f,\qquad
f\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0=\mathbb{C}[b^1_0,\ldots,b^d_0]\cdot\mathbf{1}.
\end{eqnarray*}
The vertex algebra $\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$ is freely generated by
its associated vertex algebroid (see \S\ref{VA.VAoid} and
\S\ref{VAoid.VA}).
To describe the latter, consider the affine space
$\mathbb{A}^d=\textrm{Spec}\,\mathbb{C}[b^1,\cdots,b^d]$ and identify
the functions, $1$-forms and vector fields on $\mathbb{A}^d$
with the following subquotients of $\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$: \vspace{0.03in} \\
\indent $\cdot$\;
$\mathcal{O}(\mathbb{A}^d)=\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0$ via $b^i=b^i_0\boldsymbol{1}$,
$b^i b^j=b^i_0 b^j_0\boldsymbol{1}$, etc. \vspace{0.03in} \\
\indent $\cdot$\;
$\Omega^1(\mathbb{A}^d)\subset\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1$ via
$db^i=b^i_{-1}\boldsymbol{1}$ \vspace{0.03in} \\
\indent $\cdot$\;
$\mc{T}(\mathbb{A}^d)=\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1/\Omega^1(\mathbb{A}^d)$ via
$\partial_i=\partial/\partial b^i=$ coset of $a_{i,-1}\boldsymbol{1}$ \vspace{0.03in} \\
Under these identifications, the vertex algebroid associated to
$\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$ is of the form
\begin{eqnarray*}
\big(\mathcal{O}(\mathbb{A}^d),\Omega^1(\mathbb{A}^d),\mc{T}(\mathbb{A}^d),
\bullet,\{\;\},\{\;\}_{\Omega}\big).
\end{eqnarray*}
The extended Lie algebroid structure consists of
the usual differential on functions,
Lie bracket on vector fields,
Lie derivations by vector fields on functions and $1$-forms,
and pairing between $1$-forms and vector fields.
Using the splitting
\begin{eqnarray} \label{algCDO.splitting}
s:\mc{T}(\mathbb{A}^d)\rightarrow\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1,\qquad
X=X^i\partial_i\;\mapsto\;a_{i,-1}X^i
\end{eqnarray}
the rest of the vertex algebroid structure, according to
(\ref{VA.VAoid.456}), reads as follows
\begin{align} \label{CDO.VAoid456}
X\bullet f=(\partial_j X^i)(\partial_i f)db^j,\quad
\{X,Y\}=-(\partial_j X^i)(\partial_i Y^j),\quad
\{X,Y\}_\Omega=-(\partial_k\partial_j X^i)(\partial_i Y^j)db^k
\end{align}
These expressions do not seem to have any obvious
global meaning;
however, see Theorem \ref{thm.globalCDO}.
\end{subsec}
\begin{subsec} \label{sec.CDO.Rd}
{\bf The sheaf of CDOs on $\mathbb{R}^d$.}
Now regard $b^1,\ldots,b^d$ as the coordinates of
$\mathbb{R}^d$.
The ($\mathbb{C}$-valued) smooth functions, $1$-forms
and vector fields on an open set
$W\subset\mathbb{R}^d$ form an extended Lie algebroid
just as in \S\ref{algCDO}, and the expressions in
(\ref{CDO.VAoid456}) again define a vertex
algebroid
\begin{eqnarray*}
\big(C^\infty(W),\Omega^1(W),\mc{T}(W),
\bullet,\{\;\},\{\;\}_\Omega\big).
\end{eqnarray*}
The vertex algebra it freely generates
(see \S\ref{VAoid.VA}) will be denoted by $\mc{D}^{\mathrm{ch}}(W)$.
This vertex algebra also has a family of conformal
elements of central charge $2d$, namely
\begin{eqnarray} \label{localCDO.conformal}
\partial_{i,-1}db^i+\frac{1}{2}\,T\omega,\qquad
\omega\in\Omega^1(W),\;d\omega=0.
\end{eqnarray}
When $W$ varies, we obtain a sheaf
of conformal vertex algebras $\mc{D}^{\mathrm{ch}}$ on $\mathbb{R}^d$.
\end{subsec}
\begin{defn} \label{defn.CDO}
A {\bf sheaf of CDOs} on a smooth manifold $M$ of dimension $d$
is a sheaf of vertex algebras $\mathcal{V}$ with the following
properties:
\vspace{-0.05in}
\begin{itemize}
\item[$\cdot$]
its weight-zero component is $\mathcal{V}_0=C^\infty_M$, and
\vspace{-0.08in}
\item[$\cdot$]
each point of $M$ has a neighborhood $U$ such that $(U,\mathcal{V}|_U)$
is isomorphic to $(W,\mc{D}^{\mathrm{ch}}|_W)$ for some open set $W\subset\mathbb{R}^d$.
\vspace{-0.05in}
\end{itemize}
A {\bf conformal structure} on $\mathcal{V}$ is an element
$\nu\in\mathcal{V}(M)$ such that, under each isomorphism
as postulated above, $\nu|_U\in\mathcal{V}(U)$ corresponds
to one of the conformal elements
(\ref{localCDO.conformal}) of $\mc{D}^{\mathrm{ch}}(W)$.
\end{defn}
In order to state the results on the construction
and classification of sheaves of CDOs, let us
introduce a notation that will also appear
often in the sequel.
\begin{defn} \label{tnabla}
Let $M$ be a smooth manifold.
Given a connection $\nabla$ on $TM$ and $X\in\mc{T}(M)$, define
an operator $\nabla^t X\in \Gamma(\textrm{End}\,TM)$ by
\begin{eqnarray*}
(\nabla^t X)(Y):=\nabla_X Y-[X,Y],\qquad Y\in\mc{T}(M).
\end{eqnarray*}
Notice that if $\nabla$ is torsion-free, then
$\nabla^t X=\nabla X$.
\end{defn}
\begin{theorem} \label{thm.globalCDO}
\cite{myCDO}
Let $M$ be a smooth manifold of dimension $d$.
(a) Given a connection $\nabla$ on $TM$ with curvature $R$ and
$H\in\Omega^3(M)$ satisfying $dH=\mathrm{Tr}\,(R\wedge R)$, there is
a sheaf of vertex algebroids
$\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}_\Omega\big)$
on $M$ defined by the following expressions
\begin{align}
\nonumber
X\bullet f\;\;\; &:= (\nabla X)f \\
\label{CDO.VAoid}
\{X,Y\}\;\; &:= -\mathrm{Tr}\,(\nabla^t X\cdot\nabla^t Y) \\
\nonumber
\{X,Y\}_\Omega &:=
\mathrm{Tr}\,\bigg(-\nabla(\nabla^t X)\cdot\nabla^t Y
+\nabla^t X\cdot\iota_Y R
-\iota_X R\cdot\nabla^t Y\bigg)
+\frac{1}{2}\iota_X\iota_Y H
\end{align}
and the sheaf of vertex algebras it freely generates
(see \S\ref{VAoid.VA}) is a sheaf of CDOs on $M$,
which we denote by $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$.
Up to isomorphism, every sheaf of CDOs on $M$ is
of this form.
(b) There is a one-to-one correspondence between
conformal structures
on $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ and $\omega\in\Omega^1(M)$
satisfying $d\omega=\mathrm{Tr}\, R$.
Given such $\omega$, the corresponding conformal
structure, which we denote by $\nu^\omega$, has the
local expression
\begin{eqnarray} \label{CDO.conformal}
\nu^\omega|_U
=\sigma_{i,-1}\sigma^i
+\frac{1}{2}\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma
-\Gamma^\sigma_{-2}\mathbf{1}\big)
+\sigma^i([\sigma_j,\sigma_k])
\sigma^k_{-1}(\Gamma^\sigma)^j_{\phantom{i}i}
+\frac{1}{2}\omega_{-2}\mathbf{1}
\end{eqnarray}
where $U\subset M$ is an open subset,
$\sigma=(\sigma_1,\ldots,\sigma_d)$ any $C^\infty(U)$-basis of
$\mc{T}(U)$, $(\sigma^1,\ldots,\sigma^d)$ the dual basis of
$\Omega^1(U)$, and
$\Gamma^\sigma\in\Omega^1(U)\otimes\mathfrak{gl}_d$ the connection
$1$-form of $\nabla$ with respect to $\sigma$
(i.e.~$\nabla\sigma_i=(\Gamma^\sigma)^j_{\phantom{i}i}\otimes\sigma_j$).
This conformal structure has central charge $2d=2\dim M$ and
the property that
\begin{eqnarray} \label{CDO.L1}
\nu^\omega_1\alpha=0\;\textrm{ for }\alpha\in\Omega^1(M),\qquad
\nu^\omega_1 X=\mathrm{Tr}\,\nabla^t X-\omega(X)\;\textrm{ for }X\in\mc{T}(M).
\quad\qedsymbol
\end{eqnarray}
\end{theorem}
{\it Remarks.}
(i) By Theorem \ref{thm.globalCDO}, a smooth manifold
$M$ admits sheaves of CDOs if and only if $p_1(M)$ is
trivial in de Rham cohomology, while conformal structures
always exist.
For example, if $\nabla$ is orthogonal with respect to
a Riemannian metric, then $\mathrm{Tr}\, R=0$ and a conformal
structure can be defined using, say, $\omega=0$.
However, this result generalizes to cs-manifolds
(supermanifolds with $\mathbb{C}$-valued structure sheaf)
in which case the obstruction to conformal structures
may well be nontrivial.~\cite{myCDO}
(ii) In \cite{myCDO}, we only obtained a local expression of
$\nu^\omega$ in terms of local coordinate vector fields,
but that implies the more general expression
(\ref{CDO.conformal}) by a straightforward calculation.
(iii) In the original work \cite{GMS2} as well as in
\cite{myCDO}, sheaves of CDOs and conformal structures
were constructed by gluing local data.
In the smooth case, this culminates in a description
by generators and relations (or generating fields
and OPEs) as seen above.
However, the expressions in (\ref{CDO.VAoid}) and
(\ref{CDO.conformal}) do not seem very inspiring.
Later in \S\ref{CDOP.CDOM}, we will obtain a more
conceptual description of CDOs using semi-infinite
cohomology.
\begin{theorem} \label{thm.globalCDO.iso}
\cite{myCDO}
Let $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ and $\mc{D}^{\mathrm{ch}}_{M,\nabla,H'}$ be sheaves of CDOs
on a smooth manifold $M$ constructed as in Theorem
\ref{thm.globalCDO}a;
denote by $\bullet$, $\{\;\}$, $\{\;\}_\Omega$
(resp.~$\{\;\}'_\Omega$) the structure maps determined by
$\nabla$ and $H$ (resp.~$H'$) as in (\ref{CDO.VAoid}).
(a) There is a one-to-one correspondence between isomorphisms
$\mc{D}^{\mathrm{ch}}_{M,\nabla,H}\stackrel{\sim}{\rightarrow}\mc{D}^{\mathrm{ch}}_{M,\nabla,H'}$
that restricts to the identity on $C^\infty_M$ and
$\beta\in\Omega^2(M)$ satisfying $d\beta=H'-H$.
Given such $\beta$, the corresponding isomorphism is induced by
an isomorphism of sheaves of vertex algebroids
(see \S\ref{VAoid.VA.mor})
\begin{eqnarray*}
(\mathrm{id},\Delta_\beta):
\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}_\Omega\big)
\rightarrow
\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}'_\Omega\big)
\end{eqnarray*}
where $\Delta_\beta:\mc{T}_M\rightarrow\Omega^1_M$ is given by
$\Delta_\beta(X)=\frac{1}{2}\iota_X\beta$.
(b) Every isomorphism described above respects the correspondence
in Theorem \ref{thm.globalCDO}b.
$\qedsymbol$
\end{theorem}
{\it Remarks.}
(i) By Theorems \ref{thm.globalCDO}a and \ref{thm.globalCDO.iso}a,
if $M$ admits sheaves of CDOs, their isomorphism classes form
an $H^3(M)$-torsor.
(ii) Since each sheaf of CDOs $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ is fine, for most
purposes it suffices to (and we will) work only with the vertex
algebra of global sections $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$, which will be
called an {\bf algebra of CDOs} on $M$.
The reader should keep in mind that by construction
\begin{eqnarray} \label{CDO.wt01}
\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)_0=C^\infty(M),\qquad
\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)_1=\Omega^1(M)\oplus\mc{T}(M).
\end{eqnarray}
For a description of the higher weights, see
\S\ref{freeVA.PBW}.
\begin{lemma} \label{lemma.1form.0mode}
\cite{myCDO}
Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$ on
a smooth manifold $M$.
For any $\alpha\in\Omega^1(M)$ and $X\in\mc{T}(M)$, we have
$\alpha_0 X=-\iota_X d\alpha$.
Moreover, $\alpha_0=0$ on $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$ if
and only if $d\alpha=0$.
$\qedsymbol$
\end{lemma}
\newpage
\setcounter{equation}{0}
\section{Formal Loop Group Actions on CDOs}
\label{sec.LGaction}
This section investigates the condition under which
a Lie group action on a manifold lifts to a projective
``formal loop group action'' on an algebra of CDOs.
This turns out to be a condition on the equivariant
first Pontrjagin class of the manifold.
The said action provides evidence of a conjectural
interpretation of (smooth) CDOs in terms of
``formal loops'', in the vein of \cite{KV}.
\begin{subsec} \label{sec.Gmfld}
{\bf Setting:~manifold with a Lie group action.}
Throughout this section, let $G$ be a compact connected
Lie group, $\mathfrak{g}$ its Lie algebra, and $\lambda$
an invariant symmetric bilinear form on $\mathfrak{g}$.
Recall the loop algebra
$L\mathfrak{g}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$:
writing $A\otimes t^n$ as $A_n$, the Lie bracket is
given by $[A_n,B_m]=[A,B]_{n+m}$.
Also recall that $\lambda$ determines a central extension
$\hat{\mathfrak{g}}_\lambda=L\mathfrak{g}\oplus\mathbb{C}$ with
\begin{eqnarray*}
[A_n,B_m]=[A,B]_{n+m}+n\lambda(A,B)\delta_{n+m,0},\quad
A,B\in\mathfrak{g},\;\,n,m\in\mathbb{Z}.
\end{eqnarray*}
Let $P$ be a smooth manifold with a smooth right $G$-action.
Later we will specialize to the case of a principal
bundle, but at the moment $P$ can be any right $G$-manifold.
The left $G$-action on $C^\infty(P)$ determines and is
determined by a map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(P)$.
The vector field generated by $A\in\mathfrak{g}$ will be written
as $A^P\in\mc{T}(P)$.
\end{subsec}
\begin{subsec} \label{sec.Cartan}
{\bf The equivariant de Rham complex.}
Recall that $H^*_G(P)=H^*(EG\times_G P)$ can be computed
by the Cartan model $(\Omega^*_G(P),d_G)$.~\cite{GuiSte}
The graded algebra of Cartan cochains is given by
\begin{eqnarray*}
\Omega^*_G(P)=
\bigoplus_{k\geq 0}\Omega^k_G(P),\qquad
\Omega^k_G(P)=
\bigoplus_{2i+j=k}\big(\textrm{Sym}^i\mathfrak{g}^\vee
\otimes\Omega^j(P)\big)^G.
\end{eqnarray*}
Let us adopt a convention:~regard each
$\xi\in\Omega^*_G(P)$ as a $G$-equivariant polynomial map
$\xi:\mathfrak{g}\rightarrow\Omega^*(P)$ and write its value at
$A\in\mathfrak{g}$ as $\xi_A$.
The Cartan differential then reads
\begin{eqnarray*}
(d_G\xi)_A=d\xi_A-\iota_{A^P}\xi_A.
\end{eqnarray*}
The characteristic map $H^*(BG)\rightarrow H^*_G(P)$ is
represented by the inclusion
$(\textrm{Sym}^*\mathfrak{g}^\vee)^G\hookrightarrow\Omega^*_G(P)$.
Given $\eta\in(\textrm{Sym}^*\mathfrak{g}^\vee)^G=H^*(BG)$, its image will be
denoted by $\eta(P)\in H^*_G(P)$.
\end{subsec}
\begin{subsec} \label{sec.inv.data}
{\bf CDOs with a $G$-action.}
Choose a $G$-invariant connection $\nabla$ on $TP$, with
curvature tensor $R$.
This means for $A\in\mathfrak{g}$ we have $L_{A^P}\nabla=0$,
or equivalently
\begin{eqnarray} \label{conn.Ginv}
\nabla(\nabla^t A^P)=-\iota_{A^P}R
\end{eqnarray}
in terms of Definition \ref{tnabla}.
\footnote{
For any $X\in\mc{T}(P)$ we have
$L_X\nabla=\nabla(\nabla^t X)+\iota_X R$.
This is proved as follows:
for $Y,Z\in\mc{T}(P)$
\begin{align*}
(L_X\nabla)_Y Z
&=[X,\nabla_Y Z]-\nabla_{[X,Y]}Z-\nabla_Y[X,Z] \\
&=\nabla_Y\nabla_X Z-\nabla_Y[X,Z]
-\nabla_X\nabla_Y Z+[X,\nabla_Y Z]
+\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z
-\nabla_{[X,Y]}Z \\
&=\nabla_Y(\nabla^t X)(Z)-(\nabla^t X)(\nabla_Y Z)+R_{X,Y}Z
=\big(\nabla_Y(\nabla^t X)+R_{X,Y}\big)(Z)
\end{align*}
}
Assume $p_1(P)=0$ and choose $H\in\Omega^3(P)^G$ such that
$dH=\mathrm{Tr}\,(R\wedge R)$.
By Theorem \ref{thm.globalCDO}a, $\nabla$ and $H$
determine an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(P)$, which
is freely generated by a vertex algebroid
(see also \S\ref{VAoid.VA})
\begin{eqnarray*}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
\bullet,\{\;\},\{\;\}_\Omega\big);
\end{eqnarray*}
when there is no risk of confusion, we simply write
$\mc{D}^{\mathrm{ch}}(P)$.
Clearly, the $G$-invariance of $\nabla$ and $H$
implies $G$-equivariance of the structure maps
$\bullet$, $\{\;\}$, $\{\;\}_\Omega$, so that the
$G$-action on $C^\infty(P)=\mc{D}^{\mathrm{ch}}(P)_0$ extends to
a $G$-action on $\mc{D}^{\mathrm{ch}}(P)$.
Without loss of generality, assume that $\mathrm{Tr}\, R=0$.
(For example, this is true if $\nabla$ is orthogonal
with respect to a Riemannian metric.)
Choose $\omega\in\Omega^1(P)^G$ such that $d\omega=0$.
By Theorem \ref{thm.globalCDO}b, $\omega$ determines
a $G$-invariant conformal structure $\nu^\omega$ on
$\mc{D}^{\mathrm{ch}}(P)$.
The Fourier modes of the associated Virasoro field
will be denoted by $L_n^\omega$, $n\in\mathbb{Z}$.
Soon we will make a more specific choice of $\omega$.
\end{subsec}
{\it Remark.}
The $G$-invariance of $\nabla$ and $H$, as well as that
of other geometric data to appear later, can always be
achieved by averaging over $G$ with respect to the
Haar measure.
\begin{defn} \label{formalLG.action}
Given a vertex algebra $V$,
by an {\bf inner $\hat{\mathfrak{g}}_\lambda$-action} on $V$
we simply mean a map of vertex algebras from
$V_\lambda(\mathfrak{g})$ to $V$, and it is
an {\bf inner $(\hat{\mathfrak{g}}_\lambda,G)$-action}
if its induced $\mathfrak{g}$-action on $V$ (see below)
integrates into a $G$-action.
\end{defn}
{\it Remarks.}
(i) By definition of $V_\lambda(\mathfrak{g})$ (see Example
\ref{VAoid.Lie}), any map $V_\lambda(\mathfrak{g})\rightarrow V$
is determined by its component of weight one, i.e.~a linear map
$\mathfrak{g}=V_\lambda(\mathfrak{g})_1\rightarrow V_1$.
Taking the zeroth modes then yields a map of Lie algebras
from $\mathfrak{g}$ to the inner derivations of $V$.
This is the induced $\mathfrak{g}$-action referred to above.
(ii) Now we may state the goal of this section more precisely:
find the condition under which the given $G$-action on
$\mc{D}^{\mathrm{ch}}(P)_0=C^\infty(P)$ extends to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$.
\begin{subsec} \label{CDO.Cartan}
{\bf Cartan cochains associated to CDOs.}
The $G$-action on $P$ and the vertex algebroid structure associated
to $\mc{D}^{\mathrm{ch}}(P)$ together determine two Cartan cochains of degree $4$,
namely
\begin{eqnarray} \label{Cartan4.VAoid}
\begin{array}{ll}
\chi^{2,2}\in\big(\mathfrak{g}^\vee\otimes\Omega^2(P)\big)^G,
&\chi^{2,2}_A:=\{A^P,-\}_\Omega=
\mathrm{Tr}\,(\nabla^t A^P\cdot R)-\frac{1}{2}\iota_{A^P}H
\vspace{0.03in}\vss \\
\chi^{4,0}\in\big(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P)\big)^G,
\quad
&\chi^{4,0}_A:=\{A^P,A^P\}
=-\mathrm{Tr}\,(\nabla^t A^P\cdot\nabla^t A^P)
\end{array}
\end{eqnarray}
Indeed, by (\ref{CDO.VAoid}) and (\ref{conn.Ginv}),
the operator $\{A^P,-\}_\Omega:\mc{T}(P)\rightarrow\Omega^1(P)$
may be viewed as the indicated $2$-form, and
the $G$-invariance of $\nabla$, $H$ implies
the $G$-equivariance of $\chi^{2,2}$, $\chi^{4,0}$.
Moreover, the $G$-action on $P$ and the conformal structure
$\nu^\omega$ on $\mc{D}^{\mathrm{ch}}(P)$ together determine a Cartan cochain
of degree $2$, namely
\begin{eqnarray} \label{Cartan2.conformal}
\chi^{2,0}\in\big(\mathfrak{g}^\vee\otimes C^\infty(P)\big)^G,\qquad
\chi^{2,0}_A:=L_1^\omega A^P=\mathrm{Tr}\,\nabla^t A^P-\omega(A^P)
\end{eqnarray}
in view of (\ref{CDO.L1}).
Indeed, the $G$-invariance of $\nabla$, $\omega$ implies the
$G$-equivariance of $\chi^{2,0}$.
\end{subsec}
\begin{lemma} \label{lemma.Cartan.closed}
(a) The Cartan cochain $2\chi^{2,2}+\chi^{4,0}$ is closed and
represents $8\pi^2 p_1(P)_G\in H^4_G(P)$.
(b) The Cartan cochain $\chi^{2,0}$ is exact.
In fact, it is trivial with a suitable choice of conformal
structure $\nu^\omega$.
\end{lemma}
\begin{proof}
(a) According to \cite{BGV,BT}, the Cartan cochain
$A\mapsto\mathrm{Tr}\,(R-\nabla^t A^P)^2$ is closed and represents
$-8\pi^2 p_1(P)_G$.
Our claim then follows from the calculation
\begin{eqnarray*}
\mathrm{Tr}\,(R-\nabla^t A^P)^2
=dH-2\,\mathrm{Tr}\,(\nabla^t A^P\cdot R)
+\mathrm{Tr}\,(\nabla^t A^P\cdot\nabla^t A^P)
=(d_G H)_A-2\chi^{2,2}_A-\chi^{4,0}_A
\end{eqnarray*}
where we have used $dH=\mathrm{Tr}\,(R\wedge R)$ and
(\ref{Cartan4.VAoid}).
(b) According to \cite{BGV,BT} again, the Cartan cochain
$A\mapsto\mathrm{Tr}\,(R-\nabla^t A^P)=-\mathrm{Tr}\,\nabla^t A^P$ is exact.
Since we have
$(d_G\omega)_A=d\omega-\omega(A^P)=-\omega(A^P)$, this
proves the exactness of (\ref{Cartan2.conformal}).
This also means there exists some
$\omega'\in\Omega^1(P)^G$ such that
\begin{eqnarray*}
-\mathrm{Tr}\,\nabla^t A^P
=(d_G\omega')_A
=d\omega'-\omega'(A^P)
\end{eqnarray*}
which is equivalent to two equations:
firstly $d\omega'=0$, so that $\omega'$
also determines a conformal structure
$\nu^{\omega'}$;
and secondly $L_1^{\omega'}A^P=0$.
\end{proof}
{\it Remarks.}
(i) The closedness of $2\chi^{2,2}+\chi^{4,0}$ is
equivalent to a pair of equations
\begin{eqnarray} \label{Cartan4.closed}
\phantom{WWWW}
d\chi^{2,2}_A=0,\qquad
d\chi^{4,0}_A=2\iota_{A^P}\chi^{2,2}_A,\qquad
\textrm{for }A\in\mathfrak{g}
\end{eqnarray}
which can also be verified directly from our
assumptions on $\nabla$ and $H$.
(ii) From now on we assume that the conformal
structure $\nu^\omega$ has been chosen such
that $L_1^\omega A^P=0$ for $A\in\mathfrak{g}$, and
write the associated Virasoro operators more
simply as $L_n$.
\vspace{0.08in}
{\it Preparation.}
The following sequence of results all concern
lifting the given map of Lie algebras
$\mathfrak{g}\rightarrow\mc{T}(P)$ to a linear map
of the form
\begin{eqnarray} \label{lift.CDO1}
\mathfrak{g}\,\rightarrow\,
\mc{D}^{\mathrm{ch}}(P)_1=\mc{T}(P)\oplus\Omega^1(P),\qquad
A\mapsto A^P+h^{2,1}_A
\end{eqnarray}
where $h^{2,1}:\mathfrak{g}\rightarrow\Omega^1(P)$ is
assumed to be $G$-equivariant.
In other words, $h^{2,1}$ is a Cartan cochain.
\begin{prop} \label{prop.lift.Lie}
The linear map (\ref{lift.CDO1}) induces a map of
Lie algebras as follows
\begin{eqnarray} \label{lift.Lie}
\mathfrak{g}\,\rightarrow\,\mathrm{Der}\,\mc{D}^{\mathrm{ch}}(P),\qquad
A\mapsto\big(A^P+h^{2,1}_A\big)_0
\end{eqnarray}
if and only if the closed $2$-form
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-invariant for
each $A\in\mathfrak{g}$.
\end{prop}
\begin{proof}
For $A,B\in\mathfrak{g}$, we have the following calculation
\begin{align*}
\big[\big(A^P+h^{2,1}_A\big)_0,\big(B^P+h^{2,1}_B\big)_0\big]
&=\Big([A^P,B^P]+\{A^P,B^P\}_\Omega+L_{A^P}h^{2,1}_B
-L_{B^P}h^{2,1}_A\Big)_0 \\
&=\Big([A,B]^P+h^{2,1}_{[A,B]}\Big)_0
+\Big(\iota_{B^P}\chi^{2,2}_A-L_{B^P}h^{2,1}_A\Big)_0
\end{align*}
using (\ref{freeVA.comm}), (\ref{Cartan4.VAoid}) and
the $G$-equivariance of $h^{2,1}$.
By Lemma \ref{lemma.1form.0mode}, the second term
vanishes if and only if
\begin{eqnarray*}
0=d\big(\iota_{B^P}\chi^{2,2}_A-L_{B^P}h^{2,1}_A\big)
=L_{B^P}\big(\chi^{2,2}_A-dh^{2,1}_A\big)
\end{eqnarray*}
where we have used (\ref{Cartan4.closed}).
This proves our claim.
\end{proof}
The map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(P)$ extends
in an obvious way to a map of extended Lie algebroids
$i:(\mathbb{C},0,\mathfrak{g})\rightarrow(C^\infty(P),\Omega^1(P),\mc{T}(P))$.
Now we would like to extend it further to a map of vertex
algebroids and hence a map of vertex algebras
(see Example \ref{VAoid.Lie} and \S\ref{VAoid.VA.mor}).
\begin{prop} \label{prop.lift.VA}
The linear map (\ref{lift.CDO1}) determines a map of
vertex algebroids as follows
\begin{eqnarray*}
(i,h^{2,1}):(\mathbb{C},0,\mathfrak{g},0,\lambda,0)\rightarrow
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
\bullet,\{\;\},\{\;\}_\Omega\big)
\end{eqnarray*}
and hence a map of vertex algebras
$V_\lambda(\mathfrak{g})\rightarrow\mc{D}^{\mathrm{ch}}(P)$
if and only if, for each $A\in\mathfrak{g}$, the closed
$2$-form $\chi^{2,2}_A-dh^{2,1}_A$ is $G$-horizontal
and the following equation holds:
\begin{eqnarray} \label{lambda.Cartan40}
\chi^{4,0}_A+2h^{2,1}_A(A^P)
=\lambda(A,A).
\end{eqnarray}
\end{prop}
\begin{proof}
Let $A,B\in\mathfrak{g}$.
According to Definition \ref{VAoid.mor}, the linear map
$h^{2,1}:\mathfrak{g}\rightarrow\Omega^1(P)$ defines a map
between the said vertex algebroids if and only if
\begin{align*}
\{A^P,B^P\}\;\;
&=\lambda(A,B)-h^{2,1}_A(B^P)-h^{2,1}_B(A^P) \\
\{A^P,B^P\}_\Omega
&=-L_{A^P}h^{2,1}_B+L_{B^P}h^{2,1}_A-dh^{2,1}_A(B^P)
+h^{2,1}_{[A,B]}
\end{align*}
By (\ref{Cartan4.VAoid}) the first equation is equivalent
to (\ref{lambda.Cartan40}).
By (\ref{Cartan4.VAoid}) again and the $G$-equivariance of
$h^{2,1}$, the second equation can be rewritten as
$\iota_{B^P}(\chi^{2,2}_A-dh^{2,1}_A)=0$.
This proves our claim.
In fact, the $G$-horizontality of $\chi^{2,2}_A-dh^{2,1}_A$
always implies that the left side of (\ref{lambda.Cartan40})
is locally constant, since
\begin{eqnarray*}
d\big(\chi^{4,0}_A+2\iota_{A^P}h^{2,1}_A\big)
=2\iota_{A^P}\chi^{2,2}_A-2\iota_{A^P}dh^{2,1}_A
+2L_{A^P}h^{2,1}_A
=2\iota_{A^P}(\chi^{2,2}_A-dh^{2,1}_A)
\end{eqnarray*}
by virtue of (\ref{Cartan4.closed}) and the $G$-equivariance
of $h^{2,1}$.
Therefore if $P$ is connected, the horizontal condition
guarantees that (\ref{lambda.Cartan40}) must hold for some
$\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
\end{proof}
\begin{prop} \label{prop.Gaction}
The map of Lie algebras (\ref{lift.Lie}) integrates into
a homomorphism $G\rightarrow\mathrm{Aut}\,\mc{D}^{\mathrm{ch}}(P)$
if and only if the closed $2$-form
$\chi^{2,2}_A-dh^{2,1}_A$ vanishes for each $A\in\mathfrak{g}$.
\end{prop}
\begin{proof}
First we compute the one-parameter subgroup of automorphisms
generated by the inner derivation $(A^P+h^{2,1}_A)_0$.
Since $\mc{D}^{\mathrm{ch}}(P)$ is freely generated by a vertex
algebroid, it suffices to compute the automorphisms
at weights $0$ and $1$.
For the following computations, keep (\ref{freeVA.comm})
in mind.
For $f\in C^\infty(P)$ and $\alpha\in\Omega^1(P)$ we
simply have
\begin{align}
\label{Gaction.fcn}
\big(A^P+h^{2,1}_A\big)_0^n f=(A^P)^n f
\qquad&\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}f=e^{tA}\cdot f \\
\label{Gaction.1form}
\big(A^P+h^{2,1}_A\big)_0^n\alpha=L_{A^P}^n\alpha\;\;
\qquad&\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}\alpha=e^{tA}\cdot\alpha
\end{align}
where $\cdot$ refers to the given $G$-actions.
For $X\in\mc{T}(P)$ we first have
\begin{align*}
\big(A^P+h^{2,1}_A\big)_0 X
=[A^P,X]+\{A^P,X\}_\Omega-\iota_X dh^{2,1}_A
=[A^P,X]+\iota_X(\chi^{2,2}_A-dh^{2,1}_A)
\end{align*}
by Lemma \ref{lemma.1form.0mode} and
(\ref{Cartan4.VAoid}).
Then it follows by induction and the $G$-equivariance
of $\chi^{2,2}$, $h^{2,1}$ that
\begin{align}
\big(A^P+h^{2,1}_A\big)_0^n X
&=L_{A^P}^n X
+nL_{A^P}^{n-1}\iota_X(\chi^{2,2}_A-dh^{2,1}_A),\qquad
n\geq 1 \nonumber\\
\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}X
&=e^{tA}\cdot X
+te^{tA}\cdot\iota_X(\chi^{2,2}_A-dh^{2,1}_A)
\label{Gaction.vf}
\end{align}
This finishes the computation of the automorphisms.
If $\chi^{2,2}_A-dh^{2,1}_A$ vanishes for all $A\in\mathfrak{g}$,
it follows from (\ref{Gaction.fcn})--(\ref{Gaction.vf})
that (\ref{lift.Lie}) integrates into the $G$-action described
in \S\ref{sec.inv.data}.
Conversely, assume that (\ref{lift.Lie}) integrates into
a $G$-action.
By (\ref{Gaction.vf}), $\chi^{2,2}_A-dh^{2,1}_A$ must vanish
whenever $e^A=1$.
Since every element of $G$ lies in a torus, the subset
$\{A\in\mathfrak{g}\,|\,e^A=1\}$ spans $\mathfrak{g}$, so that
$\chi^{2,2}_A-dh^{2,1}_A$ must in fact vanish for all
$A\in\mathfrak{g}$.
\end{proof}
{\it Remark.}
Notice that according to the above proof, whenever
(\ref{lift.Lie}) is integrable, the resulting $G$-action
on $\mc{D}^{\mathrm{ch}}(P)$ can only be the one described in
\S\ref{sec.inv.data}.
\begin{subsec}
{\bf Comparing the three conditions.}
Recall the conditions encountered respectively in
Propositions \ref{prop.lift.Lie}, \ref{prop.lift.VA} and
\ref{prop.Gaction}:
\begin{itemize}
\item[(i)\phantom{ii}]
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-invariant for $A\in\mathfrak{g}$
\vspace{-0.1in}
\item[(ii)\phantom{i}]
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-horizontal for $A\in\mathfrak{g}$
\vspace{-0.1in}
\item[(iii)]
$\chi^{2,2}_A-dh^{2,1}_A=0$ for $A\in\mathfrak{g}$
\end{itemize}
In general, (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i);
the second implication follows from (\ref{Cartan4.closed}).
From another point of view, a map of vertex algebras as in
Proposition \ref{prop.lift.VA} always induces a map of Lie
algebras as in Proposition \ref{prop.lift.Lie},
but the other implication may seem somewhat
surprising.
In the case $\mathfrak{g}$ is semisimple, so that
$[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, all three conditions are
equivalent.
\end{subsec}
Here is the main result of this section.
\begin{theorem} \label{thm.formalLG.action}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$
if and only if
\begin{eqnarray*}
8\pi^2 p_1(P)_G=\lambda(P).
\end{eqnarray*}
Moreover, this action is primary with respect to the chosen
conformal structure $\nu^\omega$.
\end{theorem}
\begin{proof}
Recall Definition \ref{formalLG.action} and
Example \ref{VAoid.Lie}.
First of all, an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
as described is determined by a linear map
$\mathfrak{g}\rightarrow\mc{D}^{\mathrm{ch}}(P)_1$ of the form
$A\mapsto A^P+h^{2,1}_A$, where
$h^{2,1}\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
By Propositions \ref{prop.lift.VA} and
\ref{prop.Gaction}, the precise conditions on
$h^{2,1}$ are
\begin{eqnarray} \label{formalLG.h}
\left\{\begin{array}{l}
\chi^{2,2}_A=dh^{2,1}_A \vspace{0.03in} \\
\chi^{4,0}_A=\lambda(A,A)-2h^{2,1}_A(A^P)
\end{array}\right.
\quad\iff\quad
2\chi^{2,2}+\chi^{4,0}=\lambda+2d_G h^{2,1}.
\end{eqnarray}
By Lemma \ref{lemma.Cartan.closed}a, this proves the
first claim.
The second claim is simply a restatement of Lemma
\ref{lemma.Cartan.closed}b and a subsequent remark
there.
\end{proof}
{\it Remark.}
In the sequel, $h^{2,1}$ will be called the
{\bf associated Cartan cochain} of the inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action.
\begin{example} \label{CDOG}
{\bf CDOs on a Lie group.}
Let $G$ be a compact simple Lie group.
In this discussion, $A,B$ always mean general elements
of $\mathfrak{g}=T_e G$.
The following notations will be used:
\vspace{-0.1in}
\begin{itemize}
\item[$\cdot$]
$A^\ell$ (resp.~$A^r$) is the left-invariant (resp.~right-invariant)
vector field on $G$ extending $A$;
\vspace{-0.1in}
\item[$\cdot$]
$\theta$ is the Maurer-Cartan form on $G$,
i.e.~$\theta\in\Omega^1(G)\otimes\mathfrak{g}$ such that
$\theta(A^\ell)=A$;
\vspace{-0.1in}
\item[$\cdot$]
$\lambda_{\t{\sf ad}}$ is the Killing form and
$\lambda_0=(2h^\vee)^{-1}\lambda_{\t{\sf ad}}$ the normalized Killing
form on $\mathfrak{g}$, where $h^\vee$ is the dual Coxeter number.
\vspace{-0.1in}
\end{itemize}
There is a canonical isomorphism $H^4(BG)\cong H^3(G)$,
where both spaces are one-dimensional.
The isomorphism is represented by
\begin{eqnarray*}
\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G\quad
\longmapsto\quad
-\frac{1}{6}\,\lambda(\theta\wedge[\theta\wedge\theta])
\in\Omega^3(G)
\end{eqnarray*}
and we will use $\lambda_0$, or the corresponding
$3$-form $H_0$, as the respective basis.
Now we construct a conformal algebra of CDOs on $G$.
Let $\nabla$ be the flat connection on $TG$ such that
$\nabla A^\ell=0$.
Given $s\in\mathbb{C}$, there is a closed $3$-form $sH_0$.
By Theorems \ref{thm.globalCDO}a and \ref{thm.globalCDO.iso}a,
$\nabla$ and $sH_0$ together define an algebra of CDOs
$\mc{D}^{\mathrm{ch}}_s(G)=\mc{D}^{\mathrm{ch}}_{\nabla,sH_0}(G)$, and every algebra of CDOs on $G$
is up to isomorphism of this form.
Notice that the vertex algebroid structure of
$\mc{D}^{\mathrm{ch}}_s(G)$ includes such special cases as
\begin{align}
\label{CDOG.VAoidl}
&\{A^\ell,A^\ell\}=-2h^\vee\lambda_0(A,A),
&&\hspace{-1in}
\{A^\ell,-\}_\Omega
=\frac{s}{4}\,\lambda_0(A,[\theta\wedge\theta]) \\
\label{CDOG.VAoidr}
&\{A^r,-\}=0,
&&\hspace{-1in}
\{A^r,-\}_\Omega
=\frac{s}{4}\,\lambda_0(\theta(A^r),[\theta\wedge\theta])
\end{align}
By Theorem \ref{thm.globalCDO}b, the trivial $1$-form defines
a conformal structure $\nu$ of central charge $2\dim\mathfrak{g}$.
If $J_1,J_2,\cdots$ are a basis of $\mathfrak{g}$ and
$\theta^1,\theta^2,\cdots$ the respective components of
$\theta$, we can write $\nu=J^\ell_{a,-1}\theta^a$.
Also, $\nu$ has the property that $L_1 A^\ell=L_1 A^r=0$.
First consider the action of $G$ on itself by right
multiplication.
The induced map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(G)$
sends $A$ to $A^\ell$.
Both $\nabla$ and $H_0$ are invariant under this action.
Define $h^\ell\in(\mathfrak{g}^\vee\otimes\Omega^1(G))^G$ by
$h^\ell_A=-\frac{s}{2}\lambda_0(A,\theta)$.
In view of (\ref{CDOG.VAoidl}), $h^\ell$ satisfies
(\ref{formalLG.h}) with $\lambda=(-s-2h^\vee)\lambda_0$.
Therefore $h^\ell$ is the Cartan cochain associated to
an inner $(\hat{\mathfrak{g}},G)$-action
\begin{eqnarray} \label{CDOG.formalLGl}
V_{-s-2h^\vee}(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_s(G)\,;
\end{eqnarray}
denote its image by $V_{-s-2h^\vee}(\mathfrak{g})^\ell$.
This action is primary with respect to $\nu$.
Now consider also the action of $G$ on itself by inverse
left multiplication, i.e.~each $g\in G$ acts via left
multiplication by $g^{-1}$.
The induced map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(G)$
sends $A$ to $-A^r$.
Since left- and right-invariant vector fields commute,
$\nabla$ and $H_0$ are invariant under this action as well.
Define $h^r\in(\mathfrak{g}^\vee\otimes\Omega^1(G))^G$
by $h^r_A=\frac{s}{2}\lambda_0(\theta(A^r),\theta)$.
In view of (\ref{CDOG.VAoidr}), $h^r$ satisfies
(\ref{formalLG.h}) with $\lambda=s\lambda_0$.
\footnote{
Notice that $(dh^r_A)(B^r,C^r)=\frac{s}{2}\lambda_0(A,[B,C])
=\chi^{2,2}_A(B^r,C^r)$ for $B,C\in\mathfrak{g}$, so that indeed
$dh^r_A=\chi^{2,2}_A$.
}
Therefore $h^r$ is the Cartan cochain associated to
an inner $(\hat{\mathfrak{g}},G)$-action
\begin{eqnarray} \label{CDOG.formalLGr}
V_s(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_s(G)\,;
\end{eqnarray}
denote its image by $V_s(\mathfrak{g})^r$.
This action is also primary with respect to $\nu$.
\end{example}
{\it Remark.}
For both $G$-actions, since $H^*_G(G)=H^*(\textrm{pt})$,
the condition in Theorem \ref{thm.formalLG.action}
is trivial for any $\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$,
so that in fact there are many other inner
$(\hat{\mathfrak{g}},G)$-actions on $\mc{D}^{\mathrm{ch}}_s(G)$.
\newpage
\setcounter{equation}{0}
\section{CDOs on Principal Bundles}
\label{sec.CDOP}
In this section, the object of interest is an algebra
of CDOs $\mc{D}^{\mathrm{ch}}(P)$ on the total space of a smooth
principal bundle $P\rightarrow M$, equipped
with a projective ``formal loop group action'' in
the sense of \S\ref{sec.LGaction}.
The first goal is to understand the subalgebra of
$\mc{D}^{\mathrm{ch}}(P)$ invariant under the said action.
In subsequent sections we will construct and study
modules over this invariant subalgebra.
The second (and more technical) goal is to give
an alternative description of $\mc{D}^{\mathrm{ch}}(P)$ in the case
$P\rightarrow M$ is a principal frame bundle.
This vertex algebra will play a central role in
the rest of the paper:
it is not merely a particular algebra of CDOs, but
should be regarded as a more fundamental object from
which all algebras of CDOs (as well as other related
constructions) derive.
\begin{subsec} \label{sec.prin.bdl}
{\bf Setting:~principal bundle.}
Let $G$ be again a compact connected Lie group and
$\pi:P\rightarrow M$ a smooth principal $G$-bundle.
Identify $H^*_G(P)$ with $H^*(M)$ via $\pi^*$.
Choose a connection $\Theta$ on $\pi$,
i.e. some $\Theta\in(\Omega^1(P)\otimes\mathfrak{g})^G$
such that $\Theta(A^P)=A$ for $A\in\mathfrak{g}$;
then its curvature is
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Keep in mind that $\Theta$ is equivalent to
a $G$-equivariant vector bundle decomposition
$TP=T_h P\oplus T_v P$, with
$T_h P=\ker\Theta$ and $T_v P=\ker\pi_*$;
and $\Omega$ measures the non-integrability of
the subbundle $T_h P$.
Given $X\in\mc{T}(M)$, denote its horizontal lift by
$\widetilde X\in\mc{T}_h(P)^G$.
Let us extend the notation $(\cdot)^P$
(see \S\ref{sec.Gmfld}) $C^\infty(P)$-linearly to
represent the isomorphism
$C^\infty(P)\otimes\mathfrak{g}\cong\mc{T}_v(P)$.
For $X,Y\in\mc{T}(M)$ notice that
\begin{eqnarray} \label{horlift.nonint}
[\widetilde X,\widetilde Y]-\widetilde{[X,Y]}=-\Omega(\widetilde X,\widetilde Y)^P.
\end{eqnarray}
Also recall the notations introduced in
\S\ref{sec.Cartan}.
Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)=\mc{D}^{\mathrm{ch}}_{\nabla,H}(P)$
defined as in \S\ref{sec.inv.data}.
Let $\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Recall from Definition \ref{formalLG.action} and the proof
of Theorem \ref{thm.formalLG.action} the meaning of an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ and its
associated Cartan cochain.
It will be understood without further comment that any inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ considered
below extends the given $G$-action on $\mc{D}^{\mathrm{ch}}(P)_0=C^\infty(P)$.
\end{subsec}
{\it Preparation.}
Given a representation $\rho:G\rightarrow GL(V)$,
let us denote the induced map of Lie algebras also by
$\rho$ and define an invariant symmetric bilinear form
on $\mathfrak{g}$ by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, $\lambda_{\t{\sf ad}}$ is the Killing form.
If we regard $V$ as a $G$-equivariant vector bundle over
a point and identify $H^4_G(\textrm{pt})$ with
$(\textrm{Sym}^2\mathfrak{g}^\vee)^G$, then
$-8\pi^2 p_1(V)_G=\lambda_\rho$.
\begin{lemma} \label{lemma.Gp1.p1}
$8\pi^2 p_1(P)_G=8\pi^2 p_1(M)-\lambda_{\t{\sf ad}}(P)$.
\end{lemma}
\begin{proof}
Consider the $G$-equivariant decomposition
$TP=T_h P\oplus T_v P$.
On one hand, since $T_h P\cong\pi^*TM$ and $\pi^*$
identifies $H^*_G(P)$ with $H^*(M)$, we have
$p_1(T_h P)_G=p_1(TM)$.
On the other hand, $T_v P\cong P\times\mathfrak{g}$
with $G$ acting on $\mathfrak{g}$ in the adjoint
representation, so that
$-8\pi^2 p_1(T_v P)=\lambda_{\t{\sf ad}}(P)$.
This proves the lemma.
\end{proof}
Hence Theorem \ref{thm.formalLG.action} specializes
to our current setting as follows.
\begin{corollary} \label{cor.formalLG.P}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ if
and only if
\begin{eqnarray} \label{formalLG.condition2}
\qquad
8\pi^2 p_1(M)=(\lambda+\lambda_{\t{\sf ad}})(P).
\quad\qedsymbol
\end{eqnarray}
\end{corollary}
{\it Remark.}
For convenience, we will refer to $\mc{D}^{\mathrm{ch}}(P)$ as
a {\bf principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra}
when it is equipped with an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action, or
a {\bf principal $(\hat{\mathfrak{g}},G)$-algebra} if we
do not wish to specify $\lambda$.
\begin{subsec} \label{sec.invCDO}
{\bf The invariant subalgebra.}
Suppose $\mc{D}^{\mathrm{ch}}(P)$ is a principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra, i.e.~there is
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$ defined by
some associated Cartan cochain
$h\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
Consider the centralizer subalgebra \cite{Kac,FB-Z}
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}
:=C\big(\mc{D}^{\mathrm{ch}}(P),V_\lambda(\mathfrak{g})\big)
\subset\mc{D}^{\mathrm{ch}}(P).
\end{eqnarray*}
This is the subalgebra whose fields are
$\hat{\mathfrak{g}}_\lambda$-invariant.
The weight-zero component consists of $f\in C^\infty(P)$
such that $(A^P+h_A)_0 f=A^P f=0$ for $A\in\mathfrak{g}$,
namely
\begin{eqnarray} \label{invCDO.0}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_0
=C^\infty(P)^G
=\pi^*C^\infty(M).
\end{eqnarray}
On the other hand, the weight-one component consists of
$\alpha+X\in\Omega^1(P)\oplus\mc{T}(P)$ such that
\begin{align*}
\left\{\begin{array}{l}
(A^P+h_A)_0(\alpha+X)=L_{A^P}\alpha+[A^P,X]=0 \vspace{0.03in} \\
(A^P+h_A)_1(\alpha+X)=\alpha(A^P)+\{A^P,X\}+h_A(X)=0
\end{array}\right.\qquad
\textrm{for }A\in\mathfrak{g}
\end{align*}
where we have used (\ref{freeVA.comm}), Lemma
\ref{lemma.1form.0mode} and (\ref{formalLG.h}).
It follows that there is a pullback square
\begin{eqnarray*}
\xymatrix{
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_1\ar[r]\ar[d] &
\mc{T}(P)^G\ar[d]^q \\
\Omega^1(P)^G\ar[r]^{p\phantom{WW}} &
(\mathfrak{g}^\vee\otimes C^\infty(P))^G
}
\end{eqnarray*}
where $p$, $q$ are suitable adjoints to
$(\alpha,A)\mapsto-\alpha(A^P)$ and
$(X,A)\mapsto\{A^P,X\}+h_A(X)$ respectively.
Observe that
(i) $p$ is surjective, as contraction with
$\Theta\in(\Omega^1(P)\otimes\mathfrak{g})^G$ furnishes a right inverse,
and
(ii) $\ker p$ is the space of the $G$-invariant horizontal
(i.e.~basic) $1$-forms on $P$, or in other words
$\pi^*\Omega^1(M)$.
This implies the following short exact sequence
\begin{eqnarray}\label{invCDO.1}
\xymatrix{
0\ar[r] &
\pi^*\Omega^1(M)\ar[r] &
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_1\ar[r] &
\mc{T}(P)^G\ar[r] & 0
}
\end{eqnarray}
In view of (\ref{invCDO.0}) and (\ref{invCDO.1}),
the vertex algebroid associated to
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ (see \S\ref{VA.VAoid}) is
of the form
\begin{eqnarray} \label{invCDO.VAoid}
\big(C^\infty(M),\Omega^1(M),\mc{T}(P)^G,\cdots\big).
\end{eqnarray}
Since $\mc{D}^{\mathrm{ch}}(P)$ is freely generated by its associated
vertex algebroid (see \S\ref{VAoid.VA}),
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ must contain at least
a subalgebra that is freely generated by
(\ref{invCDO.VAoid}).
However, the author does not know if
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ may or may not contain more
elements (see also Example \ref{invCDOG}).
\end{subsec}
{\it Remarks.}
(i) By the above discussion, any module over
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ contains in its lowest
weight a module over the Atiyah algebroid
$(C^\infty(M),\mc{T}(P)^G)$, which can be e.g.~the space
of sections of a vector bundle with connection
associated to $(P\rightarrow M,\Theta)$.
\footnote{ \label{invVF}
Given a $G$-representation $W$, the space of
sections of the associated vector bundle is
$(C^\infty(P)\otimes W)^G$, which is naturally
a module over $(C^\infty(M),\mc{T}(P)^G)$.
In fact, the action of $\widetilde X\in\mc{T}_h(P)^G$ can be
identified with the covariant derivative along
the corresponding $X\in\mc{T}(M)$, and the action of
$\mc{T}_v(P)^G\cong(C^\infty(P)\otimes\mathfrak{g})^G$ is
induced by the representation
$\mathfrak{g}\rightarrow\textrm{End}\,W$.
}
In subsequent sections we will construct and study
such modules over $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$.
(ii) It is noteworthy that $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$
is different from an algebra of CDOs on the base
manifold $M$ --- compare
(\ref{invCDO.0})--(\ref{invCDO.1}) and
(\ref{CDO.wt01}) --- and by comparison has more
interesting modules.
In \S\ref{CDOP.CDOM} we will see that to recover
an algebra of CDOs on $M$ from $\mc{D}^{\mathrm{ch}}(P)$ requires
the use of semi-infinite cohomology (as well as
a suitable choice of $P\rightarrow M$ plus other
geometric data).
\begin{example} \label{invCDOG}
{\bf The invariant subalgebra on a Lie group.}
This is a continuation of Example \ref{CDOG} and
notations will not be explained again.
Recall the subalgebras
$V_{-s-2h^\vee}(\mathfrak{g})^\ell,\;V_s(\mathfrak{g})^r\subset\mc{D}^{\mathrm{ch}}_s(G)$
generated respectively by $A^\ell+h^\ell_A$ and
$A^r+h^r_A$ for $A\in\mathfrak{g}$.
Consider the centralizer subalgebra
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}
=C\Big(\mc{D}^{\mathrm{ch}}_s(G),V_{-s-2h^\vee}(\mathfrak{g})^\ell\Big).
\end{eqnarray*}
By the discussion in \S\ref{sec.invCDO} and
(\ref{CDOG.VAoidr}),
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_0=\mathbb{C}$ and
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_1$ consists of elements
of the form $A^r+\alpha$ where $A\in\mathfrak{g}$ and
$\alpha\in\Omega^1(G)$ satisfies
\begin{eqnarray*}
\phantom{WWW}
\alpha(B^\ell)
=-\{B^\ell,A^r\}-h^\ell_B(A^r)
=\frac{s}{2}\,\lambda_0(B,\theta(A^r))
=h^r_A(B^\ell)\qquad
\textrm{for }B\in\mathfrak{g}
\end{eqnarray*}
i.e.~$\alpha=h^r_A$.
In other words, we have
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_i=V_s(\mathfrak{g})^r_i$
for $i=0,1$.
Since $V_s(\mathfrak{g})^r$ is freely generated by
its two lowest weights, there is an inclusion
\begin{eqnarray} \label{invCDOG.subalg}
V_s(\mathfrak{g})^r\subset\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}.
\end{eqnarray}
This means the subalgebras
$V_{-s-2h^\vee}(\mathfrak{g})^\ell$ and
$V_s(\mathfrak{g})^r$ commute with each other.
If $s\neq-h^\vee$, then both
$V_{-s-2h^\vee}(\mathfrak{g})^\ell$ and $V_s(\mathfrak{g})^r$
admit Sugawara conformal vectors \cite{Kac,FB-Z}
\begin{eqnarray*}
&\displaystyle
\nu^\ell\in V_{-s-2h^\vee}(\mathfrak{g})^\ell\;
\textrm{ of central charge }
\left(\frac{s+2h^\vee}{s+h^\vee}\right)\dim\mathfrak{g},& \\
&\displaystyle
\nu^r\in V_s(\mathfrak{g})^r\;
\textrm{ of central charge }
\left(\frac{s}{s+h^\vee}\right)\dim\mathfrak{g}.\quad&
\end{eqnarray*}
On the other hand, since $\mc{D}^{\mathrm{ch}}_s(G)$ has a conformal
vector $\nu$ of central charge $2\dim\mathfrak{g}$ and
the subalgebra $V_{-s-2h^\vee}(\mathfrak{g})^\ell$ is
generated by elements that are primary with respect
to $\nu$, the coset construction yields a conformal
vector \cite{Kac,FB-Z}
\begin{eqnarray*}
\nu-\nu^\ell\in\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}\;
\textrm{ of central charge }
\left(\frac{s}{s+h^\vee}\right)\dim\mathfrak{g}.
\end{eqnarray*}
In fact, a computation (which we omit) shows that
$\nu-\nu^\ell=\nu^r$, i.e.~the Sugawara vector of
$V_s(\mathfrak{g})^r$ is also conformal for
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}$.
This seems to suggest that the inclusion
(\ref{invCDOG.subalg}) is in fact surjective, at least
for generic values of $s$, but the author has yet to
find either a proof or a counterexample.
\footnote{
Notice that Lemma \ref{lemma.conf.gen} does not apply
here.
}
\end{example}
\begin{subsec} \label{sec.prin.bdl2}
{\bf Setting refined:~principal frame bundle.}
In addition to the data described in
\S\ref{sec.prin.bdl}, suppose there is also
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$.
This induces a Riemannian metric on $M$ and
an orthogonal connection on $TM$, which (for
simplicity) is assumed to be torsion-free.
Let $[p,v]\in TM$ denote the coset of
$(p,v)\in P\times\mathbb{R}^d$.
For $i=1,\ldots,d$, let $\mathbf{e}_i\in\mathbb{R}^d$ be
the standard basis vectors and $\tau_i\in\mc{T}_h(P)$
the tautological vector fields defined by
\begin{eqnarray*}
\tau_i|_p:=\textrm{horizontal lift of }[p,\mathbf{e}_i]
\end{eqnarray*}
which constitute a framing of $T_h P$.
For $A\in\mathfrak{g}$ and $i,j=1,\ldots,d$, we have
the Lie brackets
\begin{eqnarray} \label{tau.brackets}
[A^P,\tau_i]=\rho(A)_{ji}\tau_j,
\qquad
[\tau_i,\tau_j]=-\Omega(\tau_i,\tau_j)^P
\end{eqnarray}
where $\rho$ here denotes the induced map of Lie
algebras $\mathfrak{g}\rightarrow\mathfrak{so}_d$.
\footnote{
$[\tau_i,\tau_j]$ has no horizontal component precisely
because of our assumption that the induced connection
on $TM$ is torsion-free.
Moreover, the Jacobi identity for any
$\tau_i,\tau_j,\tau_k$ is the same as the two Bianchi
identities combined.
}
Suppose the conformal algebra of CDOs
$\mc{D}^{\mathrm{ch}}(P)=\mc{D}^{\mathrm{ch}}_{\nabla',H'}(P)$ is defined as in
\S\ref{sec.inv.data} using the following data.
(Notice the various change of notations here.)
Let $\nabla'$ be the $G$-invariant \emph{flat}
connection on $TP$ with respect to which all
$A^P$ and $\tau_i$ are parallel, and
$H'$ be a $G$-invariant \emph{closed} $3$-form
on $P$;
together they determine the vertex algebroid
\begin{align} \label{CDOP.VAoid}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
\bullet',\{\;\}',\{\;\}'_\Omega\big)
\end{align}
by which $\mc{D}^{\mathrm{ch}}(P)$ is freely generated
(see Theorem \ref{thm.globalCDO}a).
Also, the trivial $1$-form determines a conformal
vector of $\mc{D}^{\mathrm{ch}}(P)$, namely
\begin{eqnarray} \label{CDOP.conformal}
\nu'=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
\end{eqnarray}
where $t_1,t_2,\cdots$ are a basis of $\mathfrak{g}$;
$\Theta^1,\Theta^2,\cdots$ the corresponding
components of $\Theta$, so that
$\Theta=\Theta^a\otimes t_a$;
$\tau_1,\cdots,\tau_d$ the vector fields defined
above; and
$\tau^1,\cdots,\tau^d$ the $1$-forms given
by $\tau^i(\tau_j)=\delta^i_j$ and $\tau^i(A^P)=0$
(see Theorem \ref{thm.globalCDO}b).
In the rest of this section, we will provide a more
detailed description of this conformal vertex algebra
in the presence of an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action.
\end{subsec}
According to the axioms of a vertex algebroid
(Definition \ref{VAoid}), the entire structure of
(\ref{CDOP.VAoid}) is determined by the special
cases appearing in the following statement.
\begin{lemma} \label{lemma.CDOP.VAoid}
For $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$ and
$i,j=1,\ldots,d$, we have
\begin{itemize}
\item[$\cdot$]
$A^P\bullet'f=\tau_i\bullet'f=0$
\vspace{-0.05in}
\item[$\cdot$]
$\{A^P,B^P\}'=-(\lambda_{\t{\sf ad}}+\lambda_\rho)(A,B)$,\;\;
$\{A^P,\tau_i\}'=0$,\;\;
$\{\tau_i,\tau_j\}'=2\mathrm{Ric}_{ij}$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}'_\Omega
=\frac{1}{2}\iota_{A^P}\iota_{B^P}H'$,\;\;
$\{A^P,\tau_i\}'_\Omega
=\frac{1}{2}\iota_{A^P}\iota_{\tau_i}H'$,\;\;
$\{\tau_i,\tau_j\}'_\Omega=d\,\mathrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H'$
\end{itemize}
where $\mathrm{Ric}_{ij}=\rho(\Omega(\tau_i,\tau_k))_{jk}
\in C^\infty(P)$.
Moreover, we also have $\nu'_1 A^P=\nu'_1\tau_i=0$.
\end{lemma}
\begin{proof}
Recall (\ref{CDO.VAoid}) and (\ref{CDO.L1}).
For $\bullet$ the claim is clear.
Using (\ref{tau.brackets}) we compute the
following operators
\begin{align*}
&\nabla^{\prime\,t} A^P:\left\{
\begin{array}{l}
B^P\mapsto\,-[A,B]^P \vspace{0.01in} \\
\tau_i\;\;\mapsto\,-\rho(A)_{ji}\tau_j
\end{array}\right.,\hspace{-0.35in}
&&\nabla^{\prime\,t}\tau_i:\left\{
\begin{array}{l}
A^P\mapsto\,\rho(A)_{ji}\tau_j \vspace{0.01in} \\
\tau_j\;\;\mapsto\,\Omega(\tau_i,\tau_j)^P
\end{array}\right. \\
\Rightarrow\qquad\quad
&\nabla'(\nabla^{\prime\,t} A^P)=0,
&&\nabla'(\nabla^{\prime\,t}\tau_i):\left\{
\begin{array}{l}
A^P\mapsto\,0 \vspace{0.01in} \\
\tau_j\;\;\mapsto\,
(d\Omega(\tau_i,\tau_j))^P
\end{array}\right.
\end{align*}
The rest of the lemma now follows easily from
these calculations and the symmetries of the
Riemannian curvature tensor $\rho(\Omega)$.
\end{proof}
Theorem \ref{thm.formalLG.action} specializes
to our current setting as follows.
\begin{corollary} \label{cor.formalLG.P2}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$
if and only if
\begin{eqnarray*}
(\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho)(P)=0.
\end{eqnarray*}
Moreover, this action is primary with respect to
the conformal vector $\nu'$.
\end{corollary}
\begin{proof}
By assumption, $P\rightarrow M$ is a lifting
of the special orthogonal frame bundle
$F_{SO}(TM)\rightarrow M$ along
$\rho:G\rightarrow SO_d$.
Hence we have a commutative diagram
\begin{align*}
\xymatrix{
&
H^4(BSO_d)\ar[r]^{\;\;\rho^*}\ar[d] &
H^4(BG)\ar[d] \\
H^4(M)\ar[r]^{\cong\phantom{WWW}} &
H^4_{SO_d}(F_{SO}(TM))\ar[r]^{\phantom{WW}\cong} &
H^4_G(P)
}
\end{align*}
Also recall the identifications like
$H^4(BG)\cong(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Since $-8\pi^2 p_1\in H^4(BSO_d)$ can be identified
with the bilinear form $(A,B)\mapsto\mathrm{Tr}\, AB$,
its image under $\rho^*$ is $\lambda_\rho$ and
so by the diagram above we have
$-8\pi^2p_1(M)=\lambda_\rho(P)$.
Now the first claim is just a special case of
Corollary \ref{cor.formalLG.P}.
The other claim is true because $\nu'_1 A^P=0$
for $A\in\mathfrak{g}$ by Lemma \ref{lemma.CDOP.VAoid}.
\end{proof}
{\it Remark.}
In the sequel we will write
$\lambda^*=\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho$.
\begin{subsec} \label{CDOP.FLGaction}
{\bf Inner $(\hat{\mathfrak{g}}_\lambda,G)$-action on CDOs.}
Suppose $\mc{D}^{\mathrm{ch}}(P)$ is a principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra, i.e.~there is
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$, defined
by some associated Cartan cochain
$h\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
Notice that the Cartan cochains in
(\ref{Cartan4.VAoid}) may now be written as
$\chi^{2,2}=\frac{1}{2}d_G H'$ and
$\chi^{4,0}=-\lambda_{\t{\sf ad}}-\lambda_\rho$ by Lemma
\ref{lemma.CDOP.VAoid}.
Hence the condition (\ref{formalLG.h}) on $h$
is now equivalent to the equation
\begin{align} \label{CDOP.h}
d_G(H'-2h)=\lambda^*.
\end{align}
This of course agrees with Corollary
\ref{cor.formalLG.P2}.
The purpose of the next two lemmas is to
modify the description of the vertex algebra
$\mc{D}^{\mathrm{ch}}(P)$ in such a way that the inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action becomes
manifest and the data $(H',h)$ involved are
replaced by a basic $3$-form that trivializes
$\lambda^*(\Omega\wedge\Omega)$.
\end{subsec}
\begin{lemma} \label{lemma.basic.3form}
There is a natural bijection between the
following two sets of data:
(i) $(H',h)\in\Omega^3(P)^G\oplus
(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$ such
that $d_G(H'-2h)=\lambda^*$, and
(ii) $(H,\beta)\in\pi^*\Omega^3(M)\oplus\Omega^2(P)^G$
such that $dH=\lambda^*(\Omega\wedge\Omega)$ and
$\beta|_{T_h P}=0$.
\end{lemma}
\begin{proof}
Once the maps in both directions are described below,
it will be clear they are inverses to each other.
Let $\mathrm{CS}_{\lambda^*}(\Theta)
=\lambda^*(\Theta\wedge\Omega)
-\frac{1}{6}\lambda^*(\Theta\wedge[\Theta\wedge\Theta])$.
Notice that as Cartan cochains
$\mathrm{CS}_{\lambda^*}(\Theta)\in\Omega^3(P)^G$ and
$\lambda^*(-,\Theta)\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$
have the following differentials
\begin{align} \label{dCS}
d_G\,\mathrm{CS}_{\lambda^*}(\Theta)
=\lambda^*(\Omega\wedge\Omega)
-\lambda^*(-,d\Theta),\qquad
d_G\,\lambda^*(-,\Theta)
=\lambda^*(-,d\Theta)-\lambda^*.
\end{align}
{\it From (i) to (ii).}
Let $(H',h)$ be a pair as in (i).
First define a $G$-invariant $2$-form on $P$ by
\begin{align} \label{beta}
\beta(A^P,B^P)=h_A(B^P)-h_B(A^P),\qquad
\beta(A^P,\tau_i)=2h_A(\tau_i),\qquad
\beta(\tau_i,\tau_j)=0
\end{align}
for $A,B\in\mathfrak{g}$ and $i,j=1,\ldots,d$.
Indeed, the $G$-equivariance of $h$ together with
(\ref{tau.brackets}) imply the $G$-invariance of $\beta$.
Observe that the component of $d_G(H'-2h)=\lambda^*$
in $(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P))^G$ allows
us to write
\begin{align} \label{iA.beta}
\iota_{A^P}\beta=2h_A-\lambda^*(A,\Theta).
\end{align}
Then define a $G$-invariant $3$-form on $P$ by the first
expression below
\begin{align}
\label{basic.3form}
H=H'+\mathrm{CS}_{\lambda^*}(\Theta)-d\beta
=H'+\mathrm{CS}_{\lambda^*}(\Theta)
-2h+\lambda^*(-,\Theta)-d_G\beta
\end{align}
but we also have the second expression thanks to
(\ref{iA.beta}).
It follows from $d_G(H'-2h)=\lambda^*$ and
(\ref{dCS}) that $d_G H=\lambda^*(\Omega\wedge\Omega)$,
or equivalently, $H$ is horizontal (hence basic)
and satisfies $dH=\lambda^*(\Omega\wedge\Omega)$.
{\it From (ii) to (i).}
Use (\ref{basic.3form}) again:
given $(H,\beta)$ as in (ii), define the desired pair
$(H',h)$ by
\begin{align*}
H'-2h
=H-\mathrm{CS}_{\lambda^*}(\Theta)-\lambda^*(-,\Theta)
+d_G\beta.
\end{align*}
Indeed, it follows from
$d_G H=\lambda^*(\Omega\wedge\Omega)$ and (\ref{dCS})
that $d_G(H'-2h)=\lambda^*$.
\end{proof}
{\it Preparation.}
This continues with the discussion in
\S\ref{CDOP.FLGaction}.
Since the data $(H',h)$ in the definition of $\mc{D}^{\mathrm{ch}}(P)$
as a principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra
satisfy (\ref{CDOP.h}), by Lemma \ref{lemma.basic.3form}
they give rise to a $G$-invariant $2$-form $\beta$ and
a basic $3$-form $H$ satisfying
$dH=\lambda^*(\Omega\wedge\Omega)$.
Let $\Delta:\mc{T}(P)\rightarrow\Omega^1(P)$ be the
$C^\infty(P)$-linear map with
$\Delta(A^P)=-h_A$ for $A\in\mathfrak{g}$ and
$\Delta(\tau_i)=-\frac{1}{2}\iota_{\tau_i}\beta$
for $i=1,\ldots,d$.
By Lemma \ref{lemma.newVAoid}, $\Delta$ determines
an isomorphism $(\textrm{id},\Delta)$ from the vertex
algebroid (\ref{CDOP.VAoid}) to a new vertex
algebroid
\begin{align} \label{CDOP.VAoidnew}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
\bullet,\{\;\},\{\;\}_\Omega\big).
\end{align}
This in turn induces an isomorphism from $\mc{D}^{\mathrm{ch}}(P)$
to the vertex algebra freely generated by
(\ref{CDOP.VAoidnew}), which we denote by
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
By composition, there is an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$
whose associated Cartan cochain is trivial,
i.e.~it is determined by the map
$\mathfrak{g}\hookrightarrow\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ sending
$A$ simply to $A^P$.
\vspace{0.08in}
According to the axioms of a vertex algebroid
(Definition \ref{VAoid}), the entire structure of
(\ref{CDOP.VAoidnew}) is determined by the
following special cases.
\begin{lemma} \label{lemma.CDOP.VAoidnew}
For $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$ and
$i,j=1,\ldots,d$, we have
\begin{itemize}
\item[$\cdot$]
$A^P\bullet f=\tau_i\bullet f=0$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}=\lambda(A,B)$,\;\;
$\{A^P,\tau_i\}=0$,\;\;
$\{\tau_i,\tau_j\}=2\mathrm{Ric}_{ij}$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}_\Omega=\{A^P,\tau_i\}_\Omega=0$,\;\;
$\{\tau_i,\tau_j\}_\Omega=d\,\mathrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
+\lambda^*(\Omega(\tau_i,\tau_j),\Theta)$
\end{itemize}
\end{lemma}
\begin{proof}
Recall Definition \ref{VAoid.mor} for the meaning of
$(\textrm{id},\Delta):(\ref{CDOP.VAoid})\rightarrow(\ref{CDOP.VAoidnew})$
as well as Lemma \ref{lemma.CDOP.VAoid} for the
structure maps of (\ref{CDOP.VAoid}).
Firstly, the claim for $\bullet$ follows from the
$C^\infty(P)$-linearity of $\Delta$.
Consider again the map of vertex algebroid $(i,h)$ in
Proposition \ref{prop.lift.VA}.
Because of the composition
\begin{align*}
(\textrm{id},\Delta)\circ(i,h)=(i,0):
(\mathbb{C},0,\mathfrak{g},0,\lambda,0)\rightarrow
(\ref{CDOP.VAoidnew})
\end{align*}
it is clear that $\{A^P,B^P\}=\lambda(A,B)$ and
$\{A^P,B^P\}_\Omega=0$.
This is equivalent to the comment above concerning
the inner $(\hat{\mathfrak{g}}_\lambda,G)$-action on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
The other values of $\{\;\}$ are computed as follows:
\begin{align*}
&\textstyle
\{A^P,\tau_i\}
=\{A^P,\tau_i\}'+h_A(\tau_i)+\frac{1}{2}\beta(\tau_i,A^P)
=0 \\
&\textstyle
\{\tau_i,\tau_j\}
=\{\tau_i,\tau_j\}'+\frac{1}{2}\beta(\tau_i,\tau_j)
+\frac{1}{2}\beta(\tau_j,\tau_i)
=2\textrm{Ric}_{ij}
\end{align*}
using the definition of $\beta$ in (\ref{beta}).
The other values of $\{\;\}_\Omega$ are computed as
follows:
\begin{align*}
&\textstyle
\{A^P,\tau_i\}_\Omega
=\{A^P,\tau_i\}'_\Omega
+\frac{1}{2}L_{A^P}\iota_{\tau_i}\beta
-L_{\tau_i}h_A
+dh_A(\tau_i)
-\frac{1}{2}\iota_{[A^P,\tau_i]}\beta\\
&\textstyle
\phantom{\{A^P,\tau_i\}'_\Omega}
=\frac{1}{2}\iota_{A^P}\iota_{\tau_i}H'
-\iota_{\tau_i}dh_A=0 \\
&\textstyle
\{\tau_i,\tau_j\}_\Omega
=\{\tau_i,\tau_j\}'_\Omega
+\frac{1}{2}L_{\tau_i}\iota_{\tau_j}\beta
-\frac{1}{2}L_{\tau_j}\iota_{\tau_i}\beta
+\frac{1}{2}d\beta(\tau_i,\tau_j)
+h_{\Omega(\tau_i,\tau_j)}\\
&\textstyle
\phantom{\{\tau_i,\tau_j\}'_\Omega}
=d\,\textrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H'
-\frac{1}{2}\iota_{\Omega(\tau_i,\tau_j)}\beta
-\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}d\beta
+\frac{1}{2}\iota_{\Omega(\tau_i,\tau_j)}\beta
+\frac{1}{2}\lambda^*(\Omega(\tau_i,\tau_j),\Theta) \\
&\textstyle
\phantom{\{\tau_i,\tau_j\}'_\Omega}
=d\,\textrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
+\lambda^*(\Omega(\tau_i,\tau_j),\Theta)
\end{align*}
using, in order:
the $G$-invariance of $\beta$;
the component of (\ref{CDOP.h}) in
$(\mathfrak{g}^\vee\otimes\Omega^2(P))^G$;
the identity
\begin{eqnarray*}
L_{\tau_i}\iota_{\tau_j}
-L_{\tau_j}\iota_{\tau_i}
+d\iota_{\tau_j}\iota_{\tau_i}
=L_{\tau_i}\iota_{\tau_j}
-\iota_{\tau_j}d\iota_{\tau_i}
=L_{\tau_i}\iota_{\tau_j}
-\iota_{\tau_j}L_{\tau_i}
+\iota_{\tau_j}\iota_{\tau_i}d
=\iota_{[\tau_i,\tau_j]}
-\iota_{\tau_i}\iota_{\tau_j}d;
\end{eqnarray*}
the Lie bracket $[\tau_i,\tau_j]=-\Omega(\tau_i,\tau_j)^P$;
equation (\ref{iA.beta});
and finally the definition of $H$ in (\ref{basic.3form}).
\end{proof}
{\it Remark.}
Because of the bijection in Lemma \ref{lemma.basic.3form},
we know that \emph{any} basic $3$-form $H$ that
satisfies $dH=\lambda^*(\Omega\wedge\Omega)$ determines
a vertex algebroid (\ref{CDOP.VAoidnew}) as per Lemma
\ref{lemma.CDOP.VAoidnew}, and hence equivalently
a principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
\begin{corollary} \label{cor.fund.normal.comm}
In the vertex algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ we have
the following normal-ordered expansions and commutation
relations
\footnote{
The author found it interesting (and comforting)
to verify directly all Jacobi identities in sight.
}
\begin{align*}
&(fA^P)_n
=\sum_{k\geq 0}f_{n-k}A^P_k
+\sum_{k<0}A^P_k f_{n-k},\qquad
(f\tau_i)_n
=\sum_{k\geq 0}f_{n-k}\tau_{i,k}
+\sum_{k<0}\tau_{i,k}f_{n-k} \\
&[A^P_n,B^P_m]=[A,B]^P_{n+m}+n\lambda(A,B)\delta_{n+m,0},
\qquad
[A^P_n,\tau_{i,m}]=\rho(A)_{ji}\tau_{j,n+m}
\phantom{\rule[-0.05in]{.1pt}{.1pt}} \\
&\textstyle
[\tau_{i,n},\tau_{j,m}]
=-\Omega(\tau_i,\tau_j)^P_{n+m}
+(n-m)(\mathrm{Ric}_{ij})_{n+m}
+\Big(\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
+\lambda^*(\Omega(\tau_i,\tau_j),\Theta)\Big)_{n+m}
\end{align*}
for $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$, $i,j=1,\ldots,d$
and $n,m\in\mathbb{Z}$.
\end{corollary}
\begin{proof}
This follows immediately from Lemma \ref{lemma.CDOP.VAoidnew}
and Definition \ref{VAoid.VA}.
\end{proof}
\begin{lemma} \label{lemma.CDOP.conformal}
The vertex algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ has a conformal
vector
\begin{eqnarray*}
\nu=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda^*(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
Moreover, we have $\nu_1 A^P=\nu_1 \tau_i=0$ for
$A\in\mathfrak{g}$ and $i=1,\ldots,d$.
\end{lemma}
\begin{proof}
By construction the isomorphism from $\mc{D}^{\mathrm{ch}}(P)$ to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ (see {\it Preparation} before
Lemma \ref{lemma.CDOP.VAoidnew}) sends the conformal
vector $\nu'$ in (\ref{CDOP.conformal}) to
\begin{align*}
\nu
&\textstyle
=(t^P_a-h_{t_a})_{-1}\Theta^a
+(\tau_i-\frac{1}{2}\iota_{\tau_i}\beta)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a
+\tau_{i,-1}\tau^i
-(h_{t_a})_{-1}\Theta^a
-\frac{1}{2}(\iota_{\tau_i}\beta)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\Big(h_{t_a}(t_b^P)\Theta^b
+h_{t_a}(\tau_i)\tau^i\Big)_{-1}\Theta^a
-\frac{1}{2}\Big(\beta(\tau_i,t^P_a)\Theta^a
+\beta(\tau_i,\tau_j)\tau^j\Big)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda^*(t_a,t_b)\Theta^b_{-1}\Theta^a
-h_{t_a}(\tau_i)\tau^i_{-1}\Theta^a
+h_{t_a}(\tau_i)\tau^i_{-1}\Theta^a+0 \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda^*(\Theta_{-1}\Theta)
\end{align*}
where we have used, in order:
the component of (\ref{CDOP.h}) in
$(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P))^G$;
the symmetry $\alpha_{-1}\alpha'=\alpha'_{-1}\alpha$
for $1$-forms $\alpha,\alpha'$;
and the definition of $\beta$ in (\ref{beta}).
The said isomorphism also sends $\nu'_1(A^P+h_A)$ to
$\nu_1 A^P$ and $\nu'_1(\tau_i+\iota_{\tau_i}\beta)$ to
$\nu_1\tau_i$.
This proves our claims in view of Lemma
\ref{lemma.CDOP.VAoid} and (\ref{CDO.L1}).
\end{proof}
The following summarizes all our discussions since
\S\ref{sec.prin.bdl2}.
\begin{theorem} \label{thm.CDOP}
Suppose we have the following data:
a principal $G$-bundle $\pi:P\rightarrow M$,
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta$ on $\pi$ that induces
the Levi-Civita connection on $TM$;
an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$;
and a basic $3$-form $H$ on $P$ satisfying
$dH=\lambda^*(\Omega\wedge\Omega)$, where
$\Omega$ is the curvature of $\Theta$ and
$\lambda^*=\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho$.
Given these data, there is an associated vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ with generating fields
\begin{eqnarray*}
\t{\sf Y}(f,x)\textrm{ for }f\in C^\infty(P),\qquad
\t{\sf Y}(A^P,x)\textrm{ for }A\in\mathfrak{g},\qquad
\t{\sf Y}(\tau_i,x)\textrm{ for }i=1,\ldots,d
\end{eqnarray*}
whose weights are $0$, $1$ and $1$ respectively.
(For the notations see \S\ref{sec.prin.bdl} and
\S\ref{sec.prin.bdl2}.)
Notice that the vacuum is $1\in C^\infty(P)$.
The various OPEs with $\t{\sf Y}(f,x)$ have the following
leading terms
\begin{align*}
&\t{\sf Y}(g,x')\t{\sf Y}(f,x)
=\t{\sf Y}(fg,x)+\t{\sf Y}(fdg,x)(x'-x)+O((x'-x)^2),\quad
g\in C^\infty(P) \\
&\t{\sf Y}(A^P,x')\t{\sf Y}(f,x)
=\frac{\t{\sf Y}(A^P f,x)}{x'-x}+\t{\sf Y}(fA^P,x)+O(x'-x) \\
&\t{\sf Y}(\tau_i,x')\t{\sf Y}(f,x)
=\frac{\t{\sf Y}(\tau_i f,x)}{x'-x}+\t{\sf Y}(f\tau_i,x)+O(x'-x)
\end{align*}
where in fact all the second terms are meant to be taken
(plus linearity) as the definition of fields associated
to arbitrary $1$-forms and vector fields on $P$.
The OPEs between the other generating fields have the
following singular parts
\begin{align*}
&\t{\sf Y}(A^P,x')\t{\sf Y}(B^P,x)\sim
\frac{\lambda(A,B)}{(x'-x)^2}
+\frac{\t{\sf Y}([A,B]^P,x)}{x'-x},\qquad
\t{\sf Y}(A^P,x')\t{\sf Y}(\tau_i,x)\sim
\frac{\rho(A)_{ji}\t{\sf Y}(\tau_j,x)}{x'-x} \\
&\t{\sf Y}(\tau_i,x')\t{\sf Y}(\tau_j,x)\sim
\frac{2\t{\sf Y}\big(\mathrm{Ric}_{ij},\frac{x+x'}{2}\big)}{(x'-x)^2}
-\frac{\t{\sf Y}\big(\Omega(\tau_i,\tau_j)^P,x\big)}{x'-x}
+\frac{\frac{1}{2}\t{\sf Y}\big(\iota_{\tau_i}\iota_{\tau_j}H,x\big)
+\t{\sf Y}\big(\lambda^*\big(\Omega(\tau_i,\tau_j),
\Theta\big),x\big)}{x'-x}
\end{align*}
where $\mathrm{Ric}_{ij}=\rho(\Omega(\tau_i,\tau_k))_{jk}$.
Clearly the fields $\t{\sf Y}(A^P,x)$ for $A\in\mathfrak{g}$ represent
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
(see Definition \ref{formalLG.action}).
Moreover, there is a Virasoro field of central charge
$2\dim P$ given by
\begin{eqnarray*}
:\t{\sf Y}(t^P_a,x)\t{\sf Y}(\Theta^a,x):
+:\t{\sf Y}(\tau_i,x)\t{\sf Y}(\tau^i,x):
-\frac{1}{2}\,\lambda^*\big(\t{\sf Y}(\Theta,x),\t{\sf Y}(\Theta,x)\big)
\end{eqnarray*}
with respect to which all generating fields are primary.
(For the notations see \S\ref{sec.prin.bdl2}.)
$\qedsymbol$
\end{theorem}
{\it Remark.}
For convenience (and lack of imagination), we will
refer to $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ as
a {\bf principal frame $(\hat{\mathfrak{g}}_\lambda,G)$-algebra}
or, if we do not wish to specify $\lambda$,
a {\bf principal frame $(\hat{\mathfrak{g}},G)$-algebra}.
Even though this vertex algebra arises here as
a particular algebra of CDOs, it should really be regarded
as a more fundamental object for both conceptual and
aesthetic reasons.
In \S\ref{CDOP.CDOM} we will see that any algebra of CDOs
can be recovered in a natural way from a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra with $\lambda=-\lambda_{\t{\sf ad}}$;
and this is only a special case of a more general
construction associated to principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebras with different $\lambda$
(to be introduced in \S\ref{sec.ass}).
Also noteworthy is that, compared to an arbitrary algebra
of CDOs, a principal frame $(\hat{\mathfrak{g}},G)$-algebra
requires a smaller set of generating fields and possesses
arguably more appealing OPEs and Virasoro field.
\footnote{
For generating fields, $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ requires only
a finite number of canonical vector fields (that constitute
a framing of $TP$), as opposed to all vector fields.
For OPEs, compare the vertex algebroids in
Lemma \ref{lemma.CDOP.VAoidnew} and
Theorem \ref{thm.globalCDO}a.
For Virasoro fields, compare the conformal vectors in
Lemma \ref{lemma.CDOP.conformal} and
Theorem \ref{thm.globalCDO}b.
}
For these reasons, it seems desirable to find a more direct
construction of principal frame $(\hat{\mathfrak{g}},G)$-algebras,
perhaps in a future paper.
\begin{subsec} \label{sec.invCDO.can}
{\bf The invariant subalgebra.}
This discussion is a more detailed version of
\S\ref{sec.invCDO} specifically for a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
Consider the centralizer subalgebra
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}
:=C\big(\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),V_\lambda(\mathfrak{g})\big).
\end{eqnarray*}
The weight-zero component is again $C^\infty(P)^G$.
Let us describe the weight-one component below.
Consider an arbitrary element of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}_1$: some
$\alpha+\mathscr{X}+\mathscr{Y}\in\Omega^1(P)\oplus\mc{T}_h(P)\oplus\mc{T}_v(P)$
that is annihilated by $A^P_0$ and $A^P_1$ for $A\in\mathfrak{g}$.
Suppose $t_1,t_2,\cdots$ are a basis of $\mathfrak{g}$ and
$t^1,t^2,\cdots$ the dual basis of $\mathfrak{g}^\vee$.
Let us write $\mathscr{X}=\mathscr{X}^i\tau_i$ and
$\mathscr{Y}=\mathscr{Y}^a t^P_a$.
By Lemma \ref{lemma.CDOP.VAoidnew} and Definition
\ref{VAoid}, both $\{A^P,\mathscr{X}^i\tau_i\}_\Omega$ and
$\{A^P,\mathscr{Y}^a t^P_a\}_\Omega$ vanish;
then by (\ref{freeVA.comm}) the first condition on
$\alpha+\mathscr{X}+\mathscr{Y}$ reads
\begin{align*}
&A^P_0(\alpha+\mathscr{X}+\mathscr{Y})
=L_{A^P}\alpha+[A^P,\mathscr{X}]+[A^P,\mathscr{Y}]=0
\qquad\textrm{for }A\in\mathfrak{g}.
\end{align*}
Hence $\alpha$, $\mathscr{X}$, $\mathscr{Y}$ are
all $G$-invariant.
Notice that the $G$-invariance of $\mathscr{Y}$ means
$A^P\mathscr{Y}^a=-t^a([A,t_b])\mathscr{Y}^b$.
By Lemma \ref{lemma.CDOP.VAoidnew} and Definition
\ref{VAoid} again, we find that
$\{A^P,\mathscr{X}^i\tau_i\}=\rho(A)_{ji}\tau_j\mathscr{X}^i$ and
\begin{align*}
\{A^P,\mathscr{Y}^a t^P_a\}
&=\lambda(A,t_a)\mathscr{Y}^a+[A,t_a]^P\mathscr{Y}^a \\
&=\lambda(A,t_a)\mathscr{Y}^a-t^a([[A,t_a],t_b])\mathscr{Y}^b \\
&=(\lambda+\lambda_{\t{\sf ad}})(A,\mathscr{Y});
\end{align*}
then by (\ref{freeVA.comm}) the second condition on
$\alpha+\mathscr{X}+\mathscr{Y}$ reads
\begin{align*}
A^P_1(\alpha+\mathscr{X}+\mathscr{Y})
=\alpha(A^P)+\rho(A)_{ij}\tau_i\mathscr{X}^j
+(\lambda+\lambda_{\t{\sf ad}})(A,\mathscr{Y})
=0\qquad\textrm{for }A\in\mathfrak{g}.
\end{align*}
It follows from the two conditions that the $1$-form
$\alpha+(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}
+(\lambda+\lambda_{\t{\sf ad}})(\Theta,\mathscr{Y})$ is $G$-invariant
and horizontal (i.e.~basic).
Therefore $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}_1$ is
the direct sum of the following subspaces:
\begin{align} \label{invCDOP.1}
\pi^*\Omega^1(M),\quad
\big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
\mathscr{X}\in\mc{T}_h(P)^G\big\},\quad
\big\{\mathscr{Y}-(\lambda+\lambda_{\t{\sf ad}})(\Theta,\mathscr{Y}):
\mathscr{Y}\in\mc{T}_v(P)^G\big\}.
\end{align}
This provides in our current setting an explicit
description of the short exact sequence (\ref{invCDO.1})
together with a splitting.
\end{subsec}
\newpage
\setcounter{equation}{0}
\section{Associated Modules and Algebras over CDOs}
\label{sec.ass}
Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)$ on the total space
of a principal bundle $P\rightarrow M$, equipped with
a projective ``formal loop group action''.
This section introduces a construction of modules over
the invariant subalgebra of $\mc{D}^{\mathrm{ch}}(P)$ using semi-infinite
cohomology;
it is very much analogous to the construction of associated
vector bundles of $P\rightarrow M$.
In fact, the said modules should have an interpretation
as the spaces of sections of certain vector bundles over
the hypothetical ``formal loop space'' of $M$.
Examples will be studied in \S\ref{CDOP.CDOM} and
\S\ref{sec.FL.spinor}.
The semi-infinite cohomology of a centrally extended loop
algebra can be defined using the Feigin complex, which
is a vertex algebraic analogue of the Chevalley-Eilenberg
complex, but with some important differences.
For more about semi-infinite cohomology, see
e.g.~\cite{Feigin,Voronov,BD}.
\begin{subsec} \label{sec.CE}
{\bf The Chevalley-Eilenberg complex.}
Given a Lie algebra $\mathfrak{g}$, consider the supermanifold
$\Pi\mathfrak{g}$.
Let $t_1,t_2,\ldots$ be a basis of $\mathfrak{g}$;
$t^1,t^2,\ldots$ the dual basis of $\mathfrak{g}^\vee$; and
$(\phi^1,\phi^2,\ldots)$ the corresponding coordinates of
$\Pi\mathfrak{g}$.
By definition $\mathcal{O}(\Pi\mathfrak{g})=\wedge^*\mathfrak{g}^\vee$ and
$\mc{T}(\Pi\mathfrak{g})$ consists of the derivations on
$\wedge^*\mathfrak{g}^\vee$.
Consider
\begin{eqnarray*}
J,\,q\in\mc{T}(\Pi\mathfrak{g}),\qquad
\theta\in\mathcal{O}(\Pi\mathfrak{g})\otimes\mathfrak{g}
\end{eqnarray*}
corresponding respectively to the exterior degree on
$\wedge^*\mathfrak{g}^\vee$, the Chevalley-Eilenberg differential
on $\wedge^*\mathfrak{g}^\vee$, and the Maurer-Cartan form on
$\mathfrak{g}$;
they can be written in coordinates as follows
\begin{eqnarray} \label{JQ.coor}
J=\phi^a\frac{\partial}{\partial\phi^a}\,,\qquad
q=-\frac{1}{2}\,t^a([t_b,t_c])\phi^b\phi^c\frac{\partial}{\partial\phi^a}\,,
\qquad
\theta=\phi^a\otimes t_a.
\end{eqnarray}
Notice that $J=J\otimes 1$,\; $q=q\otimes 1$ and $\theta$
satisfy
\begin{eqnarray} \label{JQ.relations}
[J,q]=q,\qquad
J\theta=\theta,\qquad
[q,q]=0,\qquad
q\theta+\frac{1}{2}\,[\theta,\theta]=0.
\end{eqnarray}
Let $W$ be a $\mathfrak{g}$-module.
If $J,q,\theta$ are regarded as operators on
$\mathcal{O}(\Pi\mathfrak{g})\otimes W$ and $Q=q+\theta$,
then by (\ref{JQ.relations}) they satisfy
$[J,Q]=Q$ and $[Q,Q]=2Q^2=0$.
The Chevalley-Eilenberg complex of $\mathfrak{g}$ with
coefficients in $W$ can be written as
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes W,\,J,\,Q\Big)
\end{eqnarray*}
where $J$ is the grading operator and $Q$ is
the differential.
\end{subsec}
\begin{subsec} \label{sec.Feigin}
{\bf The Feigin complex.}
Consider the algebra of CDOs $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$, which is
a fermionic version of \S\ref{algCDO};
for details see \cite{myCDO}.
Given an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$, let $\hat{\mathfrak{g}}_\lambda$ be the centrally
extended loop algebra described in \S\ref{sec.Gmfld} and
$V_\lambda(\mathfrak{g})$ the vertex algebra in Example
\ref{VAoid.Lie}.
Now regard
\begin{eqnarray*}
J,q,\theta\textrm{ as elements of }
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_\lambda(\mathfrak{g})
\textrm{ of weight }1.
\end{eqnarray*}
The following computations in
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_\lambda(\mathfrak{g})$ make use of
(\ref{freeVA.comm}), (\ref{JQ.coor}) and
the super version of (\ref{CDO.VAoid456}):
\begin{align*}
&J_0 q
=[J,q]+\{J,q\}_\Omega=q+0=q \\
&J_0\theta
=(J_0\otimes 1)(\phi^a\otimes t_a)
=J\phi^a\otimes t_a=\phi^a\otimes t_a=\theta \\
&q_0 q
=[q,q]+\{q,q\}_\Omega
=0-t^a([t_b,t_c])\,t^b([t_a,t_d])\,\phi^d d\phi^c
=-\lambda_{\t{\sf ad}}(t_c,t_d)\,\phi^d d\phi^c \\
&\hspace{-0.1in}
\left.\begin{array}{l}
q_0\theta=(q_0\otimes 1)(\phi^a\otimes t_a)
\rule[-0.1in]{0in}{0in} \\
\theta_0 q=(\phi^a_1\otimes t_{a,-1})(q\otimes 1)
\end{array}\right\}
=q\phi^a\otimes t_a
=-\frac{1}{2}\phi^b\phi^c\otimes[t_b,t_c] \\
&\theta_0\theta
=(\phi^a_0\otimes t_{a,0}+\phi^a_{-1}\otimes t_{a,1})
(\phi^b\otimes t_b)
=\phi^a\phi^b\otimes[t_a,t_b]
-\lambda(t_a,t_b)\,\phi^b d\phi^a
\end{align*}
where $\lambda_{\t{\sf ad}}$ denotes the Killing form.
Hence $J$ and $Q=q+\theta$ satisfy
\begin{eqnarray*}
\begin{array}{rcc}
&J_0 Q=Q,&
Q_0 Q=-(\lambda_{\t{\sf ad}}+\lambda)(t_a,t_b)\,\phi^a d\phi^b
\phantom{\Big(} \\
\Rightarrow\quad&
[J_0,Q_0]=Q_0,&
\quad
[Q_0,Q_0]
=(\lambda_{\t{\sf ad}}+\lambda)(t_a,t_b)\cdot
\sum_{n\in\mathbb{Z}}n\phi^a_{-n}\phi^b_n
\end{array}
\end{eqnarray*}
where we have used (\ref{freeVA.comm}) and
(\ref{freeVA.NOP}).
In particular, $Q_0^2=0$ if and only if
$\lambda=-\lambda_{\t{\sf ad}}$.
Let $W$ be a $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module.
Notice that $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ is a module over
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})$.
The Feigin complex of $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$ with
coefficients in $W$ is
\begin{eqnarray*}
\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W,\,J_0,\,Q_0\Big)
\end{eqnarray*}
where $J_0$ is the grading operator and
$Q_0$ is the differential.
\end{subsec}
\begin{subsec} \label{sec.semiinf}
{\bf Semi-infinite cohomology.}
For any $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module $W$ as above, let
\begin{eqnarray} \label{semiinf}
H^{\frac{\infty}{2}+*}(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},W)
:=H^*\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W,\,Q_0\Big).
\end{eqnarray}
Notice that if $W$ is a vertex algebra such that the
$\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner,
i.e. induced by a map of vertex algebras
$V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})\rightarrow W$,
then (\ref{semiinf}) has the structure of
a $\mathbb{Z}$-graded vertex algebra.
(In this case, $J$ and $Q$ will also denote their own
images in $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$.)
\end{subsec}
{\it Remarks.}
(i) Unlike Lie algebra cohomology, the grading on
semi-infinite cohomology is neither bounded above nor below.
More precisely, the restriction of $J_0$ to
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_k\otimes W$ takes values between $-k$
and $\dim\mathfrak{g}+k$.
(ii) This is only a special case of semi-infinite cohomology.
For a modern exposition of the notion in the more general
setting of Tate Lie algebras, see \cite{BD}.
\begin{lemma} \label{lemma.semiinf.ops}
Let $W$ be a $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module.
(a) The $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-invariant operators on
$W$ induce grading-preserving operators on (\ref{semiinf}).
Moreover, if $W$ is a vertex algebra such that its
$\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner, then there
is a map of vertex algebras from the centralizer subalgebra
$W^{\hat{\mathfrak{g}}}=C\big(W,V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})\big)$
\cite{Kac,FB-Z} to the zeroth gradation of (\ref{semiinf}).
(b) If there is a Virasoro action on $W$ of central charge $c$
such that the $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is primary,
then it induces a grading-preserving Virasoro action on
(\ref{semiinf}) of central charge $c-2\dim\mathfrak{g}$.
Moreover, if $W$ is in fact a conformal vertex algebra such
that its $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner and
primary, then (\ref{semiinf}) is also a conformal vertex
algebra with a conformal vector in the zeroth gradation.
\end{lemma}
\begin{proof}
(a) For the first statement, suppose $F\in\textrm{End}\,W$
satisfies $[A_n,F]=0$ for $A\in\mathfrak{g}$ and $n\in\mathbb{Z}$.
Since on $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ we have
\begin{eqnarray*}
[Q_0,1\otimes F]
=[q_0\otimes 1+\phi^a_{-n}\otimes t_{a,n},\,1\otimes F]
=\phi^a_{-n}\otimes[t_{a,n},F]
=0
\end{eqnarray*}
the operator $1\otimes F$ is well-defined on (\ref{semiinf}).
Clearly it preserves the grading.
For the second statement, let $u\in W^{\hat{\mathfrak{g}}}$,
i.e.~$u\in W$ with $A_n u=0$ for $A\in\mathfrak{g}$ and $n\geq 0$.
Since in $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ we have
\begin{eqnarray*}
Q_0(\mathbf{1}\otimes u)
=(q_0\otimes 1+\phi^a_{-n}\otimes t_{a,n})(\mathbf{1}\otimes u)
=\sum_{n\geq 0}\phi^a_{-n}\mathbf{1}\otimes t_{a,n}u
=0
\end{eqnarray*}
the element $\mathbf{1}\otimes u$ represents a class
$[\mathbf{1}\otimes u]$ in (\ref{semiinf}).
Clearly $u\mapsto[\mathbf{1}\otimes u]$ is a map of
vertex algebras and the image is contained in the zeroth
gradation.
(b) The graded vertex algebra $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$ has a conformal
vector $\nu^{\Pi\mathfrak{g}}$ of central charge $-2\dim\mathfrak{g}$
(see \S\ref{sec.ass.C});
denote its Virasoro operators by $L^{\Pi\mathfrak{g}}_n$, $n\in\mathbb{Z}$.
Since $q$ and $\phi^a$ are primary, we have
\begin{eqnarray*}
[L^{\Pi\mathfrak{g}}_n,q_0]=0,\qquad
[L^{\Pi\mathfrak{g}}_n,\phi^a_m]=-(n+m)\phi^a_{n+m},\qquad
n,m\in\mathbb{Z}.
\end{eqnarray*}
Since $\nu^{\Pi\mathfrak{g}}$ belongs to the zeroth gradation,
all $L^{\Pi\mathfrak{g}}_n$ preserve the grading.
For the first statement, suppose $L^W_n\in\textrm{End}\,W$ for
$n\in\mathbb{Z}$ define a Virasoro action of central charge $c$
and satisfy $[L^W_n,A_m]=-mA_{n+m}$ for $A\in\mathfrak{g}$ and
$n,m\in\mathbb{Z}$.
Then $L^{\Pi\mathfrak{g}}_n\otimes 1+1\otimes L^W_n$ for
$n\in\mathbb{Z}$ define a grading-preserving Virasoro action on
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ of central charge $c-2\dim\mathfrak{g}$.
Since
\begin{align*}
[Q_0,\,L^{\Pi\mathfrak{g}}_n\otimes 1+1\otimes L^W_n]
&=[q_0,L^{\Pi\mathfrak{g}}_n]\otimes 1
+[\phi^a_{-m},L^{\Pi\mathfrak{g}}_n]\otimes t_{a,m}
+\phi^a_{-m}\otimes[t_{a,m},L^W_n] \\
&=0+(n-m)\phi^a_{n-m}\otimes t_{a,m}
+m\phi^a_{-m}\otimes t_{a,n+m} \\
&=0
\end{align*}
the said Virasoro action is well-defined on
(\ref{semiinf}) as well.
For the second statement, suppose $\nu^W\in W$ is
a conformal vector whose Virasoro operators $L^W_n$,
$n\in\mathbb{Z}$, satisfy $[L^W_n,A_m]=-mA_{n+m}$ for
$A\in\mathfrak{g}$ and $n,m\in\mathbb{Z}$.
Then $\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^W$
is a conformal vector of $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$
belonging to the zeroth gradation.
Since we have
\begin{align*}
Q_0(\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}
+\mathbf{1}\otimes\nu^W)
&=Q_0(L^{\Pi\mathfrak{g}}_{-2}\otimes 1
+1\otimes L^W_{-2})(\mathbf{1}\otimes\mathbf{1}) \\
&=\big[Q_0,\,L^{\Pi\mathfrak{g}}_{-2}\otimes 1
+1\otimes L^W_{-2}\big](\mathbf{1}\otimes\mathbf{1}) \\
&=0
\end{align*}
by the computation above,
$\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^W$
also represents a conformal vector of (\ref{semiinf}).
\end{proof}
The main purpose of this section is to introduce
the following construction.
As before, $G$ is a compact connected Lie group and
$\mathfrak{g}$ its Lie algebra.
Recall Definition \ref{formalLG.action} and
Corollary \ref{cor.formalLG.P}.
\begin{defn} \label{defn.ass}
Let $\pi:P\rightarrow M$ be a smooth principal $G$-bundle
and $\lambda,\lambda'\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Assume that $\lambda+\lambda'=-\lambda_{\t{\sf ad}}$ and
$(\lambda+\lambda_{\t{\sf ad}})(P)=8\pi^2 p_1(M)$.
For any principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}(P)$
and positive-energy $(\hat{\mathfrak{g}}_{\lambda'},G)$-module $W$,
we define
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,W)
:=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}(P)\otimes W\big).
\end{align*}
Notice that if $W$ is a vertex algebra such that the
$(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner, then
$\Gamma^{\mathrm{ch}}(\pi,W)$ also has the structure of a vertex algebra
(see \S\ref{sec.semiinf}).
\end{defn}
{\it Remarks.}
(i) The positive-energy condition means that there
is a diagonalizable operator $L^W_0$ on $W$ such
that $[L^W_0,A_n]=-nA_n$ for $A\in\mathfrak{g}$, $n\in\mathbb{Z}$,
and the eigenvalues of $L^W_0$ are bounded below.
Whenever $W$ admits a Virasoro action, it will be
understood that $L^W_0$ coincides with the Virasoro
operator that is usually given the same notation.
For consistency, the eigenvalues of $L^W_0$ will
also be called weights.
Let $L^{\Pi\mathfrak{g}}_0$ (resp.~$L^P_0$) be the weight
operator on $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$ (resp.~$\mc{D}^{\mathrm{ch}}(P)$).
Notice that $L^{\Pi\mathfrak{g}}_0+L^P_0+L^W_0$ commutes
with $Q_0$ (see the proof of Lemma
\ref{lemma.semiinf.ops}b), so that it induces
a weight decomposition of $\Gamma^{\mathrm{ch}}(\pi,W)$.
(ii) The above definition still makes sense without
the integrability and/or the positive-energy conditions.
On the other hand, under these two conditions, we can
say that the lowest-weight component $W_0$ of $W$ is
a $G$-representation and then the lowest-weight component of
$\Gamma^{\mathrm{ch}}(\pi,W)$ is $(C^\infty(P)\otimes W_0)^G$,
i.e. the space of sections of the associated vector
bundle $P\times_G W_0\rightarrow M$.
\begin{lemma} \label{lemma.ass}
Consider again the data in Definition \ref{defn.ass}.
(a) For any $W$ as described, $\Gamma^{\mathrm{ch}}(\pi,W)$ is a module
over $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$.
Moreover, if $W$ is a vertex algebra such that
its $(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner,
then there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,W)$.
(b) If there is a Virasoro action on $W$ of central charge
$c$ such that the $(\hat{\mathfrak{g}}_{\lambda'},G)$-action
is primary, then it induces a Virasoro action on $\Gamma^{\mathrm{ch}}(\pi,W)$
of central charge $c+2\dim M$.
Moreover, if $W$ is in fact a conformal vertex algebra such
that its $(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner and
primary, then $\Gamma^{\mathrm{ch}}(\pi,W)$ is also a conformal vertex
algebra.
\end{lemma}
\begin{proof}
Recall \S\ref{sec.invCDO}.
This follows from Lemma \ref{lemma.semiinf.ops}, together
with the fact that $\mc{D}^{\mathrm{ch}}(P)$ is a conformal vertex algebra
with central charge $2\dim P$ and a primary inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action (see Theorem
\ref{thm.formalLG.action}).
\end{proof}
\newpage
\setcounter{equation}{0}
\section{Example: Recovering Algebras of CDOs}
\label{CDOP.CDOM}
Suppose $P\rightarrow M$ is a principal frame bundle and
$\mc{D}^{\mathrm{ch}}(P)$ is an algebra of CDOs on $P$ with a suitable
``formal loop group action''.
In this section, we analyze the zeroth semi-infinite
cohomology of $\mc{D}^{\mathrm{ch}}(P)$ and in the end identify it
as an algebra of CDOs on $M$.
This provides a description of an arbitrary algebra
of CDOs that is conceptually more satisfying than
the original one given in \S\ref{sec.review}.
\begin{subsec} \label{sec.ass.C}
{\bf Goal:~a new description of algebras of CDOs.}
Consider the special case of Definition \ref{defn.ass}
associated to a principal frame
$(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ and the trivial
$(\hat{\mathfrak{g}}_0,G)$-module $\mathbb{C}$:
\begin{eqnarray} \label{ass.C}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
=H^0\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),
Q_0\Big).
\end{eqnarray}
By Lemma \ref{lemma.ass}, $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is a conformal
vertex algebra of central charge $2\dim M$.
For convenience, let us elaborate on all the data
involved and recall some relevant notations.
\begin{itemize}
\item[$\centerdot$]
Let $(t_1,t_2,\cdots)$ be a basis of $\mathfrak{g}$,
$(t^1,t^2,\cdots)$ the dual basis of $\mathfrak{g}^\vee$,
$(\phi^1,\phi^2,\cdots)$ the corresponding coordinates
of the supermanifold $\Pi\mathfrak{g}$, and
$(\partial_1,\partial_2,\cdots)$ their coordinate vector fields.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex superalgebra
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$, see \cite{Kac} or \cite{myCDO}.
Let us mention that, as a fermionic analogue of
\S\ref{algCDO}, it is generated by such elements
as $\phi^1,\phi^2,\cdots$ and $\partial_1,\partial_2,\cdots$,
and has a conformal vector
\begin{eqnarray} \label{oddg.conformal}
\nu^{\Pi\mathfrak{g}}=-\partial_{a,-1}d\phi^a
\end{eqnarray}
of central charge $-2\dim\mathfrak{g}$.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, see Theorem \ref{thm.CDOP} with
$\lambda=-\lambda_{\t{\sf ad}}$ in mind.
Let us mention that it is defined using the following
data:
a smooth principal $G$-bundle $\pi:P\rightarrow M$
together with a representation
$\rho:G\rightarrow SO(\mathbb{R}^d)$ and an isomorphism
$P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta=\Theta^a\otimes t_a$ on $\pi$
that induces the Levi-Civita connection on $TM$;
and a basic $3$-form $H$ on $P$ that satisfies
$dH=\lambda_\rho(\Omega\wedge\Omega)$, where
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Also, it has a conformal vector
\begin{eqnarray*}
\nu^P=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
(As before, $\rho$ also denotes the induced map
of Lie algebras $\mathfrak{g}\rightarrow\mathfrak{so}_d$
and $\lambda_\rho$ the invariant symmetric
bilinear form on $\mathfrak{g}$ given by
$\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
For the meaning of the other notations, see
\S\ref{sec.prin.bdl2}.)
\vspace{-0.05in}
\item[$\centerdot$]
For details of the Feigin complex appearing in (\ref{ass.C}),
see \S\ref{sec.Feigin}.
Let us mention that the grading operator $J_0$ is
determined by $J_0(\phi^a\otimes u)=\phi^a\otimes u$ and
$J_0(\partial_a\otimes u)=-\partial_a\otimes u$ for any $u$;
and the differential $Q_0=(q+\theta)_0$ is induced by
the elements
\begin{eqnarray} \label{Q.theta}
q=q\otimes\mathbf{1}
=-\frac{1}{2}\,t^a([t_b,t_c])\phi^b\phi^c\partial_a\otimes\mathbf{1},
\qquad
\theta=\phi^a\otimes t_a^P.
\end{eqnarray}
The conformal vector
$\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^P$
belongs to the zeroth gradation and is
$Q_0$-closed.
\vspace{-0.05in}
\end{itemize}
By the assumption on $\pi$, $M$ is an oriented
Riemannian manifold.
Let $\nabla$ be the Levi-Civita connection and $R$
the Riemannian curvature of $M$.
The assumption on $H$ can then be written as
$dH=\mathrm{Tr}\,(R\wedge R)$.
By Theorem \ref{thm.globalCDO}, the data $(\nabla,H)$
determine an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$
with a conformal vector $\nu=\nu^0$ of central
charge $2\dim M$.
The goal of this section is to prove that
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
\cong\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)
\end{eqnarray*}
as conformal vertex algebras.
In fact, we will analyze $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ without
using any prior knowledge of $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$,
and effectively rediscover the latter in the end.
\end{subsec}
Throughout this section we identify $\Omega^*(M)$ with
the basic subspace of $\Omega^*(P)$.
\begin{subsec} \label{sec.ass.C.0}
{\bf The component of weight zero.}
Consider the Feigin complex that appears in
(\ref{ass.C}):
\begin{eqnarray} \label{Feigin.C}
\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),\,
Q_0\Big).
\end{eqnarray}
Since its weight-zero component is simply the
Chevalley-Eilenberg complex (see \S\ref{sec.CE})
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes C^\infty(P),\,
Q_0\Big)
\end{eqnarray*}
the weight-zero component of (\ref{ass.C}) is
\begin{eqnarray} \label{ass.C.0}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_0
=H^0\big(\mathfrak{g},C^\infty(P)\big)
=C^\infty(P)^G
=C^\infty(M).
\end{eqnarray}
Understanding the rest of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ requires
more work, mostly because in higher weights the
Feigin complex (\ref{Feigin.C}) contains elements in
both positive and negative gradations.
\end{subsec}
\begin{lemma} \label{lemma.G}
Let $\gamma=\partial_a\otimes\Theta^a\in
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_1\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$.
Then we have
\begin{eqnarray*}
Q_0\gamma=\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}
+\mathbf{1}\otimes t^P_{a,-1}\Theta^a,\qquad
\gamma_0\gamma=0
\end{eqnarray*}
and therefore the anticommutators
\begin{eqnarray*}
\phantom{\Big(}
[Q_0,\gamma_0]=L^{\Pi\mathfrak{g}}_0\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_0,\qquad
[\gamma_0,\gamma_0]=2\gamma_0^2=0.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Recall the element $Q=q+\theta$ from (\ref{Q.theta}).
Let us first write
\begin{eqnarray*}
Q_0\gamma
=\big(q_0\otimes 1+\phi^a_{-1}\otimes t^P_{a,1}
+\phi^a_0\otimes t^P_{a,0}
+\phi^a_1\otimes t^P_{a,-1}\big)(\partial_b\otimes\Theta^b).
\end{eqnarray*}
Then each of these terms is computed as follows:
\begin{align*}
&q_0\partial_b\otimes\Theta^b
=\big([q,\partial_b]+\{q,\partial_b\}_\Omega\big)\otimes\Theta^b
=-t^c([t_b,t_d])\phi^d\partial_c\otimes\Theta^b \\
&\phi^a_{-1}\partial_b\otimes t^P_{a,1}\Theta^b
=-\partial_{b,-1}d\phi^a\otimes\Theta^b(t^P_a)
=-\partial_{a,-1}d\phi^a\otimes\mathbf{1} \\
&\phi^a_0\partial_b\otimes t^P_{a,0}\Theta^b
=-\partial_{b,-1}\phi^a\otimes L_{t^P_a}\Theta^b
=-\phi^a\partial_b\otimes t^b([t_a,t_c])\Theta^c \\
&\phi^a_1\partial_b\otimes t^P_{a,-1}\Theta^b
=\mathbf{1}\otimes t^P_{a,-1}\Theta^a
\end{align*}
where we have used (\ref{freeVA.comm}), the super version
of (\ref{Weyl})--(\ref{CDO.VAoid456}) and
the $G$-invariance of $\Theta=\Theta^a\otimes t_a$.
In view of (\ref{oddg.conformal}), this proves
the claimed expression for $Q_0\gamma$.
On the other hand, we have
\begin{eqnarray*}
\gamma_0\gamma
=(\partial_{a,1}\otimes\Theta^a_{-1})(\partial_b\otimes\Theta^b)
=\partial_{a,1}\partial_b\otimes\Theta^a_{-1}\Theta^b
=0
\end{eqnarray*}
thanks to the super version of (\ref{Weyl}).
\end{proof}
\begin{lemma} \label{lemma.Feigin.C.1}
The weight-one component of the Feigin complex
(\ref{Feigin.C}) is quasi-isomorphic to
the following (Chevalley-Eilenberg) subcomplex
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_h(P)_1,
\,Q_0\Big)
\end{eqnarray*}
where $\mc{D}^{\mathrm{ch}}_h(P)_1\subset\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ is
the direct sum of the subspaces
\begin{eqnarray*}
\Omega^1_h(P)
\quad\textrm{and}\quad
\Big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
\mathscr{X}=\mathscr{X}^i\tau_i\in\mc{T}_h(P)\Big\}.
\end{eqnarray*}
\end{lemma}
\begin{proof}
First we compute the operator $(t^P_{a,-1}\Theta^a)_0$ on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_k$, $k=0,1$.
Notice that (\ref{freeVA.comm})--(\ref{freeVA.NOP})
will be used repeatedly.
For $f\in C^\infty(P)$ and $\alpha\in\Omega^1(P)$, we have
\begin{align*}
&(t^P_{a,-1}\Theta^a)_0 f
=\Theta^a_0\,t^P_{a,0}f
=0 \\
&(t^P_{a,-1}\Theta^a)_0\alpha
=\big(\Theta^a_0\,t^P_{a,0}+\Theta^a_{-1}t^P_{a,1}\big)\alpha
=0+\alpha(t^P_a)\Theta^a
=\alpha_v
\end{align*}
where $\alpha_v$ means the vertical part of $\alpha$.
For $\mathscr{X}=\mathscr{X}^i\tau_i\in\mc{T}_h(P)$, we have
\begin{align*}
(t^P_{a,-1}\Theta^a)_0\mathscr{X}
&=\big(t^P_{a,-1}\Theta^a_1+\Theta^a_0\,t^P_{a,0}
+\Theta^a_{-1}t^P_{a,1}\big)\mathscr{X} \\
&=0-\iota_{[t^P_a,\mathscr{X}]}d\Theta^a
+\rho(t_a)_{ij}(\tau_i\mathscr{X}^j)\Theta^a \\
&=-\iota_{[t^P_a,\mathscr{X}]}\Omega^a
+(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}
\end{align*}
using Lemma \ref{lemma.1form.0mode} and the computation
of $\{A^P,\mathscr{X}\}$ in \S\ref{sec.invCDO.can}.
For $\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)$, we have
\begin{align*}
(t^P_{a,-1}\Theta^a)_0\mathscr{Y}
&=\big(t^P_{a,-1}\Theta^a_1
+\Theta^a_0\,t^P_{a,0}
+\Theta^a_{-1}t^P_{a,1}\big)\mathscr{Y} \\
&=\Theta^a(\mathscr{Y})t^P_a
-\iota_{[t^P_a,\mathscr{Y}]}d\Theta^a
+\big([t_a,t_b]^P\mathscr{Y}^b-\lambda_{\t{\sf ad}}(t_a,\mathscr{Y})\big)
\Theta^a \\
&=\mathscr{Y}
-\Theta^b([t^P_a,\mathscr{Y}])\iota_{t^P_b}d\Theta^a
+\Theta^b\big([[t_a,t_b]^P,\mathscr{Y}]\big)\Theta^a \\
&=\mathscr{Y}
+\Theta^b([t^P_a,\mathscr{Y}])t^a([t_b,t_c])\Theta^c
+t^c([t_a,t_b])\Theta^b([t^P_c,\mathscr{Y}])\Theta^a \\
&=\mathscr{Y}
\end{align*}
using Lemma \ref{lemma.CDOP.VAoidnew},
Lemma \ref{lemma.1form.0mode} and
the computation of $\{A^P,\mathscr{Y}\}$ in
\S\ref{sec.invCDO.can} (with $\lambda=-\lambda_{\t{\sf ad}}$ but
without the assumption that $\mathscr{Y}$ is $G$-invariant).
Now consider the null-homotopic operator on the Feigin
complex (\ref{Feigin.C}) that is defined by the
first expression below but is also equivalent to the
other expressions by Lemma \ref{lemma.G}:
\begin{align*}
e=\big[Q_0,\,\gamma_0 Q_0 \gamma_0\big]
=\big[Q_0,\gamma_0\big]^2
=\Big(L_0^{\Pi\mathfrak{g}}\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_0\Big)^2.
\end{align*}
Let $e_1$ denote its restriction to weight one.
By the calculations above, we have
\begin{eqnarray*}
\begin{array}{ll}
e_1(u_1\otimes f)=u_1\otimes f, &
u_1\in\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_1,\; f\in C^\infty(P) \\
e_1(u_0\otimes\alpha)=u_0\otimes\alpha_v, &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\alpha\in\Omega^1(P) \\
e_1(u_0\otimes\mathscr{X})=u_0\otimes(\tau_i\mathscr{X}^j)
\rho(\Theta)_{ij},\quad &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\mathscr{X}\in\mc{T}_h(P) \\
e_1(u_0\otimes\mathscr{Y})=u_0\otimes\mathscr{Y}, &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\mathscr{Y}\in\mc{T}_v(P)
\end{array}
\end{eqnarray*}
Notice that $e_1$ is an idempotent.
Therefore the image of $1-e_1$ is a quasi-isomorphic
subcomplex.
This is the subcomplex stated in the lemma.
\end{proof}
\begin{subsec} \label{sec.ass.C.1}
{\bf The component of weight one.}
By Lemma \ref{lemma.Feigin.C.1}, the weight-one component
of (\ref{ass.C}) is
\begin{align}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_1
&\cong H^0\big(\mathfrak{g},\mc{D}^{\mathrm{ch}}_h(P)_1\big)
=\mc{D}^{\mathrm{ch}}_h(P)_1^G \nonumber \\
&=\Omega^1_h(P)^G\oplus
\big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
\mathscr{X}\in\mc{T}_h(P)^G\big\} \nonumber \\
&=\Omega^1(M)\oplus
\big\{\widetilde X-(\tau_i\widetilde X^j)\rho(\Theta)_{ij}:
X\in\mc{T}(M)\big\} \label{ass.C.1}
\end{align}
\footnote{
By Lemma \ref{lemma.ass}, there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$.
Comparing (\ref{invCDOP.1}) and (\ref{ass.C.1}), we see
that the weight-one component of the said map is surjective
and its kernel is $\mc{T}_v(P)^G$ (since $\lambda=-\lambda_{\t{\sf ad}}$).
}
Keep in mind that we identify $\Omega^*(M)$ with the basic
subspace of $\Omega^*(P)$, and denote the horizontal lift
of any $X\in\mc{T}(M)$ by $\widetilde X\in\mc{T}_h(P)^G$.
Notice that by (\ref{tau.brackets}) the $G$-invariance
of $\widetilde X=\widetilde X^i\tau_i$ means
\begin{align} \label{lift.inv}
A^P\widetilde X^i=-\rho(A)_{ij}\widetilde X^j\quad\textrm{for }A\in\mathfrak{g}.
\end{align}
Also notice that the operator
$\nabla X\in\Gamma(\textrm{End}\,TM)$ lifts horizontally to
\begin{align} \label{DX.lift}
\widetilde{\nabla X}
=\big(d\widetilde X^j+\rho(\Theta)_{jk}\widetilde X^k\big)\otimes\tau_j
=(\tau_i\widetilde X^j)\,\tau^i\otimes\tau_j
\end{align}
where the first equality simply expresses the relation
between $\nabla$ and $\Theta$, and the second equality
follows from (\ref{lift.inv}).
The unusual-looking term in (\ref{ass.C.1}) can be given
a global interpretation using (\ref{DX.lift}).
\end{subsec}
\begin{subsec} \label{sec.ass.C.VAoid}
{\bf The associated vertex algebroid.}
In view of (\ref{ass.C.0}) and (\ref{ass.C.1}),
the extended Lie algebroid associated to $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
is $(C^\infty(M),\Omega^1(M),\mc{T}(M))$ with the
usual structure maps (see \S\ref{VA.VAoid}).
More precisely, we are identifying each $X\in\mc{T}(M)$
with
\begin{align*}
\phantom{WWWW}
\ell X:=\,\textrm{class of }\widetilde X-(\tau_i\widetilde X^j)\rho(\Theta)_{ij}
\;\;\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_1.
\end{align*}
\footnote{
For example, it follows from (\ref{horlift.nonint}) that
$[\ell X,\ell Y]=\ell[X,Y]$, i.e.~the Lie bracket in the
said extended Lie algebroid indeed agrees with the usual
Lie bracket on $\mc{T}(M)$.
}
Consider the vertex algebroid associated to
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ (see \S\ref{VA.VAoid} again):
\begin{eqnarray*}
\big(
C^\infty(M),\Omega^1(M),\mc{T}(M),
\bullet,\{\;\},\{\;\}_\Omega
\big).
\end{eqnarray*}
The explicit expressions of $\bullet,\{\;\},\{\;\}_\Omega$
may not seem very meaningful, but we will obtain
them below for the sole purpose of recovering the
corresponding known expressions for $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$.
Let $V^f$ be the vertex algebra freely generated by
the above vertex algebroid, and
\begin{eqnarray*}
\Phi:V^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
\end{eqnarray*}
the resulting universal map (see \S\ref{VAoid.VA}).
By construction, $\Phi$ is an isomorphism in the two
lowest weights.
In fact, we will see that $\Phi$ is an isomorphism in
all weights, which means $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is freely
generated by its associated vertex algebroid.
\end{subsec}
\begin{prop} \label{prop.ass.C.VAoid}
For $f\in C^\infty(M)$ and $X,Y\in\mc{T}(M)$, we have
\begin{align*}
\ell X\bullet f\quad
&=(\nabla X)f \\
\{\ell X,\ell Y\}\;\;
&=-\mathrm{Tr}\,(\nabla X\cdot\nabla Y) \\
\{\ell X,\ell Y\}_\Omega
&\textstyle
=\mathrm{Tr}\,\Big(-\nabla(\nabla X)\cdot\nabla Y
+\nabla X\cdot\iota_Y R
-\iota_X R\cdot\nabla Y\Big)
+\frac{1}{2}\iota_X\iota_Y H
\end{align*}
\end{prop}
\begin{proof}
Let us verify these expressions at the level of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
Recall the normal-ordered expansions and commutation
relations in Corollary \ref{cor.fund.normal.comm},
as well as those in the larger collection
(\ref{freeVA.comm})--(\ref{freeVA.NOP}).
By definition (\ref{VA.VAoid.456}), $\ell X\bullet f$
is represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align*}
\big(\widetilde X-\tau_i\widetilde X^j\,&\rho(\Theta)_{ij}\big)_{-1}f
-\big(\widetilde{fX}-\tau_i(\widetilde{fX})^j\rho(\Theta)_{ij}\big) \\
&=\big(\tau_{i,-1}\widetilde X^i_0+\widetilde X^i_{-1}\tau_{i,0}\big)f
-f\widetilde X
+\big(\tau_i(f\widetilde X^j)-f\tau_i\widetilde X^j\big)\rho(\Theta)_{ij}
\qquad\qquad \\
&=f\widetilde X+(\tau_i f)d\widetilde X^i
-f\widetilde X
+(\tau_i f)\widetilde X^j\rho(\Theta)_{ij} \\
&=(\tau_i f)\big(d\widetilde X^i+\rho(\Theta)_{ij}\widetilde X^j\big)
\end{align*}
which by (\ref{DX.lift}) proves the claim.
Now keep (\ref{lift.inv}) in mind.
By (\ref{VA.VAoid.456}), $\{\ell X,\ell Y\}$ is represented by
\begin{align*}
\big(\widetilde X-\tau_i\widetilde X^j\,&\rho(\Theta)_{ij}\big)_1
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\rho(\Theta)_{k\ell}\big) \\
&=\big(\widetilde X^i_1\tau_{i,0}+\widetilde X^i_0\tau_{i,1}\big)
\tau_{k,-1}\widetilde Y^k \\
&=-[\tau_{k,-1},\widetilde X^i_1]\,\tau_{i,0}\widetilde Y^k
-[[\tau_{i,0},\tau_{k,-1}],\widetilde X^i_1]\,\widetilde Y^k
+\widetilde X^i_0\,[\tau_{i,1},\tau_{k,-1}]\,\widetilde Y^k \\
&=-(\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)
+\widetilde Y^k\Omega(\tau_i,\tau_k)^P\widetilde X^i
-\widetilde X^i\Omega(\tau_i,\tau_k)^P\widetilde Y^k
+2\textrm{Ric}_{ik}\widetilde X^i\widetilde Y^k \\
&=-(\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)
\end{align*}
which by (\ref{DX.lift}) again proves the claim.
The computation of $\{\ell X,\ell Y\}_\Omega$ is more
tedious and will only be sketched here.
First of all, by (\ref{VA.VAoid.456}) and Lemma
\ref{lemma.1form.0mode}, $\{\ell X,\ell Y\}_\Omega$ is
represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align}
&\quad\;
\big(\widetilde X-\tau_i\widetilde X^j\,\rho(\Theta)_{ij}\big)_0
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\rho(\Theta)_{k\ell}\big)
-\big(\widetilde{[X,Y]}-\tau_i\widetilde{[X,Y]}{}^j\rho(\Theta)_{ij}\big)
\nonumber \\
&=\big(\tau_{i,-1}\widetilde X^i_1
+\widetilde X^i_0\tau_{i,0}
+\widetilde X^i_{-1}\tau_{i,1}\big)\tau_{k,-1}\widetilde Y^k
-\widetilde{[X,Y]}
\nonumber \\
&\qquad\qquad\qquad\qquad
+\iota_{\widetilde Y}d\big(\tau_i\widetilde X^j\,\rho(\Theta)_{ij}\big)
-L_{\widetilde X}\big(\tau_i\widetilde Y^j\,\rho(\Theta)_{ij}\big)
+\tau_i\widetilde{[X,Y]}{}^j\rho(\Theta)_{ij}
\label{ass.C.VAoid3}
\end{align}
With some work, it turns out the first line in
(\ref{ass.C.VAoid3}) can be rewritten as
\begin{align*}
\textstyle
-\Omega(\widetilde X,\widetilde Y)^P
-\tau_i\widetilde Y^k\,d(\tau_k\widetilde X^i)
+\frac{1}{2}\iota_{\widetilde X}\iota_{\widetilde Y}H
+\lambda^*(\Omega(\widetilde X,\widetilde Y),\Theta)
\end{align*}
while the second line in (\ref{ass.C.VAoid3}),
thanks to the first Bianchi identity, equals
\begin{align*}
\big((\tau_i\widetilde X^k)(\tau_k\widetilde Y^j)
-(\tau_i\widetilde Y^k)(\tau_k\widetilde X^j)\big)\rho(\Theta)_{ij}
+\tau_i\widetilde X^j\,\iota_{\widetilde Y}\rho(\Omega)_{ij}
-\tau_i\widetilde Y^j\,\iota_{\widetilde X}\rho(\Omega)_{ij}
-\lambda_\rho(\Omega(\widetilde X,\widetilde Y),\Theta).
\end{align*}
Notice that $\lambda^*=\lambda_\rho$ in our current
setting (see \S\ref{sec.ass.C}) and by the proof of
Lemma \ref{lemma.Feigin.C.1} we may ignore the term
$-\Omega(\widetilde X,\widetilde Y)^P$.
Then it follows from (\ref{tau.brackets}) and (\ref{lift.inv})
that $\{\ell X,\ell Y\}_\Omega$ is also represented by
\begin{align*}
\textstyle
-(\tau_\ell\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)\tau^\ell
+\tau_i\widetilde X^j\,\iota_{\widetilde Y}\rho(\Omega)_{ij}
-\tau_i\widetilde Y^j\,\iota_{\widetilde X}\rho(\Omega)_{ij}
+\frac{1}{2}\iota_{\widetilde X}\iota_{\widetilde Y}H.
\end{align*}
By (\ref{DX.lift}) again this proves our last claim.
\end{proof}
{\it Remark.}
This result recovers the vertex algebroid described in
Theorem \ref{thm.globalCDO}a (as $\nabla$ is now
torsion-free).
In other words, we have $V^f=\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$.
\begin{subsec} \label{sec.ass.C.conformal}
{\bf The conformal vector.}
According to \S\ref{sec.ass.C} and Lemma \ref{lemma.G},
the vertex algebra $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ has a conformal
vector $\nu^M$ of central charge $2d=2\dim M$,
represented by
\begin{align}
\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^P
-Q_0\gamma
&=\mathbf{1}\otimes\Big(\nu^P-t^P_{a,-1}\Theta^a\Big)
\nonumber \\
&\textstyle
=\mathbf{1}\otimes\Big(\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)\Big)
\label{ass.C.conformal}
\end{align}
The key to understanding the whole structure of
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is the observation that $\nu^M$ is
generated by the associated vertex algebroid (in a certain way).
In fact, as we will see, its representative
(\ref{ass.C.conformal}) can be expressed entirely
in terms of elements of (\ref{ass.C.1}).
\end{subsec}
For the following statement, notice that by assumption
$\pi:P\rightarrow M$ is a lifting of the special orthogonal
frame bundle $F_{SO}(TM)\rightarrow M$
(see \S\ref{sec.ass.C}).
\begin{prop} \label{prop.ass.C.conformal}
The vertex algebra $V^f$ has a conformal vector $\nu^f$
with $\Phi(\nu^f)=\nu^M$ (see \S\ref{sec.ass.C.VAoid}).
Given an open subset $U\subset M$ and a smooth
section $\sigma:U\rightarrow\pi^{-1}(U)\subset P$ of
$\pi$, there is a local expression
\begin{align*}
\textstyle
\nu^f|_U
=(\ell\sigma_i)_{-1}\sigma^i
+\frac{1}{2}\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma)
+\sigma^i([\sigma_j,\sigma_k])\,
\sigma^k_{-1}\Gamma^\sigma_{ji}
\end{align*}
\footnote{ \label{underlying.sheaf}
Recall from \S\ref{sec.review} that, like most
constructions in this paper, $V^f$ is the space of global
sections of an underlying sheaf.
}
where $(\sigma_1,\ldots,\sigma_d)$ is the
$C^\infty(U)$-basis of $\mc{T}(U)$ induced by $\sigma$;
$(\sigma^1,\ldots,\sigma^d)$ the dual basis of
$\Omega^1(U)$;
and $\Gamma^\sigma=\rho(\sigma^*\Theta)
\in\Omega^1(U)\otimes\mathfrak{so}_d$.
In particular, $\nu^f\in\mathcal{F}_{\preceq(-1;-1)}V^f$
(see \S\ref{freeVA.PBW}).
\end{prop}
\begin{proof}
Consider the smooth map $g:\pi^{-1}(U)\rightarrow G$
defined by $\sigma(\pi(p))=p\cdot g(p)$ for
$p\in\pi^{-1}(U)$.
To ease notations, we also write
$\rho(g):\pi^{-1}(U)\rightarrow SO(\mathbb{R}^d)$ simply
as $g$.
Then we have
\begin{eqnarray} \label{frame.lift}
\widetilde\sigma_i=g_{ri}\tau_r,\qquad
\sigma^i=g_{ri}\tau^r,\qquad
\tau_r=g_{ri}\widetilde\sigma_i,\qquad
\tau^r=g_{ri}\sigma^i.
\end{eqnarray}
Since $\nabla\sigma_i=\Gamma^\sigma_{ji}\otimes\sigma_j$,
it follows from (\ref{DX.lift}) and (\ref{frame.lift}) that
\begin{eqnarray} \label{dg}
dg+\rho(\Theta)\cdot g
=(\tau_k g)\tau^k
=g\cdot\Gamma^\sigma
\end{eqnarray}
where $\cdot$ denotes matrix multiplication.
The main task here is to express (\ref{ass.C.conformal})
entirely in terms of $\sigma$.
Let us first write down
\begin{align*}
\tau_{r,-1}\tau^r
&=\tau_{r,-1}g_{ri,0}\sigma^i
=(\tau_{r,-1}g_{ri})_{-1}\sigma^i
-\big(g_{ri,-1}\tau_{r,0}+g_{ri,-2}\tau_{r,1}\big)
g_{si}\tau^s \\
&\textstyle
=\widetilde\sigma_{i,-1}\sigma^i
-(\tau_r g_{si})\tau^s_{-1}dg_{ri}
-\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}dg\big)
-\frac{1}{2}\mathrm{Tr}\,\big(g^{-1}\cdot Tdg\big)
\end{align*}
using (\ref{frame.lift}),
Corollary \ref{cor.fund.normal.comm} and
the Lie derivative $L_{\tau_r}\tau^s=\rho(\Theta)_{sr}$
implied by (\ref{tau.brackets}).
Then we work on each term separately, using
(\ref{frame.lift}) and (\ref{dg}) a number of times:
\begin{align*}
\textrm{2nd term}
&=(\tau_r g_{si})\tau^s_{-1}
\big(\rho(\Theta)_{rt}g_{ti}
-g_{rj}\Gamma^\sigma_{ji}\big) \\
&=-g_{si}(\tau_r g_{ti})\tau^s_{-1}\rho(\Theta)_{rt}
+g_{si} g_{rj}(\tau_r g_{sk})\sigma^k_{-1}
\Gamma^\sigma_{ji} \\
&=-(\tau_r\widetilde\sigma_i^t)\sigma^i_{-1}\rho(\Theta)_{rt}
+g_{si} g_{rk}(\tau_r g_{sj})\sigma^k_{-1}
\Gamma^\sigma_{ji}
+\sigma^i([\widetilde\sigma_j,\widetilde\sigma_k])\sigma^k_{-1}
\Gamma^\sigma_{ji} \\
&=-(\tau_r\widetilde\sigma_i^t)\sigma^i_{-1}\rho(\Theta)_{rt}
+\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma\big)
+\sigma^i([\sigma_j,\sigma_k])\sigma^k_{-1}
\Gamma^\sigma_{ji} \\
\textrm{3rd term}
&=\lambda_\rho(\Theta_{-1}\Theta)
-\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}\cdot
g\cdot\Gamma^\sigma\big) \\
\textrm{4th term}
&\textstyle
=-\frac{1}{2}\,T\big(\mathrm{Tr}\,(g^{-1}dg)\big)
-\frac{1}{2}\mathrm{Tr}\,\big(g^{-1}\cdot dg_{-1}
\cdot g^{-1}\cdot dg\big) \\
&\textstyle
=0-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
+\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}\cdot
g\cdot\Gamma^\sigma\big)
-\frac{1}{2}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)
\end{align*}
These calculations together yield an identity
in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(\pi^{-1}(U))$:
\begin{align} \label{ass.C.conformal.local}
\textstyle
\tau_{r,-1}\tau^r
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
=\big(\widetilde\sigma_i
-\tau_r\widetilde\sigma^t_i\,\rho(\Theta)_{rt}\big)_{-1}
\sigma^i
+\frac{1}{2}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)
+\sigma^i([\sigma_j,\sigma_k])\sigma^k_{-1}
\Gamma^\sigma_{ji}.
\end{align}
The left hand side, as observed in
\S\ref{sec.ass.C.conformal}, represents
$\nu^M\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$;
and it is defined globally on $P$.
Now that the right hand side is manifestly generated by
the subspace (\ref{ass.C.1}), it defines an element
$\nu^f\in V^f$ such that $\Phi(\nu^f)=\nu^M$.
It remains to show that $\nu^f$ is a conformal
vector of $V^f$.
According to \cite{Kac,FB-Z}, this amounts to
checking:
(i) $\nu^f_{-1}=T$,
(ii) $\nu^f_0=L_0$, and
(iii) $\nu^f_2\nu^f\in\mathbb{C}$.
For (i) and (ii) it suffices to check them on
\begin{eqnarray*}
C^\infty(U)\cup\{\ell\sigma_1,\ldots,\ell\sigma_d\}
\end{eqnarray*}
because $V^f$ is locally generated by these elements.
In fact, since $\Phi(\nu^f)=\nu^M$ is known to be conformal
and $\Phi$ is an isomorphism in weights $0$ and $1$,
everything we need to check automatically holds except
for the equation $\nu^f_{-1}\ell\sigma_i=T(\ell\sigma_i)$
in weight $2$.
This equation can be verified by a straightforward calculation,
which we omit.
\end{proof}
{\it Remark.}
Since we have already observed that $V^f=\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$,
this result recovers the conformal vector $\nu^0$ described
in Theorem \ref{thm.globalCDO}b (as $\Gamma^\sigma$ is
now traceless).
\begin{corollary} \label{cor.ass.C.freeVA}
The map of vertex algebras
$\Phi:V^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
(see \S\ref{sec.ass.C.VAoid}) is an isomorphism.
\end{corollary}
\begin{proof}
By Proposition \ref{prop.ass.C.conformal},
the conformal vector $\nu^M\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ belongs to
$\mathcal{F}_{\preceq(-1;-1)}$.
Then Lemma \ref{lemma.conf.gen} applies so that
$\Phi$ is surjective.
By construction, the ideal $\ker\Phi\subset V^f$ is
trivial in weights $0$ and $1$.
Then by Proposition \ref{prop.ass.C.conformal} again
and Lemma \ref{lemma.conf.ideal}, $\ker\Phi$ is
in fact trivial in all weights.
\end{proof}
Propositions \ref{prop.ass.C.VAoid},
\ref{prop.ass.C.conformal} and
Corollary \ref{cor.ass.C.freeVA} together show
that $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})\cong\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$
as conformal vertex algebras.
This fulfills the goal of this section.
Let us summarize our work in the form of a new
description of algebras of CDOs.
\begin{theorem} \label{thm.CDO.semiinf}
Suppose $\pi:P\rightarrow M$ is a smooth prinicpal
$G$-bundle and $\rho:G\rightarrow SO(\mathbb{R}^d)$ is
a representation such that there is an isomorphism
$P\times_\rho\mathbb{R}^d\cong TM$.
Given a principal frame $(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ (see Theorem \ref{thm.CDOP}), the zeroth
semi-infinite cohomology
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
\end{eqnarray*}
is an algebra of CDOs on $M$.
Up to isomorphism, every algebra of CDOs on $M$ arises
this way.
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ as a vertex algebra is freely generated
by its weight-zero and weight-one components, which are
represented by the following subspaces of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ bijectively:
\begin{align*}
\pi^*C^\infty(M),\qquad
\pi^*\Omega^1(M)\oplus
\big\{\widetilde X-\tau_i\widetilde X^j\,\rho(\Theta)_{ij}:
X\in\mc{T}(M)\big\}.
\end{align*}
(For details of the latter representation, see the proof
of Lemma \ref{lemma.Feigin.C.1}.)
Moreover, $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ has a conformal vector
of central charge $2d=2\dim M$, represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align*}
\textstyle
\qquad
\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta).
\qquad\qedsymbol
\end{align*}
\end{theorem}
To conclude this section, let us sketch an extension of
Theorem \ref{thm.CDO.semiinf}.
\begin{subsec} \label{CDO.cs}
{\bf Generalization to supermanifolds.}
Suppose $\pi:P\rightarrow M$ and
$\rho:G\rightarrow SO(\mathbb{R}^d)$ are the same as above;
also let $\rho':G\rightarrow U(\mathbb{C}^r)$ be another
representation, $E=P\times_{\rho'}\mathbb{C}^r$ the associated
vector bundle and $\Pi E$ the corresponding cs-manifold.
\cite{QFS.susy}
The $G$-action on
$\mathcal{O}(\Pi\mathbb{R}^r)\otimes\mathbb{C}=\wedge^*(\mathbb{C}^r)^\vee$
induced by $\rho'$ lifts to an inner
$(\hat{\mathfrak{g}}_{\lambda_{\rho'}},G)$-action on
$\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)$, a fermionic analogue of
\S\ref{algCDO} (see also Definition \ref{formalLG.action}).
By Theorem \ref{thm.CDOP}, there exists a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ with
$\lambda+\lambda_{\rho'}=-\lambda_{\t{\sf ad}}$ if and only if
\begin{eqnarray*}
\lambda^*(P)=(\lambda_\rho-\lambda_{\rho'})(P)=0
\qquad\iff\qquad
p_1(M)-ch_2(E)=0.
\end{eqnarray*}
In this case, we can apply Definition \ref{defn.ass}
to construct a vertex superalgebra
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))
=H^{\frac{\infty}{2}+0}\Big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}\,,\,
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)\Big).
\end{eqnarray*}
Moreover, the $(\hat{\mathfrak{g}}_{\lambda_{\rho'}},G)$-action
on $\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)$ is primary if and only if
\begin{eqnarray*}
\rho'(\mathfrak{g})\subset\mathfrak{su}_r
\qquad\iff\qquad
c_1(E)=0.
\end{eqnarray*}
In this case, $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))$
has a conformal vector of central charge $2(d-r)$.
By a similar analysis, $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))$ is
an algebra of CDOs on $\Pi E$ in the sense of \cite{myCDO}.
In particular, in the case $\rho'=\rho_{\mathbb{C}}$
(i.e. $E=TM_{\mathbb{C}}$), both obstructions are trivial
and $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^d))$ is the smooth
chiral de Rham algebra of $M$.
\end{subsec}
\newpage
\setcounter{equation}{0}
\section{Example: Spinor Module over CDOs}
\label{sec.FL.spinor}
Given a spin manifold $M$ with a trivialization of its first
Pontrjagin form, we are interested in the space of sections
of the hypothetical ``spinor bundle on the formal loops of $M$''
as a module over its ``differential operators''.
The object of study in this section \emph{should} have the
above geometric meaning, but is more precisely an example
of the construction introduced in \S\ref{sec.ass} defined
using semi-infinite cohomology.
Following a similar strategy as in \S\ref{CDOP.CDOM}, we
analyze this spinor module so as to obtain a more explicit
description in terms of generating data and relations.
It is hoped that a deeper understanding, including the
identification of an appropriate Dirac operator, will lead to
a useful geometric theory of the Witten genus.
(This was in fact the original motivation of the paper.)
In this section, $G=\textrm{Spin}_d$ and
$\mathfrak{g}=\mathfrak{spin}_d=\mathfrak{so}_d$ where $d=2d'$ is even
(except in \S\ref{sec.Cl}).
Since $\mathfrak{so}_d$ is simple, we use the more common notation
$(\widehat{\mathfrak{so}}_d)_k$ in place of
$(\widehat{\mathfrak{so}}_d)_{k\lambda_0}$ (see \S\ref{sec.Gmfld}),
where $\lambda_0$ is the normalized Killing form,
i.e. $\lambda_0(A,B)=\frac{1}{2}\mathrm{Tr}\, AB$\, for $A,B\in\mathfrak{so}_d$.
\begin{subsec} \label{sec.Cl}
{\bf The Ramond Clifford algebra.}
Let $C\ell$ be the unital $\mathbb{Z}/2\mathbb{Z}$-graded
associative $\mathbb{C}$-algebra with the following generators
and relations
\begin{eqnarray} \label{Cl}
e_{i,n}\textrm{ odd},\;\;i=1,\ldots,d,\;\;n\in\mathbb{Z},
\qquad
[e_{i,n},e_{j,m}]
=e_{i,n}e_{j,m}+e_{j,m}e_{i,n}
=-2\delta_{ij}\delta_{n+m,0}.
\end{eqnarray}
(The notations are chosen to resemble those in \cite{spin}.)
Suppose $W$ is a $C\ell$-module with the property that
for each $w\in W$, there exists $N\in\mathbb{Z}$ such that
$e_{i,n}w=0$ for $n>N$.
Then the operators
\begin{eqnarray} \label{Cl.so}
A^{C\ell}_n
=\frac{1}{4}A_{ji}\,e_{i,n-r}e_{j,r}
\quad\textrm{for }A\in\mathfrak{so}_d,\;n\in\mathbb{Z}
\end{eqnarray}
define an $(\widehat{\mathfrak{so}}_d)_1$-action on $W$.
On the other hand, the operators
\begin{eqnarray} \label{Cl.Virasoro}
L^{C\ell}_n
=-\frac{1}{8}\sum_{r\geq 0}(2r-n)e_{i,n-r}e_{i,r}
+\frac{1}{8}\sum_{r<0}(2r-n)e_{i,r}e_{i,n-r}
+\frac{d}{16}\delta_{n,0}\quad
\textrm{for }n\in\mathbb{Z}
\end{eqnarray}
define a Virasoro action on $W$ of central charge
$d/2$.
This is in fact the Sugawara construction associated to
the above $(\widehat{\mathfrak{so}}_d)_1$-action, and
accordingly satisfies
\begin{eqnarray*}
[L^{C\ell}_n,A^{C\ell}_m]=-mA^{C\ell}_{n+m}\quad
\textrm{for }n,m\in\mathbb{Z}.
\end{eqnarray*}
The eigenvalues of $L^{C\ell}_0$ are called weights
(as in previous sections) and each $e_{i,n}$ changes
weights by $-n$.
For more details, see e.g. \cite{Fuchs}
\end{subsec}
For the rest of the section, $d=2d'$ is even.
\begin{subsec} \label{sec.S}
{\bf The spinor representation of $\widehat{\mathfrak{so}}_{2d'}$.}
Let $C\ell_0$ (resp.~$C\ell_{\geq 0}$) be the
subalgebra of $C\ell$ generated by those $e_{i,n}$ with
$n=0$ (resp.~$n\geq 0$).
The finite-dimensional Clifford algebra $C\ell_0$ has a unique
(up to isomorphism) irreducible $\mathbb{Z}/2\mathbb{Z}$-graded
representation $S_0$.
Let us also regard $S_0$ as a $C\ell_{\geq 0}$-module on which
$\{e_{i,n}\}_{n>0}$ act trivially.
Then define a $C\ell$-module by
\begin{eqnarray*}
S=C\ell\otimes_{C\ell_{\geq 0}}S_0.
\end{eqnarray*}
By (\ref{Cl}), $S$ as a vector space is spanned by elements of
the form
\begin{eqnarray*}
e_{i_p,n_p}\cdots e_{i_1,n_1}s,\qquad
n_1<0,\;\;
(n_p,i_p)<\cdots<(n_1,i_1),\;\;
s\in S_0
\end{eqnarray*}
where the indicated pairs are ordered lexicographically.
According to \S\ref{sec.Cl}, $S$ admits
a $((\widehat{\mathfrak{so}}_{2d'})_1,\textrm{Spin}_{2d'})$-action
as well as a Virasoro action of central charge $d'$, such that
the former is primary.
Notice that the element displayed above has weight
\begin{eqnarray*}
\frac{d'}{8}+|n_1|+\ldots+|n_p|.
\end{eqnarray*}
For convenience, we will write $S_k\subset S$ for
the component of weight $(d'/8)+k$.
\end{subsec}
Given a representation
$\rho:\mathfrak{so}_{2d'}\rightarrow\mathfrak{gl}(V)$, let $\lambda_\rho$
denote the invariant symmetric bilinear form on $\mathfrak{so}_{2d'}$
defined by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, notice that $\lambda_\rho=2\lambda_0$ for
the standard representation $\rho$, and
$\lambda_{\t{\sf ad}}=(4d'-4)\lambda_0$.
\begin{subsec} \label{sec.ass.S}
{\bf The spinor module over CDOs.}
Consider the special case of Definition \ref{defn.ass}
associated to a principal frame
$((\hat{\mathfrak{so}}_{2d'})_{3-4d'},\textrm{Spin}_{2d'})$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ and the
$((\hat{\mathfrak{so}}_{2d'})_1,\textrm{Spin}_{2d'})$-module $S$:
\begin{align}
\Gamma^{\mathrm{ch}}(\pi,S)
&=H^{\frac{\infty}{2}+0}\Big(
(\widehat{\mathfrak{so}}_{2d'})_{4-4d'}\,,
\,\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S\Big) \nonumber \\
&=H^0\Big(
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)
\otimes S\,,\,Q_0
\Big) \label{ass.S}
\end{align}
By Lemma \ref{lemma.ass} and \S\ref{sec.S},
$\Gamma^{\mathrm{ch}}(\pi,S)$ is a module over the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$ and
admits a Virasoro action of central charge
$5d'=\frac{5}{2}\dim M$.
For convenience, let us elaborate on all the data involved
and recall some relevant notations.
\begin{itemize}
\item[$\centerdot$]
Let $(t_1,t_2,\ldots)$ be a basis of $\mathfrak{so}_{2d'}$,
$(t^1,t^2,\ldots)$ the dual basis of $(\mathfrak{so}_{2d'})^\vee$,
$(\phi^1,\phi^2,\ldots)$ the corresponding
coordinates of the supermanifold $\Pi\mathfrak{so}_{2d'}$, and
$(\partial_1,\partial_2,\ldots)$ their coordinate vector fields.
\vspace{-0.05in}
\item[$\centerdot$]
For comments on the conformal vertex superalgebra
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})$, see \S\ref{sec.ass.C}.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, see Theorem \ref{thm.CDOP} with
$\lambda=(3-4d')\lambda_0$ in mind.
Let us mention that it is defined using the following
data:
a principal $\textrm{Spin}_{2d'}$-frame bundle
$\pi:P\rightarrow M$;
the Levi-Civita connection $\Theta=\Theta^a\otimes t_a$
on $\pi$;
and a basic $3$-form $H$ on $P$ that satisfies
$dH=\lambda_0(\Omega\wedge\Omega)$, where
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Also, there is a conformal vector
\begin{eqnarray*}
\nu^P=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda_0(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
(For the notations see \S\ref{sec.prin.bdl2}.)
\vspace{-0.05in}
\item[$\centerdot$]
For details on the Feigin complex in (\ref{ass.S}),
see \S\ref{sec.Feigin}.
Let us mention that its differential is
\begin{align} \label{ass.S.Q.theta}
Q_0
=q_0\otimes 1\otimes 1
+\phi^a_{-r}\otimes t^P_{a,r}\otimes 1
+\phi^a_{-r}\otimes 1\otimes t^{C\ell}_{a,r}
\end{align}
where $q$ is the odd vector field shown in (\ref{JQ.coor}).
Also the Virasoro operators
\begin{align*}
\qquad\qquad
L^{\Pi{\mathfrak{so}}}_n\otimes 1\otimes 1
+1\otimes L^P_n\otimes 1
+1\otimes 1\otimes L^{C\ell}_n,
\qquad n\in\mathbb{Z}
\end{align*}
preserve the gradation and commute with $Q_0$.
\vspace{-0.05in}
\end{itemize}
Let us repeat the geometric ingredients in
slightly different words:
$M$ is a Riemannian manifold with a spin structure
and a $3$-form $H$ that satisfies
$dH=\frac{1}{2}\mathrm{Tr}\,(R\wedge R)$, where $R$ is the
Riemannian curvature.
This can be viewed as the de Rham version of
a \emph{string structure}.
In the sequel, we will describe $\Gamma^{\mathrm{ch}}(\pi,S)$ more
explicitly in terms of generating data (i.e. a subspace and
some fields) and their relations.
\end{subsec}
Throughout this section we identify $\Omega^*(M)$ with
the basic subspace of $\Omega^*(P)$.
\begin{subsec} \label{sec.ass.S.0}
{\bf The component of the lowest weight.}
Consider the Feigin complex appearing in (\ref{ass.S}):
\begin{eqnarray} \label{Feigin.S}
\Big(
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)
\otimes S\,,\,Q_0
\Big).
\end{eqnarray}
Since its component of weight $d'/8$ is simply
the Chevalley-Eilenberg complex (see \S\ref{sec.CE})
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{so}_{2d'})\otimes C^\infty(P)
\otimes S_0\,,\,Q_0\Big)
\end{eqnarray*}
the component of (\ref{ass.S}) of weight $d'/8$ is
\begin{eqnarray} \label{ass.S.0}
\Gamma^{\mathrm{ch}}(\pi,S)_0
=H^0\big(\mathfrak{so}_{2d'},C^\infty(P)\otimes S_0\big)
=(C^\infty(P)\otimes S_0)^{\mathfrak{so}_{2d'}}
=S(M)
\end{eqnarray}
i.e.~the space of sections of the spinor bundle on $M$.
Understanding the rest of $\Gamma^{\mathrm{ch}}(\pi,S)$ requires more work,
just as in \S\ref{CDOP.CDOM}.
\end{subsec}
\begin{lemma} \label{lemma.G.S}
Let $\gamma=\partial_a\otimes\Theta^a\in
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})_1\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$.
For $n\in\mathbb{Z}$ we have
\begin{eqnarray*}
[Q_0,\,\gamma_n\otimes 1]
=L^{\Pi\mathfrak{so}}_n\otimes 1\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_n\otimes 1
+1\otimes\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}\,.
\end{eqnarray*}
Moreover, for $n\in\mathbb{Z}$ and
$\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}
\subset\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ we also have
\begin{eqnarray*}
[Q_0,\,
(\gamma_0(1\otimes\mathscr{Y}))_n\otimes 1]
=1\otimes\mathscr{Y}_n\otimes 1
+1\otimes\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}\,.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Recall the differential $Q_0$ from (\ref{ass.S.Q.theta}).
The first anticommutator is computed as follows
\begin{align*}
[Q_0,\,\gamma_n\otimes 1]
&=[(q\otimes 1+\phi^a\otimes t^P_a)_0,\,\gamma_n]\otimes 1
+[\phi^a_{-r},\partial_{b,n-s}]\otimes\Theta^b_s
\otimes t^{C\ell}_{a,r} \\
&=L^{\Pi\mathfrak{so}}_n\otimes 1\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_n\otimes 1
+1\otimes\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
\end{align*}
using Lemma \ref{lemma.G} and the fermionic analogue of
(\ref{Weyl}).
On the other hand, since
$\gamma_0(1\otimes\mathscr{Y})=\partial_b\otimes\mathscr{Y}^b$,
the second anticommutator can be written as
\begin{align*}
[Q_0,(\partial_b\otimes\mathscr{Y}^b)_n\otimes 1]
&=[(q\otimes 1+\phi^a\otimes t^P_a)_0,
(\partial_b\otimes\mathscr{Y}^b)_n]\otimes 1
+[\phi^a_{-r},\partial_{b,n-s}]\otimes\mathscr{Y}^b_s
\otimes t^{C\ell}_{a,r} \\
&=\big((q\otimes 1+\phi^a\otimes t^P_a)_0
(\partial_b\otimes\mathscr{Y}^b)\big)_n\otimes 1
+1\otimes\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}
\end{align*}
using again the analogue of (\ref{Weyl}).
It remains to compute the element
$(q\otimes 1+\phi^a\otimes t^P_a)_0
(\partial_b\otimes\mathscr{Y}^b)$, which is the sum of the
following:
\begin{align*}
&(q_0\partial_b)\otimes\mathscr{Y}^b
=[q,\partial_b]\otimes\mathscr{Y}^b
=-t^c([t_b,t_d])\phi^d\partial_c\otimes\mathscr{Y}^b \\
&(\phi^a_0\otimes t^P_{a,0})(\partial_b\otimes\mathscr{Y}^b)
=\phi^a\partial_b\otimes t^P_a\mathscr{Y}^b
=-t^b([t_a,t_c])\phi^a\partial_b\otimes\mathscr{Y}^c \\
&(\phi^a_1\otimes t^P_{a,-1})(\partial_b\otimes\mathscr{Y}^b)
=\delta^a_b\otimes t^P_{a,-1}\mathscr{Y}^b
=1\otimes\mathscr{Y}
\end{align*}
Indeed, the first line is similar to a computation
in the proof of Lemma \ref{lemma.G};
the second follows from the
$\mathfrak{so}_{2d'}$-invariance of $\mathscr{Y}$;
and the last follows from the analogue of
(\ref{Weyl}) and Corollary
\ref{cor.fund.normal.comm}.
\end{proof}
{\it Preparation.}
Let $C\ell(M)=(C^\infty(P)\otimes
C\ell_0)^{\mathfrak{so}_{2d'}}$ which is the same as the algebra
of $C^\infty(M)$-linear endomorphisms of $S(M)$.
Given $X\in\mc{T}(M)$, let us denote its horizontal lift by
$\widetilde X=\widetilde X^i\tau_i\in\mc{T}_h(P)^{\mathfrak{so}_{2d'}}$ (as usual) and
its Clifford action by $cX=\widetilde X^i\otimes e_{i,0}\in C\ell(M)$.
Notice that $\{cX:X\in\mc{T}(M)\}$ generates $C\ell(M)$
as an algebra.
On the other hand, each
$\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$
satisfies
\begin{align*}
\mathscr{Y}\otimes 1+\mathscr{Y}^a\otimes t_a=0
\quad\textrm{on}\quad
(C^\infty(P)\otimes S_0)^{\mathfrak{so}_{2d'}}=S(M)
\end{align*}
and hence represents an endomorphism
$c\mathscr{Y}=-\mathscr{Y}^a\otimes t_a\in C\ell(M)$.
\footnote{
To be precise, we are regarding $\mathfrak{so}_d$ as a Lie
subalgebra of $C\ell_0$ via the standard inclusion
$A\mapsto\frac{1}{4}A_{ji}e_{i,0}e_{j,0}$.
}
\begin{subsec} \label{sec.ass.S.fields}
{\bf Fields of low weights.}
Recall Lemma \ref{lemma.semiinf.ops}a for the special case
(\ref{ass.S}):
the $\widehat{\mathfrak{so}}_{2d'}$-invariant fields
on $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$, including
the vertex operators of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$,
induce fields on $\Gamma^{\mathrm{ch}}(\pi,S)$.
Let us now describe a collection of such fields
(or rather their Fourier modes);
we will later see that their actions on the subspace
$\Gamma^{\mathrm{ch}}(\pi,S)_0=S(M)$ generate the entire space
$\Gamma^{\mathrm{ch}}(\pi,S)$.
Given $f\in C^\infty(M)$ and $X\in\mc{T}(M)$, there are fields
on $\Gamma^{\mathrm{ch}}(\pi,S)$ whose Fourier modes are
\begin{align}
f_n
&:=\textrm{the operator induced by }f_n\otimes 1
\nonumber \\
\ell X_n
&:=\textrm{the operator induced by }
(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij})_n\otimes 1
\label{ass.S.gen.fields} \\
c X_n
&:=\textrm{the operator induced by }
\widetilde X^i_{n-r}\otimes e_{i,r}
\nonumber
\end{align}
Indeed, the first two are well-defined because both $f$ and
$\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}$ belong to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$
according to \S\ref{sec.invCDO.can};
and so is the last one because for $A\in\mathfrak{so}_{2d'}$ and
$m\in\mathbb{Z}$ we have
\begin{align*}
\big[A^P_m\otimes 1+1\otimes A^{C\ell}_m,\,
\widetilde X^i_{n-r}\otimes e_{i,r}\big]
&=[A^P_m,\widetilde X^i_{n-r}]\otimes e_{i,r}
+\widetilde X^i_{n-r}\otimes[A^{C\ell}_m,e_{i,r}] \\
&=(A^P\widetilde X^i)_{m+n-r}\otimes e_{i,r}
+\widetilde X^i_{n-r}\otimes(A_{ji}e_{j,m+r}) \\
&=-A_{ij}\widetilde X^j_{m+n-r}\otimes e_{i,r}
+A_{ji}\widetilde X^i_{n-r}\otimes e_{j,m+r} \\
&=0
\end{align*}
where we have used (\ref{freeVA.comm}), the
$\mathfrak{so}_{2d'}$-invariance of $\widetilde X$ and
(\ref{Cl})--(\ref{Cl.so}).
These three types of fields have weights
$0$, $1$ and $\frac{1}{2}$ respectively.
It will be convenient to also consider some other
fields generated by those in (\ref{ass.S.gen.fields}).
Given $\alpha\in\Omega^1(M)$ and
$\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$, there are fields on
$\Gamma^{\mathrm{ch}}(\pi,S)$ whose Fourier modes are
\begin{align} \label{ass.S.other.fields}
\begin{array}{rl}
\alpha_n
&:=\textrm{the operator induced by }\alpha_n\otimes 1 \\
c\mathscr{Y}_n
&:=\textrm{the operator induced by }
(\mathscr{Y}+\lambda_0(\Theta,\mathscr{Y}))_n\otimes 1
\phantom{\Big)}
\end{array}
\end{align}
These are well-defined because both $\alpha$ and
$\mathscr{Y}+\lambda_0(\Theta,\mathscr{Y})$ belong to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$ according to
\S\ref{sec.invCDO.can}.
To see that $\alpha_n$ is determined by
(\ref{ass.S.gen.fields}), it suffices to consider $\alpha=fdg$
and notice that
\begin{eqnarray} \label{ass.S.1formfield}
(fdg)_n=-\sum_{r\in\mathbb{Z}}r f_{n-r}g_r\quad
\textrm{for }f,g\in C^\infty(M)
\end{eqnarray}
by (\ref{freeVA.comm})--(\ref{freeVA.NOP}).
On the other hand, by Lemma \ref{lemma.G.S} we may represent
$c\mathscr{Y}_n$ alternatively by
\begin{align}
-\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}
+\lambda_0(\Theta,\mathscr{Y})_n\otimes 1.
\label{vertical.Cl}
\end{align}
The next lemma explains how $c\mathscr{Y}_n$ is determined by
(\ref{ass.S.gen.fields}).
\end{subsec}
\begin{lemma} \label{lemma.verfield}
For $\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$, the operator
$c\mathscr{Y}_n$ on $\Gamma^{\mathrm{ch}}(\pi,S)$ is a sum of operators of the form
\begin{align} \label{cX.cY.normal}
\sum_{r\geq 0}cX_{n-r}\,cY_r
-\sum_{r<0}cY_r\,cX_{n-r}
-2\langle X,\nabla Y\rangle_n
+\langle X,Y\rangle_n,\quad
X,Y\in\mc{T}(M)
\end{align}
where $\langle\;\rangle$ and $\nabla$ denote the Riemannian
metric and Levi-Civita connection on $TM$.
\end{lemma}
\begin{proof}
Let $\mathscr{Y}_{ij}=\mathscr{Y}^a(t_a)_{ij}\in C^\infty(P)$.
As explained above, $c\mathscr{Y}_n$ is represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by (\ref{vertical.Cl}),
which by (\ref{Cl.so}) and (\ref{freeVA.NOP}) can be
written as
\begin{align*}
\textstyle
\frac{1}{4}\,\big(\mathscr{Y}_{ij,n-r}\otimes 1\big)
\big(1\otimes e_{i,r-s}e_{j,s}
-2\Theta_{ij,r}\otimes 1\big).
\end{align*}
Since $c\mathscr{Y}=\frac{1}{4}\,\mathscr{Y}_{ij}\otimes e_{i,0}e_{j,0}$
corresponds to a $2$-vector field under the isomorphism
$C\ell(M)\cong\Gamma(\wedge^* TM)$, (\ref{vertical.Cl}) can
be expressed further as a sum of operators of the form
\begin{align} \label{Cl2.field}
\textstyle
\qquad
\frac{1}{2}\,
\big((\widetilde X^i\widetilde Y^j-\widetilde Y^i\widetilde X^j)_{n-r}
\otimes 1\big)
\big(1\otimes e_{i,r-s}e_{j,s}
-2\Theta_{ij,r}\otimes 1\big),\quad
X,Y\in\mc{T}(M)
\end{align}
where the factor of $\frac{1}{2}$ is included for
convenience.
Let us show that (\ref{Cl2.field}) represents
(\ref{cX.cY.normal}), which will then prove the lemma.
For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}).
Consider the first two terms in (\ref{cX.cY.normal}) together.
By definition (\ref{ass.S.gen.fields}) they are represented by
\begin{align*}
\sum_{r\geq 0}(\widetilde X^i_{n-r-s}\otimes e_{i,s})
(\widetilde Y^j_{r-t}\otimes e_{j,t})
-\sum_{r<0}(\widetilde Y^j_{r-t}\otimes e_{j,t})
(\widetilde X^i_{n-r-s}\otimes e_{i,s}).
\end{align*}
In view of (\ref{Cl}), let us split this sum into three parts.
The first part consists of terms with $i\neq j$:
\begin{align*}
\sum_r\sum_{i\neq j}\sum_{s,t}
\widetilde X^i_{n-r-s}\widetilde Y^j_{r-t}\otimes e_{i,s}e_{j,t}
=\sum_{i\neq j}\sum_{s,t}
(\widetilde X^i\widetilde Y^j)_{n-s-t}\otimes e_{i,s}e_{j,t}\,.
\end{align*}
The second part consists of terms with $i=j$ and
$s+t\neq 0$, which vanishes by symmetry:
\begin{align*}
\sum_r\sum_i\sum_{s+t\neq 0}
\widetilde X^i_{n-r-s}\widetilde Y^i_{r-t}\otimes e_{i,s}e_{i,t}
=\sum_i\sum_{s+t\neq 0}
(\widetilde X^i\widetilde Y^i)_{n-s-t}\otimes e_{i,s}e_{i,t}
=0\,.
\end{align*}
The last part consists of terms with $i=j$ and $s+t=0$,
which is computed as follows:
\begin{align*}
\sum_{r\geq 0}\sum_{i,s}
\widetilde X^i_{n-r-s}&\widetilde Y^i_{r+s}\otimes e_{i,s}e_{i,-s}
-\sum_{r<0}\sum_{i,s}
\widetilde X^i_{n-r-s}\widetilde Y^i_{r+s}\otimes e_{i,-s}e_{i,s} \\
&=\sum_{i,u}\widetilde X^i_{n-u}\widetilde Y^i_u\otimes
\Big(\sum_{s\geq -u}e_{i,-s}e_{i,s}
-\sum_{s>u}e_{i,-s}e_{i,s}\Big) \\
&=-\sum_{i,u}(2u+1)\widetilde X^i_{n-u}\widetilde Y^i_u\otimes 1 \\
&=-(\widetilde X^i\widetilde Y^i)_n\otimes 1
+2(\widetilde X^i d\widetilde Y^i)_n\otimes 1 \\
&=-\langle X,Y\rangle_n\otimes 1
+2\langle X,\nabla Y\rangle_n\otimes 1
-2\big(\widetilde X^i\widetilde Y^j\Theta_{ij}\big)_n\otimes 1
\end{align*}
where we have used (\ref{DX.lift}) in the last equality.
It follows from these calculations and a little reorganization
that (\ref{cX.cY.normal}) is indeed represented by
(\ref{Cl2.field}) as claimed.
\end{proof}
{\it Remark.}
Since we have
$cX\cdot cY=-\langle X,Y\rangle\otimes 1+\frac{1}{2}
(\widetilde X^i\widetilde Y^j-\widetilde Y^i\widetilde X^j)\otimes e_{i,0}e_{j,0}$ in
$C\ell(M)$, the proof above shows that if we are to associate
a field to $cX\cdot cY$, it (or rather its Fourier modes)
should be given by the first three terms in
(\ref{cX.cY.normal}).
Perhaps this defines the correct normal-ordered product of the
fields associated to $cX$ and $cY$, where the term
$-2\langle X,\nabla Y\rangle_n$ is analogous to a normal-order
constant.
\vspace{0.08in}
Now we record the relations between the lowest-weight
component $\Gamma^{\mathrm{ch}}(\pi,S)_0=S(M)$ and the fields introduced in
\S\ref{sec.ass.S.fields}, as well as the relations between
the fields themselves.
\begin{prop} \label{prop.S.fields.ground}
For $f\in C^\infty(M)$, $X\in\mc{T}(M)$ and $s\in S(M)$, we have
\begin{eqnarray*}
&f_n s=cX_n s=\ell X_n s=0\;\;\;\textrm{for}\;\;n>0 &\\
&f_0 s=fs,\qquad
cX_0 s=cX\cdot s,\qquad
\ell X_0 s=\nabla_X s&
\end{eqnarray*}
where $\cdot$ denotes Clifford multiplication and
$\nabla$ the Levi-Civita connection.
\end{prop}
\begin{proof}
The first line is true simply because $S(M)\subset\Gamma^{\mathrm{ch}}(\pi,S)$
has the lowest weight.
The second line follows immediately from (\ref{ass.S.0})
and (\ref{ass.S.gen.fields}).
\end{proof}
\begin{prop} \label{prop.S.fields.relations}
Recall the maps $\bullet$, $\{\;\}$, $\{\;\}_\Omega$ in
Proposition \ref{prop.ass.C.VAoid}.
For $f,g\in C^\infty(M)$, $X,Y\in\mc{T}(M)$ and
$n,m\in\mathbb{Z}$, we have the normal-ordered
expansions
\begin{align*}
(fg)_n=f_{n-r}g_r,\quad
c(fX)_n=f_{n-r}\,cX_r,\quad
\ell(fX)_n=\sum_{r\geq 0} f_{n-r}\,\ell X_r
+\sum_{r<0}\ell X_r f_{n-r}
-(\ell X\bullet f)_n
\end{align*}
as well as the supercommutation relations
\begin{align*}
&[f_n,g_m]=0,\qquad
[cX_n,f_m]=0,\qquad
[\ell X_n,f_m]=(Xf)_{n+m} \\
&[\ell X_n,cY_m]=c(\nabla_X Y)_{n+m},\qquad
[cX_n,cY_m]=-2\langle X,Y\rangle_{n+m} \\
&[\ell X_n,\ell Y_m]=\ell[X,Y]_{n+m}
-c\,\Omega(\widetilde X,\widetilde Y)^P_{n+m}
+(\{\ell X,\ell Y\}_\Omega)_{n+m}
+n\{\ell X,\ell Y\}_{n+m}
\end{align*}
where $\langle\;\;\rangle$ and $\nabla$ denote
the Riemannian metric and Levi-Civita connection on $TM$.
Notice that the canonical isomorphism
$\mc{T}_v(P)^{\mathfrak{so}_{2d'}}\cong\Gamma(\mathrm{End}\,TM)$
identifies $-\Omega(\widetilde X,\widetilde Y)^P$ with the Riemannian curvature
operator $R_{X,Y}=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$.
\end{prop}
\begin{proof}
Let us verify the relations on the level of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$.
For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}) and
(\ref{ass.S.gen.fields}).
The first two normal-ordered expansions are
easy, e.g.
\begin{align*}
(\widetilde{fX})^i_{n-s}\otimes e_{i,s}
=f_{n-s-t}\widetilde X^i_t\otimes e_{i,s}
=(f_{n-r}\otimes 1)(\widetilde X^i_{r-s}\otimes e_{i,s}).
\end{align*}
The remaining normal-ordered expansion follows immediately from
the equation
\begin{align*}
\widetilde{fX}-\tau_i(\widetilde{fX})^j\Theta_{ij}
=(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij})_{-1}f
-\ell X\bullet f
\end{align*}
obtained from the first calculation in the proof of Proposition
\ref{prop.ass.C.VAoid}.
For the supercommutators, the first three are easy.
The next two are computed as follows, using (\ref{DX.lift})
and (\ref{Cl}) respectively:
\begin{align*}
&\big[(\widetilde X-\tau_i\widetilde X^j\Theta_{ij})_n\otimes 1,\,
\widetilde Y^k_{m-r}\otimes e_{k,r}\big] \\
&\qquad\qquad
=\big[\widetilde X_n,\widetilde Y^k_{m-r}]
\otimes e_{k,r}
=(\widetilde X^i\tau_i\widetilde Y^k)_{n+m-r}\otimes e_{k,r}
=(\widetilde{\nabla_X Y})^k_{n+m-r}\otimes e_{k,r} \\
&\big[\widetilde X^i_{n-r}\otimes e_{i,r},\,
\widetilde Y^j_{m-s}\otimes e_{j,s}\big] \\
&\qquad\qquad
=\widetilde X^i_{n-r}\widetilde Y^j_{m-s}\otimes[e_{i,r},e_{j,s}]
=-2\widetilde X^i_{n-r}\widetilde Y^i_{m+r}\otimes 1
=-2\langle X,Y\rangle_{n+m}\otimes 1
\end{align*}
The last (super)commutator follows immediately from the
equations
\begin{align*}
&\big(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}\big)_1
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\Theta_{k\ell}\big)
=\{\ell X,\ell Y\} \\
&\big(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}\big)_0
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\Theta_{k\ell}\big)\\
&\qquad\qquad
=\big(\widetilde{[X,Y]}-\tau_i\widetilde{[X,Y]}{}^j\Theta_{ij}\big)
-\Omega(\widetilde X,\widetilde Y)^P
-\lambda_0(\Omega(\widetilde X,\widetilde Y),\Theta)
+\{\ell X,\ell Y\}_\Omega
\end{align*}
together with (\ref{ass.S.other.fields});
these equations are the results of the second and third
calculations in the proof of Proposition
\ref{prop.ass.C.VAoid},
applied to our current setting where $\lambda^*=\lambda_0$ and
$\lambda_\rho=2\lambda_0$ (see \S\ref{sec.ass.S}).
\end{proof}
The following construction is manufactured using
precisely the information about $\Gamma^{\mathrm{ch}}(\pi,S)$ we have
gathered so far:
the subspace $S(M)$,
the fields in \S\ref{sec.ass.S.fields} and
their relations in Propositions \ref{prop.S.fields.ground}
and \ref{prop.S.fields.relations}.
\begin{subsec} \label{sec.ass.S.free}
{\bf Comparing with a generators-and-relations construction.}
Let $\mathcal{U}$ be a unital $\mathbb{Z}/2\mathbb{Z}$-graded associative
algebra with generators
\begin{align*}
\textrm{even}:\quad
f_n,\;\,
\ell X_n,\;\,
\alpha_n,\;\,
c\mathscr{Y}_n;\qquad\quad
\textrm{odd}:\quad
cX_n
\end{align*}
for $f\in C^\infty(M)$, $X\in\mc{T}(M)$,
$\alpha\in\Omega^1(M)$, $\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$
and $n\in\mathbb{Z}$, such that
(i) $1_n=\delta_{n,0}$,
(ii) $f\mapsto f_n$, $X\mapsto\ell X_n$, $\cdots$ are linear and
(iii) they satisfy the supercommutation relations in Proposition
\ref{prop.S.fields.relations}.
The subalgebra $\mathcal{U}_+\subset\mathcal{U}$ generated by
$\{f_n,\,\ell X_n,\,cX_n\}_{n\geq 0}$ has an action on $S(M)$ as
described in Proposition \ref{prop.S.fields.ground}.
Let $\widetilde W^f=\mathcal{U}\otimes_{\mathcal{U}_+}S(M)$ and
$W^f=\widetilde W^f/\sim$ be the quotient obtained by imposing the
normal-ordered expansions in (\ref{ass.S.1formfield}),
Lemma \ref{lemma.verfield} and Proposition
\ref{prop.S.fields.relations}.
Define an operator $L^f_0$ on $W^f$ by
\begin{align} \label{freemod.wt}
L^f_0|_{S(M)}=\frac{d'}{8},\qquad
[L^f_0,f_n]=-nf_n,\qquad
[L^f_0,\ell X_n]=-n\,\ell X_n,\qquad
[L^f_0,cX_n]=-n\,cX_n
\end{align}
which are consistent with all the above relations;
its eigenvalues are called weights.
By construction, there is a unique linear map
\begin{eqnarray*}
\Phi:W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)
\end{eqnarray*}
that restricts to the identity on $S(M)$
\footnote{
Since the composition
$\widetilde W^f\twoheadrightarrow W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)$ restricts
to the identity on $1\otimes S(M)\subset\widetilde W^f$, the quotient
map must be injective there.
Hence we may indeed identify $S(M)$ as a subspace of $W^f$.
}
and respects all the above operators.
Clearly $\Phi$ is weight-preserving.
As we will see later, $\Phi$ is in fact an isomorphism and
thus provides a generators-and-relations description
of $\Gamma^{\mathrm{ch}}(\pi,S)$.
\end{subsec}
{\it Remark.}
The relations imposed in the construction of $W^f$ are not
subject to any higher relations.
For example, given $X,Y\in\mc{T}(M)$ and $s\in S(M)$,
the expression $[\ell X_0,cY_0]s\,$ can be evaluated by
either taking the commutator first or applying the action
on $S(M)$ first, but the results are identical automatically
due to the fact that the Levi-Civita connection is compatible
with Clifford multiplication.~\cite{spin}
Similarly, other composites of relations (e.g.~Jacobi identities)
are already implied by the geometric data.
\begin{subsec} \label{sec.ass.S.Virasoro}
{\bf The Virasoro action.}
According to \S\ref{sec.ass.S} and Lemma \ref{lemma.G.S},
the following operators
\begin{align}
\textstyle
(\tau_{i,-1}\tau^i)_n\otimes 1
+1\otimes L^{C\ell}_n
-\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
-\frac{1}{2}\lambda_0(\Theta_{-1}\Theta)_n\otimes 1,\qquad
n\in\mathbb{Z}
\label{ass.S.Virasoro}
\end{align}
on $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ induce a Virasoro action
on $\Gamma^{\mathrm{ch}}(\pi,S)$ of central charge $5d'$;
denote the induced operators by $L^M_n$.
The key to understanding the entire structure of
$\Gamma^{\mathrm{ch}}(\pi,S)$ is the observation that $L_0^M$ is
generated by the fields described in
\S\ref{sec.ass.S.fields} (in a certain way).
\end{subsec}
For the following statement, recall that $\pi:P\rightarrow M$
is by assumption a lifting of the orthogonal frame bundle of $TM$
(see \S\ref{sec.ass.S}).
\begin{prop} \label{prop.ass.S.wt}
Let $U\subset M$ be an open subset and
$\sigma:U\rightarrow\pi^{-1}(U)\subset P$ a smooth
section of $\pi$.
Both weight operators $L^f_0$ on $W^f$
(see \S\ref{sec.ass.S.free}) and $L^M_0$ on $\Gamma^{\mathrm{ch}}(\pi,S)$
have the local expression:
\footnote{
Recall from \S\ref{sec.review} that, like most
constructions in this paper, both $W^f$ and $\Gamma^{\mathrm{ch}}(\pi,S)$
are the spaces of global sections of underlying sheaves.
}
\begin{align}
&\sum_{r\geq 0}\sigma^i_{-r}(\ell\sigma_i)_r
+\sum_{r<0}(\ell\sigma_i)_r\sigma^i_{-r}
+\frac{3}{4}\,\mathrm{Tr}\,(\Gamma^\sigma_{-r}\Gamma^\sigma_r)
+\sigma^i([\sigma_j,\sigma_k])_{-r}\,\sigma^k_{-s}
(\Gamma^\sigma_{ji})_{r+s} \nonumber \\
&\qquad\quad
-\frac{1}{2}\sum_{r>0}r(c\sigma_i)_{-r}(c\sigma_i)_r
+\frac{1}{4}(c\sigma_i)_{-r}(c\sigma_j)_{-s}
(\Gamma^\sigma_{ij})_{r+s}
+\frac{d'}{8}
\label{ass.S.wt.fields}
\end{align}
where $(\sigma_1,\ldots,\sigma_{2d'})$ is the
$C^\infty(U)$-basis of $\mc{T}(U)$ induced by $\sigma$;
$(\sigma^1,\ldots,\sigma^{2d'})$ the dual basis of
$\Omega^1(U)$;
and $\Gamma^\sigma=\sigma^*\Theta\,\in
\Omega^1(U)\otimes\mathfrak{so}_{2d'}$.
\end{prop}
\begin{proof}
First consider $L^M_0$.
As explained above, $L^M_0$ is represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by
$(\ref{ass.S.Virasoro})_{n=0}$, which we would
like to split into two parts:
\begin{align*}
\textstyle
\ell'
=\big(\tau_{i,-1}\tau^i
-\frac{1}{2}\,\mathrm{Tr}\,(\Theta_{-1}\Theta)\big)_0\otimes 1,
\qquad
\ell''
=1\otimes L^{C\ell}_0
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
+\frac{1}{4}\,\mathrm{Tr}\,(\Theta_{-1}\Theta)_0\otimes 1.
\end{align*}
Since the calculation in the proof of Proposition
\ref{prop.ass.C.conformal} is valid for any principal frame
algebra, $\ell'$ can be expressed as the zeroth mode of
(\ref{ass.C.conformal.local}).
Our main task is to express $\ell''$ entirely in terms of
$\sigma$ as well.
For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}) and adopt again the
notations in the proof of Proposition \ref{prop.ass.C.conformal}.
Let $\theta=g^{-1}dg$.
Also let us introduce a (slight abuse of) notation:
\begin{align} \label{frame.Cl}
(c\sigma_i)_r
=\widetilde\sigma_{i,r-s}^j\otimes e_{j,s}
=g_{ji,r-s}\otimes e_{j,s},\qquad
i=1,\ldots,2d',\;\;r\in\mathbb{Z}.
\end{align}
By (\ref{Cl}) these operators satisfy the anticommutation
relations
\begin{align} \label{frame.Cl.comm}
[(c\sigma_i)_r,(c\sigma_j)_s]
=-2\delta_{ij}\delta_{r+s,0}.
\end{align}
Now the first term of $\ell''$ can be written as the first
line below by (\ref{Cl.Virasoro}) and (\ref{frame.Cl}), and
the subsequent rearrangement is valid because every sum in sight
is finite when applied to an arbitrary element:
\begin{align*}
1\otimes L^{C\ell}_0
&=\frac{1}{4}\bigg(
-\sum_{r\geq 0}r\,(c\sigma_j)_s(c\sigma_k)_t
+\sum_{r<0}r\,(c\sigma_k)_t(c\sigma_j)_s\bigg)
\big(g_{ij,-r-s}g_{ik,r-t}\otimes 1\big)
+\frac{d'}{8} \\
&=\frac{1}{4}\bigg(
-\sum_{t\geq 0}r\,(c\sigma_j)_s(c\sigma_k)_t
+\sum_{t<0}r\,(c\sigma_k)_t(c\sigma_j)_s
-\sum_{r\geq 0>t}
r\,[(c\sigma_j)_s,(c\sigma_k)_t] \\
&\qquad\qquad\qquad
+\sum_{r<0\leq t}
r\,[(c\sigma_j)_s,(c\sigma_k)_t]\bigg)
\big(g_{ij,-r-s}g_{ik,r-t}\otimes 1\big)
+\frac{d'}{8}
\end{align*}
For the first two sums above, write $r=t+(r-t)$ and sum
over $r$:
\begin{align*}
\textrm{1st sum}\;
&=-\frac{1}{4}\sum_{t\geq 0}
t\,(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}\sum_{t\geq 0}
(c\sigma_j)_s(c\sigma_k)_t
\big(\theta_{jk,-s-t}\otimes 1\big) \\
\textrm{2nd sum}
&=\frac{1}{4}\sum_{t<0}
t\,(c\sigma_j)_t(c\sigma_j)_{-t}
-\frac{1}{4}\sum_{t<0}
(c\sigma_k)_t(c\sigma_j)_s
\big(\theta_{jk,-s-t}\otimes 1\big)
\end{align*}
For the other two sums above, apply (\ref{frame.Cl.comm}) and
rearrange:
\begin{align*}
\textrm{3rd\,+\,4th sums}
&=\frac{1}{2}\bigg(
\sum_{r\geq 0>t}-\sum_{r<0\leq t}\bigg)\,
r\,g_{ij,-r+t}g_{ij,r-t}\otimes 1
=\frac{1}{4}\sum_u(u^2-u)\,
g_{ij,-u}g_{ij,u}\otimes 1 \\
&=-\frac{1}{4}\big(\mathrm{Tr}\,(dg^{-1})_{-1}(dg)\big)_0\otimes 1
=\frac{1}{4}\,\mathrm{Tr}\,(\theta_{-1}\theta)_0
\otimes 1
\end{align*}
These calculations together yield the following expression
for the first term of $\ell''$:
\begin{align} \label{ClL0.local}
1\otimes L^{C\ell}_0
=-\frac{1}{2}\sum_{t>0}t(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}(c\sigma_j)_s(c\sigma_k)_t
(\theta_{jk,-s-t}\otimes 1)
+\frac{1}{4}\mathrm{Tr}\,(\theta_{-1}\theta)_0\otimes 1
+\frac{d'}{8}\,.
\end{align}
On the other hand, the second term of $\ell''$ can be written
as follows by (\ref{Cl.so}) and (\ref{frame.Cl}), where
the normal ordering of $e_{i,r-s}e_{j,s}$ guarantees
the validity of the subsequent step:
\begin{align*}
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
&=\frac{1}{4}\Theta_{ij,-r}\otimes e_{i,r-s}e_{j,s}
=\frac{1}{4}\sum_{s\geq 0}
\Theta_{ij,-r}\otimes e_{i,r-s}e_{j,s}
-\frac{1}{4}\sum_{s<0}
\Theta_{ij,-r}\otimes e_{j,s}e_{i,r-s} \\
&=\frac{1}{4}\bigg(
\sum_{s\geq 0}(c\sigma_k)_t(c\sigma_\ell)_u
-\sum_{s<0}(c\sigma_\ell)_u(c\sigma_k)_t\bigg)
\big((g^{-1}\Theta)_{kj,-s-t}\,g_{j\ell,s-u}
\otimes 1\big)
\end{align*}
These sums can be handled in a similar way as in the calculation
of $1\otimes L_0^{C\ell}$.
Let us omit the details and just write down the result:
\begin{align} \label{Theta.Cl.local}
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
=\frac{1}{4}\,(c\sigma_k)_t(c\sigma_\ell)_u
\big((g^{-1}\Theta\,g)_{k\ell,-t-u}\otimes 1\big)
+\frac{1}{2}\,\mathrm{Tr}\,\big((g^{-1}\Theta\,g)_{-1}\theta\big)_0
\otimes 1.
\end{align}
Then it follows from (\ref{ClL0.local}), (\ref{Theta.Cl.local})
and (\ref{dg}) that
\begin{align*}
\ell''
=-\frac{1}{2}\sum_{t>0}
t(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}(c\sigma_j)_s(c\sigma_k)_t
(\Gamma^\sigma_{jk,-s-t}\otimes 1)
+\frac{1}{4}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)_0
\otimes 1
+\frac{d'}{8}\,.
\end{align*}
Combining this with the earlier comment on $\ell'$ and keeping
in mind (\ref{ass.S.gen.fields})--(\ref{ass.S.other.fields}),
we see that $\ell'+\ell''$ represents the operator
(\ref{ass.S.wt.fields}).
This proves the claim for $L^M_0$.
Since the expression in (\ref{ass.S.wt.fields}) is generated
by the operators in
(\ref{ass.S.gen.fields})--(\ref{ass.S.other.fields}),
it determines an operator $L'_0$ on $W^f$.
To show $L'_0=L^f_0$, we need to verify that $L'_0$ satisfies
(\ref{freemod.wt}) using only the relations in
(\ref{ass.S.1formfield}), Lemma \ref{lemma.verfield}
and Propositions
\ref{prop.S.fields.ground}--\ref{prop.S.fields.relations}.
This is straightforward (but uninteresting).
\end{proof}
\begin{corollary} \label{cor.ass.S.free}
The linear map $\Phi:W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)$
(see \S\ref{sec.ass.S.free}) is an isomorphism.
\end{corollary}
\begin{proof}
Let $f\in C^\infty(M)$, $\alpha\in\Omega^1(M)$
and $X\in\mc{T}(M)$.
By definition, $\Phi$ is an isomorphism on
the lowest weight $d'/8$.
Assume that $\Phi$ is an isomorphism on all weights up to
$(d'/8)+k-1$ for some $k>0$.
Let $u\in\Gamma^{\mathrm{ch}}(\pi,S)_k$.
For reason of weight as well as
(\ref{ass.S.1formfield}), elements of the form
\begin{align} \label{lowered}
f_n u,\;\ell X_n u,\;cX_n u\;\textrm{ for }n>0,\qquad
\alpha_n u\;\textrm{ for }n\geq 0
\end{align}
belong to the image of $\Phi$;
then by Proposition \ref{prop.ass.S.wt} so does
$ku=(L^M_0-d'/8)u$.
This proves the surjectivity of $\Phi$ on weight $(d'/8)+k$.
On the other hand, let $u\in W^f_k\cap\ker\Phi$.
Since the elements (\ref{lowered}) are also in the kernel of
$\Phi$, for reason of weight and (\ref{ass.S.1formfield}) again
they must be trivial;
then by Proposition \ref{prop.ass.S.wt} so is $ku=(L^f_0-d'/8)u$.
This proves the injectivity of $\Phi$ on weight $(d'/8)+k$.
By induction, $\Phi$ is an isomorphism on all weights.
\end{proof}
The following summarizes our analysis of $\Gamma^{\mathrm{ch}}(\pi,S)$.
\begin{theorem} \label{thm.ass.S}
Suppose $M^{2d'}$ is a Riemannian manifold with
a spin structure $\pi:P\rightarrow M$ and
a $3$-form $H$ satisfying $dH=\frac{1}{2}\mathrm{Tr}\,(R\wedge R)$,
where $R$ is the Riemannian curvature.
Let $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ be the associated principal
frame $(\widehat{\mathfrak{so}}_{2d'},\mathrm{Spin}_{2d'})$-algebra
(see Theorem \ref{thm.CDOP}), $S$ the spinor representation
of $\widehat{\mathfrak{so}}_{2d'}$, and
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,S)
=H^{\frac{\infty}{2}+0}\big(
\widehat{\mathfrak{so}}_{2d'}\,,
\,\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S\big).
\end{align*}
Then $\Gamma^{\mathrm{ch}}(\pi,S)$ contains $S(M)$ as a subspace
(see \S\ref{sec.ass.S.0}) and admits three particular types
of fields $\t{\sf Y}(f,x)$, $\t{\sf Y}(\ell X,x)$, $\t{\sf Y}(cX,x)$ associated to
$f\in C^\infty(M)$ and $X\in\mc{T}(M)$, whose Fourier modes
are represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by
the following operators: (see \S\ref{sec.ass.S.fields})
\begin{align*}
f_n\otimes 1,\qquad
(\widetilde X-\tau_i\widetilde X^j\Theta_{ij})_n\otimes 1,\qquad
\widetilde X^i_{n-r}\otimes e_{i,r},\qquad
n\in\mathbb{Z}.
\end{align*}
In fact, $\Gamma^{\mathrm{ch}}(\pi,S)$ is generated from $S(M)$
by the actions of these fields,
subject only to the relations stated in Propositions
\ref{prop.S.fields.ground}--\ref{prop.S.fields.relations}
(see also (\ref{ass.S.1formfield}) and Lemma
\ref{lemma.verfield}).
Moreover, there is a Virasoro action on $\Gamma^{\mathrm{ch}}(\pi,S)$ of central
charge $5d'$, induced by the following operators on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$: (see \S\ref{sec.ass.S.Virasoro})
\begin{align*}
\textstyle
\qquad
(\tau_{i,-1}\tau^i)_n\otimes 1
+1\otimes L^{C\ell}_n
-\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
-\frac{1}{4}\mathrm{Tr}\,(\Theta_{-1}\Theta)_n\otimes 1,\qquad
n\in\mathbb{Z}.
\qquad
\qedsymbol
\end{align*}
\end{theorem}
The generators-and-relations description of $\Gamma^{\mathrm{ch}}(\pi,S)$
allows us to define a filtration and identify the
associated graded space.
\begin{subsec} \label{ass.S.PBW}
{\bf PBW filtration.}
Given an increasing sequence of negative integers
$\mf{n}=\{n_1\leq\cdots\leq n_s<0\}$, possibly empty,
let us write
\begin{align*}
|\mf{n}|=|n_1|+\cdots+|n_s|\quad (0\textrm{ if }\mf{n}=\{\}),\qquad
\mf{n}(i)=\textrm{number of times }i\textrm{ appears in }\mf{n}
\end{align*}
and regard $\mf{n}$ as a partition of $-|\mf{n}|$.
For any nonnegative integer $w$, let $\mathscr{I}_w$ be the set
of triples $(\mf{n};\mf{m};\mf{p})$ of such sequences with
$|\mf{n}|+|\mf{m}|+|\mf{p}|=w$ and $\mf{m}(i)\leq 2d'$ for all $i<0$.
Define a partial ordering on $\mathscr{I}_w$ by declaring
that $(\mf{n};\mf{m}\;\mf{p})\prec(\mf{n}';\mf{m}';\mf{p}')$ if one of the
following is true:
\vspace{-0.05in}
\begin{itemize}
\item[$\cdot$]
$|\mf{n}|<|\mf{n}'|$, or $|\mf{n}|=|\mf{n}'|$ and $|\mf{m}|<|\mf{m}'|$
\vspace{-0.08in}
\item[$\cdot$]
$\mf{n}'$ is a proper subpartition of $\mf{n}$, $\mf{m}=\mf{m}'$
and $\mf{p}=\mf{p}'$
\vspace{-0.08in}
\item[$\cdot$]
$\mf{n}=\mf{n}'$, $\mf{m}=\mf{m}'$ and $\mf{p}$ is a proper subpartition
of $\mf{p}'$
\vspace{-0.05in}
\end{itemize}
For example in $\mathscr{I}_3$, a particular ascending chain is
\begin{align*}
(;;-2,-1)
\prec(;;-3)
\prec(;-1,-1;-1)
\prec(-2;-1;)
\prec(-1,-1;-1;)
\end{align*}
while $(;;-1,-1,-1)$ and $(-1,-1,-1;;)$ are
the unique minimal and maximal elements.
Given a sequence $\mf{n}=\{n_1\leq\cdots\leq n_s<0\}$
as above and an $s$-tuple
$\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_s)$ in
$\Omega^1(M)$, or an $s$-tuple
$\boldsymbol{X}=(X_1,\ldots,X_s)$ in $\mc{T}(M)$, let us introduce
the notations
\begin{align*}
\boldsymbol{\alpha}_\mf{n}=\alpha_{1,n_1}\cdots\alpha_{s,n_s},\quad
\boldsymbol{cX}_\mf{n}=(cX_1)_{n_1}\cdots(cX_s)_{n_s},\quad
\boldsymbol{\ell X}_\mf{n}=(\ell X_1)_{n_1}\cdots(\ell X_s)_{n_s}\quad
(1\textrm{ if }\mf{n}=\{\})
\end{align*}
which are operators on $\Gamma^{\mathrm{ch}}(\pi,S)$ (see
\S\ref{sec.ass.S.fields}).
It follows from Theorem \ref{thm.ass.S} that for $w\geq 0$
we have
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,S)_w
=\textrm{span}\Big\{
\boldsymbol{\ell X}_\mf{n}\boldsymbol{cY}_\mf{m}\boldsymbol{\alpha}_\mf{p}\, s:
(\mf{n};\mf{m}\;\mf{p})\in\mathscr{I}_w;\,
\textrm{all suitable }\boldsymbol{\alpha},\boldsymbol{X},\boldsymbol{Y};\,
s\in S(M)
\Big\}.
\end{align*}
Indeed, as $f_n=-\frac{1}{n}(df)_n$ for
$f\in C^\infty(M)$ and $n\neq 0$, Propositions
\ref{prop.S.fields.ground} and \ref{prop.S.fields.relations}
allow us to express every element of $\Gamma^{\mathrm{ch}}(\pi,S)$ in the
indicated form.
For $(\mf{n};\mf{m};\mf{p})\in\mathscr{I}_w$ consider the subspaces
\begin{align*}
\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}
&=\textrm{span}\big\{
\boldsymbol{\ell X}_{\mf{n}'}\boldsymbol{cY}_{\mf{m}'}\boldsymbol{\alpha}_{\mf{p}'} s:
(\mf{n}';\mf{m}';\mf{p}')\preceq(\mf{n};\mf{m};\mf{p})
\big\}\subset\Gamma^{\mathrm{ch}}(\pi,S)_w \\
\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}
&=\textrm{span}\big\{
\boldsymbol{\ell X}_{\mf{n}'}\boldsymbol{cY}_{\mf{m}'}\boldsymbol{\alpha}_{\mf{p}'} s:
(\mf{n}';\mf{m}';\mf{p}')\prec(\mf{n};\mf{m};\mf{p})
\big\}\subset\Gamma^{\mathrm{ch}}(\pi,S)_w
\end{align*}
For the next statement, let $O$ (and $O'$)
stand for one of the operators of the form
$\alpha_n$, $cX_n$ or $\ell X_n$ with $n<0$, and
$fO$ the corresponding operator $(f\alpha)_n$,
$c(fX)_n$ or $\ell(fX)_n$, where
$f\in C^\infty(M)$.
The subspaces $\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}$ and
$\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}$ have the following
properties:
\begin{align*}
\begin{array}{crcrl}
\textrm{(i)}&\;\;
\cdots OO'\cdots\in\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}&
\;\;\Rightarrow&
\cdots[O,O']\cdots\in\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}\phantom{,}& \\
\textrm{(ii)}&
\cdots Os\in\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}&
\;\;\Rightarrow& \phantom{\Big(}
\cdots\big((fO)s-O(fs)\big)\in\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})},&
\quad s\in S(M)
\end{array}
\end{align*}
Indeed (i) follows from the supercommutation
relations in Proposition \ref{prop.S.fields.relations}
and (ii) from the normal-ordered expansions there as
well as Proposition \ref{prop.S.fields.ground}.
Consequently, there is a natural isomorphism
\begin{align*}
&\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}/\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})} \\
&\qquad\qquad
\cong
\bigg(\bigotimes_{i<0}\textrm{Sym}^{\mf{p}(i)}\Omega^1(M)\bigg)\otimes
\bigg(\bigotimes_{i<0}\textrm{Sym}^{\mf{n}(i)}\mc{T}(M)\bigg)\otimes
\bigg(\bigotimes_{i<0}\wedge^{\mf{m}(i)}\mc{T}(M)\bigg)\otimes S(M)
\end{align*}
where all tensor, symmetric and exterior products
are over $C^\infty(M)$.
Let $q$ be a formal variable.
When all $(\mf{n};\mf{m}\;\mf{p})\in\mathscr{I}_w$ and all $w\geq 0$
are considered, we obtain
\begin{align}
&\bigoplus_{w\geq 0}\bigg(
q^w\bigoplus_{(\mf{n};\mf{m};\mf{p})\in\mathscr{I}_w}
\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}/\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}
\bigg) \nonumber \\
&\qquad\qquad\quad\cong
\bigg(\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\Omega^1(M)\bigg)
\otimes\bigg(\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\mc{T}(M)\bigg)
\otimes\bigg(\bigotimes_{k\geq 1}\wedge_{q^k}\mc{T}(M)\bigg)
\otimes S(M) \label{ass.S.AG}
\end{align}
where $\textrm{Sym}_t=\sum_{n=0}^\infty t^n\textrm{Sym}^n$
and $\wedge_t=\sum_{n=0}^{2d'}t^n\wedge^n$ as usual.
\footnote{
If the partial ordering on $\mathscr I_w$ is extended to
a total ordering, the latter will induce
a filtration on $\Gamma^{\mathrm{ch}}(\pi,S)_w$ whose associated
graded space is the coefficient of $q^w$ in
(\ref{ass.S.AG}).
}
\end{subsec}
To conclude this section, let us use an example to
illustrate a speculation about $\Gamma^{\mathrm{ch}}(\pi,S)$ that we plan
to investigate in a future paper.
\begin{subsec} \label{ass.S.sconformal}
{\bf The case of $\mathbb{R}^{2d'}$:~(0,1) superconformal structure.}
When $M=\mathbb{R}^{2d'}$ with the standard spin structure and
$H$ is the trivial $3$-form, $\Gamma^{\mathrm{ch}}(\pi,S)$ can be identified
with $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$.
Recall the definition of $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})$ from
\S\ref{sec.CDO.Rd}.
Then it is easy to check that the following operators on
$\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$
\begin{align*}
&\textstyle
L_n=\frac{1}{4}
\big((\partial_i+db^i)_{-1}(\partial_i+db^i)\big)_n
\otimes 1 \\
&\textstyle
\bar L_n=-\frac{1}{4}
\big((\partial_i-db^i)_{-1}(\partial_i-db^i)\big)_n
\otimes 1
+1\otimes L_n^{C\ell},
\qquad n\in\mathbb{Z} \\
&\textstyle
\bar G_n=\frac{1}{2}
(\partial_i-db^i)_{n-r}\otimes e_{i,r}
\end{align*}
satisfy the $(0,1)$ (Ramond) superconformal algebra
of central charges $(2d',3d')$, namely
\begin{align*}
&\textstyle
[L_n,L_m]=(n-m)L_{n+m}
+\frac{d'}{6}(n^3-n)\delta_{n,-m} \\
&[L_n,\bar L_m]=[L_n,\bar G_m]=0 \phantom{\big(}\\
&\textstyle
[\bar L_n,\bar L_m]=(n-m)\bar L_{n+m}
+\frac{d'}{4}(n^3-n)\delta_{n,-m} \\
&\textstyle
[\bar L_n,\bar G_m]=\big(\frac{n}{2}-m\big)\bar G_{n+m},
\qquad
[\bar G_n,\bar G_m]=2\bar L_{n+m}
+\frac{d'}{4}(4n^2-1)\delta_{n,-m}
\end{align*}
Notice that the diagonal Virasoro action given by
$L_n+\bar L_n$ is precisely the one described in
\S\ref{sec.ass.S.Virasoro}.
Let us consider the operator $\slashed D=2\bar G_0$ and its kernel.
Since $\slashed D^2=4(\bar L_0-d'/8)$, on $\ker\slashed D$
we have $\bar L_0=d'/8$.
Notice that $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$ can be generated from
$S(\mathbb{R}^{2d'})=C^\infty(\mathbb{R}^{2d'})\otimes S_0$ by the
actions of $(\partial_i+db^i)_n$, $(\partial_i-db^i)_n$ and $e_{i,n}$ for $n<0$.
Also notice that $\bar L_0\geq d'/8$ on $S(\mathbb{R}^{2d'})$ and
\begin{align*}
[\bar L_0,(\partial_i+db^i)_n]=0,\qquad
[\bar L_0,(\partial_i-db^i)_n]=-n(\partial_i+db^i)_n,\qquad
[\bar L_0,e_{i,n}]=-ne_{i,n}.
\end{align*}
All these imply that $\ker\slashed D$ must be contained in the
subspace generated from $S(\mathbb{R}^{2d'})$ by $(\partial_i+db^i)_n$
only.
This subspace can be identified with
\begin{align} \label{spinors.Rd.subspace}
\bigg(
\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\mc{T}(\mathbb{R}^{2d'})
\bigg)\otimes S(\mathbb{R}^{2d'})
\end{align}
where the various products are taken over
$C^\infty(\mathbb{R}^{2d'})$ and the power of $q$ corresponds to
the eigenvalue of $L_0+\bar L_0-d'/8$.
Since $\slashed D$ also commutes with $(\partial_i+db^i)_n$,
it follows that the restriction of $\slashed D$
to (\ref{spinors.Rd.subspace}) is simply a classical
twisted Dirac operator on $\mathbb{R}^{2d'}$.
Moreover, all $L_n$ commute with $\slashed D$ and hence induce
a Virasoro action on $\ker\slashed D$ of central charge $2d'$.
\end{subsec}
{\it Remark.}
The author speculates that, in general, the Virasoro action
on $\Gamma^{\mathrm{ch}}(\pi,S)$ described in \S\ref{sec.ass.S.Virasoro}
always extends to an action of the $(0,1)$
superconformal algebra (which includes a Dirac-Ramond operator
$\slashed D=2\bar G_0$);
and if $M$ is closed, a more sophisticated version of the argument
above would prove that
\begin{align*}
\textrm{Virasoro character of }\ker\slashed D
&:=q^{-d'/12}\times
\textrm{supertrace of }q^{L_0}\textrm{ on }\ker\slashed D \\
&=q^{-d'/12}\times
\textrm{supertrace of }q^{L_0+\bar L_0-d'/8}\textrm{ on }
\ker\slashed D \\
&=q^{-d'/12}\times\hat{A}\bigg(M,\,
\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}TM\otimes{\mathbb{C}}
\bigg) \\
&=\eta(q)^{-2d'}\times\textrm{Witten genus of }M
\end{align*}
where $\eta(q)$ the Dedekind $\eta$-function.
This would provide at least a partial mathematical
formulation of the original, physical interpretation
of the Witten genus.
\cite{Witten.ell,Witten.index}
It is believed that such a description of the Witten genus
may lead to various geometric applications as well as a better
understanding of its family version in elliptic cohomology.
\cite{Hopkins.ICM}
\newpage
\titleformat{\section}[block]
{\sc\large\filcenter}
{Appendix \S\thesection.}{.5em}{}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,768
|
Gerbera Jamesona (Gerbera jamesonii Bolus) – gatunek rośliny z rodziny astrowatych (Asteraceae).
Występowanie
Zasięg geograficzny obejmuje Afrykę Południową.
Morfologia
Pokrój Bylina o wysokości do 1 m.
Liście Odziomkowe, całobrzegie, o długości do 25 cm.
Kwiaty: Kształtu języczkowatego barwne w kolorach żółtym, pomarańczowym, czerwonym lub różowym.
Owoce Niełupka z krótkim puchem kielichowym.
Zastosowanie
Roślina ozdobna hodowana na kwiat cięty.
Przypisy
Bibliografia
Mutisioideae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,277
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{"url":"https:\/\/www.akalin.com\/magic-erasure-codes","text":"# The Magic of Erasure Codes\n\nFred Akalin\n@fakalin\n$\\xmapsto{\\mathtt{ComputeParity}}$\n\nCashcat pictures from @CatsAndMoney.\n\n?\n$\\xmapsto{\\mathtt{ReconstructData}}$\n\n## Erasure codes, how do they work?\n\nFirst, turn into a byte-level problem...\ncashcat0.jpg: aa b3 3f 2f 13 33 ...\ncashcat1.jpg: bb 34 25 36 3f ed ...\ncashcat2.jpg: cc 35 d3 3f ff c0 ...\n\n|\n|\nComputeParity\n|\n\\|\/\n\ncashcats.p00: 14 34 ...\ncashcats.p01: 25 53 ...\n\n\u2026and so on.\nRecord hashes, and treat corrupted files as missing.\ncashcat0.jpg: XX XX XX XX XX XX ... (sha1 mismatch)\ncashcat1.jpg: bb 34 25 36 3f ed ...\ncashcat2.jpg: ?? ?? ?? ?? ?? ?? ... (missing)\ncashcats.p00: 14 34 11 50 3f fe ...\ncashcats.p01: 25 53 23 36 1f 3d ...\n\n|\n|\nReconstructData\n|\n\\|\/\n\ncashcat0.jpg: aa b3 ...\ncashcat1.jpg: bb 34 ...\ncashcat2.jpg: cc 35 ...\n\n\u2026and so on.\n$$(p_0, p_1) = \\mathtt{ComputeParity}(d_0, d_1, d_2)$$\n$\\begin{pmatrix} p_0 \\\\ p_1 \\end{pmatrix} = P \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$ $= \\begin{pmatrix} 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix} \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n\nor\n\n\\begin{aligned} p_0 &= 1\/3 \\cdot d_0 + 1\/2 \\cdot d_1 + 1\/1 \\cdot d_2 \\\\ p_1 &= 1\/4 \\cdot d_0 + 1\/3 \\cdot d_1 + 1\/2 \\cdot d_2 \\end{aligned}\n\n\u26a0\ufe0f For now, rational numbers (fractions), not bytes. \u26a0\ufe0f\n\nGenerate parity matrix with parityCount=2 rows and dataCount=3 columns.\n\n$$P = \\begin{pmatrix} 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix}$$\n$$P[i, j] = \\frac{1}{(\\texttt{dataCount} + i) - j}$$\nfor i, j in range(0, 2) \u00d7 range(0, 3)\n\nCalled a Cauchy matrix.\n\n$\\begin{pmatrix} p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix} \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n\nAppend to identity function\u2026\n\n$\\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\\\ p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix} \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n\nRemove unknown rows\u2026\n\n$\\begin{pmatrix} \\xcancel{d_0} \\\\ d_1 \\\\ \\xcancel{d_2} \\\\ p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} \\xcancel{1} & \\xcancel{0} & \\xcancel{0} \\\\ 0 & 1 & 0 \\\\ \\xcancel{0} & \\xcancel{0} & \\xcancel{1} \\\\ 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix} \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n\nRemove unknown rows\u2026\n\n$\\begin{pmatrix} d_1 \\\\ p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1\/3 & 1\/2 & 1\/1 \\\\ 1\/4 & 1\/3 & 1\/2 \\end{pmatrix} \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n$= M \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n$$\\begin{pmatrix} d_1 \\\\ p_0 \\\\ p_1 \\end{pmatrix} = M \\cdot \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$$\nFind $M^{-1}$ such that $M^{-1} \\cdot \\begin{pmatrix} d_1 \\\\ p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix}$\n\u2022 Always exists, by special properties of Cauchy matrices!\n\u2022 Use row reduction (a.k.a. Gaussian elimination) to compute.\n$$(d_0, d_1, d_2) = \\mathtt{ReconstructData}(?, d_1, ?, p_0, p_1)$$\n$$\\begin{pmatrix} d_0 \\\\ d_1 \\\\ d_2 \\end{pmatrix} = M^{-1} \\cdot \\begin{pmatrix} d_1 \\\\ p_0 \\\\ p_1 \\end{pmatrix} = \\begin{pmatrix} -1 & -6 & 12 \\\\ 1 & 0 & 0 \\\\ -1\/6 & 3 & -4 \\end{pmatrix} \\cdot \\begin{pmatrix} d_1 \\\\ p_0 \\\\ p_1 \\end{pmatrix}$$\n\nRemaining problem\u2026\n\nHow to compute parity bytes instead of parity numbers?\n\n\u2022 Matrix elements must obey a particular interface.\n\u2022 They must belong to a field.\n\u2022 We want a field with 256 elements.\n\n## What is a field?\n\ninterface Field<T> {\nstatic Zero: T, One: T\nplus(b: T): T \/\/ $a \\oplus b$\nnegate(): T \/\/ $-a$\ntimes(b: T): T \/\/ $a \\otimes b$\nreciprocate(): T \/\/ $a^{-1}$, for $a \\ne 0$\nequals(b: T): bool \/\/ $a = b$\n}\n \/\/ $a \\ominus b$\nminus(b: T): T => this.plus(b.negate())\n\/\/ $a \\oslash b$\ndividedBy(b: T): T => this.times(b.reciprocate())\n\n\n## Field \ud83d\udcdcinterface contract\ud83d\udcdc\n\n\u2022 Identities: $a \\oplus 0 = a \\otimes 1 = a$.\n\u2022 Inverses: $a \\oplus -a = 0$, and if $a \\ne 0$,\n\ud83d\udc49$a \\otimes a^{-1} = 1$\ud83d\udc48.\n\u2022 Associativity: $(a \\oplus b) \\oplus c = a \\oplus (b \\oplus c)$, $(a \\otimes b) \\otimes c = a \\otimes (b \\otimes c)$.\n\u2022 Commutativity: $a \\oplus b = b \\oplus a$, $a \\otimes b = b \\otimes a$.\n\u2022 Distributivity: $a \\otimes (b \\oplus c) = (a \\otimes b) \\oplus (a \\otimes c)$.\n\u2022 Rational numbers (i.e., fractions) form a field: $(p\/q)^{-1} = q\/p$.\n\u2022 Integers don\u2019t form a field, since only $1$ and $-1$ have reciprocals.\n\u2022 Integers mod $p$ form a field with $p$ elements, only when $p$ is prime, e.g. field of 257 elements.\n\ud83e\udd13 Math fact \ud83e\udd13: Given an integer $0 \\lt a \\lt 257$, there is exactly one $0 \\lt b \\lt 257$ such that $(a \\cdot b) \\bmod 257 = 1\\text{.}$\n(Can replace $257$ with any prime number $p$).\nreciprocate() => {\nif (this === 0) { throw new Error(\"Division by zero\"); }\nfor (let i = 0; i < 257; i += 1) {\nif ((this * i) % 257 === 1) { return i; }\n}\nthrow new Error(\"Shouldn't get here!\");\n}\n\nIf speed is a problem, build a table.\n\u2022 Field of 257 elements gets us close\u2026\n\u2022 Problem: 256 isn\u2019t a prime number!\n\u2022 Does this mean we cannot make a field of 256 elements?\nNo!\nWe can\u2019t just do so based on standard integer arithmetic.\n\n## Binary carry-less arithmetic\n\nLet $a = 23$ and $b = 54$. Then, with binary carry-less addition,\n a = 23 = 10111b\n^ b = 54 = 110110b\n-------\n100001b\nso $a \\oplus b = 100001_{\\mathrm{b}} = 33$.\nBinary carry-less addition is just bitwise xor!\nSo is binary carry-less subtraction, since a ^ a = 0.\nLet $a = 23$ and $b = 54$. Then, with carry-less arithmetic,\n a = 23 = 10111b\n^* b = 54 = 110110b\n------------\n10111\n^ 10111\n^ 10111\n^ 10111\n------------\n1111100010b\nso $a \\otimes b = 1111100010_{\\mathrm{b}} = 994$.\nLet $a = 55$ and $b = 19$. Then, with carry-less arithmetic,\n 11b\n--------\nb = 19 = 10011b )110111b = 55 = a\n^ 10011\n-----\n10001\n^ 10011\n-----\n10b\nso $a \\oslash b = 11_{\\mathrm{b}} = 3$ with remainder $10_{\\mathrm{b}} = 3$.\nImportant subtlety: we don\u2019t stop when the remainder is less than the divisor, but when it has fewer digits than the divisor.\n\n## Properties of (carry-less) mod\n\n\u2022 mod by $256 \\le n \\lt 512$ $\\Rightarrow$ $n$ possible remainders.\n\u2022 clmod by $256 \\le n \\lt 512$ $\\Rightarrow$ $256$ possible remainders.\n\u2022 Very interesting\u2026clmod by prime number $256 \\le p \\lt 512$?\n\n## What are prime numbers?\n\n\u2022 An integer $p \\gt 1$ that cannot be expressed as $p = a \\cdot b$, for $a, b \\gt 1$.\n\u2022 Multiplication operation determines the prime numbers.\n\u2022 Want a \u201ccarry-less\u201d prime number between 256 and 512!\n\u2022 283 is such a number. (Coincidentally, it\u2019s also a regular prime.)\n\n## Field of 256 elements\n\n\u2026is just binary carry-less arithmetic mod 283.\nclass Field256Element implements Field<Field256Element> {\nstatic Zero = 0, One = 1\nplus(b) => (a ^ b)\nnegate() => a\ntimes(b): => clmod(clmul(a, b), 283)\nreciprocate() => \/\/ next slide\nequals(b) => (a === b)\n}\n\ud83e\udd13 Math fact \ud83e\udd13: Given an integer $0 \\lt a \\lt 256$, there is a $0 \\lt b \\lt 256$ such that $(a \\otimes b) \\mathbin{\\mathrm{clmod}} 283 = 1$.\nreciprocate() => {\nif (this === 0) { throw new Error(\"Division by zero\"); }\nfor (let i = 0; i < 256; i += 1) {\nif (clmod(clmul(this, i), 283) === 1) { return i; }\n}\nthrow new Error(\"Shouldn't get here!\");\n}\n\n## The full algorithm\n\n1. Do everything with Field256Element, i.e. bytes with carry-less arithmetic mod 283.\n2. In particular, compute $P$ like: \\begin{aligned} P &= \\begin{pmatrix} (3 \\oplus 0)^{-1} & (3 \\oplus 1)^{-1} & (3 \\oplus 2)^{-1} \\\\ (4 \\oplus 0)^{-1} & (4 \\oplus 1)^{-1} & (4 \\oplus 2)^{-1} \\end{pmatrix} \\\\ &= \\begin{pmatrix} 3^{-1} & 2^{-1} & 1 \\\\ 4^{-1} & 5^{-1} & 6^{-1} \\end{pmatrix} = \\begin{pmatrix} 246 & 141 & 1 \\\\ 203 & 82 & 123 \\end{pmatrix} \\end{aligned}\n3. Then proceed as before.\nThank you!\n\ud83d\udd74\ud83c\udffc","date":"2021-05-18 14:13:41","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\": 0.9965876936912537, \"perplexity\": 6769.128082103642}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989637.86\/warc\/CC-MAIN-20210518125638-20210518155638-00231.warc.gz\"}"}
| null | null |
Dallas Auto Transport
March 22, 2013 By Autos In Transit
As one of the premier auto transport specialists moving vehicles in and out of the Dallas-Fort Worth Metroplex, we are experts on Texas car shipping. We transport thousands of vehicles in and out of Texas each year, to and from storage facilities (terminal-to-terminal transport) and direct locations (door-to-door transport).
We pride ourselves on staying abreast of events and other news related to the area's automotive industry. One such example is the Dallas Auto Show, which took place on March 13-17 at the Dallas Convention Center. Our cost-effective transport solutions allow those shipping beautiful and rare cars to auto shows to know they can get a premium service for less expense than they might expect. We believe, like they do, that car transport should be simple and efficient.
700+ Vehicles Wow Dallas Show Attendees
Those attending the Dallas Auto Show were in for some eye-popping automotive excitement. The event was vast, with over 700 vehicles on display from more than 30 carmakers. Many of the cars and trucks were shipped to the Star state from another state. For lengthy transports, customer support is crucial, which is why we have such a strong name for ourselves.
Ford's F-150 Atlas, a concept vehicle that may give us a sense of what the pickup will look like in its 2015 model, was center stage. Chevrolet, meanwhile, sought to grab our attention with its limited-edition Hot Wheels Camaro, designed after the toy car brand. Accura was showcasing its MDX, a prototype for an SUV the company is planning to manufacture in the United States.
A variety of 2014 models were offered for viewing at the event. The most notable among the 2014 bunch were two SUV redesigns, one from Jeep and one from Subaru. Jeep was showing off its new and improved Grand Cherokee, while Subaru was providing a look at the all-new Forester. Kia, meanwhile, made its first appearance in the full-size luxury segment with its 2014 Cadenza.
Other Featured Elements
High-end automakers, including Maserati, Lamborghini, and Rolls-Royce, had vehicles at the show as well. The Dallas Auto Show was not just about looking, though. Interactivity was possible, with test driving options from nine different manufacturers. The entertainment world made an appearance at the event as well. Throughout the weekend, stars of the truTV reality show Lizard Lick Towing were meeting with fans and signing autographs. By many accounts, the show was an incredible success.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,881
|
import psutil
from vnpy.trader.vtFunction import loadIconPath
from vnpy.trader.vtGlobal import globalSetting
from vnpy.trader.uiBasicWidget import *
########################################################################
class MainWindow(QtWidgets.QMainWindow):
"""主窗口"""
signalStatusBar = QtCore.Signal(type(Event()))
#----------------------------------------------------------------------
def __init__(self, mainEngine, eventEngine):
"""Constructor"""
super(MainWindow, self).__init__()
self.mainEngine = mainEngine
self.eventEngine = eventEngine
l = self.mainEngine.getAllGatewayDetails()
self.gatewayNameList = [d['gatewayName'] for d in l]
self.widgetDict = {} # 用来保存子窗口的字典
# 获取主引擎中的上层应用信息
self.appDetailList = self.mainEngine.getAllAppDetails()
self.initUi()
self.loadWindowSettings('custom')
#----------------------------------------------------------------------
def initUi(self):
"""初始化界面"""
self.setWindowTitle('VnTrader')
self.initCentral()
self.initMenu()
self.initStatusBar()
#----------------------------------------------------------------------
def initCentral(self):
"""初始化中心区域"""
widgetMarketM, dockMarketM = self.createDock(MarketMonitor, vtText.MARKET_DATA, QtCore.Qt.RightDockWidgetArea)
widgetLogM, dockLogM = self.createDock(LogMonitor, vtText.LOG, QtCore.Qt.BottomDockWidgetArea)
widgetErrorM, dockErrorM = self.createDock(ErrorMonitor, vtText.ERROR, QtCore.Qt.BottomDockWidgetArea)
widgetTradeM, dockTradeM = self.createDock(TradeMonitor, vtText.TRADE, QtCore.Qt.BottomDockWidgetArea)
widgetOrderM, dockOrderM = self.createDock(OrderMonitor, vtText.ORDER, QtCore.Qt.RightDockWidgetArea)
widgetPositionM, dockPositionM = self.createDock(PositionMonitor, vtText.POSITION, QtCore.Qt.BottomDockWidgetArea)
widgetAccountM, dockAccountM = self.createDock(AccountMonitor, vtText.ACCOUNT, QtCore.Qt.BottomDockWidgetArea)
widgetTradingW, dockTradingW = self.createDock(TradingWidget, vtText.TRADING, QtCore.Qt.LeftDockWidgetArea)
self.tabifyDockWidget(dockTradeM, dockErrorM)
self.tabifyDockWidget(dockTradeM, dockLogM)
self.tabifyDockWidget(dockPositionM, dockAccountM)
dockTradeM.raise_()
dockPositionM.raise_()
# 连接组件之间的信号
widgetPositionM.itemDoubleClicked.connect(widgetTradingW.closePosition)
# 保存默认设置
self.saveWindowSettings('default')
#----------------------------------------------------------------------
def initMenu(self):
"""初始化菜单"""
# 创建菜单
menubar = self.menuBar()
# 设计为只显示存在的接口
gatewayDetails = self.mainEngine.getAllGatewayDetails()
sysMenu = menubar.addMenu(vtText.SYSTEM)
for d in gatewayDetails:
if d['gatewayType'] == GATEWAYTYPE_FUTURES:
self.addConnectAction(sysMenu, d['gatewayName'], d['gatewayDisplayName'])
sysMenu.addSeparator()
for d in gatewayDetails:
if d['gatewayType'] == GATEWAYTYPE_EQUITY:
self.addConnectAction(sysMenu, d['gatewayName'], d['gatewayDisplayName'])
sysMenu.addSeparator()
for d in gatewayDetails:
if d['gatewayType'] == GATEWAYTYPE_INTERNATIONAL:
self.addConnectAction(sysMenu, d['gatewayName'], d['gatewayDisplayName'])
sysMenu.addSeparator()
for d in gatewayDetails:
if d['gatewayType'] == GATEWAYTYPE_BTC:
self.addConnectAction(sysMenu, d['gatewayName'], d['gatewayDisplayName'])
sysMenu.addSeparator()
for d in gatewayDetails:
if d['gatewayType'] == GATEWAYTYPE_DATA:
self.addConnectAction(sysMenu, d['gatewayName'], d['gatewayDisplayName'])
sysMenu.addSeparator()
sysMenu.addAction(self.createAction(vtText.CONNECT_DATABASE, self.mainEngine.dbConnect, loadIconPath('database.ico')))
sysMenu.addSeparator()
sysMenu.addAction(self.createAction(vtText.EXIT, self.close, loadIconPath('exit.ico')))
# 功能应用
appMenu = menubar.addMenu(vtText.APPLICATION)
for appDetail in self.appDetailList:
function = self.createOpenAppFunction(appDetail)
action = self.createAction(appDetail['appDisplayName'], function, loadIconPath(appDetail['appIco']))
appMenu.addAction(action)
# 帮助
helpMenu = menubar.addMenu(vtText.HELP)
helpMenu.addAction(self.createAction(vtText.CONTRACT_SEARCH, self.openContract, loadIconPath('contract.ico')))
helpMenu.addSeparator()
helpMenu.addAction(self.createAction(vtText.RESTORE, self.restoreWindow, loadIconPath('restore.ico')))
helpMenu.addAction(self.createAction(vtText.ABOUT, self.openAbout, loadIconPath('about.ico')))
helpMenu.addSeparator()
helpMenu.addAction(self.createAction(vtText.TEST, self.test, loadIconPath('test.ico')))
#----------------------------------------------------------------------
def initStatusBar(self):
"""初始化状态栏"""
self.statusLabel = QtWidgets.QLabel()
self.statusLabel.setAlignment(QtCore.Qt.AlignLeft)
self.statusBar().addPermanentWidget(self.statusLabel)
self.statusLabel.setText(self.getCpuMemory())
self.sbCount = 0
self.sbTrigger = 10 # 10秒刷新一次
self.signalStatusBar.connect(self.updateStatusBar)
self.eventEngine.register(EVENT_TIMER, self.signalStatusBar.emit)
#----------------------------------------------------------------------
def updateStatusBar(self, event):
"""在状态栏更新CPU和内存信息"""
self.sbCount += 1
if self.sbCount == self.sbTrigger:
self.sbCount = 0
self.statusLabel.setText(self.getCpuMemory())
#----------------------------------------------------------------------
def getCpuMemory(self):
"""获取CPU和内存状态信息"""
cpuPercent = psutil.cpu_percent()
memoryPercent = psutil.virtual_memory().percent
return vtText.CPU_MEMORY_INFO.format(cpu=cpuPercent, memory=memoryPercent)
#----------------------------------------------------------------------
def addConnectAction(self, menu, gatewayName, displayName=''):
"""增加连接功能"""
if gatewayName not in self.gatewayNameList:
return
def connect():
self.mainEngine.connect(gatewayName)
if not displayName:
displayName = gatewayName
actionName = vtText.CONNECT + displayName
connectAction = self.createAction(actionName, connect,
loadIconPath('connect.ico'))
menu.addAction(connectAction)
#----------------------------------------------------------------------
def createAction(self, actionName, function, iconPath=''):
"""创建操作功能"""
action = QtWidgets.QAction(actionName, self)
action.triggered.connect(function)
if iconPath:
icon = QtGui.QIcon(iconPath)
action.setIcon(icon)
return action
#----------------------------------------------------------------------
def createOpenAppFunction(self, appDetail):
"""创建打开应用UI的函数"""
def openAppFunction():
appName = appDetail['appName']
try:
self.widgetDict[appName].show()
except KeyError:
appEngine = self.mainEngine.appDict[appName]
self.widgetDict[appName] = appDetail['appWidget'](appEngine, self.eventEngine)
self.widgetDict[appName].show()
return openAppFunction
#----------------------------------------------------------------------
def test(self):
"""测试按钮用的函数"""
# 有需要使用手动触发的测试函数可以写在这里
pass
#----------------------------------------------------------------------
def openAbout(self):
"""打开关于"""
try:
self.widgetDict['aboutW'].show()
except KeyError:
self.widgetDict['aboutW'] = AboutWidget(self)
self.widgetDict['aboutW'].show()
#----------------------------------------------------------------------
def openContract(self):
"""打开合约查询"""
try:
self.widgetDict['contractM'].show()
except KeyError:
self.widgetDict['contractM'] = ContractManager(self.mainEngine)
self.widgetDict['contractM'].show()
#----------------------------------------------------------------------
def closeEvent(self, event):
"""关闭事件"""
reply = QtWidgets.QMessageBox.question(self, vtText.EXIT,
vtText.CONFIRM_EXIT, QtWidgets.QMessageBox.Yes |
QtWidgets.QMessageBox.No, QtWidgets.QMessageBox.No)
if reply == QtWidgets.QMessageBox.Yes:
for widget in self.widgetDict.values():
widget.close()
self.saveWindowSettings('custom')
self.mainEngine.exit()
event.accept()
else:
event.ignore()
#----------------------------------------------------------------------
def createDock(self, widgetClass, widgetName, widgetArea):
"""创建停靠组件"""
widget = widgetClass(self.mainEngine, self.eventEngine)
dock = QtWidgets.QDockWidget(widgetName)
dock.setWidget(widget)
dock.setObjectName(widgetName)
dock.setFeatures(dock.DockWidgetFloatable|dock.DockWidgetMovable)
self.addDockWidget(widgetArea, dock)
return widget, dock
#----------------------------------------------------------------------
def saveWindowSettings(self, settingName):
"""保存窗口设置"""
settings = QtCore.QSettings('vn.trader', settingName)
settings.setValue('state', self.saveState())
settings.setValue('geometry', self.saveGeometry())
#----------------------------------------------------------------------
def loadWindowSettings(self, settingName):
"""载入窗口设置"""
settings = QtCore.QSettings('vn.trader', settingName)
# 这里由于PyQt4的版本不同,settings.value('state')调用返回的结果可能是:
# 1. None(初次调用,注册表里无相应记录,因此为空)
# 2. QByteArray(比较新的PyQt4)
# 3. QVariant(以下代码正确执行所需的返回结果)
# 所以为了兼容考虑,这里加了一个try...except,如果是1、2的情况就pass
# 可能导致主界面的设置无法载入(每次退出时的保存其实是成功了)
try:
self.restoreState(settings.value('state').toByteArray())
self.restoreGeometry(settings.value('geometry').toByteArray())
except AttributeError:
pass
#----------------------------------------------------------------------
def restoreWindow(self):
"""还原默认窗口设置(还原停靠组件位置)"""
self.loadWindowSettings('default')
self.showMaximized()
########################################################################
class AboutWidget(QtWidgets.QDialog):
"""显示关于信息"""
#----------------------------------------------------------------------
def __init__(self, parent=None):
"""Constructor"""
super(AboutWidget, self).__init__(parent)
self.initUi()
#----------------------------------------------------------------------
def initUi(self):
""""""
self.setWindowTitle(vtText.ABOUT + 'VnTrader')
text = u"""
Developed by Traders, for Traders.
License:MIT
Website:www.vnpy.org
Github:www.github.com/vnpy/vnpy
"""
label = QtWidgets.QLabel()
label.setText(text)
label.setMinimumWidth(500)
vbox = QtWidgets.QVBoxLayout()
vbox.addWidget(label)
self.setLayout(vbox)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,898
|
Oshkosh, Wis.— Continental Girbau Inc. (Continental), in Oshkosh, Wis., recently welcomed Dave "Mac" Mc Allister, of Berlin, Wis., as director of services and product development. Mc Allister, who fills a newly created position at Continental, oversees customer and technical service areas for all Continental subsidiaries. He is also a member of Continental's seven-person Executive Committee.
Mc Allister, who has managed technical, engineering or development teams over the past 23 years, holds a decade of experience related directly to the commercial laundry industry. Since 2000, he's climbed the ranks at Magnum Products LLC (now Generac® Mobile Products), in Berlin, Wis. During that time, he directed engineering and product development efforts.
In his new role at Continental, Mc Allister manages the customer service and technical service departments; innovates strategies to help the company meet strategic goals; implements tactical operations plans for enhanced customer experiences; and monitors Girbau Industrial work procedures and schedules.
Mc Allister holds an associate's degree in automotive technologies from Fox Valley Technical College, in Appleton, Wis.
To find out more about Continental Girbau visit their website or call 800-256-1073.
Continental Girbau Inc. is the largest of 15 subsidiaries of the Girbau Group, based in Vic, Spain. Girbau laundry products – marketed throughout 90 countries worldwide – meet rigorous environmental and safety standards established by the International Organization for Standardization (ISO). Girbau S.A. holds both ISO9001 and ISO14001 certifications. Ever focused on laundry efficiency, Continental Girbau is a member of the U.S. Green Building Council (USGB), a 501(c)(3) nonprofit that developed the Leadership in Energy and Environmental Design (LEED) Green Building Rating System.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 965
|
\section*{Abstract}
\label{sec:Abstract}
The calculation of the anharmonic modes of small to medium sized
molecules for assigning experimentally measured frequencies to the
corresponding type of molecular motions is computationally challenging
at sufficiently high levels of quantum chemical theory. Here, a
practical and affordable way to calculate coupled-cluster quality
anharmonic frequencies using second order vibrational perturbation
theory (VPT2) from machine-learned models is presented. The approach,
referred to as ``NN + VPT2'', uses a high-dimensional neural network
(PhysNet) to learn potential energy surfaces (PESs) at different
levels of theory from which harmonic and VPT2 frequencies can be
efficiently determined. The NN + VPT2 approach is applied to eight
small to medium sized molecules (H$_2$CO, trans-HONO, HCOOH, CH$_3$OH,
CH$_3$CHO, CH$_3$NO$_2$, CH$_3$COOH and CH$_3$CONH$_2$) and
frequencies are reported from NN-learned models at the
MP2/aug-cc-pVTZ, CCSD(T)/aug-cc-pVTZ and CCSD(T)-F12/aug-cc-pVTZ-F12
levels of theory. For the largest molecules and at the highest levels
of theory, transfer learning (TL) is used to determine the necessary
full-dimensional, near-equilibrium PESs. Overall, NN+VPT2 yields
anharmonic frequencies to within 20 cm$^{-1}$ of experimentally
determined frequencies for close to 90 \% of the modes for the highest
quality PES available and to within 10 cm$^{-1}$ for more than 60 \%
of the modes. For the MP2 PESs only $\sim 60$~\% of the NN+VPT2
frequencies were within 20~cm$^{-1}$ of the experiment, with outliers
up to $\sim 150$~cm$^{-1}$ compared with experiment. It is also
demonstrated that the approach allows to provide correct assignments
for strongly interacting modes such as the OH bending and the OH
torsional modes in formic acid monomer and the CO-stretch and OH-bend
mode in acetic acid.
\section{Introduction}
\label{sec:Introduction}
Vibrational spectroscopy is a sensitive probe for identifying
molecules or to follow conformational and structural changes in the
gas phase and in solution. One essential task in the practical use of
vibrational spectroscopy is the assignment of a measured frequency to
its corresponding type of molecular motion. Based on this information
it is also possible to predict changes both in the characterization of
these motions and their influence on the frequencies themselves. In
spectrally congested regions, as in the frequency range between 1200
cm$^{-1}$ and 1700 cm$^{-1}$, these assignments are particularly
challenging due to couplings between the different degrees of freedom.
Similarly, force field parametrization relies on fitting computed
normal mode frequencies to the correctly assigned band positions from
experiment. In practice it would, however, be preferable to use
anharmonic computed frequencies in force field development because
normal modes are already based on the harmonic oscillator
assumption. It is for such tasks that computational approaches are
particularly valuable. \\
\noindent
An accurate description of the vibrational dynamics and IR
spectroscopy remains a challenging problem in molecular
spectroscopy\cite{qu2019quantum}. Often, these calculations require
accurate, full-dimensional potential energy surfaces (PESs) for which
machine learning (ML) methods have gained a lot of
attention\cite{unke2020high}. ML potentials are used to generate
statistical models for energies and forces based on molecular
structures from extensive \textit{ab initio} data. The resulting
potentials can reproduce the reference data with unprecedented
accuracies\cite{meuwly2020ml,qu2021acac} (energies and forces with
errors in the range of $10^{-2} - 10^{-5}$~kcal/mol and $10^{-1} -
10^{-3}$~kcal/mol/\r{A}, harmonic frequencies are obtained within
$\sim 0.5$~cm$^{-1}$) and thus are superior to \textit{ab initio}
potentials due to their efficiency.\\
\noindent
Predictions of accurate anharmonic frequencies which compare
sufficiently well with experiment remain a challenge to overcome and
allow assignments of the vibrations and interpretation of
spectroscopic
features.\cite{nejad2020glycolic,mata2017benchmarking,barone2014fully}
Recent studies\cite{qu2018ir,qu2018quantum} illustrate that IR spectra
determined from molecular dynamics (MD) simulations are unable to
capture the full anharmonic behaviour of the molecule, especially for
high-frequency modes. This motivates the search for alternative
approaches, given that a sufficient number of reference calculations,
at high levels of theory, have become
possible\cite{qu2018high,koner2020permutationally,meuwly2020ml}.\\
\noindent
This work presents a practical approach to calculate anharmonic
frequencies from accurate, machine-learned potentials for small (4
atoms) to medium-sized (9 atoms) molecules. The ML-PESs are used as an
external energy function to quantum chemistry software (here the
Gaussian package\cite{g09}), to determine energies, forces, force
constants, dipole moments and dipole moment derivatives for a
vibrational perturbation theory (VPT2)
analysis\cite{barone2005anharmonic}. This has the advantage that a
direct comparison of results from explicit normal mode and VPT2
calculations, at a given level of theory from the electronic structure
code, and the results from using the ML-PES is possible. Moreover, ML
models can be systematically improved by including additional data
and/or by using data from higher levels of quantum chemical
theory. For this, transfer learning (TL) schemes, which have been
shown to be data-efficient alternatives
\cite{taylor2009transfer,pan2009survey,meuwly2020ml}, can be used to
achieve higher quality PESs and will be explored here, too. \\
\noindent
The present work reports on a systematic study of anharmonic
frequencies for H$_2$CO, trans-HONO, HCOOH, CH$_3$OH, CH$_3$CHO,
CH$_3$NO$_2$, CH$_3$COOH and CH$_3$CONH$_2$ based on machine-learned
models employing reference data from electronic structure calculations
at different levels of theory. The machine-learned models will then be
used for harmonic and VPT2 calculations which can also be compared
with results from experiments. Furthermore, TL to higher levels of
theory is explored. As an example, for the largest molecules a
sufficient number of energies and forces (estimated to be around
$10^4$ or larger) at the highest levels of theory, such as CCSD(T), is
usually not feasible. Hence, a relevant question is whether by
starting from a robust MP2-learned PES one can `transfer learn' to a
CCSD(T)-quality PES from a considerably smaller set of reference
energies and gradients at this higher level of theory which also
yields improved anharmonic frequencies compared with the MP2
level. Earlier work suggests that TL can result in substantial
improvements in accuracy and
data-efficiency\cite{meuwly2020ml}. Finally, the study will also
discuss how the cost for generating an accurate ML model (i.e. the
cost of structure sampling, training and evaluating the ML model)
compares with a single, 'straight' \textit{ab initio} VPT2
calculation. \\
\noindent
First, the methods and data generation strategies are presented. This
is followed by an assessment of the harmonic and anharmonic modes from
the ML-PESs compared with reference electronic structure calculations
at different levels of theory. Also, where available, comparison with
experimental results is made. Then, the computational efficiency of
the approach chosen is considered. Finally,
the results are discussed in a broader context and conclusions are drawn.
\section{Computational Methods}
\label{sec:Methods}
\subsection{Data sets: sampling and quantum chemical methods}
Data sets at three levels of theory, including
MP2/aug-cc-pVTZ\cite{moller1934note, kendall1992electron},
CCSD(T)/aug-cc-pVTZ\cite{pople1987quadratic, purvis1982full,
kendall1992electron}, and
CCSD(T)-F12/aug-cc-pVTZ-F12\cite{adler2007simple,
peterson2008systematically} (referred to as ``MP2'', ``CCSD(T)'',
and ``CCSD(T)-F12'' for convenience in the following), were
generated. All single point electronic structure calculations,
including energies, forces and dipole moments required for ML, as well
as harmonic frequency calculations, were carried out using
MOLPRO\cite{MOLPRO}. Data sets at the MP2 level of theory were
generated for all molecules, for CCSD(T) they were generated for
molecules with $N_{\rm atom} \leq 6$, and data sets at the CCSD(T)-F12
level of theory were generated for molecules with $N_{\rm atom} \leq
5$.\\
\noindent
As molecules of different sizes are considered, the number of
\textit{ab initio} calculations for the data base was scaled
accordingly. Here, $(3N-6)\cdot 600$ geometries were sampled for each
molecule and the optimized geometry was added. Generation of the
reference geometries was based on normal mode
sampling\cite{smith2017ani}: First, the molecules were optimized at
the MP2 level of theory, and the normal mode vectors were determined
alongside with the corresponding harmonic force constants. Then,
distorted (non-equilibrium) conformations were obtained by randomly
displacing the atoms along linear combinations of normal mode vectors.
This sampling was carried out at different temperatures (here $T =10$,
50, 100, 300, 500, 1000, 1500 and 2000~K). The total number of
geometries was evenly divided between the temperatures ,
i.e. $((3N-6)\cdot 600)/8$ geometries were generated for each $T$.\\
\noindent
For three molecules, HONO, CH$_3$CHO and CH$_3$COOH, the initial data
set was enlarged by including additional geometries as the PhysNet
models predicted normal mode frequencies with larger errors than for
the other molecules. For HONO, an additional 2805 structures were
added. The geometries were sampled from $NVT$ simulations run at
1000~K using the semiempirical GFN2-xTB method\cite{bannwarth2019gfn2}
and extended with structures along particular normal modes. Moreover,
because the structures evaluated in the VPT2 calculations (i.e. the
atoms are slightly displaced from the equilibrium geometry for the
calculation of numerical derivatives) can be extracted, these were
added as well (henceforth, ``VPT2 geometries''). For CH$_3$CHO, a
total of 1072 additional structures were added. These correspond to
VPT2 geometries and geometries along particular normal modes. For
CH$_3$COOH, 109 additional, VPT2 geometries are added.\\
\subsection{Machine learning: PhysNet}
The representation of the PESs is based on a neural network (NN) using
the PhysNet architecture.\cite{MM.physnet:2019} A detailed description
of the NN architecture is given in Reference~\citenum{MM.physnet:2019}
and only the salient features and those used in the present work are
given below. PhysNet is a high-dimensional NN of the ``message
passing'' type\cite{gilmer2017neural} and was applied recently to
different molecular
systems\cite{rivero2019reactive,brickel2019reactive,mm.ht:2020,mm.atmos:2020,sweeny2020thermal,meuwly2020ml}. The
NN learns a feature vector that encodes the local chemical environment
of each atom $i$ for the prediction of total molecular energies,
atomic forces and molecular dipole moments. The feature vector
initially contains information about the nuclear charges $Z_i$ and Cartesian
coordinates $\bm{r}_i$ and is iteratively refined (``learned'') during
training. The total energy of a molecule with arbitrary geometry
includes long-range electrostatics and dispersion interactions and is
given by
\begin{align}
E = \sum_{i=1}^N E_i +
k_e\sum_{i=1}^N\sum_{j>i}\frac{q_iq_j}{r_{ij}} + E_{\rm D3}.
\end{align}
where $E_i$ are the atomic energy contributions, $E_{\rm D3}$ is
Grimme's D3 dispersion correction \cite{grimme2010consistent}, $q_i$
are partial charges, $r_{ij}$ are interatomic distances, $k_e$ is
Coulomb's constant and $N$ is the total number of atoms. Note that the
partial charges are adjusted to assure charge conservation and the
Coulomb term is damped for short distances to avoid numerical
instabilities, see Ref.~\citenum{MM.physnet:2019} for details. Besides
the energy, PhysNet also predicts molecular dipole moments from
partial charges according to $\bm{\mu} = \sum_{i=1}^N q_i \bm{r_i}$
and analytical derivatives of $E$ with respect to the Cartesian
coordinates of the atoms are obtained by reverse mode automatic
differentiation\cite{baydin2017automatic}.\\
\noindent
For the present work, PhysNet was adapted to additionally predict
analytical derivatives of the dipole moment $\bm{\mu}$ and second
order derivatives of $E$ with respect to Cartesian coordinates
(i.e. Hessians). This was easily achieved using
Tensorflow\cite{tensorflow2015-whitepaper} utilities.\\
\noindent
The data sets containing $(3N-6)\cdot 600 + 1$ data points were split
according to 85/10/5~\% into training/validation/test sets for the
fitting of the ML model. The training set always contained the
optimized geometry. PES representations are obtained by adapting the
PhysNet parameters to best describe reference energies, forces, and
dipole moments from explicit quantum chemical calculations. The
optimization is carried out using
AMSGrad\cite{reddi2019convergence}. The relative contribution of the
different error terms is controlled by weighting hyperparameters
giving the force errror a higher weight/importance compared to the
energy error. All hyperparameters of the NN architecture and its
optimization approach were set to the values presented in
Reference~\citenum{MM.physnet:2019}, except for the cutoff radius
$r_{\rm cut}$ for interactions in the NN which was set to 6~\AA. Such
a cutoff was sufficient to include all interactions in all
molecules.\\
\subsection{Vibrational second-order perturbation theory}
Second-order vibrational perturbation theory (VPT2) is used to include
anharmonic and mode coupling effects into spectroscopic
properties\cite{nielsen1951vibration}. Many of the common quantum
chemistry packages, such Gaussian\cite{g09,barone2005anharmonic} or
CFOUR\cite{cfour}, include implementations for the calculation of
infrared frequencies and intensities including anharmonic effects
directly from \textit{ab initio} data. VPT2 assumes that the potential
energy of a system can be expressed as a quartic force field given by
\begin{align}\label{eq:vpt2_forcefield}
V = \frac{1}{2}\sum\omega_i\hat{q_i}^2 + \frac{1}{3!}\sum \phi_{ijk} \hat{q_i}
\hat{q_j} \hat{q_k} + \frac{1}{4!} \sum\phi_{ijkl} \hat{q_i} \hat{q_j} \hat{q_k} \hat{q_l}.
\end{align}
\noindent
Here, $\omega_i$ is a harmonic frequency, $\hat{q_i}$ are (reduced
dimensionless) normal mode coordinates, not to be confused with the
partial charges $q_i$, and $\phi_{ijk}$ and $\phi_{ijkl}$ are third-
and fourth-order derivatives of the potential $V$ with respect to
normal mode coordinates
\cite{yu2015vibrational,barone2005anharmonic}. The first term in
Equation~\ref{eq:vpt2_forcefield} corresponds to the harmonic part of
the potential while the remaining terms describe anharmonic
effects. Using expressions from earlier
work\cite{yu2015vibrational,barone2005anharmonic,bloino2012general,bloino2015vpt2}
and omitting kinetic/rotational terms, the cubic and quartic force
constants can be used to obtain anharmonic constants:
\begin{align}
16\chi_{ii} &= \phi_{iiii} - \sum_j
\frac{\left(8\omega_i^2-3\omega_j^2\right)\phi^2_{iij}}{\omega_j\left(4\omega_i^2
- \omega_j^2\right)}\\
4\chi_{ij} &= \phi_{iijj} - \sum_k
\frac{\phi_{iik}\phi_{jjk}}{\omega_k^2} + \sum_k
\frac{2\omega_k\left(\omega_i^2 + \omega_j^2 -
\omega_k^2\right)\phi_{ijk}^2}{\Delta_{ijk}}\\ \Delta_{ijk} &=
\left(\omega_i + \omega_j - \omega_k\right)\left(\omega_i +
\omega_j + \omega_k\right)\left(\omega_i - \omega_j +
\omega_k\right)\left(\omega_i - \omega_j - \omega_k\right)
\end{align}
From the resulting anharmonic constants $\chi$ the anharmonic
fundamental frequencies $\nu_i$ are obtained according to
\begin{align}
\nu_i = \omega_i + 2\chi_{ii} + \frac{1}{2}\sum_{i\neq j} \chi_{ij}.
\label{eq:vpt2_freq}
\end{align}
In addition, overtones, combination bands and zero-point energies can
be determined, see e.g. Ref.~\citenum{yu2015vibrational}.\\
\noindent
In this work, the generalized VPT2\cite{barone2005anharmonic}
implementation (GVPT2) in the Gaussian\cite{g09} suite is used to
determine anharmonic frequencies. Specific settings including which
VPT2 model to use, step-size for numerical differentiation and
resonance thresholds are retained at their default values. Gaussian's
standardized interface initialized by the ``External'' keyword is used
to call an external script (i.e. the PhysNet potential), which
produces an energy, gradients and Hessians for a given
molecule/geometry. The output from the external potential is recovered
from a standard text file by Gaussian where the the cubic and quartic
terms are determined by numerical derivatives and used for a VPT2
calculation.\\
\section{Results}
\label{sec:Results}
The performance of the ML models is assessed by three points:
i) Out-of-sample errors for energies ($\Delta E$) and forces ($\Delta F$)
for which geometries of a separate test set are used. These errors
quantify to what extent the ML models are capable of interpolating
between training points.
ii) Comparing harmonic frequencies determined
from the trained PhysNet model and conventional \textit{ab initio}
harmonic frequencies calculations at the same level of theory. These
harmonic frequencies also affect the anharmonic (VPT2) frequencies,
see Eq.~\ref{eq:vpt2_freq}), and
iii) Comparing VPT2 frequencies from the
PhysNet model with those from Gaussian at the MP2 level of theory and
with those from experiment, respectively. The comparison with
\textit{ab initio} MP2 VPT2 frequencies quantifies how closely the
PhysNet PES reproduces the \textit{ab initio} potential around the
minimum.\\
\subsection{Out-of-sample errors}
All the PhysNet models were evaluated on separate test sets by means of
MAEs and root mean squared errors (RMSEs) for energies and forces
between reference calculations and predictions by PhysNet, see
Figures~\ref{fig:energy_learning} and \ref{fig:force_learning}. MAEs
(squares) and the RMSEs (triangles) at different levels of theory are
color coded. For each of the molecules and levels of theory two
independent PhysNet models (opaque and transparent symbols) were
trained on the same data to assess consistency and
reproducibility. The lowest MAE($E$) is $\approx 0.0005$~kcal/mol (for
H$_2$CO) whereas the largest is $\approx 0.0218$~kcal/mol (for
CH$_3$CONH$_2$) indicating that errors scale with system size. These
MAEs correspond to about 0.003 and 0.03~\% of the energy range spanned
by the data sets (16.3~kcal/mol for H$_2$CO and 72.6~kcal/mol for
CH$_3$CONH$_2$), respectively, and all $R^2$ coefficients are above
0.99998 (see Table~S2). From
Figure~\ref{fig:energy_learning} it is apparent that two independently
trained models (opaque and transparent symbols on the same vertical
line) can still differ appreciably (MAEs($E$) of $\sim 0.0013$ and
$\sim 0.0072$~kcal/mol for H$_2$CO trained on MP2 data, see
Tab.~S2) although both models are still of remarkable
quality.\\
\begin{figure}[h!]
\centering \includegraphics[width=0.9\textwidth]{energy_error.eps}
\caption{The out-of-sample MAEs (squares) and RMSEs (triangles) of the
energy for the different molecules. The levels of theory are
color-coded and opaque and transparent symbols represent two PhysNet
models trained independently on the same data. A general trend
showing an increased error with increasing system size is
visible. The lowest MAE is $\sim 0.0005$~kcal/mol (H$_2$CO, CCSD(T))
and the highest is $\sim 0.0218$~kcal/mol (CH$_3$CONH$_2$, MP2). All
PhysNet models predict the independent test set with chemical
accuracy (better than 1~kcal/mol) and all out-of-sample performance
measures are summarized in Table~S2.}
\label{fig:energy_learning}
\end{figure}
\noindent
The appreciable variations between the two models trained on the same
data (opaque and transparent symbols, respectively) are notably
smaller for the force errors (i.e. the opaque and transparent squares
are very close to each other), MAE($F$) and RMSE($F$), see
Figure~\ref{fig:force_learning}. Earlier work\cite{meuwly2020ml}
showed that this is a consequence of the different weights
(hyperparameters) in the loss function of PhysNet. During training,
the higher weight of the force leads to a slight deterioration of the
energies while the forces are still improving. The higher weight on
the forces, however, is motivated by the fact that accurate
derivatives are required for the present work. Similar to the errors
in the energies $\Delta E$, the force errors $\Delta F$ tend to
increase with increasing system size. The force errors for H$_2$CO are
notably lower compared to the other molecules. Because PhysNet is
invariant with respect to permutation of equivalent atoms, the C$_{\rm
2v}$ symmetry of H$_2$CO could be responsible for the lower
out-of-sample errors. All test sets are predicted with energy errors
considerably better than chemical accuracy and, according to earlier
studies using PhysNet, are within the expected range (see
e.g. Refs.~\citenum{meuwly2020ml} and \citenum{mm.ht:2020} for H$_2$CO
and for larger molecules, respectively).\\
\begin{figure}[h!]
\centering \includegraphics[width=0.9\textwidth]{force_error.eps}
\caption{The out-of-sample MAEs (squares) and RMSEs (triangles) of the
forces for the different molecules. The levels of theory are
color-coded, and the opaque and transparent symbols represent two
PhysNet models trained independently on the same data. A general
trend showing an increased error with increasing system size is
visible. The lowest MAE is $\sim 0.0013$~kcal/mol/\r{A} (H$_2$CO,
MP2) and the highest is $\sim 0.0400$~kcal/mol/\r{A}
(CH$_3$CONH$_2$, MP2). All out-of-sample performance measures can be
found in Tab.~S2.}
\label{fig:force_learning}
\end{figure}
\subsection{Harmonic frequencies}
Harmonic frequencies computed from the PhysNet representations are
useful to test how well the ML model reproduces the region close to
the minimum of the PES compared with frequencies determined directly
from the \textit{ab initio} calculations. Also, they provide the
anharmonic frequencies through Eq. \ref{eq:vpt2_freq} which will be
essential for the calculation of VPT2
frequencies. Figure~\ref{fig:harm_freq} compares the reference
\textit{ab initio} harmonic frequencies from MOLPRO\cite{MOLPRO} with
those from PhysNet after optimizing the structure of each molecule
with the respective energy function. From the two models trained
independently on the same data, only the frequencies from the best
PhysNet model are discussed. The complete list of PhysNet and
\textit{ab initio} harmonic frequencies for all molecules and at all
levels of theory are reported in
Tables~S3-S5,
S7-S9,
S11-S13,
S15, S16,
S18, S20,
S22, and S24.\\
\noindent
For the smaller molecules (Figure~\ref{fig:harm_freq}A), all harmonic
frequencies from the PhysNet models are within 1~cm$^{-1}$ (mostly $<
0.5$~cm$^{-1}$) of the reference {\it ab initio} calculations and the
MAEs($\omega$) range from 0.08 to 0.21~cm$^{-1}$. Similarly, for the
larger molecules (Figure~\ref{fig:harm_freq}B) most of the harmonic
frequencies are within 1~cm$^{-1}$ of the reference values. Single
larger differences are found for the low frequency modes with errors
of 1.23 and 2.01~cm$^{-1}$ for CH$_3$NO$_2$ and CH$_3$CONH$_2$,
respectively. The remaining PhysNet frequencies reproduce the {\it ab
initio} harmonic frequencies with errors smaller than 1~cm$^{-1}$
and MAEs($\omega$) are between 0.09 and 0.28~cm$^{-1}$. Note that all
the optimized structures are true minima (i.e. no imaginary
frequencies were found).\\
\begin{figure}[h!]
\centering \includegraphics[width=1.0\textwidth]{harm_freq_comb.eps}
\caption{The accuracy of the PhysNet harmonic frequencies is shown
with respect to the appropriate reference \textit{ab initio}
values. Here, $\Delta\omega$ corresponds to $\omega_{\rm
ref}-\omega_{\rm PhysNet}$ and the figure is divided into two
windows for clarity. For the small molecules (A), irrespective of
the level of theory, all frequencies are reproduced to within less
than 1~cm$^{-1}$. For the larger molecules (B), deviations larger
than 1~cm$^{-1}$ are found. Largest errors are found for
CH$_3$NO$_2$ (1.23~cm$^{-1}$) and for CH$_3$CONH$_2$
(2.01~cm$^{-1}$). Tables containing the PhysNet and \textit{ab
initio} harmonic frequencies can be found in
Tabs.~S3-S5,
S7-S9,
S11-S13,
S15, S16,
S18, S20,
S22, S24.}
\label{fig:harm_freq}
\end{figure}
\subsection{VPT2 frequencies}
Next the quality of the VPT2 frequencies for the best (based on the
MAE($\nu$)) PhysNet models is assessed. For this, VPT2 frequencies
computed from PhysNet models trained on MP2 data are compared to their
\textit{ab initio} counterparts, see
Figure~\ref{fig:mp2_anharm_freq}. This comparison was not done for the
higher levels of theory because the VPT2 method in Gaussian is only
available for levels with analytical second derivatives\cite{g09} and
because the calculations get very expensive for the larger
molecules. The majority of the MP2 VPT2 frequencies are reproduced to
within better than 5~cm$^{-1}$ with single larger differences of up to
$\sim 10$~cm$^{-1}$. It is apparent that the PhysNet VPT2 frequencies
for the smaller molecules (Figure~\ref{fig:mp2_anharm_freq}A) are more
accurate - with MAEs($\nu$) between 0.95 and 1.55~cm$^{-1}$ - than the
frequencies of the larger molecules
(Figure~\ref{fig:mp2_anharm_freq}B). For the larger molecules
MAEs($\nu$) between 1.14 and 2.71~cm$^{-1}$ are found. The good
agreement between \textit{ab initio} and PhysNet VPT2 frequencies
further confirms the high quality of the ML potentials as third and
fourth order derivatives of the potential are very sensitive to the
shape and accuracy of the PES. Note that for two molecules
(CH$_3$NO$_2$ and CH$_3$CONH$_2$) one and two negative VPT2
frequencies are found for the lowest frequency modes from PhysNet as
well as from the \textit{ab initio} calculations. This is a deficiency
of VPT2 and can be explained by Equation~\ref{eq:vpt2_freq}. If the
perturbation $2\chi_{ii} +\frac{1}{2}\sum_{i\neq j} \chi_{ij}$ exceeds
the harmonic frequency $\omega_i$ (and is negative), which can occur
in particular for small $\omega_i$, the VPT2 frequency can become
negative. These frequencies are reported in
Tables~S23 and S25. Even
though a direct comparison of the higher level PhysNet VPT2 to the
\textit{ab initio} reference is intractable, it seems reasonable to
assume that the coupled cluster quality PhysNet models reach similar
accuracies. This is further supported by the findings for the harmonic
frequencies at the CCSD(T) and CCSD(T)-F12 levels.\\
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth]{mp2_anharm_freq.eps}
\caption{The accuracy of the PhysNet MP2 VPT2 frequencies is shown
with respect to their MP2 \textit{ab initio} values. Here,
$\Delta\nu$ corresponds to $\nu_{\rm MP2}-\nu_{\rm PhysNet}$ and the
figure is divided into two windows for clarity. For most of the
molecules and frequencies, the reference anharmonic frequencies are
reproduced with deviations smaller than $10$~cm$^{-1}$ (or even
5~cm$^{-1}$ for the small molecules). The anharmonic frequencies of
CH$_3$NO$_2$ and CH$_3$CONH$_2$ having negative frequencies are not
shown. Tables containing the PhysNet and \textit{ab initio} VPT2
frequencies can be found in
Tabs.~S6,
S10,
S14,
S17,
S19, S21,
S23, S25.}
\label{fig:mp2_anharm_freq}
\end{figure}
\noindent
The (anharmonic) PhysNet VPT2 frequencies can also be compared
directly to experimental data, see Figure~\ref{fig:anharm_freq}. For
the small molecules good agreement is found and the frequencies
converge towards the experimental values when going from MP2 to
CCSD(T)-F12. VPT2 frequencies from the PhysNet model for H$_2$CO
trained on CCSD(T)-F12 data reproduce the experimental
data\cite{herndon2005determination} with a maximum deviation of $\sim
20$~cm$^{-1}$ and a MAE($\nu$) of $\sim 4$~cm$^{-1}$. Similar trends
are also visible for the other CCSD(T)-F12 models where a MAE($\nu$)
of $\sim 7$~cm$^{-1}$ ($\sim 4$~cm$^{-1}$) is found for HONO (HCOOH),
see Table~\ref{tab:all_mae_vpt2_exp}. In going from MP2 to CCSD(T)-F12
the MAE reduces by a factor of almost five. For CH$_3$OH the highest
quality PhysNet model is trained on CCSD(T) reference
calculations. Here, larger deviations of up to $\sim 58$~cm$^{-1}$ for
single vibrations and a MAE($\nu$) = 14.30~cm$^{-1}$ are found.\\
\noindent
For the larger molecules (Figure~\ref{fig:anharm_freq}B) the PhysNet
models are only trained on MP2 reference data. As judged from the
improvements between VPT2 calculations at the MP2, CCSD(T), and
CCSD(T)-F12 levels with experiment for H$_2$CO, HONO and HCOOH, for
PhysNet trained on MP2 data only modest agreement between computed and
experimentally observed frequencies is expected for the larger
molecules. VPT2 modes from the PhysNet models trained on MP2 data for
the larger molecules (Figure~\ref{fig:anharm_freq}B) with frequencies
below $\sim 2000$~cm$^{-1}$ are centered around the experimental
values with a MAE($\nu$) between 6 and 30~cm$^{-1}$. The higher
frequencies, however, tend to be overestimated by PhysNet trained on
data at the MP2 level of theory (MAE($\nu$) between 31 and
97~cm$^{-1}$). Because VPT2 frequencies from PhysNet and \textit{ab
initio} MP2 calculations agree well (see
Figure~\ref{fig:mp2_anharm_freq}) it is concluded that the (large)
errors are not caused by the ML method itself but rather are a
deficiency of the MP2 level of theory and/or of the VPT2 approach when
compared with experiment. Nevertheless, for completeness, the MAEs
with respect to experiment are given in
Table~\ref{tab:all_mae_vpt2_exp}. As it becomes computationally
unfeasible to calculate a comprehensive CCSD(T) data set containing
some thousand data points for the larger molecules, an alternative
approach to improve the ML models is required. This is explored
next.\\
\begin{figure}[h!]
\centering \includegraphics[width=1.0\textwidth]{anharm_freq.eps}
\caption{VPT2 frequencies from PhysNet compared with experimental
values\cite{herndon2005determination, guilmot1993rovibrational,
tew2016ab, serrallach1974methanol, wiberg1995acetaldehyde,
goubet2015standard, wells1941infra, ganeshsrinivas1996simulation}.
Here, $\Delta\nu = \nu_{\rm exp}-\nu_{\rm PhysNet}$ and the figure
is divided into two panels for smaller (panel A) and larger (panel
B) molecules. For H$_2$CO good agreement is achieved with
experiment, where the frequencies converge towards the experiment
going from MP2 to CCSD(T)-F12 with a maximum deviation of $\sim
20$~cm$^{-1}$. A similar trend is also visible for HCOOH and
CH$_3$OH. In panel B, larger deviations are visible especially for
high frequencies. Tables
S6,
S10,
S14,
S17,
S19, S21,
S23, S25 report the
PhysNet, MP2 \textit{ab initio} VPT2, and experimental frequencies.
For HCOOH the literature suggests different frequencies for the OH
bending mode, see Refs.~\cite{millikan1957fam,tew2016ab} (1229 and
1302~cm$^{-1}$, respectively). Here, a frequency of 1302 is used,
following Ref.~\citenum{tew2016ab}.}
\label{fig:anharm_freq}
\end{figure}
\subsection{Transfer learning to coupled cluster quality}
As the accuracy of the MP2 based models is limited and as it is
computationally too expensive to obtain comprehensive coupled cluster
quality training data sets for the larger molecules, alternative
methods have to be considered. Among NN based methods TL can be used
to exploit correlations between data at different levels of theory
\cite{smith2018outsmarting}. TL uses the knowledge, acquired when
learning how to solve a task A, as a starting point to learn how to
solve a related, but different task
B\cite{pan2009survey,taylor2009transfer}. Methods related to TL are
$\Delta$-machine learning~\cite{DeltaPaper2015} or multi-fidelity
learning~\cite{batra2019multifidelity} which become increasingly
popular in quantum chemical applications
\cite{smith2018outsmarting,mm.ht:2020,nandi2021delta,mezei2020noncovalent}. Also,
TL bears similarities with potential morphing techniques which exploit
the fact that the overall shapes of PESs at sufficiently high levels
of theory are related and can be transformed by virtue of suitable
coordinate
transformations\cite{meuwly1999morphing,bowman1991simple}.\\
\noindent
Here, TL is used to obtain coupled cluster quality PESs for the larger
molecules based on models trained at the MP2 level. For this, the MP2
models are used as a good initial guess and only a fraction of the
data set, calculated at the CCSD(T) level, needs to be provided for
TL. To transfer learn the models, the parameters of a PhysNet model
trained on MP2 data is used to initialize the PhysNet training. Then,
PhysNet is trained (i.e. the parameters are adjusted) using energies,
forces and dipole moments from the higher level of theory. The
training is performed following the same approach and using the same
hyperparameters as when training a model from scratch, except that the
learning rate is decreased to values between $10^{-5} - 10^{-4}$.\\
\noindent
First, TL is applied to H$_2$CO which serves as a ``toy model''
because PhysNet models at different levels of theory are readily
available and allow for direct comparison. In other words, the TL
model based on MP2 data and retrained with input from CCSD(T)-F12 can
be directly compared with PhysNet trained entirely on CCSD(T)-F12
reference data. For this, the MP2 model is used as initial guess and
transfer learned to CCSD(T)-F12 quality using 188 data points (151 of
the original data set extended with 37 VPT2 geometries) and tested on
the remaining 3450 structures from the original CCSD(T)-F12 data
set. The TL model shows slightly lower errors for energies (MAE($E$) =
0.0004~kcal/mol) compared to the model trained from scratch (MAE($E$)
= 0.0009~kcal/mol)), whereas the forces are slightly less accurate
with MAE($F$) of 0.0066 and 0.0020~kcal/mol/{\AA} for TL and from
scratch, respectively. For the harmonic frequencies, all reference
CCSD(T)-F12 values are reproduced with deviations smaller than
0.3~cm$^{-1}$ and with a MAE($\omega$) = 0.1~cm$^{-1}$, which is
similar for a model trained from scratch, see
Tables~S5 and
S26. The most relevant measure of
performance is the direct comparison of VPT2 to experimental
frequencies, especially when the high level \textit{ab initio}
calculation of harmonic frequencies become too
expensive. Figure~\ref{fig:tl_h2co} shows the deviation of a transfer
learned PhysNet model, a PhysNet model trained on CCSD(T)-F12 data and
a model trained on MP2 data (the ``base model'' for TL) with respect
to the experimental H$_2$CO frequencies. VPT2 frequencies from
NN$_{\rm TL}$ are very close to those from NN$_{\rm CCSD(T)-F12}$
except for one high frequency mode and clearly superior to those from
NN$_{\rm MP2}$ throughout when compared with experiment.\\
\begin{figure}[h]
\centering \includegraphics[width=1.0\textwidth]{anharm_freq_tl_h2co.eps}
\caption{Comparison of VPT2 frequencies from PhysNet trained on MP2
(NN$_{\rm MP2}$), CCSD(T)-F12 (NN$_{\rm CCSD(T)-F12}$) and TL
(NN$_{\rm TL}$) from MP2 to CCSD(T)-F12 with the experiment
\cite{herndon2005determination} for H$_2$CO. Here, $\Delta\nu =
\nu_{\rm exp}-\nu_{\rm PhysNet}$. The VPT2 frequencies on the TL-PES
are within 8~cm$^{-1}$ to the values obtained from a model trained
from scratch yielding an improvement in comparison to an MP2 model
(its ``starting point'').}
\label{fig:tl_h2co}
\end{figure}
\noindent
TL is used to obtain high quality PESs for all the models for
which only MP2 reference data are available (i.e. CH$_3$CHO,
CH$_3$NO$_2$, CH$_3$COOH and CH$_3$CONH$_2$). For each of the
molecules around 5~\% or less of the original, lower level, data set
is recalculated at the CCSD(T) level. The TL data sets contain
geometries sampled using the normal mode approach (evenly split among
the different temperatures), VPT2 geometries and the MP2 optimized
geometry. They contained CCSD(T) information (energies, gradients and
dipole moments) on 262, 452, 542 and 632 CCSD(T) geometries for
CH$_3$CHO, CH$_3$NO$_2$, CH$_3$COOH and CH$_3$CONH$_2$,
respectively. The TL data sets are again split randomly according to
85/10/5~\% into training/validation/test sets for TL. As a
consequence, the TL models are tested only on a small number of
molecular geometries. \\
\noindent
For the larger molecules assessing the performance of a TL model is
more difficult. On the one hand, the rather small number of geometries
in the TL set are mostly used for training and validation. Thus, only
a small fraction ($< 50$ geometries) remains for testing the models
and might give less meaningful results statistically. The evaluation
of the separate test set for all molecules yields MAEs($E$) smaller
than 0.03~kcal/mol and MAE($F$) smaller than 0.02~kcal/mol/{\AA} (compare
to Figure~\ref{fig:energy_learning} and \ref{fig:force_learning}) and
are within the realm of what was expected. The MAEs and RMSEs of
energies and forces for all TL models are listed in
Table~S28 for completeness.\\
\noindent
A second measure of performance are the harmonic frequencies which can
be compared with those from direct \textit{ab initio} calculations.
Even though optimizations and normal mode calculations at the CCSD(T)
level become computationally expensive rather quickly for larger
molecules, they were performed for all the larger molecules CH$_3$CHO,
CH$_3$NO$_2$, CH$_3$COOH and CH$_3$CONH$_2$. For CH$_3$CHO,
CH$_3$NO$_2$ and CH$_3$COOH the harmonic frequencies on the TL PhysNet
models compare to within MAE($\omega$) = 0.2~cm$^{-1}$, MAE($\omega$)
= 0.3~cm$^{-1}$ and MAE($\omega$) = 0.1~cm$^{-1}$ with the explicit
calculations at the CCSD(T) level of theory, see
Tables~S29 to
S31. These errors are also within those
achieved by the models trained from scratch. Slightly larger errors
are found for CH$_3$CONH$_2$ (MAE($\omega$) = 1.1~cm$^{-1}$, see
Table~S32).\\
\begin{figure}[h]
\centering \includegraphics[width=1.0\textwidth]{anharm_freq_TL.eps}
\caption{Comparison of the VPT2 frequencies from PhysNet trained on
MP2 (NN$_{\rm MP2}$, blue squares) and TL to CCSD(T) (NN$_{\rm TL}$,
red asterisks) with experimental values. Here, $\Delta\nu = \nu_{\rm
exp}-\nu_{\rm PhysNet}$. The TL models yield overall improved VPT2
frequencies, especially for the high frequency modes. A rather large
deviation is found for the frequency around 1300~cm$^{-1}$ for
NN$_{\rm TL}$ for CH$_3$COOH. A red circle marks a surprisingly high
error, compared to the other modes. It is possible that this
discrepancy is caused by a misassignment of experimental modes, see
text.}
\label{fig:tl_large_molecules}
\end{figure}
\noindent
Finally, the performance of the models transfer learned to CCSD(T) can
be assessed by comparing the VPT2 frequencies with experiment, see
Figure~\ref{fig:tl_large_molecules}. As a reference, the VPT2
frequencies from PhysNet models trained on MP2 data (blue squares) are
reported together with the TL models. It is apparent that the TL
models are closer to experiment especially for the high frequency
modes and all MAEs($\nu$) with respect to experiment are reduced, see
Table~\ref{tab:all_mae_vpt2_exp}. For CH$_3$CHO
the MAE($\nu$) is reduced from 15.3 to 5.5 cm$^{-1}$, corresponding to
a factor of almost 3, while for the remaining three molecules this
factor is closer to 2. Good agreement is found for CH$_3$CHO,
CH$_3$NO$_2$ and CH$_3$COOH while the picture is less clear for
CH$_3$CONH$_2$.\\
\begin{table}[h]
\begin{tabular}{lccc|c}
\toprule
\textbf{MAE($\nu$) [cm$^{-1}$]} & \textbf{MP2}& \textbf{CCSD(T)} & \textbf{CCSD(T)-F12} & \textbf{scaled MP2/CCSD(T)}\\
\midrule
H$_2$CO & 18.40 & 9.04 & 3.98/{\bf 6.95} & 48.22/37.74\\
HONO & 26.98 & 5.56 & 6.74 & 38.44/17.31\\
HCOOH & 17.61 & 9.06 & 3.59 & 32.99/26.31\\
CH$_3$OH & 27.28 & 14.30 & -- & 23.47/19.60 \\
CH$_3$CHO & 15.27 & {\bf 5.47} & -- & 29.04/21.90\\
CH$_3$COOH & 16.25 & {\bf 10.83} &-- & 21.39/19.74 \\
CH$_3$NO$_2$ & 28.56 &{\bf 15.38} & -- & 34.27/25.28 \\
CH$_3$CONH$_2$ & 47.23 &{\bf 25.13} &-- & 43.46/34.81\\
\bottomrule
\end{tabular}
\caption{MAEs of the PhysNet VPT2 frequencies with respect to
experimental values\cite{herndon2005determination,
guilmot1993rovibrational, tew2016ab, serrallach1974methanol,
wiberg1995acetaldehyde, goubet2015standard, wells1941infra,
ganeshsrinivas1996simulation}. The level of theory of the training
data is given in the header and the results of the transfer learned
PhysNet models are highlighted in bold. An additional column shows
the MAEs of the ``conventional'' approach where harmonic frequencies
are scaled by an empirical factor\cite{cccbdb} (see also
Figure~S1 for an illustration of the scaled
frequencies compared with experiment).}
\label{tab:all_mae_vpt2_exp}
\end{table}
\subsection{Timings}
By using \textit{ab initio} calculations (see Table~S1 for cost of \textit{ab initio} energy, forces and dipole
moment calculations) and a PhysNet representation of the underlying
PES, it is possible to determine VPT2 frequencies at considerably
higher levels of theory for larger molecules than is possible from
straight {\it ab initio} evaluations. This raises the question: how do
the computational efficiencies of the different approaches compare?\\
\noindent
The following points are considered:
\begin{itemize}
\item {\it Feasibility at a given \textit{ab initio} level:} VPT2
calculations at the MP2 level have been performed for all molecules
considered. The calculation times ranged from few CPU hours for the
small molecules (H$_2$CO: 2.5~h, HONO: 5~h), to multiple CPU hours
for HCOOH/CH$_3$OH (14~h) and CH$_3$CHO (60~h) up to several days
for the largest molecules (CH$_3$COOH: 190~h, CH$_3$NO$_2$: 103~h,
CH$_3$CONH$_2$:~298~h). However, compared with experiment, VPT2
frequencies at the MP2 level of theory were found to be inferior to
CCSD(T) with unsigned deviations of up to $~150$~cm$^{-1}$, see
Table~\ref{tab:all_mae_vpt2_exp}. While a full CCSD(T) VPT2
treatment for the smallest molecules would be possible, this will be
too costly for the larger structures. It was
reported\cite{jacobsen2013anharmonic} that a VPT2 calculation for
H$_2$CO at various levels of theory is around 10 times more
expensive than the calculation of harmonic frequencies. This is also
found in the present work. Even though this factor will increase for
larger molecules (as more derivatives need to be evaluated) it can
be used to tentatively assess the cost of a CCSD(T) VPT2 treatment
of a larger molecule. The calculation of harmonic frequencies of
CH$_3$CONH$_2$ at the CCSD(T) level is found to take more than 20
CPU days which suggests that a CCSD(T) VPT2 calculation would take
more than 200 CPU days.\\
\item {\it Data Generation and Learning of PhysNet:} The time for
generating a PhysNet model includes computation of a data set
(\textit{ab initio} calculations of energies, forces and dipole
moments), training and validating the NN. The most costly steps are
the training ($\leq 5$~days) and, depending on the level of theory,
the calculation of the reference data. As an example, generating the
most expensive data set (5401 HCOOH geometries at the CCSD(T)-F12
level: $\sim 130$~CPU min per calculation) takes around two days
assuming that 250 calculations can be performed in parallel (the
``real time'' of the calculations might be longer depending on I/O
performance). The PhysNet VPT2 calculation then is the least
expensive step, taking less than 1~hour (H$_2$CO: 18 min,
CH$_3$CONH$_2$: 58 min).\\
\noindent
As an explicit example, data generation at the MP2 level, NN
training and VPT2 calculations for CH$_3$COOH or CH$_3$CONH$_2$ are
considered. With a conservative estimate for generating
reference data, fitting and testing of the PhysNet model of 7
days, a direct \textit{ab initio} VPT2 calculation at the MP2
level of theory exceeds the cost of the NN + VPT2 approach for
these two molecules by a factor of 1.1 (10 \%) and 1.7 (70 \%),
respectively. For larger molecules this factor increases further as
it does also for higher-level methods. In addition, the approach
pursued here has the added benefit of a full-dimensional,
near-equilibrium PES that can also be used for
MD simulations, albeit some additional reference calculations may be
required.\\
\item {\it Crossover between NN + VPT2 versus ab initio:} It is of
interest to ask at what point does the NN + VPT2 approach become
advantageous over the \textit{ab initio} approach in terms of
overall computational cost. For the higher levels of theory the cost
of \textit{ab initio} VPT2 calculations scales less favourably than
single (energy, forces and dipole moment) calculations due to the
repetitive evaluation of first- to fourth-order derivatives of the
potential energy. If the cost for an \textit{ab initio} VPT2
calculation is estimated\cite{jacobsen2013anharmonic} as $10*t_{\rm
harm}$ (where $t_{\rm harm}$ is the time to calculate harmonic
frequencies), then a PhysNet model becomes more favourable already
for HCOOH at the CCSD(T) level ($t_{\rm harm} = 30$~h). The
preference of the NN + VPT2 approach for larger molecules and higher
levels is even more apparent when using TL. As an example, the
CCSD(T) data set for CH$_3$CONH$_2$ (632 data points; the
calculations including energy, forces and dipole moments take $\sim
9$ CPU hours on average while the real time was closer to 15~h) is
generated in less than 2 days, and the TL can be performed with
costs on the order of 1~hour. Together with the time for reference
data generation at the MP2 level, training and evaluating of the
base NN, the total estimated time for VPT2 frequencies for
CH$_3$CONH$_2$ at the CCSD(T) level on the order of 10 CPU days,
compared with an estimated 200 CPU days from a brute force
calculation. Empirically, this is close to linear scaling, whereas
at the \textit{ab initio} level the scaling increases formally from
$N^5$ for MP2 to $N^7$ for CCSD(T)\cite{friesner2005ab}.\\
\end{itemize}
\noindent
In summary, while at the MP2 level the NN + VPT2 approach is
computationally superior only for the two largest molecules, it
outperforms the \textit{ab initio} approach at higher levels for all
but tetra-atomic molecules. The feasibility of the NN + VPT2
approach for larger molecules and higher levels is strengthened
when employing TL.\\
\section{Discussion}
The results presented so far have established that a ML model
yields the same normal mode and VPT2 frequencies as direct evaluations
from electronic structure calculations and comparison between
computations and experiments is favourable (unsigned deviations
between $< 0.5$ and 21~cm$^{-1}$ for the CCSD(T)-F12 models and
between $< 0.5$ and 88~cm$^{-1}$ for the CCSD(T) models, including
TL). Alternatively, calculated normal mode frequencies are often
scaled by empirical factors to compare directly with
experiment.\cite{scott1996harmonic} Table~\ref{tab:all_mae_vpt2_exp}
summarizes the mean absolute errors between experimentally measured
frequencies, those from VPT2 calculations at the three levels of
theory, and the scaled harmonic frequencies from MP2 and CCSD(T)
calculations (last column). Scaled MP2 and CCSD(T) frequencies differ
on average by 20 cm$^{-1}$ to 50 cm$^{-1}$ from those measured
experimentally. This is comparable to the difference between
(anharmonic) VPT2 calculations at the MP2 level on the PhysNet-PES and
experiment. On the other hand, VPT2 calculations on the CCSD(T) and
CCSD(T)-F12 PESs are in better agreement (average MAE of 10 cm$^{-1}$
and 5 cm$^{-1}$) and clearly outperform results from MP2 calculations
(scaled harmonic and VPT2) and scaled harmonic CCSD(T) frequencies.\\
\noindent
Application of TL showed that the harmonic frequencies of the TL
models reach MAE($\omega$) with respect to reference comparable to
models trained from scratch. The small additional cost for obtaining a
high-quality PES from a lower-level PES and the rapid scaling of the
costs for obtaining \textit{ab initio} harmonic frequencies motivates
the use of TL for the determination of high quality harmonic
frequencies for which fully-dimensional \textit{ab initio} PESs are
computationally too expensive.\\
\noindent
Comparison of accurate (anharmonic) VPT2 frequencies with experiment
can provide additional insight. Such comparisons depend on the quality
of the computations and that of the experiment. One particularly
appealing possibility is to compare and potentially reassign
individual modes. As an example, the OH bending vibration of monomeric
formic acid is considered. The early infrared studies were carried out
in the late 1950s by Milliken and Pitzer \cite{millikan1957fam} and by
Miyazawa and Pitzer \cite{miyazawa1959fam}, where the OH bending
vibration was assigned to a signal at 1229~cm$^{-1}$. Later work (see
e.g. Refs. \citenum{freytes2002overtone} or ~\citenum{reva1994ir} and
references therein) reported an OH bending frequency of 1223~cm$^{-1}$
consistent with the earlier work. An OH bending frequency of
1223~cm$^{-1}$ is $\sim 80$~cm$^{-1}$ lower than what is found from
VPT2 calculations on the PhysNet PES at the CCSD(T)-F12 level of
theory which finds it at 1302~cm$^{-1}$.\\
\noindent
Recent theoretical work determined vibrational
frequencies from vibrational configuration interaction (VCI)
calculations on a global CCSD(T)(F12*)/cc-pVTZ-F12 PES for cis- and
trans-formic acid and tried to re-assign experimentally determined
fundamental, overtone and combination
bands.\cite{tew2016ab,freytes2002overtone} These calculations
proposed to assign the fundamental OH bend to an experimentally
measured frequency at 1306.2~cm$^{-1}$ and the first overtone of the
OH torsion to a frequency at 1223~cm$^{-1}$, converse to previous
assignments.\cite{freytes2002overtone} Due to the rather close
spacing, these two vibrations are in strong 1:2 Fermi resonance. This
assignment (see Figure~\ref{fig:anharm_freq}) is consistent with the
VPT2 calculations on the PhysNet(CCSD(T)-F12) PES results which find
the OH bend at 1301.9~cm$^{-1}$. In addition, the first overtone of
the OH torsion can also be calculated using PhysNet + VPT2 and
is found at 1220.2~cm$^{-1}$, only 3 cm$^{-1}$ away from an
experimentally determined signal. Given the agreement of the two
recent high-level computational treatments, based on very different
approaches, and the fact that experiment indeed finds vibrational
bands around 1220 and 1300 cm$^{-1}$, a reassignment of the
fundamental OH bend to a frequency of 1306.2 cm$^{-1}$ is supported.\\
\noindent
A similar situation arises for CH$_3$COOH
(Figure~\ref{fig:tl_large_molecules}, red circle), where the signal at
1325.5 ~cm$^{-1}$ from VPT2 calculation on the TL PhysNet(CCSD(T)) PES
disagrees by $\sim 60$~cm$^{-1}$ from the experimentally measured
frequency at 1266~cm$^{-1}$.\cite{goubet2015standard} While (mostly)
older
studies\cite{wilmshurst1956infrared,haurie1965spectres,berney1970infrared,maccoas2003rotational}
assigned the C-O stretch to a frequency around 1260~cm$^{-1}$ and the
OH bend to a lower frequency around 1180~cm$^{-1}$, more recent work
came to a converse assignment\cite{goubet2015standard,marechal1987ir}
($\nu_7= 1266$~cm$^{-1}$ and $\nu_8= 1184$~cm$^{-1}$, following the
notation in Reference~\citenum{goubet2015standard}). Goubet et
al.\cite{goubet2015standard} noted that $\nu_7$ and $\nu_8$ are not
well predicted by theoretical calculations at the anharmonic level
(RI-MP2/aVQZ harmonic frequencies corrected with VPT2 corrections
obtained at the B98/aVQZ level following $\nu_{\rm RI-MP2} =
\omega_{\rm RI-MP2} - \left(\omega_{B98} - \nu_{B98}\right)$) and
ascribed the discrepancy to Fermi resonances between $\nu_7$ and
$\nu_8$ with the first overtone of $\nu_{16}$ (A$''$ fundamental at
642~cm$^{-1}$) and $\nu_{11}$ (A$''$ fundamental at
581.5~cm$^{-1}$). Similarly, a large deviation of --60~cm$^{-1}$
between TL PhysNet(CCSD(T)) and $\nu_7$ is found in this study
compared with --61~cm$^{-1}$ in
Reference~\citenum{goubet2015standard}.\\
\noindent
Besides Fermi resonance as an explanation for the discrepancies,
little work has been done on overtone and combination
bands. Reference~\citenum{olbert2011raman}, however, suggested an
alternative assignment of two bands at 1324.4 and 1259.4~cm$^{-1}$
previously observed in the IR spectra of acetic acid isolated in Ar
matrix \cite{berney1970infrared,macoas2004photochemistry}. They
proposed to reassign the frequency at 1324.4~cm$^{-1}$ to $\nu_7$ and
1259.4~cm$^{-1}$ to the first overtone of $\nu_{16}$ based on VPT2
calculations at the B3LYP/6-311++G(2d,2p) level and deuteration
experiments. This new assignment is supported by the NN + VPT2 results
yielding frequencies of 1325.5 and 1250.2~cm$^{-1}$ for $\nu_7$ and
$2\nu_{16}$, respectively. These present findings, for formic acid and
acetic acid, encourage further theoretical work on overtones and
combination bands.\\
\section{Conclusion and Outlook}
\label{sec:Conclusion}
The combined NN + VPT2 approach together with TL is shown to provide
accurate anharmonic frequencies at high levels of electronic structure
theory. The PhysNet code was adapted to additionally predict
analytical derivatives of the dipole moment $\bm{\mu}$ and second
order derivatives of $E$ with respect to Cartesian coordinates
(i.e. Hessians). Because for many high-level electronic structure
methods analytical second derivatives are not explicitly implemented,
the present NN-based approach is advantageous both, in terms of
accuracy and computational efficiency. The present method can be
systematically improved by using data at higher levels of theory,
explicitly or by using TL, and allows to obtain VPT2 frequencies at
levels of theory for which \textit{ab initio} VPT2 calculation are
impractical.\\
\noindent
The current study can be extended to include IR intensities for
fundamentals, combination bands and overtones. Moreover, the approach
can be extended to investigate larger molecules and assist in
(re)assigning experimental IR spectra. It is also shown that for the
smallest molecules and the lowest level of theory considered
(MP2/aug-cc-pVTZ) direct evaluation of harmonic and VPT2 frequencies
is computationally more efficient than the combined NN + VPT2
approach. However, for molecules with 6 and more atoms and for the
higher levels of quantum chemical treatment (CCSD(T) and higher) the
NN + VPT2 approach is computationally considerably more
efficient. Finally, the VPT2 frequencies at these highest levels of
theory compare favourably (to within a few cm$^{-1}$) with
experimentally determined frequencies. One additional application for
high-level VPT2 calculations is the accurate determination of
dimerization energies, such as H-bonded complexes, for which accurate
anharmonic zero point vibrational energies are
required\cite{suhm2012dissociation}. Also, using DFT calculations for
the reference ML model in TL to higher levels of theory is an
attractive prospect to further improve the efficiency of the present
method. Finally, additional improvements may be achieved by using
sophisticated sampling or learning approaches such as active
learning.\cite{smith2018less}
\section*{Data Availability Statement}
The PhysNet codes are available at
\url{https://github.com/MMunibas/PhysNet}, and the VibML dataset
containing the reference data can be downloaded from Zenodo
\url{https://doi.org/10.5281/zenodo.4585449}.\\
\section*{Acknowledgments}
This work was supported by the Swiss National Science Foundation
through grants 200021-117810, 200020-188724 and the NCCR MUST, and the
University of Basel.
\section{Data sets}
\begin{table}[h]
\begin{tabular}{ccccc}
\toprule
\#& \textbf{Molec} & \textbf{MP2/AVTZ} & \textbf{CCSD(T)/AVTZ} & \textbf{CCSD(T)-F12/AVTZ-F12}\\
\midrule
1& H$_2$CO & 0.5 min & 5.5 min & 18.5 min \\
2& HONO & 1.0 min & 27.0 min & 88.0 min \\
3& HCOOH & 1.5 min & 33.5 min & 130.0 min \\
4& CH$_3$OH & 1.0 min & 21.5 min & 70.0 min \\
5& CH$_3$CHO & 4.5 min & 102.0 min & 255.0 min \\
6& CH$_3$NO$_2$ & 5.5 min & 268.0 min & 876.0 min \\
7& CH$_3$COOH & 9.0 min & 380.0 min & ?? \\
8& CH$_3$CONH$_2$ & 11.0 min & 600.0 min & ??\\
\bottomrule
\end{tabular}
\caption{CPU timings for single processor jobs for the ab initio calculations using the given levels
of theory for a single calculation of energy, gradients and dipole
moments with tightened convergence criteria. For entries with ?? the calculation failed either due to
memory or space issues. Note that for example a CCSD(T)-F12 calculation for CH$_3$CHO needed about 350 GB of
space. For molecules 1 to 3 the calculation on CCSD(T)-F12, for
molecule 4 the calculations on CCSD(T) and for molecules 5 to 8 the calculations on the MP2 level of theory were
performed. While the timing calculations were performed on a Intel(R) Xeon(R) CPU E5-2630 v4 @ 2.20GHz for consistency, the remaining calculations were
performed on mixed computer architectures.}\label{sitab: timings}
\end{table}
\section{Results}
\subsection{Energy- and force-errors}
\begin{sidewaystable}[h]
\resizebox{\linewidth}{!}{%
\begin{tabular}{lcccccccccccccccc}
\toprule
& \multicolumn{2}{c}{{\bf H$_2$CO}} & \multicolumn{2}{c}{{\bf HONO}}& \multicolumn{2}{c}{{\bf HCOOH}} & \multicolumn{2}{c}{{\bf CH$_3$OH}} & \multicolumn{2}{c}{{\bf CH$_3$CHO}}& \multicolumn{2}{c}{{\bf CH$_3$COOH}}& \multicolumn{2}{c}{{\bf CH$_3$NO$_2$}}& \multicolumn{2}{c}{{\bf CH$_3$CONH$_2$}}\\
{\bf MP2} & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$& NN$_1$ & NN$_2$\\
\midrule
EMAE: & 7.19 & 1.26 & 1.92 & 2.47 & 20.42 & 10.43 & 7.90 & 2.93 & 1.36 & 15.13 & 10.35 & 21.05 & 8.48 & 12.26 & 21.83 & 22.30 \\
ERMSE: & 7.19 & 1.26 & 3.92 & 7.01 & 20.46 & 10.47 & 7.98 & 3.13 & 3.54 & 15.51 & 10.52 & 21.20 & 8.54 & 12.47 & 22.96 & 24.36 \\
FMAE: & 1.42 & 1.41 & 5.79 & 5.64 & 6.75 & 6.92 & 7.44 & 7.24 & 13.25 & 13.01 & 13.50 & 14.06 & 9.65 & 8.55 & 36.49 & 40.02 \\
FRMSE: & 3.66 & 3.44 & 23.32 & 27.01 & 29.83 & 18.33 & 27.02 & 29.08 & 53.37 & 47.07 & 38.41 & 43.00 & 73.22 & 67.00 & 106.02 & 128.84 \\
1-R2: & 5.4E-6 & 1.7E-7 & 1.0E-8 & 3.0E-8 & 1.4E-5 & 3.6E-6 & 6.0E-7 & 9.0E-8 & 3.0E-7 & 5.8E-6 & 2.3E-6 & 9.1E-6 & 2.4E-6 & 5.2E-6 & 3.2E-6 & 3.6E-6 \\
\midrule
{\bf CCSD(T)} & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$& NN$_1$ & NN$_2$\\
\midrule
EMAE: & 0.47 & 1.21 & 1.56 & 2.36 & 21.90 & 0.63 & 8.67 & 5.04 & & & & & & & & \\
ERMSE: & 0.49 & 1.22 & 4.71 & 5.54 & 21.91 & 0.87 & 8.74 & 5.15 & & & & & & & & \\
FMAE: & 1.48 & 1.49 & 4.87 & 5.74 & 5.93 & 6.73 & 6.56 & 7.32 & & & & & & & & \\
FRMSE: & 4.25 & 4.19 & 14.94 & 15.21 & 21.64 & 27.07 & 22.61 & 23.59 & & & & & & & & \\
1-R2: & 3.0E-8 & 1.6E-7 & 2.0E-8 & 2.0E-8 & 1.6E-5 & 2.0E-8 & 7.5E-7 & 2.6E-7 & & & & & & & & \\
\midrule
{\bf CCSD(T)-F12} & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$ & NN$_1$ & NN$_2$& NN$_1$ & NN$_2$\\
\midrule
EMAE: & 0.90 & 2.44 & 6.96 & 5.90 & 3.13 & 7.60 & & & & & & & & & & \\
ERMSE: & 1.00 & 2.49 & 11.27 & 6.81 & 3.25 & 7.68 & & & & & & & & & & \\
FMAE: & 2.02 & 2.43 & 5.09 & 5.15 & 6.24 & 5.69 & & & & & & & & & & \\
FRMSE: & 5.13 & 6.44 & 15.97 & 17.34 & 26.35 & 23.51 & & & & & & & & & & \\
1-R2: & 1.1E-7 & 6.7E-7 & 9.0E-8 & 3.0E-8 & 3.5E-7 & 2.0E-6 & & & & & & & & & & \\
\bottomrule
\end{tabular}}
\caption{Performance measures of two PhysNet models trained independently on
the same \textit{ab initio} data and evaluated on the test set. The MAEs and RMSEs are
given in kcal/mol(/\AA) and multiplied by a factor of 1000 for clarity. }\label{sitab:all_mae}
\end{sidewaystable}
\clearpage
\subsection{H$_2$CO}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 1196.65 & 1196.65 & 1196.70 & 0.05 & 0.05 \\
2 & 1266.71 & 1266.70 & 1266.64 & 0.07 & 0.06 \\
3 & 1539.99 & 1539.91 & 1539.91 & 0.08 & 0.00 \\
4 & 1752.61 & 1752.47 & 1752.51 & 0.10 & 0.04 \\
5 & 2973.23 & 2973.11 & 2973.27 & 0.04 & 0.16 \\
6 & 3047.04 & 3047.10 & 3047.30 & 0.26 & 0.20 \\
\bottomrule
\textbf{MAE:} & 0.10 & 0.09 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of H$_2$CO calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:h2co_harm_mp2}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)) & NN2(CCSD(T)) & CCSD(T) & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 1181.00 & 1181.02 & 1181.05 & 0.05 & 0.03 \\
2 & 1261.67 & 1261.64 & 1261.53 & 0.14 & 0.11 \\
3 & 1529.54 & 1529.43 & 1529.42 & 0.12 & 0.01 \\
4 & 1764.82 & 1764.73 & 1764.87 & 0.05 & 0.14 \\
5 & 2932.10 & 2931.96 & 2932.13 & 0.03 & 0.17 \\
6 & 2999.94 & 2999.79 & 3000.07 & 0.13 & 0.28 \\
\bottomrule
\textbf{MAE:} & 0.09 & 0.13 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of H$_2$CO calculated from PhysNet models trained on
CCSD(T) data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:h2co_harm_cc}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode &NN1(CCSD(T)-F12) & NN2(CCSD(T)-F12) & CCSD(T)-F12 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 1186.52 & 1186.50 & 1186.53 & 0.01 & 0.03 \\
2 & 1268.15 & 1268.17 & 1268.08 & 0.07 & 0.09 \\
3 & 1532.76 & 1532.72 & 1532.67 & 0.09 & 0.05 \\
4 & 1776.40 & 1776.43 & 1776.53 & 0.13 & 0.10 \\
5 & 2933.37 & 2933.57 & 2933.75 & 0.38 & 0.18 \\
6 & 3005.45 & 3005.69 & 3005.75 & 0.30 & 0.06 \\
\bottomrule
\textbf{MAE:} & 0.17 & 0.08 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of H$_2$CO calculated from PhysNet models trained on
CCSD(T)-F12 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:h2co_harm_ccf12}
\end{table}
\begin{sidewaystable}[h]
\resizebox{\linewidth}{!}{%
\begin{tabular}{lrrrrrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & NN1(CCSD(T)) & NN1(CCSD(T)) & NN1(CCSD(T)-F12) & NN1(CCSD(T)-F12) & Exp \\
\midrule
1 & 1180.04 & 1179.94 & 1180.23 & 1163.73 & 1163.52 & 1166.36 & 1166.73 & 1167.00 \\
2 & 1246.72 & 1246.67 & 1246.71 & 1240.62 & 1241.36 & 1246.01 & 1245.61 & 1249.00 \\
3 & 1507.55 & 1507.47 & 1507.95 & 1493.89 & 1494.91 & 1498.43 & 1498.36 & 1500.00 \\
4 & 1719.02 & 1716.10 & 1720.94 & 1729.94 & 1731.88 & 1745.32 & 1744.66 & 1746.00 \\
5 & 2823.82 & 2823.11 & 2826.67 & 2775.38 & 2779.58 & 2783.01 & 2778.06 & 2782.00 \\
6 & 2862.29 & 2859.67 & 2862.61 & 2819.51 & 2821.49 & 2825.99 & 2821.91 & 2843.00 \\
\bottomrule
\textbf{MAE:} & 18.49 & 18.40 & & 10.65 & 9.04 & 3.98 & 5.28 & \\
\end{tabular}}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of H$_2$CO calculated using PhysNet (NN1 and NN2)
trained on MP2, CCSD(T) and CCSD(T)-F12 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{herndon2005determination}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment.}\label{sitab:h2co_ah}
\end{sidewaystable}
\clearpage
\subsection{HONO}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 586.44 & 586.45 & 586.53 & 0.09 & 0.08 \\
2 & 602.45 & 602.40 & 602.38 & 0.07 & 0.02 \\
3 & 805.28 & 805.18 & 805.20 & 0.08 & 0.02 \\
4 & 1283.39 & 1283.37 & 1283.28 & 0.11 & 0.09 \\
5 & 1659.77 & 1659.76 & 1659.81 & 0.04 & 0.05 \\
6 & 3754.55 & 3754.40 & 3754.95 & 0.40 & 0.55 \\
\bottomrule
\textbf{MAE:} & 0.13 & 0.13 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HONO calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hono_harm_mp2}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)) & NN2(CCSD(T)) & CCSD(T) & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 565.06 & 565.02 & 564.84 & 0.22 & 0.18 \\
2 & 617.31 & 617.34 & 617.20 & 0.11 & 0.14 \\
3 & 815.84 & 815.85 & 815.79 & 0.05 & 0.06 \\
4 & 1305.94 & 1305.95 & 1305.94 & 0.00 & 0.01 \\
5 & 1715.12 & 1715.15 & 1715.35 & 0.23 & 0.20 \\
6 & 3759.58 & 3759.68 & 3760.30 & 0.72 & 0.62 \\
\bottomrule
\textbf{MAE:} & 0.22 & 0.20 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HONO calculated from PhysNet models trained on
CCSD(T) data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hono_harm_cc}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)-F12) & NN2(CCSD(T)-F12) & CCSD(T)-F12 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 573.12 & 573.07 & 573.02 & 0.10 & 0.05 \\
2 & 631.54 & 631.55 & 631.40 & 0.14 & 0.15 \\
3 & 831.14 & 831.23 & 831.14 & 0.00 & 0.09 \\
4 & 1315.91 & 1315.85 & 1315.76 & 0.15 & 0.09 \\
5 & 1733.36 & 1733.39 & 1733.55 & 0.19 & 0.16 \\
6 & 3774.84 & 3774.96 & 3775.54 & 0.70 & 0.58 \\
\bottomrule
\textbf{MAE:} & 0.21 & 0.19 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HONO calculated from PhysNet models trained on
CCSD(T)-F12 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hono_harm_ccf12}
\end{table}
\clearpage
\begin{sidewaystable}[h]
\resizebox{\linewidth}{!}{%
\begin{tabular}{lrrrrrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & NN1(CCSD(T)) & NN1(CCSD(T)) & NN1(CCSD(T)-F12) & NN1(CCSD(T)-F12) & Exp \\
\midrule
1 & 551.59 & 551.72 & 551.35 & 533.67 & 533.93 & 541.10 & 542.37 & 543.88 \\
2 & 566.02 & 566.12 & 565.97 & 595.98 & 596.32 & 608.55 & 608.92 & 595.62 \\
3 & 766.25 & 765.86 & 765.83 & 785.91 & 786.93 & 799.70 & 799.04 & 790.12 \\
4 & 1235.12 & 1230.53 & 1233.45 & 1260.08 & 1260.91 & 1267.74 & 1267.88 & 1263.21 \\
5 & 1637.07 & 1636.84 & 1637.31 & 1691.60 & 1691.48 & 1709.88 & 1710.24 & 1699.76 \\
6 & 3580.86 & 3581.00 & 3577.17 & 3580.01 & 3581.84 & 3590.28 & 3586.80 & 3590.77 \\
\bottomrule
\textbf{MAE:} & 26.98 & 27.83 & & 6.14 & 5.56 & 6.74 & 7.14 & \\
\end{tabular}}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of HONO calculated using PhysNet (NN1 and NN2)
trained on MP2, CCSD(T) and CCSD(T)-F12 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{guilmot1993rovibrational}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment.}\label{sitab:hono_ah}
\end{sidewaystable}
\clearpage
\subsection{HCOOH}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 625.93 & 625.97 & 625.86 & 0.07 & 0.11 \\
2 & 675.16 & 675.14 & 675.28 & 0.12 & 0.14 \\
3 & 1058.72 & 1058.68 & 1058.77 & 0.05 & 0.09 \\
4 & 1130.58 & 1130.91 & 1130.60 & 0.02 & 0.31 \\
5 & 1301.64 & 1301.70 & 1301.56 & 0.08 & 0.14 \\
6 & 1409.12 & 1409.07 & 1408.91 & 0.21 & 0.16 \\
7 & 1793.39 & 1793.17 & 1793.29 & 0.10 & 0.12 \\
8 & 3123.89 & 3123.92 & 3123.94 & 0.05 & 0.02 \\
9 & 3740.49 & 3740.31 & 3740.56 & 0.07 & 0.25 \\
\bottomrule
\textbf{MAE:} & 0.08 & 0.15 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HCOOH calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hcooh_harm_mp2}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)) & NN2(CCSD(T)) & CCSD(T) & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 626.52 & 626.68 & 626.46 & 0.06 & 0.22 \\
2 & 664.46 & 664.43 & 664.94 & 0.48 & 0.51 \\
3 & 1050.94 & 1050.93 & 1050.99 & 0.05 & 0.06 \\
4 & 1131.48 & 1131.52 & 1131.27 & 0.21 & 0.25 \\
5 & 1310.97 & 1311.12 & 1310.78 & 0.19 & 0.34 \\
6 & 1404.92 & 1404.91 & 1404.71 & 0.21 & 0.20 \\
7 & 1802.52 & 1802.56 & 1802.63 & 0.11 & 0.07 \\
8 & 3088.01 & 3087.86 & 3087.61 & 0.40 & 0.25 \\
9 & 3741.66 & 3741.54 & 3741.84 & 0.18 & 0.30 \\
\bottomrule
\textbf{MAE:} & 0.21 & 0.24 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HCOOH calculated from PhysNet models trained on
CCSD(T) data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hcooh_harm_cc}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)-F12) & NN2(CCSD(T)-F12) & CCSD(T)-F12 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 630.68 & 630.66 & 630.56 & 0.12 & 0.10 \\
2 & 672.06 & 672.06 & 672.14 & 0.08 & 0.08 \\
3 & 1055.17 & 1055.17 & 1055.23 & 0.06 & 0.06 \\
4 & 1136.51 & 1136.47 & 1136.26 & 0.25 & 0.21 \\
5 & 1315.68 & 1315.68 & 1315.48 & 0.20 & 0.20 \\
6 & 1407.20 & 1407.08 & 1406.99 & 0.21 & 0.09 \\
7 & 1811.85 & 1811.79 & 1811.81 & 0.04 & 0.02 \\
8 & 3092.10 & 3091.79 & 3091.90 & 0.20 & 0.11 \\
9 & 3755.50 & 3755.53 & 3755.69 & 0.19 & 0.16 \\
\bottomrule
\textbf{MAE:} & 0.15 & 0.11 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of HCOOH calculated from PhysNet models trained on
CCSD(T)-F12 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:hcooh_harm_ccf12}
\end{table}
\clearpage
\begin{sidewaystable}[h]
\resizebox{\linewidth}{!}{%
\begin{tabular}{lrrrrrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & NN1(CCSD(T)) & NN1(CCSD(T)) & NN1(CCSD(T)-F12) & NN1(CCSD(T)-F12) & Exp \\
\midrule
1 & 619.28 & 618.68 & 619.54 & 620.37 & 619.30 & 626.05 & 625.78 & 626.16 \\
2 & 642.69 & 641.62 & 641.78 & 630.86 & 630.52 & 637.65 & 637.58 & 640.72 \\
3 & 1037.06 & 1036.44 & 1036.38 & 1028.34 & 1028.38 & 1033.08 & 1033.04 & 1033.47 \\
4 & 1097.01 & 1096.67 & 1098.18 & 1099.54 & 1098.65 & 1104.11 & 1104.70 & 1104.85 \\
5 & 1220.49 & 1219.25 & 1220.63 & 1294.92 & 1293.36 & 1301.99 & 1301.89 & 1306.20 \\
6 & 1380.81 & 1380.24 & 1380.94 & 1375.30 & 1374.86 & 1377.35 & 1378.23 & 1380.00 \\
7 & 1758.17 & 1757.89 & 1760.53 & 1765.98 & 1768.89 & 1774.44 & 1774.60 & 1776.83 \\
8 & 2959.50 & 2960.15 & 2967.80 & 2923.05 & 2922.24 & 2924.61 & 2927.30 & 2942.00 \\
9 & 3554.88 & 3555.83 & 3554.90 & 3557.15 & 3562.98 & 3565.70 & 3565.34 & 3570.50 \\
\bottomrule
\textbf{MAE:} & 17.62 & 17.61 & & 9.47 & 9.06 & 3.97 & 3.59 & \\
\end{tabular}}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of HCOOH calculated using PhysNet (NN1 and NN2)
trained on MP2, CCSD(T) and CCSD(T)-F12 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{tew2016ab}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment.}\label{sitab:hcooh_ah}
\end{sidewaystable}
\clearpage
\clearpage
\subsection{CH$_3$OH}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode& NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 289.86 & 289.85 & 289.38 & 0.48 & 0.47 \\
2 & 1056.62 & 1056.61 & 1056.64 & 0.02 & 0.03 \\
3 & 1088.65 & 1088.59 & 1088.63 & 0.02 & 0.04 \\
4 & 1183.93 & 1183.91 & 1183.87 & 0.06 & 0.04 \\
5 & 1371.59 & 1371.60 & 1371.57 & 0.02 & 0.03 \\
6 & 1491.97 & 1491.96 & 1492.01 & 0.04 & 0.05 \\
7 & 1525.40 & 1525.42 & 1525.41 & 0.01 & 0.01 \\
8 & 1535.71 & 1535.79 & 1535.80 & 0.09 & 0.01 \\
9 & 3053.97 & 3054.05 & 3053.94 & 0.03 & 0.11 \\
10 & 3125.24 & 3125.24 & 3125.08 & 0.16 & 0.16 \\
11 & 3182.86 & 3182.88 & 3182.91 & 0.05 & 0.03 \\
12 & 3859.73 & 3859.80 & 3859.62 & 0.11 & 0.18 \\
\bottomrule
\textbf{MAE:} & 0.09 & 0.10 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$OH calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3oh_harm_mp2}
\end{table}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(CCSD(T)) & NN2(CCSD(T)) & CCSD(T)& $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 286.35 & 286.33 & 285.95 & 0.40 & 0.38 \\
2 & 1053.43 & 1053.41 & 1053.38 & 0.05 & 0.03 \\
3 & 1082.20 & 1082.18 & 1082.11 & 0.09 & 0.07 \\
4 & 1175.88 & 1175.85 & 1175.72 & 0.16 & 0.13 \\
5 & 1379.10 & 1379.11 & 1379.01 & 0.09 & 0.10 \\
6 & 1483.92 & 1483.91 & 1483.86 & 0.06 & 0.05 \\
7 & 1512.18 & 1512.20 & 1512.16 & 0.02 & 0.04 \\
8 & 1522.63 & 1522.65 & 1522.67 & 0.04 & 0.02 \\
9 & 3010.68 & 3010.64 & 3010.43 & 0.25 & 0.21 \\
10 & 3068.92 & 3068.88 & 3068.65 & 0.27 & 0.23 \\
11 & 3128.02 & 3127.96 & 3127.97 & 0.05 & 0.01 \\
12 & 3843.44 & 3843.41 & 3843.27 & 0.17 & 0.14 \\
\bottomrule
\textbf{MAE:} & 0.14 & 0.12 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$OH calculated from PhysNet models trained on
CCSD(T) data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3oh_harm_cc}
\end{table}
\clearpage
\begin{table}[]
\begin{tabular}{lrrrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & NN1(CCSD(T)) & NN1(CCSD(T)) & Exp \\
\midrule
1 & 240.84 & 241.49 & 240.64 & 236.97 & 238.08 & 271.50 \\
2 & 1031.16 & 1030.51 & 1030.29 & 1026.41 & 1027.04 & 1033.50 \\
3 & 1069.23 & 1069.65 & 1069.25 & 1063.32 & 1063.78 & 1074.50 \\
4 & 1154.53 & 1154.90 & 1155.02 & 1145.68 & 1145.95 & 1145.00 \\
5 & 1321.89 & 1321.99 & 1321.98 & 1321.31 & 1322.05 & 1332.00 \\
6 & 1457.42 & 1457.78 & 1457.30 & 1447.89 & 1448.52 & 1454.50 \\
7 & 1483.39 & 1483.11 & 1483.22 & 1469.21 & 1468.55 & 1465.00 \\
8 & 1490.11 & 1489.77 & 1489.48 & 1475.75 & 1475.62 & 1479.50 \\
9 & 2998.34 & 2997.26 & 2996.01 & 2827.05 & 2829.22 & 2844.20 \\
10 & 3000.01 & 3001.00 & 2996.40 & 2911.42 & 2912.12 & 2970.00 \\
11 & 3048.63 & 3048.07 & 3046.42 & 2987.57 & 2988.73 & 2999.00 \\
12 & 3688.76 & 3686.33 & 3687.39 & 3666.53 & 3667.99 & 3681.50 \\
\bottomrule
\textbf{MAE:} & 27.57 & 27.28 & & 15.07 & 14.30 & \\
\end{tabular}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of CH$_3$OH calculated using PhysNet (NN1 and NN2)
trained on MP2 and CCSD(T) data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{serrallach1974methanol}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment. Note that the lowest experimental
frequency was obtained from Ar-matrix measurements.}\label{sitab:ch3oh_ah}
\end{table}
\clearpage
\subsection{CH$_3$CHO}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 160.92 & 160.91 & 160.54 & 0.38 & 0.37 \\
2 & 506.62 & 506.67 & 506.58 & 0.04 & 0.09 \\
3 & 779.28 & 779.25 & 779.24 & 0.04 & 0.01 \\
4 & 905.34 & 905.34 & 905.34 & 0.00 & 0.00 \\
5 & 1136.34 & 1136.34 & 1136.37 & 0.03 & 0.03 \\
6 & 1143.70 & 1143.72 & 1143.64 & 0.06 & 0.08 \\
7 & 1390.46 & 1390.48 & 1390.38 & 0.08 & 0.10 \\
8 & 1426.61 & 1426.58 & 1426.75 & 0.14 & 0.17 \\
9 & 1481.35 & 1481.32 & 1481.20 & 0.15 & 0.12 \\
10 & 1493.68 & 1493.73 & 1493.54 & 0.14 & 0.19 \\
11 & 1763.36 & 1763.45 & 1763.86 & 0.50 & 0.41 \\
12 & 2955.29 & 2955.30 & 2954.41 & 0.88 & 0.89 \\
13 & 3069.58 & 3069.48 & 3069.68 & 0.10 & 0.20 \\
14 & 3149.19 & 3149.18 & 3149.53 & 0.34 & 0.35 \\
15 & 3199.78 & 3199.83 & 3199.44 & 0.34 & 0.39 \\
\bottomrule
\textbf{MAE:} & 0.22 & 0.23 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$CHO calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3cho_harm_mp2}
\end{table}
\clearpage
\begin{table}[]
\begin{tabular}{lrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & Exp \\
\midrule
1 & 150.37 & 151.46 & 149.94 & 143.80 \\
2 & 505.60 & 505.45 & 505.96 & 508.80 \\
3 & 766.03 & 766.42 & 765.11 & 764.10 \\
4 & 867.46 & 868.57 & 867.05 & 865.90 \\
5 & 1110.82 & 1110.70 & 1109.98 & 1097.80 \\
6 & 1115.91 & 1116.88 & 1116.41 & 1113.80 \\
7 & 1352.20 & 1351.93 & 1351.80 & 1352.60 \\
8 & 1403.31 & 1403.48 & 1403.23 & 1394.90 \\
9 & 1437.73 & 1438.49 & 1437.76 & 1433.50 \\
10 & 1445.30 & 1444.24 & 1445.08 & 1436.30 \\
11 & 1728.74 & 1726.97 & 1726.61 & 1746.00 \\
12 & 2735.11 & 2735.32 & 2735.51 & 2715.40 \\
13 & 2966.08 & 2965.81 & 2964.93 & 2923.20 \\
14 & 3012.46 & 3011.77 & 3012.12 & 2964.30 \\
15 & 3064.91 & 3060.87 & 3059.86 & 3014.30 \\
\bottomrule
\textbf{MAE:} & 15.27 & 15.32 & & \\
\end{tabular}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of CH$_3$CHO calculated using PhysNet (NN1 and NN2)
trained on MP2 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{wiberg1995acetaldehyde}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment.}\label{sitab:ch3cho_ah}
\end{table}
\clearpage
\clearpage
\subsection{CH$_3$COOH}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 77.46 & 77.49 & 77.20 & 0.26 & 0.29 \\
2 & 422.71 & 422.66 & 422.69 & 0.02 & 0.03 \\
3 & 549.67 & 549.67 & 549.71 & 0.04 & 0.04 \\
4 & 583.21 & 583.18 & 583.15 & 0.06 & 0.03 \\
5 & 661.91 & 661.90 & 662.07 & 0.16 & 0.17 \\
6 & 873.85 & 873.76 & 873.74 & 0.11 & 0.02 \\
7 & 1008.24 & 1008.20 & 1008.18 & 0.06 & 0.02 \\
8 & 1075.99 & 1075.94 & 1075.86 & 0.13 & 0.08 \\
9 & 1206.09 & 1205.93 & 1205.89 & 0.20 & 0.04 \\
10 & 1342.18 & 1342.04 & 1342.07 & 0.11 & 0.03 \\
11 & 1422.08 & 1421.89 & 1421.91 & 0.17 & 0.02 \\
12 & 1492.58 & 1492.45 & 1492.36 & 0.22 & 0.09 \\
13 & 1501.13 & 1500.97 & 1500.98 & 0.15 & 0.01 \\
14 & 1809.90 & 1809.91 & 1810.19 & 0.29 & 0.28 \\
15 & 3097.30 & 3097.21 & 3097.34 & 0.04 & 0.13 \\
16 & 3180.81 & 3180.82 & 3181.01 & 0.20 & 0.19 \\
17 & 3223.21 & 3223.25 & 3223.17 & 0.04 & 0.08 \\
18 & 3751.61 & 3751.59 & 3751.57 & 0.04 & 0.02 \\
\bottomrule
\textbf{MAE:} & 0.13 & 0.09 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$COOH calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3cooh_harm_mp2}
\end{table}
\clearpage
\begin{table}[]
\begin{tabular}{lrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & Exp \\
\midrule
1 & 74.53 & 73.11 & 77.93 & -- \\
2 & 422.52 & 423.21 & 422.94 & 424.00 \\
3 & 536.36 & 538.21 & 536.39 & 534.50 \\
4 & 574.91 & 575.17 & 575.61 & 581.50 \\
5 & 638.93 & 641.25 & 639.50 & 642.00 \\
6 & 855.20 & 855.37 & 855.10 & 847.00 \\
7 & 987.08 & 987.15 & 987.46 & 991.00 \\
8 & 1047.38 & 1045.69 & 1047.31 & 1049.00 \\
9 & 1159.13 & 1161.15 & 1159.17 & 1184.00 \\
10 & 1320.53 & 1322.89 & 1320.83 & 1266.00 \\
11 & 1377.70 & 1378.64 & 1378.53 & 1384.50 \\
12 & 1438.15 & 1437.89 & 1439.69 & 1430.00 \\
13 & 1449.95 & 1450.01 & 1450.29 & 1430.00 \\
14 & 1780.63 & 1787.68 & 1780.90 & 1792.00 \\
15 & 2984.12 & 2989.62 & 2989.40 & 2944.00 \\
16 & 3037.04 & 3040.77 & 3040.41 & 2996.00 \\
17 & 3082.90 & 3080.65 & 3080.19 & 3051.00 \\
18 & 3574.65 & 3567.07 & 3568.62 & 3585.50 \\
\bottomrule
\textbf{MAE:} & 16.25 & 16.67 & & \\
\end{tabular}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of CH$_3$COOH calculated using PhysNet (NN1 and NN2)
trained on MP2 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{goubet2015standard}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment. The lowest frequency was
not reported in Ref.~\citenum{goubet2015standard}.}\label{sitab:ch3cooh_ah}
\end{table}
\clearpage
\newpage
\subsection{CH$_3$NO$_2$}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $|\Delta 1|$ & $|\Delta 2|$ \\
\midrule
1 & 27.68 & 27.54 & 28.91 & 1.23 & 1.37 \\
2 & 478.64 & 478.70 & 478.65 & 0.01 & 0.05 \\
3 & 610.36 & 610.29 & 610.43 & 0.07 & 0.14 \\
4 & 669.74 & 669.73 & 669.67 & 0.07 & 0.06 \\
5 & 940.48 & 940.47 & 940.48 & 0.00 & 0.01 \\
6 & 1127.26 & 1127.31 & 1127.28 & 0.02 & 0.03 \\
7 & 1148.84 & 1148.94 & 1148.99 & 0.15 & 0.05 \\
8 & 1411.99 & 1411.96 & 1412.12 & 0.13 & 0.16 \\
9 & 1430.56 & 1430.56 & 1430.54 & 0.02 & 0.02 \\
10 & 1491.99 & 1492.04 & 1491.90 & 0.09 & 0.14 \\
11 & 1502.66 & 1502.71 & 1502.67 & 0.01 & 0.04 \\
12 & 1745.37 & 1745.15 & 1745.72 & 0.35 & 0.57 \\
13 & 3115.32 & 3115.35 & 3115.24 & 0.08 & 0.11 \\
14 & 3221.42 & 3221.49 & 3221.29 & 0.13 & 0.20 \\
15 & 3247.85 & 3247.85 & 3247.61 & 0.24 & 0.24 \\
\bottomrule
\textbf{MAE:} & 0.17 & 0.21 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$NO$_2$ calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3no2_harm_mp2}
\end{table}
\clearpage
\begin{table}[]
\begin{tabular}{lrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & Exp \\
\midrule
1 & -84.94 & -55.88 & -47.12 & -- \\
2 & 479.22 & 478.01 & 477.77 & 479.00 \\
3 & 593.12 & 594.60 & 593.54 & 599.00 \\
4 & 661.89 & 662.60 & 663.10 & 647.00 \\
5 & 922.30 & 925.33 & 923.87 & 921.00 \\
6 & 1104.10 & 1106.03 & 1105.09 & 1097.00 \\
7 & 1120.33 & 1119.07 & 1119.84 & 1153.00 \\
8 & 1385.86 & 1385.15 & 1385.96 & 1384.00 \\
9 & 1394.18 & 1394.87 & 1395.09 & 1413.00 \\
10 & 1447.35 & 1448.15 & 1448.14 & 1449.00 \\
11 & 1448.21 & 1450.30 & 1449.79 & 1488.00 \\
12 & 1713.45 & 1726.72 & 1727.56 & 1582.00 \\
13 & 3011.56 & 3008.23 & 3006.22 & 2965.00 \\
14 & 3086.54 & 3087.68 & 3086.01 & 3048.00 \\
15 & 3107.14 & 3101.98 & 3106.89 & 3048.00 \\
\bottomrule
\textbf{MAE:} & 28.56 & 29.12 & & \\
\end{tabular}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of CH$_3$NO$_2$ calculated using PhysNet (NN1 and NN2)
trained on MP2 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{wells1941infra}. The MP2
frequencies are added as the VPT2 calculation is feasible in Gaussian. The MAEs are given
with respect to experiment. The lowest frequency was
not reported in Ref.~\citenum{wells1941infra} and was negative in the VPT2 calculation.}\label{sitab:ch3no2_ah_mp2}
\end{table}
\clearpage
\subsection{CH$_3$CONH$_2$}
\begin{table}[]
\begin{tabular}{lrrrrr}
\toprule
Mode & NN1(MP2) & NN2(MP2) & MP2 & $\Delta 1$ & $\Delta 2$ \\
\midrule
1 & 33.70 & 31.89 & 33.19 & 0.51 & 1.30 \\
2 & 137.63 & 134.99 & 139.64 & 2.01 & 4.65 \\
3 & 427.85 & 428.12 & 427.97 & 0.12 & 0.15 \\
4 & 521.86 & 522.31 & 522.32 & 0.46 & 0.01 \\
5 & 547.79 & 547.63 & 547.83 & 0.04 & 0.20 \\
6 & 659.90 & 660.50 & 660.40 & 0.50 & 0.10 \\
7 & 861.13 & 861.06 & 861.05 & 0.08 & 0.01 \\
8 & 989.55 & 989.63 & 989.39 & 0.16 & 0.24 \\
9 & 1060.85 & 1060.61 & 1060.44 & 0.41 & 0.17 \\
10 & 1119.95 & 1119.96 & 1119.90 & 0.05 & 0.06 \\
11 & 1353.38 & 1353.30 & 1353.21 & 0.17 & 0.09 \\
12 & 1412.89 & 1412.92 & 1412.90 & 0.01 & 0.02 \\
13 & 1492.30 & 1492.32 & 1492.32 & 0.02 & 0.00 \\
14 & 1510.95 & 1510.80 & 1510.96 & 0.01 & 0.16 \\
15 & 1623.80 & 1624.11 & 1623.89 & 0.09 & 0.22 \\
16 & 1765.77 & 1765.60 & 1765.94 & 0.17 & 0.34 \\
17 & 3086.67 & 3086.85 & 3086.75 & 0.08 & 0.10 \\
18 & 3174.52 & 3174.68 & 3174.53 & 0.01 & 0.15 \\
19 & 3198.69 & 3198.06 & 3198.43 & 0.26 & 0.37 \\
20 & 3619.17 & 3619.10 & 3618.80 & 0.37 & 0.30 \\
21 & 3769.75 & 3769.86 & 3769.35 & 0.40 & 0.51 \\
\bottomrule
\textbf{MAE:} & 0.28 & 0.44 & & & \\
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$CONH$_2$ calculated from PhysNet models trained on
MP2 data and compared to their reference \textit{ab initio} values. The MAEs of the
PhysNet predictions with respect to reference are given.}\label{sitab:ch3conh2_harm_mp2}
\end{table}
\clearpage
\begin{table}[]
\begin{tabular}{lrrrr}
\toprule
Mode & NN1(MP2) & NN1(MP2) & MP2 & Exp \\
\midrule
1 & -2337.85 & -2548.69 & -2425.72 & -- \\
2 & -206.36 & -204.12 & -213.48 & 269.00 \\
3 & 453.25 & 457.15 & 451.29 & 427.00 \\
4 & 479.72 & 473.40 & 477.92 & 507.00 \\
5 & 574.42 & 576.91 & 574.20 & 548.00 \\
6 & 701.19 & 706.62 & 700.92 & 625.00 \\
7 & 839.33 & 840.76 & 840.39 & 858.00 \\
8 & 981.95 & 984.00 & 979.28 & 965.00 \\
9 & 1037.24 & 1038.03 & 1033.17 & 1040.00 \\
10 & 1056.99 & 1054.48 & 1054.59 & 1134.00 \\
11 & 1342.26 & 1344.85 & 1340.62 & 1319.00 \\
12 & 1370.49 & 1370.80 & 1371.24 & 1385.00 \\
13 & 1444.48 & 1443.41 & 1443.50 & 1432.00 \\
14 & 1464.30 & 1459.85 & 1463.58 & 1433.00 \\
15 & 1556.28 & 1556.57 & 1552.95 & 1600.00 \\
16 & 1718.97 & 1726.61 & 1725.92 & 1733.00 \\
17 & 2996.35 & 2990.34 & 2994.98 & 2860.00 \\
18 & 3005.78 & 2993.60 & 3008.41 & 2900.00 \\
19 & 3068.72 & 3072.59 & 3070.81 & 2967.00 \\
20 & 3496.12 & 3504.90 & 3501.75 & 3450.00 \\
21 & 3646.50 & 3665.08 & 3657.46 & 3550.00 \\
\bottomrule
\textbf{MAE:} & 47.23 & 48.40 & & \\
\end{tabular}
\caption{VPT2 anharmonic frequencies (in cm$^{-1}$) of CH$_3$CONH$_2$ calculated using PhysNet (NN1 and NN2)
trained on MP2 data. They are compared to their reference \textit{\textit{ab initio}}
values (MP2) as well as with experiment \cite{ganeshsrinivas1996simulation}. The MP2
frequenices are added as the VPT2 calculation is feasible in Gaussian.
Only 20 modes are assigned.
Note that four frequencies were obtained in argon matrix (269, 427, 1432, 1433).}\label{sitab:ch3conh2_ah_mp2}
\end{table}
\clearpage
\section{Transfer learning}
\subsection{H$_2$CO}
\begin{table}[]
\begin{tabular}{rrr}
\toprule
NN(TL) & CCSD(T)-F12 & $|\Delta|$\\
\midrule
1186.45 & 1186.53 & 0.08 \\
1268.20 & 1268.08 & 0.12 \\
1532.73 & 1532.67 & 0.06 \\
1776.55 & 1776.53 & 0.02 \\
2934.04 & 2933.75 & 0.29 \\
3006.04 & 3005.75 & 0.29 \\
\bottomrule
\end{tabular}
\caption{Normal mode frequencies of H$_2$CO calculated from a TL model trained on
CCSD(T)-F12 data and compared to their reference \textit{ab initio} values.
All frequencies are given in cm$^{-1}$. }\label{sitab:tl_h2co_ccf12_harm}
\end{table}
\begin{table}[]
\begin{tabular}{llll}
\toprule
NN(TL) & Exp & $|\Delta|$ \\
\midrule
1167.39 & 1167.00 & 0.39 \\
1244.93 & 1249.00 & 4.07 \\
1497.82 & 1500.00 & 2.18 \\
1745.22 & 1746.00 & 0.78 \\
2770.34 & 2782.00 & 11.66 \\
2820.39 & 2843.00 & 22.61 \\
\bottomrule
\end{tabular}
\caption{VPT2 anharmonic frequencies of H$_2$CO calculated using a TL PhysNet model
trained on CCSD(T)-F12 data. They are compared with experiment \cite{herndon2005determination}.
All frequencies are given in cm$^{-1}$.}
\end{table}
\clearpage
\subsection{TL: Energy- and force-errors}
\begin{table}[h]
\begin{tabular}{lcccc}
\toprule
& \multicolumn{4}{c}{\bf NN(TL)} \\
& CH$_3$CHO & CH$_3$NO$_2$ & CH$_3$COOH & CH$_3$CONH$_2$\\
\midrule
EMAE: & 0.0196 & 0.0228 & 0.0027 & 0.0097 \\
ERMSE: & 0.0197 & 0.023 & 0.0029 & 0.0099 \\
FMAE: & 0.0119 & 0.0183 & 0.0103 & 0.0161 \\
FRMSE: & 0.0307 & 0.0366 & 0.0188 & 0.0401 \\
\bottomrule
\end{tabular}
\caption{Out-of-sample errors of the TL models to CCSD(T) quality.
The energy errors are given in kcal/mol and the force errors are given
in kcal/mol/\AA. }\label{sitab:tl_oos_errors}
\end{table}
\clearpage
\subsection{CH$_3$CHO}
\begin{table}[]
\begin{tabular}{rrr}
\toprule
NN(TL) & CCSD(T) & $|\Delta|$ \\
\midrule
158.34 & 157.97 & 0.37 \\
503.66 & 503.57 & 0.09 \\
775.07 & 774.97 & 0.10 \\
896.09 & 896.02 & 0.07 \\
1130.36 & 1130.31 & 0.05 \\
1136.02 & 1136.00 & 0.02 \\
1388.27 & 1388.30 & 0.03 \\
1420.92 & 1420.99 & 0.07 \\
1474.26 & 1474.25 & 0.01 \\
1484.77 & 1484.90 & 0.13 \\
1777.53 & 1777.36 & 0.17 \\
2919.35 & 2919.94 & 0.59 \\
3030.98 & 3031.38 & 0.40 \\
3098.85 & 3099.27 & 0.42 \\
3150.54 & 3151.01 & 0.47 \\
\bottomrule
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$CHO calculated from a TL model trained on
CCSD(T) data and compared to their reference \textit{ab initio} values.}\label{sitab:tl_ch3cho_cc_harm}
\end{table}
\clearpage
\subsection{CH$_3$NO$_2$}
\begin{table}[]
\begin{tabular}{rrr}
\toprule
NN(TL) & CCSD(T) & $|\Delta|$ \\
\midrule
30.33 & 27.39 & 2.94 \\
474.41 & 474.28 & 0.13 \\
610.43 & 610.26 & 0.17 \\
662.28 & 662.25 & 0.03 \\
929.12 & 929.13 & 0.01 \\
1114.68 & 1114.60 & 0.08 \\
1144.23 & 1144.29 & 0.06 \\
1410.41 & 1410.48 & 0.07 \\
1425.35 & 1425.33 & 0.02 \\
1477.67 & 1478.01 & 0.34 \\
1490.72 & 1491.32 & 0.60 \\
1613.79 & 1613.72 & 0.07 \\
3082.80 & 3082.90 & 0.10 \\
3179.75 & 3179.78 & 0.03 \\
3207.18 & 3207.13 & 0.05 \\
\bottomrule
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$NO$_2$ calculated from a TL model trained on
CCSD(T) data and compared to their reference \textit{ab initio} values.}\label{sitab:tl_ch3no2_cc_harm}
\end{table}
\clearpage
\subsection{CH$_3$COOH}
\begin{table}[]
\begin{tabular}{rrr}
\toprule
NN(TL) & CCSD(T) & $|\Delta|$ \\
\midrule
80.67 & 80.21 & 0.46 \\
418.84 & 418.77 & 0.07 \\
542.51 & 542.63 & 0.12 \\
582.77 & 582.64 & 0.13 \\
656.54 & 656.58 & 0.04 \\
865.99 & 865.99 & 0.00 \\
1006.75 & 1006.91 & 0.16 \\
1073.32 & 1073.92 & 0.60 \\
1216.03 & 1215.93 & 0.10 \\
1347.96 & 1347.97 & 0.01 \\
1421.21 & 1421.17 & 0.04 \\
1484.75 & 1484.64 & 0.11 \\
1491.69 & 1491.78 & 0.09 \\
1816.71 & 1816.86 & 0.15 \\
3059.07 & 3059.07 & 0.00 \\
3131.34 & 3131.39 & 0.05 \\
3175.04 & 3175.00 & 0.04 \\
3756.19 & 3756.31 & 0.12 \\
\bottomrule
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$COOH calculated from a TL model trained on
CCSD(T) data and compared to their reference \textit{ab initio} values.}\label{sitab:tl_ch3cooh_cc_harm}
\end{table}
\clearpage
\subsection{CH$_3$CONH$_2$}
\begin{table}[]
\begin{tabular}{rrr}
\toprule
NN(TL) & CCSD(T) & $|\Delta|$ \\
\midrule
45.28 & 47.22 & 1.94 \\
250.38 & 240.57 & 9.81 \\
419.92 & 418.61 & 1.31 \\
511.15 & 511.53 & 0.38 \\
550.94 & 551.26 & 0.32 \\
641.74 & 639.95 & 1.79 \\
853.02 & 853.02 & 0.00 \\
985.36 & 984.86 & 0.50 \\
1058.77 & 1059.14 & 0.37 \\
1128.27 & 1127.94 & 0.33 \\
1345.44 & 1345.26 & 0.18 \\
1411.63 & 1411.86 & 0.23 \\
1484.27 & 1484.28 & 0.01 \\
1500.92 & 1500.59 & 0.33 \\
1630.04 & 1630.02 & 0.02 \\
1768.18 & 1768.24 & 0.06 \\
3046.36 & 3045.80 & 0.56 \\
3119.82 & 3118.47 & 1.35 \\
3156.39 & 3157.45 & 1.06 \\
3594.26 & 3595.34 & 1.08 \\
3730.94 & 3732.00 & 1.06 \\
\bottomrule
\end{tabular}
\caption{Normal mode frequencies (in cm$^{-1}$) of CH$_3$CONH$_2$ calculated from a TL model trained on
CCSD(T) data and compared to their reference \textit{ab initio} values.}\label{sitab:tl_ch3conh2_cc_harm}
\end{table}
\clearpage
\section{Scaled harmonic frequencies}
\begin{figure}[h]
\centering \includegraphics[width=0.9\textwidth]{scaledharm_freq_comb.eps
\caption{The accuracy of the scaled \textit{ab initio} harmonic frequencies is shown with respect to the experimental
values. The figure is divided into two windows for clarity. }
\label{sifig:scaledharm_freq}
\end{figure}
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,187
|
There are not one but two different types of leg length discrepancies, congenital and acquired. Congenital means you are born with it. One leg is anatomically shorter in comparison to the other. As a result of developmental stages of aging, the human brain picks up on the step pattern and identifies some difference. The entire body typically adapts by dipping one shoulder over to the "short" side. A difference of less than a quarter inch isn't blatantly excessive, does not need Shoe Lifts to compensate and normally won't have a serious effect over a lifetime.
Leg length inequality goes largely undiagnosed on a daily basis, yet this issue is very easily remedied, and can eradicate quite a few cases of low back pain.
Therapy for leg length inequality commonly consists of Shoe Lifts. Many are very inexpensive, normally priced at below twenty dollars, compared to a custom orthotic of $200 or maybe more. Differences over a quarter inch can take their toll on the spine and should probably be compensated for with a heel lift. In some cases, the shortage can be so extreme that it requires a full lift to both the heel and sole of the shoe.
Lower back pain is easily the most widespread ailment affecting people today. Over 80 million men and women are afflicted by back pain at some stage in their life. It's a problem that costs companies huge amounts of money year after year because of lost time and productivity. Fresh and superior treatment methods are constantly sought after in the hope of decreasing the economic influence this condition causes.
People from all corners of the earth suffer the pain of foot ache as a result of leg length discrepancy. In these types of situations Shoe Lifts might be of very beneficial. The lifts are capable of alleviating any discomfort and pain in the feet. Shoe Lifts are recommended by numerous skilled orthopaedic practitioners".
So as to support the body in a balanced fashion, your feet have a critical part to play. In spite of that, it's often the most overlooked region in the body. Many people have flat-feet meaning there is unequal force placed on the feet. This will cause other parts of the body such as knees, ankles and backs to be affected too. Shoe Lifts guarantee that ideal posture and balance are restored.
A hammertoe is a deformity of the second, third or fourth toe in which the toe becomes bent at the middle joint; hence, it resembles a hammer. Claw toe and mallet toe are related conditions. While a hammer toe is contracted at the first toe joint, a mallet toe is contracted at the second toe joint, and a claw toe is contracted at both joints. According to the 2012 National Foot Health Assessment conducted by the NPD Group for the Institute for Preventive Foot Health, 3 percent of U.S. adults age 21 and older (about 7 million people) have experienced hammer toe or claw toe. The condition is significantly more prevalent in females than in males.
Shoes that narrow toward the toe may make your forefoot look smaller. But they also push hammertoes the smaller toes into a flexed (bent) position. The toes rub against the shoe, leading to the formation of corns and calluses, which further aggravate the condition. A higher heel forces the foot down and squishes the toes against the shoe, increasing the pressure and the bend in the toe. Eventually, the toe muscles become unable to straighten the toe, even when there is no confining shoe.
Bunions most commonly affect women. Some studies report that bunion symptoms occur nearly 10 times more frequently in women. It has been suggested that tight-fitting shoes, especially high-heel and narrow-toed shoes, might increase the risk for bunion formation. Tight footwear certainly is a factor in precipitating the pain and swelling of bunions. Complaints of bunions are reported to be more prevalent in people who wear shoes than in barefoot people. Other risk factors for the development of bunions include abnormal formation of the bones of the foot at birth (congenital) and arthritic diseases such as rheumatoid arthritis. In some cases, repetitive stresses to the foot can lead to bunion formation. Bunions are common in ballet dancers.
While bunions may be considered cosmetically undesirable, they are not necessarily painful. In cases where the individual has minor discomfort that can be eased by wearing wider shoes made of soft leather and/or with the aid of spacers-padding placed between the toes to correct alignment-further treatment may not be necessary. (Anti-inflammatory agents can be used to alleviate temporary discomfort at the site of the bursa.) For those who continue to experience pain on a daily basis and who cannot wear most types of shoe comfortably, surgical treatment may be the best choice.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,128
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The Cape Town producer heavyweight Hipe has opened the doors of his studio to aspiring local producers. Hipe founded Hiperdelic in 2014 and has since worked with the likes of Proverb, Youngsta, Ill Skillz and Nonku Phiri.
find the right fit" says Hipe.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,863
|
The Great Rift Business Development Organization is a non-profit organization which promotes job creation, economic growth, sustainable development and improved standards in Power County, Idaho and Aberdeen, Idaho by working in partnership with businesses, universities, communities and our workforce. The organization provides a variety of services which encourage economic development, technology, entrepreneurship, business development, business retention, and business expansion. The GRBDO also is involved in community projects which will improve the quality of life and business climate in the communities served.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,651
|
{"url":"https:\/\/socratic.org\/questions\/58c9f86a11ef6b0898c63740","text":"# Question #63740\n\n$7000. #### Explanation: Here your capital (p) =$5000, Time invested (T) = 8 Years and Rate of Interest (R) = 5% . We don't know total interest (I) = ?\n$\\Rightarrow I = \\frac{5000 \\times 5 \\times 8}{100}$\n$\\Rightarrow I = 2000$\nSo, total value of investment will be = $5000+$2000 = \\$7000.","date":"2020-01-19 05:35:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 2, \"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\": 0.5034883618354797, \"perplexity\": 2301.5940247726126}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250594209.12\/warc\/CC-MAIN-20200119035851-20200119063851-00048.warc.gz\"}"}
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{"url":"http:\/\/math.stackexchange.com\/questions\/147896\/if-r-is-commutative-and-j-lhd-i-lhd-r-does-it-follow-that-j-lhd-r","text":"# If $R$ is commutative, and $J\\lhd I\\lhd R,$ does it follow that $J\\lhd R?$\n\n$\\lhd$ will stand for \"is an ideal of\" in this post.\n\nLet $R$ be a commutative ring, $J\\lhd I\\lhd R$. Does it follow that $J\\lhd R?$\n\nI don't think it does, but I'm having difficulty finding a counterexample. An example for a non-commutative $R$ is here.\n\nI've tried several things, but at random really, so I don't think it makes sense to post it here.\n\n-\nAnd just to make sure that I understand: we are not talking about normal subgroups of an abelian group here, right? \u2013\u00a0Thomas May 21 '12 at 21:55\n@Thomas I'm not sure I understand the question... Isn't every subgroup of an abelian group normal? Anyway, the question is about rings. \u2013\u00a0user23211 May 21 '12 at 21:58\nAbsolutely right. And you did write \"ring\". It most be a work injury that whenever I see a $\\lhd$ I think normal subgroup... oops \u2013\u00a0Thomas May 21 '12 at 22:01\n@Thomas I think $\\lhd$ is a pretty standard notation for \"is an ideal of\". Have you not encountered it? If it's not as common as I thought, I'll edit the question to make clear what I mean. \u2013\u00a0user23211 May 21 '12 at 22:05\nI actually have never seen it used for ideals before, but you don't have to edit your question. Looking at it again, it is clear what you are asking about. \u2013\u00a0Thomas May 21 '12 at 22:10\n\nConsider everyone's favourite commutative ring $R=\\mathbb{Z}[x]$. Let $I=\\langle x^2\\rangle\\triangleleft R$ and $J\\subseteq I$ be the subset of those polynomials that don't contain a $x^3$ term. Clearly $J$ is an ideal of $I$, since you can't produce a $x^3$ term from terms of degree greater than 2, but $J$ isn't an ideal of $R$.\nWhat happens if the ring must have $1$. Is the statement also false? \u2013\u00a0user136266 Apr 21 '15 at 11:29","date":"2016-07-28 20:29:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8515806794166565, \"perplexity\": 190.1775476047581}, \"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-2016-30\/segments\/1469257828322.57\/warc\/CC-MAIN-20160723071028-00219-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
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Prepare for your camping menu with these 30 Camping Tin Foil Dinners! These recipes contain everything from breakfast, dinner, and dessert recipes!
You all certainly love your camping recipes!
These are a few of our personal favorites.
Most are super simple recipes. Add all of your ingredients to a foil pack and cook!
Did I mention how easy the clean up is?? Wad up that paper and toss it into the garbage.
All super important things to consider when you are in the great outdoors where resources are limited.
Use heavy duty foil. The last thing you want to happen when you are handling a tin foil dinner is to have the foil give out or rip. Heavy duty foil ensures that your dinner will stay safe and sound during the cooking process.
Know how to fold a tin foil packet. You want to fold your packet in a way that allows the cooking to be even. Follow this super simple tutorial. As a result, you will have perfect recipes every time!
Prep before you leave. Chop your veggies, slice the chicken, or even put the packets together before you head out. This makes for a more enjoyable camping experience. Less stress while you cook means more time to enjoy your vacation!
Don't skip the non-stick spray!
So without dragging this on further… Here are the recipes!
30 Camping Tin Foil Dinners Round Up!
Love these recipes? You will LOVE my cookbook! Foiled! Easy, Tasty, Tin Foil Meals is packed with great ideas for the great outdoors! Pick up your copy now on Amazon and prepare for a delicious camping season!
If you're looking for some great inspiration on what kind of meals to add to your camping menu make sure to check out my cookbook! Foiled! Easy, Tasty, Tin Foil Meals is packed with great ideas from breakfast to desserts! Make sure to give me a shout out on social media if you give any of them a try! I'd love to feature you and your cooking adventures!
Thank you, Lauren and Vanessa, for featuring my corn on the cob here among these other mouthwatering camping foil dinners. You made my day!
Lauren and Vanessa are actually my contributors. :0) I (Jesseca) am happy to feature your amazing recipes!
I love this post! I make a mean tin foil dinner, but I always need new ideas. This looks like the place to start!
These look so yummy!!! Can you bake them in the oven?
Such a great post! Thanks for featuring my Sausage & Vegetable Tinfoil Dinner!
Thank YOU for letting me feature it.
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{
"redpajama_set_name": "RedPajamaC4"
}
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{"url":"https:\/\/discourse.julialang.org\/t\/ann-gridap-jl-0-15-comes-with-major-enhancements\/51971","text":"# [ANN] Gridap.jl 0.15 comes with major enhancements!\n\nAfter months of work, the Gridap developers are happy to announce the Gridap v0.15 release !\n\nThis new version comes with major improvements in the user API (and also in the library internals). The most exciting novelty is how the user writes the weak form. Now, it should be as easy as \u201ccopying\u201d it from the paper. Gridap will automatically generate an efficient Finite Element assembly loop for you, thanks to the awesome Julia JIT compiler.\n\n## Examples\n\nHere is how the weak form of some well-known PDEs can be specified with the new Gridap API.\n\n### Poisson equation with a source term and a Neumann boundary condition\n\n(see complete code here)\n\nf(x) = 1.0\nh(x) = 3.0\na(u,v) = \u222b( \u2207(v)\u22c5\u2207(u) )*d\u03a9\nl(v) = \u222b( v*f )*d\u03a9 + \u222b( v*h )*d\u0393\n\n\n### Linear elasticity without body forces\n\n(see complete code here)\n\nE = 70.0e9;\n\u03bd = 0.33\n\u03bb = (E*\u03bd)\/((1+\u03bd)*(1-2*\u03bd))\n\u03bc = E\/(2*(1+\u03bd))\n\u03c3(\u03b5) = \u03bb*tr(\u03b5)*one(\u03b5) + 2*\u03bc*\u03b5\na(u,v) = \u222b( \u03b5(v) \u2299 (\u03c3\u2218\u03b5(u)) )*d\u03a9\nl(v) = 0\n\n\n### Incompressible Navier-Stokes equation\n\n(see complete code here)\n\nRe = 10.0\nres((u,p),(v,q)) = \u222b( v\u22c5(Re*u\u22c5\u2207(u)) + \u2207(v)\u2299\u2207(u) - (\u2207\u22c5v)*p + q*(\u2207\u22c5u) )*d\u03a9\n\n\n### Poisson equation discretized with DG\n\n(see complete code here)\n\na(u,v) =\n\u222b( \u2207(v)\u22c5\u2207(u) )*d\u03a9 +\n\u222b( (\u03b3\/h)*v*u - v*(n_\u0393\u22c5\u2207(u)) - (n_\u0393\u22c5\u2207(v))*u )*d\u0393 +\n\u222b( (\u03b3\/h)*jump(v*n_\u039b)\u22c5jump(u*n_\u039b) -\njump(v*n_\u039b)\u22c5mean(\u2207(u)) -\nmean(\u2207(v))\u22c5jump(u*n_\u039b) )*d\u039b\n\nl(v) =\n\u222b( v*f )*d\u03a9 +\n\u222b( (\u03b3\/h)*v*u - (n_\u0393\u22c5\u2207(v))*u )*d\u0393\n\n\n### Stokes equation with an equal order DG\n\n(see complete code here)\n\na((u,p),(v,q)) =\n\u222b( \u2207(v)\u2299\u2207(u) - \u2207(q)\u22c5u + v\u22c5\u2207(p) )*d\u03a9 +\n\u222b( (\u03b3\/h)*v\u22c5u - v\u22c5(n_\u0393\u22c5\u2207(u)) - (n_\u0393\u22c5\u2207(v))\u22c5u + 2*(q*n_\u0393)\u22c5u )*d\u0393 +\n\u222b( (\u03b3\/h)*jump(v\u2297n_\u039b)\u2299jump(u\u2297n_\u039b) -\njump(v\u2297n_\u039b)\u2299mean(\u2207(u)) -\nmean(\u2207(v))\u2299jump(u\u2297n_\u039b) +\n(\u03b30*h)*jump(q*n_\u039b)\u22c5jump(p*n_\u039b) +\njump(q*n_\u039b)\u22c5mean(u) -\nmean(v)\u22c5jump(p*n_\u039b)\n)*d\u039b\n\nl((v,q)) =\n\u222b( v\u22c5f + q*g )*d\u03a9 +\n\u222b( (\u03b3\/h)*v\u22c5u - (n_\u0393\u22c5\u2207(v))\u22c5u + (q*n_\u0393)\u22c5u )*d\u0393\n\n\n## How to get started?\n\nIf you are further interested in the project, visit the Gridap.jl repository. Consider to give us an !\n\nIf you want to start learning how to solve PDEs with Gridap.jl, visit our Tutorials repository.\n\nYou are welcome to interact with the Gridap community via our Gitter chat.\n\n## Do you want to collaborate?\n\nWe are open to scientific collaborations with research groups willing to use Gridap for their research. We can collaborate, e.g., to add new functionality to the library and to write scientific publications resulting from it.\n\n51 Likes\n\nJust stumbled onto this\u2026 Just wanted to say that this looks brilliant!\n\nQuick question: A lot of epidemic models are naturally written as integro-differential equations e.g. time since infection models where the rate of infectious contacts might be proportional to\n\nR(t) \\int_0^\\infty I(t,\\tau) w(\\tau) d\\tau\n\nWhere the generation time distribution w(\\tau) governs how infectious contacts distribute over time since an individual was infected \\tau, and the instantaneous reproductive number R(t) covers changes in transmission over calendar time.\n\nDo you think that Gridap.jl can easily solve these models?\n\nHi @SamBrand!\n\nGridap is designed to solve PDEs via various types of Finite Element methods.\n\nFor ODEs check out DifferentialEquations.jl\n\nOfftopic, but do you have a particular resource in studying integro-differential equations within the context of epidemiology?\n\nI\u2019d say no. You can. try ApproxFun though\n\nCheers!\n\nDifferentialEquations.jl ecosystem is brilliant, but the time-since-infection model sort of falls inbetween ODEs and PDEs.\n\nA vanilla version of time-since-infection could be written as:\n\n\\partial_t S(t) = - \\frac{S(t)}{N} R(t) \\int_0^\\infty I(t,\\tau) w(\\tau) d \\tau, \\\\ \\partial_t I(t,\\tau) + \\partial_\\tau I(t,\\tau) = 0, \\\\ I(t,0) = \\frac{S(t)}{N} R(t) \\int_0^\\infty I(t,\\tau)w(\\tau) d \\tau .\n\nI have solved these sorts of equations using just standard discretisation. But always wondered if you could just drop them into a more sophisticated FEM solver\u2026\n\nHi @affans .\n\nThe classic reference for me wrt to this sort of epi model is O. Diekmann, J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases (Wiley, 2000).\n\nAlthough, just for your awareness, they use the equivalent formulation where you model the incidence (i.e. rate of new infections) and integrate over the past rather than the current density of people who were infected in the past. So the the equations look slightly different.\n\n2 Likes","date":"2021-01-19 15:55:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7147089242935181, \"perplexity\": 9802.182524771502}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703519395.23\/warc\/CC-MAIN-20210119135001-20210119165001-00481.warc.gz\"}"}
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\section{Introduction}
After the establishment of universality and scaling in equilibrium, \cite{Cardy_book,Sachdev_book} particularly with the invention of the renormalization group,\cite{Wilson1971a,Wilson1971b} their extension to out-of-equilibrium systems has become a major area of research in physics.
Notwithstanding considerable progress for classical out-of-equilibrium systems,\cite{Hohenberg1977} out-of-equilibrium dynamics for quantum
systems remains an active frontier. In recent years, the field of dynamical quantum phase transitions\cite{Heyl2013,Heyl2014,Heyl2015} (DQPT) has
shed new light on the out-of-equilibrium quantum many-body criticality. The concept of DQPT relies on an intuitive analogy between the thermal partition function and the overlap of an initial state and its time-evolved self in the wake of a quench, referred to as the \textit{Loschmidt amplitude} in this context. One can thus construct the Loschmidt return rate, which is proportional to the logarithm of this overlap, as a dynamical analog of the thermal free energy, with evolution time standing for complexified inverse temperature. As such, nonanalyticities in the return rate would occur at \textit{critical times} much the same way a thermal phase transition manifests itself as a nonanalyticity in the thermal free energy at a critical temperature.
Even though quite a few of the original questions raised at the onset of the field of DQPT have now been answered, in the quest to attain these answers far more questions have arisen.\cite{Heyl_review} For example, in the initial study of quenches in the one-dimensional transverse-field Ising model,\cite{Heyl2013} the singularities in the return rate were found only after quenches across the equilibrium quantum critical point.\cite{Heyl_review} Soon afterwards it became clear that this was not the necessary condition for DQPT. Rather, one can define
the concept of a \textit{dynamical} quantum critical point, which may in some cases coincide with the equilibrium one, that separates quenches with DQPT from those without.\cite{Karrasch2013,Andraschko2014,Vajna2014}
Yet later it was observed that in long-range quantum spin chains with power-law interactions,\cite{Halimeh2017a,Zauner2017} the DQPT occur regardless of the quench. Nevertheless, a meaningful dynamical critical point can still be defined, this time separating qualitatively
different types of evolution after the quench, which notably depends on whether the order parameter oscillates about zero in the time-evolved quenched system.\cite{Halimeh2017a,Zauner2017,Homrighausen2017,Lang2018a,Lang2018b} This in turn spurred further study of the
steady states reached after the quench, their relationship with the DQPT occurring in the evolution towards them, and the effects of
quasiparticle excitations, which is still ongoing.\cite{Halimeh2018,Hashizume2018,Vanderstraeten2018,Liu2018,Jafari2019,Defenu2019}
The large body of theoretical work on DQPT in closed clean quantum many-body systems in the wake of a quantum quench has recently been supported by experimental realizations in ion-trap\cite{Jurcevic:2017be} and ultracold-atom\cite{Flaeschner2018} setups. Naturally, the robustness of DQPT has since been investigated in other paradigms of quantum many-body physics, such as Floquet systems\cite{Kosior2018a,Kosior2018b,Yang2019} where a novel type of ``Floquet singularities'' appear as intrinsic features of periodic time modulation, and also in disordered systems,\cite{Gurarie2018} where singularities arise in the return rate of random-field classical Ising models. This latter result directly inspires the investigation of DQPT in many-body localized (MBL) quantum systems\cite{Huse2015,Altman2015,Abanin2017} as the next frontier in this field. Indeed, MBL systems are drastically alien to the models that have thus far been utilized in the study of DQPT. On the one hand, so-called \textit{emergent integrability} due to the failure of MBL systems to thermalize after a quantum quench may perhaps allow one to consider that DQPT behavior should not be ruled out given its prevalence in clean integrable models. On the other hand, how the emergent exotic MBL phases, if at all, are related to possible DQPT nonanalyticities is a completely open question. Furthermore, since the exact origin of DQPT is still not fully understood, with a quasiparticle origin thus far the most convincing argument,\cite{Halimeh2018,Jafari2019,Defenu2019} studying DQPT in MBL systems can possibly shed further light on what exactly the necessary and sufficient conditions are for singularities to arise in the return rate.
The Loschmidt amplitude is intimately connected to the partition function of the corresponding system
evaluated at imaginary temperature as we elaborate below.
In turn, the imaginary-temperature partition function measures the spectral form factor, or the correlations between energy levels in a quantum system.\cite{Gurarie2018} DQPT would then be equivalent to having strong correlations between energy levels across significant energy intervals much larger than the level spacing. We would expect those correlations in integrable and close-to-integrable systems, as well as in some MBL systems, but not in chaotic quantum systems. This line of inquiry further motivates studying MBL systems for signatures of DQPT in them.
In this work, we provide analytic and numerical evidence showing that quantum many-body systems that support MBL can indeed show rich DQPT behavior.
The rest of the paper is organized as follows: In Sec.~\ref{sec:PartFunc}, we review the imaginary-temperature boundary and full partition functions, and present a candidate quantum MBL model in whose partition functions singularities arise. In Sec.~\ref{sec:lbits}, we use the $l$-bits formalism to determine the conditions under which disordered quantum models will exhibit singularities in their imaginary-temperature full partition function. We present and discuss the DQPT in our candidate model in Sec.~\ref{sec:results}. We conclude in Sec.~\ref{sec:conc} and provide an outlook on future work. Additional details on the $l$-bits analysis are provided in Appendix~\ref{sec:app}.
\section{Partition functions and models}\label{sec:PartFunc}
We would like to study the probability that a quantum system placed initially in a state $\ket{\psi}$, after evolving for time $t$, returns back to its original state $\ket{\psi}$. This quantity can
be written as
\begin{align}
Z(t) = \bra{\psi} e^{-i H t} \ket{\psi} = \sum_{\alpha} \left| c_\alpha \right|^2 e^{-i E_\alpha t}.
\end{align}
Here $c_\alpha$ are the overlap coefficients between the state $\ket{\psi}$ and the eigenstates $\ket{\psi_\alpha}$ of the Hamiltonian $H$ corresponding
to the energy levels $E_\alpha$. $Z(t)$ is a boundary partition function construed in the seminal DQPT work of Ref.~\onlinecite{Heyl2013} as a dynamical analog of the thermal partition function. Consequently, the Loschmidt return rate,
\begin{equation} \label{eq:lerr} \lambda(t)=-\frac{1}{N}\ln|Z(t)|^2, \end{equation}
is a dynamical analog of the thermal free energy, with $N$ standing for system size.
General principles of statistical physics dictate that for large systems $Z(t)$ may be well approximated by the thermal finite-time partition function
\begin{equation} {\cal Z} (t,T)= \sum_\alpha e^{-E_\alpha \left(it + \frac 1 T \right)},
\end{equation} with the appropriately chosen temperature $T$, such that
\begin{equation} \left< \psi \right| H \left| \psi \right> =\frac{\sum_\alpha E_\alpha e^{- \frac{E_\alpha}{T}}}{\sum_\alpha e^{- \frac{E_\alpha}{T}}}.
\end{equation}
Among these, the $T\to\infty$ partition function is especially suitable for study:
\begin{align}
\mathcal{Z}(t) =\lim_{T \rightarrow \infty} {\cal Z}(t,T) = \sum_\alpha e^{-i E_\alpha t}.
\end{align}
Depending on the choice of $\ket{\psi}$, in some cases $Z(t)$ and ${\cal Z}(t)$ may even coincide, specifically when $\left| \psi \right>$
is proportional to the sum
$\sum_\alpha \left| \psi_\alpha \right>$.
At the same time, $\left| {\cal Z}\right|^2$ is the so-called spectral form factor of the system,
\begin{align} \left| \mathcal{Z}(t) \right|^2 = \sum_{\alpha,\beta} e^{i (E_\beta-E_\alpha) t}.
\end{align}
The Fourier transform of the spectral form factor over time $t$ is simply the correlation between the density of states at energies separated by
a certain energy interval,
\begin{align} && \int \frac{dt}{2\pi} \left| {\cal Z}(t) \right|^2 e^{i \omega t} =\sum_{\alpha \beta} \delta(\omega-E_\beta+E_\alpha)
\cr
&&= \int dE \, \rho\left( E + \omega/2 \right) \, \rho\left(E- \omega / 2 \right). \end{align}
Here,
\begin{align} \rho(E) = \sum_\alpha \delta \left( E - E_\alpha \right)
\end{align} is the density of states.
A nonanalyticity in ${\cal Z}(t)$ at a certain time $t_c$ is therefore a reflection of strong correlations existing between energy levels at the energy interval
$\omega_c \sim 1/t_c$.
It might be quite surprising to expect these correlations to persist over energy intervals far larger than the level spacing for an interacting quantum many-body system. Indeed, an extreme example of a generic quantum system is provided by the Random Matrix Theory (RMT). Within that theory, average spectral form factors are known to have a single nonanaliticity at $t=t_c$ equal to the inverse average level spacing.\cite{Mehta} Level spacings in generic quantum many-body theories are exponentially small in system size, resulting in an exponentially large time at which the potential nonanalytic behavior occurs. These types of singularities would then be nonobservable.\cite{footnote}
In view of this argument, it is quite remarkable that a number of models already discussed in the literature display singularities at times of the order of inverse couplings in the Hamiltonian. In other words, in these models the density of energy levels correlates across energy intervals that cover a large number of energy levels. Some of these models are integrable. It is fair to declare that in this context the existence of subtle correlations in energy levels is not surprising. Some others are nonintegrable, however.
We will not attempt here to analyze the origin of energy level correlations in these models. Instead we would like to look at systems which look as different from RMT as possible. RMT is supposed to represent generic ``chaotic" quantum systems with Wigner-Dyson level statistics. Quite distinct from these are systems with Poisson level statistics, including integrable models. A whole separate class of quantum systems with Poisson level statistics are MBL models. Here, we would like to discuss the appearance of singularities
in ${\cal Z}(t)$ and $Z(t)$ of MBL systems.
Let us concentrate on models of one-dimensional spin-$1/2$ chains of length $N$ with various random interaction terms in the Hamiltonian. Those are known to be many-body localized when disorder is strong enough.\cite{Alet2018} An example of such an MBL model can be the Ising model with random bonds and random fields
\begin{align} \label{eq:mb1} H = \sum_{n=1}^{N-1} J_n \sigma^z_n \sigma^z_{n+1} + \sum_{n=1}^N \left( h^\perp_n \sigma^x_n + h^\parallel_n \sigma^z_n \right).
\end{align}
Here $\sigma^z_n$ and $\sigma^x_n$ are Pauli matrix operators acting on spins which reside on sites $n$ of a one-dimensional lattice. All $J_n$, $h^\perp_n$ and $h^\parallel_n$ are independent random variables.
We shall present a series of analytic and numerical arguments supporting the idea that a certain class of these one-dimensional spin-$1/2$ MBL models will have singularities in their ${\cal Z}(t)$. In particular, we will demonstrate that while ${\cal Z}(t)$ is not singular in the model given by~\eqref{eq:mb1}, the following
MBL Hamiltonian
\begin{align} \label{eq:mb2} H = h \sum_{n=1}^N \sigma^z_n + \sum_{n=1}^{N-2} J_n \sigma^x_n \sigma^x_{n+1} \sigma^x_{n+2},
\end{align}
with $J_n$ being independent random variables, features singularities in its ${\cal Z}(t)$ as well as in its return probability $Z(t)$ defined for suitable initial states $\left| \psi \right>$.
\section{$l$-bits and return rate}\label{sec:lbits}
Locally conserved quantities called $l$-bits in the context of many-body localization play an important role in our arguments, so let us
review their definition. As nicely argued in Ref.~\onlinecite{Abanin2015} for any spin-$1/2$ Hamiltonian $H$ it is always possible to construct a set of mutually commuting operators $\tau_n^z$ whose square is identity $(\tau_n^z)^2=1$, which commute with any spin-$1/2$ Hamiltonian $H$. A straightforward technique to do that would be to diagonalize the Hamiltonian $H$ such as the one above. In other words we use the natural basis of spin tensor product states and write $H = U \Lambda U^\dagger$ where $U$ is a unitary matrix mapping the product state basis onto the basis of eigenstates of $H$, and $\Lambda$ is diagonal. Define then \begin{align} \tau_n^z = U\sigma^z_n U^\dagger.
\end{align} By construction, these $\tau_n^z$ all square to 1, commute with each other, and all commute with the Hamiltonian.
Furthermore, it is always possible to rewrite the Hamiltonian in terms of $\tau_n^z$ according to
\begin{eqnarray} \label{eq:lbits} H &=& K^{(0)}+\sum_n K^{(1)}_n \tau_n^z + \sum_{nm} K^{(2)}_{nm} \tau_n^z \tau^z_{m}
\cr && +\sum_{nml} K^{(3)}_{nml} \tau_n^z \tau^z_{m} \tau^z_l + \dots.
\end{eqnarray}
It is obvious that the total number of the coefficients $K$ in~\eqref{eq:lbits} is $2^N$, same as the number of eigenvalues of $H$, so they are sufficient to fully parametrize the Hamiltonian. These coefficients can all be found using
\begin{align} \label{eq:lcoef} K^{s}_{n_1 \dots n_s} = 2^{-N} \hbox{tr}\, \left( H \tau^z_{n_1} \dots \tau^z_{n_s} \right).
\end{align}
By itself this construction is too general to be of much use. However, for MBL Hamiltonians the expansion~\eqref{eq:lbits} simplifies
significantly. $U$ is defined up to the permutations of eigenvalues of the Hamiltonian. It can be argued that with the appropriate choice of $U$, the operators $\tau_n^z$ become local, that is, they can be written as a linear combination of terms involving products of spin operators $\sigma^x_m$, $\sigma^y_m$, and $\sigma^z_m$ on sites $m$ nearby $n$. At the same time, the series~\eqref{eq:lbits} also become local, in that the magnitudes of the coefficients $K^{(s)}_{n_1 \dots n_s}$ drop off exponentially with $s$, as well as with the increasing separations between the lattice sites $n_1$, $n_2$, $\dots$, $n_s$ in these coefficients. In this regime, $\tau_n^z$ are usually referred to as $l$-bits, with $l$ standing for localized. The coefficients $K^{(s)}$ are random, being some complicated combination of the random interaction coefficients of the original Hamiltonian.
The description in terms of $l$-bits allows us to rewrite the imaginary-temperature partition function in a very straightforward fashion:
\begin{eqnarray} \label{eq:classic} {\cal Z}(t) & = & e^{-i t K^{(0)}} \times \cr && \sum_{\tau = \pm 1} e^{-i t \sum_n K^{(1)}_n \tau_n - i t \sum_{nm} K^{(2)}_{nm} \tau_n \tau_m - \dots} .
\end{eqnarray}
Here, the variables $\tau_n$ are eigenvalues of the $l$-bits $\tau^z_n$ taking values $\pm 1$. We can use~\eqref{eq:classic} as a starting point to calculate
partition functions of one-dimensional spin-$1/2$ MBL models. Furthermore, since in the MBL phase the coefficients $K^{(s)}$ quickly go to zero with increasing $s$, it is sufficient to retain just a few terms in the series in the exponential of~\eqref{eq:classic}, which is what we will rely on below.
Since the coefficients $K^{(s)}_{n_1 \dots n_s}$ are random, it is important to decide which quantities can be averaged over many sets of these random coefficients. The quantity ${\cal Z}(t)$ is not self-averaging. This means that computing it over a particular set of $K^{(s)}_{n_1 \dots n_s}$ taken from the original randomly generated interaction coefficients in the MBL Hamiltonian is not the same as averaging it over many realizations of them. On the contrary, the return rate \begin{align}\label{eq:PFRR} r(t) =-\frac{1}{N}\ln\frac{\left|
{\cal Z} \right|^2}{2^{2N}}, \end{align} defined analogously to the Loschmidt return rate~\eqref{eq:lerr}, is a self-averaging quantity well defined in the ``thermodynamic limit" of infinite chain $N \rightarrow \infty$. It is $r(t)$ that can be averaged over many realizations. Its average value should coincide with its typical value computed for a particular set of $K^{(s)}_{n_1 \dots n_s}$. Various constants present in the definition of $r(t)$ are there merely for normalization
purposes.
In the work of Ref.~\onlinecite{Gurarie2018}, the partition function was evaluated for the model given by~\eqref{eq:classic} with $K^{(1)}_n = h_n$ random and
$K^{(2)}_{nm} =( \delta_{n, m-1} + \delta_{n,m+1} ) J$ representing constant nearest-neighbor interactions. In other words, the following model
was studied:
\begin{align} \label{eq:old} {\cal Z} (t) = \sum_{\tau = \pm 1} e^{- i t \sum_n h_n \tau_n - i t J \sum_{n=1}^{N-1} \tau_n \tau_{n+1}},
\end{align} with $h_n$ random independent variables. It was found, both numerically and analytically, that the resulting function $r(t)$ is singular at the points in time $t_n = n \pi/(2J)$, with $n$ an arbitrary integer. Each singularity was found to be of the type $|t-t_n| \ln |t-t_n|$.
At the same time, numerical evidence indicated that if $J$ in~\eqref{eq:old} is promoted to a bond-dependent random variable $J_n$, then $r(t)$ is featureless and has no singularities. We expect it to be true generally: a generic model~\eqref{eq:classic} with all the coefficients being
independently random variables will feature no singularities and in fact for large system size its ${\cal Z}(t)$ should self-average to a time-independent constant.
We would like to generalize beyond~\eqref{eq:old}. In the next section we present evidence supporting the conjecture that as long as there is at least one nonrandom interaction coefficient in~\eqref{eq:classic}, with the
rest being random, the return rate $r(t)$ always features singularities, at positions $t_n=n \pi/(2K)$, where $n$ is an arbitrary integer and $K$ is the nonrandom interaction coefficient in~\eqref{eq:classic}.
A generic MBL model can be expected to map onto~\eqref{eq:classic} with all interaction coefficients random. However, it is also natural to expect
that models should exist whose mapping to~\eqref{eq:classic} feature at least one nonrandom coefficient. Those models would then have singularities in their return rate. Furthermore, a possibility should not be entirely discounted that~\eqref{eq:classic} with all coefficients random but correlated in a certain
way would also feature singularities, and that MBL models exist which map onto these kinds of $l$-bit models. In the next section we present further examples of random $l$-bit models with singularities in their return rate, and demonstrate the existence
of singularities in the MBL model given by~\eqref{eq:mb2}.
\section{Results and discussion}\label{sec:results}
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth]{RandomBondConstantField2Periods.pdf}
\caption{Return rate $r(t)$ evaluated for~\eqref{eq:randombond2} for $N=45000$ showing clear singularities. Beyond the largest time shown here, the return rate repeats periodically.}
\label{introfig}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth]{RandomTripleBondConstantField2Periods.pdf}
\caption{Same as Fig.~\ref{introfig} but for~\eqref{eq:randombond3}, again with $N=45000$. $r(t)$ appears qualitatively different from
that in Fig.~\ref{introfig} but features the same singularities at $th=\pi n/2$. The return rate repeats periodically beyond the largest time shown here.}
\label{introfig2}
\end{figure}
An important remark which simplifies further analysis lies in the observation, already discussed in Ref.~\onlinecite{Gurarie2018}, that a random variable $e^{-i t h_n \sigma}$, for $\sigma = \pm 1$ and $t$ sufficiently large, regardless of the probability distribution for $h_n$, is well approximated by the variable $e^{-i f_n \sigma}$ where $f_n$ is now taken as uniformly distributed on the interval $f_n \in [-\pi, \pi]$. This should be fairly obvious: the latter variable is uniformly distributed over a unit circle in the complex plane, while the former approaches this distribution at large enough $t$.
We would now like to illustrate the principle of a single nonrandom coefficient in~\eqref{eq:classic} leading to singularities by considering for example
\begin{align} \label{eq:randombond1} {\cal Z}(t) = \sum_{\tau = \pm 1} e^{- i t h \sum_n \tau_n - i t \sum_{n=1}^{N-1} J_n \tau_n \tau_{n+1}},
\end{align}
with $J_n$ random. In line with the observation in the previous paragraph, we instead study
\begin{align} \label{eq:randombond2} {\cal Z}(t) = \sum_{\tau= \pm 1} e^{- i t h \sum_n \tau_n - i \sum_{n=1}^{N-1} f_n \tau_n \tau_{n+1}},
\end{align}
with $f_n$ independent random variables uniformly distributed over the interval $[-\pi, \pi]$.
The advantage of this representation of ${\cal Z}(t)$ is that~\eqref{eq:randombond2} is automatically periodic in $t$, as is clear by inspection. At the same time, the original model~\eqref{eq:randombond1} approaches~\eqref{eq:randombond2} at sufficiently large $t$, $t \gtrsim 2\pi/h$. This can also be easily verified numerically.
Fig.~\ref{introfig} shows $r(t)$ evaluated for~\eqref{eq:randombond2} for $N=45000$ sites and $th$ ranging from $0$ to $\pi$. $r(t)$ continues periodically beyond the displayed range of $t$. To produce this result, we took advantage of the standard transfer matrix calculation
of the partition function, which allowed us to easily go to fairly large system sizes. The self-averaging property of $r(t)$ is obvious in Fig.~\ref{introfig}: even though only one realization of disorder is taken, the curve is smooth.
Similarly we can evaluate, for example, the partition function
\begin{align} \label{eq:randombond3} {\cal Z}(t) = \sum_{\sigma = \pm 1} e^{- i t h \sum_n \tau_n - i \sum_{n=1}^{N-2} f_n \tau_n \tau_{n+1}
\tau_{n+2}},
\end{align} with the result shown in Fig.~\ref{introfig2}.
This shows a new feature at $t h = \pi/4 + \pi n/2$, however the more interesting singularity still remains at $t h = \pi n/2$. All this is compatible with the observation that one nonrandom coefficient in~\eqref{eq:classic} is sufficient to generate periodically repeating singularities in ${\cal Z}(t)$.
Ref.~\onlinecite{Gurarie2018} presented a detailed analysis of the singularities in~\eqref{eq:old} using analytic techniques. That analysis is no longer available for the more intricate models of~\eqref{eq:randombond2} and~\eqref{eq:randombond3}, with random bonds instead of random fields. Instead, we have studied singularities in $r(t)$ close
to $t_n = \pi n/(2 h)$ in those models numerically. We find that these singularities are not in the universality class of~\eqref{eq:old}. Instead, they are of the power-law type
\begin{align} r(t) \sim \left| t - t_n \right|^\nu,\end{align}
with $\nu \approx 0.2$, as shown in Fig.~\ref{fit} for the model~\eqref{eq:randombond2}. The analysis of singularities in \eqref{eq:randombond3}
also produces $\nu \approx 0.2$.
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth]{fit.pdf}
\caption{(Color online). Scaling of singularities of $r(t)$ in the vicinity of $t_n=\pi n/(2h)$ for model~\eqref{eq:randombond2}
indicating a power-law behavior $\propto|t-t_n|^\nu$ with $\nu \approx 0.2$.}
\label{fit}
\end{figure}
We have studied a number of other models of the type~\eqref{eq:classic} with one nonrandom interaction coupling and the rest random, and they all feature singularities supporting the conjecture stated above. It is possible, however, that in those other models the singularities belong to other
universality classes with distinct values of the exponent $\nu$. Studying this would be an interesting direction of further research.
Given that all spin-$1/2$ MBL systems map into~\eqref{eq:classic}, it is natural to expect that among them systems can be found whose map into~\eqref{eq:classic} does not produce all random and independent couplings. Those will necessarily feature singularities in their partition function ${\cal Z}(t)$. Identifying them however might not be easy. We propose~\eqref{eq:mb2} as the candidate model for this purpose.
\begin{figure}[tb]
\centering
\centering
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{hZ_JiXXX_PartitionFunction.pdf}\\
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{FigLERR.pdf}
\caption{(Color online). Partition-function return rate $r(t)$ given in~\eqref{eq:PFRR} (upper panel) and Loschmidt return rate $\lambda(t)$ given in~\eqref{eq:lerr} for the quantum model~\eqref{eq:mb2}, with each return rate averaged over $1000$ realizations of disorder. The initial state used for calculating $\lambda(t)$ is the fully $x$-polarized product state $\ket{X}$. It is clear that in the weak-disorder regime $h/J_0\gg1$ singularities appear in both return rates. The singularities seem to become less pronounced for $h/J_0\ll1$.}
\label{mbs}
\end{figure}
\begin{figure}[tb]
\centering
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{FigNoCusps.pdf}\\
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{FigNoCuspsLERR.pdf}
\caption{(Color online). Partition-function and Loschmidt return rates (top and bottom panels, respectively), each averaged over $1000$ disorder realizations, for the quantum model~\eqref{eq:mb1}, where $J_n,h_n^\perp\in[-J_0,J_0]$ and $h_n^\parallel\in[-h_0^\parallel,h_0^\parallel]$ are independent random variables. The initial state used for $\lambda(t)$ is $\ket{X}$. No singularities appear in either return rate for this model.}
\label{nocusps}
\end{figure}
Fig.~\ref{mbs} shows the result of evaluating the partition-function return rate $r(t)$~\eqref{eq:PFRR} and the Loschmidt return rate $\lambda(t)$~\eqref{eq:lerr}, calculated with $\ket{\psi} = \ket{X}$ being the state representing all spins polarized in the
$x$-direction, for the model~\eqref{eq:mb2} for $N=12$ sites by exact diagonalization, averaged over $1000$ realizations of disorder, necessarily due to the relatively short length of the chain. Random variables $J_n$ are taken to be uniformly distributed on the interval $[-J_0, J_0]$. Singularities are apparent in both return rates. Note that the periodicity of ${\cal Z}(t)$ is $\pi/h$. In fact, in the absence of disorder $J_0=0$, the return rate is easy to evaluate as
\begin{align} r(t) = -\frac{1}{N}\ln\frac{\left|
{\cal Z} \right|^2}{2^{2N}}= - \ln \left| \cos(h t) \right|.
\end{align}
We see that with disorder $r(t)$ retains the periodicity of its disorder-free version, but in addition clearly develops singularities, as desired. We emphasize that not all disordered many-body models exhibit singularities in the return rate as~\eqref{eq:mb2} does. For comparison, we
can examine the return rates for the model~\eqref{eq:mb1}. Fig.~\ref{nocusps} clearly shows absence of any singularities in it (Loschmidt return rate is calculated with the same $\ket{\psi} = \ket{X}$ as above). And indeed,
we can argue that this model maps into an $l$-bit Hamiltonian with all random couplings.
We can further elucidate the model~\eqref{eq:mb2} by exploring the limiting cases of strong and weak disorder. In the strong disorder case, $h/J_0 \ll 1$, we can construct the $l$-bit Hamiltonian perturbatively; cf.~\eqref{eq:lbits}. Carrying out this procedure (see Appendix~\ref{sec:app}) in the second order
in $h/J_0$ does not produce any nonrandom couplings in~\eqref{eq:lbits}. This is in line with singularities disappearing for small $h/J_0$ in
Fig.~\ref{mbs}.
The limit of weak disorder, $h/J_0 \gg 1$, is much more subtle. At $J_0=0$ the model~\eqref{eq:mb2} results in many degenerate levels, separated by energy interval $2h$ (hence the periodicity of the partition function of $2\pi/(2h) = \pi/h$). Once disorder is turned on, each such level splits into a band. One can expect each band to be many-body localized. Indeed, the effective Hamiltonian within each band, which can be obtained
via second-order perturbation theory in $J_0/h$, is proportional to $J_0^2$, with its ratio to $J_0^2$ being $J_0$-independent. Thus the
nature of the MBL eigenfunctions of this system does not depend on disorder strength as it is taken to zero, and no perturbation theory can be useful to analyze this phase in this limit.
The precise structure of the $l$-bit Hamiltonian is difficult to determine in this regime. It is in this regime that the model~\eqref{eq:mb2} features singularities seen in Fig.~\ref{mbs}. We note the absence of the exact periodicity in Fig.~\ref{mbs}. That indicates that the appropriate $l$-bit Hamiltonian our model maps into cannot have simply a single nonrandom coefficient with the rest random and independent. That by itself
would produce a periodic-in-time $r(t)$ at large enough $t$. Rather the $l$-bit Hamiltoinan should have a more complicated structure involving correlations between its coefficients, going beyond the simple examples of~\eqref{eq:old} or \eqref{eq:randombond1}.
Whether the singularities disappear at some critical value of $h/J_0$ or are present at all values albeit getting weaker as $h/J_0$ gets smaller
cannot be explored with the methods currently available to us. The second scenario would imply that these singularities cannot be investigated
in perturbation theory over the small parameter $h/J_0$, consistent with the arguments in Appendix~\ref{sec:app}. Ultimately, MBL phases
are notoriously difficult to analyze using analytic techniques. It is therefore not surprising that we have to rely mostly on the numerics to analyze
our system.
\begin{figure}[tb]
\centering
\centering
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{hZ_JiXXX_LiZZZ_PartitionFunction.pdf}\\
\hspace{-.25 cm}
\includegraphics[width=.46\textwidth]{FigLERR2.pdf}
\caption{(Color online). Partition-function and Loschmidt return rates (top and bottom panels, respectively) for model~\eqref{eq:mb3} for fixed $h/J_0=4$ and two disorder strengths $L_0/h=0$ and $0.05$. A finite $L_0$ quickly washes out the singularities. Each return rate is averaged over $1000$ disorder configurations, with $\ket{X}$ the initial state used to compute $\lambda(t)$.}
\label{fig:triple}
\end{figure}
Finally, we observe that adding extra random terms to the Hamiltonian~\eqref{eq:mb2} can take us out of the class of MBL
Hamiltonians with singular return rates, even if the first term in~\eqref{eq:mb2} remains nonrandom.
Consider for example
\begin{align}\label{eq:mb3} H = h \sum_{n=1}^N \sigma^z_n + \sum_{n=1}^{N-2} \big(J_n \sigma^x_n \sigma^x_{n+1} \sigma^x_{n+2}+L_n \sigma^z_n \sigma^z_{n+1} \sigma^z_{n+2}\big),
\end{align} with $L_n$ random uniformly distributed on the interval $[-L_0, L_0]$.
This Hamiltonian does not feature singularities in its return rate as shown in Fig.~\ref{fig:triple}, even for tiny $L_0/h = 0.05$. Clearly, only a certain subclass of MBL models represent systems with DQPT. In line with the arguments above, the Hamiltonian in~\eqref{eq:mb3} must map onto an
$l$-bit Hamiltonian with random and essentially uncorrelated interaction coefficients.
\section{Conclusion and outlook}\label{sec:conc}
In summary, we have investigated DQPT in quantum many-body systems with disorder. Using a mapping to $l$-bits, we have determined the conditions for which singularities appear in the return rate for such models, and presented candidates for which the return rate displays singularities for small to moderate disorder strength, and which may survive even at large disorder strength. Our results show that DQPT persist in quantum MBL systems, and are not restricted to clean quantum many-body models on which hitherto their investigation has been focused. In light of the search for an origin of DQPT, our conclusions confirm that a Landau equilibrium quantum phase transition is neither a necessary nor a sufficient condition for dynamical criticality to arise in the return rate. From an MBL point of view, this opens up several questions related to what MBL phases can imply about dynamical criticality. Even though this is well understood in traditional Landau phases and has been extensively studied in one-dimensional quantum Ising chains with various interaction ranges,\cite{Halimeh2017b,Halimeh2017a,Zunkovic2018,Zauner2017} two-dimensional models,\cite{Hashizume2018,Schmitt2015} and mean-field models,\cite{Homrighausen2017,Lang2018a,Lang2018b,Weidinger2017} little is known about how different MBL phases can alter the kind or presence of singularities in the return rate. In the other direction, one can wonder what the singularities in the return rate can tell us about the equilibrium, possibly MBL, phase. In Ref.~\onlinecite{Halimeh2018}, it is illustrated how one can determine the equilibrium physics, including Landau phases and the type of quasiparticle excitations in the spectrum of the quench Hamiltonian, directly from the return rate after a quantum-quench sweep, but no protocol was provided for discerning whether an equilibrium phase can be MBL. It would be interesting to further investigate this, and not necessarily just from the point of view of the return rate. Indeed, recently it has been shown how universal equilibrium scaling functions can be deduced at short times from spin-spin correlations after a quantum quench to the vicinity of a critical point.\cite{Karl2017} Additionally, it is worth mentioning that the MBL transition has been observed in experiments with interacting fermions in one-dimensional quasirandom optical lattices through the relaxation dynamics of the initial state.\cite{Schreiber2015} Our conclusions would in principle be amenable for observation in such experiments given that DQPT have also been observed in setups of spin-polarized fermionic atoms in driven optical lattices.\cite{Flaeschner2018}
Finally, questions remain on the possible universality classes of the DQPT observed in this work, and on general principles of
mapping to $l$-bits Hamiltonians resulting in DQPT. We leave these open questions for future work.
\section*{Acknowledgments}
The authors acknowledge stimulating discussions with Dmitry Abanin, Maksym Serbyn, and Daniele Trapin. VG would also like to acknowledge support from the NSF Grant No. PHY-1521080.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,375
|
The guh-webserver is part of the [guh](http://www.guh.guru) project. It serves three purposes:
- Provide a JSON/REST-API for the [guh core](https://github.com/guh/guh) to trigger remote procedure calls via HTTP (instead of the core's UNIX socket).
- Provide a websocket connection to push live updates (events, state changes, etc.) from the core to the client.
- Serve the static files of the [web interface](https://github.com/guh/guh-webinterface).
This project is written in [Go](https://golang.org/) and depends on these libraries:
- [guh-libgo](https://github.com/guh/guh-libgo): A thin wrapper around the guh core RPC-API.
- [martini](https://github.com/go-martini/martini): A very convenient framework for web service creation.
- [martini-render](https://github.com/martini-contrib/render): A martini middleware for easy JSON, XML and HTML responses.
- [gorilla websocket](https://github.com/gorilla/websocket): A complete implementation of the websocket protocol.
- [toml](https://github.com/BurntSushi/toml): Parses TOML files, used for configuration.
- Various other packages from the standard lib.
## Installation
You have multiple options to install the guh-webserver, but it is recommended that you use our repositories. [See this wiki for detailed guides.](https://github.com/guh/guh/wiki/Install)
## Getting started
To start the web server simply type
./guh-webserver
You may configure the process with the following flags:
- `ip`: The IP of the web server (default `0.0.0.0`)
- `port`: The port of the web server (default `3000`)
- `guh_ip`: The IP of the guhd server (default `127.0.0.1`)
- `guh_port`: The port of the guhd server (default `1234`)
- `static_folder`: The folder containing the static files of guh-webinterface (default: `./public`)
- `conf_path`: A path pointing at a config file (default: `/etc/guh/guh-webserver.conf`)
For example:
./guh-webserver --guh_ip=192.168.0.2 --guh_port=80
Here is an example for `guh-webserver.conf`:
IP = "127.0.0.1"
Port = 3000
GuhIP = "192.168.0.3"
GuhPort = 1234
StaticFolder = "/my/cusom/public"
### Introspection
#### Vendors
#### Supported devices
#### Configured devices
### Adding new devices
#### Configured devices
#### Discovered devices
First retrieve a list of discovered devices like so:
GET /api/v1/device_classes/{XXX-XXX}/discover.json
You will receive a response like this:
[
{
"id": "{YYY-YYY}",
"title": "some title",
"description": "some description"
},
...
]
This represents a list of discovered devices. Each discovered device will have an `id` (it's a device descriptor ID), which can be used to add the device:
POST /api/v1/devices.json
Payload:
{
"device": {
"deviceClassId": "{XXX-XXX}",
"deviceDescriptorId": "{YYY-YYY}",
"deviceParams": [
{"name": "aaa", value: "bbb"}
]
}
}
#### Pairing devices
Adding a device that requires pairing is quite similar to adding discovered devices. First retrieve a list of discovered devices:
GET /api/v1/device_classes/{XXX-XXX}/discover.json
You will receive a list of discovered devices. By inspecting the device's setup method attribute you will find which kind of pairing the device will require SetupMethodJustAdd, SetupMethodDisplayPin, SetupMethodEnterPin or SetupMethodPushButton).
Now you can prompt your user for the pin, show them a pin or ask them to push a button on the device. Once the user is ready you can add the device:
POST /api/v1/devices.json
Payload:
{
"device": {
"deviceClassId": "{XXX-XXX}",
"deviceDescriptorId": "{YYY-YYY}",
"deviceParams": [
{"name": "aaa", value: "bbb"}
]
}
}
Example Response:
{
"displayMessage": "...",
"pairingTransactionId": "{ZZZ-ZZZ}",
"setupMethod": "..."
}
Now you can proceed to confirm the pairing by making this request:
POST /api/v1/devices/confirm_pairing.json
Payload:
{
params: {
"pairingTransactionId": "{ZZZ-ZZZ}"
}
}
#### Editing a device
To update a device make a request like this:
PUT /api/v1/devices/{XXX-XXX}.json
The process of editing a device is analogous to creating one.
#### Removing a device
To remove a device make a request like this:
DELETE /api/v1/devices/{XXX-XXX}.json
In case of success you will get an empty response with a HTTP status `204`. The removal might fail if the device is used as part of a rule. In this case you will get a status `500` and an error message. If you provide a wrong device ID you will get `404`.
### Rules
#### Listing Rules
GET /api/v1/rules.json
#### Retrieving a specific rule
GET /api/v1/rules/{XXX-XXX}.json
#### Finding a rule, containing a device
POST /api/v1/rules/find.json
Payload:
{
"deviceId": "{YYY-YYY}"
}
#### Creating a rule
POST /api/v1/rules.json
Payload:
#### Removing a rule
DELETE /api/v1/rules/{XXX-XXX}.json
### Websockets
You can connect to the webserver at `ws://[IP]:[PORT]/ws` to receive live updates from the guh core. Internally your connection will be treated like a HTTP request and then upgraded to a permanent websocket connection.
After a successful connection you will receive messages from the core pertaining to events, state changes and log entries.
#### Examples
**EventTriggered**
{
"id": 0,
"notification": "Events.EventTriggered",
"params": {
"event": {
"deviceId": "{cf004d13-5e8b-4dda-9c7f-856ebe84d458}",
"eventTypeId": "{798553bc-45c7-42eb-9105-430bddb5d9b7}",
"params": [
{
"name": "value",
"value": 56
}
]
}
}
}
**StateChanged**
{
"id": 1,
"notification": "Devices.StateChanged",
"params": {
"deviceId": "{cf004d13-5e8b-4dda-9c7f-856ebe84d458}",
"stateTypeId": "{798553bc-45c7-42eb-9105-430bddb5d9b7}",
"value": 56
}
}
**LogEntry**
{
"id": 2,
"notification": "Logging.LogEntryAdded",
"params": {
"logEntry": {
"deviceId": "{cf004d13-5e8b-4dda-9c7f-856ebe84d458}",
"eventType": "LoggingEventTypeTrigger",
"loggingLevel": "LoggingLevelInfo",
"source": "LoggingSourceStates",
"timestamp": 1431737348181,
"typeId": "{798553bc-45c7-42eb-9105-430bddb5d9b7}",
"value": "56"
}
}
}
The webserver will also inform you if it has lost the connection to the core and after it has reconnected.
Lost connection:
{"channel":"core_connection","data":null,"event":"disconnected"}
After a successful reconnection:
{"channel":"core_connection","data":null,"event":"connected"}
### License & Copyright
Copyright (c) 2015 guh
This software may be modified and distributed under the terms of the MIT license. See the LICENSE file for details.
### TODO
- [ ] Check if errors really trickle through (sometimes there seem to be problems with the scope)
- [ ] Refactor code and rename structs properly
- [ ] Vendor / VendorService
- [ ] Parse the request body by default and reduce code duplication
- [x] Properly handle deviceErrors etc. in responses
- [ ] Refactor code to use something like "RegisterEndPoint()" to be able to autogenerate meta information (required params, relationship to guh core, etc.) about the API
- [x] Convert ENV params to real command line params
- [x] Params for guh-ip, guh-port, config-path, port
- [x] Serve static files from configurable directory (http://stackoverflow.com/a/14187941/641032)
#### Endpoints
- [x] get /core/introspect.json
- [x] JSONRPC.Introspect
- [x] get /core/version.json
- [x] JSONRPC.Version
- [x] get /devices.json
- [x] Devices.GetConfiguredDevices
- [x] get /devices/:id.json
- [x] Devices.GetConfiguredDevices (filtered)
- [x] delete /devices/:id.json
- [x] Devices.RemoveConfiguredDevice
- [x] post /devices.json
- [x] Devices.AddConfiguredDevice
- [x] Devices.PairDevice
- [x] post /devices/confirm_pairing.json
- [x] Devices.ConfirmPairing
- [x] post /devices/:device_id/execute/:id.json
- [x] Actions.ExecuteAction
- [x] get /devices/:id/states.json
- [x] Devices.GetStateValues
- [x] get /devices/:id/states/:id.json
- [x] Devices.GetStateValue
- [x] get /device_classes.json
- [x] Devices.GetSupportedDevices
- [x] get /device_classes/:id.json
- [x] Devices.GetSupportedDevices (filtered)
- [x] get /device_classes/:device_class_id/action_types.json
- [x] Devices.GetActionTypes
- [ ] get /device_classes/:device_class_id/action_types/:id.json
- [ ] Actions.GetActionType
- [x] get /device_classes/:device_class_id/state_types.json
- [x] Devices.GetStateTypes
- [ ] get /device_classes/:device_class_id/state_types/:id.json
- [ ] States.GetStateType
- [x] get /device_classes/:device_class_id/event_types.json
- [x] Devices.GetEventTypes
- [ ] get /device_classes/:device_class_id/event_types/:id.json
- [ ] Events.GetEventType
- [x] get /device_classes/:id/discover.json
- [x] Devices.GetDiscoveredDevices
- [x] get /rules.json
- [x] Rules.GetRules
- [x] post /rules/find.json
- [x] Rules.FindRules
- [x] get /rules/:id.json
- [x] Rules.GetRuleDetails
- [x] post /rules.json
- [x] Rules.AddRule
- [x] patch /rules/:id/disable.json
- [x] Rules.DisableRule
- [x] patch /rules/:id/enable.json
- [x] Rules.EnableRule
- [x] delete /rules/:id.json
- [x] Rules.RemoveRule
- [x] get /vendors.json do
- [x] Devices.GetSupportedVendors
- [x] get /vendors/:id.json
- [x] Devices.GetSupportedVendors (filtered)
- [x] get /vendor/:id/device_classes.json
- [x] Devices.GetSupportedDevices (filtered)
- [ ] get /plugins.json
- [ ] Devices.GetPlugins
- [ ] get /plugins/:id.json
- [ ] Devices.GetPluginConfiguration
- [ ] put /devices/:id.json
- [ ] Devices.SetPluginConfiguration
- [ ] get /log.json
- [ ] Logging.GetLogEntries
- [x] get /ws
- [x] JSONRPC.SetNotificationStatus
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,222
|
import {RouterException} from "./RouterException";
export class TooManyRequestsException extends RouterException {
constructor(message: any, error?: Error) {
message = message || "Too Many Requests";
super(429, message, error);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,480
|
Бури (від — «селянин») або африканери (від — «африканець») — біле населення Південно-Африканської Республіки та Намібії, що є нащадками нідерландських, бельгійських, французьких і німецьких колоністів у цих країнах.
Разом із нащадками переселенців з Великої Британії, бури складають біле населення Південно-Африканської Республіки (ПАР) та Намібії на відміну від чорношкірого корінного населення.
Застарілі назви: «кейп-голандці» та «білі африканці».
Мова
Бури (африканери) розмовляють мовою африкаанс, однією з офіційних державних мов ПАР. Африкаанс походить від нідерландської.
Господарство
Бури традиційно займаються сільсьским господарством, індивідуальним фермерством.
Історія
Культура
Характерним музичним стилем є буремюзік.
Посилання
Селянський народ або ж Boerevolk
Див. також
Народи Південно-Африканської Республіки
Народи Намібії
Народи Африки
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 80
|
Hi i am looking to create a news ticker. However i dont want to set the news items myself, i would like them to be 'read' from a website.
Is this possible?, could someone point me in the direction of a program/ code that could do this?
Someone on another forum mentioned iframes would this work?
say i made a ticker and posted the code here would that help?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,443
|
Interview: Grindhouse star Rose McGowan
Posted on Monday, April 2nd, 2007 by Peter Sciretta
On March 25th, we had the opportunity to talk with most of the stars of Grindhouse. We will be posting the interviews leading up until the film's release on April 6th 2007.
We sat down with Rose McGowan (Scream, Jawbreaker) to talk about her role as the gun legged stripper in Robert Rodriguez' Planet Terror. McGowan's character was also featured in Tarantino's Death Proof.
Rose McGowan talks Black Oasis
Posted on Sunday, April 1st, 2007 by Peter Sciretta
A couple weeks back it was announced that Rose McGowan would star in a biopic of B-movie actress Susan Cabot. We spoke to McGowan about that film, and she revealed a lot about the woman, the story, and what we can expect from Black Oasis:
It's great, I'm really excited. I'm attached to Black Oasis, which is coming up. It's the story of Susan Cabot and it's very strange that she is this B-movie star in these Roger Corman films like Wasp Woman and she had an insane life. Kind of really tragic in many ways. She was really short, probably about four foot ten, I'm definitely taller than that. She would go around in eight inch platforms. She was obsessed and thought that her career wasn't reaching a certain level because of her height. She actually went out with King Hussein, and they were going to get married until he found out her real name was Harriet Shapiro and that she was born Jewish. So he could marry her. And then she had a kid and she became really obsessed with him being strong and tall and he was going to be all the things she could never be but he was born of dwarfism. And she put him through all these experimental treatments that would make him grow tall. And the guy who wrote it and is directing it is Stephan Elliott, who wrote and directed The Adventures of Priscilla, Queen of the Desert. And the woman producing it is awesome. The production company is great. Hilary Shor just produced Children of Men, so they have really great people attached. Also she starts going crazier, and her life, as she gets crazier and crazier she starts thinking that she's in all the movies she was in. All these insane movies are happening. So I get to go back to all these sets and be in all these crazy movies. That's going to be fantastic, I'm very excited.
Black Oasis will shoot this fall, and is tentatively scheduled for a 2009 release. See Rose in Grindhouse, which hits theaters on April 6th 2007.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,552
|
package org.evm.biz.user.web;
import java.util.List;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import org.evm.core.consts.MessageType;
import org.evm.core.entity.PageResult;
import org.evm.core.exception.SmartFunctionException;
import org.evm.core.web.AbstractMultiController;
import org.evm.biz.log.annotation.SystemControllerLog;
import org.evm.biz.user.entity.UserVO;
import org.evm.biz.user.service.IUserDbService;
import org.springframework.stereotype.Controller;
import org.springframework.web.servlet.ModelAndView;
@Controller
public class UserController extends AbstractMultiController {
IUserDbService userDbService;
public void setUserDbService(IUserDbService userDbService) {
this.userDbService = userDbService;
}
@SystemControllerLog(description = "查询用户")
public void ajaxFindUserList(HttpServletRequest request, HttpServletResponse response) throws Exception {
String returnMsgContent = "sucess";
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
bindPageinfo(request, whereCause);
PageResult pageResult = null;
int res = 0;
try {
pageResult = userDbService.findUsersPage(whereCause);
List row = pageResult.getRows();
if (row != null) {
for (int i = 0; i < row.size(); i++) {
UserVO user = (UserVO) row.get(i);
user.setRoleName(user.getRoleListName());
}
}
} catch (SmartFunctionException e1) {
returnMsgContent = e1.getMessage();
res = -1;
} catch (Exception e) {
// TODO Auto-generated catch block
returnMsgContent = "查询用户户信息异常!" + e.getMessage();
res = -1;
logger.error(returnMsgContent, e);
}
if (res < 0) {
super.ReturnAjaxResult(response, pageResult, returnMsgContent, MessageType.error);
} else {
super.ReturnAjaxResult(response, pageResult, returnMsgContent, MessageType.noMessage);
}
}
public void ajaxInsertUser(HttpServletRequest request, HttpServletResponse response) throws Exception {
int res = 0;
String returnMsgContent = "sucess";
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
bindPageinfo(request, whereCause);
// 待修改获得方式
whereCause.setInsUser(super.getLoginUserId(request));
whereCause.setUpdUser(super.getLoginUserId(request));
try {
UserVO vo = userDbService.getUserByName(whereCause.getUname());
if (vo != null) {
returnMsgContent = "用户名已经存在,请选择其它用户名!";
res = -2;
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.error);
return;
}
res = userDbService.insertUser(whereCause);
} catch (SmartFunctionException e1) {
returnMsgContent = e1.getMessage();
res = -1;
} catch (Exception e) {
// TODO Auto-generated catch block
returnMsgContent = "插入失败,请联系管理员!" + e.getMessage();
res = -1;
logger.error(returnMsgContent, e);
}
if (res < 0) {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.error);
} else {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.noMessage);
}
}
public void ajaxDeleteUser(HttpServletRequest request, HttpServletResponse response) throws Exception {
int res = 0;
String returnMsgContent = "sucess";
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
// 待修改 获取方式
whereCause.setInsUser(super.getLoginUserId(request));
whereCause.setUpdUser(super.getLoginUserId(request));
try {
res = userDbService.deleteUser(whereCause);
} catch (SmartFunctionException e1) {
returnMsgContent = e1.getMessage();
res = -1;
} catch (Exception e) {
// TODO Auto-generated catch block
returnMsgContent = "刪除用户信息异常!" + e.getMessage();
logger.error(returnMsgContent, e);
res = -1;
}
if (res < 0) {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.error);
} else {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.noMessage);
}
}
public void ajaxUpdateUser(HttpServletRequest request, HttpServletResponse response) throws Exception {
int res = 0;
String returnMsgContent = "sucess";
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
// 待修改 获取方式
whereCause.setInsUser(super.getLoginUserId(request));
whereCause.setUpdUser(super.getLoginUserId(request));
try {
res = userDbService.updateUser(whereCause);
} catch (SmartFunctionException e1) {
returnMsgContent = e1.getMessage();
res = -1;
} catch (Exception e) {
// TODO Auto-generated catch block
returnMsgContent = "修改用户信息异常!" + e.getMessage();
logger.error(returnMsgContent, e);
res = -1;
}
if (res < 0) {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.error);
} else {
super.ReturnAjaxResult(response, res, returnMsgContent, MessageType.noMessage);
}
}
public void ajaxFindUserById(HttpServletRequest request, HttpServletResponse response) throws Exception {
String returnMsgContent = "sucess";
int res = 0;
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
UserVO resulst = null;
try {
resulst = userDbService.getUserById(whereCause.getUid());
} catch (Exception e) {
// TODO Auto-generated catch block
returnMsgContent = "查询用户异常,请联系管理员!";
logger.error(returnMsgContent, e);
res = -1;
}
super.ReturnAjaxResult(response, resulst);
}
public ModelAndView gotoUpdatePage(HttpServletRequest request, HttpServletResponse response) throws Exception {
ModelAndView mv = new ModelAndView("updUser");
UserVO whereCause = new UserVO();
bindObject(request, whereCause);
mv.addObject("updObj", whereCause);
return mv;
}
public ModelAndView returnManagePage(HttpServletRequest request, HttpServletResponse response) throws Exception {
ModelAndView mv = new ModelAndView("userManage");
return mv;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,088
|
{"url":"https:\/\/www.datasciencecentral.com\/performance-evaluation-of-cloud-computing-platforms-for-machine\/","text":"# Performance evaluation of cloud computing platforms for Machine Learning\n\nA use case on Logistic regression training\n\nOver the last few years there are several efforts for more powerful computing platforms to face the challenges imposed by emerging applications like machine learning. General purpose CPUs have been developed specialized ML modules, GPUs and FPGAs with specialized engines are around the corner. Several startups develop novel ASICs specialized for ML applications and Deep Neural networks.\n\nIn this article we perform a comparison of 3 different platforms available on the cloud (general purpose CPUs, GPUs and FPGAs). We evaluate the performance in terms of total execution time, accuracy and cost.\n\nFor this benchmark we have selected logistic regression as it is one of the most widely used algorithm for classification. Logistic Regression is used for building predictive models for many complex pattern-matching and classification problems. It is used widely in such diverse areas as bioinformatics, finance and data analytics. Logistic regression measures the relationship between the categorical dependent variable and one or more independent variables by estimating probabilities using a logistic function.\n\nLogistic Regression was chosen because it\u2019s arguably the most popular algorithm for building predictive analytics use cases and its iterative procedure when fitting the model, which allows us to extract better results on this comparison.\n\nIn this case we evaluate the training of the MNIST dataset with 10 classes and 1 million dataset.\n\nThe purpose of these notebooks is to compare and evaluate the performance of a Logistic Regression model using three implementations, Python\u2019s Sklearn package (alongside Intel\u2019s Math Kernel Library), Rapids cuml library and InAccel Sklearn-like package. The first one is the widely used Data Science library, Intel\u2019s MKL is a cpu math processing accelerated framework, while the others are newer solutions built on top of GPU and FPGA accelerators respectively.\n\nWe compared the training in 4 different platforms:\n\n1. Reference: Intel Xeon Skylake SP (r5.2xlarge with original code)\n2. MKL: Intel Xeon Skylake SP (r5.2x large using MKL libraries)\n3. GPU: NVIDIA\u00ae V100 Tensor Core (p3.2x large RAPIDS library)\n4. FPGA: FPGA (f1.2x using InAccel ML suite)\n\nIn the case of general purpose CPUs we use both the original code (without optimized libraries) and the Intel MKL library for optimization of ML kernels. In the case of GPUs, we use the RAPIDS framework and in the case of FPGAs we use our own ML suite for logistic regression\u00a0available from InAccel.\n\nThe following figure depicts the performance of each platform (total execution time). As you can see in terms of performance, GPU achieves the best performance compared to the rest of platforms. However, the accuracy in this case is only 73% while the rest of the platforms can achieve up to 88% accuracy. So in terms of accuracy, FPGAs using the InAccel ML suite can achieve the optimum performance and very high accuracy.\n\nTotal execution time for ML training of logistic regression (MNIST). In parenthesis, the accuracy achieved for each platform\/algorithm.\n\nHowever, cost is also very important for enterprises and data scientists. In this case we compare the performance vs cost trade off using these 4 platforms. The cost of each platform is shown below:\n\n\u2022 r5.2xlarge: $0.504\/h \u2022 p3.2xlarge:$3.06\/h\n\u2022 f1.2xlarge: \\$1.65\/h\n\nIn the following figure we show the performance (total execution time) and total cost for the ML training for these 4 platforms.\n\nPerformance vs Cost for the training of Logistic regression using MNIST (In the parenthesis you can see the accuracy each model achieved)\n\nAs you can see in this figure, FPGAs on the cloud (f1.2xlarge on this case with InAccel ML suite) achieves the best combination in terms of performance-accuracy and cost. Optimized libraries for GPP (MKL) achieve the most cost-efficient solution but the performance is not as high as using accelerators. GPUs can achieve better performance but the cost is much higher and in this case the accuracy is not acceptable in many applications.","date":"2023-03-25 10:23:13","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\": 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\": 0.5044277310371399, \"perplexity\": 1424.1881530450407}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945323.37\/warc\/CC-MAIN-20230325095252-20230325125252-00038.warc.gz\"}"}
| null | null |
Q: I need help updating my form to show new database records Basically, I have a WinForm in Visual Studio, that can add, delete, and view records from a database.
What I need to do is after adding, deleting, or editing a record, it needs to refresh the databindings or the database so it pulls accurate data from the database.
I've already tried Refresh() and Update() and tried closing and showing the form again, which it doesn't do.
This is the code I use for adding a record into the database.
private void kaykay_Click(object sender, EventArgs e)
{
con.Open();
OleDbCommand cmd = con.CreateCommand();
cmd.CommandType = CommandType.Text;
cmd.CommandText =
"insert into RM_DATA
(`Protokol No`,
`Küpe No`,
`Cinsi`,
`Türü`,
`Cinsiyeti`,
`Alındığı Yer`,
`Ekip`,
`Alınma Tarihi`,
`Taburcu Tarihi`,
`İlgilisi `,
`Telefon`,
`Açıklama`,
`Mikro Çip`,
`Resim`
) values(
'" + protokolno.Text + "',
'" + kupeno.Text + "',
'" + turu.Text + "',
'" + cinsi.Text + "',
'" + cinsiyeti.Text + "',
'" + alyer.Text + "',
'" + alekip.Text + "',
'" + dateTimePicker1.Text + "',
'" + dateTimePicker2.Text + "',
'" + ilgilisi.Text + "',
'" + ilgilisitelno.Text + "',
'" + aciklama.Text + "',
'" + mikrocip.Text + "',
'" + textBox1.Text + "')";
cmd.ExecuteNonQuery();
con.Close();
MessageBox.Show("Kayit Basariyla Girildi");
}
What I need it to do is to update the form after the messagebox shows up.
It doesn't give me any errors.
EDIT: I should clarify, I am not using datagridview.
A: This method for Insert, update and delete DisplayData() :
private void DisplayData()
{
con.Open();
DataTable dt=new DataTable();
SqlDataAdapter adapt=new SqlDataAdapter("select * from tbl_Record",con);
adapt.Fill(dt);
dataGridView1.DataSource = dt;
con.Close();
}
private void kaykay_Click(object sender, EventArgs e)
{
con.Open();
OleDbCommand cmd = con.CreateCommand();
cmd.CommandType = CommandType.Text;
cmd.CommandText =
"insert into RM_DATA
(`Protokol No`,
`Küpe No`,
`Cinsi`,
...
'" + textBox1.Text + "')";
cmd.ExecuteNonQuery();
con.Close();
MessageBox.Show("Kayit Basariyla Girildi");
DisplayData();
}
A: You can use Timer_tick to update your database basically you can call show data base function in timer-tick so it will show again your database every x seconds and it will be updated
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,807
|
Maxim Sozontovich Berezovsky (transcrições alternativas: Maxim Berezovski, Maksim Berezovsky ou Maksym Berezovsky, , ; cerca de 1745(?) — 2 de Abril de 1777) foi um compositor, cantor de ópera, baixista e violinista de Hlukhiv (Glukhov em russo), na Ucrânia, no Hetmanato Cossaco, então parte do Império Russo.
Influência cultural
O filme de 1983 de Andrei Tarkovsky, Nostalghia, é "um comentário sobre o exílio contado através da vida de Berezovsky".
Compositores da Era clássica
Mortos em 1777
Nascidos na década de 1740
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,113
|
Robert Lücken (ur. 30 kwietnia 1985 r. w Amsterdamie) – holenderski wioślarz, brązowy medalista igrzysk olimpijskich, mistrz świata i Europy.
Igrzyska olimpijskie
Na igrzyskach zadebiutował w 2016 roku w Rio de Janeiro, występując w zawodach ósemek. Skład osady uzupełnili: Dirk Uittenbogaard, Boaz Meylink, Kaj Hendriks, Boudewijn Röell, Olivier Siegelaar, Tone Wieten, Mechiel Versluis i Peter Wiersum jako sternik. W eliminacjach zajęli drugie miejsce, przez co musieli uczestniczyć w repasażach. W nich awansowali do finału, dopływając do mety na drugiej pozycji. W finale zajęli trzecie miejsce, zdobywając brązowy medal.
Przypisy
Linki zewnętrzne
Profil zawodnika na stronie MKOL
Holenderscy wioślarze
Holenderscy medaliści olimpijscy
Medaliści Letnich Igrzysk Olimpijskich 2016
Wioślarze na Letnich Igrzyskach Olimpijskich 2016
Uczestnicy Mistrzostw Europy w Wioślarstwie 2010
Medaliści Mistrzostw Świata w Wioślarstwie 2009
Medaliści Mistrzostw Świata w Wioślarstwie 2013
Medaliści Mistrzostw Europy w Wioślarstwie 2013
Ludzie urodzeni w Amsterdamie
Urodzeni w 1985
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,009
|
1.This product should not be used by the following persons?
Teenagers less than 18 years old.
Persons who have demonstrated sensitivity to nicotine.
2.Because of the light,the real item may seems a little different from the pictures showed.
3.How to use it:Click the botton in succession within two seconds. And press on the botton while you are smoking.
4.We have pre-delivery of goods by the person double-check, no quality problems in the confirmation after shipment, so please rest assured that quality problems .
5.If you have any questions or need advice, feel free to contact us .
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section*{Introduction}
A remarkable fact, discovered by Wang in \cite{wa}, is that the symmetric group $S_n$ has a quantum analogue $\mathcal Q_n$. For $n\geq 4$ this quantum group is bigger than $S_n$, and fits into Woronowicz's formalism in \cite{wo}.
The quantum group $\mathcal Q_n$ is best understood via its representation theory: with suitable definitions, it appears as the Tannakian realisation of the Temperley-Lieb algebra (\cite{ba2}). This elucidates a number of questions regarding the Cayley graph, fusion rules, amenability, etc. More generally, this puts $\mathcal Q_n$ into the framework of free quantum groups of Van Daele and Wang (\cite{vdw}), where a whole machinery, inspired by work of Gromov, Jones, Voiculescu, Wassermann, Weingarten is now available.
The study of $\mathcal Q_n$, and of free quantum groups in general, focuses now on more technical aspects: matrix models (\cite{bc}, \cite{bm}), ergodic actions (\cite{bdv}, \cite{va}), harmonic analysis (\cite{vv}, \cite{ve}).
The other thing to do is to study subgroups of $\mathcal Q_n$. This was started independently by the authors in \cite{ba1}, \cite{ba2} and \cite{bi1}, \cite{bi2}, and continued in the joint paper \cite{bb}. The notion that emerges from this work is that of quantum automorphism group of a vertex-transitive graph.
In this paper we describe quantum automorphism groups of vertex-transitive graphs having $n\leq 11$ vertices, with one graph omitted. This enhances previous classification work from \cite{ba1}, \cite{ba2}, \cite{bb}, where we have $n\leq 9$, also with one graph omitted.
Needless to say, in this classification project the value of $n$ is there only to show how far our techniques go.
The four main features of the present work are:
(1) Product operations. We have general decomposition results for Cartesian and lexicographic products. These are motivated by the graphs $\texttt{Pr}(C_5),\texttt{Pr}(K_5)$ and $C_{10}(4)$, which appear at $n=10$.
(2) The discrete torus. Here $n=9$. We prove that its quantum group is equal to its classical group, namely $S_3\wr{\mathbb Z}_2$. This answers a question left open in \cite{ba2}, \cite{bb}, and provides the first example of a graph having a usual wreath product as quantum symmetry group.
(3) Circulant graphs. It is known from \cite{ba1} that the $n$-cycle with $n\neq 4$ has quantum symmetry group $D_n$. This is extended in \cite{ba2} to a bigger class of circulant graphs. Here we further enlarge the list of such graphs, with an ad-hoc proof for $C_{10}(2)$, which appears at $n=10$.
(4) The Petersen graph. This appears at $n=10$, and the corresponding quantum group seems to be different from the known ones.
Our other techniques do not apply here:
it cannot be written as a graph product, and is not a circulant graph.
Neither could we carry a direct analysis as in the torus case
because of the complexity
of some computations.
However we prove that the corresponding quantum group
is not isomorphic to $\mathcal Q_5$.
In other words, we might have here a ``new'' quantum group. However, we don't have a proof, and the question is left open.
As a conclusion, we have two questions:
(I) First is to decide whether the Petersen graph produces or not a new quantum group. If it does, this would probably change a bit the landscape: in the big table at the end, based on work since Wang's paper \cite{wa}, all quantum groups can be obtained from ${\mathbb Z}_n,S_n,\mathcal Q_n$.
(II) A good question is to try to characterize graphs having no quantum symmetry. This paper provides many new examples, and we have found some more by working on the subject, but so far we were unable to find a conceptual result here.
\smallskip
The paper is organized as follows. Sections 1, 2 are quite detailed preliminary sections, the whole paper, or at least the ideas involved in it, being intended to be accessible to non-specialists.
Sections 3, 4, 5, 6 deal with different kinds of graphs, once again in a quite self-contained way. In Section 7 we present the classification result, in the form of a big, independent table. In the last section we present a technical result about the
quantum group of the Petersen graph.
\section{Quantum permutation groups}
In this paper we use the following simplified version of Woronowicz's
compact quantum groups \cite{wo}, which is the only
one we need when dealing with quantum symmetries of
classical finite spaces.
\begin{definition} A Hopf ${\mathbb C}^*$-algebra is a ${\mathbb C}^*$-algebra $A$ with unit, endowed with morphisms
\begin{eqnarray*}
\Delta&:&A\to A\otimes A\cr
\varepsilon&:&A\to{\mathbb C}\cr
S&:&A\to A^{op}
\end{eqnarray*}
satisfying the usual axioms for a comultiplication, counit and antipode, along with the extra condition $S^2=id$.
\end{definition}
The more traditional terminology
for such an object is that of a
"universal Hopf $\mathbb C^*$-algebra of Kac type''.
The universality condition refers to the fact that the counit
and antipode are assumed to be defined on the whole $\mathbb C^*$-algebra
$A$ (in full generality, these are only defined on a
dense Hopf $*$-subalgebra) and the Kac condition
refers to the condition $S^2= id$.
We warn the reader that the Hopf $\mathbb C^*$-algebras we consider here
are not Hopf algebras in the usual sense
(the tensor product in the definition is a $\mathbb C^*$-tensor product). However, they possess canonically
defined dense Hopf $*$-subalgebras, from which they
can be reconstructed using the universal $C^*$-completion procedure.
See the survey paper \cite{mava}.
\medskip
The first example is with a compact group $G$. We can consider the algebra of continuous functions $A={\mathbb C}(G)$, with operations
\begin{eqnarray*}
\Delta(f)&=&(g,h)\to f(gh)\cr
\varepsilon(f)&=&f(1)\cr
S(f)&=&g\to f(g^{-1})
\end{eqnarray*}
where we use the canonical identification $A\otimes A={\mathbb C} (G\times G)$.
The second example is with a discrete group $\Gamma$. We have here the algebra $A={\mathbb C}^*(\Gamma)$, obtained from the usual group algebra ${\mathbb C} [\Gamma]$ by the universal $\mathbb C^*$-completion procedure, with operations
\begin{eqnarray*}
\Delta(g)&=&g\otimes g\cr
\varepsilon(g)&=&1\cr
S(g)&=&g^{-1}
\end{eqnarray*}
where we use the canonical embedding $\Gamma\subset A$.
In general, associated to an arbitrary Hopf ${\mathbb C}^*$-algebra $A$ are a compact quantum group $G$ and a discrete quantum group $\Gamma$, according to the following heuristic formulae:
$$A={\mathbb C}(G)={\mathbb C}^*(\Gamma)$$
$$G=\widehat{\Gamma}$$
$$\Gamma=\widehat{G}$$
These formulae are made into precise statements in the first section of Woronowicz'
seminal paper \cite{wo}.
They are pieces of Pontryagin duality
for locally compact quantum groups, whose latest version is given in \cite{kuva}.
The compact quantum group morphisms are defined in the usual manner:
if $A= \mathbb C(G)$ and $B= \mathbb C(H)$ are Hopf $\mathbb C^*$-algebras,
a quantum group morphism $H \rightarrow G$ arises from a Hopf $\mathbb C^*$-algebra morphism $\mathbb C(G) \rightarrow \mathbb C(H)$, and we say that
$H$ is a quantum subgroup of $G$ if the corresponding
morphism $\mathbb C(G) \rightarrow \mathbb C(H)$ is surjective.
We refer to \cite{wa0} for more details on the compact quantum
group language.
\smallskip
A square matrix $u= (u_{ij}) \in M_n(A)$ is said to be multiplicative if
$$
\Delta(u_{ij})=\sum u_{ik}\otimes u_{kj} \quad {\rm and} \quad
\varepsilon(u_{ij})=\delta_{ij}$$
Multiplicative matrices correspond to corepresentations
of the Hopf $\mathbb C^*$-algebra $A$, that is, to representations
of the compact quantum group $G$
with $A = \mathbb C(G)$.
Such a multiplicative matrix $u$ will also be interpreted
as a linear map $\mathbb C^n \longrightarrow \mathbb C^n \otimes A$.
In this paper we are essentially interested in the
following special type of multiplicative matrices.
\begin{definition}
A magic unitary matrix is a square matrix,
all of whose entries are projections and all of whose rows and columns are partitions of unity.
\end{definition}
Here we say that a finite family of projections
is a partition of unity if these projections are pairwise orthogonal
and if their sum equals 1.
\smallskip
As a first example, consider a finite group $G$ acting on a finite set $X$. The characteristic functions
$$p_{ij}=\chi\{\sigma\in G\mid \sigma(j)=i\}$$
form a magic unitary matrix, because the corresponding sets form partitions of $G$, when $i$ or $j$ varies. We have the following formulae for ${\mathbb C}(G)$:
\begin{eqnarray*}
\Delta(p_{ij})&=&\sum p_{ik}\otimes p_{kj}\cr
\varepsilon(p_{ij})&=&\delta_{ij}\cr
S(p_{ij})&=&p_{ji}
\end{eqnarray*}
and therefore $p=(p_{ij})$ is a multiplicative matrix.
In the particular case of the symmetric group $S_n$ acting on $\{1,\ldots ,n\}$, the Stone-Weierstrass theorem shows that entries of $p$ generate ${\mathbb C}(S_n)$. This suggests the following construction, due to Wang (\cite{wa}).
\begin{definition} The $\mathbb C^*$-algebra
$A_s(n)$ is the universal ${\mathbb C}^*$-algebra generated by $n^2$ elements $u_{ij}$, with relations making $u$ into a magic unitary matrix,
and with morphisms
\begin{eqnarray*}
\Delta(u_{ij})&=&\sum u_{ik}\otimes u_{kj}\cr
\varepsilon(u_{ij})&=&\delta_{ij}\cr
S(u_{ij})&=&u_{ji}
\end{eqnarray*}
as comultiplication, counit and antipode, making it into a
Hopf $\mathbb C^*$-algebra.
\end{definition}
This Hopf $\mathbb C^*$-algebra was discovered by Wang \cite{wa}.
The corresponding compact quantum group is denoted
$\mathcal Q_n$ and we call it the quantum permutation
group or quantum symmetric group.
This is motivated by the fact that
the algebra $A_s(n)$ is the biggest Hopf $\mathbb C^*$-algebra coacting on
the algebra $\mathbb C^n$, which is to say that the quantum group $\mathcal Q_n$
is the biggest one acting on
$\{1,\ldots ,n\}$. The coaction $u : \mathbb C^n \longrightarrow \mathbb C^n \otimes A_s(n)$
is defined on Dirac masses by
$$u(\delta_i)=\sum \delta_j\otimes u_{ji}$$
and verification of axioms of coactions, as well as proof of universality, is by direct computation. See \cite{wa}.
We have a surjective morphism of Hopf ${\mathbb C}^*$-algebras
$$A_s(n)\to {\mathbb C}(S_n)$$
mapping $u_{ij}$ to $p_{ij}$ for any $i,j$. This morphism expresses the fact that the compact quantum group corresponding to $A_s(n)$ contains $S_n$.
This map is an isomorphism for $n=2,3$, as known from \cite{ba2}, \cite{wa}, and explained in section 3 below. At $n=4$ we have Wang's matrix
$$u=\begin{pmatrix}p&1-p&0&0\cr 1-p&p&0&0\cr 0&0&q&1-q\cr 0&0&1-q&q \end{pmatrix}$$
with $p,q$ free projections, which shows that
there exists an epimorphism
$A_s(4) \to \mathbb C^*(\mathbb Z_2 * \mathbb Z_2)$
and hence
$A_s(n)$ is not commutative and is infinite dimensional. The same remains true for any $n\geq 4$.
\section{Quantum automorphism groups of graphs}
Consider a finite graph $X$.
In this paper this means that we have a finite set of vertices,
and certain pairs of distinct vertices are connected by unoriented edges.
It is convenient to assume that the vertex set is $\{1,\ldots ,n\}$.
\begin{definition}
The adjacency matrix of $X$ is the matrix
$$d\in M_n(0,1)$$
given by $d_{ij}=1$ if $i,j$ are connected by an edge, and $d_{ij}=0$ if not.
\end{definition}
The adjacency matrix is symmetric, and has $0$ on the diagonal. In fact, graphs having vertex set $\{1,\ldots ,n\}$ are in one-to-one correspondence with $n\times n$ symmetric 0--1 matrices having $0$ on the diagonal.
The quantum automorphism group of $X$ is obtained as an appropriate subgroup of the quantum permutation group of $\{1,\ldots ,n\}$. At level of Hopf $\mathbb C^*$-algebras, this means taking an appropriate quotient of $A_s(n)$.
\begin{definition}
Associated to a finite graph $X$ is the $\mathbb C^*$-algebra
$$A(X)=A_s(n)/<du=ud>$$
where $n$ is the number of vertices, and $d$ is the adjacency matrix.
\end{definition}
Since a permutation of the set $X$ is a graph automorphism
if and only if the corresponding permutation matrix
commutes with the adjacency matrix, it is reasonable
to say that the quantum group corresponding to $A(X)$
is the quantum automorphism group of $X$.
In this way we have a commutative diagram of Hopf ${\mathbb C}^*$-algebras
$$\begin{matrix}
A_s(n)&\ &\rightarrow&\ &A(X)\cr
\ \cr \downarrow&\ &\ &\
&\downarrow\cr \ \cr
{\mathbb C}(S_n)&\ &\rightarrow&\ &{\mathbb C}(G)
\end{matrix}$$
where $G=G(X)$ is the usual automorphism group of $X$, with the kernel of the right arrow being the commutator ideal of $A(X)$. Moreover, for a graph without edges we get indeed $A_s(n)$, and we have the formula
$$A(X)=A(X^c)$$
where $X^c$ is the complement of $X$. See \cite{ba2}, \cite{bb} for details.
The defining equations $ud = du$ of $A(X)$ means that
$d$, considered as a linear map $\mathbb C^n \rightarrow \mathbb C^n$,
is a morphism in the category of corepresentations of $A(X)$,
i.e. a morphism in the category of representations
of the quantum group dual to $A(X)$. General properties of the
representation category of a compact quantum group
(see e.g. \cite{wo}) now ensure that the spectral projections
occurring in the spectral decomposition of $d$ are
corepresentations morphisms, and hence the corresponding
eigensubspaces are subcorepresentations. This key fact will be
used freely in the paper.
\smallskip
The following notion will play a central role in this paper.
\begin{definition}
We say that $X$ has no quantum symmetry if
$$A(X)={\mathbb C}(G)$$
where $G=G(X)$ is the usual automorphism group of $X$.
\end{definition}
This is the same as saying that $A(X)$ is commutative, because by the above considerations, ${\mathbb C}(G)$ is its biggest commutative quotient.
\medskip
We are particularly interested in the case of graphs $X$ having the property that $G$ acts transitively on the set of vertices.
These graphs were called homogeneous in previous work \cite{ba2}, \cite{bb},
but we use here the following more traditional terminology.
\begin{definition}
The graph $X$ is called vertex-transitive if for any two vertices $i,j$ there is $\sigma\in G(X)$ such that $\sigma(i)=j$.
\end{definition}
Each section of the paper ends with a small table, gathering information about vertex-transitive graphs having $\leq 11$ vertices. These small tables are to be put together in a single big table, at the end.
\smallskip
What we know so far is that we have
$$A(K_n)=A_s(n)$$
where $K_n$ is the complete graph having $n$ vertices. Moreover, we already mentioned that for $n=2,3$ the arrow
$$A_s(n)\to {\mathbb C}(S_n)$$
is an isomorphism, and for $n\geq 4$ it is not.
This information is summarized in the following table.
\begin{center}\begin{tabular}[t]{|l|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
2&$K_2$&$ {{\mathbb Z}_2}$&$ {{\mathbb Z}_2}$\\
\hline
3&$K_3$&$ {S_3}$&$ {S_3}$\\
\hline
$n\geq4$&$K_n$&$ S_n$&$ \mathcal Q_n$\\
\hline
\end{tabular}\end{center}
\medskip
Here in the right column $\mathcal Q_n$ with $n\geq 4$ is the compact quantum group associated to $A_s(n)$.
\section{Circulant graphs}
A graph with $n$ vertices is called circulant if its automorphism group contains a cycle of length $n$ (and hence in particular a
copy of the cyclic group ${\mathbb Z}_n$). We are particularly interested in connected circulant graphs, which are the cycles with chords.
\begin{definition}
The graph $C_n(k_1,\ldots ,k_r)$, where
$$1<k_1<\ldots <k_r\leq [n/2]$$
are integers, is obtained by drawing the $n$-cycle $C_n$, then connecting all pairs of vertices at distance $k_i$, for any $i$.
\end{definition}
As basic examples, we have the $n$-cycle $C_n$, corresponding to the value $r=0$, and the $2n$-cycle with diagonals, $C_n^+=C_{2n}(n)$.
Observe that $K_n$ is a cycle with chords as well.
The adjacency matrix of a cycle with chords, denoted as usual $d$, is a circulant matrix. We use the following basic fact.
\begin{proposition}
We have $d(\xi^s)=2f(s)\xi^s$, where
$$f(s)=\sum_{i=0}^r \cos\left(\frac{2k_is\pi}{n}\right)
\quad \rm{(with} \ k_0=1\rm{)}$$
and $\xi$ is the vector whose coordinates are
$1,\omega , \ldots , \omega^{n-1}$ in the canonical basis of $\mathbb C^n$,
with $\omega = e^{\frac{2i\pi}{n}}$.
\end{proposition}
This tells us that we have the following eigenspaces for $d$:
\begin{eqnarray*}
V_0&=&{\mathbb C} 1\cr
V_1&=&{\mathbb C}\xi +{\mathbb C}\xi^{n-1}\cr
V_2&=&{\mathbb C}\xi^2+{\mathbb C}\xi^{n-2}\cr
\ldots&&\ldots\cr
V_{m}&=&{\mathbb C}\xi^{m}+{\mathbb C}\xi^{n-m}\cr
\end{eqnarray*}
where $m=[n/2]$ and all sums are direct, except maybe for the last one, which depends on the parity of $n$.
The fact that these eigenspaces correspond or not to different eigenvalues depends of course on $f$.
We use the following result from \cite{ba2}, whose proof is briefly explained, because several versions of it will appear throughout the paper.
\begin{theorem}
If $n\neq 4$ and the associated function
$$f:\{1,2,\ldots,[n/2]\}\to{\mathbb R}$$
is injective, then $C_n(k_1,\ldots ,k_r)$ has no quantum symmetry.
\end{theorem}
\begin{proof}
Since ${\mathbb C}\xi\oplus{\mathbb C}\xi^{n-1}$ is invariant, the coaction can be written as
$${v}(\xi)=\xi\otimes a+\xi^{n-1}\otimes b$$
for some $a,b$. By taking powers and using $n\neq 4$ we get by induction
$${v}(\xi^s)=\xi^s\otimes a^s+\xi^{n-s}\otimes b^s$$
for any $s$, along with the relations $ab=-ba$ and $ab^2=ba^2=0$.
Now from ${v}(\xi^*)={v}(\xi)^*$ we get $b^*=b^{n-1}$, so $(ab)(ab)^*=ab^na^*=0$. Thus $ab=ba=0$, so $A(X)$ is commutative and we are done.
\end{proof}
\begin{corollary}
The following graphs have no quantum symmetry:
\begin{enumerate}
\item The cycles $C_n$ with $n\neq 4$.
\item The cycles with diagonals $C_8^+,C_{10}^+$.
\item The cycles with chords $C_9(3),C_{11}(2),C_{11}(3)$.
\end{enumerate}
\end{corollary}
\begin{proof} (1) follows from the fact that $f$ is decreasing, hence injective. As for (2) and (3), the corresponding 5 functions are given by
\begin{eqnarray*}
C_8^+&:&-0.29,1,-1.7,0\cr
C_{10}^+&:&-0.19,1.3,-1.3,0.19,-2\cr
C_9(3)&:&0.26,-0.32,0.5,-1.43\cr
C_{11}(2)&:&1.25,-0.23,-1.10,-0.79,-0.11\cr
C_{11}(3)&:&0.69,-0.54,0.27,0.18.-1.61
\end{eqnarray*}
with $0.01$ error, so they are injective, and Theorem 3.1 applies.
\end{proof}
The graphs in Corollary 3.1 have usual symmetry group $D_n$, where $n$ is the number of vertices. We don't know if $G=D_n$ with $n\neq 4$ implies that the graph has no quantum symmetry. However, we are able to prove this for $n\leq 11$: graphs satisfying $G=D_n$ are those in Corollary 3.1, plus the cycle with chords $C_{10}(2)$, discussed below.
\begin{theorem}
The graph $C_{10}(2)$ has no quantum symmetry.
\end{theorem}
\begin{proof}
The function $f$ is given by
$$f(s)=\cos\left(\frac{s\pi}{5}\right)+\cos\left(\frac{2s\pi}{5}\right)$$
and we have $f(1)=-f(3)\simeq 1.11$, $f(2)=f(4)=-0.5$ and $f(5)=0$. Thus the list of eigenspaces is:
\begin{eqnarray*}
V_0&=&{\mathbb C} 1\cr
V_1&=&{\mathbb C}\xi\oplus{\mathbb C}\xi^{9}\cr
V_2&=&{\mathbb C}\xi^2\oplus{\mathbb C}\xi^4\oplus{\mathbb C}\xi^{6}\oplus{\mathbb C}\xi^{8}\cr
V_3&=&{\mathbb C}\xi^3\oplus{\mathbb C}\xi^{7}\cr
V_5&=&{\mathbb C}\xi^5
\end{eqnarray*}
Since coactions preserve eigenspaces, we can write
$${v}(\xi)=\xi\otimes a+\xi^{9}\otimes b$$
for some $a,b$. Taking the square of this relation gives
$${v}(\xi^2)=\xi^2\otimes a^2+\xi^{8}\otimes b^2+1\otimes (ab+ba)$$
and once again since ${v}$ preserves eigenspaces, we get $ab=-ba$. Taking now the cube of the above relation gives
\begin{eqnarray*}
{v}(\xi^3)&=&\xi^3\otimes a^3+\xi^{7}\otimes b^3
+\xi\otimes ba^2 +\xi^{9}\otimes ab^{2}
\end{eqnarray*}
and once again since ${v}$ preserves eigenspaces, we get:
$$ab^2=0= ba^2$$
With the relations $ab=-ba$ and $ab^2=ba^2=0$ in hand, we get by induction the formula
$${v}(\xi^s)=\xi^s\otimes a^s+\xi^{n-s}\otimes b^s$$
and we can conclude by using adjoints, as in proof of Theorem 3.1.
\end{proof}
For graphs having $n\leq 11$ vertices, results in this section are summarized in the following table.
\begin{center}\begin{tabular}[t]{|l|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
$n\geq5$&$C_n$&$ D_n$&$ D_n$\\
\hline
8&$C_8,C_8^+$&$ D_8$&$ D_8$\\
\hline
9&$C_9,C_9(3)$&$ D_9$&$ D_9$\\
\hline
10&$C_{10},C_{10}(2),C_{10}^+$&$ D_{10}$&$ D_{10}$\\
\hline
11&$C_{11},C_{11}(2),C_{11}(3)$&$ D_{11}$&$ D_{11}$\\
\hline
\end{tabular}\end{center}
\medskip
As already mentioned, we don't know if these computations are particular cases of some general result.
\section{Products of graphs}
For a finite graph $X$, it is convenient to use the notation
$$X=(X,\sim)$$
where the $X$ on the right is the set of vertices, and where we write $i\sim j$ when two vertices $i,j$ are connected by an edge.
\begin{definition}
Let $X,Y$ be two finite graphs.
\begin{enumerate}
\item
The direct product $X\times Y$ has vertex set $X\times Y$, and edges
$$(i,\alpha)\sim(j,\beta)\Longleftrightarrow i\sim j,\, \alpha\sim\beta.$$
\item
The Cartesian product $X\square Y$ has vertex set $X\times Y$, and edges
$$(i,\alpha)\sim(j,\beta)\Longleftrightarrow i=j,\, \alpha\sim\beta\mbox{ \rm{or} }i\sim j,\alpha=\beta.$$
\end{enumerate}
\end{definition}
The direct product is the usual one in a categorical sense. As for the Cartesian product, this is a natural one from a geometric viewpoint: for instance a product by a segment gives a prism.
\begin{definition}
The prism having basis $X$ is $\mathtt{Pr}(X)=K_2\square X$.
\end{definition}
We have embeddings of usual symmetry groups
$$G(X) \times G(Y) \subset G(X \times Y)$$
$$G(X) \times G(Y) \subset G(X \square Y)$$
which have the following quantum analogues.
\begin{proposition}
We have surjective morphisms of Hopf ${\mathbb C}^*$-algebras
$$A(X \times Y) \longrightarrow A(X) \otimes A(Y)$$
$$A(X \square Y) \longrightarrow A(X) \otimes A(Y).$$
\end{proposition}
\begin{proof}
We use the canonical identification
$${\mathbb C}(X \times Y)={\mathbb C}(X) \otimes {\mathbb C}(Y)$$
given by $\delta_{(i,\alpha)}=\delta_i\otimes\delta_\alpha$. The adjacency matrices are given by
$$d_{X \times Y} = d_X \otimes d_Y$$
$$d_{X \square Y} = d_X \otimes 1 + 1 \otimes d_Y$$
so if $u$ commutes with $d_X$ and $v$ commutes with $d_Y$, the matrix
$$u\otimes v=(u_{ij}v_{\alpha\beta})_{(i\alpha,j\beta)}$$
is a magic unitary that
commutes with both $d_{X\times Y}$ and $d_{X\square Y}$. This gives morphisms as in the statement, and surjectivity follows by summing over $i$ and $\beta$.
\end{proof}
\begin{theorem}
Let $X$ and $Y$ be finite connected regular graphs. If their spectra $\{\lambda\}$ and $\{\mu\}$ do not contain $0$ and satisfy
$$\{ \lambda_i/\lambda_j\} \cap \{\mu_k/\mu_l\}
= \{1\}$$
then $A(X \times Y)=A(X) \otimes A(Y)$. Also, if their spectra satisfy
$$\{\lambda_i - \lambda_j \} \cap \{\mu_k - \mu_l\}
= \{0\}$$
then $A(X \square Y)=A(X) \otimes A(Y)$.
\end{theorem}
\begin{proof}
We follow \cite{ba2}, where the first statement is proved. Let $\lambda_1$ be the valence of $X$. Since $X$ is regular
we have $\lambda_1 \in {\rm Sp}(X)$, with $1$ as eigenvector,
and since $X$ is connected $\lambda_1$ has multiplicity one.
Hence if $P_1$ is the orthogonal projection onto
${\mathbb C}1$, the spectral decomposition of $d_X$ is of the following form:
$$d_X = \lambda_1 P_1 + \sum_{i\not=1}\lambda_i P_i$$
We have a similar formula for $d_Y$:
$$d_Y = \mu_1 Q_1 + \sum_{j\not=1}\mu_j Q_j$$
This gives the following formulae for products:
$$d_{X\times Y}=\sum_{ij}(\lambda_i\mu_j)P_{i}\otimes Q_{j}$$
$$d_{X \square Y} = \sum_{i,j}(\lambda_i + \mu_i)P_i \otimes Q_j$$
Here projections form partitions of unity, and the scalar are distinct, so these are spectral decomposition formulae. We can conclude as in \cite{ba2}.
The universal coactions will commute with any of the spectral
projections, and hence with both $P_1 \otimes 1$ and $1 \otimes Q_1$.
In both cases the universal coaction $v$ is the tensor product of
its restrictions to the images of $P_1\otimes 1$
(i.e. $1 \otimes \mathbb C(Y)$) and of $1\otimes Q_1$
(i.e. $\mathbb C(X) \otimes 1$).
\end{proof}
\begin{corollary}
\
\begin{enumerate}
\item We have $A(K_m \times K_n)=A(K_m) \otimes A(K_n)$ for $m \not = n$.
\item We have $A(\mathtt{Pr}(K_n))={\mathbb C}({{\mathbb Z}_2})\otimes A_s(n)$, for any $n$.
\item We have $A(\mathtt{Pr}(C_n))={\mathbb C}(D_{2n})$, for $n$ odd.
\item We have $A(\mathtt{Pr}(C_4))={\mathbb C}({{\mathbb Z}_2})\otimes A_s(4)$.
\end{enumerate}
\end{corollary}
\begin{proof}
The spectra of graphs involved are ${\rm Sp}(K_2)=\{-1,1\}$ and
\begin{eqnarray*}
{\rm Sp}(K_n)&=&\{ -1,\ n-1\}\cr
{\rm Sp}(C_n)&=&\{2\cos (2k\pi /n)\mid k=1,\ldots ,n\}
\end{eqnarray*}
so the first three assertions follow from Theorem 4.1. We have
$$\texttt{Pr}(C_4)=K_2\times K_4$$
(this graph is the cube)
and the fourth assertion follows from the first one.
\end{proof}
We get the following table, the product operation
$\times$ on quantum groups being the one dual
to the tensor product of Hopf $\mathbb C^*$-algebras.
\begin{center}\begin{tabular}[t]{|l|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
8&$\texttt{Pr}(C_4)$&$S_4\times{{\mathbb Z}_2}$&$\mathcal Q_4\times{{\mathbb Z}_2}$\\
\hline
10&$\texttt{Pr}(C_5)$&$D_{10}$&$D_{10}$\\
\hline
10&$\texttt{Pr}(K_5)$&$S_5\times{{\mathbb Z}_2}$&$\mathcal Q_5\times{{\mathbb Z}_2}$\\
\hline
\end{tabular}\end{center}
\section{The torus graph}
Theorem 4.1 doesn't apply to the case $X=Y$, and the problem of computing the algebras $A(X\times X)$ and $A(X\square X)$ appears.
At level of classical symmetry groups, there is no simple formula describing $G(X\times X)$ and $G(X\square X)$. Thus we have reasons to believe that the above problem doesn't have a simple solution either.
A simpler question is to characterize graphs $X$ such that $X\times X$ or $X\square X$ has no quantum symmetry. We don't have a general result here, but we are able however to deal with the case $X=K_3$.
\begin{definition}
The graph $\mathtt{Torus}$ is the graph $K_3\times K_3=K_3\square K_3$.
\end{definition}
The result below answers a question asked in \cite{ba2}, \cite{bb}. It also provides the first example of graph having a classical wreath product as quantum symmetry group.
\begin{theorem}
The graph $\mathtt{Torus}$ has no quantum symmetry.
\end{theorem}
\begin{proof}
The spectrum of $K_3$ is known to be
$${\rm Sp}(K_3)=\{ -1,2\}$$
with corresponding eigenspaces given by
\begin{eqnarray*}
F_2&=&{\mathbb C} 1\cr
F_{-1}&=&{\mathbb C}\xi\oplus{\mathbb C}\xi^2
\end{eqnarray*}
where $\xi$ is the vector formed by third roots of unity.
Tensoring the adjacency matrix of $K_3$ with itself gives
$${\rm Sp}(\texttt{Torus}) = \{-2,1,4\}$$
with corresponding eigenspaces given by
\begin{eqnarray*}
E_4&=&{\mathbb C}\xi_{00}\cr
E_{-2}&=&{\mathbb C}\xi_{10}\oplus {\mathbb C}\xi_{01}\oplus {\mathbb C}\xi_{20} \oplus {\mathbb C}\xi_{02}\cr
E_{1}&=&{\mathbb C}\xi_{11} \oplus{\mathbb C}\xi_{12} \oplus{\mathbb C}\xi_{21} \oplus{\mathbb C}\xi_{22}
\end{eqnarray*}
where we use the notation $\xi_{ij}=\xi^i\otimes \xi^j$.
The universal coaction $v$ preserves eigenspaces, so we have
\begin{eqnarray*}
v(\xi_{10})&=& \xi_{10} \otimes a + \xi_{01} \otimes b + \xi_{20} \otimes c
+ \xi_{02} \otimes d\cr
v(\xi_{01})&=&\xi_{10} \otimes\alpha + \xi_{01} \otimes \beta +
\xi_{20} \otimes \gamma + \xi_{02} \otimes \delta
\end{eqnarray*}
for some $a,b,c,d,\alpha,\beta,\gamma,\delta \in A$. Taking the square of $v(\xi_{10})$ gives
$$v(\xi_{20})=\xi_{20} \otimes a^2 + \xi_{02} \otimes b^2 + \xi_{10} \otimes c^2
+\xi_{01} \otimes d^2$$
along with relations coming from eigenspace preservation:
$$ab = -ba, \ ad=-da, \ bc = -cb , \ cd = -dc$$
$$ac+ca = -(bd+db)$$
Now since $a,b$ anticommute, their squares have to commute.
On the other hand, by applying $v$ to the equality $\xi_{10}^*=\xi_{20}$, we get the following formulae for adjoints:
$$a^* = a^2, \ b^*=b^2, \ c^* = c^2, \ d^* = d^2$$
The commutation relation $a^2b^2=b^2a^2$ reads now $a^*b^*=b^*a^*$, and by taking adjoints we get $ba=ab$. Together with $ab=-ba$ this gives:
$$ab=ba=0$$
The same method applies to $ad,bc,cd$, and we end up with:
$$ab=ba=0,\ ad=da =0, \ bc=cb =0, \ cd = dc =0$$
We apply now $v$ to the equality $1=\xi_{10}\xi_{20}$. We get that $1$ is the sum of $16$ terms, all of them of the form $\xi_{ij}\otimes P$, where $P$ are products between $a,b,c,d$ and their squares. Due to the above formulae 8 terms vanish, and the $8$ remaining ones produce the formula
$$1=a^3 +b^3 +c^3 +d^3$$
along with relations coming from eigenspace preservation:
$$ac^2=ca^2=bd^2=db^2=0$$
Now from $ac^2=0$ we get $a^2c^2=0$, and by taking adjoints this gives $ca=0$. The same method applies to $ac,bd,db$, and we end up with:
$$ac=ca=0,\ bd=db=0$$
In the same way one shows that $\alpha,\beta,\gamma,\delta$
pairwise commute:
$$\alpha\beta=\beta\alpha=\ldots =\gamma\delta=\delta\gamma=0$$
It remains to show that $a,b,c,d$ commute with $\alpha,\beta,\gamma,\delta$. For, we apply $v$ to the following equality:
$$\xi_{10}\xi_{01}=\xi_{01}\xi_{10}$$
We get an equality between two sums having 16 terms each, and by using
again eigenspace preservation we get the following formulae relating the corresponding 32 products $a\alpha, \alpha a$ etc.:
$$a\alpha = 0 = \alpha a , \ b\beta =0 = \beta b , \
c\gamma = 0 = \gamma c , \ d \delta =0 = \delta d,$$
$$a\gamma + c\alpha + b \delta + d\beta = 0 =
\alpha c + \gamma a + \beta d + \delta b,$$
$$a\beta +b \alpha = \alpha b + \beta a, \
b \gamma + c\beta = \beta c + \gamma b ,$$
$$c\delta + d \gamma = \gamma d + \delta c , \
a \delta + d\alpha = \alpha d + \delta a$$
Multiplying the first equality in the second row on the left by $a$ and on the right by $\gamma$ gives $a^2\gamma^2 =0$, and by taking adjoints we get $\gamma a=0$. The same method applies to the other 7 products involved in the second row, so all 8 products involved in the second row vanish:
$$a\gamma =c\alpha=b\delta = d\beta= \alpha c=\gamma a
=\beta d=\delta b=0$$
We use now the first equality in the third row. Multiplying it on the left by $a$ gives $a^2\beta=a\beta a$, and multiplying it on the right by $a$ gives $a\beta a=\beta a^2$. Thus we get the commutation relation $a^2\beta=\beta a^2$.
On the other hand from $a^3+b^3+c^3+d^3=1$ we get $a^4=a$, so:
$$a\beta = a^4 \beta = a^2a^2 \beta = \beta a^2 a^2 = \beta a$$
One shows in a similar manner that the missing commutation formulae $a\delta = \delta a$ etc. hold as well. Thus $A$ is commutative.
\end{proof}
\begin{center}\begin{tabular}[t]{|l|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
9&$\texttt{Torus}$&$S_3\wr{{\mathbb Z}_2}$&$S_3\wr{{\mathbb Z}_2}$\\
\hline
\end{tabular}\end{center}
\medskip
\section{Lexicographic products}
Let $X$ and $Y$ be two finite graphs. Their lexicographic product is obtained by putting a copy of $X$ at each vertex of $Y$:
\begin{definition}
The lexicographic product $X\circ Y$ has vertex set $X\times Y$, and edges are given by
$$(i,\alpha)\sim(j,\beta)\Longleftrightarrow \alpha\sim\beta\mbox{ \rm{or} }\alpha=\beta,\,
i\sim j.$$
\end{definition}
The terminology comes from a certain similarity with the ordering of usual words, which is transparent when iterating $\circ$.
The simplest example
is with $X\circ X_n$, where $X_n$
is the graph having $n$ vertices and no edges:
the graph $X\circ X_n$ is the graph consisting of $n$ disjoint copies of $X$.
\begin{definition}
$nX$ is the disjoint union of $n$ copies of $X$.
\end{definition}
When $X$ is connected, we have an isomorphism
$$G(nX)=G(X)\wr S_n$$
where $\wr$ is a wreath product. In other words, we have:
$$G(X\circ X_n)=G(X)\wr G(X_n)$$
In the general case, we have the following embedding of usual symmetry groups:
$$G(X)\wr G(Y)\subset G(X\circ Y)$$
The quantum analogues of these results use the notion of free wreath product from \cite{bi2, bb}. In the following definition, a pair
$(A,u)$ is what we call a quantum permutation group in \cite{bb}:
$A$ is a Hopf $\mathbb C^*$-algebra
and $u$ is a multiplicative magic unitary matrix.
\begin{definition}
The free wreath product of $(A,u)$ and $(B,v)$ is
$$A*_wB=(A^{*n}*B)/<[u_{ij}^{(a)},v_{ab}]=0>$$
where $n$ is the size of $v$, with magic unitary matrix $w_{ia,jb}=u_{ij}^{(a)}v_{ab}$.
\end{definition}
In other words, $A*_wB$ is the universal $\mathbb C^*$-algebra generated by $n$ copies of $A$ and a copy of $B$, with the $a$-th copy of $A$ commuting with the $a$-th row of $v$, for any $a$. The Hopf $\mathbb C^*$-algebra structure on $A *_w B$
is the unique one making $w$ into a multiplicative matrix.
With this definition, we have the following result (\cite{bb}).
\begin{theorem}
If $X$ is connected we have $A(nX)=A(X)*_wA_s(n)$.
\end{theorem}
Note that the embedding $A(X)^{*n}\hookrightarrow A(X)*_wA_s(n)$ ensures
that $A(nX)$ is an infinite-dimensional algebra whenever
$n \geq 2$ and $G(X)$ is non trivial.
In the general case,
we have the following quantum analogue of the embedding result for $G(X)\wr G(Y)$.
\begin{proposition}
We have a surjective morphism of Hopf ${\mathbb C}^*$-algebras
$$A(X\circ Y) \longrightarrow A(X) *_w A(Y).$$
\end{proposition}
\begin{proof}
We use the canonical identification
$${\mathbb C}(X \times Y)={\mathbb C}(X) \otimes {\mathbb C}(Y)$$
given by $\delta_{(i,\alpha)}=\delta_i\otimes\delta_\alpha$. The adjacency matrix of $X\circ Y$ is
$$d_{X\circ Y} = d_X \otimes 1 + \mathbb I \otimes d_Y$$
where $\mathbb I$ is the square matrix filled with $1$'s.
Let $u,v$ be the magic unitary matrices of $A(X),A(Y)$. The magic unitary matrix of $A(X)*_wA(Y)$ is given by
$$w_{ia,jb}= u_{ij}^{(a)}v_{ab}$$
and from the fact that $u$ commutes with $d_X$ (and $\mathbb I$)
and $v$ commutes with $d_Y$, we get that $w$ commutes with $d_{X\circ Y}$. This gives a morphism as in the statement, and surjectivity follows by summing over $i$ and $b$.
\end{proof}
\begin{theorem}
Let $X,Y$ be regular graphs, with $X$ connected. If their spectra $\{\lambda_i\}$ and $\{\mu_j\}$ satisfy the condition
$$\{ \lambda_1-\lambda_i\mid i\neq 1 \} \cap \{-n\mu_j\} = \emptyset$$
where $n$ and $\lambda_1$ are the order and valence of $X$, then $A(X \circ Y)=A(X) *_w A(Y)$.
\end{theorem}
\begin{proof}
We denote by $P_i,Q_j$ the spectral projections corresponding to $\lambda_i,\mu_j$. Since $X$ is connected we have $P_1=\frac{1}{n}\,{\mathbb I}$, and we get:
\begin{eqnarray*}
d_{X\circ Y}
&=&d_X\otimes 1+{\mathbb I}\otimes d_Y\cr
&=&\left(\sum_i\lambda_iP_i\right)\otimes\left(\sum_jQ_j\right)+\left(nP_1\right)\otimes \left(\sum_i\mu_jQ_j\right)\cr
&=&\sum_j(\lambda_1+n\mu_j)(P_1 \otimes Q_j) + \sum_{i\not=1}\lambda_i (P_i\otimes 1)
\end{eqnarray*}
In this formula projections form a partition of unity and scalars are distinct, so this is the spectral decomposition of $d_{X\circ Y}$.
Let $W$ be the universal coaction on
$X\circ Y$. Then $W$ must commute with all spectral projections, and in particular:
$$[W,P_1 \otimes Q_j]=0$$
Summing over $j$ gives $[W, P_1 \otimes 1]=0$, so $1\otimes {\mathbb C}(Y)$ is invariant under the coaction. The corresponding restriction of $W$ gives a coaction of $A(X\circ Y)$
on $1 \otimes {\mathbb C}(Y)$, say
$$W(1 \otimes e_a) = \sum_b 1 \otimes e_b \otimes y_{ba}$$
where $y$ is a magic unitary. On the other hand we can write
$$W(e_i \otimes 1) = \sum_{jb} e_j \otimes e_b \otimes x_{ji}^b$$
and by multiplying by the previous relation we get:
\begin{eqnarray*}
W(e_i \otimes e_a)
&=&\sum_{jb} e_j \otimes e_b \otimes
y_{ba}x_{ji}^b\cr
&=& \sum_{jb} e_j \otimes e_b \otimes x_{ji}^b y_{ba}
\end{eqnarray*}
This shows that coefficients of $W$ have the following form:
$$W_{jb,ia} = y_{ba} x_{ji}^b=x_{ji}^b y_{ba}$$
Consider now the matrix $x^b=(x_{ij}^b)$. Since $W$ is a morphism of algebras, each row of $x^b$ is a partition of unity. Also using the antipode, we have
\begin{eqnarray*}
S\left(\sum_jx_{ji}^{b}\right)
&=&S\left(\sum_{ja}x_{ji}^{b}y_{ba}\right)\cr
&=&S\left(\sum_{ja}W_{jb,ia}\right)\cr
&=&\sum_{ja}W_{ia,jb}\cr
&=&\sum_{ja}x_{ij}^ay_{ab}\cr
&=&\sum_ay_{ab}\cr
&=&1
\end{eqnarray*}
so we conclude that $x^b$ is a magic unitary.
We check now that $x^a,y$ commute with $d_X,d_Y$. We have
$$(d_{X\circ Y})_{ia,jb} = (d_X)_{ij}\delta_{ab} + (d_Y)_{ab}$$
so the two products between $W$ and $d_{X\circ Y}$ are given by:
$$(Wd_{X\circ Y})_{ia,kc}=\sum_j W_{ia,jc} (d_X)_{jk} + \sum_{jb}W_{ia,jb}(d_Y)_{bc}$$
$$(d_{X\circ Y}W)_{ia,kc}=\sum_j (d_X)_{ij} W_{ja,kc} + \sum_{jb}(d_Y)_{ab}W_{jb,kc}$$
Now since $W$ commutes with $d_{X\circ Y}$, the terms on the right are equal, and by summing over $c$ we get:
$$\sum_j x_{ij}^a(d_X)_{jk} + \sum_{cb} y_{ab}(d_Y)_{bc}
= \sum_{j} (d_X)_{ij}x_{jk}^a + \sum_{cb} (d_Y)_{ab}y_{bc}$$
The graph $Y$ being regular, the second sums in both terms are equal to the valency of $Y$, so we get $[x^a,d_X]=0$.
Now once again from the formula coming from commutation of $W$ with $d_{X\circ Y}$, we get $[y,d_Y] =0$.
Summing up, the coefficients of $W$ are of the form
$$W_{jb,ia}=x_{ji}^by_{ba}$$
where $x^b$ are magic unitaries commuting with $d_X$, and $y$ is a magic unitary commuting with $d_Y$. This gives a morphism
$$A(X)*_wA(Y) \longrightarrow A(X\circ Y)$$
mapping $u_{ji}^{(b)}\to x_{ji}^b$
and $v_{ba}\to y_{ba}$, which is inverse to the morphism in the previous proposition.
\end{proof}
\begin{corollary}
We have $A(C_{10}(4))= {\mathbb C}({\mathbb Z}_2)*_w{\mathbb C}(D_5)$.
\end{corollary}
\begin{proof}
We have isomorphisms
$$C_{10}(4)=C_{10}(4,5)^c=K_2\circ C_5$$
and Theorem 6.2 applies to the product on the right.
\end{proof}
Together with Theorem 6.1, this corollary gives the following table,
where ${\,\wr_*\,}$ is defined by ${\mathbb C}(G{\,\wr_*\,}H)={\mathbb C}(G)*_w{\mathbb C}(H)$.
\begin{center}\begin{tabular}[t]{|l|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
4&$2K_2$&${\mathbb Z}_2\wr{\mathbb Z}_2$&${{\mathbb Z}_2}{\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
6&$2K_3$&${S_3}\wr{{\mathbb Z}_2}$&${S_3}{\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
6&$3K_2$&${{\mathbb Z}_2}\wr{S_3}$&${{\mathbb Z}_2}{\,\wr_*\,}{S_3}$\\
\hline
8&$2K_4$&$S_4\wr{{\mathbb Z}_2}$&$\mathcal Q_4{\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
8&$2C_4$&$({\mathbb Z}_2\wr{\mathbb Z}_2)\wr{{\mathbb Z}_2}$&$({\mathbb Z}_2{\,\wr_*\,}{\mathbb Z}_2){\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
8&$4K_2$&${{\mathbb Z}_2}\wr S_4$&${{\mathbb Z}_2}{\,\wr_*\,}\mathcal Q_4$\\
\hline
9&$3K_3$&${S_3}\wr{S_3}$&${S_3}{\,\wr_*\,}{S_3}$\\
\hline
10&$2C_5$&$D_5\wr{{\mathbb Z}_2}$&$D_5{\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
10&$2K_5$&$S_5\wr{{\mathbb Z}_2}$&$\mathcal Q_5{\,\wr_*\,}{{\mathbb Z}_2}$\\
\hline
10&$5K_2$&${{\mathbb Z}_2}\wr S_5$&${{\mathbb Z}_2}{\,\wr_*\,}\mathcal Q_5$\\
\hline
10&$C_{10}(4)$&${{\mathbb Z}_2}\wr D_5$&${{\mathbb Z}_2}{\,\wr_*\,} D_5$\\
\hline
\end{tabular}\end{center}
\medskip
\section{Classification table}
We are now in position of writing down a big table. We first recall the graph notations used in the paper.
\begin{definition}
We use the following notations.
\begin{enumerate}
\item Basic graphs:
- the complete graph having $n$ vertices is denoted $K_n$.
- the disjoint union of $n$ copies of $X$ is denoted $nX$.
- the prism having basis $X$ is denoted $\mathtt{Pr}(X)$.
\item Circulant graphs:
- the $n$-cycle is denoted $C_n$.
- the $2n$-cycle with diagonals is denoted $C_{2n}^+$.
- the $n$-cycle with chords of length $k$ is denoted $C_n(k)$.
\item Special graphs:
- the triangle times itself is denoted $\mathtt{Torus}$.
- the Petersen graph is denoted $\mathtt{Petersen}$.
\end{enumerate}
\end{definition}
As for quantum group notations, these have to be taken with care, because quantum groups do not really exist etc. Here they are.
\begin{definition}
We use the following notations.
- ${\mathbb Z}_n,D_n,S_n$ are the cyclic, dihedral and symmetric groups.
- $\mathcal Q_n$ is the quantum permutation group.
- $\times,\wr,{\,\wr_*\,}$ are the product, wreath product and free wreath product.
\end{definition}
The vertex-transitive graphs of order less than 11, modulo complementation, are given by the following table.
\vfill\eject
\begin{center}\begin{tabular}[t]{|l|l|l|l|}
\hline
Order & Graph & Classical group & Quantum group\\
\hline \hline
2&$K_2$&$ {{\mathbb Z}_2}$&$ {{\mathbb Z}_2}$\\
\hline\hline
3&$K_3$&${{S_3}}$&${{S_3}}$\\
\hline\hline
4 & $2K_2$& ${\mathbb Z}_2\wr{\mathbb Z}_2$ & ${{\mathbb Z}_2}{\,\wr_*\,} {{\mathbb Z}_2}$ \\
\hline
4 & $K_4$ & $S_4$ & $\mathcal Q_4$ \\
\hline
\hline
5 & $C_5$ & $D_5$ & $D_5$ \\
\hline
5 & $K_5$ & $S_5$ & $\mathcal Q_5$\\
\hline \hline
6 & $C_6$ & $D_6$ & $D_6$ \\
\hline
6 & $2K_3$ & ${{S_3}}\wr{{{\mathbb Z}_2}}$ & ${{S_3}}{\,\wr_*\,}{{{\mathbb Z}_2}}$ \\
\hline
6 & $3K_2$ & ${{{\mathbb Z}_2}}\wr {{S_3}}$ & ${{{\mathbb Z}_2}}{\,\wr_*\,} {{S_3}}$ \\
\hline
6 & $K_6$ & $S_6$ & $\mathcal Q_6$ \\
\hline \hline
7 & $C_7$ & $D_7$ & $D_7$ \\
\hline
7 & $K_7$ & $S_7$ & $\mathcal Q_7$\\
\hline \hline
8 & $C_8$, $C_8^+$& $D_8$ & $D_8$\\
\hline
8 & $\texttt{Pr}(C_4)$ &
$S_4 \times {{{\mathbb Z}_2}}$ & $\mathcal Q_4\times{{{\mathbb Z}_2}}$ \\
\hline
8 & $2K_4$ & $S_4\wr{{{\mathbb Z}_2}}$ & $\mathcal Q_4{\,\wr_*\,}{{{\mathbb Z}_2}}$ \\
\hline
8 & $2C_4$& $({\mathbb Z}_2\wr{\mathbb Z}_2)\wr{{{\mathbb Z}_2}}$
& $({{{\mathbb Z}_2}}{\,\wr_*\,}{{{\mathbb Z}_2}}){\,\wr_*\,}{{{\mathbb Z}_2}}$ \\
\hline
8 & $4K_2$& ${{{\mathbb Z}_2}}\wr S_4$ & ${{{\mathbb Z}_2}}{\,\wr_*\,} \mathcal Q_4$ \\
\hline
8 & $K_8$ & $S_8$ & $\mathcal Q_8$ \\
\hline \hline
9 & $C_9$, $C_9(3)$ & $D_9$ & $D_9$ \\
\hline
9 & $\texttt{Torus}$& ${{S_3}}\wr{{{\mathbb Z}_2}}$ & ${{S_3}}\wr{{{\mathbb Z}_2}}$ \\
\hline
9 & $3K_3$ & ${{S_3}}\wr {{S_3}}$ & ${{S_3}}{\,\wr_*\,} {{S_3}}$ \\
\hline
9 & $K_9$ & $S_9$ & $\mathcal Q_9$ \\
\hline \hline
10 & $C_{10}$, $C_{10}(2)$, $C_{10}^+$, $\texttt{Pr}(C_5)$ & $D_{10}$ & $D_{10}$\\
\hline
10 & $\texttt{Petersen}$ & $S_5$ & $?$ \\
\hline
10 & $\texttt{Pr}(K_5)$ & $S_5 \times {{{\mathbb Z}_2}}$ &
$\mathcal Q_5\times{{{\mathbb Z}_2}}$ \\
\hline
10 & $C_{10}(4)$& ${{\mathbb Z}_2}\wr D_5$ & ${{\mathbb Z}_2}{\,\wr_*\,} D_5$\\
\hline
10 & $2C_5$ & $D_5\wr{{{\mathbb Z}_2}}$ & $D_5{\,\wr_*\,}{{{\mathbb Z}_2}}$ \\
\hline
10 & $2K_{5}$ & $S_5\wr{{{\mathbb Z}_2}}$ & $\mathcal Q_5{\,\wr_*\,}{{{\mathbb Z}_2}}$\\
\hline
10 & $5K_2$ & ${{{\mathbb Z}_2}}\wr S_5$ & ${{{\mathbb Z}_2}}{\,\wr_*\,}\mathcal Q_5$ \\
\hline
10 & $K_{10}$ & $S_{10}$ & $\mathcal Q_{10}$ \\
\hline \hline
11 & $C_{11}$, $C_{11}(2)$, $C_{11}(3)$& $D_{11}$ & $D_{11}$ \\
\hline
11 & $K_{11}$ & $S_{11}$ & $\mathcal Q_{11}$ \\
\hline
\end{tabular}\end{center}
\vfill\eject
Here the first three columns are well-known, and can be found in various books or websites. The last one collects results in this paper.
By using the equality $D_n={\mathbb Z}_n\rtimes {\mathbb Z}_2$, we reach the conclusion in the abstract: with one possible
exception, all quantum groups in the right column can be obtained from ${\mathbb Z}_n,S_n,\mathcal Q_n$ by using the operations $\times,\rtimes,\wr,{\,\wr_*\,}$.
The exceptional situation is that of the Petersen graph,
which might give a new quantum group.
We discuss it in the next section.
\section{The Petersen graph}
The techniques of the previous sections do not apply to the Petersen graph,
which is not a circulant graph and cannot be written as a graph product.
Also we could not carry a direct analysis similar to the one of the torus
because of the complexity of some computations.
The usual symmetry group is $S_5$, so
in view of the results in our classification table, we have at least two natural
candidates for the quantum symmetry group of the Petersen graph: $S_5$ and $\mathcal Q_5$.
\begin{theorem}
The quantum automorphism group of the Petersen graph has an irreducible
5-dimensional representation. In particular it is not
isomorphic to the quantum symmetric group $\mathcal Q_5$.
\end{theorem}
\begin{proof}
Let $G$ be the quantum automorphism group of the Petersen graph, denoted here $\texttt{P}$. We have an inclusion $S_5 \subset G$. It is well-known
that
$${\rm Sp}(\mathtt{P}) = \{4, -2 , 1 \}$$
and that the corresponding eigenspaces have dimensions
$1,4,5$. These eigenspaces furnish representations of
$G$ and of $S_5$. It is straightforward to compute the character
of the permutation representation of $S_5$ on $\mathbb C(\texttt{P})$, and then
using the character table of $S_5$
(see e.g. \cite{fh}), we see
that $\mathbb C(\texttt{P})$ is the direct sum of
$3$ irreducible representations of $S_5$. These have to be the previous
eigenspaces, and in particular the $5$-dimensional one is an irreducible
representation of $S_5$, and of $G$.
On the other hand, it is known from \cite{ba0}
that $\mathcal Q_5$ has no irreducible 5-dimensional representation. Thus the quantum groups $G$ and $\mathcal Q_5$ are not
isomorphic.
\end{proof}
The question now is: does the Petersen graph have quantum symmetry?
In other words, is $A(\texttt{P})$ commutative?
The above result seems to indicate that if $A(\texttt{P})$ is not commutative,
we probably will have a new quantum permutation group.
|
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{"url":"http:\/\/gmatclub.com\/forum\/geometry-ii-q3-q9-86884.html?fl=similar","text":"Find all School-related info fast with the new School-Specific MBA Forum\n\n It is currently 26 Aug 2016, 05:29\n\nGMAT Club Daily Prep\n\nThank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\nEvents & Promotions\n\nEvents & Promotions in June\nOpen Detailed Calendar\n\nGeometry - II q3 & q9\n\nAuthor Message\nCurrent Student\nJoined: 13 Jul 2009\nPosts: 145\nLocation: Barcelona\nSchools: SSE\nFollowers: 17\n\nKudos [?]: 363 [0], given: 22\n\nGeometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n15 Nov 2009, 11:17\nQuote:\nIf points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?\n\n1. $$AB^2 = BC^2 + AC^2$$\n2. $$\\angle CAB$$ equals 30 degrees\n\n(C) 2008 GMAT Club - Geometry - II#3\n\n* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\n* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\n* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\n* EACH statement ALONE is sufficient\n* Statements (1) and (2) TOGETHER are NOT sufficient\nStatements(1) and (2) combined are sufficient. S1 and S2 define the shape of the triangle. Because this triangle can be inscribed in a circle of radius 1, we also know the size of the triangle. There is only one triangle with a given shape and size.\n\nI understand that there's only one possible triangle with that given shape, but how can you know the area?\n\nAlso:\n\nQuote:\nIf the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?\n\n1. The perimeter of the rectangle is 84\n2. The shorter side of the rectangle is 2\n\n(C) 2008 GMAT Club - Geometry - II#9\n\n* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\n* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\n* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\n* EACH statement ALONE is sufficient\n* Statements (1) and (2) TOGETHER are NOT sufficient\n\nStatement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nStatement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nWe know the sides but what about the angle between the diagonal and the longer side?\n_________________\n\nPerformance: Gmat | Toefl\nContributions: The Idioms Test | All Geometry Formulas | Friendly Error Log | GMAT Analytics\nMSc in Management: All you need to know | Student Lifestyle | Class Profiles\n\nManager\nJoined: 13 Oct 2009\nPosts: 115\nLocation: USA\nSchools: IU KSB\nFollowers: 4\n\nKudos [?]: 73 [0], given: 66\n\nRe: Geometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n15 Nov 2009, 13:36\nIf points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?\n\n1. $$AB^2 = BC^2 + AC^2$$\n2. $$\\angle CAB$$ equals 30 degrees\n\n(C) 2008 GMAT Club - Geometry - II#3\n\n* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\n* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\n* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\n* EACH statement ALONE is sufficient\n* Statements (1) and (2) TOGETHER are NOT sufficient\nStatements(1) and (2) combined are sufficient. S1 and S2 define the shape of the triangle. Because this triangle can be inscribed in a circle of radius 1, we also know the size of the triangle. There is only one triangle with a given shape and size.\n\nI understand that there's only one possible triangle with that given shape, but how can you know the area?\n\nStatement 1 suggests that AB is the diameter of the circle but doesn't specifically tells about the position of C. So area can not be determined. Not Sufficient.\n\nStatement 2 : SUFFIECIENT\n\nAlso:\n\nIf the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?\n\n1. The perimeter of the rectangle is 84\n2. The shorter side of the rectangle is 2\n\n(C) 2008 GMAT Club - Geometry - II#9\n\n* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\n* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\n* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\n* EACH statement ALONE is sufficient\n* Statements (1) and (2) TOGETHER are NOT sufficient\n\nStatement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nStatement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nWe know the sides but what about the angle between the diagonal and the longer side?\n\nAnswer D. If you know the sides of a right angle triangle then angle between any of the two sides can be calculated. Since this is a DS question you don't really have to calculate it.\nSVP\nJoined: 29 Aug 2007\nPosts: 2492\nFollowers: 66\n\nKudos [?]: 692 [0], given: 19\n\nRe: Geometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n15 Nov 2009, 14:14\nsaruba wrote:\nQuote:\nIf points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?\n\n1. $$AB^2 = BC^2 + AC^2$$\n2. $$\\angle CAB$$ equals 30 degrees\n\n1 says that the triangle si a right angle triangle.\n2 says one of the size of the angle is 30 degree.\n\n1 and 2 say taht the triangle right nagle traingle with angel sizes 30:60:90. Such triangle is possible only with the hypoteneous as diameter. So:\n\nh = d = 2\nb (or l) = 1\nl (or b) = sqrt3.......Suff\n\nThats C.\n_________________\n\nGmat: http:\/\/gmatclub.com\/forum\/everything-you-need-to-prepare-for-the-gmat-revised-77983.html\n\nGT\n\nSVP\nJoined: 29 Aug 2007\nPosts: 2492\nFollowers: 66\n\nKudos [?]: 692 [0], given: 19\n\nRe: Geometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n15 Nov 2009, 14:25\nsaruba wrote:\nIf the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?\n\n1. The perimeter of the rectangle is 84\n2. The shorter side of the rectangle is 2\n\nStatement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nStatement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).\n\nWe know the sides but what about the angle between the diagonal and the longer side?[\/quote]\n\nGiven that lb = 80\n\n1: 2(l+b) =84\nl+b = 42\n80\/b + b = 42\nl^2 - 42l + 80 = 0\nl (l-40) - 40 (l - 40) = 0\n(l-40)^2\nl = 40\nb = 2\n\nWith the values of l anf b, we can identify the size of the angle between diagnal and longer side.\n\nHow? Need to use triagomatic approach. Our concern is taht if we know the sides of the rectangle, then we can get the size of the desired angle. Suff...\n\n2. If the shorter side of the rectangle is 2, then longer si 40. Suff.\n\nThats D...\n_________________\n\nGmat: http:\/\/gmatclub.com\/forum\/everything-you-need-to-prepare-for-the-gmat-revised-77983.html\n\nGT\n\nIntern\nJoined: 27 Sep 2009\nPosts: 6\nFollowers: 0\n\nKudos [?]: 0 [0], given: 1\n\nRe: Geometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n29 May 2011, 09:18\nif $$AB^2 = BC^2 + AC^2$$\n\nis not means that AB is a hypotenuse ?? if yes doesn't it represent the diameter of the circle?\nI was thinking that A should be the answer.\ncan any please correct me .\nMath Forum Moderator\nJoined: 20 Dec 2010\nPosts: 2021\nFollowers: 157\n\nKudos [?]: 1558 [0], given: 376\n\nRe: Geometry - II q3 & q9\u00a0[#permalink]\n\nShow Tags\n\n29 May 2011, 10:35\npjain01 wrote:\nif $$AB^2 = BC^2 + AC^2$$\n\nis not means that AB is a hypotenuse ??\nYes. AB is a hypotenuse.\n\nif yes doesn't it represent the diameter of the circle?\nYes. It does.\n\nI was thinking that A should be the answer.\nMany right triangles can be formed with hypotenuse=2. Draw a circle. Considering AB as diameter, and C as any point on the circle, try drawing multiple triangles. Each of these triangles will be right angled triangle with different areas. Thus, this statement is not sufficient to find the area of the triangle.\n\ncan any please correct me .\n\nAlso discussed here:\narea-of-triangle-inside-a-circle-111134.html\n_________________\nRe: Geometry - II q3 & q9 \u00a0 [#permalink] 29 May 2011, 10:35\nSimilar topics Replies Last post\nSimilar\nTopics:\n1 Number properties Q9 2 02 Jan 2010, 08:25\n9 Geometry II = #12 11 05 Aug 2009, 18:41\n13 m04 q9 23 20 Dec 2008, 19:09\nM19 Q9 5 12 Nov 2008, 21:38\n46 M06 Q9 20 12 Nov 2008, 14:52\nDisplay posts from previous: Sort by\n\nGeometry - II q3 & q9\n\nModerator: Bunuel\n\n Powered by phpBB \u00a9 phpBB Group and phpBB SEO Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2016-08-26 12:29:47","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\": 0.40501517057418823, \"perplexity\": 1482.914928015427}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982295494.5\/warc\/CC-MAIN-20160823195815-00062-ip-10-153-172-175.ec2.internal.warc.gz\"}"}
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Srednje (cyr. Средње) – wieś w Bośni i Hercegowinie, w Republice Serbskiej, w mieście Sarajewo Wschodnie, w gminie Pale. W 2013 roku liczyła 3 mieszkańców.
Przypisy
Miejscowości w gminie Pale
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{"url":"https:\/\/pennylane.readthedocs.io\/en\/latest\/code\/api\/pennylane.qchem.excitations.html","text":"# qml.qchem.excitations\u00b6\n\nexcitations(electrons, orbitals, delta_sz=0)[source]\n\nGenerate single and double excitations from a Hartree-Fock reference state.\n\nSingle and double excitations can be generated by acting with the operators $$\\hat T_1$$ and $$\\hat T_2$$ on the Hartree-Fock reference state:\n\n$\\begin{split}&& \\hat{T}_1 = \\sum_{r \\in \\mathrm{occ} \\\\ p \\in \\mathrm{unocc}} \\hat{c}_p^\\dagger \\hat{c}_r \\\\ && \\hat{T}_2 = \\sum_{r>s \\in \\mathrm{occ} \\\\ p>q \\in \\mathrm{unocc}} \\hat{c}_p^\\dagger \\hat{c}_q^\\dagger \\hat{c}_r \\hat{c}_s.\\end{split}$\n\nIn the equations above the indices $$r, s$$ and $$p, q$$ run over the occupied (occ) and unoccupied (unocc) spin orbitals and $$\\hat c$$ and $$\\hat c^\\dagger$$ are the electron annihilation and creation operators, respectively.\n\nParameters\n\u2022 electrons (int) \u2013 Number of electrons. If an active space is defined, this is the number of active electrons.\n\n\u2022 orbitals (int) \u2013 Number of spin orbitals. If an active space is defined, this is the number of active spin-orbitals.\n\n\u2022 delta_sz (int) \u2013 Specifies the selection rules sz[p] - sz[r] = delta_sz and sz[p] + sz[p] - sz[r] - sz[s] = delta_sz for the spin-projection sz of the orbitals involved in the single and double excitations, respectively. delta_sz can take the values $$0$$, $$\\pm 1$$ and $$\\pm 2$$.\n\nReturns\n\nlists with the indices of the spin orbitals involved in the single and double excitations\n\nReturn type\n\ntuple(list, list)\n\nExample\n\n>>> electrons = 2\n>>> orbitals = 4\n>>> singles, doubles = excitations(electrons, orbitals)\n>>> print(singles)\n[[0, 2], [1, 3]]\n>>> print(doubles)\n[[0, 1, 2, 3]]\n\n\nUsing PennyLane\n\nDevelopment\n\nAPI","date":"2022-06-25 13:57:48","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\": 0.40842747688293457, \"perplexity\": 2921.874474123475}, \"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\/1656103035636.10\/warc\/CC-MAIN-20220625125944-20220625155944-00192.warc.gz\"}"}
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{"url":"http:\/\/www2.macaulay2.com\/Macaulay2\/doc\/Macaulay2-1.19\/share\/doc\/Macaulay2\/RelativeCanonicalResolution\/html\/_line__Bundle__From__Points__And__Multipliers.html","text":"lineBundleFromPointsAndMultipliers -- Computes basis of a line bundle from the 2g points P_i, Q_i and the multipliers\n\nSynopsis\n\n\u2022 Usage:\nlineBundleFromPointsAndMultipliers(multL,P,Q,k)\n\u2022 Inputs:\n\u2022 multL, a list, containing the normalized multipliers a_i (normalized means that b_i=1)\n\u2022 P, , the coordinate matrix of g points in P^1\n\u2022 Q, , the coordinate matrix of g points in P^1\n\u2022 k, an integer, defining the degree of the line bundle\n\u2022 Outputs:\n\u2022 f, , a basis of sections of the line bundle defined by the points P_i, Q_i and the multipliers\n\nDescription\n\nIf C is a g-nodal canonical curve with normalization $\\nu:\\ P^1 \\to P^{g-1}$ then a line bundle L of degree k on C is given by $\\nu^*(O_{P^1}(k))\\cong L$ and gluing data $\\frac{b_j}{a_j}:O_{P^1}\\otimes kk(P_j)\\to O_{P^1}\\otimes kk(Q_j)$. Given 2g points P_i, Q_i and the multipliers (a_i,b_i) we can compute a basis of sections of L as a kernel of the matrix $A=(A)_{ij}$ with $A_{ij}=b_iB_j(P_i)-a_iB_j(Q_i)$ where $B_j:P^1\\to kk,\\ (p_0:p_1)\\to p_0^{k-j}p_1^j$.","date":"2023-02-01 16:09:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9528772830963135, \"perplexity\": 1408.8420180383043}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499946.80\/warc\/CC-MAIN-20230201144459-20230201174459-00132.warc.gz\"}"}
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Honkia specializes in Rapid Prototyping and Low Volume Production. Taking advantages of the labor cost in mainland China and expertise in product prototyping to production, we offer rapid custom manufacturing services to global customers economically and efficiently.
We not only make prototypes and parts, also help and work with customers in right direction to save money and time. Our sales engineers have strong engineering background and excellent English capability to work with customers. Our rapid manufacturing services include: CNC Machining, SLA/SLS Rapid Prototyping, Vacuum Casting, Plastic Injection Molding, Reaction Injection Molding, Sheet Metal Fabrication, Surface Finishing.
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Q: Is it possible to detect position (footer, sidebar) of HTML element when parsing document? I'm trying to develop a script (in php) that anlayzes web pages and wondering if it's possible to detect the position of an image or a link on a page, whether it's located on the a) top/bottom of the page or b) on the left/right side and whether top half/bottom half of the page. I was wondering if there's any accurate method of parsing the HTML and detecting accurately the position of these elements as they would be displayed on the page.
A: There is no fool prove method. Cos different browsers will render differently on different screens. This is a procedure best suited to using javascript running on the client browser.
What is the reason you need the position on the servers for?
A few ways to implement it (off the top of my head)...
1) Spawn IE or firefox (or any browser) in the background, to get the location using javascript that will then pass it to the php scripts?
2) Download and compile the website library and write a php module (plugin) to handle this?
3) Have the position be calculated in javascript on the client end, and use AJAX to pass the position back to the server?
4) Redo the web page, so that we don need to care about the position?
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\section{ Introduction}\label{s:intro}
Multipole expansions are used throughout QM, advanced MM, and large-scale electrostatics calculations. They appear as derivatives of $r^{-1}$ in the Green's function solution to Poisson's equation. Applications include the computation of angular interaction matrix elements (themselves based on spherical harmonic functions) in quantum codes,\cite{gfere97,dchan08,gbelk13} fast $O(N)$ implicit solvent calculations in molecular dynamics\cite{abord03,mschn07,mschn07b}
and fluid flows,\cite{gdass02,fcruz11} describing molecular polarization\cite{stone,bjezi94,pren03,ckram13,ranan13}, and boundary representations
of dielectric polarization in molecular cavities.\cite{jappl89,ecanc97,jbard11,jbard12}
Despite their wide-ranging applications, actual use of these expansions is hindered
by the tenuous connections between spectral (spherical harmonic) and
real-space (Cartesian) formalisms.\cite{bjezi94}
Essentially all serious applications make use of
manually tabulated Cartesian forms, often implemented by hand.\cite{jappl89,asawa89,cpozr92,gdass02,pren03,sturz11,klore12}
This strategy is error-prone, time-consuming and useful only for low-order modes.
This leads, for example, to widely-used and highly-regarded quantum chemistry
codes that only support angular basis functions up to order 6,\cite{nwchem}
and simulation methods and codes for which even implementations at the
dipole\cite{atouk00} or quadrupole\cite{pren02,dcase05} level are great advancements.
In addition, there is a gap in understanding translations from harmonics
to representations relying completely on
point-charges,\cite{ranan13} or on explicit supersymmetric tensors.\cite{pren02,ljaco09}
The straightforward route for a point-charge
representation of multipoles up to order $p-1$
requires $2^p$ charges,\cite{asawa89,dchan08}
while a tensor representation requires $3^p$ tensor elements.
In this work we show space-optimal representations for point-charges and Cartesian tensors,
and provide simple translations between these three formalisms.
The ubiquitous success of spherical harmonics stems from the
separation of source and destination points ($x$ and $y$) in the expansion
of the potential function,
\begin{align}
\frac{1}{|y-x|} &= \sum_{l=0}^\infty \sum_{m=-l}^l O_{lm}(y) M_{lm}(x) , \quad |y| < |x| \\
\intertext{where $x$ and $y$ are three-dimensional vectors, and}
O_{lm}(x) &= |x|^l (l+|m|)! P_{lm}(\cos \theta_x) e^{-i m \phi_x} \label{e:corresp} \\
M_{lm}(x) &= |x|^{-l-1} (l-|m|)!^{-1} P_{lm}(\cos \theta_x) e^{i m \phi_x}
\end{align}
This simplified form is due to Ref.~\cite{cwhit94}.
Here, $i$, is the unit imaginary number and $P_{lm}$ are the associated Legendre polynomials.
We contend that for almost all real applications, this decomposition goes one step too far.
Rather, it is preferable to stop at the unadorned Legendre polynomials,
\begin{equation}
P_l(\hat x \cdot \hat y) = \sum_{m=-l}^l O_{lm}(\hat x) M_{lm}(\hat y)
,
\end{equation}
where $\hat x$ represents a three-dimensional unit vector, $|\hat x| = 1$.
The $P_l$ are polynomials of degree $l$ in the vector inner-product, $\hat x\cdot \hat y$,
and are obviously symmetric to interchanging $x$ and $y$.
No angles need to be defined or used.
In preference to the pair, $O$ and $M$, we then use scaled Legendre
polynomials,
\begin{equation}
L_n(x,y) \equiv \frac{|x|^n}{|y|^{n+1}} P_n(\hat x \cdot \hat y)
\label{e:harm}
.
\end{equation}
When scaled by $|y|^{2n+1}$, the $L_n$ are polynomials in the inner products,
$x\cdot x$, $y\cdot y$, and $x\cdot y$, symmetric to interchange of $x$ and $y$.
They can be generated by a two-term recurrence [since $L_n(x,y) = F_n(x,y; -1)$],
\begin{align}
F_n(x,y;\alpha) &\equiv (-x\cdot\partial_y)^n |y|^\alpha/n! \label{e:rec} \\
&= |y|^\alpha, \quad n=0 \notag \\
&= -\alpha \frac{x\cdot y}{y\cdot y} |y|^\alpha, \quad n=1 \notag \\
&= \frac{2n-2-\alpha}{n} \frac{x\cdot y}{y\cdot y} F_{n-1}(x,y;\alpha) \\
& \;\; + \frac{\alpha+2-n}{n} \frac{x\cdot x}{y\cdot y} F_{n-2}(x,y;\alpha)
.
\end{align}
They express successive derivatives of $|y|^{-1}$, so that
\begin{align}
\frac{1}{|x-y|} = e^{-x\cdot\partial_y} |y|^{-1} = \sum_{n=0}^\infty L_n(x,y) \label{e:Pexp}
.
\end{align}
\subsection{ Definition of the Quadrature Representation}
Identities previously phrased in terms of harmonics find a much simpler expression
using the reproducing kernel (Fig.~\ref{f:K}),\cite{cahre09}
\begin{equation}
K(x,y) = \sum_{n=0}^{p-1} \frac{2n+1}{4\pi} L_n(x,y) \label{e:K}
.
\end{equation}
Its status as an effective identity kernel can be shown using the orthogonality
of the Legendre polynomials on the surface of the unit sphere ($S$),\cite{cahre09}
\begin{equation}
\int_S P_n(\hat x \cdot \hat r) P_m(\hat r \cdot \hat y) \; d^2\hat r = \delta_{nm} \frac{4\pi}{2 n + 1} P_n(\hat x\cdot \hat y)
.\label{e:ortho}
\end{equation}
As $\hat y$ is varied, the set of Legendre polynomials up to order $p-1$ span the set of all $\sum_{n=0}^{p-1} 2n+1 = p^2$ polynomials in $\hat x$ from degrees $0$ through
$p-1$ on the unit sphere.\cite{cahre09
\begin{figure}
\includegraphics[width=0.45\textwidth]{geom.pdf}
\caption{Illustration of quadrature representation of the multipole expansion. The source at
position $y$ is transferred to the quadrature points, $R\hat r_i$ (using Eq.~\ref{e:rep}).
Its potential inside the sphere is calculated from those effective charges (Eq.~\ref{e:Pexp}).
Normal vector directions (pointing out of the sphere) are defined for boundary
integrals.}\label{f:K}
\end{figure}
Because $K$ generates polynomials, functions on $S$ can be represented
in terms of their values at a small set of quadrature points.
We make use of standard quadrature formulas prescribing
a set of $N \sim \tfrac{3}{2}p^2$ roots, $\hat r_i$, and associated weights, $w^0_i$, capable
of integrating all polynomials up to order $2p-2$.\cite{cahre09}
Using such a quadrature set, for polynomial functions $\sigma$ of degree less than $p$,
\begin{equation}
\sigma(\hat x) = \int K(\hat x, \hat y) \sigma(\hat y) \; dy
= \sum_{i=1}^N w^0_i \sigma(\hat r_i) K(\hat x, \hat r_i) .
\label{e:rep}
\end{equation}
We will see that the coefficients $w^0_i \sigma(\hat r_i)$ act as the effective
charges on the spherical surface.
The numerical computations presented in Sections~\ref{s:fmm} and~\ref{s:sphere} make use of
the Lebedev quadrature set.\cite{vlebe77}
The choice of scale with $|x|$ and $|y|$ in Eq.~\ref{e:K} prescribes a default behavior off the
surface of the unit sphere coinciding with that of Ref.~\citenum{cwhit94}:
\begin{align}
\int_S K(x,\hat y) O_{lm}(\hat y) \; d^2\hat y &= O_{lm}(x) \\
\int_S M_{lm}(\hat x) K(\hat x, y) \; d^2\hat x &= M_{lm}(y)
.
\end{align}
These formulas can be used to directly translate formulas using spherical harmonics
back into Legendre polynomials, as is done in Appendix~\ref{s:harm}.
Each choice of the coordinate axes for spherical harmonics
of order $n$ generates $2n+1$ harmonics, which correspond to
a particular basis for the $2n+1$ degenerate eigenfunctions of
$P_n(r_i\cdot r_j)$. According to Eq.~\ref{e:ortho}, each of the
$2n+1$ vectors has eigenvalue $4\pi/(2n+1)$. Working directly
with $P_n$ avoids decomposition to this basis, removing angular
variables from the resulting expressions.
The required separation of source and destination points ($x$ and $y$
in Fig.~\ref{f:K}) is easily achieved using the scaled reproducing kernel,
\begin{align}
(-y&\cdot\partial_x)^n |x|^{-1}/n! = L_n(y,x) \notag \\
&= \int_S K(y/R, \hat r) L_n(R \hat r, x) \; d^2\hat r \label{e:outer2} \\
\intertext{or}
&= \int_S L_n(y, R \hat r) K(\hat r, x/R) \; d^2\hat r \label{e:inner2}
.
\end{align}
Equation~\ref{e:outer2} generates the {\em outer} expansion by collecting the
integration over sources inside the bounding sphere ($|y| < R$) into an effective
surface charge distribution,
\begin{align}
\sigma_o(R \hat r) &\equiv \int \rho(y) K(y/R, \hat r) \; d^3y
. \label{e:outer} \\
\intertext{The source's $n$-th order contribution to the potential at point $x$ is then}
\int L_n(y,x) \rho(y) \; d^3y &= \int_S L_n(R\hat r, x) \sigma_o(R\hat r) \; d^2\hat r
. \label{e:Gouter} \\
\intertext{Likewise, Eq.~\ref{e:inner2} generates the {\em inner} expansion when collecting
the integration over sources outside the bounding sphere, $|x| > R$,}
\sigma_i(R \hat r) &\equiv \int \rho(x) K(\hat r, x/R) \; d^3y
. \label{e:inner} \\
\intertext{with the $n$-th order potential inside the sphere given by}
\int L_n(x,y) \rho(x) \; d^3x &= \int_S L_n(x, R \hat r) \sigma_i(R \hat r) \; d^2\hat r
. \label{e:Ginner}
\end{align}
\subsection{ Energy in the Quadrature Representation}
The interaction energy between two sets of point sources ($y$ inside and $x$
outside of a sphere of radius $R$) is written in a multipole expansion as
\begin{equation}
E = \sum_{x,y} \frac{q_x q_y}{|x - y|} = \frac{1}{R} \sum_{x,y,n=0} q_x q_y L_n(y/R,x/R)
\end{equation}
Using both the outer expansion projecting the enclosed charges
onto a surrounding sphere ($L_n(y,x) \to \int_S K(y/R,\hat r) L_n(R\hat r, x) \; d^2\hat r$, Eq.~\ref{e:outer2}),
and the inner expansion projecting the outer charges onto the same sphere
($L_n(R\hat r, x) \to \int L_n(R\hat r, R\hat s) K(\hat s, x/R) d^2\hat s$, Eq.~\ref{e:inner2}),
\begin{align}
E &= \frac{1}{R} \sum_{x,y,i,j,n=0} L_n(\hat r_i, \hat r_j) w^0_i w^0_j \notag \\
& \qquad \times q_y K(y/R,\hat r_i) q_x K(\hat r_j, x/R) \notag \\
&= \frac{1}{R} \sum_{i,j,n=0} w^0_i \sigma_o(R\hat r_i) w^0_j \sigma_i(R\hat r_j) L_n(\hat r_i, \hat r_j)
.
\end{align}
The second step uses the definitions in Eq.~\ref{e:outer} and Eq~\ref{e:inner}.
In most cases, expressions involving the potential are simpler.
In addition, when $R$ is large, the energy is described equally well
by treating the spherical quadrature points as point charge sources. This can be derived by
inserting the moment shifting formula (Eq;~\ref{e:ishift}) for
$\sigma_i(R \hat r_j) \to \sum_k K(\hat r_j, \frac{R_x \hat r_k - t}{R}) w^0_k \sigma_o(R\hat r_k;t)$
and summing over $j$ to show
\begin{align}
E &= \sum_{i,k,n} w^0_i \sigma_o(R\hat r_i) w^0_k\sigma_o(R_x \hat r_k; t) L_n(R\hat r_i, R_x \hat r_k - t) \\
&\simeq \sum_{i,k} \frac{w^0_i \sigma_o(R\hat r_i)w^0_k \sigma_o(R_x \hat r_k; t) }{|R_x r_k -t -R \hat r_i|}
\end{align}
\subsection{ Outline}
Much of the work focused on real-space (Cartesian) representations of the
expansion (Eq.~\ref{e:Pexp}) uses a tensor notation. The set of three-dimensional tensors of orders n, $\sum_i q_i r_i^{(n)}$, from $n=0$ up to order $n=p-1$ is termed a `Cartesian polytensor.'
Appendix~\ref{s:tens} shows a direct correspondence between Cartesian polytensors
and polynomial functions with maximum degree $p-1$.
It is shown that outer and inner products between supersymmetric tensors
have much simpler statements in terms of polynomial multiplication and differentiation.
Moreover, the set of trace-free Cartesian polytensors\cite{jappl89,pren03} is
exactly identified with the set of polynomials on the unit sphere, $\sigma(\hat x)$.
In combination with the representation
theorem, Eq.~\ref{e:rep}, these identities completely connect tensor and
harmonic representations to a novel, quadrature representation of polytensor space.
Section~\ref{s:fmm} illustrates the simplicity of the quadrature method by testing
moment shift formulas used during the fast multipole method.
Surprisingly, these have the same form as the initial moment fitting.
Moreover, it shows that the weights for the quadrature representations exactly
coincide with point charges that reproduce those multipole moments
at sufficient distance from the bounding sphere.
As expected by the exact correspondence to
spherical harmonics, numerical results show that the
error of representing random point sources scales with distance
identically to traditional spherical harmonic multipoles.
Section~\ref{s:sphere} provides exact quadratures for the single and
double-layer potentials on a spherical surface. Numerical results in
Sec.~\ref{s:sphere} demonstrate a singularity representation for multiple spheres
interacting through a perfect, irrotational fluid.
The conclusion summarizes the connections created here
and outlines new applications and simplifications that can be
tackled in future work.
\section{ Conversion between Quadrature and Cartesian Tensors}
Although the quadrature representation is a complete, self-contained
basis for expressing multipole moments and externally imposed fields,
comparison with existing literature requires translation between Cartesian
representations.
Writing the order-$n$ Cartesian multipole moment tensor as
$M^{(n)} \equiv \int y^{(n)} \rho(y)\; d^3y$,
the directional moment is defined as the complete ($n$-way) tensor contraction
between $M^{(n)}$ and a test vector, $r$,
\begin{equation*}
r^{(n)} \tdot{n} M^{(n)} = \int (r\cdot y)^n \rho(y) \; d^3y
.
\end{equation*}
Complete information about the polytensor, $\{M^{(n)}, n=0, \ldots, p-1\}$ is
contained in the set of directional moments against the quadrature points, $\{\hat r_i\}$
(see Appendix~\ref{s:tens} for details). These directional moments
can be substituted wherever powers of ($\hat r\cdot y$) appear
in the moment matching equation~\ref{e:outer}.
Conversion back to Cartesian form is also simple, since the quadrature
weights were defined to reproduce polynomial integrals, which include
all (trace-free) tensors,
\begin{equation}
\int \mathcal D_n r^{(n)} \rho(r) \; d^3r = \sum_i w^0_i \sigma(\hat r_i) r_i^{(n)},\, n < p
.
\end{equation}
Here, $\mathcal D_n$ is the de-tracer projection of Ref.~\citenum{jappl89}
defined by
\begin{equation}
\hat x^{(n)} \tdot{n} \mathcal D_n \hat y^{(n)} = \frac{n!}{(2n-1)!!} P_n(\hat x\cdot \hat y)
\label{e:D}
\end{equation}
These formulas show that the conversion to quadrature form is one way to
project a polynomial into a trace-free form.
Alternatively, trace-free Cartesian tensors simplify Eq.~\ref{e:outer}, since
Eq.~\ref{e:D} can be used in $K$, (Eq.~\ref{e:K})
\begin{align}
\sigma_o(R\hat r) = \sum_n &\frac{(2n+1)}{4\pi}\frac{(2n-1)!!}{n!} \notag \\
&\times \hat r^{(n)} \tdot{n} \mathcal D_n M^{(n)} R^{-n}
.
\end{align}
This sum includes only the leading terms of $L_n$ at each order, $n$.
\section{ FMM Operations}\label{s:fmm}
In this section, we elaborate the translation formulas for the quadrature representation
of multipoles. We achieve greater generality in comparison with
previous Cartesian formulations\cite{jmaki99},
by working directly with the moment space, rather than the potential. In comparison
with black-box fast multipole methods,\cite{lying04} the availability of spherical quadrature rules justifies the use of Legendre polynomials because they represent moment space --
regardless of their specific role in the Poisson problem. For a detailed introduction
to the fast multipole method, see Ref.~\cite{yliu06}.
The outer expansion is defined by the moments of the enclosed charge
distribution. The moments of the shifted distribution should coincide with those of the original.
This can be shown from
\begin{equation}
\sigma_o(R_1\hat r; x_1) = \int_S K\Big(\frac{R_0\hat s + x_0-x_1}{R_1}, \hat r\Big) \sigma_o(R_0\hat s; x_0) \; d^2\hat s
.\label{e:shift}
\end{equation}
The notation here is that $\sigma_o(R_1\hat r; x_1)$ represents the outer expansion
about the origin $x_1$, defined on the set of points at a distance $R_1$ from $x_1$.
Since both outer expansions are polynomials of maximum degree $p-1$ in $R \hat r$,
the reproducing kernel, $K$ is able to duplicate them from any appropriate set
of quadrature points.
The inner expansion is defined by the derivatives of the potential about the origin.
This makes translation troublesome because the expansion about a new point
does not necessarily need to obey the trace-free condition.
An implicit argument can be used to show that propagating the inner expansion to the potential
on a small sphere around the new expansion point can be used to find charges on
a new enclosing sphere that duplicate this potential up to the same order
as the original expansion. The net result for the inner expansion is the same
as the initial fitting,
\begin{align}
\sigma_i(R_1 \hat r; x_0+t)
&= \int K(\hat r, \tfrac{R_0\hat s - t}{R_1}) \sigma_i(R_0 \hat s; x_0) \; d^2\hat s
\intertext{or}
&= \int K(\hat r, \tfrac{R_0\hat s - t}{R_1}) \sigma_o(R_0 \hat s; x_0) \; d^2\hat s \label{e:ishift}
.
\end{align}
Both outer $\to$ inner and inner $\to$ inner shift operators give the same equation.
In fact, all shift operators arriving at an outer expansion are identical to
the initial moment matching (Eq.~\ref{e:outer}). All shift operators arriving at an inner
expansion are also identical to the inverse moment matching (Eq.~\ref{e:inner}).
Appendix~\ref{s:harm} shows that these formulas are identical to
traditional expressions involving spherical harmonics.
The only caveat that appears here is that the kernel, $K(\hat r, \hat s - t)$, contains
singularities when $|t| = 1$. This problem enters because the
shifting formula is only well-defined when $t$
does not approach the sphere bounding the source charge distribution.
This motivates the introduction of the scaled distributions in Eq.~\ref{e:outer}. The
use of unscaled moments in shift formulas based on spherical-harmonics,
while convergent for finite sums, might therefore be expected to show numerical
issues in the limit of large expansion order, $p$.\cite{jmaki99,cless12}
\begin{figure}
\includegraphics[width=0.45\textwidth]{mom.pdf}
\includegraphics[width=0.45\textwidth]{mom_c.pdf}
\caption{Pre-factors for the multipole representation error.
The residual is scaled by $r^{p+1}$ (outer expansion, labeled $+p$) or $r^{-p}$ (inner expansion, labeled $-p$).
The left panel shows the error of exact quadrature using Eq.~\ref{e:Gext} or Eq.~\ref{e:Gint},
while the right shows the error of summing $1/r$, treating the quadrature points as point sources. Point charge sources make excellent representations of multipole sources and external fields.}\label{f:racc}
\end{figure}
Figure~\ref{f:racc} shows that the numerical error of the quadrature representation, including
appropriate bounding radii, have the same power-law decrease in error
as previously shown for spherical harmonics.
It plots the accuracy of representing the electrostatic potential
as a function of distance from the bounding sphere.
A set of 4000 point charges assigned from a uniform distribution
between -1 and +1 were placed uniformily in the cube $[-\frac{\sqrt 3}{3},\frac{\sqrt 3}{3}]^{(3)}$.
The residual error per point shown was averaged over $100$ charge distributions
and over the 86 points of the Lebedev quadrature grid of order 15 on the surface
of the evaluation sphere at varying distance, $r$, from the bounding sphere.
The inner expansion has an inverted geometry, with
the sources scaled by $1/|x|^2$ to put them outside the sphere,
and the evaluation test points likewise inverted to the inside, shrinking toward $r=0$.
For comparison, the simple summation of $\sum_i w_i/ (r-R\hat r_i)$ using the quadrature
weights as charges is shown in the right panel of Fig.~\ref{f:racc}.
The pre-factor for both error curves converges at a radius above 3, showing that
the quadrature-based representation gives highly accurate point charges
that simultaneously represent all multipoles. Practically, this solves the problem
of placing $O(p^2)$ discrete charges to mimic all multipolar moments up to
arbitrary order posed in Refs.~\cite{asawa89,gfere97,ranan13}.
This is also the reason for the symmetry of the shifting formulas -- the
weights are arrived at through the same process of fitting the inner or outer expansion.
\begin{figure*}
\includegraphics[width=0.8\textwidth]{trans.pdf}
\includegraphics[width=0.45\textwidth]{mom_t.pdf}
\includegraphics[width=0.45\textwidth]{mom_it.pdf}
\caption{Absolute multipole representation error in
the potential for shifted source distributions in the outer expansion ($|x|=2$, left panel)
or shifted evaluation spheres in the inner expansion ($|y|=1/2$, right panel).
Graphics above each indicate the source locations, distributed randomly
throughout the shaded areas,
and evaluation points using dashed lines. The dotted circle indicates the
bounding sphere at $R=1$.
Both are plotted as a function of $\cos\theta = \hat r\cdot \hat t$, where $\hat t$ is the shift direction.
The magnitude of the shifts are shown in the figure legend.
Expansion orders are 2, 5, and 8 as in Fig.~\ref{f:racc}. Increasing expansion orders
show up as groups of curves with decreasing error.}\label{f:tacc}
\end{figure*}
Figure~\ref{f:tacc} shows the error in multipole shifting operations, plotted as a function
of angle from the shift direction. For the outer expansion, the test points
are fixed at $R=2$, while the source distribution is shifted to the right by 0.2, 0.6, and 0.8.
The locations of the source points were scaled down for each shift to maintain contact of the rightmost
face of the source cube with the unit bounding sphere.
Summation of $\sum_i w_i/(r-\hat r_i)$ from the representation weights had nearly identical error (not shown).
For the inner expansion, the source points were inverted to lie outside the sphere and remain fixed.
The evaluation sphere started at the origin with $R=1/2$, and was successively
shifted in the $x$-direction by 0.1, 0.2, 0.3, and 0.4, with the radius scaled down to maintain
contact between the rightmost point of original evaluation sphere and the shifted version.
The error of the inner expansion shows much more variation as a function of the cosine with respect
to the shift direction, since the distance to the unit sphere is more quickly varying than in
the outer expansion geometry. The scaling of the outer expansion with expansion order, $p$,
also appears somewhat better because the source locations were scaled down
with increasingly large shifts in that geometry.
\section{ Singularity Methods}\label{s:sphere}
The boundary integral method utilizes the fact that in regions, $\Omega$,
where $\mathcal L \phi = 0$,
the potential can be expressed as a boundary integral using the divergence theorem,
\begin{align}
\phi(x) = \int_{\partial\Omega} \phi(y) &n\cdot p(y)\partial_{y} G(x-y) \notag \\
&- G(x-y) n\cdot p(y) \partial_{y} \phi(y) d^2y
. \label{e:BEM}
\end{align}
Here $n$ is the inward-pointing normal vector on the surface at point $y$, and a
Sturm-Liouville form has been
assumed for $\mathcal L G \equiv \partial\cdot(p \partial G) - u G = \delta(r)$. The normal
vector directions are shown in Fig.~\ref{f:K}.
This equality makes it possible to solve $\mathcal L \phi = 0$
with constant potential or constant flux boundary conditions.\cite{cpozr92}
Boundary integrals over polynomial distributions on the sphere, $\sigma$, can
be computed exactly by the summations,
\begin{widetext}
\begin{align}
\int_{S} |R\hat r-x|^{-1} \sigma(R\hat r) \; d^2\hat r &= \sum_{m=0,i}^{p-1} L_m(R\hat r_i, x) w_i^\sigma \label{e:Gext} \\
\int_{S} \left[ \hat r\cdot\partial_r |R\hat r-x|^{-1}\right] \sigma(R\hat r) \; d^2\hat r &= \sum_{m=0,i}^{p-1} \frac{m}{R} L_m(R\hat r_i, x) w_i^\sigma
\label{e:Fext} \\
\int_{S} |R\hat r-y|^{-1} \sigma(R\hat r) \; d^2\hat r &= \sum_{m=0,i}^{p-1} L_m(y, R \hat r_i) w_i^\sigma \label{e:Gint} \\
\int_{S} \left[ \hat r\cdot\partial_r |R\hat r-y|^{-1} \right] \sigma(\hat r) \; d^2\hat r &= -\sum_{m=0,i}^{p-1} \frac{m+1}{R} L_m(y, R\hat r_i) w_i^\sigma
\label{e:Fint}
\end{align}
\end{widetext}
For $|y| < R$ and $|x| > R$, and where $w_i^\sigma \equiv w^0_i \sigma(R \hat r_i)$.
These can be proven by expanding $|r-x|^{-1}$ in orthogonal polynomial spaces, $L_n$, to give an integral over $S$
and then writing that integral as a quadrature, exact for polynomial $\sigma$ of degree less than $p$.
Equations~\ref{e:Gext} and~\ref{e:Gint} are just the inner and outer expansions.
Equations~\ref{e:Fext} and~\ref{e:Fint} give the complete, singular surface integral,
not just the Cauchy principal value part.\cite{cpozr92}
This can be seen by noting that the jump discontinuity in $\hat r\cdot\partial \phi$ as
both $x$ and $y$ approach the surface point, $y$ is
\begin{align}
&F_\text{ext}\big |_{x=y} - F_\text{int} \\
&= R^2 \int_S \hat r\cdot\partial_r \frac{1}{|R\hat r-x|} \sigma(R\hat r) - \hat r\cdot\partial_r \frac{1}{|\hat r-y|} \sigma(R \hat r) \; d^2\hat r \notag \\
&= 4\pi R \sum_n \frac{2n+1}{4\pi} L_n(R \hat y, R \hat r_i) w^0_i \sigma(R\hat r_i) = 4\pi \sigma(y)
.
\end{align}
The last equality comes from recognizing the expression for the reproducing kernel (Eq.~\ref{e:K}),
which scales as $R^n$ in its first argument and $R^{-n-1}$ in its second.
For a set of spheres moving through an incompressible, irrotational fluid,
the flow at every point in the fluid can be written as the derivative of a potential,
\begin{align}
v(x) &\equiv -\partial \Phi(x) \label{e:vel} \\
\partial^2 \Phi(x) &= 0 \notag
.
\end{align}
Each point on a spherical boundary gives a constant field condition,
\begin{equation}
n\cdot v_0 = -n\cdot\partial \Phi(x). \label{e:bound}
\end{equation}
The solution can be found numerically using a point-based, quadrature representation of $\Phi$,
so that Eq.~\ref{e:BEM} becomes the linear equation,
$(I-F)\cdot\Phi = G\cdot (n\cdot v_0)$.
\begin{figure}
\includegraphics[width=0.45\textwidth]{sph_coll.png} \\
\includegraphics[width=0.45\textwidth]{sph_conv.pdf}
\caption{Multipole representation of potential flow for three spheres in the $z=0$ plane (upper panel, $p=8$). Color indicates the potential field. The flow velocity was rendered using line integral convolution.
The lower panel shows the surface-averaged boundary error divided into contributions
from the larger and smaller spheres.}\label{f:sacc}
\end{figure}
Figure~\ref{f:sacc}'s top panel shows the motion of three spheres through an
incompressible 3D fluid.
Point singularities represented using quadrature multipole expansions on the surfaces of
the spheres show exponential convergence.
The smaller spheres have radius 1, are located at $x=-1$ and $y=\pm 3/2$ and are traveling with unit velocity in the $+x$ direction. The larger sphere has radius 3, is located on the $y$ axis at $x=4$,
and is traveling with unit velocity in the $-x$ direction.
The velocity at every point in the fluid is calculated using Eq.~\ref{e:vel}
from a sum of the three multipoles (outer expansion of Eqns.~\ref{e:Gext}
and~\ref{e:Fext}). Weights on the quadrature set over each sphere were solved
to fix the boundary condition, Eq.~\ref{e:bound} at each point in the quadrature set.
Error in the normal derivative boundary condition (lower panel of Figure~\ref{f:sacc})
was evaluated at the spherical surfaces
using a larger Lebedev quadrature grid of order 59 with 1202 points.
As the quadrature order ($p$) is increased,
more points are added to the multipolar expansion on each sphere,
and the discrepancy between the
computed and imposed normal velocity decreases exponentially.
\section{ Discussion}
An isomorphism between symmetric Cartesian tensors and polynomial functions is used to show an optimal representation
for polytensors in terms of real weights, $w_i$, on a fixed set of basis vectors, $\{r_i\}$. This equivalence
is compactly expressed in terms of equivalent representations of the moment integrals,
\begin{align*}
r^{(n)}\tdot{n} \int x^{(n)} \rho(x) \; d^3x &= \int (r\cdot x)^n \rho(x) \; d^3x \\
& = \sum_i^{N} (r\cdot r_i)^n w_i
.
\end{align*}
Three numerical results were presented. First, we showed the numerical error in matching
spherical moments of a cloud of discrete point charges.
We compared the exact Eq.~\ref{e:Gext} with the numerical summation over the scaled
quadrature points, $R\hat r_i$. These made use of Lebedev quadrature rules\cite{vlebe77} tabulated
up to order 131 by Burkardt.\cite{jburk}
Next, we showed that the expressions for translating the origin of
inner and outer moment expansions have the same error as the corresponding
translation formulas using spherical harmonics. Appendix~\ref{s:harm}
gives further details on the mathematical translation.
Finally, we presented convergence results for FMM solutions to potential
flow between multiple interacting spheres.
The central role of the quadrature representation
in all these methods allows us to skip over consideration of an intermediate
spherical harmonic representation.
When used in a Taylor expansion of a scale-invariant Green's function, $G(c r) = c^{\alpha} G(r)$,
the $r$-dependence of the solutions are fixed, and only a subset of the full moment space is required.
That subspace is equivalent to the set of polynomials
on the unit sphere. This condition reduces the $\binom{n+2}{n}$ polynomials at
each total degree $n$ to only $2n+1$.
Approximation of integrals involving $G$ up to order $p-1$ thus requires only $N=p^2$ coefficients.
Convenient expressions for finding these coefficients from a source distribution, translation operators, and integrals involving expansions of the Poisson kernel, $|r|^{-1}$ were given.
Numerical results verified that the outer expansions have error $r^{-p-1}$,
and the inner expansions have error $r^p$,
consistent with the corresponding spherical harmonic formulas.
The computational scaling of na\"{\i}vely evaluating the moment
translation formula Eq.~\ref{e:shift} is $p^4$. This cost
scales as $p^3$ for spherical harmonic translations that perform rotations to align
the translation axis with the azimuthal reference axis, $\hat z$.\cite{ahaig11}.
Similar savings can be realized using the quadrature-based scheme,
when using equally spaced points
along rings spaced vertically according to Gauss-Legendre\cite{edarv00}
or Gauss-Jacobi\cite{mreut09} quadrature.
Fast, $O(p^2 \log p)$, transformations between harmonics and the Gauss-Jacobi
quadrature representation are available.\cite{mtyge08,mreut09}
Further, rotations that make use of these transformations can
achieve $O(p^3)$ scaling or, using approximate algorithms for
matrix sparsification, $O(p^2 \log p)$.\cite{edarv00}
Since the matrix in Eq.~\ref{e:shift}, $[K (\frac{R_0\hat r_j + t}{R_1}, \hat r_i )]_{ij}$
has as many unique elements as unique cosines, $\hat r_i\cdot \hat r_j$,
quadrature rules like those above containing
$O(p)$ vertical rings have a translation scaling as $O(p^2)$
after rotation.
The Cartesian representations found in this work generalize and extend analogous results
found using manually tabulated spherical harmonics. The decomposition of Legendre polynomials
into spherical harmonics (Eq.~\ref{e:harm}) shows that the harmonics correspond to
particular choices for the eigenfunctions of $P_n$. To work with this set,
the matrices $[P_n(\hat r_i, \hat r_j)\sqrt{w^0_iw^0_j}]$ can be numerically diagonalized.
This gives rise to $2n+1$ eigenvectors for each $P_n$, with the same eigenvalue,
$4\pi / (2n+1)$.
Using the path from tensors to quadrature points to harmonics,
results found using Cartesian polytensors can be translated to
spherical harmonics and vice-versa.
Moreover, Cartesian tensor contractions which were na\"{\i}vely $O(3^p)$ have
been reduced to an efficient summation over an $O(p^2)$ point set.
For differential equations not respecting scale-invariance, the representation of $\rho$
via the polynomial form of its moments still provides a simple path for deriving accurate numerical methods.
Using any $\sum_n \binom{n+2}{n}$ points for which the space of directional moments,
$[c_K(r_i; r_j)]$, has full
rank, a reproducing kernel and polynomial interpolation formulas can be found through numerical inversion
of $[c_K(r_i; r_j)]$.\cite{cdebo92} Of course, considering only the moments actually appearing in the expansion of $G$,
using an optimal quadrature rule, and accelerating the evaluation of the moments by exploiting symmetry
will reduce the computational overhead of this exercise.
Real-space expressions are the most desirable starting point for deriving
fast algorithms. First, all parts of the computation have a regular, vector structure,
making simplifying optimizations for parallel hardware.
Second, high-level form of the matrix operations and lack of angular coordinates
makes derivations much simpler.
Not only are the full set of (already well-developed and efficient) spectral methods available for fast
implementation, but also new symmetry-based, sparse factorizations remain to be explored.\cite{segne01}
Sparse factorization is at the heart of fast Fourier transformation methods.
Finally, stable, $O(p^3)$ rotation formulas using point-based evaluation of the reproducing kernel (Eq.~\ref{e:K})
have been shown and critically tested in Ref.~\cite{cless12}. Even still, a large fraction of computation
time in that work was spent on evaluating spherical harmonic functions on the set of rotated real-space points.
Either of these optimizations would reduce the overall scaling to $O(p^3)$,
recovering the scaling of rotation-based translations for spherical harmonics.\cite{lgree97}
\section{ Conclusions}
The representations found in this work greatly simplify the analysis and numerical use
of distribution functions on a sphere.
Many of the optimizations for spherical harmonics rely on the
symmetry of the spherical harmonic functions. Future work should
consider alternate quadrature sets that directly exploit real-space symmetry
to reduce the scaling of these methods with expansion order.
Working with distribution functions on the sphere also presents a new approach to
tensor analysis, traditional boundary value problems and extensions of black-box
multipole methods to potentials governed by more complicated PDEs.
Three novel results that come from this connection are
space-optimal, vector-based representations for Cartesian polytensors,
an exact representation of point multipoles using discrete point charges,
and simplification of the moment shifting formulas for FMM.
\section*{Acknowledgments}
Support for this work was provided by the USF Foundation.
Line integral convolution routines used here were developed by Anne Archibald based on the work of Cabral, Brian and Laeith Leedom (SIGGRAPH '93: 263-270, 1993), and wrapped in scipy\cite{scipy} by David Huard.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,156
|
1426 Main Street, Suite 6 | Walpole, MA 02081
Accidents & Personal Injury
Pain and Injuries
Chiropractic champions the idea of a holistic approach to health and illness, recognizing the body's inherent ability to heal itself during times of physical injury, mental / emotional, or environmental stress.
Our Healing Methods
What is Holistic Health?
Neurological Integration System (NIS) and Functional Neurology
Functional Nutrition Assessment
Active Release Techniques (ART)
Holistic means addressing the whole person which includes all of the 8 areas.
Active Release Techniques (ART) is a hands-on therapy designed to restore the normal textures and tension to soft tissue structures, allowing full function. Muscles, tendons, ligaments and nerve entrapment sites are the soft tissues on which ART works. ART is used to break down scar tissue and adhesions that form within or between these soft tissues. This revolutionary therapy can be applied to the spine, upper, and lower extremities only by a credentialed provider of Active Release Techniques.
An ART practitioner uses their knowledge of anatomy, neurology, and biomechanics to correctly diagnose and treat injuries. What attracts so many professional athletes and weekend warriors to ART is the quick results obtained without surgery.
Active Release Techniques History
ART has been developed, refined, and patented by P. Michael Leahy, DC, CCSP. Dr. Leahy noticed that his patients' symptoms seemed to be related to changes in their soft tissue that could be felt by hand. By observing how muscles, fascia, tendons, ligaments and nerves responded to different types of work, Dr. Leahy was able to consistently resolve over 90% of his patients' problems. He now teaches and certifies health care providers all over the world to use ART.
We can discover and correct problems or imbalances in their early stages before they cause symptoms or show up on conventional lab testing. This is the essence of preventive healthcare. Learn More
Monday 8:30 am-12:00 pm, 3:00-6:30 pm
Wednesday 8:30 am-12:00 pm, 3:00-6:30 pm
Friday 8:30 am-12:00 pm
Saturday (Every Other)8:30 am-12:00 pm
We are located in the Bristol Square plaza on Route 1A in Walpole. We are close to Norfolk, Foxboro, Sharon, Norwood, Wrentham, and Medfield.
© 2023 Holistic Center at Bristol Square. Site developed by WaterSpout Consulting
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 290
|
Q: How can I read from and write to a local directory from any self-hosted Wordpress blog page using PHP? I want to read and write text files to /data/filename.txt using php. This functionality works fine on the main page, but if I click the Previous Entries button (which switches to page 2), the php script does not seem to be present (as the Flash files are unable to access it).
A: If I understand your site correctly, your Flash tries to load data from http://www.consolebias.com/text_poll.php. You probably coded this link as just text_poll.php in your Flash file. This works on the homepage, but not on a page that has a URL that looks like it comes from another directory, like http://consolebias.com/page/2 (even if it is a "fake" directory, the browser, where Flash runs, doesn't know that). Here, the Flash object tries to load the data from http://consolebias.com/page/text_poll.php, which doesn't exist.
If you change the reference of text_poll.php to /text_poll.php, it will always load it from the root, and it should work.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,888
|
package im.actor.model.network.connection;
import com.google.j2objc.annotations.ObjectiveCName;
import java.util.ArrayList;
import im.actor.model.NetworkProvider;
import im.actor.model.network.ConnectionCallback;
import im.actor.model.network.ConnectionEndpoint;
import im.actor.model.network.CreateConnectionCallback;
public class ManagedNetworkProvider implements NetworkProvider {
private final AsyncConnectionFactory factory;
// Persisting pending connections to avoiding GC
@SuppressWarnings("MismatchedQueryAndUpdateOfCollection")
private final ArrayList<ManagedConnection> pendingConnections = new ArrayList<ManagedConnection>();
@ObjectiveCName("initWithFactory:")
public ManagedNetworkProvider(AsyncConnectionFactory factory) {
this.factory = factory;
}
@Override
public void createConnection(int connectionId, int mtprotoVersion, int apiMajorVersion, int apiMinorVersion, ConnectionEndpoint endpoint, ConnectionCallback callback, final CreateConnectionCallback createCallback) {
final ManagedConnection managedConnection = new ManagedConnection(connectionId, mtprotoVersion,
apiMajorVersion, apiMinorVersion, endpoint, callback, new ManagedConnectionCreateCallback() {
@Override
public void onConnectionCreated(ManagedConnection connection) {
createCallback.onConnectionCreated(connection);
synchronized (pendingConnections) {
pendingConnections.remove(connection);
}
}
@Override
public void onConnectionCreateError(ManagedConnection connection) {
createCallback.onConnectionCreateError();
synchronized (pendingConnections) {
pendingConnections.remove(connection);
}
}
}, factory);
synchronized (pendingConnections) {
pendingConnections.add(managedConnection);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,260
|
«Граф Монте-Кристо» () — мини-сериал совместного производства Франции, Италии и Германии, режиссёра Жозе Дайана, снятый в 1998 году по одноимённому роману Александра Дюма. Главные роли исполнили Жерар Депардьё и Орнелла Мути.
Сюжет
В 1815 году, в день своего обручения с красавицей Мерседес, молодой капитан дальнего плавания Эдмон Дантес был без суда и следствия заточен в тюрьму Замка Иф. В заговоре против него участвовали три человека: Мондего, имевший виды на невесту Эдмона; Данглар, мечтавший занять его место на капитанском мостике, и королевский прокурор Вильфор, вынесший обвинительный приговор.
Двадцать лет провел Дантес в подземельях замка, прежде чем ему удалось бежать. На острове Монте-Кристо он нашёл сказочные сокровища, о которых ему рассказал сосед по камере. Свободный и богатый, Дантес год путешествовал по Востоку.
Но настал час для возмездия, и бывший пленник уже на дороге в Париж — под маской загадочного графа Монте-Кристо.
Несметное богатство и изысканные манеры позволили Монте-Кристо шутя войти в высшее парижское общество. Ни один из трех заговорщиков не узнал его. Только в сердце Мерседес, ставшей законной супругой негодяя Мондего, шевельнулось подозрение о возвращении любимого, которого она похоронила в своих мыслях более 20 лет назад…
То под видом итальянского священника, то под маской английского лорда, Монте-Кристо входит в жизни своих врагов, чтобы найти их наиболее уязвимые места и нанести удар наверняка.
Чтобы отомстить Мерседес, Монте-Кристо предлагает руку красавице — вдове Камилле де ля Ришарде. Однако игра в ревность причиняет графу не меньше боли, чем Мерседес: она осталась для него такой же любимой женщиной, какой была всегда.
В ролях
Расхождения с оригиналом
Хотя сериал длинный и охватывает много событий, он тем не менее имеет некоторые расхождения с сюжетом книги. Сюжет до заточения Эдмона в замок Иф показан лишь отрывком — моментом ареста и свиданием с Вильфором, а далее лишь упоминается на словах. Бертуччо не имеет никакого отношения к Бенедетто (а сундук обнаружил Кадрусс); в фильме внебрачный сын прокурора, Бенедетто, зовётся Туссеном, а граф не превращает его в виконта Андреа Кавальканти (соответственно, убрана сюжетная линия, связанная с Андреа), также убран суд над Бенедетто, где он раскрывает, что Вильфор его отец, — фактически его роль свелась лишь к убийству Кадрусса и самому факту существования. Камилла де ла Ришарде, в экранизации являющаяся любовницей Монте-Кристо и музой (вместо Гайде), в романе отсутствует. Сама Гайде была освобождена сразу же после выкупа графом (ближе к концу событий в отличие от книги), а в дальнейшем она сошлась с Францем д`Эпине. Франц также почти полностью убран — его функция свелась к явке на свадьбу к Валентине и уходу, а также к праву быть секундантом Альбера. Также в фильме отсутствуют Эдуард де Вильфор (Элоиза отравляла всех только ради себя), Эжени Данглар с подругой Луизой д'Армильи (сюжетная линия убрана), Жюли Моррель с мужем (функции легли на Максимилиана), Люсьен Дебрэ (сюжетная линия убрана), Рауль де Шато-Рено, Батистен, Али (функции обоих исполняет Бертуччо, гораздо более близкий дружески к графу и посвящённый в его тайны). Почти полностью убрана сюжетная линия в Риме (граф, Альбер, Франц, Луиджи Вампа), которая свелась к быстрому похищению Альбера и не менее быстрому его освобождению за счёт помощи графа в отмене казни. Полностью отсутствуют подземный дворец графа на острове Монте-Кристо. Более подробно раскрыто взаимодействие с домом «Томпсон и Френч». Также существуют серьёзное расхождением с финалом. В финале Дантес жертвует своё состояние монахам и возвращается в Марсель, к Мерседес, тогда как в конце романа граф уплывает вместе с Гайде на корабле, оставив остров Монте-Кристо и всё своё состояние Максимилиану и Валентине.
Ссылки
Экранизации произведений Александра Дюма (отца)
Фильмы Жозе Дайан
Приключенческие фильмы Италии
Телесериалы Италии 1998 года
Приключенческие фильмы Франции
Приключенческие фильмы Германии
Телевизионные мини-сериалы Франции
Телевизионные мини-сериалы Италии
Телесериалы по алфавиту
Телесериалы Франции 1998 года
Драматические телесериалы Франции
Исторические телесериалы Франции
Исторические телесериалы Германии
Исторические телесериалы о XIX веке
Исторические фильмы Италии
Телесериалы Германии 1998 года
Побег из тюрьмы в искусстве
Телевизионные мини-сериалы 1990-х годов
Телесериалы TF1
Телесериалы DD Productions
Телесериалы GMT Productions
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,321
|
Die `@while`-Anweisung nimmt einen SASS-Script-Ausdruck und liefert wiederholt die Styles, die in der Verschachtelung stehen, bis die While-Bedingung nicht mehr erfüllt ist. Dies kann verwendet werden, um komplexere Schleifen abzubilden, als eine `@for`-Anweisung das kann, auch wenn dies selten notwendig ist. Zum Beispiel:
```scss
$i: 6;
@while $i > 0 {
.item-#{$i} { width: 2em * $i; }
$i: $i - 2;
}
```
wird kompiliert zu:
```css
.item-6 {
width: 12em; }
.item-4 {
width: 8em; }
.item-2 {
width: 4em; }
```
# ÜBUNG
Schreibe Vorgaben für die Elemente `h1` bis `h6`, die ihre `font-size` auf `24px` minus `3px` mal die Überschriftenebene (heading level) setzen, d. h. `h1` hätte eine `font-size` von `24px - 3px * 1`, also `21px`, wenn man diese `@while`-Anweisung verwendet. Vergleiche mit dem Code der vorherigen Übung.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,973
|
package object sbt extends sbt.std.TaskExtra with sbt.internal.util.Types with sbt.ProcessExtra
with sbt.internal.librarymanagement.impl.DependencyBuilders with sbt.io.PathExtra with sbt.ProjectExtra
with sbt.internal.librarymanagement.DependencyFilterExtra with sbt.BuildExtra with sbt.TaskMacroExtra
with sbt.ScopeFilter.Make {
type Setting[T] = Def.Setting[T]
type ScopedKey[T] = Def.ScopedKey[T]
type SettingsDefinition = Def.SettingsDefinition
type File = java.io.File
type URI = java.net.URI
type URL = java.net.URL
object CompileOrder {
val JavaThenScala = xsbti.compile.CompileOrder.JavaThenScala
val ScalaThenJava = xsbti.compile.CompileOrder.ScalaThenJava
val Mixed = xsbti.compile.CompileOrder.Mixed
}
type CompileOrder = xsbti.compile.CompileOrder
implicit def maybeToOption[S](m: xsbti.Maybe[S]): Option[S] =
if (m.isDefined) Some(m.get) else None
def uri(s: String): URI = new URI(s)
def file(s: String): File = new File(s)
def url(s: String): URL = new URL(s)
final val ThisScope = Scope.ThisScope
final val GlobalScope = Scope.GlobalScope
import sbt.librarymanagement.{ Configuration, Configurations => C }
final val Compile = C.Compile
final val Test = C.Test
final val Runtime = C.Runtime
final val IntegrationTest = C.IntegrationTest
final val Default = C.Default
final val Docs = C.Docs
final val Sources = C.Sources
final val Provided = C.Provided
// java.lang.System is more important, so don't alias this one
// final val System = C.System
final val Optional = C.Optional
def config(s: String): Configuration = C.config(s)
import language.experimental.macros
def settingKey[T](description: String): SettingKey[T] = macro std.KeyMacro.settingKeyImpl[T]
def taskKey[T](description: String): TaskKey[T] = macro std.KeyMacro.taskKeyImpl[T]
def inputKey[T](description: String): InputKey[T] = macro std.KeyMacro.inputKeyImpl[T]
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,465
|
Transcribed from the 1887 Cassell & Company by David Price, email
ccx074@pglaf.org
[Picture: Book cover]
CASSELL'S NATIONAL LIBRARY.
* * * * *
AN
ESSAY UPON PROJECTS.
* * * * *
BY
DANIEL DEFOE.
[Picture: Decorative graphic]
CASSELL & COMPANY, Limited:
_LONDON_, _PARIS_, _NEW YORK & MELBOURNE_.
1887
INTRODUCTION.
DEFOE'S "Essay on Projects" was the first volume he published, and no
great writer ever published a first book more characteristic in
expression of his tone of thought. It is practical in the highest
degree, while running over with fresh speculation that seeks everywhere
the well-being of society by growth of material and moral power. There
is a wonderful fertility of mind, and almost whimsical precision of
detail, with good sense and good humour to form the groundwork of a happy
English style. Defoe in this book ran again and again into sound
suggestions that first came to be realised long after he was dead. Upon
one subject, indeed, the education of women, we have only just now caught
him up. Defoe wrote the book in 1692 or 1693, when his age was a year or
two over thirty, and he published it in 1697.
Defoe was the son of James Foe, of St. Giles's, Cripplegate, whose family
had owned grazing land in the country, and who himself throve as a meat
salesman in London. James Foe went to Cripplegate Church, where the
minister was Dr. Annesley. But in 1662, a year after the birth of Daniel
Foe, Dr. Annesley was one of the three thousand clergymen who were driven
out of their benefices by the Act of Uniformity. James Foe was then one
of the congregation that followed him into exile, and looked up to him as
spiritual guide when he was able to open a meeting-house in Little St.
Helen's. Thus Daniel Foe, not yet De Foe, was trained under the
influence of Dr. Annesley, and by his advice sent to the Academy at
Newington Green, where Charles Morton, a good Oxford scholar, trained
young men for the pulpits of the Nonconformists. In later days, when
driven to America by the persecution of opinion, Morton became
Vice-President of Harvard College. Charles Morton sought to include in
his teaching at Newington Green a training in such knowledge of current
history as would show his boys the origin and meaning of the
controversies of the day in which, as men, they might hereafter take
their part. He took pains, also, to train them in the use of English.
"We were not," Defoe said afterwards, "destitute of language, but we were
made masters of English; and more of us excelled in that particular than
of any school at that time."
Daniel Foe did not pass on into the ministry for which he had been
trained. He said afterwards, in his "Review," "It was my disaster first
to be set apart for, and then to be set apart from, the honour of that
sacred employ." At the age of about nineteen he went into business as a
hose factor in Freeman's Court, Cornhill. He may have bought succession
to a business, or sought to make one in a way of life that required no
capital. He acted simply as broker between the manufacturer and the
retailer. He remained at the business in Freeman's Court for seven
years, subject to political distractions. In 1683, still in the reign of
Charles the Second, Daniel Foe, aged twenty-two, published a pamphlet
called "Presbytery Roughdrawn." Charles died on the 6th of February,
1685. On the 14th of the next June the Duke of Monmouth landed at Lyme
with eighty-three followers, hoping that Englishmen enough would flock
about his standard to overthrow the Government of James the Second, for
whose exclusion, as a Roman Catholic, from the succession to the throne
there had been so long a struggle in his brother's reign. Daniel Foe
took leave of absence from his business in Freeman's Court, joined
Monmouth, and shared the defeat at Sedgmoor on the 6th of July. Judge
Jeffreys then made progress through the West, and Daniel Foe escaped from
his clutches. On the 15th of July Monmouth was executed. Daniel Foe
found it convenient at that time to pay personal attention to some
business affairs in Spain. His name suggests an English reading of a
Spanish name, Foà, and more than once in his life there are indications
of friends in Spain about whom we know nothing. Daniel Foe went to Spain
in the time of danger to his life, for taking part in the rebellion of
the Duke of Monmouth, and when he came back he wrote himself De Foe. He
may have heard pedigree discussed among his Spanish friends; he may have
wished to avoid drawing attention to a name entered under the letter F in
a list of rebels. He may have played on the distinction between himself
and his father, still living, that one was Mr. Foe, the other Mr. D. Foe.
He may have meant to write much, and wishing to be a friend to his
country, meant also to deprive punsters of the opportunity of calling him
a Foe. Whatever his chief reason for the change, we may be sure that it
was practical.
In April, 1687, James the Second issued a Declaration for Liberty of
Conscience in England, by which he suspended penal laws against all Roman
Catholics and Nonconformists, and dispensed with oaths and tests
established by the law. This was a stretch of the king's prerogative
that produced results immediately welcome to the Nonconformists, who sent
up addresses of thanks. Defoe saw clearly that a king who is thanked for
overruling an unwelcome law has the whole point conceded to him of right
to overrule the law. In that sense he wrote, "A Letter containing some
Reflections on His Majesty's Declaration for Liberty of Conscience," to
warn the Nonconformists of the great mistake into which some were
falling. "Was ever anything," he asked afterwards, "more absurd than
this conduct of King James and his party, in wheedling the Dissenters;
giving them liberty of conscience by his own arbitrary dispensing
authority, and his expecting they should be content with their religious
liberty at the price of the Constitution?" In the letter itself he
pointed out that "the king's suspending of laws strikes at the root of
this whole Government, and subverts it quite. The Lords and Commons have
such a share in it, that no law can be either made, repealed, or, which
is all one, suspended, but by their consent."
In January, 1688, Defoe having inherited the freedom of the City of
London, took it up, and signed his name in the Chamberlain's book, on the
26th of that month, without the "de," "Daniel Foe." On the 5th of
November, 1688, there was another landing, that of William of Orange, in
Torbay, which threatened the government of James the Second. Defoe again
rode out, met the army of William at Henley-on-Thames, and joined its
second line as a volunteer. He was present when it was resolved, on the
13th of February, 1689, that the flight of James had been an abdication;
and he was one of the mounted citizens who formed a guard of honour when
William and Mary paid their first visit to Guildhall.
Defoe was at this time twenty-eight years old, married, and living in a
house at Tooting, where he had also been active in foundation of a
chapel. From hose factor he had become merchant adventurer in trade with
Spain, and is said by one writer of his time to have been a "civet-cat
merchant." Failing then in some venture in 1692, he became bankrupt, and
had one vindictive creditor who, according to the law of those days, had
power to shut him in prison, and destroy all power of recovering his loss
and putting himself straight with the world. Until his other creditors
had conquered that one enemy, and could give him freedom to earn money
again and pay his debts—when that time came he proved his sense of
honesty to much larger than the letter of the law—Defoe left London for
Bristol, and there kept out of the way of arrest. He was visible only on
Sunday, and known, therefore, as "the Sunday Gentleman." His lodging was
at the Red Lion Inn, in Castle Street. The house, no longer an inn,
still stands, as numbers 80 and 81 in that street. There Defoe wrote
this "Essay on Projects." He was there until 1694, when he received
offers that would have settled him prosperously in business at Cadiz, but
he held by his country. The cheek on free action was removed, and the
Government received with favour a project of his, which is not included
in the Essay, "for raising money to supply the occasions of the war then
newly begun." He had also a project for the raising of money to supply
his own occasions by the establishment of pantile works, which proved
successful. Defoe could not be idle. In a desert island he would, like
his Robinson Crusoe, have spent time, not in lamentation, but in steady
work to get away.
H. M.
AUTHOR'S PREFACE.
TO DALBY THOMAS, ESQ.,
_One of the Commission's for Managing His majesty's Duties on Glass_,
_&c._
SIR,
THIS preface comes directed to you, not as commissioner, &c., under whom
I have the honour to serve his Majesty, nor as a friend, though I have
great obligations of that sort also, but as the most proper judge of the
subjects treated of, and more capable than the greatest part of mankind
to distinguish and understand them.
Books are useful only to such whose genius are suitable to the subject of
them; and to dedicate a book of projects to a person who had never
concerned himself to think that way would be like music to one that has
no ear.
And yet your having a capacity to judge of these things no way brings you
under the despicable title of a projector, any more than knowing the
practices and subtleties of wicked men makes a man guilty of their
crimes.
The several chapters of this book are the results of particular thoughts
occasioned by conversing with the public affairs during the present war
with France. The losses and casualties which attend all trading nations
in the world, when involved in so cruel a war as this, have reached us
all, and I am none of the least sufferers; if this has put me, as well as
others, on inventions and projects, so much the subject of this book, it
is no more than a proof of the reason I give for the general projecting
humour of the nation.
One unhappiness I lie under in the following book, viz.: That having kept
the greatest part of it by me for near five years, several of the
thoughts seem to be hit by other hands, and some by the public, which
turns the tables upon me, as if I had borrowed from them.
As particularly that of the seamen, which you know well I had contrived
long before the Act for registering seamen was proposed. And that of
educating women, which I think myself bound to declare, was formed long
before the book called "Advice to the Ladies" was made public; and yet I
do not write this to magnify my own invention, but to acquit myself from
grafting on other people's thoughts. If I have trespassed upon any
person in the world, it is upon yourself, from whom I had some of the
notions about county banks, and factories for goods, in the chapter of
banks; and yet I do not think that my proposal for the women or the
seamen clashes at all, either with that book, or the public method of
registering seamen.
I have been told since this was done that my proposal for a commission of
inquiries into bankrupt estates is borrowed from the Dutch; if there is
anything like it among the Dutch, it is more than ever I knew, or know
yet; but if so, I hope it is no objection against our having the same
here, especially if it be true that it would be so publicly beneficial as
is expressed.
What is said of friendly societies, I think no man will dispute with me,
since one has met with so much success already in the practice of it. I
mean the Friendly Society for Widows, of which you have been pleased to
be a governor.
Friendly societies are very extensive, and, as I have hinted, might be
carried on to many particulars. I have omitted one which was mentioned
in discourse with yourself, where a hundred tradesmen, all of several
trades, agree together to buy whatever they want of one another, and
nowhere else, prices and payments to be settled among themselves; whereby
every man is sure to have ninety-nine customers, and can never want a
trade; and I could have filled up the book with instances of like nature,
but I never designed to fire the reader with particulars.
The proposal of the pension office you will soon see offered to the
public as an attempt for the relief of the poor; which, if it meets with
encouragement, will every way answer all the great things I have said of
it.
I had wrote a great many sheets about the coin, about bringing in plate
to the Mint, and about our standard; but so many great heads being upon
it, with some of whom my opinion does not agree, I would not adventure to
appear in print upon that subject.
Ways and means also I have laid by on the same score: only adhering to
this one point, that be it by taxing the wares they sell, be it by taxing
them in stock, be it by composition—which, by the way, I believe is the
best—be it by what way soever the Parliament please, the retailers are
the men who seem to call upon us to be taxed; if not by their own
extraordinary good circumstances, though that might bear it, yet by the
contrary in all other degrees of the kingdom.
Besides, the retailers are the only men who could pay it with least
damage, because it is in their power to levy it again upon their
customers in the prices of their goods, and is no more than paying a
higher rent for their shops.
The retailers of manufactures, especially so far as relates to the inland
trade, have never been taxed yet, and their wealth or number is not
easily calculated. Trade and land has been handled roughly enough, and
these are the men who now lie as a reserve to carry on the burden of the
war.
These are the men who, were the land tax collected as it should be, ought
to pay the king more than that whole Bill ever produced; and yet these
are the men who, I think I may venture to say, do not pay a twentieth
part in that Bill.
Should the king appoint a survey over the assessors, and indict all those
who were found faulty, allowing a reward to any discoverer of an
assessment made lower than the literal sense of the Act implies, what a
register of frauds and connivances would be found out!
In a general tax, if any should be excused, it should be the poor, who
are not able to pay, or at least are pinched in the necessary parts of
life by paying. And yet here a poor labourer, who works for twelve pence
or eighteen pence a day, does not drink a pot of beer but pays the king a
tenth part for excise; and really pays more to the king's taxes in a year
than a country shopkeeper, who is alderman of the town, worth perhaps two
or three thousand pounds, brews his own beer, pays no excise, and in the
land-tax is rated it may be at £100, and pays £1 4s. per annum, but
ought, if the Act were put in due execution, to pay £36 per annum to the
king.
If I were to be asked how I would remedy this, I would answer, it should
be by some method in which every man may be taxed in the due proportion
to his estate, and the Act put in execution, according to the true intent
and meaning of it, in order to which a commission of assessment should be
granted to twelve men, such as his Majesty should be well satisfied of,
who should go through the whole kingdom, three in a body, and should make
a new assessment of personal estates, not to meddle with land.
To these assessors should all the old rates, parish books, poor rates,
and highway rates, also be delivered; and upon due inquiry to be made
into the manner of living, and reputed wealth of the people, the stock or
personal estate of every man should be assessed, without connivance; and
he who is reputed to be worth a thousand pounds should be taxed at a
thousand pounds, and so on; and he who was an overgrown rich tradesman of
twenty or thirty thousand pounds estate should be taxed so, and plain
English and plain dealing be practised indifferently throughout the
kingdom; tradesmen and landed men should have neighbours' fare, as we
call it, and a rich man should not be passed by when a poor man pays.
We read of the inhabitants of Constantinople, that they suffered their
city to be lost for want of contributing in time for its defence, and
pleaded poverty to their generous emperor when he went from house to
house to persuade them; and yet when the Turks took it, the prodigious
immense wealth they found in it, made them wonder at the sordid temper of
the citizens.
England (with due exceptions to the Parliament, and the freedom wherewith
they have given to the public charge) is much like Constantinople; we are
involved in a dangerous, a chargeable, but withal a most just and
necessary war, and the richest and moneyed men in the kingdom plead
poverty; and the French, or King James, or the devil may come for them,
if they can but conceal their estates from the public notice, and get the
assessors to tax them at an under rate.
These are the men this commission would discover; and here they should
find men taxed at £500 stock who are worth £20,000. Here they should
find a certain rich man near Hackney rated to-day in the tax-book at
£1,000 stock, and to-morrow offering £27,000 for an estate.
Here they should find Sir J— C— perhaps taxed to the king at £5,000
stock, perhaps not so much, whose cash no man can guess at; and
multitudes of instances I could give by name without wrong to the
gentlemen.
And, not to run on in particulars, I affirm that in the land-tax ten
certain gentlemen in London put together did not pay for half so much
personal estate, called stock, as the poorest of them is reputed really
to possess.
I do not inquire at whose door this fraud must lie; it is none of my
business.
I wish they would search into it whose power can punish it. But this,
with submission, I presume to say: The king is thereby defrauded and
horribly abused, the true intent and meaning of Acts of Parliament
evaded, the nation involved in debt by fatal deficiencies and interests,
fellow-subjects abused, and new inventions for taxes occasioned.
The last chapter in this book is a proposal about entering all the seamen
in England into the king's pay—a subject which deserves to be enlarged
into a book itself; and I have a little volume of calculations and
particulars by me on that head, but I thought them too long to publish.
In short, I am persuaded, was that method proposed to those gentlemen to
whom such things belong, the greatest sum of money might be raised by it,
with the least injury to those who pay it, that ever was or will be
during the war.
Projectors, they say, are generally to be taken with allowance of
one-half at least; they always have their mouths full of millions, and
talk big of their own proposals. And therefore I have not exposed the
vast sums my calculations amount to; but I venture to say I could procure
a farm on such a proposal as this at three millions per annum, and give
very good security for payment—such an opinion I have of the value of
such a method; and when that is done, the nation would get three more by
paying it, which is very strange, but might easily be made out.
In the chapter of academies I have ventured to reprove the vicious custom
of swearing. I shall make no apology for the fact, for no man ought to
be ashamed of exposing what all men ought to be ashamed of practising.
But methinks I stand corrected by my own laws a little, in forcing the
reader to repeat some of the worst of our vulgar imprecations, in reading
my thoughts against it; to which, however, I have this to reply:
First, I did not find it easy to express what I mean without putting down
the very words—at least, not so as to be very intelligible.
Secondly, why should words repeated only to expose the vice, taint the
reader more than a sermon preached against lewdness should the
assembly?—for of necessity it leads the hearer to the thoughts of the
fact. But the morality of every action lies in the end; and if the
reader by ill-use renders himself guilty of the fact in reading, which I
designed to expose by writing, the fault is his, not mine.
I have endeavoured everywhere in this book to be as concise as possible,
except where calculations obliged me to be particular; and having avoided
impertinence in the book, I would avoid it too, in the preface, and
therefore shall break off with subscribing myself,
Sir,
Your most obliged, humble servant
D. F.
AUTHOR'S INTRODUCTION.
NECESSITY, which is allowed to be the mother of invention, has so
violently agitated the wits of men at this time that it seems not at all
improper, by way of distinction, to call it the Projecting Age. For
though in times of war and public confusions the like humour of invention
has seemed to stir, yet, without being partial to the present, it is, I
think, no injury to say the past ages have never come up to the degree of
projecting and inventing, as it refers to matters of negotiation and
methods of civil polity, which we see this age arrived to.
Nor is it a hard matter to assign probable causes of the perfection in
this modern art. I am not of their melancholy opinion who ascribe it to
the general poverty of the nation, since I believe it is easy to prove
the nation itself, taking it as one general stock, is not at all
diminished or impoverished by this long, this chargeable war, but, on the
contrary, was never richer since it was inhabited.
Nor am I absolutely of the opinion that we are so happy as to be wiser in
this age than our forefathers; though at the same time I must own some
parts of knowledge in science as well as art have received improvements
in this age altogether concealed from the former.
The art of war, which I take to be the highest perfection of human
knowledge, is a sufficient proof of what I say, especially in conducting
armies and in offensive engines. Witness the now ways of rallies,
fougades, entrenchments, attacks, lodgments, and a long _et cetera_ of
new inventions which want names, practised in sieges and encampments;
witness the new forts of bombs and unheard-of mortars, of seven to ten
ton weight, with which our fleets, standing two or three miles off at
sea, can imitate God Almighty Himself and rain fire and brimstone out of
heaven, as it were, upon towns built on the firm land; witness also our
new-invented child of hell, the machine which carries thunder, lightning,
and earthquakes in its bowels, and tears up the most impregnable
fortification.
But if I would search for a cause from whence it comes to pass that this
age swarms with such a multitude of projectors more than usual,
who—besides the innumerable conceptions, which die in the bringing forth,
and (like abortions of the brain) only come into the air and dissolve—do
really every day produce new contrivances, engines, and projects to get
money, never before thought of; if, I say, I would examine whence this
comes to pass, it must be thus:
The losses and depredations which this war brought with it at first were
exceeding many, suffered chiefly by the ill-conduct of merchants
themselves, who did not apprehend the danger to be really what it was:
for before our Admiralty could possibly settle convoys, cruisers, and
stations for men-of-war all over the world, the French covered the sea
with their privateers and took an incredible number of our ships. I have
heard the loss computed, by those who pretended they were able to guess,
at above fifteen millions of pounds sterling, in ships and goods, in the
first two or three years of the war—a sum which, if put into French,
would make such a rumbling sound of great numbers as would fright a weak
accountant out of his belief, being no less than one hundred and ninety
millions of _livres_. The weight of this loss fell chiefly on the
trading part of the nation, and, amongst them, on the merchants; and
amongst them, again, upon the most refined capacities, as the insurers,
&c. And an incredible number of the best merchants in the kingdom sunk
under the load, as may appear a little by a Bill which once passed the
House of Commons for the relief of merchant-insurers, who had suffered by
the war with France. If a great many fell, much greater were the number
of those who felt a sensible ebb of their fortunes, and with difficulty
bore up under the loss of great part of their estates. These, prompted
by necessity, rack their wits for new contrivances, new inventions, new
trades, stocks, projects, and anything to retrieve the desperate credit
of their fortunes. That this is probable to be the cause will appear
further thus. France (though I do not believe all the great outcries we
make of their misery and distress—if one-half of which be true, they are
certainly the best subjects in the world) yet without question has felt
its share of the losses and damages of the war; but the poverty there
falling chiefly on the poorer sort of people, they have not been so
fruitful in inventions and practices of this nature, their genius being
quite of another strain. As for the gentry and more capable sort, the
first thing a Frenchman flies to in his distress is the army; and he
seldom comes back from thence to get an estate by painful industry, but
either has his brains knocked out or makes his fortune there.
If industry be in any business rewarded with success it is in the
merchandising part of the world, who indeed may more truly be said to
live by their wits than any people whatsoever. All foreign negotiation,
though to some it is a plain road by the help of custom, yet is in its
beginning all project, contrivance, and invention. Every new voyage the
merchant contrives is a project; and ships are sent from port to port, as
markets and merchandises differ, by the help of strange and universal
intelligence—wherein some are so exquisite, so swift, and so exact, that
a merchant sitting at home in his counting-house at once converses with
all parts of the known world. This and travel make a true-bred merchant
the most intelligent man in the world, and consequently the most capable,
when urged by necessity, to contrive new ways to live. And from hence, I
humbly conceive, may very properly be derived the projects, so much the
subject of the present discourse. And to this sort of men it is easy to
trace the original of banks, stocks, stock-jobbing, assurances, friendly
societies, lotteries, and the like.
To this may be added the long annual inquiry in the House of Commons for
ways and means, which has been a particular movement to set all the heads
of the nation at work; and I appeal, with submission, to the gentlemen of
that honourable House, if the greatest part of all the ways and means out
of the common road of land taxes, polls, and the like, have not been
handed to them from the merchant, and in a great measure paid by them
too.
However, I offer this but as an essay at the original of this prevailing
humour of the people; and as it is probable, so it is also possible to be
otherwise, which I submit to future demonstration.
Of the several ways this faculty of projecting have exerted itself, and
of the various methods, as the genius of the authors has inclined, I have
been a diligent observer and, in most, an unconcerned spectator, and
perhaps have some advantage from thence more easily to discover the _faux
pas_ of the actors. If I have given an essay towards anything new, or
made discovery to advantage of any contrivance now on foot, all men are
at the liberty to make use of the improvement; if any fraud is
discovered, as now practised, it is without any particular reflection
upon parties or persons.
Projects of the nature I treat about are doubtless in general of public
advantage, as they tend to improvement of trade, and employment of the
poor, and the circulation and increase of the public stock of the
kingdom; but this is supposed of such as are built on the honest basis of
ingenuity and improvement, in which, though I will allow the author to
aim primarily at his own advantage, yet with the circumstances of public
benefit added.
Wherefore it is necessary to distinguish among the projects of the
present times between the honest and the dishonest.
There are, and that too many, fair pretences of fine discoveries, new
inventions, engines, and I know not what, which—being advanced in notion,
and talked up to great things to be performed when such and such sums of
money shall be advanced, and such and such engines are made—have raised
the fancies of credulous people to such a height that, merely on the
shadow of expectation, they have formed companies, chose committees,
appointed officers, shares, and books, raised great stocks, and cried up
an empty notion to that degree that people have been betrayed to part
with their money for shares in a new nothing; and when the inventors have
carried on the jest till they have sold all their own interest, they
leave the cloud to vanish of itself, and the poor purchasers to quarrel
with one another, and go to law about settlements, transferrings, and
some bone or other thrown among them by the subtlety of the author to lay
the blame of the miscarriage upon themselves. Thus the shares at first
begin to fall by degrees, and happy is he that sells in time; till, like
brass money, it will go at last for nothing at all. So have I seen
shares in joint-stocks, patents, engines, and undertakings, blown up by
the air of great words, and the name of some man of credit concerned, to
£100 for a five-hundredth part or share (some more), and at last dwindle
away till it has been stock-jobbed down to £10, £12, £9, £8 a share, and
at last no buyer (that is, in short, the fine new word for
nothing-worth), and many families ruined by the purchase. If I should
name linen manufactures, saltpetre-works, copper mines, diving engines,
dipping, and the like, for instances of this, I should, I believe, do no
wrong to truth, or to some persons too visibly guilty.
I might go on upon this subject to expose the frauds and tricks of
stock-jobbers, engineers, patentees, committees, with those Exchange
mountebanks we very properly call brokers, but I have not gaul enough for
such a work; but as a general rule of caution to those who would not be
tricked out of their estates by such pretenders to new inventions, let
them observe that all such people who may be suspected of design have
assuredly this in their proposal: your money to the author must go before
the experiment. And here I could give a very diverting history of a
patent-monger whose cully was nobody but myself, but I refer it to
another occasion.
But this is no reason why invention upon honest foundations and to fair
purposes should not be encouraged; no, nor why the author of any such
fair contrivances should not reap the harvest of his own ingenuity. Our
Acts of Parliament for granting patents to first inventors for fourteen
years is a sufficient acknowledgment of the due regard which ought to be
had to such as find out anything which may be of public advantage; new
discoveries in trade, in arts and mysteries, of manufacturing goods, or
improvement of land, are without question of as great benefit as any
discoveries made in the works of nature by all the academies and royal
societies in the world.
There is, it is true, a great difference between new inventions and
projects, between improvement of manufactures or lands (which tend to the
immediate benefit of the public, and employing of the poor), and projects
framed by subtle heads with a sort of a _deceptio visus_ and legerdemain,
to bring people to run needless and unusual hazards: I grant it, and give
a due preference to the first. And yet success has so sanctified some of
those other sorts of projects that it would be a kind of blasphemy
against fortune to disallow them. Witness Sir William Phips's voyage to
the wreck; it was a mere project; a lottery of a hundred thousand to one
odds; a hazard which, if it had failed, everybody would have been ashamed
to have owned themselves concerned in; a voyage that would have been as
much ridiculed as Don Quixote's adventure upon the windmill. Bless us!
that folks should go three thousand miles to angle in the open sea for
pieces of eight! Why, they would have made ballads of it, and the
merchants would have said of every unlikely adventure, "It, was like
Phips's wreck-voyage." But it had success, and who reflects upon the
project?
"Nothing's so partial as the laws of fate,
Erecting blockheads to suppress the great.
Sir Francis Drake the Spanish plate-fleet won;
He had been a pirate if he had got none.
Sir Walter Raleigh strove, but missed the plate,
And therefore died a traitor to the State.
Endeavour bears a value more or less,
Just as 'tis recommended by success:
The lucky coxcomb ev'ry man will prize,
And prosp'rous actions always pass for wise."
However, this sort of projects comes under no reflection as to their
honesty, save that there is a kind of honesty a man owes to himself and
to his family that prohibits him throwing away his estate in
impracticable, improbable adventures; but still some hit, even of the
most unlikely, of which this was one of Sir William Phips, who brought
home a cargo of silver of near £200,000 sterling, in pieces of eight,
fished up out of the open sea, remote from any shore, from an old Spanish
ship which had been sunk above forty years.
THE HISTORY OF PROJECTS.
WHEN I speak of writing a History of Projects, I do not mean either of
the introduction of, or continuing, necessary inventions, or the
improvement of arts and sciences before known, but a short account of
projects and projecting, as the word is allowed in the general
acceptation at this present time; and I need not go far back for the
original of the practice.
Invention of arts, with engines and handicraft instruments for their
improvement, requires a chronology as far back as the eldest son of Adam,
and has to this day afforded some new discovery in every age.
The building of the Ark by Noah, so far as you will allow it a human
work, was the first project I read of; and, no question, seemed so
ridiculous to the graver heads of that wise, though wicked, age that poor
Noah was sufficiently bantered for it: and, had he not been set on work
by a very peculiar direction from heaven, the good old man would
certainly have been laughed out of it as a most senseless ridiculous
project.
The building of Babel was a right project; for indeed the true definition
of a project, according to modern acceptation, is, as is said before, a
vast undertaking, too big to be managed, and therefore likely enough to
come to nothing. And yet, as great as they are, it is certainly true of
them all, even as the projectors propose: that, according to the old
tale, if so many eggs are hatched, there will be so many chickens, and
those chickens may lay so many eggs more, and those eggs produce so many
chickens more, and so on. Thus it was most certainly true that if the
people of the Old World could have built a house up to heaven, they
should never be drowned again on earth, and they only had forgot to
measure the height; that is, as in other projects, it only miscarried, or
else it would have succeeded.
And yet, when all is done, that very building, and the incredible height
it was carried, is a demonstration of the vast knowledge of that infant
age of the world, who had no advantage of the experiments or invention of
any before themselves.
"Thus when our fathers, touched with guilt,
That huge stupendous staircase built;
We mock, indeed, the fruitless enterprise
(For fruitless actions seldom pass for wise),
But were the mighty ruins left, they'd show
To what degree that untaught age did know."
I believe a very diverting account might be given of this, but I shall
not attempt it. Some are apt to say with Solomon, "No new thing happens
under the sun; but what is, has been:" yet I make no question but some
considerable discovery has been made in these latter ages, and inventions
of human origin produced, which the world was ever without before, either
in whole or in part; and I refer only to two cardinal points, the use of
the loadstone at sea, and the use of gunpowder and guns: both which, as
to the inventing part, I believe the world owes as absolutely to those
particular ages as it does the working in brass and iron to Tubal Cain,
or the inventing of music to Jubal, his brother. As to engines and
instruments for handicraftsmen, this age, I daresay, can show such as
never were so much as thought of, much less imitated before; for I do not
call that a real invention which has something before done like it—I
account that more properly an improvement. For handicraft instruments, I
know none owes more to true genuine contrivance, without borrowing from
any former use, than a mechanic engine contrived in our time called a
knitting-frame, which, built with admirable symmetry, works really with a
very happy success, and may be observed by the curious to have a more
than ordinary composition; for which I refer to the engine itself, to be
seen in every stocking-weaver's garret.
I shall trace the original of the projecting humour that now reigns no
farther back than the year 1680, dating its birth as a monster then,
though by times it had indeed something of life in the time of the late
civil war. I allow, no age has been altogether without something of this
nature, and some very happy projects are left to us as a taste of their
success; as the water-houses for supplying of the city of London with
water, and, since that, the New River—both very considerable
undertakings, and perfect projects, adventured on the risk of success.
In the reign of King Charles I. infinite projects were set on foot for
raising money without a Parliament: oppressing by monopolies and privy
seals; but these are excluded our scheme as irregularities, for thus the
French are as fruitful in projects as we; and these are rather stratagems
than projects. After the Fire of London the contrivance of an engine to
quench fires was a project the author was said to get well by, and we
have found to be very useful. But about the year 1680 began the art and
mystery of projecting to creep into the world. Prince Rupert, uncle to
King Charles II., gave great encouragement to that part of it that
respects engines and mechanical motions; and Bishop Wilkins added as much
of the theory to it as writing a book could do. The prince has left us a
metal called by his name; and the first project upon that was, as I
remember, casting of guns of that metal and boring them—done both by a
peculiar method of his own, and which died with him, to the great loss of
the undertaker, who to that purpose had, with no small charge, erected a
water-mill at Hackney Marsh, known by the name of the Temple Mill, which
mill very happily performed all parts of the work; and I have seen some
of those guns on board the Royal Charles, a first-rate ship, being of a
reddish colour, different either from brass or copper. I have heard some
reasons of state assigned why that project was not permitted to go
forward; but I omit them, because I have no good authority for them.
After this we saw a floating-machine, to be wrought with horses, for the
towing of great ships both against wind and tide; and another for the
raising of ballast, which, as unperforming engines, had the honour of
being made, exposed, tried, and laid by before the prince died.
If thus we introduce it into the world under the conduct of that prince,
when he died it was left a hopeless brat, and had hardly any hand to own
it, till the wreck-voyage before noted, performed so happily by Captain
Phips, afterwards Sir William, whose strange performance set a great many
heads on work to contrive something for themselves. He was immediately
followed by my Lord Mordant, Sir John Narborough, and others from several
parts, whose success made them soon weary of the work.
The project of the Penny Post, so well known and still practised, I
cannot omit, nor the contriver, Mr. Dockwra, who has had the honour to
have the injury done him in that affair repaired in some measure by the
public justice of the Parliament. And, the experiment proving it to be a
noble and useful design, the author must be remembered, wherever mention
is made of that affair, to his very great reputation.
It was, no question, a great hardship for a man to be master of so fine a
thought, that had both the essential ends of a project in it (public good
and private want), and that the public should reap the benefit and the
author be left out; the injustice of which, no doubt, discouraged many a
good design. But since an alteration in public circumstances has
recovered the lost attribute of justice, the like is not to be feared.
And Mr. Dockwra has had the satisfaction to see the former injury
disowned, and an honourable return made, even by them who did not the
injury, in bare respect to his ingenuity.
A while before this several people, under the patronage of some great
persons, had engaged in planting of foreign colonies (as William Penn,
the Lord Shaftesbury, Dr. Cox, and others) in Pennsylvania, Carolina,
East and West Jersey, and the like places, which I do not call projects,
because it was only prosecuting what had been formerly begun. But here
began the forming of public joint-stocks, which, together with the East
India, African, and Hudson's Bay Companies, before established, begot a
new trade, which we call by a new name stock-jobbing, which was at first
only the simple occasional transferring of interest and shares from one
to another, as persons alienated their estates; but by the industry of
the Exchange brokers, who got the business into their hands, it became a
trade, and one perhaps managed with the greatest intrigue, artifice, and
trick that ever anything that appeared with a face of honesty could be
handled with; for while the brokers held the box, they made the whole
Exchange the gamesters, and raised and lowered the prices of stocks as
they pleased, and always had both buyers and sellers who stood ready
innocently to commit their money to the mercy of their mercenary tongues.
This upstart of a trade, having tasted the sweetness of success which
generally attends a novel proposal, introduces the illegitimate wandering
object I speak of, as a proper engine to find work for the brokers. Thus
stock-jobbing nursed projecting, and projecting, in return, has very
diligently pimped for its foster-parent, till both are arrived to be
public grievances, and indeed are now almost grown scandalous.
OF PROJECTORS.
MAN is the worst of all God's creatures to shift for himself; no other
animal is ever starved to death; nature without has provided them both
food and clothes, and nature within has placed an instinct that never
fails to direct them to proper means for a supply; but man must either
work or starve, slave or die. He has indeed reason given him to direct
him, and few who follow the dictates of that reason come to such unhappy
exigences; but when by the errors of a man's youth he has reduced himself
to such a degree of distress as to be absolutely without three
things—money, friends, and health—he dies in a ditch, or in some worse
place, a hospital.
Ten thousand ways there are to bring a man to this, and but very few to
bring him out again.
Death is the universal deliverer, and therefore some who want courage to
bear what they see before them, hang themselves for fear; for certainly
self-destruction is the effect of cowardice in the highest extreme.
Others break the bounds of laws to satisfy that general law of nature,
and turn open thieves, house-breakers, highwaymen, clippers, coiners,
&c., till they run the length of the gallows, and get a deliverance the
nearest way at St. Tyburn.
Others, being masters of more cunning than their neighbours, turn their
thoughts to private methods of trick and cheat, a modern way of thieving
every jot as criminal, and in some degree worse than the other, by which
honest men are gulled with fair pretences to part from their money, and
then left to take their course with the author, who skulks behind the
curtain of a protection, or in the Mint or Friars, and bids defiance as
well to honesty as the law.
Others, yet urged by the same necessity, turn their thoughts to honest
invention, founded upon the platform of ingenuity and integrity.
These two last sorts are those we call projectors; and as there was
always more geese than swans, the number of the latter are very
inconsiderable in comparison of the former; and as the greater number
denominates the less, the just contempt we have of the former sort
bespatters the other, who, like cuckolds, bear the reproach of other
people's crimes.
A mere projector, then, is a contemptible thing, driven by his own
desperate fortune to such a strait that he must be delivered by a
miracle, or starve; and when he has beat his brains for some such miracle
in vain, he finds no remedy but to paint up some bauble or other, as
players make puppets talk big, to show like a strange thing, and then cry
it up for a new invention, gets a patent for it, divides it into shares,
and they must be sold. Ways and means are not wanting to swell the new
whim to a vast magnitude; thousands and hundreds of thousands are the
least of his discourse, and sometimes millions, till the ambition of some
honest coxcomb is wheedled to part with his money for it, and then
(_nascitur ridiculus mus_) the adventurer is left to carry on the
project, and the projector laughs at him. The diver shall walk at the
bottom of the Thames, the saltpetre maker shall build Tom T—d's pond into
houses, the engineers build models and windmills to draw water, till
funds are raised to carry it on by men who have more money than brains,
and then good-night patent and invention; the projector has done his
business and is gone.
But the honest projector is he who, having by fair and plain principles
of sense, honesty, and ingenuity brought any contrivance to a suitable
perfection, makes out what he pretends to, picks nobody's pocket, puts
his project in execution, and contents himself with the real produce as
the profit of his invention.
OF BANKS.
BANKS, without question, if rightly managed are, or may be, of great
advantage, especially to a trading people, as the English are; and, among
many others, this is one particular case in which that benefit appears:
that they bring down the interest of money, and take from the goldsmiths,
scriveners, and others, who have command of running cash, their most
delicious trade of making advantage of the necessities of the merchant in
extravagant discounts and premiums for advance of money, when either
large customs or foreign remittances call for disbursements beyond his
common ability; for by the easiness of terms on which the merchant may
have money, he is encouraged to venture further in trade than otherwise
he would do. Not but that there are other great advantages a Royal Bank
might procure in this kingdom, as has been seen in part by this; as
advancing money to the Exchequer upon Parliamentary funds and securities,
by which in time of a war our preparations for any expedition need not be
in danger of miscarriage for want of money, though the taxes raised be
not speedily paid, nor the Exchequer burthened with the excessive
interests paid in former reigns upon anticipations of the revenue; landed
men might be supplied with moneys upon securities on easier terms, which
would prevent the loss of multitudes of estates, now ruined and devoured
by insolent and merciless mortgagees, and the like. But now we unhappily
see a Royal Bank established by Act of Parliament, and another with a
large fund upon the Orphans' stock; and yet these advantages, or others,
which we expected, not answered, though the pretensions in both have not
been wanting at such time as they found it needful to introduce
themselves into public esteem, by giving out prints of what they were
rather able to do than really intended to practise. So that our having
two banks at this time settled, and more erecting, has not yet been able
to reduce the interest of money, not because the nature and foundation of
their constitution does not tend towards it, but because, finding their
hands full of better business, they are wiser than by being slaves to old
obsolete proposals to lose the advantage of the great improvement they
can make of their stock.
This, however, does not at all reflect on the nature of a bank, nor of
the benefit it would be to the public trading part of the kingdom,
whatever it may seem to do on the practice of the present. We find four
or five banks now in view to be settled. I confess I expect no more from
those to come than we have found from the past, and I think I make no
broach on either my charity or good manners in saying so; and I reflect
not upon any of the banks that are or shall be established for not doing
what I mention, but for making such publications of what they would do.
I cannot think any man had expected the Royal Bank should lend money on
mortgages at 4 per cent. (nor was it much the better for them to make
publication they would do so from the beginning of January next after
their settlement), since to this day, as I am informed, they have not
lent one farthing in that manner.
Our banks are indeed nothing but so many goldsmiths' shops, where the
credit being high (and the directors as high) people lodge their money;
and they—the directors, I mean—make their advantage of it. If you lay it
at demand, they allow you nothing; if at time, 3 per cent.; and so would
any goldsmith in Lombard Street have done before. But the very banks
themselves are so awkward in lending, so strict, so tedious, so
inquisitive, and withal so public in their taking securities, that men
who are anything tender won't go to them; and so the easiness of
borrowing money, so much designed, is defeated. For here is a private
interest to be made, though it be a public one; and, in short, it is only
a great trade carried on for the private gain of a few concerned in the
original stock; and though we are to hope for great things, because they
have promised them, yet they are all future that we know of.
And yet all this while a bank might be very beneficial to this kingdom;
and this might be so, if either their own ingenuity or public authority
would oblige them to take the public good into equal concern with their
private interest.
To explain what I mean; banks, being established by public authority,
ought also, as all public things are, to be under limitations and
restrictions from that authority; and those limitations being regulated
with a proper regard to the ease of trade in general, and the improvement
of the stock in particular, would make a bank a useful, profitable thing
indeed.
First, a bank ought to be of a magnitude proportioned to the trade of the
country it is in, which this bank is so far from that it is no more to
the whole than the least goldsmith's cash in Lombard Street is to the
bank, from whence it comes to pass that already more banks are
contriving. And I question not but banks in London will ere long be as
frequent as lotteries; the consequence of which, in all probability, will
be the diminishing their reputation, or a civil war with one another. It
is true, the Bank of England has a capital stock; but yet, was that stock
wholly clear of the public concern of the Government, it is not above a
fifth part of what would be necessary to manage the whole business of the
town—which it ought, though not to do, at least to be able to do. And I
suppose I may venture to say above one-half of the stock of the present
bank is taken up in the affairs of the Exchequer.
I suppose nobody will take this discourse for an invective against the
Bank of England. I believe it is a very good fund, a very useful one,
and a very profitable one. It has been useful to the Government, and it
is profitable to the proprietors; and the establishing it at such a
juncture, when our enemies were making great boasts of our poverty and
want of money, was a particular glory to our nation, and the city in
particular. That when the Paris Gazette informed the world that the
Parliament had indeed given the king grants for raising money in funds to
be paid in remote years, but money was so scarce that no anticipations
could be procured; that just then, besides three millions paid into the
Exchequer that spring on other taxes by way of advance, there was an
overplus-stock to be found of £1,200,000 sterling, or (to make it speak
French) of above fifteen millions, which was all paid voluntarily into
the Exchequer. Besides this, I believe the present Bank of England has
been very useful to the Exchequer, and to supply the king with
remittances for the payment of the army in Flanders, which has also, by
the way, been very profitable to itself. But still this bank is not of
that bulk that the business done here requires, nor is it able, with all
the stock it has, to procure the great proposed benefit, the lowering the
interest of money: whereas all foreign banks absolutely govern the
interest, both at Amsterdam, Genoa, and other places. And this defect I
conceive the multiplicity of banks cannot supply, unless a perfect
understanding could be secured between them.
To remedy this defect, several methods might be proposed. Some I shall
take the freedom to hint at:—
First, that the present bank increase their stock to at least five
millions sterling, to be settled as they are already, with some small
limitations to make the methods more beneficial.
Five millions sterling is an immense sum; to which add the credit of
their cash, which would supply them with all the overplus-money in the
town, and probably might amount to half as much more; and then the credit
of running bills, which by circulating would, no question, be an
equivalent to the other half: so that in stock, credit, and bank-bills
the balance of their cash would be always ten millions sterling—a sum
that everybody who can talk of does not understand.
But then to find business for all this stock, which, though it be a
strange thing to think of, is nevertheless easy when it comes to be
examined. And first for the business; this bank should enlarge the
number of their directors, as they do of their stock, and should then
establish several sub-committees, composed of their own members, who
should have the directing of several offices relating to the distinct
sorts of business they referred to, to be overruled and governed by the
governor and directors in a body, but to have a conclusive power as to
contracts. Of these there should be—
One office for loan of money for customs of goods, which by a plain
method might be so ordered that the merchant might with ease pay the
highest customs down, and so, by allowing the bank 4 per cent. advance,
be first sure to secure the £10 per cent. which the king allows for
prompt payment at the Custom House, and be also freed from the
troublesome work of finding bondsmen and securities for the money—which
has exposed many a man to the tyranny of extents, either for himself or
his friend, to his utter ruin, who under a more moderate prosecution had
been able to pay all his debts, and by this method has been torn to
pieces and disabled from making any tolerable proposal to his creditors.
This is a scene of large business, and would, in proportion, employ a
large cash, and it is the easiest thing in the world to make the bank the
paymaster of all the large customs, and yet the merchant have so
honourable a possession of his goods, as may be neither any diminution to
his reputation or any hindrance to their sale.
As, for example, suppose I have 100 hogsheads of tobacco to import, whose
customs by several duties come to £1,000, and want cash to clear them. I
go with my bill of loading to the bank, who appoint their officer to
enter the goods and pay the duties, which goods, so entered by the bank,
shall give them title enough to any part, or the whole, without the
trouble of bills of sale, or conveyances, defeasances, and the like. The
goods are carried to a warehouse at the waterside, where the merchant has
a free and public access to them, as if in his own warehouse and an
honourable liberty to sell and deliver either the whole (paying their
disburse) or a part without it, leaving but sufficient for the payment,
and out of that part delivered, either by notes under the hand of the
purchaser, or any other way, he may clear the same, without any
exactions, but of £4 per cent., and the rest are his own.
The ease this would bring to trade, the deliverance it would bring to the
merchants from the insults of goldsmiths, &c., and the honour it would
give to our management of public imposts, with the advantages to the
Custom House itself, and the utter destruction of extortion, would be
such as would give a due value to the bank, and make all mankind
acknowledge it to be a public good. The grievance of exactions upon
merchants in this case is very great, and when I lay the blame on the
goldsmiths, because they are the principal people made use of in such
occasions, I include a great many other sorts of brokers and
money-jobbing artists, who all get a snip out of the merchant. I myself
have known a goldsmith in Lombard Street lend a man £700 to pay the
customs of a hundred pipes of Spanish wines; the wines were made over to
him for security by bill of sale, and put into a cellar, of which the
goldsmith kept the key; the merchant was to pay £6 per cent. interest on
the bond, and to allow £10 percent. premium for advancing the money.
When he had the wines in possession the owner could not send his cooper
to look after them, but the goldsmith's man must attend all the while,
for which he would be paid 5s. a day. If he brought a customer to see
them, the goldsmith's man must show them. The money was lent for two
months. He could not be admitted to sell or deliver a pipe of wine out
single, or two or three at a time, as he might have sold them; but on a
word or two spoken amiss to the goldsmith (or which he was pleased to
take so), he would have none sold but the whole parcel together. By this
usage the goods lay on hand, and every month the money remained the
goldsmith demanded a guinea per cent. forbearance, besides the interest,
till at last by leakage, decay, and other accidents, the wines began to
lessen. Then the goldsmith begins to tell the merchant he is afraid the
wines are not worth the money he has lent, and demands further security,
and in a little while, growing higher and rougher, he tells him he must
have his money. The merchant—too much at his mercy, because he cannot
provide the money—is forced to consent to the sale; and the goods, being
reduced to seventy pipes sound—wine and four unsound (the rest being sunk
for filling up), were sold for £13 per pipe the sound, and £3 the
unsound, which amounted to £922 together.
£ s. d.
The cooper's bill came to 30 0 0
The cellarage a year and a half to 18 0 0
Interests on the bond to 63 0 0
The goldsmith's men for attendance 8 0 0
Allowance for advance of the money and 74 0 0
forbearance
193 0 0
Principal money borrowed 700 0 0
893 0 0
Due to the merchant 29 0 0
922 0 0
By the moderatest computation that can be, these wines cost the merchant
as follows:—
_First Cost with Charges on Board_. £ s. d.
In Lisbon 15 mille reis per pipe is 1,500 475 0 0
mille reis; exchange, at 6s. 4d. per mille
rei
Freight to London, then at £3 per ton 150 0 0
Assurance on £500 at 2 per cent. 10 0 0
Petty charges 5 0 0
640 0 0
So that it is manifest by the extortion of this banker, the poor man lost
the whole capital with freight and charges, and made but £29 produce of a
hundred pipes of wine.
One other office of this bank, and which would take up a considerable
branch of the stock, is for lending money upon pledges, which should have
annexed to it a warehouse and factory, where all sorts of goods might
publicly be sold by the consent of the owners, to the great advantage of
the owner, the bank receiving £4 per cent. interest., and 2 per cent.
commission for sale of the goods.
A third office should be appointed for discounting bills, tallies, and
notes, by which all tallies of the Exchequer, and any part of the
revenue, should at stated allowances be ready money to any person, to the
great advantage of the Government, and ease of all such as are any ways
concerned in public undertakings.
A fourth office for lending money upon land securities at 4 per cent.
interest, by which the cruelty and injustice of mortgagees would be
wholly restrained, and a register of mortgages might be very well kept,
to prevent frauds.
A fifth office for exchanges and foreign correspondences.
A sixth for inland exchanges, where a very large field of business lies
before them.
Under this head it will not be improper to consider that this method will
most effectually answer all the notions and proposals of county banks;
for by this office they would be all rendered useless and unprofitable,
since one bank of the magnitude I mention, with a branch of its office
set apart for that business, might with ease manage all the inland
exchange of the kingdom.
By which such a correspondence with all the trading towns in England
might be maintained, as that the whole kingdom should trade with the
bank. Under the direction of this office a public cashier should be
appointed in every county, to reside in the capital town as to trade (and
in some counties more), through whose hands all the cash of the revenue
of the gentry and of trade should be returned on the bank in London, and
from the bank again on their cashier in every respective county or town,
at the small exchange of 0.5 per cent., by which means all loss of money
carried upon the road, to the encouragement of robbers and ruining of the
country, who are sued for those robberies, would be more effectually
prevented than by all the statutes against highwaymen that are or can be
made.
As to public advancings of money to the Government, they may be left to
the directors in a body, as all other disputes and contingent cases are;
and whoever examines these heads of business apart, and has any judgment
in the particulars, will, I suppose, allow that a stock of ten millions
may find employment in them, though it be indeed a very great sum.
I could offer some very good reasons why this way of management by
particular offices for every particular sort of business is not only the
easiest, but the safest, way of executing an affair of such variety and
consequence; also I could state a method for the proceedings of those
private offices, their conjunction with and dependence on the general
court of the directors, and how the various accounts should centre in one
general capital account of stock, with regulations and appeals; but I
believe them to be needless—at least, in this place.
If it be objected here that it is impossible for one joint-stock to go
through the whole business of the kingdom, I answer, I believe it is not
either impossible or impracticable, particularly on this one account:
that almost all the country business would be managed by running bills,
and those the longest abroad of any, their distance keeping them out, to
the increasing the credit, and consequently the stock of the bank.
_Of the Multiplicity of Banks_.
What is touched at in the foregoing part of this chapter refers to one
bank royal to preside, as it were, over the whole cash of the kingdom:
but because some people do suppose this work fitter for many banks than
for one, I must a little consider that head. And first, allowing those
many banks could, without clashing, maintain a constant correspondence
with one another, in passing each other's bills as current from one to
another, I know not but it might be better performed by many than by one;
for as harmony makes music in sound, so it produces success in business.
A civil war among merchants is always the rain of trade: I cannot think a
multitude of banks could so consist with one another in England as to
join interests and uphold one another's credit, without joining stocks
too; I confess, if it could be done, the convenience to trade would be
visible.
If I were to propose which way these banks should be established, I
answer, allowing a due regard to some gentlemen who have had thoughts of
the same (whose methods I shall not so much as touch upon, much less
discover; my thoughts run upon quite different methods, both for the fund
and the establishment).
Every principal town in England is a corporation, upon which the fund may
be settled, which will sufficiently answer the difficult and chargeable
work of suing for a corporation by patent or Act of Parliament.
A general subscription of stock being made, and by deeds of settlement
placed in the mayor and aldermen of the city or corporation for the time
being, in trust, to be declared by deeds of uses, some of the directors
being always made members of the said corporation, and joined in the
trust; the bank hereby becomes the public stock of the town (something
like what they call the _rentes_ of the town-house in France), and is
managed in the name of the said corporation, to whom the directors are
accountable, and they back again to the general court.
For example: suppose the gentlemen or tradesmen of the county of Norfolk,
by a subscription of cash, design to establish a bank. The subscriptions
being made, the stock is paid into the chamber of the city of Norwich,
and managed by a court of directors, as all banks are, and chosen out of
the subscribers, the mayor only of the city to be always one; to be
managed in the name of the corporation of the city of Norwich, but for
the uses in a deed of trust to be made by the subscribers, and mayor and
aldermen, at large mentioned. I make no question but a bank thus settled
would have as firm a foundation as any bank need to have, and every way
answer the ends of a corporation.
Of these sorts of banks England might very well establish fifteen, at the
several towns hereafter mentioned. Some of which, though they are not
the capital towns of the counties, yet are more the centre of trade,
which in England runs in veins, like mines of metal in the earth:
Canterbury. Salisbury. Exeter. Bristol. Worcester. Shrewsbury.
Manchester. Newcastle-upon-Tyne. Leeds, or Halifax, or York. Warwick
or Birmingham. Oxford or Reading. Bedford. Norwich. Colchester.
Every one of these banks to have a cashier in London, unless they could
all have a general correspondence and credit with the bank royal.
These banks in their respective counties should be a general staple and
factory for the manufactures of the said county, where every man that had
goods made, might have money at a small interest for advance, the goods
in the meantime being sent forward to market, to a warehouse for that
purpose erected in London, where they should be disposed of to all the
advantages the owner could expect, paying only 1 per cent. commission.
Or if the maker wanted credit in London either for Spanish wool, cotton,
oil, or any goods, while his goods were in the warehouse of the said
bank, his bill should be paid by the bank to the full value of his goods,
or at least within a small matter. These banks, either by correspondence
with each other, or an order to their cashier in London, might with ease
so pass each other's bills that a man who has cash at Plymouth, and wants
money at Berwick, may transfer his cash at Plymouth to Newcastle in
half-an-hour's time, without either hazard, or charge, or time, allowing
only 0.5 per cent. exchange; and so of all the most distant parts of the
kingdom. Or if he wants money at Newcastle, and has goods at Worcester
or at any other clothing town, sending his goods to be sold by the
factory of the bank of Worcester, he may remit by the bank to Newcastle,
or anywhere else, as readily as if his goods were sold and paid for and
no exactions made upon him for the convenience he enjoys.
This discourse of banks, the reader is to understand, to have no relation
to the present posture of affairs, with respect to the scarcity of
current money, which seems to have put a stop to that part of a stock we
call credit, which always is, and indeed must be, the most essential part
of a bank, and without which no bank can pretend to subsist—at least, to
advantage.
A bank is only a great stock of money put together, to be employed by
some of the subscribers, in the name of the rest, for the benefit of the
whole. This stock of money subsists not barely on the profits of its own
stock (for that would be inconsiderable), but upon the contingencies and
accidents which multiplicity of business occasions. As, for instance, a
man that comes for money, and knows he may have it to-morrow; perhaps he
is in haste, and won't take it to-day: only, that he may be sure of it
to-morrow, he takes a memorandum under the hand of the officer, that he
shall have it whenever he calls for it, and this memorandum we call a
bill. To-morrow, when he intended to fetch his money, comes a man to him
for money, and, to save himself the labour of telling, he gives him the
memorandum or bill aforesaid for his money; this second man does as the
first, and a third does as he did, and so the bill runs about a mouth,
two or three. And this is that we call credit, for by the circulation of
a quantity of these bills, the bank enjoys the full benefit of as much
stock in real value as the suppositious value of the bills amounts to;
and wherever this credit fails, this advantage fails; for immediately all
men come for their money, and the bank must die of itself: for I am sure
no bank, by the simple improvement of their single stock, can ever make
any considerable advantage.
I confess, a bank who can lay a fund for the security of their bills,
which shall produce first an annual profit to the owner, and yet make
good the passant bill, may stand, and be advantageous, too, because there
is a real and a suppositious value both, and the real always ready to
make good the suppositious: and this I know no way to bring to pass but
by land, which, at the same time that it lies transferred to secure the
value of every bill given out, brings in a separate profit to the owner;
and this way no question but the whole kingdom might be a bank to itself,
though no ready money were to be found in it.
I had gone on in some sheets with my notion of land being the best bottom
for public banks, and the easiness of bringing it to answer all the ends
of money deposited with double advantage, but I find myself happily
prevented by a gentleman who has published the very same, though since
this was wrote; and I was always master of so much wit as to hold my
tongue while they spoke who understood the thing better than myself.
Mr. John Asgill, of Lincoln's Inn, in a small tract entitled, "Several
Assertions proved, in order to create another Species of Money than Gold
and Silver," has so distinctly handled this very case, with such strength
of argument, such clearness of reason, such a judgment, and such a style,
as all the ingenious part of the world must acknowledge themselves
extremely obliged to him for that piece.
At the sight of which book I laid by all that had been written by me on
that subject, for I had much rather confess myself incapable of handling
that point like him, than have convinced the world of it by my
impertinence.
OF THE HIGHWAYS.
IT is a prodigious charge the whole nation groans under for the repair of
highways, which, after all, lie in a very ill posture too. I make no
question but if it was taken into consideration by those who have the
power to direct it, the kingdom might be wholly eased of that burden, and
the highways be kept in good condition, which now lie in a most shameful
manner in most parts of the kingdom, and in many places wholly
unpassable, from whence arise tolls and impositions upon passengers and
travellers, and, on the other hand, trespasses and encroachments upon
lands adjacent, to the great damage of the owners.
The rate for the highways is the most arbitrary and unequal tax in the
kingdom: in some places two or three rates of sixpence per pound in the
year; in others the whole parish cannot raise wherewith to defray the
charge, either by the very bad condition of the road or distance of
materials; in others the surveyors raise what they never expend; and the
abuses, exactions, connivances, frauds, and embezzlements are
innumerable.
The Romans, while they governed this island, made it one of their
principal cares to make and repair the highways of the kingdom, and the
chief roads we now use are of their marking out; the consequence of
maintaining them was such, or at least so esteemed, that they thought it
not below them to employ their legionary troops in the work; and it was
sometimes the business of whole armies, either when in winter quarters or
in the intervals of truce or peace with the natives. Nor have the Romans
left us any greater tokens of their grandeur and magnificence than the
ruins of those causeways and street-ways which are at this day to be seen
in many parts of the kingdom, some of which have by the visible remains
been discovered to traverse the whole kingdom, and others for more than a
hundred miles are to be traced from colony to colony, as they had
particular occasion. The famous highway or street called Watling Street,
which some will tell you began at London Stone, and passing that very
street in the City which we to this day call by that name, went on west
to that spot where Tyburn now stands, and then turned north-west in so
straight a line to St. Albans that it is now the exactest road (in one
line for twenty miles) in the kingdom; and though disused now as the
chief, yet is as good, and, I believe, the best road to St. Albans, and
is still called the Streetway. From whence it is traced into Shropshire,
above a hundred and sixty miles, with a multitude of visible antiquities
upon it, discovered and described very accurately by Mr. Cambden. The
Fosse, another Roman work, lies at this day as visible, and as plain a
high causeway, of above thirty feet broad, ditched on either side, and
coped and paved where need is—as exact and every jot as beautiful as the
king's new road through Hyde Park, in which figure it now lies from near
Marshfield to Cirencester, and again from Cirencester to the Hill, three
miles on this side Gloucester, which is not less than twenty-six miles,
and is made use of as the great road to those towns, and probably has
been so for a thousand years with little repairs.
If we set aside the barbarity and customs of the Romans as heathens, and
take them as a civil government, we must allow they were the pattern of
the whole world for improvement and increase of arts and learning,
civilising and methodising nations and countries conquered by their
valour; and if this was one of their great cares, that consideration
ought to move something. But to the great example of that generous
people I will add three arguments:—
1. It is useful, and that as it is convenient for carriages, which in a
trading country is a great help to negotiation, and promotes universal
correspondence, without which our inland trade could not be managed. And
under this head I could name a thousand conveniences of a safe, pleasant,
well-repaired highway, both to the inhabitant and the traveller, but I
think it is needless.
2. It is easy. I question not to make it appear it is easy to put all
the highroads, especially in England, in a noble figure; large, dry, and
clean; well drained, and free from floods, unpassable sloughs, deep
cart-ruts, high ridges, and all the inconveniences they now are full of;
and, when once done, much easier still to be maintained so.
3. It may be cheaper, and the whole assessment for the repairs of
highways for ever be dropped or applied to other uses for the public
benefit.
Here I beg the reader's favour for a small digression.
I am not proposing this as an undertaker, or setting a price to the
public for which I will perform it, like one of the projectors I speak
of, but laying open a project for the performance, which, whenever the
public affairs will admit our governors to consider of, will be found so
feasible that no question they may find undertakers enough for the
performance; and in this undertaking age I do not doubt but it would be
easy at any time to procure persons at their own charge to perform it for
any single county, as a pattern and experiment for the whole kingdom.
The proposal is as follows:—First, that an Act of Parliament be made with
liberty for the undertakers to dig and trench, to cut down hedges and
trees, or whatever is needful for ditching, draining and carrying off
water, cleaning, enlarging and levelling the roads, with power to lay
open or enclose lands; to encroach into lands; dig, raise, and level
fences; plant and pull up hedges or trees (for the enlarging, widening,
and draining the highways), with power to turn either the roads or
watercourses, rivers and brooks, as by the directors of the works shall
be found needful, always allowing satisfaction to be first made to the
owners of such lands (either by assigning to them equivalent lands or
payment in money, the value to be adjusted by two indifferent persons to
be named by the Lord Chancellor or Lord Keeper for the time being), and
no watercourse to be turned from any water-mill without satisfaction
first made both to the landlord and tenant.
But before I proceed, I must say a word or two to this article.
The chief, and almost the only, cause of the deepness and foulness of the
roads is occasioned by the standing water, which (for want of due care to
draw it off by scouring and opening ditches and drains, and other
watercourses, and clearing of passages) soaks into the earth, and softens
it to such a degree that it cannot bear the weight of horses and
carriages; to prevent which, the power to dig, trench, and cut down, &c.,
mentioned above will be of absolute necessity. But because the liberty
seems very large, and some may think it is too great a power to be
granted to any body of men over their neighbours, it is answered:—
1. It is absolutely necessary, or the work cannot be done, and the doing
of the work is of much greater benefit than the damage can amount to.
2. Satisfaction to be made to the owner (and that first, too, before the
damage be done) is an unquestionable equivalent; and both together, I
think, are a very full answer to any objection in that case.
Besides this Act of Parliament, a commission must be granted to fifteen
at least, in the name of the undertakers, to whom every county shall have
power to join ten, who are to sit with the said fifteen so often and so
long as the said fifteen do sit for affairs relating to that county,
which fifteen, or any seven of them, shall be directors of the works, to
be advised by the said ten, or any five of them, in matters of right and
claim, and the said ten to adjust differences in the countries, and to
have right by process to appeal in the name either of lords of manors, or
privileges of towns or corporations, who shall be either damaged or
encroached upon by the said work. All appeals to be heard and determined
immediately by the said Lord Chancellor, or commission from him, that the
work may receive no interruption.
This commission shall give power to the said fifteen to press waggons,
carts, and horses, oxen and men, and detain them to work a certain
limited time, and within certain limited space of miles from their own
dwellings, and at a certain rate of payment. No men, horses, or carts to
be pressed against their consent during the times of hay-time or harvest,
or upon market-days, if the person aggrieved will make affidavit he is
obliged to be with his horses or carts at the said markets.
It is well known to all who have any knowledge of the condition the
highways in England now lie in that in most places there is a convenient
distance land left open for travelling, either for driving of cattle, or
marching of troops of horse, with perhaps as few lanes or defiles as in
any countries. The cross-roads, which are generally narrow, are yet
broad enough in most places for two carriages to pass; but, on the other
hand, we have on most of the highroads a great deal, if waste land thrown
in (as it were, for an overplus to the highway), which, though it be used
of course by cattle and travellers on occasion, is indeed no benefit at
all either to the traveller as a road or to the poor as a common, or to
the lord of the manor as a waste; upon it grows neither timber nor grass,
in any quantity answerable to the land, but, though to no purpose, is
trodden down, poached, and overrun by drifts of cattle in the winter, or
spoiled with the dust in the summer. And this I have observed in many
parts of England to be as good land as any of the neighbouring
enclosures, as capable of improvement, and to as good purpose.
These lands only being enclosed and manured, leaving the roads to
dimensions without measure sufficient, are the fund upon which I build
the prodigious stock of money that must do this work. These lands (which
I shall afterwards make an essay to value), being enclosed, will be
either saleable to raise money, or fit to exchange with those gentlemen
who must part with some land where the ways are narrow, always reserving
a quantity of these lands to be let out to tenants, the rent to be paid
into the public stock or bank of the undertakers, and to be reserved for
keeping the ways in the same repair, and the said bank to forfeit the
lands if they are not so maintained.
Another branch of the stock must be hands (for a stock of men is a stock
of money), to which purpose every county, city, town, and parish shall be
rated at a set price, equivalent to eight years' payment, for the repair
of highways, which each county, &c., shall raise, not by assessment in
money, but by pressing of men, horses, and carriages for the work (the
men, horses, &c., to be employed by the directors); in which case all
corporal punishments—as of whippings, stocks, pillories, houses of
correction, &c.—might be easily transmitted to a certain number of days'
work on the highways, and in consideration of this provision of men the
country should for ever after be acquitted of any contribution, either in
money or work, for repair of the highways—building of bridges excepted.
There lie some popular objections against this undertaking; and the first
is (the great controverted point of England) enclosure of the common,
which tends to depopulation, and injures the poor.
2. Who shall be judges or surveyors of the work, to oblige the
undertakers to perform to a certain limited degree?
For the first, "the enclosure of the common"—a clause that runs as far as
to an encroachment upon Magna Charta, and a most considerable branch of
the property of the poor—I answer it thus:—
1. The lands we enclose are not such as from which the poor do indeed
reap any benefit—or, at least, any that is considerable.
2. The bank and public stock, who are to manage this great undertaking,
will have so many little labours to perform and offices to bestow, that
are fit only for labouring poor persons to do, as will put them in a
condition to provide for the poor who are so injured, that can work; and
to those who cannot, may allow pensions for overseeing, supervising, and
the like, which will be more than equivalent.
3. For depopulations, the contrary should be secured, by obliging the
undertakers, at such and such certain distances, to erect cottages, two
at least in a place (which would be useful to the work and safety of the
traveller), to which should be an allotment of land, always sufficient to
invite the poor inhabitant, in which the poor should be tenant for life
gratis, doing duty upon the highway as should be appointed, by which, and
many other methods, the poor should be great gainers by the proposal,
instead of being injured.
4. By this erecting of cottages at proper distances a man might travel
over all England as through a street, where he could never want either
rescue from thieves or directions for his way.
5. This very undertaking, once duly settled, might in a few years so
order it that there should be no poor for the common; and, if so, what
need of a common for the poor? Of which in its proper place.
As to the second objection, "Who should oblige the undertakers to the
performance?" I answer—
1. Their Commission and charter should become void, and all their stock
forfeit, and the lands enclosed and unsold remain as a pledge, which
would be security sufficient.
2. The ten persons chosen out of every county should have power to
inspect and complain, and the Lord Chancellor, upon such complaint, to
make a survey, and to determine by a jury, in which case, on default,
they shall be obliged to proceed.
3. The lands settled on the bank shall be liable to be extended for the
uses mentioned, if the same at any time be not maintained in the
condition at first provided, and the bank to be amerced upon complaint of
the country.
These and other conditions, which on a legal settlement to be made by
wiser heads than mine might be thought on, I do believe would form a
constitution so firm, so fair, and so equally advantageous to the
country, to the poor, and to the public, as has not been put in practice
in these later ages of the world. To discourse of this a little in
general, and to instance in a place perhaps that has not its fellow in
the kingdom—the parish of Islington, in Middlesex. There lies through
this large parish the greatest road in England, and the most frequented,
especially by cattle for Smithfield market; this great road has so many
branches, and lies for so long a way through the parish, and withal has
the inconvenience of a clayey ground, and no gravel at hand, that,
modestly speaking, the parish is not able to keep it in repair; by which
means several cross-roads in the parish lie wholly unpassable, and carts
and horses (and men too) have been almost buried in holes and sloughs;
and the main road itself has for many years lain in a very ordinary
condition, which occasioned several motions in Parliament to raise a toll
at Highgate for the performance of what it was impossible the parish
should do, and yet was of so absolute necessity to be done. And is it
not very probable the parish of Islington would part with all the waste
land upon their roads, to be eased of the intolerable assessment for
repair of the highway, and answer the poor, who reap but a small benefit
from it, some other way? And yet I am free to affirm that for a grant of
waste and almost useless land, lying open to the highway (those lands to
be improved, as they might easily be), together with the eight years'
assessment to be provided in workmen, a noble, magnificent causeway might
be erected, with ditches on either side, deep enough to receive the
water, and drains sufficient to carry it off, which causeway should be
four feet high at least, and from thirty to forty feet broad, to reach
from London to Barnet, paved in the middle, to keep it coped, and so
supplied with gravel and other proper materials as should secure it from
decay with small repairing.
I hope no man would be so weak now as to imagine that by lands lying open
to the road, to be assigned to the undertakers, I should mean that all
Finchley Common should be enclosed and sold for this work; but, lest
somebody should start such a preposterous objection, I think it is not
improper to mention, that wherever a highway is to be carried over a
large common, forest, or waste, without a hedge on either hand for a
certain distance, there the several parishes shall allot the directors a
certain quantity of the common, to lie parallel with the road, at a
proportioned number of feet to the length and breadth of the said
road—consideration also to be had to the nature of the ground; or else,
giving them only room for the road directly shall suffer them to inclose
in any one spot so much of the said common as shall be equivalent to the
like quantity of land lying by the road. Thus where the land is good and
the materials for erecting a causeway near, the less land may serve; and
on the contrary, the more; but in general allowing them the quantity of
land proportioned to the length of the causeway, and forty rods in
breadth: though where the land is poor, as on downs and plains, the
proportion must be considered to be adjusted by the country.
Another point for the dimensions of roads should be adjusted; and the
breadth of them, I think, cannot be less than thus:
From London every way ten miles the high post-road to be built full forty
feet in breadth and four feet high, the ditches eight feet broad and six
feet deep, and from thence onward thirty feet, and so in proportion.
Cross-roads to be twenty feet broad, and ditches proportioned; no lanes
and passes less than nine feet without ditches.
The middle of the high causeways to be paved with stone, chalk, or
gravel, and kept always two feet higher than the sides, that the water
might have a free course into the ditches; and persons kept in constant
employ to fill up holes, let out water, open drains, and the like, as
there should be occasion—a proper work for highwaymen and such
malefactors, as might on those services be exempted from the gallows.
It may here be objected that eight years' assessment to be demanded down
is too much in reason to expect any of the poorer sort can pay; as, for
instance, if a farmer who keeps a team of horse be at the common
assessment to work a week, it must not be put so hard upon any man as to
work eight weeks together. It is easy to answer this objection.
So many as are wanted, must be had; if a farmer's team cannot be spared
without prejudice to him so long together, he may spare it at sundry
times, or agree to be assessed, and pay the assessment at sundry
payments; and the bank may make it as easy to them as they please.
Another method, however, might be found to fix this work at once. As
suppose a bank be settled for the highways of the county of Middlesex,
which as they are, without doubt, the most used of any in the kingdom, so
also they require the more charge, and in some parts lie in the worst
condition of any in the kingdom.
If the Parliament fix the charge of the survey of the highways upon a
bank to be appointed for that purpose for a certain term of years, the
bank undertaking to do the work, or to forfeit the said settlement.
As thus: suppose the tax on land and tenements for the whole county of
Middlesex does, or should be so ordered as it might, amount to £20,000
per annum more or less, which it now does, and much more, including the
work of the farmers' teams, which must be accounted as money, and is
equivalent to it, with some allowance to be rated for the city of London,
&c., who do enjoy the benefit, and make the most use of the said roads,
both for carrying of goods and bringing provisions to the city, and
therefore in reason ought to contribute towards the highways (for it is a
most unequal thing that the road from Highgate to Smithfield Market, by
which the whole city is, in a manner, supplied with live cattle, and the
road by those cattle horribly spoiled, should lie all upon that one
parish of Islington to repair); wherefore I will suppose a rate for the
highways to be gathered through the city of London of £10,000 per annum
more, which may be appointed to be paid by carriers, drovers, and all
such as keep teams, horses, or coaches, and the like, or many ways, as is
most equal and reasonable; the waste lands in the said county, which by
the consent of the parishes, lords of the manors, and proprietors shall
be allowed to the undertakers, when inclosed and let out, may (the land
in Middlesex generally letting high) amount to £5,000 per annum more.
If, then, an Act of Parliament be procured to settle the tax of £30,000
per annum for eight years, most of which will be levied in workmen and
not in money, and the waste lands for ever, I dare be bold to offer that
the highways for the whole county of Middlesex should be put into the
following form, and the £5,000 per annum land be bound to remain as a
security to maintain them so, and the county be never burdened with any
further tax for the repair of the highways.
And that I may not propose a matter in general, like begging the
question, without demonstration, I shall enter into the particulars how
it may be performed, and that under these following heads of articles:
1. What I propose to do to the highways.
2. What the charge will be.
3. How to be raised.
4. What security for performance.
5. What profit to the undertaker.
1. _What I propose to do to the highways_.—I answer first, not repair
them; and yet secondly, not alter them—that is, not alter the course they
run; but perfectly build them as a fabric. And, to descend to the
particulars, it is first necessary to note which are the roads I mean,
and their dimensions.
First, the high post-roads, and they are for the county of Middlesex as
follows:
From London to Miles.
Staines, which is 15
Colebrook is from Hounslow 5
Uxbridge 15
Bushey, the Old Street-way 10
Barnet, or near it 9
Waltham Cross, in Ware Road 11
Bow 2
67
Besides these, there, are cross-roads, bye-roads, and lanes, which must
also be looked after; and that some of them may be put into condition,
others may be wholly slighted and shut up, or made drift-ways,
bridle-ways, or foot-ways, as may be thought convenient by the counties.
The cross-roads of most repute are as follows:
From To Miles.
London Hackney, Old Ford, and Bow 5
Hackney Dalston and Islington 2
Ditto Hornsey, Muswell Hill, to Whetstone 8
Tottenham The Chase, Southgate, &c., called Green 6
Lanes
Enfield Wash Enfield Town, Whetstone, Totteridge, to 10
Edgworth
London Hampstead, Hendon, and Edgworth 8
Edgworth Stanmore, to Pinner, to Uxbridge 8
London Harrow and Pinner Green 11
Ditto Chelsea, Fulham 4
Brentford Thistleworth, Twittenham, and Kingston 6
Kingston Staines, Colebrook, and Uxbridge 17
Ditto Chertsey Bridge 5
90
Overplus miles 50
140
And because there may be many parts of the crossroads which cannot be
accounted in the number abovementioned, or may slip my knowledge or
memory, I allow an overplus of 50 miles, to be added to the 90 miles
above, which together make the cross-roads of Middlesex to be 140 miles.
For the bye-lanes such as may be slighted need nothing but to be ditched
up; such as are for private use of lands, for carrying off corn, and
driving cattle, are to be looked after by private hands.
But of the last sort, not to be accounted by particulars, in the small
county of Middlesex we cannot allow less in cross-bye-lanes, from village
to village, and from dwelling-houses which stand out of the way to the
roads, than 1,000 miles.
So in the whole county I reckon up—
Miles.
Of the high post-road 67
Of cross-roads less public 140
Of bye-lanes and passes 1,000
1,207
These are the roads I mean, and thus divided under their several
denominations.
To the question, what I would do to them I answer—
(1). For the sixty-seven miles of high post-road I propose to throw up a
firm strong causeway well-bottomed, six feet high in the middle and four
feet on the side, faced with brick or stone, and crowned with gravel,
chalk, or stone, as the several counties they are made through will
afford, being forty-four feet in breadth, with ditches on either side
eight feet broad and four feet deep; so the whole breadth will be sixty
feet, if the ground will permit.
At the end of every two miles, or such like convenient distances, shall
be a cottage erected, with half an acre of ground allowed, which shall be
given gratis, with one shilling per week wages, to such poor man of the
parish as shall be approved, who shall, once at least every day, view his
walk, to open passages for the water to run into the ditches, to fill up
holes or soft places.
Two riders shall be allowed to be always moving the rounds, to view
everything out of repair, and make report to the directors, and to see
that the cottagers do their duty.
(2). For the 140 miles of cross-road a like causeway to be made, but of
different dimensions—the breadth twenty feet, if the ground will allow
it; the ditches four feet broad, three feet deep; the height in the
middle three feet, and on the sides one foot, or two where it may be
needful; to be also crowned with gravel, and one shilling per week to be
allowed to the poor of every parish, the constables to be bound to find a
man to walk on the highway every division for the same purpose as the
cottagers do on the greater roads.
Posts to be set up at every turning to note whither it goes, for the
direction of strangers, and how many miles distant.
(3). For the 1,000 miles of bye-lanes, only good and sufficient care to
keep them in repair as they are, and to carry the water off by clearing
and cutting the ditches, and laying materials where they are wanted.
This is what I propose to do to them, and what, if once performed, I
suppose all people would own to be an undertaking both useful and
honourable.
2. The second question I propose to give an account of is, _What the
charge will be_, which I account thus.
The work of the great causeway I propose, shall not cost less than ten
shillings per foot (supposing materials to be bought, carriage, and men's
labour to be all hired), which for sixty-seven miles in length is no less
than the sum of £176,880; as thus:
Every mile accounted at 1,760 yards, and three feet to the yard, is 5,280
feet, which at ten shillings per foot is £2,640 per mile, and that,
again, multiplied by sixty-seven, makes the sum of £176,880, into which I
include the charge of water-courses, mills to throw off water where
needful, drains, &c.
To this charge must be added, ditching to inclose land for thirty
cottages, and building thirty cottages at £40 each, which is £1,200.
The work of the smaller causeway I propose to finish at the rate of a
shilling per foot, which being for 149 miles in length, at 5,280 feet per
mile, amounts to £36,960.
Ditching, draining, and repairing 1,000 miles, Supposed at three
shillings per rod, as for 320,000 rods, is £48,000, which, added to the
two former accounts, is thus:
£
The high post-roads, or the great causeway 178,080
The small causeway 36,960
Bye-lanes, &c. 48,000
£263,040
If I were to propose some measures for the easing this charge, I could
perhaps lay a scheme down how it may be performed for less than one-half
of this charge.
As first, by a grant of the court at the Old Bailey whereby all such
criminals as are condemned to die for smaller crimes may, instead of
transportation, be ordered a year's work on the highways; others, instead
of whippings, a proportioned time, and the like; which would, by a
moderate computation, provide us generally a supply of 200 workmen, and
coming in as fast as they go off; and let the overseers alone to make
them work.
Secondly, by an agreement with the Guinea Company to furnish 200 <DW64>s,
who are generally persons that do a great deal of work; and all these are
subsisted very reasonably out of a public storehouse.
Thirdly, by carts and horses to be bought, not hired, with a few able
carters; and to the other a few workmen that have judgment to direct the
rest, and thus I question not the great causeway shall be done for four
shillings per foot charge; but of this by-the-bye.
Fourthly, a liberty to ask charities and benevolences to the work.
3. To the question, _How this money shall be raised_. I think if the
Parliament settle the tax on the county for eight years at £30,000 per
annum, no man need ask how it shall be raised . . . It will be easy
enough to raise the money; and no parish can grudge to pay a little
larger rate for such a term, on condition never to be taxed for the
highways any more.
Eight years' assessment at £30,000 per annum is enough to afford to
borrow the money by way of anticipation, if need be; the fund being
secured by Parliament, and appropriated to that use and no other.
4. As to _What security for performance_.
The lands which are inclosed may be appropriated by the same Act of
Parliament to the bank and undertakers, upon condition of performance,
and to be forfeit to the use of the several parishes to which they
belong, in case upon presentation by the grand juries, and reasonable
time given, any part of the roads in such and such parishes be not kept
and maintained in that posture they are proposed to be. Now the lands
thus settled are an eternal security to the country for the keeping the
roads in repair; because, they will always be of so much value over the
needful charge as will make it worth while to the undertakers to preserve
their title to them; and the tenure of them being so precarious as to be
liable to forfeiture on default, they will always be careful to uphold
the causeways.
Lastly, _What profit to the undertakers_. For we must allow them to
gain, and that considerably, or no man would undertake such a work.
To this I propose: first, during the work, allow them out of the stock
£3,000 per annum for management.
After the work is finished, so much of the £5,000 per annum as can be
saved, and the roads kept in good repair, let be their own; and if the
lands secured be not of the value of £5,000 a year, let so much of the
eight years' tax be set apart as may purchase land to make them up; if
they come to more, let the benefit be to the adventurers.
It may be objected here that a tax of £30,000 for eight years will come
in as fast as it can well be laid out, and so no anticipations will be
requisite; for the whole work proposed cannot be probably finished in
less time; and, if so,
The charge of the county amounts to £240,000
The lands saved eight years' revenue 40,000
£280,000
which is £13,000 more than the charge; and if the work be done so much
cheaper, as is mentioned, the profit to the undertaker will be
unreasonable.
To this I say I would have the undertakers bound to accept the salary of
£3,000 per annum for management, and if a whole year's tax can be spared,
either leave it unraised upon the country, or put it in bank to be
improved against any occasion—of building, perhaps, a great bridge; or
some very wet season or frost may so damnify the works as to make them
require more than ordinary repair. But the undertakers should make no
private advantage of such an overplus; there might be ways enough found
for it.
Another objection lies against the possibility of inclosing the lands
upon the waste, which generally belongs to some manor, whose different
tenures may be so cross, and so otherwise encumbered, that even the lords
of those manors, though they were willing, could not convey them.
This may be answered in general, that an Act of Parliament is omnipotent
with respect to titles and tenures of land, and can empower lords and
tenants to consent to what else they could not; as to particulars, they
cannot be answered till they are proposed; but there is no doubt but an
Act of Parliament may adjust it all in one head.
What a kingdom would England be if this were performed in all the
counties of it! And yet I believe it is feasible, even in the worst. I
have narrowly deserved all the considerable ways in that unpassable
county of Sussex, which (especially in some parts in the wild, as they
very properly call it, of the county) hardly admits the country people to
travel to markets in winter, and makes corn dear at market because it
cannot be brought, and cheap at the farmer's house because he cannot
carry it to market; yet even in that county would I undertake to carry on
this proposal, and that to great advantage, if backed with the authority
of an Act of Parliament.
I have seen in that horrible country the road, sixty to a hundred yards
broad, lie from side to side all poached with cattle, the land of no
manner of benefit, and yet no going with a horse, but at every step up to
the shoulders, full of sloughs and holes, and covered with standing
water. It costs them incredible sums of money to repair them; and the
very places that are mended would fright a young traveller to go over
them. The Romans mastered this work, and by a firm causeway made a
highway quite through this deep country, through Darkin in Surrey to
Stansted, and thence to Okeley, and so on to Arundel; its name tells us
what it was made of (for it was called Stone Street), and many visible
parts of it remain to this day.
Now would any lord of a manor refuse to allow forty yards in breadth out
of that road I mentioned, to have the other twenty made into a firm,
fair, and pleasant causeway over that wilderness of a country?
Or would not any man acknowledge that putting this country into a
condition for carriages and travellers to pass would be a great work?
The gentlemen would find the benefit of it in the rent of their land and
price of their timber; the country people would find the difference in
the sale of their goods, which now they cannot carry beyond the first
market town, and hardly thither; and the whole county would reap an
advantage a hundred to one greater than the charge of it. And since the
want we feel of any convenience is generally the first motive to
contrivance for a remedy, I wonder no man over thought of some expedient
for so considerable a defect.
OF ASSURANCES.
ASSURANCES among merchants, I believe, may plead prescription, and have
been of use time out of mind in trade, though perhaps never so much a
trade as now.
It is a compact among merchants. Its beginning being an accident to
trade, and arose from the disease of men's tempers, who, having run
larger adventures in a single bottom than afterwards they found
convenient, grew fearful and uneasy; and discovering their uneasiness to
others, who perhaps had no effects in the same vessel, they offer to bear
part of the hazard for part of the profit: convenience made this a
custom, and custom brought it into a method, till at last it becomes a
trade.
I cannot question the lawfulness of it, since all risk in trade is for
gain, and when I am necessitated to have a greater cargo of goods in such
or such a bottom than my stock can afford to lose, another may surely
offer to go a part with me; and as it is just if I give another part of
the gain, he should run part of the risk, so it is as just that if he
runs part of my risk, he should have part of the gain. Some object the
disparity of the premium to the hazard, when the insurer runs the risk of
£100 on the seas from Jamaica to London for 40s., which, say they, is
preposterous and unequal. Though this objection is hardly worth
answering to men of business, yet it looks something fair to them that
know no better; and for the information of such, I trouble the reader
with a few heads:
First, they must consider the insurer is out no stock.
Secondly, it is but one risk the insurer runs; whereas the assured has
had a risk out, a risk of debts abroad, a risk of a market, and a risk of
his factor, and has a risk of a market to come, and therefore ought to
have an answerable profit.
Thirdly, if it has been a trading voyage, perhaps the adventurer has paid
three or four such premiums, which sometimes make the insurer clear more
by a voyage than the merchant. I myself have paid £100 insurances in
those small premiums on a voyage I have not gotten £50 by; and I suppose
I am not the first that has done so either.
This way of assuring has also, as other arts of trade have, suffered some
improvement (if I may be allowed that term) in our age; and the first
step upon it was an insurance office for houses, to insure them from
fire. Common fame gives the project to Dr. Barebone—a man, I suppose,
better known as a builder than a physician. Whether it were his, or
whose it was, I do not inquire; it was settled on a fund of ground rents,
to answer in case of loss, and met with very good acceptance.
But it was soon followed by another, by way of friendly society, where
all who subscribe pay their quota to build up any man's house who is a
contributor, if it shall happen to be burnt. I won't decide which is the
best, or which succeeded best, but I believe the latter brings in most
money to the contriver.
Only one benefit I cannot omit which they reap from these two societies
who are not concerned in either; that if any fire happen, whether in
houses insured or not insured, they have each of them a set of lusty
fellows, generally watermen, who being immediately called up, wherever
they live, by watchmen appointed, are, it must be confessed, very active
and diligent in helping to put out the fire.
As to any further improvement to be made upon assurances in trade, no
question there may; and I doubt not but on payment of a small duty to the
government the king might be made the general insurer of all foreign
trade, of which more under another head.
I am of the opinion also that an office of insurance erected to insure
the titles of lands, in an age where they are so precarious as now, might
be a project not unlikely to succeed, if established on a good fund. But
I shall say no more to that, because it seems to be a design in hand by
some persons in town, and is indeed no thought of my own.
Insuring of life I cannot admire; I shall say nothing to it but that in
Italy, where stabbing and poisoning is so much in vogue, something may be
said for it, and on contingent annuities; and yet I never knew the thing
much approved of on any account.
OF FRIENDLY SOCIETIES.
ANOTHER branch of insurance is by contribution, or (to borrow the term
from that before mentioned) friendly societies; which is, in short, a
number of people entering into a mutual compact to help one another in
case any disaster or distress fall upon them.
If mankind could agree, as these might be regulated, all things which
have casualty in them might be secured. But one thing is particularly
required in this way of assurances: none can be admitted but such whose
circumstances are (at least, in some degree) alike, and so mankind must
be sorted into classes; and as their contingencies differ, every
different sort may be a society upon even terms; for the circumstances of
people, as to life, differ extremely by the age and constitution of their
bodies and difference of employment—as he that lives on shore against him
that goes to sea, or a young man against an old man, or a shopkeeper
against a soldier, are unequal. I do not pretend to determine the
controverted point of predestination, the foreknowledge and decrees of
Providence. Perhaps, if a man be decreed to be killed in the trenches,
the same foreknowledge ordered him to list himself a soldier, that it
might come to pass, and the like of a seaman. But this I am sure,
speaking of second causes, a seaman or a soldier are subject to more
contingent hazards than other men, and therefore are not upon equal terms
to form such a society; nor is an annuity on the life of such a man worth
so much as it is upon other men: therefore if a society should agree
together to pay the executor of every member so much after the decease of
the said member, the seamen's executors would most certainly have an
advantage, and receive more than they pay. So that it is necessary to
sort the world into parcels—seamen with seamen, soldiers with soldiers,
and the like.
Nor is this a new thing; the friendly society must not pretend to assume
to themselves the contrivance of the method, or think us guilty of
borrowing from them, when we draw this into other branches; for I know
nothing is taken from them but the bare words, "friendly society," which
they cannot pretend to be any considerable piece of invention either.
I can refer them to the very individual practice in other things, which
claims prescription beyond the beginning of the last age, and that is in
our marshes and fens in Essex, Kent, and the Isle of Ely; where great
quantities of land being with much pains and a vast charge recovered out
of the seas and rivers, and maintained with banks (which they call
walls), the owners of those lands agree to contribute to the keeping up
those walls and keeping out the sea, which is all one with a friendly
society; and if I have a piece of land in any level or marsh, though it
bounds nowhere on the sea or river, yet I pay my proportion to the
maintenance of the said wall or bank; and if at any time the sea breaks
in, the damage is not laid upon the man in whose land the breach
happened, unless it was by his neglect, but it lies on the whole land,
and is called a "level lot."
Again, I have known it practised in troops of horse, especially when it
was so ordered that the troopers mounted themselves; where every private
trooper has agreed to pay, perhaps, 2d. per diem out of his pay into a
public stock, which stock was employed to remount any of the troop who by
accident should lose his horse.
Again, the sailors' contribution to the Chest at Chatham is another
friendly society, and more might be named.
To argue against the lawfulness of this would be to cry down common
equity as well as charity: for as it is kind that my neighbour should
relieve me if I fall into distress or decay, so it is but equal he should
do so if I agreed to have done the same for him; and if God Almighty has
commanded us to relieve and help one another in distress, surely it must
be commendable to bind ourselves by agreement to obey that command; nay,
it seems to be a project that we are led to by the divine rule, and has
such a latitude in it that for aught I know, as I said, all the disasters
in the world might be prevented by it, and mankind be secured from all
the miseries, indigences, and distresses that happen in the world. In
which I crave leave to be a little particular.
First general peace might be secured all over the world by it, if all the
powers agreed to suppress him that usurped or encroached upon his
neighbour. All the contingencies of life might be fenced against by this
method (as fire is already), as thieves, floods by land, storms by sea,
losses of all sorts, and death itself, in a manner, by making it up to
the survivor.
I shall begin with the seamen; for as their lives are subject to more
hazards than others, they seem to come first in view.
_Of Seamen_.
Sailors are _les enfants perdus_, "the forlorn hope of the world;" they
are fellows that bid defiance to terror, and maintain a constant war with
the elements; who, by the magic of their art, trade in the very confines
of death, and are always posted within shot, as I may say, of the grave.
It is true, their familiarity with danger makes them despise it (for
which, I hope, nobody will say they are the wiser); and custom has so
hardened them that we find them the worst of men, though always in view
of their last moment.
I have observed one great error in the custom of England relating to
these sort of people, and which this way of friendly society would be a
remedy for:
If a seaman who enters himself, or is pressed into, the king's service be
by any accident wounded or disabled, to recompense him for the loss, he
receives a pension during life, which the sailors call "smart-money," and
is proportioned to their hurt, as for the loss of an eye, arm, leg, or
finger, and the like: and as it is a very honourable thing, so it is but
reasonable that a poor man who loses his limbs (which are his estate) in
the service of the Government, and is thereby disabled from his labour to
get his bread, should be provided for, and not suffer to beg or starve
for want of those limbs he lost in the service of his country.
But if you come to the seamen in the merchants' service, not the least
provision is made: which has been the loss of many a good ship, with many
a rich cargo, which would otherwise have been saved.
And the sailors are in the right of it, too. For instance, a merchant
ship coming home from the Indies, perhaps very rich, meets with a
privateer (not so strong but that she might fight him and perhaps get
off); the captain calls up his crew, tells them, "Gentlemen, you see how
it is; I don't question but we may clear ourselves of this caper, if you
will stand by me." One of the crew, as willing to fight as the rest, and
as far from a coward as the captain, but endowed with a little more wit
than his fellows, replies, "Noble captain, we are all willing to fight,
and don't question but to beat him off; but here is the case: if we are
taken, we shall be set on shore and then sent home, and lose perhaps our
clothes and a little pay; but if we fight and beat the privateer, perhaps
half a score of us may be wounded and lose our limbs, and then we are
undone and our families. If you will sign an obligation to us that the
owners or merchants shall allow a pension to such as are maimed, that we
may not fight for the ship, and go a-begging ourselves, we will bring off
the ship or sink by her side; otherwise I am not willing to fight, for my
part." The captain cannot do this; so they strike, and the ship and
cargo are lost.
If I should turn this supposed example into a real history, and name the
ship and the captain that did so, it would be too plain to be
contradicted.
Wherefore, for the encouragement of sailors in the service of the
merchant, I would have a friendly society erected for seamen; wherein all
sailors or seafaring men, entering their names, places of abode, and the
voyages they go upon at an office of insurance for seamen, and paying
there a certain small quarterage of 1s. per quarter, should have a sealed
certificate from the governors of the said office for the articles
hereafter mentioned:
I.
If any such seaman, either in fight or by any other accident at sea, come
to be disabled, he should receive from the said office the following sums
of money, either in pension for life, or ready money, as he pleased:
For the loss of £ or
£ per annum for life
An eye 25 2
Both eyes 100 8
One leg 50 4
Both legs 80 6
Right hand 80 6
Left hand 50 4
Right arm 100 8
Left arm 80 6
Both hands 160 12
Both arms 200 16
Any broken arm, or leg, or thigh, towards the cure £10
If taken by the Turks, £50 towards his ransom.
If he become infirm and unable to go to sea or maintain himself by age or
sickness £6 per annum.
To their wives if they are killed or drowned £50
* * * * *
In consideration of this, every seaman subscribing to the society shall
agree to pay to the receipt of the said office his quota of the sum to be
paid whenever, and as often as, such claims are made, the claims to be
entered into the office and upon sufficient proof made, the governors to
regulate the division and publish it in print.
For example, suppose 4,000 seamen subscribe to this society, and after
six months—for no man should claim sooner than six months—a merchant's
ship having engaged a privateer, there comes several claims together, as
thus—
A was wounded and lost one leg £50
B blown up with powder, and has lost an eye 25
C had a great shot took off his arm 100
D with a splinter had an eye struck out 25
E was killed with a great shot; to be paid to his wife 50
£250
The governors hereupon settle the claims of these persons, and make
publication "that whereas such and such seamen, members of the society,
have in an engagement with a French privateer been so and so hurt, their
claims upon the office, by the rules and agreement of the said office,
being adjusted by the governors, amounts to £250, which, being equally
divided among the subscribers, comes to 1s. 3d. each, which all persons
that are subscribers to the said office are desired to pay in for their
respective subscriptions, that the said wounded persons may be relieved
accordingly, as they expect to be relieved if the same or the like
casualty should befall them."
It is but a small matter for a man to contribute, if he gave 1s. 3d. out
of his wages to relieve five wounded men of his own fraternity; but at
the same time to be assured that if he is hurt or maimed he shall have
the same relief, is a thing so rational that hardly anything but a
hare-brained follow, that thinks of nothing, would omit entering himself
into such an office.
I shall not enter further into this affair, because perhaps I may give
the proposal to some persons who may set it on foot, and then the world
may see the benefit of it by the execution.
II.—FOR WIDOWS.
The same method of friendly society, I conceive, would be a very proper
proposal for widows.
We have abundance of women, who have been bred well and lived well,
ruined in a few years, and perhaps left young with a houseful of children
and nothing to support them, which falls generally upon the wives of the
inferior clergy, or of shopkeepers and artificers.
They marry wives with perhaps £300 to £1,000 portion, and can settle no
jointure upon them. Either they are extravagant and idle, and waste it;
or trade decays; or losses or a thousand contingencies happen to bring a
tradesman to poverty, and he breaks. The poor young woman, it may be,
has three or four children, and is driven to a thousand shifts, while he
lies in the Mint or Friars under the dilemma of a statute of bankruptcy;
but if he dies, then she is absolutely undone, unless she has friends to
go to.
Suppose an office to be erected, to be called an office of insurance for
widows, upon the following conditions:
Two thousand women, or their husbands for them, enter their names into a
register to be kept for that purpose, with the names, age, and trade of
their husbands, with the place of their abode, paying at the time of
their entering 5s. down with 1s. 4d. per quarter, which is to the setting
up and support of an office with clerks and all proper officers for the
same; for there is no maintaining such without charge. They receive
every one of them a certificate sealed by the secretary of the office,
and signed by the governors, for the articles hereafter mentioned:
If any one of the women become a widow at any time after six months from
the date of her subscription, upon due notice given, and claim made at
the office in form as shall be directed, she shall receive within six
mouths after such claim made the sum of £500 in money without any
deductions, saving some small fees to the officers, which the trustees
must settle, that they may be known.
In consideration of this, every woman so subscribing obliges herself to
pay, as often as any member of the society becomes a widow, the due
proportion or share, allotted to her to pay towards the £500 for the said
widow, provided her share does not exceed the sum of 5s.
No seamen's or soldiers' wives to be accepted into such a proposal as
this, on the account before mentioned, because the contingencies of their
lives are not equal to others—unless they will admit this general
exception, supposing they do not die out of the kingdom.
It might also be an exception that if the widow that claimed had really,
_bonâ fide_, left her by her husband to her own use, clear of all debts
and legacies, £2,000, she should have no claim, the intent being to aid
the poor, not add to the rich. But there lie a great many objections
against such an article, as—
1. It may tempt some to forswear themselves.
2. People will order their wills so as to defraud the exception.
One exception must be made, and that is, either very unequal matches (as
when a woman of nineteen marries an old man of seventy), or women who
have infirm husbands—I mean, known and publicly so; to remedy which two
things are to be done:
1. The office must have moving officers without doors, who shall inform
themselves of such matters, and if any such circumstances appear, the
office should have fourteen days' time to return their money and declare
their subscriptions void.
2. No woman whose husband had any visible distemper should claim under a
year after her subscription.
One grand objection against this proposal is, how you will oblige people
to pay either their subscription or their quarterage.
To this I answer, by no compulsion (though that might be performed too),
but altogether voluntary; only with this argument to move it, that if
they do not continue their payments, they lose the benefit of their past
contributions.
I know it lies as a fair objection against such a project as this, that
the number of claims are so uncertain that nobody knows what they engage
in when they subscribe, for so many may die annually out of two thousand
as may make my payment £20 or £25 per annum; and if a woman happen to pay
that for twenty years, though she receives the £500 at last, she is a
great loser; but if she dies before her husband, she has lessened his
estate considerably, and brought a great loss upon him.
First, I say to this that I would have such a proposal as this be so fair
and so easy, that if any person who had subscribed found the payments too
high and the claims fall too often, it should be at their liberty at any
time, upon notice given, to be released, and stand obliged no longer;
and, if so, _volenti non fit injuria_. Every one knows best what their
own circumstances will bear.
In the next place, because death is a contingency no man can directly
calculate, and all that subscribe must take the hazard; yet that a
prejudice against this notion may not be built on wrong grounds, let us
examine a little the probable hazard, and see how many shall die annually
out of 2,000 subscribers, accounting by the common proportion of burials
to the number of the living.
Sir William Petty, in his political arithmetic, by a very ingenious
calculation, brings the account of burials in London to be one in forty
annually, and proves it by all the proper rules of proportioned
computation; and I will take my scheme from thence.
If, then, one in forty of all the people in England die, that supposes
fifty to die every year out of our two thousand subscribers; and for a
woman to contribute 5s. to every one, would certainly be to agree to pay
£12 10s. per annum. upon her husband's life, to receive £500 when he
died, and lose it if she died first; and yet this would not be a hazard
beyond reason too great for the gain.
But I shall offer some reasons to prove this to be impossible in our
case: first, Sir William Petty allows the city of London to contain about
a million of people, and our yearly bill of mortality never yet amounted
to 25,000 in the most sickly years we have had (plague years excepted);
sometimes but to 20,000, which is but one in fifty. Now it is to be
considered here that children and ancient people make up, one time with
another, at least one-third of our bills of mortality, and our assurances
lie upon none but the middling age of the people, which is the only age
wherein life is anything steady; and if that be allowed, there cannot die
by his computation above one in eighty of such people every year; but
because I would be sure to leave room for casualty, I will allow one in
fifty shall die out of our number subscribed.
Secondly, it must be allowed that our payments falling due only on the
death of husbands, this one in fifty must not be reckoned upon the two
thousand, for it is to be supposed at least as many women shall die as
men, and then there is nothing to pay; so that one in fifty upon one
thousand is the most that I can suppose shall claim the contribution in a
year, which is twenty claims a year at 5s. each, and is £5 per annum.
And if a woman pays this for twenty years, and claims at last, she is
gainer enough, and no extraordinary loser if she never claims at all.
And I verily believe any office might undertake to demand at all
adventures not above £6 per annum, and secure the subscriber £500 in case
she come to claim as a widow.
I forbear being more particular on this thought, having occasion to be
larger in other prints, the experiment being resolved upon by some
friends who are pleased to think this too useful a project not to be put
in execution, and therefore I refer the reader to the public practice of
it.
I have named these two cases as special experiments of what might be done
by assurances in way of friendly society; and I believe I might, without
arrogance, affirm that the same thought might be improved into methods
that should prevent the general misery and poverty of mankind, and at
once secure us against beggars, parish poor, almshouses, and hospitals;
and by which not a creature so miserable or so poor but should claim
subsistence as their due, and not ask it of charity.
I cannot believe any creature so wretchedly base as to beg of mere
choice, but either it must proceed from want or sordid prodigious
covetousness; and thence I affirm there can be no beggar but he ought to
be either relieved or punished, or both. If a man begs for more
covetousness without want, it is a baseness of soul so extremely sordid
as ought to be used with the utmost contempt, and punished with the
correction due to a dog. If he begs for want, that want is procured by
slothfulness and idleness, or by accident; if the latter, he ought to be
relieved; if the former, he ought to be punished for the cause, but at
the same time relieved also, for no man ought to starve, let his crime be
what it will.
I shall proceed, therefore, to a scheme by which all mankind, be he never
so mean, so poor, so unable, shall gain for himself a just claim to a
comfortable subsistence whosoever age or casualty shall reduce him to a
necessity of making use of it. There is a poverty so far from being
despicable that it is honourable, when a man by direct casualty, sudden
Providence, and without any procuring of his own, is reduced to want
relief from others, as by fire, shipwreck, loss of limbs, and the like.
These are sometimes so apparent that they command the charity of others;
but there are also many families reduced to decay whose conditions are
not so public, and yet their necessities as great. Innumerable
circumstances reduce men to want; and pressing poverty obliges some
people to make their cases public, or starve; and from thence came the
custom of begging, which sloth and idleness has improved into a trade.
But the method I propose, thoroughly put in practice, would remove the
cause, and the effect would cease of course.
Want of consideration is the great reason why people do not provide in
their youth and strength for old age and sickness; and the ensuing
proposal is, in short, only this—that all persons in the time of their
health and youth, while they are able to work and spare it, should lay up
some small inconsiderable part of their gettings as a deposit in safe
hands, to lie as a store in bank to relieve them, if by age or accident
they come to be disabled, or incapable to provide for themselves; and
that if God so bless them that they nor theirs never come to need it, the
overplus may be employed to relieve such as shall.
If an office in the same nature with this were appointed in every county
in England, I doubt not but poverty might easily be prevented, and
begging wholly suppressed.
THE PROPOSAL IS FOR A PENSION OFFICE.
THAT an office be erected in some convenient place, where shall be a
secretary, a clerk, and a searcher, always attending.
That all sorts of people who are labouring people and of honest repute,
of what calling or condition soever, men or women (beggars and soldiers
excepted), who, being sound of their limbs and under fifty years of age,
shall come to the said office and enter their names, trades, and places
of abode into a register to be kept for that purpose, and shall pay down
at the time of the said entering the sum of sixpence, and from thence one
shilling per quarter, shall every one have an assurance under the seal of
the said office for these following conditions:
1. Every such subscriber, if by any casualty (drunkenness and quarrels
excepted) they break their limbs, dislocate joints, or are dangerously
maimed or bruised, able surgeons appointed for that purpose shall take
them into their care, and endeavour their cure gratis.
2. If they are at any time dangerously sick, on notice given to the said
office able physicians shall be appointed to visit them, and give their
prescriptions gratis.
3. If by sickness or accident, as aforesaid, they lose their limbs or
eyes, so as to be visibly disabled to work, and are otherwise poor and
unable to provide for themselves, they shall either be cured at the
charge of the office, or be allowed a pension for subsistence during
life.
4. If they become lame, aged, bedrid, or by real infirmity of body are
unable to work, and otherwise incapable to provide for themselves, on
proof made that it is really and honestly so they shall be taken into a
college or hospital provided for that purpose, and be decently maintained
during life.
5. If they are seamen, and die abroad on board the merchants' ships they
were employed in, or are cast away and drowned, or taken and die in
slavery, their widows shall receive a pension during their widowhood.
6. If they were tradesmen and paid the parish rates, if by decay and
failure of trade they break and are put in prison for debt, they shall
receive a pension for subsistence during close imprisonment.
7. If by sickness or accidents they are reduced to extremities of
poverty for a season, on a true representation to the office they shall
be relieved as the governors shall see cause.
It is to be noted that in the fourth article such as by sickness and age
are disabled from work, and poor, shall be taken into the house and
provided for; whereas in the third article they who are blind or have
lost limbs, &c., shall have pensions allowed them.
The reason of this difference is this:
A poor man or woman that has lost his hand, or leg, or sight, is visibly
disabled, and we cannot be deceived; whereas other infirmities are not so
easily judged of, and everybody would be claiming a pension, when but few
will demand being taken into a hospital but such as are really in want.
And that this might be managed with such care and candour as a design
which carries so good a face ought to be, I propose the following method
for putting it into practice:
I suppose every undertaking of such a magnitude must have some principal
agent to push it forward, who must manage and direct everything, always
with direction of the governors.
And first I will suppose one general office erected for the great
parishes of Stepney and Whitechapel; and as I shall lay down afterwards
some methods to oblige all people to come in and subscribe, so I may be
allowed to suppose here that all the inhabitants of those two large
parishes (the meaner labouring sort, I mean) should enter their names,
and that the number of them should be 100,000, as I believe they would be
at least.
First, there should be named fifty of the principal inhabitants of the
said parishes (of which the church-wardens for the time being, and all
the justices of the peace dwelling in the bounds of the said parish, and
the ministers resident for the time being, to be part) to be governors of
the said office.
The said fifty to be first nominated by the Lord Mayor of London for the
time being, and every vacancy to be supplied in ten days at farthest by
the majority of voices of the rest.
The fifty to choose a committee of eleven, to sit twice a week, of whom
three to be a quorum; with a chief governor, a deputy-governor, and a
treasurer.
In the office, a secretary with clerks of his own, a registrar and two
clerks, four searchers, a messenger (one in daily attendance under
salary), a physician, a surgeon, and four visitors.
In the hospital, more or less (according to the number of people
entertained), a housekeeper, a steward, nurses, a porter, and a chaplain.
For the support of this office, and that the deposit money might go to
none but the persons and uses for whom it is paid, and that it might not
be said officers and salaries was the chief end of the undertaking (as in
many a project it has been), I propose that the manager or undertaker,
whom I mentioned before, be the secretary, who shall have a clerk allowed
him, whose business it shall be to keep the register, take the entries,
and give out the tickets (sealed by the governors and signed by himself),
and to enter always the payment of quarterage of every subscriber. And
that there may be no fraud or connivance, and too great trust be not
reposed in the said secretary, every subscriber who brings his quarterage
is to put it into a great chest, locked up with eleven locks, every
member of the committee to keep a key, so that it cannot be opened but in
the presence of them all; and every time a subscriber pays his
quarterage, the secretary shall give him a sealed ticket thus [Christmas
96] which shall be allowed as the receipt of quarterage for that quarter.
_Note_.—The reason why every subscriber shall take a receipt or ticket
for his quarterage is because this must be the standing law of the
office—that if any subscribers fail to pay their quarterage, they shall
never claim after it until double so much be paid, nor not at all that
quarter, whatever befalls them.
The secretary should be allowed to have 2d. for every ticket of entry he
gives out, and ld. for every receipt he gives for quarterage, to be
accounted for as follows:
One-third to himself in lieu of salary, he being to pay three clerks out
of it.
One-third to the clerks and other officers among them.
And one-third to defray the incident charge of the office.
_Thus calculated_. Per annum.
£ _s._ _d._
100,000 subscribers paying 1d. each 1,666 3 4
every quarter is
One-third
To the secretary per annum and three 555 7 9
clerks
One-third
£ per
annum.
To a registrar 100
To a clerk 50
To four searchers 100
To a physician 100
To a surgeon 100
To four visitors 100
550 0 0
One-third to incident charges, such as
To ten committee-men, 260
5s. each sitting, twice
per week is
To a clerk of 50
committees
To a messenger 40
A house for the office 40
A house for the 100
hospital
Contingencies 70
15_s._ 7_d._ 560 15 7
£1,666 3 4
All the charge being thus paid out of such a trifle as ld. per quarter,
the next consideration is to examine what the incomes of this
subscription may be, and in time what may be the demands upon it.
£ s. d.
If 100,000 persons subscribe, they pay 2,500 0 0
down at their entering each 6d., which
is
And the first year's payment is in stock 20,000 0 0
at 1s. per quarter
It must be allowed that under three 175 0 0
months the subscriptions will not be
well complete; so the payment of
quarterage shall not begin but from the
day after the books are full, or shut
up; and from thence one year is to pass
before any claim can be made; and the
money coming in at separate times, I
suppose no improvement upon it for the
first year, except of the £2,500, which,
lent to the king on some good fund at £7
per cent. interest, advances the first
year
The quarterage of the second year, 19,800 0 0
abating for 1,000 claims
And the interest of the first year's 1,774 10 0
money at the end of the second year,
lent to the king, as aforesaid, at 7 per
cent. interest, is
The quarterage of the third year, 19,400 0 0
abating for claims
The interest of former cash to the end 3,284 8 0
of the third year
Income of three years £66,933 18 0
_Note_.—Any persons may pay 2s. up to 5s. quarterly, if they please, and
upon a claim will be allowed in proportion.
To assign what shall be the charge upon this, where contingency has so
great a share, is not to be done; but by way of political arithmetic a
probable guess may be made.
It is to be noted that the pensions I propose to be paid to persons
claiming by the third, fifth, and sixth articles are thus: every person
who paid 1s. quarterly shall receive 12d. weekly, and so in proportion
every 12d. paid quarterly by any one person to receive so many shillings
weekly, if they come to claim a pension.
The first year no claim is allowed; so the bank has in stock completely
£22,500. From thence we are to consider the number of claims.
Sir William Petty, in his "Political Arithmetic," supposes not above one
in forty to die per annum out of the whole number of people; and I can by
no means allow that the circumstances of our claims will be as frequent
as death, for these reasons:
1. Our subscriptions respect all persons grown and in the prime of their
age; past the first, and providing against the last, part of danger (Sir
William's account including children and old people, which always make up
one-third of the bills of mortality).
2. Our claims will fall thin at first for several years; and let but the
money increase for ten years, as it does in the account for three years,
it would be almost sufficient to maintain the whole number.
3. Allow that casualty and poverty are our debtor side; health,
prosperity, and death are the creditor side of the account; and in all
probable accounts those three articles will carry off three fourth-parts
of the number, as follows: If one in forty shall die annually (as no
doubt they shall, and more), that is 2,500 a year, which in twenty years
is 50,000 of the number; I hope I may be allowed one-third to be out of
condition to claim, apparently living without the help of charity, and
one third in health and body, and able to work; which, put together, make
83,332; so it leaves 16,668 to make claims of charity and pensions in the
first twenty years, and one-half of them must, according to Sir William
Petty, die on our hands in twenty years; so there remains but 8,334.
But to put it out of doubt, beyond the proportion to be guessed at, I
will allow they shall fall thus:
The first year, we are to note, none can claim; and the second year the
number must be very few, but increasing: wherefore I suppose
£
One in every 500 shall claim the second year, which is 500
200; the charge whereof is
One in every 100 the third year is 1,000; the charge 2,500
Together with the former 200 500
£3,500
To carry on the calculation.
£ _s._ _d._
We find the stock at the end of the 66,933 18 0
third year
The quarterage of the fourth year, 19,000 0 0
abating as before
Interest of the stock 4,882 17 6
The quarterage of the fifth year 18,600 0 0
Interest of the stock 6,473 0 0
£115,889 15 6
The charge 3,000 0 0
2,000 to fall the fourth year 5,000 0 0
And the old continued 3,500 0 0
2,000 the fifth year 5,000 0 0
The old continued 11,000 0 0
£27,500 0 0
By this computation the stock is increased above the charge in five years
£89,379 15s. 6d.; and yet here are sundry articles to be considered on
both sides of the account that will necessarily increase the stock and
diminish the charge:
First, in the five years' time 6,200 3,400 0 0
having claimed charity, the number being
abated for in the reckoning above for
stock, it may be allowed new
subscriptions will be taken in to keep
the number full, which in five years
amounts to
Their sixpences is 115 0 0
£3,555 0 0
Which added to £115,889 15s. 6d. augments 119,444 15 6
be stock to
Six thousand two hundred persons claiming 4,000 0 0
help, which falls, to be sure, on the
aged and infirm, I think, at a modest
computation, in five years' time 500 of
them may be dead, which, without allowing
annually, we take at an abatement of
£4,000 out of the charge
Which reduces the charge to 23,500 0 0
Besides this, the interest of the quarterage, which is supposed in the
former account to lie dead till the year is out, which cast up from
quarter to quarter, allowing it to be put out quarterly, as it may well
be, amounts to, by computation for five years, £5,250.
From the fifth year, as near as can be computed, the number of pensioners
being so great, I make no doubt but they shall die off the hands of the
undertaker as fast as they shall fall in, excepting, so much difference
as the payment of every year, which the interest of the stock shall
supply.
_For example_:
£ _s._ _d._
At the end of the fifth year the stock 94,629 15 6
in hand
The payment of the sixth year 20,000 0 0
Interest of the stock 5,408 4 0
£120,037 19 6
Allow an overplus charge for keeping in 10,000 0 0
the house, which will be dearer than
pensions, £10,000 per annum
Charge of the sixth year 22,500 0 0
Balance in cash 87,537 19 6
£120,037 19 6
This also is to be allowed—that all those persons who are kept by the
office in the house shall have employment provided for them, whereby no
persons shall be kept idle, the works to be suited to every one's
capacity without rigour, only some distinction to those who are most
willing to work; the profits of the said work to the stock of the house.
Besides this, there may great and very profitable methods be found out to
improve the stock beyond the settled interest of 7 per cent., which
perhaps may not always be to be had, for the Exchequer is not always
borrowing money; but a bank of £80,000, employed by faithful hands, need
not want opportunities of great, and very considerable improvement.
Also it would be a very good object for persons who die rich to leave
legacies to, which in time might be very well supposed to raise a
standing revenue to it.
I will not say but various contingencies may alter the charge of this
undertaking, and swell the claims beyond proportion further than I extend
it; but all that, and much more, is sufficiently answered in the
calculations by above £80,000 in stock to provide for it.
As to the calculation being made on a vast number of subscribers, and
more than, perhaps, will be allowed likely to subscribe, I think the
proportion may hold good in a few as well as in a great many; and perhaps
if 20,000 subscribed, it might be as effectual. I am indeed willing to
think all men should have sense enough to see the usefulness of such a
design, and be persuaded by their interest to engage in it; but some men
have less prudence than brutes, and will make no provision against age
till it comes; and to deal with such, two ways might be used by authority
to compel them.
1. The churchwardens and justices of peace should send the beadle of the
parish, with an officer belonging to this office, about to the poorer
parishioners to tell them that, since such honourable provision is made
for them to secure themselves in old age from poverty and distress, they
should expect no relief from the parish if they refused to enter
themselves, and by sparing so small a part of their earnings to prevent
future misery.
2. The churchwardens of every parish might refuse the removal of persons
and families into their parish but upon their having entered into this
office.
3. All persons should be publicly desired to forbear giving anything to
beggars, and all common beggars suppressed after a certain time; for this
would effectually suppress beggary at last.
And, to oblige the parishes to do this on behalf of such a project, the
governor of the house should secure the parish against all charges coming
upon them from any person who did subscribe and pay the quarterage, and
that would most certainly oblige any parish to endeavour that all the
labouring meaner people in the parish should enter their names; for in
time it would most certainly take all the poor in the parish off of their
hands.
I know that by law no parish can refuse to relieve any person or family
fallen into distress; and therefore to send them word they must expect no
relief, would seem a vain threatening. But thus far the parish may do:
they shall be esteemed as persons who deserve no relief, and shall be
used accordingly; for who indeed would ever pity that man in his distress
who at the expense of two pots of beer a month might have prevented it,
and would not spare it?
As to my calculations, on which I do not depend either, I say this: if
they are probable, and that in five years' time a subscription of a
hundred thousand persons would have £87,537 19s. 6d. in cash, all charges
paid, I desire any one but to reflect what will not such a sum do. For
instance, were it laid out in the Million Lottery tickets, which are now
sold at £6 each, and bring in £1 per annum for fifteen years, every
£1,000 so laid out pays back in time £2,500, and that time would be as
fast as it would be wanted, and therefore be as good as money; or if laid
out in improving rents, as ground-rents with buildings to devolve in
time, there is no question but a revenue would be raised in time to
maintain one-third part of the number of subscribers, if they should come
to claim charity.
And I desire any man to consider the present state of this kingdom, and
tell me, if all the people of England, old and young, rich and poor, were
to pay into one common bank 4s. per annum a head, and that 4s. duly and
honestly managed, whether the overplus paid by those who die off, and by
those who never come to want, would not in all probability maintain all
that should be poor, and for ever banish beggary and poverty out of the
kingdom.
OF WAGERING.
WAGERING, as now practised by politics and contracts, is become a branch
of assurances; it was before more properly a part of gaming, and as it
deserved, had but a very low esteem; but shifting sides, and the war
providing proper subjects, as the contingencies of sieges, battles,
treaties, and campaigns, it increased to an extraordinary reputation, and
offices were erected on purpose which managed it to a strange degree and
with great advantage, especially to the office-keepers; so that, as has
been computed, there was not less gaged on one side and other, upon the
second siege of Limerick, than two hundred thousand pounds.
How it is managed, and by what trick and artifice it became a trade, and
how insensibly men were drawn into it, an easy account may be given.
I believe novelty was the first wheel that set it on work, and I need
make no reflection upon the power of that charm: it was wholly a new
thing, at least upon the Exchange of London; and the first occasion that
gave it a room among public discourse, was some persons forming wagers on
the return and success of King James, for which the Government took
occasion to use them as they deserved.
I have heard a bookseller in King James's time say, "That if he would
have a book sell, he would have it burnt by the hand of the common
hangman;" the man, no doubt, valued his profit above his reputation; but
people are so addicted to prosecute a thing that seems forbid, that this
very practice seemed to be encouraged by its being contraband.
The trade increased, and first on the Exchange and then in coffee-houses
it got life, till the brokers, those vermin of trade, got hold of it, and
then particular offices were set apart for it, and an incredible resort
thither was to be seen every day.
These offices had not been long in being, but they were thronged with
sharpers and setters as much as the groom-porters, or any gaming-ordinary
in town, where a man had nothing to do but to make a good figure and
prepare the keeper of the office to give him a credit as a good man, and
though he had not a groat to pay, he should take guineas and sign
polities, till he had received, perhaps, £300 or £400 in money, on
condition to pay great odds, and then success tries the man; if he wins
his fortune is made; if not, he's a better man than he was before by just
so much money, for as to the debt, he is your humble servant in the
Temple or Whitehall.
But besides those who are but the thieves of the trade, there is a method
as effectual to get money as possible, managed with more appearing
honesty, but no less art, by which the wagerer, in confederacy with the
office-keeper, shall lay vast sums, great odds, and yet be always sure to
win.
For example: A town in Flanders, or elsewhere, during the war is
besieged; perhaps at the beginning of the siege the defence is vigorous,
and relief probable, and it is the opinion of most people the town will
hold out so long, or perhaps not be taken at all: the wagerer has two or
three more of his sort in conjunction, of which always the office-keeper
is one; and they run down all discourse of the taking the town, and offer
great odds it shall not be taken by such a day. Perhaps this goes on a
week, and then the scale turns; and though they seem to hold the same
opinion still, yet underhand the office-keeper has orders to take all the
odds which by their example was before given against the taking the town;
and so all their first-given odds are easily secured, and yet the people
brought into a vein of betting against the siege of the town too. Then
they order all the odds to be taken as long as they will run, while they
themselves openly give odds, and sign polities, and oftentimes take their
own money, till they have received perhaps double what they at first
laid. Then they turn the scale at once, and cry down the town, and lay
that it shall be taken, till the length of the first odds is fully run;
and by this manage, if the town be taken they win perhaps two or three
thousand pounds, and if it be not taken, they are no losers neither.
It is visible by experience, not one town in ten is besieged but it is
taken. The art of war is so improved, and our generals are so wary, that
an army seldom attempts a siege, but when they are almost sure to go on
with it; and no town can hold out if a relief cannot be had from abroad.
Now, if I can by first laying £500 to £200 with A, that the town shall
not be taken, wheedle in B to lay me £5,000 to £2,000 of the same; and
after that, by bringing down the vogue of the siege, reduce the wagers to
even-hand, and lay £2,000 with C that the town shall not be taken; by
this method, it is plain—
If the town be not taken, I win £2,200 and lose £2,000.
If the town be taken, I win £5,000 and lose £2,500.
This is gaming by rule, and in such a knot it is impossible to lose; for
if it is in any man's or company of men's power, by any artifice to alter
the odds, it is in their power to command the money out of every man's
pocket, who has no more wit than to venture.
OF FOOLS.
OF all persons who are objects of our charity, none move my compassion
like those whom it has pleased God to leave in a full state health and
strength, but deprived of reason to act for themselves. And it is, in my
opinion, one of the greatest scandals upon the understanding of others to
mock at those who want it. Upon this account I think the hospital we
call Bedlam to be a noble foundation, a visible instance of the sense our
ancestors had of the greatest unhappiness which can befall humankind;
since as the soul in man distinguishes him from a brute, so where the
soul is dead (for so it is as to acting) no brute so much a beast as a
man. But since never to have it, and to have lost it, are synonymous in
the effect, I wonder how it came to pass that in the settlement of that
hospital they made no provision for persons born without the use of their
reason, such as we call fools, or, more properly, naturals.
We use such in England with the last contempt, which I think is a strange
error, since though they are useless to the commonwealth, they are only
so by God's direct providence, and no previous fault.
I think it would very well become this wise age to take care of such; and
perhaps they are a particular rent-charge on the great family of mankind,
left by the Maker of us all, like a younger brother, who though the
estate be given from him, yet his father expected the heir should take
some care of him.
If I were to be asked, Who ought in particular to be charged with this
work? I would answer in general those who have a portion of understanding
extraordinary. Not that I would lay a tax upon any man's brains, or
discourage wit by appointing wise men to maintain fools; but, some
tribute is due to God's goodness for bestowing extraordinary gifts; and
who can it be better paid to than such as suffer for want of the same
bounty?
For the providing, therefore, some subsistence for such that natural
defects may not be exposed:
It is proposed that a fool-house be erected, either by public authority,
or by the city, or by an Act of Parliament, into which all that are
naturals or born fools, without respect or distinction, should be
admitted and maintained.
For the maintenance of this, a small stated contribution, settled by the
authority of an Act of Parliament, without any damage to the persons
paying the same, might be very easily raised by a tax upon learning, to
be paid by the authors of books:
Every book that shall be printed in folio, from 40 sheets £5
and upwards, to pay at the licensing (for the whole
impression)
Under 40 sheets 40s.
Every quarto 20s.
Every octavo of 10 sheets and upward 20s.
Every octavo under 10 sheets, and every bound book in 12mo 10s.
Every stitched pamphlet 2s.
Reprinted copies the same rates.
This tax to be paid into the Chamber of London for the space of twenty
years, would, without question, raise a fund sufficient to build and
purchase a settlement for this house.
I suppose this little tax being to be raised at so few places as the
printing-presses, or the licensers of books, and consequently the charge
but very small in gathering, might bring in about £1,500 per annum for
the term of twenty years, which would perform the work to the degree
following:
The house should be plain and decent (for I don't think the ostentation
of buildings necessary or suitable to works of charity), and be built
somewhere out of town for the sake of the air.
The building to cost about £1,000, or, if the revenue exceed, to cost
£2,000 at most, and the salaries mean in proportion.
In the House. Per annum.
A steward £30
A purveyor 20
A cook 20
A butler 20
Six women to assist the cook and clean the house, £4 24
each
Six nurses to tend the people, £3 each 18
A chaplain 20
£152
A hundred alms-people at £8 per annum, diet, &c. 800
£952
The table for the officers, and contingencies, and 500
clothes for the alms-people, and firing, put together
An auditor of the accounts, a committee of the governors, and two
clerks.
Here I suppose £1,500 per annum revenue, to be settled upon the house,
which, it is very probable might be raised from the tax aforesaid. But
since an Act of Parliament is necessary to be had for the collecting this
duty, and that taxes for keeping of fools would be difficultly obtained,
while they are so much wanted for wise men, I would propose to raise the
money by voluntary charity, which would be a work that would leave more
honour to the undertakers than feasts and great shows, which our public
bodies too much diminish their stocks with.
But to pass all suppositious ways, which are easily thought of, but
hardly procured, I propose to maintain fools out of our own folly. And
whereas a great deal of money has been thrown about in lotteries, the
following proposal would very easily perfect our work.
A CHARITY-LOTTERY.
That a lottery be set up by the authority of the Lord Mayor and Court of
Aldermen, for a hundred thousand tickets, at twenty shillings each, to be
drawn by the known way and method of drawing lotteries, as the
million-lottery was drawn, in which no allowance to be made to anybody,
but the fortunate to receive the full sum of one hundred thousand pounds
put in, without discount, and yet this double advantage to follow:
1. That an immediate sum of one hundred thousand pounds shall be raised
and paid into the Exchequer for the public use.
2. A sum of above twenty thousand pounds be gained, to be put into the
hands of known trustees, to be laid out in a charity for the maintenance
of the poor.
That as soon as the money shall be come in, it shall be paid into the
Exchequer, either on some good fund, if any suitable, or on the credit of
the Exchequer; and that when the lottery is drawn, the fortunate to
receive tallies or bills from the Exchequer for their money, payable at
four years.
The Exchequer receives this money, and gives out tallies according to the
prizes, when it is drawn, all payable at four years; and the interest of
this money for four years is struck in tallies proportioned to the
maintenance; which no parish would refuse that subsisted them wholly
before.
I make no question but that if such a hospital was erected within a mile
or two of the city, one great circumstance would happen, viz., that the
common sort of people, who are very much addicted to rambling in the
fields, would make this house the customary walk, to divert themselves
with the objects to be seen there, and to make what they call sport with
the calamity of others, as is now shamefully allowed in Bedlam.
To prevent this, and that the condition of such, which deserves pity, not
contempt, might not be the more exposed by this charity, it should be
ordered: that the steward of the house be in commission of the peace
within the precincts of the house only, and authorised to punish by
limited fines or otherwise any person that shall offer any abuse to the
poor alms-people, or shall offer to make sport at their condition.
If any person at reading of this should be so impertinent as to ask to
what purpose I would appoint a chaplain in a hospital of fools, I could
answer him very well by saying, for the use of the other persons,
officers, and attendants in the house. But besides that, pray, why not a
chaplain for fools, as well as for knaves, since both, though in a
different manner, are incapable of reaping any benefit by religion,
unless by some invisible influence they are made docile; and since the
same secret power can restore these to their reason, as must make the
other sensible, pray why not a chaplain? Idiots indeed were denied the
communion in the primitive churches, but I never read they were not to be
prayed for, or were not admitted to hear.
If we allow any religion, and a Divine Supreme Power, whose influence
works invisibly on the hearts of men (as he must be worse than the people
we talk of, who denies it), we must allow at the same time that Power can
restore the reasoning faculty to an idiot, and it is our part to use the
proper means of supplicating Heaven to that end, leaving the disposing
part to the issue of unalterable Providence.
The wisdom of Providence has not left us without examples of some of the
most stupid natural idiots in the world who have been restored to their
reason, or, as one would think, had reason infused after a long life of
idiotism; perhaps, among other wise ends, to confute that sordid
supposition that idiots have no souls.
OF BANKRUPTS.
THIS chapter has some right to stand next to that of fools, for besides
the common acceptation of late, which makes every unfortunate man a fool,
I think no man so much made a fool of as a bankrupt.
If I may be allowed so much liberty with our laws, which are generally
good, and above all things are tempered with mercy, lenity, and freedom,
this has something in it of barbarity; it gives a loose to the malice and
revenge of the creditor, as well as a power to right himself, while it
leaves the debtor no way to show himself honest. It contrives all the
ways possible to drive the debtor to despair, and encourages no new
industry, for it makes him perfectly incapable of anything but starving.
This law, especially as it is now frequently executed, tends wholly to
the destruction of the debtor, and yet very little to the advantage of
the creditor.
1. The severities to the debtor are unreasonable, and, if I may so say,
a little inhuman, for it not only strips him of all in a moment, but
renders him for ever incapable of helping himself, or relieving his
family by future industry. If he escapes from prison, which is hardly
done too, if he has nothing left, he must starve or live on charity; if
he goes to work no man dare pay him his wages, but he shall pay it again
to the creditors; if he has any private stock left for a subsistence he
can put it nowhere; every man is bound to be a thief and take it from
him; if he trusts it in the hands of a friend he must receive it again as
a great courtesy, for that friend is liable to account for it. I have
known a poor man prosecuted by a statute to that degree that all he had
left was a little money which he knew not where to hide; at last, that he
might not starve, he gives it to his brother who had entertained him; the
brother, after he had his money quarrels with him to get him out of his
house, and when he desires him to let him have the money lent him, gives
him this for answer, I cannot pay you safely, for there is a statute
against you; which run the poor man to such extremities that he destroyed
himself. Nothing is more frequent than for men who are reduced by
miscarriage in trade to compound and set up again and get good estates;
but a statute, as we call it, for ever shuts up all doors to the debtor's
recovery, as if breaking were a crime so capital that he ought to be cast
out of human society and exposed to extremities worse than death. And,
which will further expose the fruitless severity of this law, it is easy
to make it appear that all this cruelty to the debtor is so far,
generally speaking, from advantaging the creditors, that it destroys the
estate, consumes it in extravagant charges, and unless the debtor be
consenting, seldom makes any considerable dividends. And I am bold to
say there is no advantage made by the prosecuting of a statute with
severity, but what might be doubly made by methods more merciful. And
though I am not to prescribe to the legislators of the nation, yet by way
of essay I take leave to give my opinion and my experience in the
methods, consequences, and remedies of this law.
All people know, who remember anything of the times when that law was
made, that the evil it was pointed at was grown very rank, and breaking
to defraud creditors so much a trade, that the parliament had good reason
to set up a fury to deal with it; and I am far from reflecting on the
makers of that law, who, no question, saw it was necessary at that time.
But as laws, though in themselves good, are more or less so, as they are
more or less seasonable, squared, and adapted to the circumstances and
time of the evil they are made against; so it were worth while (with
submission) for the same authority to examine:
1. Whether the length of time since that act was made has not given
opportunity to debtors,
(1) To evade the force of the act by ways and shifts to avoid the
power of it, and secure their estates out of the reach of it.
(2) To turn the point of it against those whom it was made to relieve.
Since we see frequently now that bankrupts desire statutes, and procure
them to be taken out against themselves.
2. Whether the extremities of this law are not often carried on beyond
the true intent and meaning of the act itself by persons who, besides
being creditors, are also malicious, and gratify their private revenge by
prosecuting the offender, to the ruin of his family.
If these two points are to be proved, then I am sure it will follow that
this act is now a public grievance to the nation, and I doubt not but
will be one time or other repealed by the same wise authority which made
it.
1. Time and experience has furnished the debtors with ways and means to
evade the force of this statute, and to secure their estate against the
reach of it, which renders it often insignificant, and consequently, the
knave against whom the law was particularly bent gets off, while he only
who fails of mere necessity, and whose honest principle will not permit
him to practise those methods, is exposed to the fury of this act. And
as things are now ordered, nothing is more easy than for a man to order
his estate so that a statute shall have no power over it, or at least but
a little.
If the bankrupt be a merchant, no statute can reach his effects beyond
the seas; so that he has nothing to secure but his books, and away he
goes into the Friars. If a Shopkeeper, he has more difficulty: but that
is made easy, for there are men and carts to be had whose trade it is,
and who in one night shall remove the greatest warehouse of goods or
cellar of wines in the town and carry them off into those nurseries of
rogues, the Mint and Friars; and our constables and watch, who are the
allowed magistrates of the night, and who shall stop a poor little
lurking thief, that it may be has stole a bundle of old clothes, worth
five shilling, shall let them all pass without any disturbance, and
hundred honest men robbed of their estates before their faces, to the
eternal infamy of the justice of the nation.
And were a man but to hear the discourse among the inhabitants of those
dens of thieves, when they first swarm about a new-comer to comfort him,
for they are not all hardened to a like degree at once. "Well," says the
first, "come, don't be concerned, you have got a good parcel of goods
away I promise you, you need not value all the world." "All! would I had
done so," says another, "I'd a laughed at all my creditors." "Ay," says
the young proficient in the hardened trade, "but my creditors!" "Hang
the creditors!" says a third; "why, there's such a one, and such a one,
they have creditors too, and they won't agree with them, and here they
live like gentlemen, and care not a farthing for them. Offer your
creditors half a crown in the pound, and pay it them in old debts, and if
they won't take it let them alone; they'll come after you, never fear
it." "Oh! but a statute," says he again. "Oh! but the devil," cries the
Minter. "Why, 'tis the statutes we live by," say they; "why, if it were
not for statutes, creditors would comply, and debtors would compound, and
we honest fellows here of the Mint would be starved. Prithee, what need
you care for a statute? A thousand statutes can't reach you here." This
is the language of the country, and the new-comer soon learns to speak
it; for I think I may say, without wronging any man, I have known many a
man go in among them honest, that is, without ill design, but I never
knew one come away so again. Then comes a graver sort among this black
crew (for here, as in hell, are fiends of degrees and different
magnitude), and he falls into discourse with the new-comer, and gives him
more solid advice. "Look you, sir, I am concerned to see you melancholy;
I am in your circumstance too, and if you'll accept of it, I'll give you
the best advice I can," and so begins the grave discourse.
The man is in too much trouble not to want counsel, so he thanks him, and
he goes on:—"Send a summons to your creditors, and offer them what you
can propose in the pound (always reserving a good stock to begin the
world again), which if they will take, you are a free man, and better
than you were before; if they won't take it, you know the worst of it,
you are on the better side of the hedge with them: if they will not take
it, but will proceed to a statute, you have nothing to do but to oppose
force with force; for the laws of nature tell you, you must not starve;
and a statute is so barbarous, so unjust, so malicious a way of
proceeding against a man, that I do not think any debtor obliged to
consider anything but his own preservation, when once they go on with
that." "For why," says the old studied wretch, "should the creditors
spend your estate in the commission, and then demand the debt of you too?
Do you owe anything to the commission of the statute?" "No," says he.
"Why, then," says he, "I warrant their charges will come to £200 out of
your estate, and they must have 10s. a day for starving you and your
family. I cannot see why any man should think I am bound in conscience
to pay the extravagance of other men. If my creditors spend £500 in
getting in my estate by a statute, which I offered to surrender without
it, I'll reckon that £500 paid them, let them take it among them, for
equity is due to a bankrupt as well as to any man, and if the laws do not
give it us, we must take it."
This is too rational discourse not to please him, and he proceeds by this
advice; the creditors cannot agree, but take out a statute; and the man
that offered at first it may be 10s. in the pound, is kept in that cursed
place till he has spent it all and can offer nothing, and then gets away
beyond sea, or after a long consumption gets off by an act of relief to
poor debtors, and all the charges of the statute fall among the
creditors. Thus I knew a statute taken out against a shopkeeper in the
country, and a considerable parcel of goods too seized, and yet the
creditors, what with charges and two or three suits at law, lost their
whole debts and 8s. per pound contribution money for charges, and the
poor debtor, like a man under the surgeon's hand, died in the operation.
2. Another evil that time and experience has brought to light from this
act is, when the debtor himself shall confederate with some particular
creditor to take out a statute, and this is a masterpiece of plot and
intrigue. For perhaps some creditor honestly received in the way of
trade a large sum of money of the debtor for goods sold him when he was
_sui juris_, and he by consent shall own himself a bankrupt before that
time, and the statute shall reach back to bring in an honest man's
estate, to help pay a rogue's debt. Or a man shall go and borrow a sum
of money upon a parcel of goods, and lay them to pledge; he keeps the
money, and the statute shall fetch away the goods to help forward the
composition. These are tricks I can give too good an account of, having
more than once suffered by the experiment. I could give a scheme, of
more ways, but I think it is needless to prove the necessity of laying
aside that law, which is pernicious to both debtor and creditor, and
chiefly hurtful to the honest man whom it was made to preserve.
The next inquiry is, whether the extremities of this law are not often
carried on beyond the true intent and meaning of the act itself, for
malicious and private ends to gratify passion and revenge?
I remember the answer a person gave me, who had taken out statutes
against several persons, and some his near relations, who had failed in
his debt; and when I was one time dissuading him from prosecuting a man
who owed me money as well as him, I used this argument with him:—"You
know the man has nothing left to pay." "That's true," says he; "I know
that well enough." "To what purpose, then," said I, "will you prosecute
him?" "Why, revenge is sweet," said he. Now a man that will prosecute a
debtor, not as a debtor, but by way of revenge, such a man is, I think,
not intentionally within the benefit of our law.
In order to state the case right, there are four sorts of people to be
considered in this discourse; and the true case is how to distinguish
them,
1. There is the honest debtor, who fails by visible necessity, losses,
sickness, decay of trade, or the like.
2. The knavish, designing, or idle, extravagant debtor, who fails
because either he has run out his estate in excesses, or on purpose to
cheat and abuse his creditors.
3. There is the moderate creditor, who seeks but his own, but will omit
no lawful means to gain it, and yet will hear reasonable and just
arguments and proposals.
4. There is the rigorous severe creditor, that values not whether the
debtor be honest man or knave, able or unable, but will have his debt,
whether it be to be had or no, without mercy, without compassion, full of
ill language, passion, and revenge.
How to make a law to suit to all these is the case. That a necessary
favour might be shown to the first, in pity and compassion to the
unfortunate, in commiseration of casualty and poverty, which no man is
exempt from the danger of. That a due rigour and restraint be laid upon
the second, that villainy and knavery might not be encouraged by a law.
That a due care be taken of the third, that men's estates may as far as
can be secured to them. And due limits set to the last, that no man may
have an unlimited power over his fellow-subjects, to the ruin of both
life and estate.
All which I humbly conceive might be brought to pass by the following
method, to which I give the title of
_A Court of Inquiries_.
This court should consist of a select number of persons, to be chosen
yearly out of the several wards of the City by the Lord Mayor and Court
of Aldermen, and out of the several Inns of Court by the Lord Chancellor,
or Lord Keeper, for the time being, and to consist of,
A President, } To be chosen by the rest, and
named every year also.
A Secretary,
A Treasurer,
A judge of causes for the proof of debts.
Fifty-two citizens, out of every ward two; of which number to be
twelve merchants.
Two lawyers (barristers at least) out of each of the Inns of Court.
That a Commission of Inquiry into bankrupts' estates be given to these,
confirmed and settled by Act of Parliament, with power to hear, try, and
determine causes as to proof of debts, and disputes in accounts between
debtor and creditor, without appeal.
The office for this court to be at Guildhall, where clerks should be
always attending, and a quorum of the commissioners to sit _de die in
diem_, from three to six o'clock in the afternoon.
To this court every man who finds himself pressed by his affairs, so that
he cannot carry on his business, shall apply himself as follows:—
He shall go to the secretary's office, and give in his name, with this
short petition:—
To the Honourable the President and Commissioners of His Majesty's
Court of Inquiries. The humble petition of A. B., of the Parish of —
in the —
Haberdasher.
Showeth
That your petitioner being unable to carry on his business, by reason
of great losses and decay of trade, and being ready and willing to
make a full and entire discovery of his whole estate, and to deliver
up the same to your honours upon oath, as the law directs for the
satisfaction of his creditors, and having to that purpose entered his
name into the books of your office on the — of this instant.
Your petitioner humbly prays the protection of this Honourable Court.
And shall ever pray, &c.
The secretary is to lay this petition before the commissioners, who shall
sign it of course; and the petitioner shall have an officer sent home
with him immediately, who shall take possession of his house and goods,
and an exact inventory of everything therein shall be taken at his
entrance by other officers also, appointed by the court; according to
which inventory the first officer and the bankrupt also shall be
accountable.
This officer shall supersede even the Sheriff in possession, excepting by
an extent for the king; only with this provision:—
That if the Sheriff be in possession by warrant on judgment obtained by
due course of law, and without fraud or deceit, and, _bonâ fide_, in
possession before the debtor entered his name in the office, in such case
the plaintiff to have a double dividend allotted to his debt; for it was
the fault of the debtor to let execution come upon his goods before he
sought for protection; but this not to be allowed upon judgment
confessed.
If the Sheriff be in possession by _fieri facias_ for debt immediately
due to the king, the officer, however, shall quit his possession to the
commissioners, and they shall see the king's debt fully satisfied before
any division be made to the creditors.
The officers in this case to take no fee from the bankrupt, nor to use
any indecent or uncivil behaviour to the family (which is a most
notorious abuse now permitted to the sheriff's officers), whose fees I
have known, on small executions, on pretence of civility, amount to as
much as the debt, and yet behave themselves with unsufferable insolence
all the while.
This officer being in possession, the goods may be removed, or not
removed; the shop shut up or not shut up; as the bankrupt upon his
reasons given to the commissioners may desire.
The inventory being taken, the bankrupt shall have fourteen days' time,
and more if desired, upon showing good reasons to the commissioners, to
settle his books and draw up his accounts; and then shall deliver up all
his books, together with a full and true account of his whole estate,
real and personal, to which account he shall make oath, and afterwards to
any particular of it, if the commissioners require.
After this account given in, the commissioners shall have power to
examine upon oath all his servants, or any other person; and if it
appears that he has concealed anything, in breach of his oath, to punish
him, as is hereafter specified.
Upon a fair and just surrender of all his estate and effects, _bonâ
fide_, according to the true intent and meaning of the act, the
commissioners shall return to him in money, or such of his goods as he
shall choose, at a value by a just appraisement, £5 per cent. of all the
estate he surrendered, together with a full and free discharge from all
his creditors.
The remainder of the estate of the debtor to be fairly and equally
divided among the creditors, who are to apply themselves to the
commissioners. The commissioners to make a necessary inquiry into the
nature and circumstances of the debts demanded, that no pretended debt be
claimed for the private account of the debtor; in order to which inquiry
they shall administer the following oath to the creditor, for the proof
of the debt.
I, A. B., do solemnly swear and attest that the account hereto
annexed is true and right, and every article therein rightly and
truly stated and charged in the names of the persons to whom they
belong; and that there is no person or name named, concealed, or
altered in the said account by me, or by my knowledge, order, or
consent. And that the said — does really and _bonâ fide_ owe and
stand indebted to me for my own proper account the full sum of —
mentioned in the said account, and that for a fair and just value
made good to him, as by the said account expressed; and also that I
have not made or known of any private contract, promise, or agreement
between him the said — (or any body for him) and me, or any person
whatsoever.
So help me God.
Upon this oath, and no circumstances to render the person suspected, the
creditor shall have an unquestioned right to his dividend, which shall be
made without the delays and charges that attend the commissions of
bankrupts. For,
1. The goods of the debtor shall upon the first meeting of the creditors
be either sold in parcels, as they shall agree, or divided among them in
due proportion to their debts.
2. What debts are standing out, the debtors shall receive summonses from
the commissioners, to pay by a certain time limited; and in the meantime
the secretary is to transmit accounts to the persons owing it, appointing
them a reasonable time to consent or disprove the account.
And every six months a just dividend shall be made among the creditors of
the money received; and so, if the effects lie abroad, authentic
procurations shall be signed by the bankrupt to the commissioners, who
thereupon correspond with the persons abroad, in whose hands such effects
are, who are to remit the same as the commissioners order; the dividend
to be made, as before, every six months, or oftener, if the court see
cause.
If any man thinks the bankrupt has so much favour by these articles, that
those who can dispense with an oath have an opportunity to cheat their
creditors, and that hereby too much encouragement is given to men to turn
bankrupt; let them consider the easiness of the discovery, the difficulty
of a concealment, and the penalty on the offender.
1. I would have a reward of 30 per cent. be provided to be paid to any
person who should make discovery of any part of the bankrupt's estate
concealed by him, which would make discoveries easy and frequent.
2. Any person who should claim any debt among the creditors, for the
account of the bankrupt, or his wife or children, or with design to
relieve them out of it, other or more than is, _bonâ fide_, due to him
for value received, and to be made out; or any person who shall receive
in trust, or by deed of gift, any part of the goods or other estate of
the bankrupt, with design to preserve them for the use of the said
bankrupt, or his wife or children, or with design to conceal them from
the creditors, shall forfeit for every such act £500, and have his name
published as a cheat, and a person not fit to be credited by any man.
This would make it very difficult for the bankrupt to conceal anything.
3. The bankrupt having given his name, and put the officer into
possession, shall not remove out of the house any of his books; but
during the fourteen days' time which he shall have to settle the accounts
shall every night deliver the books into the hands of the officer; and
the commissioners shall have liberty, if they please, to take the books
the first day, and cause duplicates to be made, and then to give them
back to the bankrupt to settle the accounts.
4. If it shall appear that the bankrupt has given in a false account,
has concealed any part of his goods or debts, in breach of his oath, he
shall be set in the pillory at his own door, and be imprisoned during
life without bail.
5. To prevent the bankrupt concealing any debts abroad, it should be
enacted that the name of the bankrupt being entered at the office, where
every man might search gratis, should be publication enough; and that
after such entry, no discharge from the bankrupt should be allowed in
account to any man, but whoever would adventure to pay any money to the
said bankrupt or his order should be still debtor to the estate, and pay
it again to the commissioners.
And whereas wiser heads than mine must be employed to compose this law,
if ever it be made, they will have time to consider of more ways to
secure the estate for the creditors, and, if possible, to tie the hands
of the bankrupt yet faster.
This law, if ever such a happiness should arise to this kingdom, would be
a present remedy for a multitude of evils which now we feel, and which
are a sensible detriment to the trade of this nation.
1. With submission, I question not but it would prevent a great number
of bankrupts, which now fall by divers causes. For,
(1.) It would effectually remove all crafty designed breakings, by
which many honest men are ruined. And
(2.) Of course 'twould prevent the fall of those tradesmen who are
forced to break by the knavery of such.
2. It would effectually suppress all those sanctuaries and refuges of
thieves, the Mint, Friars, Savoy, Rules, and the like; and that these two
ways:—
(1.) Honest men would have no need of it, here being a more safe,
easy, and more honourable way to get out of trouble.
(2.) Knaves should have no protection from those places, and the Act
be fortified against those places by the following clauses, which I
have on purpose reserved to this head.
Since the provision this court of inquiries makes for the ease and
deliverance of every debtor who is honest is so considerable, 'tis most
certain that no man but he who has a design to cheat his creditors will
refuse to accept of the favour; and therefore it should be enacted,
That if any man who is a tradesman or merchant shall break or fail, or
shut up shop, or leave off trade, and shall not either pay or secure to
his creditors their full and whole debts, twenty shillings in the pound,
without abatement or deduction; or shall convey away their books or
goods, in order to bring their creditors to any composition; or shall not
apply to this office as aforesaid, shall be guilty of felony, and upon
conviction of the same shall suffer as a felon, without benefit of
clergy.
And if any such person shall take sanctuary either in the Mint, Friars,
or other pretended privilege place, or shall convey thither any of their
goods as aforesaid, to secure them from their creditors, upon complaint
thereof made to any of His Majesty's Justices of the Peace, they shall
immediately grant warrants to the constable, &c., to search for the said
persons and goods, who shall be aided and assisted by the trained bands,
if need be, without any charge to the creditors, to search for, and
discover the said persons and goods; and whoever were aiding in the
carrying in the said goods, or whoever knowingly received either the
goods or the person, should be also guilty of felony.
For as the indigent debtor is a branch of the commonwealth which deserves
its care, so the wilful bankrupt is one of the worst sort of thieves.
And it seems a little unequal that a poor fellow who for mere want steals
from his neighbour some trifle shall be sent out of the kingdom, and
sometimes out of the world, while a sort of people who defy justice, and
violently resist the law, shall be suffered to carry men's estates away
before their faces, and no officers to be found who dare execute the law
upon them.
Any man would be concerned to hear with what scandal and reproach
foreigners do speak of the impotence of our constitution in this point;
that in a civilised Government, as ours is, the strangest contempt of
authority is shown that can be instanced in the world.
I may be a little the warmer on this head, on account that I have been a
larger sufferer by such means than ordinary. But I appeal to all the
world as to the equity of the case. What the difference is between
having my house broken up in the night to be robbed, and a man coming in
good credit, and with a proffer of ready money in the middle of the day,
and buying £500 of goods, and carrying them directly from my warehouse
into the Mint, and the next day laugh at me, and bid me defiance; yet
this I have seen done. I think 'tis the justest thing in the world that
the last should be esteemed the greater thief, and deserves most to be
hanged.
I have seen a creditor come with his wife and children, and beg of the
debtor only to let him have part of his own goods again, which he had
bought, knowing and designing to break. I have seen him with tears and
entreaties petition for his own, or but some of it, and be taunted and
sworn at, and denied by a saucy insolent bankrupt. That the poor man has
been wholly ruined by the cheat. It is by the villainy of such many an
honest man is undone, families starved and sent a begging, and yet no
punishment prescribed by our laws for it.
By the aforesaid commission of inquiry all this might be most effectually
prevented, an honest, indigent tradesman preserved, knavery detected and
punished; Mints, Friars, and privilege-places suppressed, and without
doubt a great number of insolencies avoided and prevented; of which many
more particulars might be insisted upon, but I think these may be
sufficient to lead anybody into the thought; and for the method, I leave
it to the wise heads of the nation, who know better than I how to state
the law to the circumstances of the crime.
OF ACADEMIES.
WE have in England fewer of these than in any part of the world, at least
where learning is in so much esteem. But to make amends, the two great
seminaries we have are, without comparison, the greatest, I won't say the
best, in the world; and though much might be said here concerning
universities in general, and foreign academies in particular, I content
myself with noting that part in which we seem defective. The French, who
justly value themselves upon erecting the most celebrated academy of
Europe, owe the lustre of it very much to the great encouragement the
kings of France have given to it. And one of the members making a speech
at his entrance tells you that it is not the least of the glories of
their invincible monarch to have engrossed all the learning of the world
in that sublime body.
The peculiar study of the academy of Paris has been to refine and correct
their own language, which they have done to that happy degree that we see
it now spoken in all the courts of Christendom, as the language allowed
to be most universal.
I had the honour once to be a member of a small society, who seemed to
offer at this noble design in England. But the greatness of the work,
and the modesty of the gentlemen concerned, prevailed with them to desist
an enterprise which appeared too great for private hands to undertake.
We want, indeed, a Richelieu to commence such a work. For I am persuaded
were there such a genius in our kingdom to lead the way, there would not
want capacities who could carry on the work to a glory equal to all that
has gone before them. The English tongue is a subject not at all less
worthy the labour of such a society than the French, and capable of a
much greater perfection. The learned among the French will own that the
comprehensiveness of expression is a glory in which the English tongue
not only equals but excels its neighbours; Rapin, St. Evremont, and the
most eminent French authors have acknowledged it. And my lord Roscommon,
who is allowed to be a good judge of English, because he wrote it as
exactly as any ever did, expresses what I mean in these lines:—
"For who did ever in French authors see
The comprehensive English energy?
The weighty bullion of one sterling line,
Drawn to French wire would through whole pages shine."
"And if our neighbours will yield us, as their greatest critic has done,
the preference for sublimity and nobleness of style, we will willingly
quit all pretensions to their insignificant gaiety."
It is great pity that a subject so noble should not have some as noble to
attempt it. And for a method, what greater can be set before us than the
academy of Paris? Which, to give the French their due, stands foremost
among all the great attempts in the learned part of the world.
The present King of England, of whom we have seen the whole world writing
panegyrics and encomiums, and whom his enemies, when their interest does
not silence them, are apt to say more of than ourselves; as in the war he
has given surprising instances of a greatness of spirit more than common:
so in peace, I daresay, with submission, he shall never have an
opportunity to illustrate his memory more than by such a foundation. By
which he shall have opportunity to darken the glory of the French king in
peace, as he has by his daring attempts in the war.
Nothing but pride loves to be flattered, and that only as it is a vice
which blinds us to our own imperfections. I think princes as
particularly unhappy in having their good actions magnified as their evil
actions covered. But King William, who has already won praise by the
steps of dangerous virtue, seems reserved for some actions which are
above the touch of flattery, whose praise is in themselves.
And such would this be. And because I am speaking of a work which seems
to be proper only for the hand of the king himself, I shall not presume
to carry on this chapter to the model, as I have done in other subjects.
Only thus far:
That a society be erected by the king himself, if his Majesty thought
fit, and composed of none but persons of the first figure in learning;
and it were to be wished our gentry were so much lovers of learning that
birth might always be joined with capacity.
The work of this society should be to encourage polite learning, to
polish and refine the English tongue, and advance the so much neglected
faculty of correct language, to establish purity and propriety of style,
and to purge it from all the irregular additions that ignorance and
affectation have introduced; and all those innovations in speech, if I
may call them such, which some dogmatic writers have the confidence to
foster upon their native language, as if their authority were sufficient
to make their own fancy legitimate.
By such a society I daresay the true glory of our English style would
appear; and among all the learned part of the world be esteemed, as it
really is, the noblest and most comprehensive of all the vulgar languages
in the world.
Into this society should be admitted none but persons eminent for
learning, and yet none, or but very few, whose business or trade was
learning. For I may be allowed, I suppose, to say we have seen many
great scholars mere learned men, and graduates in the last degree of
study, whose English has been far from polite, full of stiffness and
affectation, hard words, and long unusual coupling of syllables and
sentences, which sound harsh and untuneable to the ear, and shock the
reader both in expression and understanding.
In short, there should be room in this society for neither clergyman,
physician, nor lawyer. Not that I would put an affront upon the learning
of any of those honourable employments, much less upon their persons.
But if I do think that their several professions do naturally and
severally prescribe habits of speech to them peculiar to their practice,
and prejudicial to the study I speak of, I believe I do them no wrong.
Nor do I deny but there may be, and now are, among some of all those
professions men of style and language, great masters of English, whom few
men will undertake to correct; and where such do at any time appear,
their extraordinary merit should find them a place in this society; but
it should be rare, and upon very extraordinary occasions that such be
admitted.
I would therefore have this society wholly composed of gentlemen; whereof
twelve to be of the nobility, if possible, and twelve private gentlemen,
and a class of twelve to be left open for mere merit, let it be found in
who or what sort it would, which should lie as the crown of their study,
who have done something eminent to deserve it. The voice of this society
should be sufficient authority for the usage of words, and sufficient
also to expose the innovations of other men's fancies; they should
preside with a sort of judicature over the learning of the age, and have
liberty to correct and censure the exorbitance of writers, especially of
translators. The reputation of this society would be enough to make them
the allowed judges of style and language, and no author would have the
impudence to coin without their authority. Custom, which is now our best
authority for words, would always have its original here, and not be
allowed without it. There should be no more occasion to search for
derivations and constructions, and 'twould be as criminal then to coin
words as money.
The exercises of this society would be lectures on the English tongue,
essays on the nature, original, usage, authorities, and differences of
words, or the propriety, parity, and cadence of style, and of the
politeness and manner in writing; reflections upon irregular usages, and
corrections of erroneous customs in words; and, in short, everything that
would appear necessary to the bringing our English tongue to a due
perfection, and our gentlemen to a capacity of writing like themselves;
to banish pride and pedantry, and silence the impudence and impertinence
of young authors, whose ambition is to be known, though it be by their
folly.
I ask leave here for a thought or two about that inundation custom has
made upon our language and discourse by familiar swearing; and I place it
here, because custom has so far prevailed in this foolish vice that a
man's discourse is hardly agreeable without it; and some have taken upon
them to say it is pity it should not be lawful, it is such a grace in a
man's speech, and adds so much vigour to his language.
I desire to be understood right, and that by swearing I mean all those
cursory oaths, curses, execrations, imprecations, asseverations, and by
whatsoever other names they are distinguished, which are used in
vehemence of discourse, in the mouths almost of all men more or less, of
what sort soever.
I am not about to argue anything of their being sinful and unlawful, as
forbid by divine rules; let the parson alone to tell you that, who has,
no question, said as much to as little purpose in this case as in any
other. But I am of the opinion that there is nothing so impertinent, so
insignificant, so senseless, and foolish as our vulgar way of discourse
when mixed with oaths and curses, and I would only recommend a little
consideration to our gentlemen, who have sense and wit enough, and would
be ashamed to speak nonsense in other things, but value themselves upon
their parts, I would but ask them to put into writing the commonplaces of
their discourse, and read them over again, and examine the English, the
cadence, the grammar of them; then let then turn them into Latin, or
translate them into any other language, and but see what a jargon and
confusion of speech they make together.
Swearing, that lewdness of the tongue, that scum and excrement of the
mouth, is of all vices the most foolish and senseless. It makes a man's
conversation unpleasant, his discourse fruitless, and his language
nonsense.
It makes conversation unpleasant, at least to those who do not use the
same foolish way of discourse, and, indeed, is an affront to all the
company who swear not as he does; for if I swear and curse in company I
either presume all the company likes it or affront them who do not.
Then it is fruitless; for no man is believed a jot the more for all the
asseverations, damnings, and swearings he makes. Those who are used to
it themselves do not believe a man the more because they know they are so
customary that they signify little to bind a man's intention, and they
who practise them not have so mean an opinion of those that do as makes
them think they deserve no belief.
Then, they are the spoilers and destroyers of a man's discourse, and turn
it into perfect nonsense; and to make it out I must descend a little to
particulars, and desire the reader a little to foul his mouth with the
brutish, sordid, senseless expressions which some gentlemen call polite
English, and speaking with a grace.
Some part of them indeed, though they are foolish enough, as effects of a
mad, inconsiderate rage, are yet English; as when a man swears he will do
this or, that, and it may be adds, "God damn him he will;" that is, "God
damn him if he don't." This, though it be horrid in another sense, yet
may be read in writing, and is English: but what language is this?
"Jack, God damn me, Jack, how dost do? How hast thou done this long
time, by God?" And then they kiss; and the other, as lewd as himself,
goes on:—
"Dear Tom, I am glad to see thee with all my heart, let me die. Come,
let us go take a bottle, we must not part so; pr'ythee let's go and be
drunk by God."
This is some of our new florid language, and the graces and delicacies of
style, which if it were put into Latin, I would fain know which is the
principal verb.
But for a little further remembrance of this impertinence, go among the
gamesters, and there nothing is more frequent than, "God damn the dice,"
or "God damn the bowls."
Among the sportsmen it is, "God damn the hounds," when they are at a
fault; or, "God damn the horse," if he baulks a leap. They call men
"sons of —," and "dogs," and innumerable instances may be given of the
like gallantry of language, grown now so much a custom.
It is true, custom is allowed to be our best authority for words, and it
is fit it should be so; but reason must be the judge of sense in
language, and custom can never prevail over it. Words, indeed, like
ceremonies in religion, may be submitted to the magistrate; but sense,
like the essentials, is positive, unalterable, and cannot be submitted to
any jurisdiction; it is a law to itself; it is ever the same; even an Act
of Parliament cannot alter it.
Words, and even usages in style, may be altered by custom, and
proprieties in speech differ according to the several dialects of the
country, and according to the different manner in which several languages
do severally express themselves.
But there is a direct signification of words, or a cadence in expression,
which we call speaking sense; this, like truth, is sullen and the same,
ever was and will be so, in what manner, and in what language soever it
is expressed. Words without it are only noise, which any brute can make
as well as we, and birds much better; for words without sense make but
dull music. Thus a man may speak in words, but perfectly unintelligible
as to meaning; he may talk a great deal, but say nothing. But it is the
proper position of words, adapted to their significations, which makes
them intelligible, and conveys the meaning of the speaker to the
understanding of the hearer; the contrary to which we call nonsense; and
there is a superfluous crowding in of insignificant words, more than are
needful to express the thing intended, and this is impertinence; and that
again, carried to an extreme, is ridiculous.
Thus when our discourse is interlined with needless oaths, curses, and
long parentheses of imprecations, and with some of very indirect
signification, they become very impertinent; and these being run to the
extravagant degree instanced in before, become perfectly ridiculous and
nonsense, and without forming it into an argument, it appears to be
nonsense by the contradictoriness; and it appears impertinent by the
insignificancy of the expression.
After all, how little it becomes a gentleman to debauch his mouth with
foul language, I refer to themselves in a few particulars.
This vicious custom has prevailed upon good manners too far; but yet
there are some degrees to which it has not yet arrived.
As, first, the worst slaves to this folly will neither teach it to nor
approve of it in their children. Some of the most careless will indeed
negatively teach it by not reproving them for it; but sure no man ever
ordered his children to be taught to curse or swear.
2. The grace of swearing has not obtained to be a mode yet among the
women: "God damn ye" does not fit well upon a female tongue; it seems to
be a masculine vice, which the women are not arrived to yet; and I would
only desire those gentlemen who practice it themselves to hear a woman
swear: it has no music at all there, I am sure; and just as little does
it become any gentleman, if he would suffer himself to be judged by all
the laws of sense or good manners in the world.
It is a senseless, foolish, ridiculous practice; it is a mean to no
manner of end; it is words spoken which signify nothing; it is folly
acted for the sake of folly, which is a thing even the devil himself
don't practice. The devil does evil, we say, but it is for some design,
either to seduce others, or, as some divines say, from a principle of
enmity to his Maker. Men steal for gain, and murder to gratify their
avarice or revenge; whoredoms and ravishments, adulteries and sodomy, are
committed to please a vicious appetite, and have always alluring objects;
and generally all vices have some previous cause, and some visible
tendency. But this, of all vicious practices, seems the most nonsensical
and ridiculous; there is neither pleasure nor profit, no design pursued,
no lust gratified, but is a mere frenzy of the tongue, a vomit of the
brain, which works by putting a contrary upon the course of nature.
Again, other vices men find some reason or other to give for, or excuses
to palliate. Men plead want to extenuate theft, and strong provocations
to excuse murders, and many a lame excuse they will bring for whoring;
but this sordid habit even those that practise it will own to be a crime,
and make no excuse for it; and the most I could ever hear a man say for
it was that he could not help it.
Besides, as it is an inexcusable impertinence, so it is a breach upon
good manners and conversation, for a man to impose the clamour of his
oaths upon the company he converses with; if there be any one person in
the company that does not approve the way, it is an imposing upon him
with a freedom beyond civility.
To suppress this, laws, Acts of Parliament, and proclamations are baubles
and banters, the laughter of the lewd party, and never had, as I could
perceive, any influence upon the practice; nor are any of our magistrates
fond or forward of putting them in execution.
It must be example, not penalties, must sink this crime; and if the
gentlemen of England would once drop it as a mode, the vice is so foolish
and ridiculous in itself, it would soon grow odious and out of fashion.
This work such an academy might begin, and I believe nothing would so
soon explode the practice as the public discouragement of it by such a
society; where all our customs and habits, both in speech and behaviour,
should receive an authority. All the disputes about precedency of wit,
with the manners, customs, and usages of the theatre, would be decided
here; plays should pass here before they were acted, and the critics
might give their censures and damn at their pleasure; nothing would ever
die which once received life at this original. The two theatres might
end their jangle, and dispute for priority no more; wit and real worth
should decide the controversy, and here should be the infallible judge.
The strife would then be only to do well,
And he alone be crowned who did excel.
Ye call them Whigs, who from the church withdrew,
But now we have our stage dissenters too,
Who scruple ceremonies of pit and box,
And very few are sound and orthodox,
But love disorder so, and are so nice,
They hate conformity, though 'tis in vice.
Some are for patent hierarchy; and some,
Like the old Gauls, seek out for elbow room;
Their arbitrary governors disown,
And build a conventicle stage of their own.
Fanatic beaux make up the gaudy show,
And wit alone appears incognito.
Wit and religion suffer equal fate;
Neglect of both attends the warm debate.
For while the parties strive and countermine,
Wit will as well as piety decline.
Next to this, which I esteem as the most noble and most useful proposal
in this book, I proceed to academies for military studies, and because I
design rather to express my meaning than make a large book, I bring them
all into one chapter.
I allow the war is the best academy in the world, where men study by
necessity and practice by force, and both to some purpose, with duty in
the action, and a reward in the end; and it is evident to any man who
knows the world, or has made any observations on things, what an
improvement the English nation has made during this seven years' war.
But should you ask how clear it first cost, and what a condition England
was in for a war at first on this account—how almost all our engineers
and great officers were foreigners, it may put us in mind how necessary
it is to have our people so practised in the arts of war that they may
not be novices when they come to the experiment.
I have heard some who were no great friends to the Government take
advantage to reflect upon the king, in the beginning of his wars in
Ireland, that he did not care to trust the English, but all his great
officers, his generals, and engineers were foreigners. And though the
case was so plain as to need no answer, and the persons such as deserved
none, yet this must be observed, though it was very strange: that when
the present king took possession of this kingdom, and, seeing himself
entering upon the bloodiest war this age has known, began to regulate his
army, he found but very few among the whole martial part of the nation
fit to make use of for general officers, and was forced to employ
strangers, and make them Englishmen (as the Counts Schomberg, Ginkel,
Solms, Ruvigny, and others); and yet it is to be observed also that all
the encouragement imaginable was given to the English gentlemen to
qualify themselves, by giving no less than sixteen regiments to gentlemen
of good families who had never been in any service and knew but very
little how to command them. Of these, several are now in the army, and
have the rewards suitable to their merit, being major-generals,
brigadiers, and the like.
If, then, a long peace had so reduced us to a degree of ignorance that
might have been dangerous to us, had we not a king who is always followed
by the greatest masters in the world, who knows what peace and different
governors may bring us to again?
The manner of making war differs perhaps as much as anything in the
world; and if we look no further back than our civil wars, it is plain a
general then would hardly be fit to be a colonel now, saving his capacity
of improvement. The defensive art always follows the offensive; and
though the latter has extremely got the start of the former in this age,
yet the other is mightily improving also.
We saw in England a bloody civil war, where, according to the old temper
of the English, fighting was the business. To have an army lying in such
a post as not to be able to come at them was a thing never heard of in
that war; even the weakest party would always come out and fight (Dunbar
fight, for instance); and they that were beaten to-day would fight again
to-morrow, and seek one another out with such eagerness, as if they had
been in haste to have their brains knocked out. Encampments,
intrenchments, batteries, counter-marchings, fortifying of camps, and
cannonadings were strange and almost unknown things; and whole campaigns
were passed over, and hardly any tents made use of. Battles, surprises,
storming of towns, skirmishes, sieges, ambuscades, and beating up
quarters was the news of every day. Now it is frequent to have armies of
fifty thousand men of a side stand at bay within view of one another, and
spend a whole campaign in dodging (or, as it is genteelly called,
observing) one another, and then march off into winter quarters. The
difference is in the maxims of war, which now differ as much from what
they were formerly as long perukes do from piqued beards, or as the
habits of the people do now from what they then were. The present maxims
of the war are:
"Never fight without a manifest advantage."
"And always encamp so as not to be forced to it."
And if two opposite generals nicely observe both these rules, it is
impossible they should ever come to fight.
I grant that this way of making war spends generally more money and less
blood than former wars did; but then it spins wars out to a greater
length; and I almost question whether, if this had been the way of
fighting of old, our civil war had not lasted till this day. Their maxim
was:
"Wherever you meet your enemy, fight him."
But the case is quite different now; and I think it is plain in the
present war that it is not he who has the longest sword, so much as he
who has the longest purse, will hold the war out best. Europe is all
engaged in the war, and the men will never be exhausted while either
party can find money; but he who finds himself poorest must give out
first; and this is evident in the French king, who now inclines to peace,
and owns it, while at the same time his armies are numerous and whole.
But the sinews fail; he finds his exchequer fail, his kingdom drained,
and money hard to come at: not that I believe half the reports we have
had of the misery and poverty of the French are true; but it is manifest
the King of France finds, whatever his armies may do, his money won't
hold out so long as the Confederates, and therefore he uses all the means
possible to procure a peace, while he may do it with the most advantage.
There is no question but the French may hold the war out several years
longer; but their king is too wise to let things run to extremity. He
will rather condescend to peace upon hard terms now than stay longer, if
he finds himself in danger to be forced to worse.
This being the only digression I design to be guilty of, I hope I shall
be excused it.
The sum of all is this: that, since it is so necessary to be in a
condition for war in a time of peace, our people should be inured to it.
It is strange that everything should be ready but the soldier: ships are
ready, and our trade keeps the seamen always taught, and breeds up more;
but soldiers, horsemen, engineers, gunners, and the like must be bred and
taught; men are not born with muskets on their shoulders, nor
fortifications in their heads; it is not natural to shoot bombs and
undermine towns: for which purpose I propose a
_Royal Academy for Military Exercises_.
The founder the king himself; the charge to be paid by the public, and
settled by a revenue from the Crown, to be paid yearly.
I propose this to consist of four parts:
1. A college for breeding up of artists in the useful practice of all
military exercises; the scholars to be taken in young, and be maintained,
and afterwards under the king's care for preferment, as their merit and
His Majesty's favour shall recommend them; from whence His Majesty would
at all times be furnished with able engineers, gunners, fire-masters.
bombardiers, miners, and the like.
The second college for voluntary students in the same exercises; who
should all upon certain limited conditions be entertained, and have all
the advantages of the lectures, experiments, and learning of the college,
and be also capable of several titles, profits, and settlements in the
said college, answerable to the Fellows in the Universities.
The third college for temporary study, into which any person who is a
gentleman and an Englishman, entering his name and conforming to the
orders of the house, shall be entertained like a gentleman for one whole
year gratis, and taught by masters appointed out of the second college.
The fourth college, of schools only, where all persons whatsoever for a
small allowance shall be taught and entered in all the particular
exercises they desire; and this to be supplied by the proficients of the
first college.
I could lay out the dimensions and necessary incidents of all this work,
but since the method of such a foundation is easy and regular from the
model of other colleges, I shall only state the economy of the house.
The building must be very large, and should rather be stately and
magnificent in figure than gay and costly in ornament: and I think such a
house as Chelsea College, only about four times as big, would answer it;
and yet, I believe, might be finished for as little charge as has been
laid out in that palace-like hospital.
The first college should consist of one general, five colonels, twenty
captains.
Being such as graduates by preferment, at first named by the founder; and
after the first settlement to be chosen out of the first or second
colleges; with apartments in the college, and salaries.
£ per ann.
The general 300
The colonels 100
The captains 60
2,000 scholars, among whom shall be the following degrees:
allowed £ per ann.
Governors 100 10
Directors 200 5
Exempts 200 5
Proficients 500
Juniors 1,000
The general to be named by the founder, out of the colonels; the colonels
to be named by the general, out of the captains; the captains out of the
governors; the governors from the directors; and the directors from the
exempts; and so on.
The juniors to be divided into ten schools; the schools to be thus
governed: every school has
100 juniors, in 10 classes.
Every class to have 2 directors.
100 classes of juniors is 1,000
Each class 2 directors 200
1,200
The proficients to be divided into five schools:
Every school to have ten classes of 10 each.
Every class 2 governors.
50 classes of proficients is 500
Each class 2 governors is 100
600
The exempts to be supernumerary, having a small allowance, and maintained
in the college till preferment offer.
The second college to consist of voluntary students, to be taken in,
after a certain degree of learning, from among the proficients of the
first, or from any other schools, after such and such limitations of
learning; who study at their own charge, being allowed certain
privileges; as—
Chambers rent-free on condition of residence.
Commons gratis, for certain fixed terms.
Preferment, on condition of a term of years' residence.
Use of libraries, instruments, and lectures of the college.
This college should have the following preferments, with salaries
£ per ann.
A governor 200
A president 100
50 college-majors 50
200 proficients 10
500 voluntary students, without allowance.
The third and fourth colleges, consisting only of schools for temporary
study, may be thus:
The third—being for gentlemen to learn the necessary arts and exercises
to qualify them for the service of their country, and entertaining them
one whole year at the public charge—may be supposed to have always one
thousand persons on its hands, and cannot have less than 100 teachers,
whom I would thus order:
Every teacher shall continue at least one year, but by allowance two
years at most; shall have £20 per annum extraordinary allowance; shall be
bound to give their constant attendance; and shall have always five
college-majors of the second college to supervise them, who shall command
a month, and then be succeeded by five others, and, so on—£10 per annum
extraordinary to be paid them for their attendance.
The gentlemen who practise to be put to no manner of charge, but to be
obliged strictly to the following articles:
1. To constant residence, not to lie out of the house without leave of
the college-major.
2. To perform all the college exercises, as appointed by the masters,
without dispute.
3. To submit to the orders of the house.
To quarrel or give ill-language should be a crime to be punished by way
of fine only, the college-major to be judge, and the offender be put into
custody till he ask pardon of the person wronged; by which means every
gentleman who has been affronted has sufficient satisfaction.
But to strike challenge, draw, or fight, should be more severely
punished; the offender to be declared no gentleman, his name posted up at
the college-gate, his person expelled the house, and to be pumped as a
rake if ever he is taken within the college-walls.
The teachers of this college to be chosen, one half out of the exempts of
the first college, and the other out of the proficients of the second.
The fourth college, being only of schools, will be neither chargeable nor
troublesome, but may consist of as many as shall offer themselves to be
taught, and supplied with teachers from the other schools.
The proposal, being of so large an extent, must have a proportionable
settlement for its maintenance; and the benefit being to the whole
kingdom, the charge will naturally lie upon the public, and cannot well
be less, considering the number of persons to be maintained, than as
follows.
FIRST COLLEGE.
£ per ann.
The general 300
5 colonels at £100 per ann. each 500
20 captains at 60 ,, 1,200
100 governors at 10 ,, 1,000
200 directors at 5 ,, 1,000
200 exempts at 5 ,, 1,000
2,000 heads for subsistence, at £20 per head per 40,000
ann., including provision, and all the officers'
salaries in the house, as butlers, cooks, purveyors,
nurses, maids, laundresses, stewards, clerks,
servants, chaplains, porters, and attendants, which
are numerous.
SECOND COLLEGE.
A governor 200
A president 100
50 college-majors at £50 per ann. each 2,500
200 proficients at 10 2,000
Commons for 500 students during times of exercises at 2,500
£5 per ann. each
200 proficients' subsistence, reckoning as above 4,000
THIRD COLLEGE.
The gentlemen here are maintained as gentlemen, and 25,000
are to have good tables, who shall therefore have an
allowance at the rate of £25 per head, all officers
to be maintained out of it; which is
100 teachers, salary and subsistence ditto 4,500
50 college-majors at £10 per ann. is 500
Annual charge 86,300
The building to cost 50,000
Furniture, beds, tables, chairs, linen, &c. 10,000
Books, instruments, and utensils for experiments 2,000
So the immediate charge would be 62,000
The annual charge 86,300
To which add the charges of exercises and experiments 3,700
90,000
The king's magazines to furnish them with 500 barrels of gunpowder per
annum for the public uses of exercises and experiments.
In the first of these colleges should remain the governing part, and all
the preferments to be made from thence, to be supplied in course from the
other; the general of the first to give orders to the other, and be
subject only to the founder.
The government should be all military, with a constitution for the same
regulated for that purpose, and a council to hear and determine the
differences and trespasses by the college laws.
The public exercises likewise military, and all the schools be
disciplined under proper officers, who are so in turn or by order of the
general, and continue but for the day.
The several classes to perform several studies, and but one study to a
distinct class, and the persons, as they remove from one study to
another, to change their classes, but so as that in the general exercises
all the scholars may be qualified to act all the several parts as they
may be ordered.
The proper studies of this college should be the following:
Geometry. Bombarding.
Astronomy. Gunnery.
History. Fortification.
Navigation. Encamping.
Decimal arithmetic. Intrenching.
Trigonometry. Approaching.
Dialing. Attacking.
Gauging. Delineation.
Mining. Architecture.
Fireworking. Surveying.
And all arts or sciences appendices to such as these, with exercises for
the body, to which all should be obliged, as their genius and capacities
led them, as:
1. Swimming; which no soldier, and, indeed, no man whatever, ought to be
without.
2. Handling all sorts of firearms.
3. Marching and counter-marching in form.
4. Fencing and the long-staff.
5. Riding and managing, or horsemanship.
6. Running, leaping, and wrestling.
And herewith should also be preserved and carefully taught all the
customs, usages, terms of war, and terms of art used in sieges, marches
of armies and encampments, that so a gentleman taught in this college
should be no novice when he comes into the king's armies, though he has
seen no service abroad. I remember the story of an English gentleman, an
officer at the siege of Limerick, in Ireland, who, though he was brave
enough upon action, yet for the only matter of being ignorant in the
terms of art, and knowing not how to talk camp language, was exposed to
be laughed at by the whole army for mistaking the opening of the
trenches, which he thought had been a mine against the town.
The experiments of these colleges would be as well worth publishing as
the acts of the Royal Society. To which purpose the house must be built
where they may have ground to cast bombs, to raise regular works, as
batteries, bastions, half-moons, redoubts, horn-works, forts, and the
like; with the convenience of water to draw round such works, to exercise
the engineers in all the necessary experiments of draining and mining
under ditches. There must be room to fire great shot at a distance, to
cannonade a camp, to throw all sorts of fireworks and machines that are,
or shall be, invented; to open trenches, form camps, &c.
Their public exercises will be also very diverting, and more worth while
for any gentleman to see than the sights or shows which our people in
England are so fond of.
I believe as a constitution might be formed from these generals, this
would be the greatest, the gallantest and the most useful foundation in
the world. The English gentry would be the best qualified, and
consequently best accepted abroad, and most useful at home of any people
in the world; and His Majesty should never more be exposed to the
necessity of employing foreigners in the posts of trust and service in
his armies.
And that the whole kingdom might in some degree be better qualified for
service, I think the following project would be very useful:
When our military weapon was the long-bow, at which our English nation in
some measure excelled the whole world, the meanest countryman was a good
archer; and that which qualified them so much for service in the war was
their diversion in times of peace, which also had this good effect—that
when an army was to be raised they needed no disciplining: and for the
encouragement of the people to an exercise so publicly profitable an Act
of Parliament was made to oblige every parish to maintain butts for the
youth in the country to shoot at.
Since our way of fighting is now altered, and this destructive engine the
musket is the proper arms for the soldier, I could wish the diversion
also of the English would change too, that our pleasures and profit might
correspond. It is a great hindrance to this nation, especially where
standing armies are a grievance, that if ever a war commence, men must
have at least a year before they are thought fit to face an enemy, to
instruct them how to handle their arms; and new-raised men are called raw
soldiers. To help this—at least, in some, measure—I would propose that
the public exercises of our youth should by some public encouragement
(for penalties won't do it) be drawn off from the foolish boyish sports
of cocking and cricketing, and from tippling, to shooting with a firelock
(an exercise as pleasant as it is manly and generous) and swimming, which
is a thing so many ways profitable, besides its being a great
preservative of health, that methinks no man ought to be without it.
1. For shooting, the colleges I have mentioned above, having provided
for the instructing the gentry at the king's charge, that the gentry, in
return of a favour, should introduce it among the country people, which
might easily be done thus:
If every country gentleman, according to his degree, would contribute to
set-up a prize to be shot for by the town he lives in or the
neighbourhood, about once a year, or twice a year, or oftener, as they
think fit; which prize not single only to him who shoots nearest, but
according to the custom of shooting.
This would certainly set all the young men in England a-shooting, and
make them marksmen; for they would be always practising, and making
matches among themselves too, and the advantage would be found in a war;
for, no doubt, if all the soldiers in a battalion took a true level at
their enemy there would be much more execution done at a distance than
there is; whereas it has been known how that a battalion of men has
received the fire of another battalion, and not lost above thirty or
forty men; and I suppose it will not easily be forgotten how, at the
battle of Agrim, a battalion of the English army received the whole fire
of an Irish regiment of Dragoons, but never knew to this day whether they
had any bullets or no; and I need appeal no further than to any officer
that served in the Irish war, what advantages the English armies made of
the Irish being such wonderful marksmen.
Under this head of academies I might bring in a project for an
_Academy for Women_.
I have often thought of it as one of the most barbarous customs in the
world, considering us as a civilised and a Christian country, that we
deny the advantages of learning to women. We reproach the sex every day
with folly and impertinence, while I am confident, had they the
advantages of education equal to us, they would be guilty of less than
ourselves.
One would wonder indeed how it should happen that women are conversable
at all, since they are only beholding to natural parts for all their
knowledge. Their youth is spent to teach them to stitch and sew, or make
baubles. They are taught to read indeed, and perhaps to write their
names, or so, and that is the height of a woman's education. And I would
but ask any who slight the sex for their understanding, What is a man (a
gentleman, I mean) good for that is taught no more?
I need not give instances, or examine the character of a gentleman with a
good estate, and of a good family, and with tolerable parts, and examine
what figure he makes for want of education.
The soul is placed in the body like a rough diamond, and must be
polished, or the lustre of it will never appear. And it is manifest that
as the rational soul distinguishes us from brutes, so education carries
on the distinction, and makes some less brutish than others. This is too
evident to need any demonstration. But why, then, should women be denied
the benefit of instruction? If knowledge and understanding had been
useless additions to the sex, God Almighty would never have given them
capacities, for He made nothing needless: besides, I would ask such what
they can see in ignorance that they should think it a necessary ornament
to a woman. Or, How much worse is a wise woman than a fool? or, What has
the woman done to forfeit the privilege of being taught? Does she plague
us with her pride and impertinence? Why did we not let her learn, that
she might have had more wit? Shall we upbraid women with folly, when it
is only the error of this inhuman custom that hindered them being made
wiser?
The capacities of women are supposed to be greater and their senses
quicker than those of the men; and what they might be capable of being
bred to is plain from some instances of female wit which this age is not
without, which upbraids us with injustice, and looks as if we denied
women the advantages of education for fear they should vie with the men
in their improvements.
To remove this objection, and that women might have at least a needful
opportunity of education in all sorts of useful learning, I propose the
draft of an academy for that purpose.
I know it is dangerous to make public appearances of the sex; they are
not either to be confined or exposed: the first will disagree with their
inclinations, and the last with their reputations; and therefore it is
somewhat difficult; and I doubt a method proposed by an ingenious lady,
in a little book called, "Advice to the Ladies," would be found
impracticable. For, saving my respect to the sex, the levity which
perhaps is a little peculiar to them (at least in their youth) will not
bear the restraint; and I am satisfied nothing but the height of bigotry
can keep up a nunnery. Women are extravagantly desirous of going to
heaven, and will punish their pretty bodies to get thither; but nothing
else will do it, and even in that case sometimes it falls out that nature
will prevail.
When I talk therefore of an academy for women I mean both the model, the
teaching, and the government different from what is proposed by that
ingenious lady, for whose proposal I have a very great esteem, and also a
great opinion of her wit; different, too, from all sorts of religious
confinement, and, above all, from vows of celibacy.
Wherefore the academy I propose should differ but little from public
schools, wherein such ladies as were willing to study should have all the
advantages of learning suitable to their genius.
But since some severities of discipline more than ordinary would be
absolutely necessary to preserve the reputation of the house, that
persons of quality and fortune might not be afraid to venture their
children thither, I shall venture to make a small scheme by way of essay.
The house I would have built in a form by itself, as well as in a place
by itself.
The building should be of three plain fronts, without any jettings or
bearing-work, that the eye might at a glance see from one coin to the
other; the gardens walled in the same triangular figure, with a large
moat, and but one entrance.
When thus every part of the situation was contrived as well as might be
for discovery, and to render intriguing dangerous, I would have no
guards, no eyes, no spies set over the ladies, but shall expect them to
be tried by the principles of honour and strict virtue.
And if I am asked why, I must ask pardon of my own sex for giving this
reason for it:
I am so much in charity with women, and so well acquainted with men, that
it is my opinion there needs no other care to prevent intriguing than to
keep the men effectually away. For though inclination, which we prettily
call love, does sometimes move a little too visibly in the sex, and
frailty often follows, yet I think verily custom, which we miscall
modesty, has so far the ascendant over the sex that solicitation always
goes before it.
"Custom with women, 'stead of virtue, rules;
It leads the wisest, and commands the fools;
For this alone, when inclinations reign,
Though virtue's fled, will acts of vice restrain.
Only by custom 'tis that virtue lives,
And love requires to be asked before it gives.
For that which we call modesty is pride:
They scorn to ask, and hate to be denied.
'Tis custom thus prevails upon their want;
They'll never beg what, asked, they easily grant.
And when the needless ceremony's over,
Themselves the weakness of the sex discover.
If, then, desires are strong, and nature free,
Keep from her men and opportunity.
Else 'twill be vain to curb her by restraint;
But keep the question off, you keep the saint."
In short, let a woman have never such a coming principle, she will let
you ask before she complies—at least, if she be a woman of any honour.
Upon this ground I am persuaded such measures might be taken that the
ladies might have all the freedom in the world within their own walls,
and yet no intriguing, no indecencies, nor scandalous affairs happen; and
in order to this, the following customs and laws should be observed in
the colleges, of which I would propose one at least in every county in
England, and about ten for the city of London.
After the regulation of the form of the building as before;
1. All the ladies who enter into the house should set their hands to the
orders of the house, to signify their consent to submit to them.
2. As no woman should be received but who declared herself willing, and
that it was the act of her choice to enter herself, so no person should
be confined to continue there a moment longer than the same voluntary
choice inclined her.
3. The charges of the house being to be paid by the ladies, every one
that entered should have only this incumbrance—that she should pay for
the whole year, though her mind should change as to her continuance.
4. An Act of Parliament should make it felony, without clergy, for any
man to enter by force or fraud into the house, or to solicit any woman,
though it were to marry, while she was in the house. And this law would
by no means be severe, because any woman who was willing to receive the
addresses of a man might discharge herself of the house when she pleased;
and, on the contrary, any woman who had occasion might discharge herself
of the impertinent addresses of any person she had an aversion to by
entering into the house.
In this house the persons who enter should be taught all sorts of
breeding suitable to both their genius and their quality, and, in
particular, music and dancing, which it would be cruelty to bar the sex
of, because they are their darlings; but, besides this, they should be
taught languages, as particularly French and Italian; and I would venture
the injury of giving a woman more tongues than one.
They should, as a particular study, be taught all the graces of speech,
and all the necessary air of conversation, which our common education is
so defective in that I need not expose it. They should be brought to
read books, and especially history, and so to read as to make them
understand the world, and be able to know and judge of things when they
hear of them.
To such whose genius would lead them to it I would deny no sort of
learning: but the chief thing in general is to cultivate the
understandings of the sex, that they may be capable of all sorts of
conversation; that, their parts and judgments being improved, they may be
as profitable in their conversation as they are pleasant.
Women, in my observation, have little or no difference in them but as
they are, or are not, distinguished by education. Tempers indeed may in
some degree influence them, but the main distinguishing part is their
breeding.
The whole sex are generally quick and sharp; I believe I may be allowed
to say generally so; for you rarely see them lumpish and heavy when they
are children, as boys will often be. If a woman be well bred, and taught
the proper management of her natural wit, she proves generally very
sensible and retentive; and, without partiality, a woman of sense and
manners is the finest and most delicate part of God's creation, the glory
of her Maker, and the great instance of His singular regard to man (His
darling creature), to whom He gave the best gift either God could bestow
or man receive; and it is the most sordid piece of folly and ingratitude
in the world to withhold from the sex the due lustre which the advantages
of education gives to the natural beauty of their minds.
A woman well bred and well taught, furnished with the additional
accomplishments of knowledge and behaviour, is a creature without
comparison; her society is the emblem of sublimer enjoyments; her person
is angelic, and her conversation heavenly; she is all softness and
sweetness, peace, love, wit, and delight; she is every way suitable to
the sublimest wish, and the man that has such a one to his portion has
nothing to do but to rejoice in her, and be thankful.
On the other hand, suppose her to be the very same woman, and rob her of
the benefit of education, and it follows thus:
If her temper be good, want of education makes her soft and easy.
Her wit, for want of teaching, makes her impertinent and talkative.
Her knowledge, for want of judgment and experience, makes her fanciful
and whimsical.
If her temper be bad, want of breeding makes her worse, and she grows
haughty, insolent, and loud.
If she be passionate, want of manners makes her termagant and a scold,
which is much at one with lunatic.
If she be proud, want of discretion (which still is breeding) makes her
conceited, fantastic, and ridiculous.
And from these she degenerates to be turbulent, clamorous, noisy, nasty,
and "the devil."
Methinks mankind for their own sakes (since, say what we will of the
women, we all think fit one time or other to be concerned with them)
should take some care to breed them up to be suitable and serviceable, if
they expected no such thing as delight from them. Bless us! what care do
we take to breed up a good horse, and to break him well! And what a
value do we put upon him when it is done!—and all because he should be
fit for our use. And why not a woman?—since all her ornaments and
beauty, without suitable behaviour, is a cheat in nature, like the false
tradesman who puts the best of his goods uppermost, that the buyer may
think the rest are of the same goodness.
Beauty of the body, which is the women's glory, seems to be now unequally
bestowed, and nature (or, rather, Providence) to lie under some scandal
about it, as if it was given a woman for a snare to men, and so make a
kind of a she-devil of her: because, they say, exquisite beauty is rarely
given with wit, more rarely with goodness of temper, and never at all
with modesty. And some, pretending to justify the equity of such a
distribution, will tell us it is the effect of the justice of Providence
in dividing particular excellences among all His creatures, "Share and
share alike, as it were," that all might for something or other be
acceptable to one another, else some would be despised.
I think both these notions false; and yet the last, which has the show of
respect to Providence, is the worst; for it supposes Providence to be
indigent and empty, as if it had not wherewith to furnish all the
creatures it had made, but was fain to be parsimonious in its gifts, and
distribute them by piece-meal, for fear of being exhausted.
If I might venture my opinion against an almost universal notion, I would
say most men mistake the proceedings of Providence in this case, and all
the world at this day are mistaken in their practice about it. And,
because the assertion is very bold, I desire to explain myself.
That Almighty First Cause which made us all is certainly the fountain of
excellence, as it is of being, and by an invisible influence could have
diffused equal qualities and perfections to all the creatures it has
made, as the sun does its light, without the least ebb or diminution to
Himself; and has given indeed to every individual sufficient to the
figure His providence had designed him in the world.
I believe it might be defended if I should say that I do suppose God has
given to all mankind equal gifts and capacities, in that He has given
them all souls equally capable; and that the whole difference in mankind
proceeds either from accidental difference in the make of their bodies,
or from the foolish difference of education.
1. _From accidental difference in bodies_.—I would avoid discoursing
here of the philosophical position of the soul in the body: but if it be
true, as philosophers do affirm, that the understanding and memory is
dilated or contracted according to the accidental dimensions of the organ
through which it is conveyed, then, though God has given a soul as
capable to me as another, yet if I have any natural defect in those parts
of the body by which the soul should act, I may have the same soul
infused as another man, and yet he be a wise man and I a very fool. For
example, if a child naturally have a defect in the organ of hearing, so
that he could never distinguish any sound, that child shall never be able
to speak or read, though it have a soul capable of all the
accomplishments in the world. The brain is the centre of the soul's
actings, where all the distinguishing faculties of it reside; and it is
observable, a man who has a narrow contracted head, in which there is not
room for the due and necessary operations of nature by the brain, is
never a man of very great judgment; and that proverb, "A great head and
little wit," is not meant by nature, but is a reproof upon sloth; as if
one should, by way of wonder say, "Fie, fie, you that have a great head
have but little wit; that's strange! that must certainly be your own
fault." From this notion I do believe there is a great matter in the
breed of men and women; not that wise men shall always get wise children:
but I believe strong and healthy bodies have the wisest children; and
sickly, weakly bodies affect the wits as well as the bodies of their
children. We are easily persuaded to believe this in the breeds of
horses, cocks, dogs, and other creatures; and I believe it is as visible
in men.
But to come closer to the business; the great distinguishing difference
which is seen in the world between men and women is in their education;
and this is manifested by comparing it with the difference between one
man or woman and another.
And herein it is that I take upon me to make such a bold assertion, that
all the world are mistaken in their practice about women: for I cannot
think that God Almighty ever made them so delicate, so glorious
creatures, and furnished them with such charms, so agreeable and so
delightful to mankind, with souls capable of the same accomplishments
with men, and all to be only stewards of our houses, cooks, and slaves.
Not that I am for exalting the female government in the least: but, in
short, I would have men take women for companions, and educate them to be
fit for it. A woman of sense and breeding will scorn as much to encroach
upon the prerogative of the man as a man of sense will scorn to oppress
the weakness of the woman. But if the women's souls were refined and
improved by teaching, that word would be lost; to say, "the weakness of
the sex," as to judgment, would be nonsense; for ignorance and folly
would be no more to be found among women than men. I remember a passage
which I heard from a very fine woman; she had wit and capacity enough, an
extraordinary shape and face, and a great fortune, but had been
cloistered up all her time, and, for fear of being stolen, had not had
the liberty of being taught the common necessary knowledge of women's
affairs; and when she came to converse in the world her natural wit made
her so sensible of the want of education that she gave this short
reflection on herself:
"I am ashamed to talk with my very maids," says she, "for I don't know
when they do right or wrong: I had more need go to school than be
married."
I need not enlarge on the loss the defect of education is to the sex, nor
argue the benefit of the contrary practice; it is a thing will be more
easily granted than remedied: this chapter is but an essay at the thing,
and I refer the practice to those happy days, if ever they shall be, when
men shall be wise enough to mend it.
OF A COURT MERCHANT.
I ASK pardon of the learned gentlemen of the long robe if I do them any
wrong in this chapter, having no design to affront them when I say that
in matters of debate among merchants, when they come to be argued by
lawyers at the bar, they are strangely handled. I myself have heard very
famous lawyers make sorry work of a cause between the merchant and his
factor; and when they come to argue about exchanges, discounts, protests,
demurrages, charter-parties, freights, port-charges, assurances,
barratries, bottomries, accounts current, accounts in commission, and
accounts in company, and the like, the solicitor has not been able to
draw a brief, nor the counsel to understand it. Never was young parson
more put to it to make out his text when he is got into the pulpit
without his notes than I have seen a counsel at the bar when he would
make out a cause between two merchants. And I remember a pretty history
of a particular case, by way of instance, when two merchants, contending
about a long factorage account, that had all the niceties of
merchandising in it, and labouring on both sides to instruct their
counsel, and to put them in when they were out, at last they found them
make such ridiculous stuff of it that they both threw up the cause and
agreed to a reference, which reference in one week, without any charge,
ended all the dispute, which they had spent a great deal of money in
before to no purpose.
Nay, the very judges themselves (no reflection upon their learning) have
been very much at a loss in giving instructions to a jury, and juries
much more to understand them; for, when all is done, juries, which are
not always, nor often indeed, of the wisest men, are, to be sure, in
umpires in causes so nice that the very lawyer and judge can hardly
understand them.
The affairs of merchants are accompanied with such variety of
circumstances, such new and unusual contingencies, which change and
differ in every age, with a multitude of niceties and punctilios (and
those, again, altering as the customs and usages of countries and states
do alter), that it has been found impracticable to make any laws that
could extend to all cases. And our law itself does tacitly acknowledge
its own imperfection in this case, by allowing the custom of merchants to
pass as a kind of law in cases of difficulty.
Wherefore it seems to me a most natural proceeding that such affairs
should be heard before, and judged by, such as by known experience and
long practice in the customs and usages of foreign negotiation are of
course the most capable to determine the same.
Besides the reasonableness of the argument there are some cases in our
laws in which it is impossible for a plaintiff to make out his case, or a
defendant to make out his plea; as, in particular, when his proofs are
beyond seas (for no protests, certifications, or procurations are allowed
in our courts as evidence); and the damages are infinite and
irretrievable by any of the proceedings of our laws.
For the answering all these circumstances, a court might be erected by
authority of Parliament, to be composed of six judges commissioners, who
should have power to hear and decide as a court of equity, under the
title of a "Court Merchant."
The proceedings of this court should be short, the trials speedy, the
fees easy, that every man might have immediate remedy where wrong is
done. For in trials at law about merchants' affairs the circumstances of
the case are often such as the long proceedings of courts of equity are
more pernicious than in other cases; because the matters to which they
are generally relating are under greater contingencies than in other
cases, as effects in hands abroad, which want orders, ships, and seamen
lying at demurrage and in pay, and the like.
These six judges should be chosen of the most eminent merchants of the
kingdom, to reside in London, and to have power by commission to summon a
council of merchants, who should decide all cases on the hearing, of both
parties, with appeal to the said judges.
Also to delegate by commission petty councils of merchants in the most
considerable ports of the kingdom for the same purpose.
The six judges themselves to be only judges of appeal; all trials to be
heard before the council of merchants by methods and proceedings singular
and concise.
The council to be sworn to do justice, and to be chosen annually out of
the principal merchants of the city.
The proceedings here should be without delay; the plaintiff to exhibit
his grievance by way of brief, and the defendant to give in his answer,
and a time of hearing to be appointed immediately.
The defendant by motion shall have liberty to put off hearing upon
showing good cause, not otherwise.
At hearing, every man to argue his own cause if he pleases, or introduce
any person to do it for him.
Attestations and protests from foreign parts, regularly procured and
authentically signified in due form, to pass in evidence; affidavits in
due form likewise attested and done before proper magistrates within the
king's dominions, to be allowed as evidence.
The party grieved may appeal to the six judges, before whom they shall
plead by counsel, and from their judgment to have no appeal.
By this method infinite controversies would be avoided and disputes
amicably ended, a multitude of present inconveniences avoided, and
merchandising matters would in a merchant-like manner be decided by the
known customs and methods of trade.
OF SEAMEN.
IT is observable that whenever this kingdom is engaged in a war with any
of its neighbours two great inconveniences constantly follow: one to the
king and one to trade.
1. That to the king is, that he is forced to press seamen for the
manning of his navy, and force them involuntarily into the service: which
way of violently dragging men into the fleet is attended with sundry ill
circumstances, as:
(1.) Our naval preparations are retarded, and our fleets always late for
want of men, which has exposed them not a little, and been the ruin of
many a good and well-laid expedition.
(2.) Several irregularities follow, as the officers taking money to
dismiss able seamen, and filling up their complement with raw and
improper persons.
(3.) Oppressions, quarrellings, and oftentimes murders, by the rashness
of press-masters and the obstinacy of some unwilling to go.
(4.) A secret aversion to the service from a natural principle, common
to the English nation, to hate compulsion.
(5.) Kidnapping people out of the kingdom, robbing houses, and picking
pockets, frequently practised under pretence of pressing, as has been
very much used of late.
With various abuses of the like nature, some to the king, and some to the
subject.
2. To trade. By the extravagant price set on wages for seamen, which
they impose on the merchant with a sort of authority, and he is obliged
to give by reason of the scarcity of men, and that not from a real want
of men (for in the height of a press, if a merchant-man wanted men, and
could get a protection for them, he might have any number immediately,
and none without it, so shy were they of the public service).
The first of these things has cost the king above three millions sterling
since the war, in these three particulars:
1. Charge of pressing on sea and on shore, and in small craft employed
for that purpose.
2. Ships lying in harbour for want of men, at a vast charge of pay and
victuals for those they had.
3. Keeping the whole navy in constant pay and provisions all the winter,
for fear of losing the men against summer, which has now been done
several years, besides bounty money and other expenses to court and
oblige the seamen.
The second of these (viz., the great wages paid by the merchant) has cost
trade, since the war, above twenty millions sterling. The coal trade
gives a specimen of it, who for the first three years of the war gave £9
a voyage to common seamen, who before sailed for 36s.; which, computing
the number of ships and men used in the coal trade, and of voyages made,
at eight hands to a vessel, does, modestly accounting, make £89,600
difference in one year in wages to seamen in the coal trade only.
For other voyages the difference of sailors' wages is 50s, per month and
55s. per month to foremast-men, who before went for 26s. per month;
besides subjecting the merchant to the insolence of the seamen, who are
not now to be pleased with any provisions, will admit no half-pay, and
command of the captains even what they please; nay, the king himself can
hardly please them.
For cure of these inconveniences it is the following project is proposed,
with which the seamen can have no reason to be dissatisfied, nor are not
at all injured; and yet the damage sustained will be prevented, and an
immense sum of money spared, which is now squandered away by the
profuseness and luxury of the seamen. For if prodigality weakens the
public wealth of the kingdom in general, then are the seamen but ill
commonwealths-men, who are not visibly the richer for the prodigious sums
of money paid them either by the king or the merchant.
The project is this: that by an Act of Parliament an office or court be
erected, within the jurisdiction of the Court of Admiralty, and subject
to the Lord High Admiral, or otherwise independent, and subject only to a
parliamentary authority, as the commission for taking and stating the
public accounts.
In this court or office, or the several branches of it (which, to that
end, shall be subdivided and placed in every sea-port in the kingdom),
shall be listed and entered into immediate pay all the seamen in the
kingdom, who shall be divided into colleges or chambers of sundry
degrees, suitable to their several capacities, with pay in proportion to
their qualities; as boys, youths, servants, men able and raw, midshipmen,
officers, pilots, old men, and pensioners.
The circumstantials of this office:
1. No captain or master of any ship or vessel should dare to hire or
carry to sea with him any seamen but such as he shall receive from the
office aforesaid.
2. No man whatsoever, seaman or other, but applying himself to the said
office to be employed as a sailor, should immediately enter into pay, and
receive for every able seaman 24s. per month, and juniors in proportion;
to receive half-pay while unemployed, and liberty to work for themselves:
only to be at call of the office, and leave an account where to be found.
3. No sailor could desert, because no employment would be to be had
elsewhere.
4. All ships at their clearing at the Custom House should receive a
ticket to the office for men, where would be always choice rather than
scarcity, who should be delivered over by the office to the captain or
master without any trouble or delay; all liberty of choice to be allowed
both to master and men, only so as to give up all disputes to the
officers appointed to decide.
_Note_.—By this would be avoided the great charge captains and owners are
at to keep men on board before they are ready to go; whereas now the care
of getting men will be over, and all come on board in one day: for, the
captain carrying the ticket to the office, he may go and choose his men
if he will; otherwise they will be sent on board to him, by tickets sent
to their dwellings to repair on board such a ship.
5. For all these men that the captain or master of the ship takes he
shall pay the office, not the seamen, 28s. per month (which 4s. per month
overplus of wages will be employed to pay the half-pay to the men out of
employ), and so in proportion of wages for juniors.
6. All disputes concerning the mutinying of mariners, or other matters
of debate between the captains and men, to be tried by way of appeal in a
court for that purpose to be erected, as aforesaid.
7. All discounting of wages and time, all damages of goods, averages,
stopping of pay, and the like, to be adjusted by stated and public rules
and laws in print, established by the same Act of Parliament, by which
means all litigious suits in the Court of Admiralty (which are infinite)
would be prevented.
8. No ship that is permitted to enter at the Custom House and take in
goods should ever be refused men, or delayed in the delivering them above
five days after a demand made and a ticket from the Custom House
delivered (general cases, as arrests and embargoes, excepted).
_The Consequences of this Method_.
1. By this means the public would have no want of seamen, and all the
charges and other inconveniences of pressing men would be prevented.
2. The intolerable oppression upon trade, from the exorbitance of wages
and insolence of mariners, would be taken off.
3. The following sums of money should be paid to the office, to lie in
bank as a public fund for the service of the nation, to be disposed of by
order of Parliament, and not otherwise; a committee being a ways
substituted in the intervals of the session to audit the accounts, and a
treasury for the money, to be composed of members of the House, and to be
changed every session of Parliament:
(1). Four shillings per month wages advanced by the merchants to the
office for the men, more than the office pays them.
(2). In consideration of the reducing men's wages, and consequently
freights, to the former prices (or near them), the owners of ships or
merchants shall pay at the importation of all goods forty shillings per
ton freight, to be stated upon all goods and ports in proportion;
reckoning it on wine tonnage from Canaries as the standard, and on
special freights in proportion to the freight formerly paid, and half the
said price in times of peace.
_Note_.—This may well be done, and no burden; for if freights are reduced
to their former prices (or near it), as they will be if wages are so too,
then the merchant may well pay it: as, for instance, freight from Jamaica
to London, formerly at £6 10s. per ton, now at £18 and £20; from
Virginia, at £5 to £6 10s., now at £14, £16, and £17; from Barbadoes, at
£6, now at £16; from Oporto, at £2, now at £6; and the like.
The payment of the above-said sums being a large bank for a fund, and it
being supposed to be in fair hands and currently managed, the merchants
shall further pay upon all goods shipped out, and shipped on board from
abroad, for and from any port of this kingdom, £4 per cent. on the real
value, _bonâ fide_; to be sworn to if demanded. In consideration whereof
the said office shall be obliged to pay and make good all losses,
damages, averages, and casualties whatsoever, as fully as by the custom
of assurances now is done, without any discounts, rebates, or delays
whatsoever; the said £4 per cent. to be stated on the voyage to the
Barbadoes, and enlarged or taken off, in proportion to the voyage, by
rules and laws to be printed and publicly known.
Reserving only, that then, as reason good, the said office shall have
power to direct ships of all sorts, how and in what manner, and how long
they shall sail with or wait for convoys; and shall have power (with
limitations) to lay embargoes on ships, in order to compose fleets for
the benefit of convoys.
These rules, formerly noted, to extend to all trading by sea, the
coasting and home-fishing trade excepted; and for them it should be
ordered—
First, for coals; the colliers being provided with men at 28s. per month,
and convoys in sufficient number, and proper stations from Tynemouth Bar
to the river, so as they need not go in fleets, but as wind and weather
presents, run all the way under the protection of the men-of-war, who
should be continually cruising from station to station, they would be
able to perform their voyage, in as short time as formerly, and at as
cheap pay, and consequently could afford to sell their coals at 17s. per
chaldron, as well as formerly at 15s.
Wherefore there should be paid into the treasury appointed at Newcastle,
by bond to be paid where they deliver, 10s. per chaldron, Newcastle
measure; and the stated price at London to be 27s. per chaldron in the
Pool, which is 30s. at the buyer's house; and is so far from being dear,
a time of war especially, as it is cheaper than ever was known in a war;
and the officers should by proclamation confine the seller to that price.
In consideration also of the charge of convoys, the ships bringing coals
shall all pay £1 per cent. on the value of the ship, to be agreed on at
the office; and all convoy-money exacted by commanders of ships shall be
relinquished, and the office to make good all losses of ships, not goods,
that shall be lost by enemies only.
These heads, indeed, are such as would need some explication, if the
experiment were to be made; and, with submission, would reduce the seamen
to better circumstances; at least, it would have them in readiness for
any public service much easier than by all the late methods of
encouragement by registering seamen, &c.
For by this method all the seamen in the kingdom should be the king's
hired servants, and receive their wages from him, whoever employed them;
and no man could hire or employ them but from him. The merchant should
hire them of the king, and pay the king for them; nor would there be a
seaman in England out of employ—which, by the way, would prevent their
seeking service abroad. If they were not actually at sea they would
receive half-pay, and might be employed in works about the yards, stores,
and navy, to keep all things in repair.
If a fleet or squadron was to be fitted out they would be manned in a
week's time, for all the seamen in England would be ready. Nor would
they be shy of the service; for it is not an aversion to the king's
service, nor it is not that the duty is harder in the men-of-war than the
merchant-men, nor it is not fear of danger which makes our seamen lurk
and hide and hang back in a time of war, but it is wages is the matter:
24s. per month in the king's service, and 40s. to 50s. per month from the
merchant, is the true cause; and the seaman is in the right of it, too;
for who would serve his king and country, and fight, and be knocked on
the head at 24s. per month that can have 50s. without that hazard? And
till this be remedied, in vain are all the encouragements which can be
given to seamen; for they tend but to make them insolent, and encourage
their extravagance.
Nor would this proceeding be any damage to the seamen in general; for
24s. per month wages, and to be kept in constant service (or half-pay
when idle), is really better to the seaman than 45s. per month, as they
now take it, considering how long they often lie idle on shore out of
pay; for the extravagant price of seamen's wages, though it has been an
intolerable burden to trade, has not visibly enriched the sailors, and
they may as well be content with 24s. per month now as formerly.
On the other hand, trade would be sensibly revived by it, the intolerable
price of freights would be reduced, and the public would reap an immense
benefit by the payments mentioned in the proposal; as—
1. 4s. per month upon the wages of all the seamen employed by the
merchant (which if we allow 200,000 seamen always in employ, as there
cannot be less in all the ships belonging to England) is £40,000 per
month.
2. 40s. per ton freight upon all goods imported.
3. 4 per cent. on the value of all goods exported or imported.
4. 10s. per chaldron upon all the coals shipped at Newcastle, and 1 per
cent. on the ships which carry them.
What these four articles would pay to the Exchequer yearly it would be
very difficult to calculate, and I am too near the end of this book to
attempt it: but I believe no tax ever given since this war has come near
it.
It is true, out of this the public would be to pay half-pay to the seamen
who shall be out of employ, and all the losses, and damages on goods and
ships; which, though it might be considerable, would be small, compared
to the payment aforesaid: for as the premium of 4 per cent. is but small,
so the safety lies upon all men being bound to insure. For I believe any
one will grant me this: it is not the smallness of a premium ruins the
insurer, but it is the smallness of the quantity he insures; and I am not
at all ashamed to affirm that, let but a premium of £4 per cent. be paid
into one man's hand for all goods imported and exported, and any man may
be the general insurer of the kingdom, and yet that premium can never
hurt the merchant either.
So that the vast revenue this would raise would be felt nowhere: neither
poor nor rich would pay the more for coals; foreign goods would be
brought home cheaper, and our own goods carried to market cheaper; owners
would get more by ships, merchants by goods; and losses by sea would be
no loss at all to anybody, because repaid by the public stock.
Another unseen advantage would arise by it: we should be able to outwork
all our neighbours, even the Dutch themselves, by sailing as cheap and
carrying goods as cheap in a time of war as in peace—an advantage which
has more in it than is easily thought of, and would have a noble
influence upon all our foreign trade. For what could the Dutch do in
trade if we could carry our goods to Cadiz at 50s. per ton freight, and
they give £8 or £10 and the like in other places? Whereby we could be
able to sell cheaper or get more than our neighbours.
There are several considerable clauses might be added to this proposal
(some of great advantage to the general trade of the kingdom, some to
particular trades, and more to the public), but I avoid being too
particular in things which are but the product of my own private opinion.
If the Government should ever proceed to the experiment, no question but
much more than has been hinted at would appear; nor do I see any great
difficulty in the attempt, or who would be aggrieved at it; and there I
leave it, rather wishing than expecting to see it undertaken.
THE CONCLUSION.
UPON a review of the several chapters of this book I find that, instead
of being able to go further, some things may have suffered for want of
being fully expressed; which if any person object against, I only say, I
cannot now avoid it. I have endeavoured to keep to my title, and offered
but an essay; which any one is at liberty to go on with as they please,
for I can promise no supplement. As to errors of opinion, though I am
not yet convinced of any, yet I nowhere pretend to infallibility.
However, I do not willingly assert anything which I have not good grounds
for. If I am mistaken, let him that finds the error inform the world
better, and never trouble himself to animadvert upon this, since I assure
him I shall not enter into any pen-and-ink contest on the matter.
As to objections which may lie against any of the proposals made in this
book, I have in some places mentioned such as occurred to my thoughts. I
shall never assume that arrogance to pretend no other or further
objections may be raised; but I do really believe no such objection can
be raised as will overthrow any scheme here laid down so as to render the
thing impracticable. Neither do I think but that all men will
acknowledge most of the proposals in this book would be of as great, and
perhaps greater, advantage to the public than I have pretended to.
As for such who read books only to find out the author's _faux pas_, who
will quarrel at the meanness of style, errors of pointing, dulness of
expression, or the like, I have but little to say to them. I thought I
had corrected it very carefully, and yet some mispointings and small
errors have slipped me, which it is too late to help. As to language, I
have been rather careful to make it speak English suitable to the manner
of the story than to dress it up with exactness of style, choosing rather
to have it free and familiar, according to the nature of essays, than to
strain at a perfection of language which I rather wish for than pretend
to be master of.
* * * * *
* * * * *
Printed by Cassell and Company, Limited, La Bella Sauvage, London, E.C.
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 9,334
|
<?php
/**
* This file is part of the Minty templating library.
* (c) Dániel Buga <daniel@bugadani.hu>
*
* For licensing information see the LICENSE file.
*/
namespace Minty\Compiler\Operators;
use Minty\Compiler\Compiler;
use Minty\Compiler\Node;
use Minty\Compiler\Nodes\ArrayIndexNode;
use Minty\Compiler\Nodes\OperatorNode;
use Minty\Compiler\Nodes\VariableNode;
use Minty\Compiler\Operator;
class ConditionalOperator extends Operator
{
public function __construct()
{
}
public function operators()
{
throw new \BadMethodCallException('Conditional operator is handled differently.');
}
public function compile(Compiler $compiler, OperatorNode $node)
{
$left = $node->getChild(OperatorNode::OPERAND_LEFT);
$compiler->add('(');
if ($this->isPropertyAccessOperator($left)) {
$root = $left;
while ($this->isPropertyAccessOperator($root)) {
$root = $root->getChild(OperatorNode::OPERAND_LEFT);
}
if ($root instanceof VariableNode) {
$compiler
->add('isset(')
->compileNode($root)
->add(') &&');
}
$left->addData('mode', 'has');
$left->compile($compiler);
$compiler->add(' && ');
$left->addData('mode', 'get');
$left->compile($compiler);
} elseif ($left instanceof VariableNode) {
$compiler
->add('isset(')
->compileNode($left)
->add(') && ')
->compileNode($left);
} elseif ($left instanceof ArrayIndexNode) {
$variable = $left->getChild('identifier');
$keys = [$left->getChild('key')];
while ($variable instanceof ArrayIndexNode) {
/** @var $right OperatorNode */
$keys[] = $variable->getChild('key');
$variable = $variable->getChild('identifier');
}
$arguments = [$variable, array_reverse($keys)];
$compiler
->add('isset(')
->compileNode($variable)
->add(') && $context->hasProperty')
->compileArgumentList($arguments)
->add(' && $context->getProperty')
->compileArgumentList($arguments);
} else {
$compiler->compileNode($left);
}
$compiler->add(') ? (');
if ($node->hasChild(OperatorNode::OPERAND_MIDDLE)) {
$compiler->compileNode($node->getChild(OperatorNode::OPERAND_MIDDLE));
} else {
$compiler->compileNode($node->getChild(OperatorNode::OPERAND_LEFT));
}
$compiler->add(') : (')
->compileNode($node->getChild(OperatorNode::OPERAND_RIGHT))
->add(')');
}
/**
* @param Node $operand
*
* @return bool
*/
private function isPropertyAccessOperator(Node $operand)
{
if (!$operand instanceof OperatorNode) {
return false;
}
return $operand->getOperator() instanceof PropertyAccessOperator;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 439
|
\section{\bf Introduction}\label{intro}\par
Given the many successes of Einstein's classical theory of
general relativity~\cite{abs,mtw,sw1},
the fact that the only accepted complete treatment of quantum general
relativity, superstring theory~\cite{gsw,jp}, involves
\footnote{ Recently, the loop quantum gravity approach~\cite{lpqg1}
has been advocated by several authors, but it has still unresolved
theoretical issues of principle, unlike the superstring theory.
Like the superstring theory, loop quantum gravity
introduces a fundamental length
, the Planck length, as the smallest distance in the theory. This
is a basic modification of Einstein's theory.}
many hitherto unseen degrees of freedom, some at
masses well-beyond the Planck mass, is
even more of an acute issue, as we have to wonder if such degrees of
freedom are anything more than a mathematical artifact?
The situation is reminiscent of the old string theory~\cite{schwarz}
of hadrons, which was ultimately superseded by the fundamental
point particle field theory of QCD~\cite{qcd1}.\par
Accordingly, in the recent literature, several authors have attempted
to apply well-tested methods from the Standard Model~\cite{qcd1,sm1} (SM)
physics arena to quantum
gravitational physics: in Refs.~\cite{dono1}, the famous low energy expansion technique from chiral perturbation theory for QCD has been used to address
quantum gravitational effects in the large distance regime,
in Refs.~\cite{sola} renormalization group methods in curved space-time
have been used to address astrophysical and cosmological
(low energy) effects and in Refs.
~\cite{laut,reuter2,litim} the asymptotic safety fixed-point
approach of Weinberg~\cite{wein1} has been used
to address the bad UV behavior of quantum general relativity
whereas in Refs.~\cite{bw1,bw2,bw3,bw4} the
new resummed quantum gravity approach
(RQG) has also been used to address the bad UV behavior
of quantum general relativity (QGR). The ultimate check on these developments, which are not mutually
exclusive, will be the confrontation with experimental data. In this vein,
we focus in the following on an important issue that arises when semi-classical
arguments are applied to massive black hole solutions of Einstein's theory.\par
More precisely, Hawking~\cite{hawk1} has pointed-out that a massive black hole
emits thermal radiation with a temperature known as the Bekenstein-Hawking
temperature~\cite{hawk1,bek-hawk}. This result is well accepted by now.
This raises the question as to what is the final state of the Hawking
evaporation process? In Ref.~\cite{reuter2},
it was shown that an originally
massive black hole emits Hawking radiation until its mass reaches
a critical mass $M_{cr}\sim M_{Pl}$, at which the
Bekenstein-Hawking temperature vanishes and the evaporation process stops.
Here, $M_{Pl}$ is the Planck mass, $1.22\times 10^{19}GeV$.
This would in principle
leave a Planck scale remnant as the final state of the Hawking process.\par
Specifically, in Ref.~\cite{reuter2}, the running Newton constant
was found to be
\begin{equation}
G(r)=\frac{G_Nr^3}{r^3+\tilde{\omega}G_N\left[r+\gamma G_N M\right]}
\label{rnG}
\end{equation}
for a central body of mass $M$ where $\gamma$ is a phenomenological
parameter~\cite{reuter2} satisfying $0\le\gamma\le\frac{9}{2}$,
$\tilde{\omega}=\frac{118}{15\pi}$ and $G_N$ is the Newton constant
at zero momentum transfer.
The respective lapse function in the metric class
\begin{equation}
ds^2 = f(r)dt^2-f(r)^{-1}dr^2 - r^2d\Omega^2
\end{equation}
is then taken to be
\begin{equation}
\begin{split}
f(r)&=1-\frac{2G(r)M}{r}\cr
&= \frac{B(x)}{B(x)+2x^2}|_{x=\frac{r}{G_NM}},
\end{split}
\label{reuter1}
\end{equation}
where
\begin{equation}
B(x)=x^3-2x^2+\Omega x+\gamma \Omega
\end{equation}
for
\begin{equation}
\Omega=\frac{\tilde\omega}{G_NM^2}=\frac{\tilde\omega M_{Pl}^2}{M^2}.
\end{equation}
This leads to the conclusions that~\cite{reuter2}
for $M<M_{cr}$ there is no horizon in the metric in the system and
that for $M\downarrow M_{cr}$ the Bekenstein-Hawking temperature
vanishes, leaving a Planck scale remnant, where
\begin{equation}
M_{cr}=\left[\frac{\tilde{\omega}}{\Omega_{cr}G_N}\right]^{\frac{1}{2}}
\label{mcrit}
\end{equation}
for
\begin{equation}
\Omega_{cr} = \frac{1}{8}(9\gamma+2)\sqrt{\gamma+2}\sqrt{9\gamma+2}-\frac{27}{8}\gamma^2-\frac{9}{2}\gamma+\frac{1}{2}.
\end{equation}
For reference, we see that for the range $0<\gamma<\frac{9}{2}$ from
Ref.~\cite{reuter2}
we have $1>\Omega_{cr}\gtrsim .2$.
The source of these results can be seen
to be the fixed-point behavior for the running Newton constant
in momentum space found in
Ref.~\cite{reuter2},
\begin{equation}
G(k) = \frac{G_N}{1+\omega G_Nk^2}
\end{equation}
where $\omega\sim 1$ depends on the the precise details of the
IR momentum cut-off in the blocking procedure used in Ref.~\cite{reuter2}.
As we have shown in Refs.~\cite{bw1,bw2,bw3}, our RQG theory gives the same
fixed-point behavior for $G(k)$ so that we would naively conclude that
we should have the same black hole physics phenomenology as that
described above for Ref.~\cite{reuter2}. Indeed, we have shown~\cite{bw2,bw3}
that for massive elementary particles, the classical conclusion that
they should be black holes is obviated by our rigorous quantum
loop effects, which do not contain any unknown phenomenological parameters.
We note as well that the results in Ref.~\cite{maart}, obtained
in a simple toy model using loop quantum gravity methods~\cite{lpqg1},
also support the
conclusion that, for masses below a critical value, black holes do not form;
the authors in Ref.~\cite{maart} are unable to specify the precise value
of this critical mass.\par
However, as we have shown in Ref.~\cite{bw1,bw2,bw3}, for elementary massive
particles, and this Bonnano-Reuter Planck scale remnant would indeed be such a
massive object with a mass smaller than many of the fundamental
excitations in the superstring theory for example, quantum loop
effects, resummed to all orders in $\kappa=\sqrt{8\pi G_N}$,
lead to the Newton potential
\begin{equation}
\Phi_{N}(r)= -\frac{G_N M_{cr}}{r}(1-e^{-ar})
\label{newtn}
\end{equation}
where the constant $a$ depends on the masses of the
fundamental particles in the
universe. We take here the latter particles to be those
in the SM and its extension
as suggested by the theory of electroweak symmetry breaking~\cite{ewsymb}
and the theory of grand unification~\cite{guts}. For the upper
bound on $a$ we use, we will not need to speculate about what particles
may exist beyond those in the SM
; and, for the
SM particles we use
the known rest masses~\cite{pdg2002,pdg2004}\footnote{ For the neutrinos,
we use the estimate $m_\nu\sim 3eV$~\cite{neut}.} as well as the value $m_H\cong 120GeV$
for the mass of the physical Higgs particle -- the latter is known to be
greater than 114.4GeV with 95\% CL~\cite{lewwg}.
\par
More precisely, when the graphs Figs. 1 and 2 are computed
in our resummed quantum gravity
\begin{figure}
\begin{center}
\epsfig{file=fig3cn.eps,width=77mm,height=38mm}
\end{center}
\caption{\baselineskip=7mm The graviton((a),(b)) and its ghost((c)) one-loop contributions to the graviton propagator. $q$ is the 4-momentum of the graviton.}
\label{fig1}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{file=fig1cn.eps,width=77mm,height=38mm}
\end{center}
\caption{\baselineskip=7mm The scalar one-loop contribution to the
graviton propagator. $q$ is the 4-momentum of the graviton.}
\label{fig2}
\end{figure}
theory as presented in Refs.~\cite{bw1,bw2,bw3}, the coefficient $c_{2,eff}$
in eq.(12) of Ref.~\cite{bw3} becomes here, summing over the SM particles
in the presence of the
recently measured small
cosmological constant~\cite{cosm1}, which implies the
gravitational infrared cut-off of $m_g\cong 3.1\times 10^{-33}$eV,
\begin{equation}
c_{2,eff}=\sum_j n_j I(\lambda_c(j))
\label{ceff}
\end{equation}
where we define~\cite{bw3} $n_j$ as the effective number of degrees of freedom
for particle $j$ and the integral $I$ is given by
\begin{equation}
I(\lambda_c)\cong \int^{\infty}_{0}dx x^3(1+x)^{-4-\lambda_c x}
\end{equation}
with the further definition $\lambda_c(j)=\frac{2m_j^2}{\pi M_{Pl}^2}$
where the value of $m_j$ is the rest mass of particle $j$
when that is nonzero. When the rest mass of particle $j$ is zero,
the value of $m_j$ turns-out to be~\cite{elswh}
$\sqrt{2}$ times the gravitational infrared cut-off
mass~\cite{cosm1}. We further note that, from the
exact one-loop analysis of Ref.\cite{thvelt1}, it also follows
that the value of $n_j$ for the graviton and its attendant ghost is $42$.
For $\lambda_c\rightarrow 0$, we have found the approximate representation
\begin{equation}
I(\lambda_c)\cong \ln\frac{1}{\lambda_c}-\ln\ln\frac{1}{\lambda_c}-\frac{\ln\ln\frac{1}{\lambda_c}}{\ln\frac{1}{\lambda_c}-\ln\ln\frac{1}{\lambda_c}}-\frac{11}{6}.
\end{equation}
\par
We wish to combine our result in (\ref{newtn}) with the
result for $G(r)$ in (\ref{rnG})
from Ref.~\cite{reuter2}. We do this by omitting
from the $c_{2,eff}$ the contributions from the graviton and its ghost,
as these are presumably already taken into account in $G(r)$ in (\ref{rnG}),
and by replacing $G_N$ in (\ref{newtn}) with the running result
$G(r)$ from (\ref{rnG}).
Thus our improved Newton potential reads
\begin{equation}
\Phi_{N}(r)= -\frac{G(r) M_{cr}}{r}(1-e^{-ar}),
\label{newtnrn}
\end{equation}
where now, with
\begin{equation}
c_{2,eff} \cong 1.41\times 10^{4}
\end{equation}
and, from eq.(8) in Ref.~\cite{bw3},
\begin{equation}
a \cong (\frac{360\pi M_{Pl}^2}{c_{2,eff}})^{\frac{1}{2}}
\end{equation}
we have that
\begin{equation}
a \cong 0.283 M_{Pl}.
\end{equation}
Since the result from Ref.~\cite{reuter2} for $G(r)$
is based on analyzing the pure Einstein theory with no matter, it only
contains the effects of pure gravity loops whereas, if we omit the
graviton and its ghost loops from our result for $c_{2,eff}$,
our result for $a$ in (\ref{newtnrn}) only contains matter loops.
Hence, there really is no double counting of effects in (\ref{newtnrn}).\par
As we have explained elsewhere~\cite{bw3}, if we use the connection
between $k$ and r that is employed in Ref.~\cite{reuter2} and restrict
our result for $c_{2,eff}$ to pure graviton and its ghost loops, we recover
the results of Ref.~\cite{reuter2} for $G(k)$ and $G(r)$ with the similar
value of the coefficient of $k^2$ in the denominator of $G(k)$, for example.
Thus, we can arrive at our result in (\ref{newtnrn}) independent
of the exact renormalization group equation (ERGE)
arguments in Ref.~\cite{reuter2}.
A more detailed version of such an analysis will appear
elsewhere~\cite{elswh}.\par
We also stress here that, when one uses the ERGE for a theory, one
obtains the flow of the coupling parameters in the theory. To get to
the exact S-matrix, and derive a formula for the Newton potential
for example, one then has to employ the corresponding
improved Feynman rules for example. Thus, in the analysis in Ref.~\cite{reuter2}and in Ref.~\cite{perc}, which extends the ERGE analysis of Ref.~\cite{reuter2}
to include matter fields, one finds the results for the behavior of
the couplings at the analyzed asymptotically safe fixed point. The
corresponding computation of the S-matrix near the fixed point
with the attendant improved running couplings is fully consistent
with our results in (\ref{newtn}),~(\ref{newtnrn})~\cite{elswh}.\par
At the critical value $M_{cr}$, the function
$B(x)+2x^2=x^3+\Omega_{cr} x+\gamma\Omega_{cr}$ just
equals $2x^2$ at $x=x_{cr}$,
producing there a double zero of $B(x)$ and of the lapse function
$f(r)=1+2\Phi_N$.
When we introduce our improvement into the lapse
function via $G(r)\rightarrow G(r)(1-e^{-ar})$, the effect is to reduce
the size of the coefficient of $-2x^2$ in $B(x)$ to $-2\xi x^2$
where $\xi=\xi(x)=1-e^{-aG_NM_{cr}x} < 1$ and thereby to remove
the double zero at $x_{cr}$.
The respective monotone behaviors of the polynomials
$x^3+\Omega_{cr} x+ \gamma\Omega_{cr}$
and $2\xi(x_{cr})x^2$ then allow us to conclude that the lapse function
remains positive and does not vanish as $x\downarrow 0$, i.e.,
our quantum loop
effects have obviated the horizon of the would-be Planck scale remnant
so that the entire mass of the would-be Planck scale remnant
is made accessible to our universe by our quantum loop effects.
This result holds for all choices of the parameter $\gamma$ in
the range specified by Ref.~\cite{reuter2}.
We note the nature of the way the results in Ref.~\cite{reuter2}
and our result in (\ref{newtnrn}) are to be
combined: first one carries out the
analysis in Ref.~\cite{reuter2} and shows that the originally massive
black hole evaporates by Hawking radiation down to the
critical mass $M_{cr}$; then, in this regime of masses, the
Schwarzschild radius is in the Planck scale regime, wherein the
calculation in (\ref{newtnrn}) is applicable to show that the
horizon at $M_{cr}$ is in fact absent. One can not simply
use the result in (\ref{newtnrn}) for all values of
$M$ because it is only valid in the deep UV. Above, we have
used a step function at $x=x_{cr}$ to turn-on our
improvement for $x\le x_{cr}$.\par
This is still only a rather approximate way of combining our result in
(\ref{newtnrn}) and the result (\ref{reuter1}) of Ref.~\cite{reuter2}
and it leaves open the question as to the sensitivity of our conclusions
to the nature of the approximation.
In principle each result is a representation of the quantum loop effects
on the lapse function if we interpret these effects in terms
their manifestations on the effective value of Newton's constant
as it has been done in Ref.~\cite{reuter2}:
\begin{equation}
f(r)=1-\frac{2G_{eff}(r)M}{r}
\end{equation}
where we take either $G_{eff}(r)$ from (\ref{rnG}) or
we use (\ref{newtnrn}) to get $G_{eff}=G_N(1-e^{-ar})$
where now we must set $a=0.210M_{Pl}$ to reflect the effects
of pure gravity loops. The former choice is valid for
very large r, so it applies to very massive black holes
outside of their horizons whereas the latter choice should be
applicable to the deep UV at or below the Planck scale.
A better approximation is then, after the originally very massive
black hole has, via the analysis of Ref.~\cite{reuter2}, Hawking
radiated down to a size approaching the Planck size, to join the two
continuously at some intermediate value of r by determining
the outermost solution, $r_>$, of the equation
\begin{equation}
1-\frac{2G(r)M}{r}=1-\frac{2G_N(1-e^{-ar})M}{r}
\end{equation}
where $G(r)$ is given above by (\ref{rnG}),
and to use the RHS of the latter equation for $f(r)$
for $r<r_>$. For example, for $\Omega= 0.2$, we find $r_>\cong 27.1/M_{Pl}$,
so that, whereas the result (\ref{reuter1}) would give an
outer horizon at $x_+\cong 1.89$ for $\gamma=0$~\footnote{We follow Ref.~\cite{reuter2} and ask for self-consistency in the determination of
$\gamma$ and leads us to the choice $\gamma=0$ here.}, we get
$x_+\cong 1.15$ when we do this continuous combination; moreover, the
inner horizon implied by (\ref{reuter1}) at $x_-\cong 0.106$
moves to negative values of x so that it ceases to exist.
The Bekenstein-Hawking temperature
in this continuous combination
remains positive for all $x>0$ because the two equations
\begin{eqnarray}
0&=&x-2+2e^{-\frac{yx}{2}},\nonumber\\
0&=&1-ye^{-\frac{yx}{2}},
\label{sys1}
\end{eqnarray}
where $y=2\sigma/\Omega^{\frac{1}{2}}$ for
$\sigma=(a/M_{Pl})\sqrt{\tilde\omega}$, require as well
\begin{equation}
1=ye^{-(y-1)}.
\label{sys2}
\end{equation}
The expression on the RHS of this latter equation
has a maximum for $y\ge 0$ at $y=1$ and this maximum is
just $1$, the constant on the LHS of the same equation.
The only positive
solution to (\ref{sys2}) is then $y=1$, or $\Omega=4\sigma^2$.
This corresponds to x=0 in the (\ref{sys1}), which contradicts
the assumption therein that $x>0$. Hence, we see that
the outer horizon just approaches $x=0$ for
$\Omega\rightarrow 4\sigma^2\equiv\Omega'_{cr}$
and that, at this point, the derivative of the lapse function
is positive. The mass $M'_{cr}$ implied by $\Omega'_{cr}$
is $2.38~M_{Pl}$. In other words, originally massive black holes
emit Hawking radiation until they reach the point $M'_{cr}\sim M_{Pl}$
at which their horizon vanishes, in complete agreement with our more
approximate treatment above.\par
The intriguing question is that, after reaching the final mass $M'_{cr}$,
which is now made accessible by quantum loops, how will that mass
manifest itself? Depending on the value of its baryon number, we can expect
that there is non-zero probability for its decay into to just
two body final states,
such as two nucleons, resulting in cosmic rays with energy $E=\frac{1}{2}M'_{cr}\cong 1.2M_{Pl}$.
More complicated decays would populate the cosmic ray spectrum with
energy $E<\frac{1}{2}M'_{cr}\cong 1.2M_{Pl}$.
Such cosmic rays
may help to explain the current data~\cite{cosmicray,westerhoff}
on cosmic rays
with energies exceeding $10^{19}$eV.
\par
In sum, all of the mass
of the originally very massive black hole is ultimately made accessible
to our universe by quantum loop effects. This conclusion agrees with some
recent results by Hawking~\cite{hawk2}.\par
\section*{Note Added}
We point out that the map given in Ref.~\cite{reuter2} for the
phenomenological distance correlation for the respective infrared cut-off
$k$ is based on standard arguments from quantum mechanics and the
parameter $\gamma$ encodes a large part of the phenomenological
aspects of that correlation. In Refs.~\cite{bw1,bw2,bw3}, the variable
$k$ is the Fourier conjugate of the position 4-vector $x$ so that
the connection from function space of $\vec k$ space to that of $\vec r$
space is given by standard Fourier transformation with no phenomenological
parameters,i.e., the result in (\ref{newtn}) does not have any sensitivity
to parameters such as $\gamma$. This underscores the correctness
of (\ref{newtnrn}) and the main conclusion we draw from it: for all choices of $\gamma$, the Planck scale remnant has its horizon
obviated by quantum loop effects.
\section*{Acknowledgments}
We thank Prof. S. Jadach for useful discussions.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,791
|
{"url":"http:\/\/www.ece.rice.edu\/~aca\/Pub.html","text":"Selected publications of A.C. Antoulas\n\n1. New results on the algebraic theory of linear systems: the solution of the cover problems Linear Algebra and Its Applications, Special issue on Linear Systems and Control, edited by P.A. Fuhrmann, 50, pp. 1-43 (1983).\n\nAbstract. A wide class of control problems can be formulated in terms of subspace inclusions called the cover problems. This paper presents a general solution of the these problems both in state-space and polynomial terms. The unifying element turns out to be the partial realization problem. The contribution of this paper is that of parametrizing all solutions keeping track of their complexity at the same time.\n\n2. A new approach to synthesis problems in linear systems IEEE Transactions on Automatic Control, AC-30, pp. 465-474 (1985).\n\nAbstract. A very general synthesis problem is formulated in terms of a nonlinear equation with rational entries and solved. The novelty consists in the fact that all admissible compensators are parametrized and classified according to their complexity. The key for doing this is on the one hand, the use of polynomial matrix factorizations, and on the other the conversion of the problem to an appropriate partial realization problem.\n\n3. On recursiveness and related topics in linear systems IEEE Transactions on Automatic Control, AC-31, pp. 1121-1135 (1986).\n\nAbstract. This paper is concerned with a cascade decomposition of linear systems. It explores the close relationships between a number of topics like: matrix linear fractional representations, nested feedback interconnections formal power series, recursive realization of matrix impulse response sequences matrix continued fractions a matrix Euclidean algorithm, etc. As a corollary the celebrated Berlekamp-Massey algorithm is extended to the matrix case. The natural tool for achieving this is a certain polynomial unimodular matrix of size p+m which can be associated to every system with m inputs and p outputs.\n\n4. On the scalar rational interpolation problem with B.D.O. Anderson, IMA J. of Mathematical Control and Information, Special Issue on Parametrization problems, edited by D. Hinrichsen and J.C. Willems, 3, pp. 61-88 (1986).\n\nAbstract. Rational interpolation is treated as a generalization of (partial) realization. It is shown that the Loewner matrix an old tool for studying the interpolation problem generalizes the Hankel matrix in this respect. Moreover, the parametrization of all solutions to the rational interpolation problem with the complexity (McMillan degree) as parameter is achieved for the first time.\n\n5. Rational interpolation and the Euclidean Algorithm Linear Algebra and Its Applications, 108, pp. 157-171 (1988).\n\nAbstract. The (scalar) rational interpolation problem with a different degree of complexity (namely the sum of the numerator and the denominator degrees as opposed to the maximum between the numerator and the denominator degrees) is considered. It is shown that the Euclidean algorithm and the degrees of the successive quotients, provide the key to parametrizing all solutions where the complexity is the parameter defined above. This clarifies many aspects of the Pade and the Cauchy approximation problems not well understood in the literature.\n\n6. On the stable rational interpolation problem with B.D.O. Anderson, Linear Algebra and Its Applications, Special Issue on Linear Control Theory, 122\/123\/124, pp. 301-329 (1989).\n\nAbstract. Various classical results in rational interpolation theory are revisited and reinterpreted using the Loewner matrix introduced above. An important result in this regard is the fact that the celebrated Nevanlinna-Pick algorithm consists of nothing more than unconstrained minimal interpolation of the original set of data together with a mirror-image set of data.\n\n7. The cascade structure in system theory in Three decades of mathematical system theory, edited by H. Nijmeijer and J.M. Schumacher, Springer Lecture Notes in Control and Information Science, 135, pp. 1-18 (1989).\n\nAbstract. It is pointed out that many seemingly diverse results in system theory can be explained in a unified way using the cascade interconnection of two-port systems as a tool. In particular this refers to passive network synthesis results like Darlington synthesis and Inverse Scattering on the one hand, and recursive realization results on the other.\n\n8. State space and polynomial approaches to rational interpolation with B.D.O. Anderson, in Progress in Systems and Control Theory III: Realization and Modeling in System Theory, M.A. Kaashoek, J.H. van Schuppen and A.C.M. Ran Editors, Birkh\u00e4user (1990), pp. 73-82.\n\nAbstract. Further insight is provided on the use of the Loewner matrix in the study of the scalar rational interpolation problem. In particular all solutions are constructed both in a state-space and a polynomial setting.\n\n9. Rational interpolation and state-variable realizations B.D.O. Anderson and A.C. Antoulas, Linear Algebra and Its Applications, Special Issue on Matrix problems, 137\/138, pp. 479-509 (1990).\n\nAbstract. The construction of solutions of the matrix rational interpolation problem in state-space form, is derived using the (block) Loewner matrix. This theory generalizes the construction of realizations by means of the (block) Hankel matrix.\n\n10. On minimal realizations Systems and Control Letters, 14: 319-324 (1990).\n\nAbstract. The machinery introduced in the recursiveness paper 25 is shown to lead to a new test for minimality of (both scalar and matrix) realizations based on the degrees of the entries of an appropriate Bezout identity.\n\n11. On the solution of the minimal rational interpolation problem with J.A. Ball, J. Kang, and J.C. Willems, Linear Algebra and Its Applications, Special Issue on Matrix Problems, 137\/138: 511-573 (1990).\n\nAbstract. This paper provides for the first time, the solution of a general matrix rational interpolation problem with the (McMillan) degree as measure of complexity; the computation of minimal interpolants follows as a corollary. This is done by defining directly from the data a pair of matrices. All admissible interpolant degrees are then obtained from the reachability indices of this pair. Moreover the corresponding linear dependencies of the columns of the reachability matrix of this pair provide a parametrization of all interpolants of appropriate complexity. These results are extended to a very general bitangential interpolation problem by the appropriate definition of a second pair of matrices.\n\n12. Rational interpolation and Prony's method with J.C. Willems, Analysis and Optimization of Systems, J.L. Lions and A. Bensoussan, eds, Springer-Verlag, Lecture Notes in Control and Info. Science, 144: 297-306 (1990).\n\nAbstract. Using the tools introduced in the above paper the widely used Prony's method of data fitting is re-examined. It is actually shown that the concept of reachability allows a complete characterization of all solutions to that problem. It also illustrates how to avoid mis-using this method (often encountered in the literature).\n\n13. Mathematical System Theory: The influence of R.E. Kalman edited book, Springer-Verlag Berlin Heidelberg, 605 pages (1992).\n\nAbstract. This book is a Festschrift containing a collection of papers written by leaders in the field, surveying the influence of R. E. Kalman's ideas in mathematical system theory. It was published on the occasion of Kalman's 60th birthday.\n\n14. The realization problem: Linear deterministic systems with T. Matsuo and Y. Yamamoto, in Mathematical System Theory, A.C. Antoulas Editor, pp. 191-212, Springer-Verlag (1992).\n\nAbstract. The realization problem for linear deterministic systems is reviewed both for finite- and infinite-dimensional systems. This is the problem of constructing the state from impulse response data.\n\n15. A behavioral approach to linear exact modeling with J.C. Willems, IEEE Transactions on Automatic Control, AC-38: 1776-1802 (1993).\n\nAbstract. Using the modeling framework introduced by the second author results obtained earlier by the first author on the recursiveness of the realization problem, are generalized to exact modeling of linear systems based on arbitrary time series. The main tool for achieving this sweeping generalization is again a polynomial matrix of size p+m which is attached to any system with m inputs and p outputs.\n\n16. Noise identification in approximate modeling with Y. Yamamoto (in japanese), Journal of the Society of Instrument and Control Engineers (SICE), 32: 718-722 (1993).\n\nAbstract. The problem of identifying all possible noise terms in a given set of data is addressed. The answer termed generalized least squares (GLS) turns out to be a generalization of the well known least squares (LS) and of the total least squares (TLS) schemes.\n\n17. A nonlinear approach to the aircraft tracking problem with R.H. Bishop, Journal of Guidance Control, and Dynamics, 17: 1124-1130 (1994).\n\nAbstract. This paper grew out of work done for Oerlikon-Contraves AG, Z\u00fcrich. It describes the application of recent advances in non-linear filtering to the problem of tracking a maneuvering aircraft. In comparison to the extended Kalman Filter currently in use the advantages of the new so-called geometric filter, are guaranteed convergence and greatly reduced computational time requirements.\n\n18. Recursive modeling of discrete-time time series in IMA volume on Linear Algebra for Control, P. van Dooren and B.W. Wyman Editors, Springer Verlag, vol. 62: 1-20 (1993).\n\nAbstract. The approach to modeling inspired by the behavioral framework consists in treating all measurements on an equal footing not distinguishing between inputs and outputs. Consequently the initial search is for autonomous models. In the linear time invariant case, the main result guarantees the existence of a minimal complexity autonomous generating model; this means that all other models can be explicitly constructed from this generating model. Among them in most cases the so-called input-output controllable models are of interest. The main purpose of this paper is to show how these models can be constructed in a an easy-to-implement recursive way.\n\n19. A unifying framework for linear exact modeling Proc. 12th IFAC Triennial World Congress, Sydney, Pergamon Press (1993).\n\nAbstract. Exact modeling problems satisfying constraints of complexity stability and norm boundedness, are cast and solved into a common framework. The data are exponential trajectories. Rational interpolation turns out to be a special case of this modeling problem. The main tool is the most powerful model a central concept in the behavioral framework and its representations, called the generating systems.\n\n20. A new approach to modeling for control Linear Algebra and Its Applications, Third Special Issue on Systems and Control, 203-204: 45-65 (1994).\n\nAbstract. The behavioral approach to exact modeling leads to the concept of generating systems. It turns out that all exact models are parametrized by means of a feedback interconnection of a generating system with an arbitrary system. This approach is extended in the present paper to the approximate modeling case. The main tool for achieving this is the Hankel norm approximation applied to the generating system. This set-up has the additional feature that it allows us to address at the same time the control problem with little additional effort.\n\n21. The cascade structure for lossless systems with M. Mansour, Comptes Rendus de l'Acad\u00e9mie des Sciences Paris, 319, S\u00e9rie II, p. 1001-1008 (1994).\n\nAbstract. It is shown that a certain decomposition of a stable system into a cascade of stable two-port subsystems has the property that the elimination of any number of these subsystems preserves stability. This is equivalent to the elimination of rows and columns from the associated Mansour matrix. The decomposition also gives a natural way for separating fast and slow modes of the original system.\n\n22. On the approximation of Hankel matrices in Operators Systems and Linear Algebra, edited by U. Helmke, D. Pr\u00e4tzel-Wolters and E. Zerz, pages 17-23, Teubner Verlag (1997).\n\nAbstract. {\\rm In this paper we examine the problem of optimal approximation in the 2-norm of a finite square Hankel matrix by a Hankel matrix of rank one and provide a necessary and sufficient condition for its solvability.\n\n23. On behaviors rational interpolation, and the Lanczos algorithm with E.J. Grimme and D.C. Sorensen, Proc. 13th IFAC Triennial World Congress, San Francisco, Pergamon Press (1996).\n\nAbstract. This paper explores the interconnections between two methods which can be used to obtain rational interpolants. The first method: the behavioral approach, constructs a generating system in the frequency domain which explains a given data set. The second method: the rational Lanczos algorithm, can be used to construct a rational interpolant for a system defined by (potentially very high-order) state-space equations. This paper works to merge the theoretical attributes of the behavioral approach with the theoretical and computational properties of rational Lanczos. As a result it lays the foundation for the computation of reduced-order stabilizing controllers through rational interpolation.\n\n24. Approximation of linear operators in the 2-norm Linear Algebra and Its Applications, Special Issue on Challenges in Matrix Theory, 278: 309-316 (1998).\n\nAbstract. The problems of approximating in the $2$-induced norm linear oparators which are (1) finite-dimensional unstructured, and (2) infinite-dimensional structured (Hankel) have been solved. The solutions of these two problems exhibit striking similarities. These similarities suggest the search of a unifying framework for the approximation of linear operators in the 2-induced norm.\n\n25. Optimizing the EV of the data matrix: a case study in non-smooth analysis with A. Astolfi, Report EE Dept., Imperial College London, 1998.\n\nAbstract. In this work we study some properties of the extrema of the EV's associated to the data covariance matrix arising in the robust identification problem for discrete time finite dimensional linear systems. It is well known that the EV's are continuous but in general, non-differentiable functions of the parameters of the matrix. As a consequence the maximization of a pre-specified EV cannot be performed using tools from smooth analysis and is typically performed numerically using LMI's methods. Nevertheless we show that non-differentiable extrema have a simple interpretation enabling their detection. Finally the convexity properties of the EV's of the data covariance matrix are studied.\n\n26. Controllability and Observability with E.D. Sontag and Y. Yamamoto, in Wiley Encyclopedia of Electrical and Electronics Engineering, edited by J.G. Webster, volume 4: 264-281 (1999).\n\n27. Approximation of linear dynamical systems in Wiley Encyclopedia of Electrical and Electronics Engineering, edited by J.G. Webster, volume 11: 403-422 (1999).\n\nAbstract. The last two articles are Invited papers prepared for the Wiley Encyclopedia of Electrical and Electronics Engineering.\n\n28. On the choice of inputs in identification for robust control with B.D.O. Anderson, Automatica, 35: 1009-1031 (1999).\n\nAbstract. The thesis that noisy identification has close ties to the study of the singular value decomposition of perturbed matrices is investigated. In particular by assuming an upper bound on the norm of the perturbation one can obtain a convex parametrization of an uncertain family of systems which contains the system generating the data. In this approach the second-smallest singular value of the data matrix becomes a quantity of great importance as it provides an upper bound for the size of the uncertain family. This yields a new tool leading to the design of input functions which are optimal or persistently exciting from the point of view of identification for robust control.\n\n29. Approximation of large-scale dynamical systems A.C. Antoulas, Draft, to appear, SIAM Press (2003).\n\nAbstract. This textbook presents two classes of approximation methods for linear dynamical systems which lead to explicit and easy-to-implement algorithms and have great potential for applications. The important feature of the first approach is that the approximation error can be quantified while for the second its numerical stability.\n\n30. A survey of model reduction methods for large-scale systems A.C. Antoulas, D.C. Sorensen, and S. Gugercin, Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing,'' Contemporary Mathematics, AMS publications, 2001.\n\nAbstract. An overview of model reduction methods and a comparison of the resulting algorithms are presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better.\n\n31. Lyapunov, Lanczos, and Inertia A.C. Antoulas and D.C. Sorensen, Linear Algebra and Its Applications, 326: 137-150 (2001).\n\nAbstract. We present a new proof of the inertia result associated with Lyapunov equations. Furthermore we present a connection between the Lyapunov equation and the Lanczos process which is closely related to the Schwarz form of a matrix. We provide a method for reducing a general matrix to Schwarz form in a finite number of steps (O(n3)). Hence, we provide a finite method for computing inertia without computing eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.\n\n32. Approximation of large-scale dynamical systems: An overview A.C. Antoulas and D.C. Sorensen, Technical Report, February 2001.\n\nAbstract. In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.\n\n33. The Sylvester equation and approximate balanced reduction D.C. Sorensen and A.C. Antoulas, Fourth Special Issue on Linear Systems and Control, Edited by V. Blondel, D. Hinrichsen, J. Rosenthal, and P.M. van Dooren, Linear Algebra and Its Applications, {\\bf 351-352}: 671-700 (2002).\n\nAbstract. The purpose of this paper is to investigate methods for the iterative computation of approximately balanced reduced order systems. The resulting approach is both completely automatic once an error tolerance is specified and yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad\\'e via Lanczos) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the {\\it cross gramian} of the system. The cross gramian is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation with some new insights. Our method produces a rank $k$ approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the orignal system. It is well suited to large scale problems because there are no matrix factorizations of the large (sparse) system matrix. Only matrix-vector products are required.\n\n34. On the decay rate of the Hankel singular values and related issues A.C. Antoulas, D.C. Sorensen, and Y. Zhou, Systems and Control Letters, vol 46 (5), pp. 323-342 (2002).\n\nAbstract. This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues like the Hankel singular values, are decaying rapidly. This fact has motivated the development of several algorithms for computing low rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low rank Cauchy matrix determined by the poles of the system.\n\n35. Frequency domain representation and singular value decomposition A.C. Antoulas, UNESCO EOLSS (Encyclopedia for the Life Sciences), Contribution 6.43.13.4, June 2001.\n\nAbstract. This contribution reviews the external and the internal representations of linear time-invariant systems. This is done both in the time and the frequency domains. The realization problem is then discussed. Given the importance of norms in robust control and model reduction, the final part of this contribution is dedicated to the definition and computation of various norms. Again, the interplay between time and frequency domain norms is emphasized.\n\n36. The issue of consistency in identification for robust control S. Gugercin, A.C. Antoulas, and H.P. Zhang, Technical Report, July 2001.\n\nAbstract. Given measured data generated by a discrete-time linear system we propose a model consisting of a linear, time-invariant system affected by norm-bounded perturbation. Under mild assumptions, the plants belonging to the resulting uncertain family form a convex set. The approach depends on two key parameters: an a priori given bound of the perturbation, and the input used to generate the data. It turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs. Two model validation problems in this identification scheme are explored, namely the worst and the best invalidation problems. It turns out that while the former is a max-min optimization problem subject to a spherical constraint, the latter is a quadratic optimization problem with a quadratic and a convex constraint.\n\n37. A modified low-rank Smith method for large-scale Lyapunov equations S. Gugercin, D.C. Sorensen, and A.C. Antoulas, Numerical Algorithms, Vol. 32, Issue 1, pp. 27-55, January 2003.\n\nAbstract. In this note we present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by Penzl, the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually much lower than in the original cyclic low-rank Smith method. The modified method never requires more columns than the original one. Upper bounds are established for the errors of the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. 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WrestlingFigs > Wrestling News Posts > EXCLUSIVE INTERVIEW WITH PAUL HEYMAN @ UGO
EXCLUSIVE INTERVIEW WITH PAUL HEYMAN @ UGO
A MUST READ…….
Paul Heyman Interview
By Matthew Randazzo V
"Everything they've done with Edge this year has just been spectacular. Even things that I didn't agree with them doing, it didn't matter because Edge's performances are so on the money, no one can compare to him."
Paul Heyman is a pro wrestling icon, one of the last great ringside managers, arguably the most innovative and copied booker of the modern era, and the founder of the original, revolutionary incarnation of Extreme Championship Wrestling. As the visionary behind ECW in the 1990's, Heyman broke all existing wrestling conventions and introduced American audiences to the likes of Rey Mysterio, Eddy Guerrero, Rob Van Dam, Sabu, and countless others. After his long and frequently acrimonious tour of duty as a World Wrestling Entertainment writer, announcer, and manager came to an end in 2007, Heyman has sought to revolutionize online entertainment with his Heyman Hustle video-blog/talk show/online bully pulpit. In this exclusive interview with UGO, the king of controversy, hype, and hustle is at his uncensored best, working himself into an ecstatic lather promoting his new show and unleashing his one-of-a-kind vitriol on everyone from Jerry Lawler to strippers in Toronto to presidential candidate John McCain.
UGO: Since leaving WWE in 2007, you've been best known for writing columns and producing the video blog The Heyman Hustle for The Sun, a top UK newspaper, at www.thesun.co.uk and www.HeymanHustle.com. In a few sentences, make your pitch for why pro wrestling fans familiar with your work from ECW, WCW, and WWE should watch the Hustle.
Paul Heyman: I'd humbly suggest that if they don't watch now, they're going to regret it later. We're on the cutting edge of a new platform, as broadband television is not just emerging, it's exploding. The Heyman Hustle is, by far, the most popular broadband television show on Sun-TV. We're blowing away everyone else's numbers. In the month of July, we more than doubled our nearest competitor. So obviously, we're earning an audience. If someone is enticed to watch because of my work in wrestling, you'll see some of the same trademarks ... innovative production techniques, cool music, larger than life personas, and a dialogue that doesn't fit the same stagnant pattern people are used to. Besides, if you haven't checked it out by now, you're way behind a very cool crowd watching our show.
UGO: On a recent WWE 24/7 roundtable, Jerry Lawler admitted to intentionally breaking your jaw in June of 1987 for refusing to take an unprotected fall off a scaffold in a Scaffold Match. Are you surprised by Lawler's admission? What is your response to Lawler bragging in public that he took advantage of your trust to physically assault you for refusing to perform such a wildly dangerous stunt?
Paul Heyman: Well, he broke my jaw BEFORE the scaffold match. He's always taken liberties with other people, especially those who by the nature of the situation had to place some trust in him. F*** him. I don't spend any time in my life thinking about him. He's in the past where he belongs. And no, I'm not surprised he's still talking about me. If half HIS locker room banged MY wife, I guess I'd hold a grudge, too.
UGO: One meaning of the word "hustler" is "BS Artist," a description that many in wrestling, unfairly or not, have ascribed to you. In your opinion, who was the most talented BS artist you ever met in wrestling? Who was the least?
Paul Heyman: You ask that question as if I didn't realize the play on words. Who was the most talented BS artist? Well, there was this one stripper in Toronto that made everyone believe they were the first celebrity she ever went to bed with. She was fantastic! She bedded more wrestlers and athletes than Jim Barnett in his heyday. God Bless that woman, she provided a great deal of comfort to a lot of road weary individuals. Who was the least talented? I have several in mind, but they're BS wasn't any good, so it never got them anywhere, and therefore you never heard of them!
UGO: You were involved in an attempt to purchase the MMA promotion Strikeforce that did not come to fruition. If you were in charge of a major MMA promotion, how would you promote MMA differently from Dana White of UFC or Gary Shaw of Elite XC? What can MMA promoters learn from pro wrestling?
Paul Heyman: If I made the decision to do something in MMA, the last thing I'd ever do is tip my hand as to what I'd do differently or the same as Dana White. Competing with UFC is a full time job, 24/7 and then some. Dana White is not some investor looking to capitalize on a craze. He is someone with Vince McMahon's work ethic and dedication and desire and craving to be and stay number one. You want to take on UFC, you're making an enemy out of a workaholic who enjoys the addiction to hard work and competition. As for what UFC in particular can learn from pro wrestling, I'd suggest pro wrestling has more to learn at this stage from UFC.
UGO: Clearly, your former on-screen pupil Brock Lesnar is one of the breakout MMA stars of 2008. Are there any currently unheralded MMA fighters who strike you as having similar superstar potential and, if so, why?
Paul Heyman: A ton of them. There are a lot of very marketable personalities in MMA right now. I think UFC missed a great opportunity when Affliction signed Chris Horodecki. This kid is Brad Pitt in "Fight Club." I think Frank Mir's promos before and after the Lesnar fight showed a great personality and a guy who can talk people into paying to see his matches. Mir has a huge upside if he can continue to win. Everyone raves about Urijah Faber and rightfully so. He ate through Jeff Curran, Chance Farrar, Dominic Cruz, Joe Pearson, he's just an amazing fighter with a great look and a personality that plays so well in media appearances. Untapped as far as his potential as a branded superstar and a marketed commodity goes. There's a lot of great fighters out there. There's also a lot of very marketable ones as well.
UGO: You've received a great deal of praise for your writing as a pro wrestling booker. What writers do you admire and enjoy? What are your favorite films and TV shows?
Paul Heyman: I'm a big fan of people who can either frame a complex situation in a way that anyone can understand ... or people who take a radical viewpoint, and assign ration and reason to it. I find Cornel West to be very interesting. I don't agree with almost anything he says, but I sure respect and admire his ability to present the case for what he believes in. I'm a great admirer of Henry Rollins. I think he's one of the most underrated social commentators out there, and wish he had a bigger platform and wider audience to hear his provocative ramblings. I enjoy reading Bill Clinton's or James Carville's viewpoints. I think a lot of people would be surprised at how profound Martin Luther King's commentaries were, even though those writings are 40-some-odd years old. But he had more than just a dream, he had a vision of a society at peace with itself, not just people with each other. It depends WHERE you look. John Lennon? Some of Pearl Jam's early work? Read some of Tupac's lyrics, and not just in the commercialized songs. I admire people who can put together words that cause people to think about a subject, and question their own instinctual first reaction.
As for my favorite films, I admire films from a number of different genres. Love the characters in True Romance. Love everything about The Professional with Gary Oldman, Jean Reno, and Natalie Portman. Cry everytime I watch Angels with Dirty Faces with James Cagney. I can always watch Jaws, or The Departed, or Casablanca, or The Godfather, or Romeo Must Die, or High Plains Drifter, or The Taking of Pelham 123. I'll watch a Marx Brothers movie any time I get the chance.
As for television, I grew up watching Johnny Carson. [I] think The West Wing was brilliant, and The Sopranos blows everything away. Except, of course, that ECW television show. Now THAT was some classic entertainment!
UGO: In the year since you've been officially gone from World Wrestling Entertainment, have there been any programs/angles that made you stop and think, "Wow, they're really doing this right!" Any that were especially bad or mishandled?
Paul Heyman: Everything they've done with Edge this year has just been spectacular. Even things that I didn't agree with them doing, it didn't matter because Edge's performances are so on the money, no one can compare to him. He's just far and away BETTER than everyone else. His range of emotions, his delivery, his work ethic is just incomparable right now. He's at the top of the game, and he's awe-inspiring. As for things they've done that are especially bad or mishandled, I don't think my opinion is relevant on that subject in this forum because I wasn't there when the decisions were made, and don't know the circumstances under which the trigger was pulled to implement a storyline or call a finish or determine an angle.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,013
|
\section{Measuring coalgebras and measuring comodules}
Although measuring coalgebras (and dual coalgebras in particular)
have been around for a long time, I will develop the theory of
measuring coalgebras and measuring comodules in parallel, as the
first serves as an accessible model for the second.
\begin{Definition}{\rm Measuring coalgebras. If $A$ and $B$ are algebras over a
field ${\bf k}$ a {\em measuring coalgebra} is a coalgebra $C$ over
${\bf k}$ with comultiplication
\begin{equation}
\Delta : C \to C \otimes C, \quad \Delta c = \displaystyle{\sum_{(c)}} c_{(2)} \otimes c_{(1)}
\end{equation}
and counit $\epsilon: C \to {\bf k}$ together with a linear map,
called a {\em measuring map}
\begin{equation}
f: C \to {\rm Hom}_{{\bf k}}(A,B)
\end{equation}
such that
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\phi c(aa') = \displaystyle{\sum_{(c)}} \phi c_{(2)}(a)\phi c_{(1)}(a')$
\item $\phi c(1_A) = \epsilon(c)1_B$
\end{enumerate}
for $a, a'$ in $A$, $1_A, 1_B$, the appropriate identity elements.
The map $\phi$ is said to measure.
Statements i and ii are equivalent to the statement that the
transpose map
\begin{equation}
\phi: A \to {\rm Hom}_{{\bf k}}(C, B)
\end{equation}
is an algebra homomorphism where the multiplication in
${\rm Hom}_{{\bf k}}(C,B)$ is given by
\begin{equation}
\mu\bullet \nu (c) = \displaystyle{\sum_{(c)}}\mu(c_{(2)})\nu(c_{(1)})
\end{equation}
with identity
\begin{equation}
1(c) = \epsilon(c)1_B.
\end{equation}
The following proposition summarizes results about measuring
coalgebras described in [4].
}\end{Definition}
\begin{Proposition} \hfil\break {\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Given algebras $A$, $B$, there is a category of measuring
coalgebras $C(A,B)$ whose objects are measuring coalgebras
$(C,\phi)$ and whose maps \hfil $r: (C,\phi) \to (C',\phi')$ are
coalgebra maps $r:C \to C'$ such that the diagram
\setlength{\unitlength}{0.7cm}
\begin{picture}(12,5)
\put(3,3.5){$C$}
\put(4,3.5){\vector(1,0){5}}
\put(10,3.5){${\rm Hom}_{{\bf k}}(A,B)$}
\put(4,3){\vector(1,-1){1.8}}
\put(8.0,1.0){\vector(1,1){1.8}}
\put(6.5,0.5){$C'$}
\end{picture}
commutes.
\item The subcategory of finite dimensional measuring coalgebras is
dense in $C(A,B)$. Essentially, every measuring coalgebra is a
limit of finite dimensional subcoalgebras. (For a discussion of
density see [6], chapter 5.)
\item The category $C(A,B)$ has a final object, $(P(A,B), \pi)$ called the
universal measuring coalgebra.
Thus there is a correspondences of sets
\begin{equation}
\mbox{Coalgebra maps} (C, P(A,B)) \longleftrightarrow \mbox{ Algebra maps} (A,{\rm Hom}_{{\bf k}}(C,B)).
\end{equation}
\item If $A_i, i, = 1, 2, 3$ are algebras there is a map
$ m:C(A_2,A_3)\times C(A_1,A_2) \to C(A_1,A_3)$.
In particular, $P(A,A)$ is a bialgebra.
\item The universal measuring coalgebra $P(A,A)$ is a bialgebra.
\end{enumerate}
Proofs. Full proofs can be found in [4] However, as the presentation in [4] is
highly categorical and more general than is necessary here, direct proofs of ii
and iii are indicated here.
ii.To establish density it is sufficient to show that every element $c$ of a
coalgebra $C$ is contained in a finite dimensional subcoalgebra
$C_1$. As this result is the essential property of coalgebras, the
proof, from [9] p46 is repeated here.
Let $C' = {\rm Hom}_{{\bf k}}(C,k)$ be the dual algebra and consider the action of $C'$
on $C$ given by
\begin{equation}
c'\cdot c = \displaystyle{\sum_{(c)}} c'(c_{(2)})c_{(1)}
\end{equation}
Evidently the $C'$ module $V$ generated by $c$ is finite
dimensional, and
\begin{equation}
C' \to \mbox{End}(V)
\end{equation}
is an algebra homomorphism of cofinite dimensional kernel $J$.
Let $J^{\perp}$ be the subspace of $C$ on which $J$ is identically zero. Finally
notice that $J^{\perp} \to {\rm Hom}(C'/J, {\bf k})$ is finite dimensional
and $c$ is in $J^{\perp}$. A subcoalgebra of a measuring coalgebra is
itself a measuring coalgebra(with the restriction of the measuring
map), hence the result.
iii. This depends on two categorical properties of coalgebras.
a) Arbitrary coproducts exist in the category of coalgebras and
coalgebra maps.
b) Coequalizers also exist in this category.
The construction of $P(A,B)$ proceeds as follows. Consider the
collection $\{(C_{\lambda},\phi_{\lambda})\}$ of finite dimensional
measuring coalgebras. Form the coproduct $\sqcup_{\lambda}
C\lambda$. This is a measuring coalgebra. Now consider the set
$\{\rho({\lambda},\mu)\}$ of maps $\rho({\lambda},\mu):C_{\lambda}
\to C_{\mu}$ of finite dimensional measuring coalgebras. Form
$\sqcup_{\rho(\lambda,\mu)}C_{\lambda}$. This is also a measuring
coalgebra. There are two maps
\begin{equation}
\alpha,\beta:\sqcup_{\rho(\lambda,\mu)} C_{\lambda} \to \sqcup_{\lambda}C_{\lambda}.
\end{equation}
On $C_{\lambda}$, $\alpha$ is just the inclusion
$C_{\lambda} \to \sqcup{}_{\lambda}C_{\lambda}$
while $\beta$ is the composition of
$\rho(\lambda,\mu)$ with the inclusion $C_{\mu} \to \sqcup{}_{\lambda}C_{\lambda}$.
The claim is that the coequalizer $P(A,B)$ has the desired universal
property. If $(D,\psi)$ is a measuring coalgebra then $D$ is the
union of finite dimensional subcoalgebras $D_{\nu}$. Evidently
there is a map $r_{\nu}:D_{\nu} \to P(A,B)$ and this map is unique.
The uniqueness of $r_{\nu}$ guarantees that the map $\rho:D
\to P(A,B)$ given by
$\rho(d) = r_{\nu}(d)$ if $d$ is in $D_{\nu}$ is well defined.
}\end{Proposition}
\begin{Examples}\label{EXAMP}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $P(A,B)$ is intended to generalize the set of
algebra homomorphisms from $A$ to $B$, and so it does. Let
$C_0 = {\bf k} g$ be the one dimensional coalgebra with
$\Delta g = g\otimes g$, $\epsilon(g) = 1$.
Then a map $\phi: C_0 \to {\rm Hom}_{{\bf k}}(A,B)$ measures if and only
if $\phi(g)$ is an algebra homomorphism. Thus $P(A,B)$ contains
all algebra homomorphisms.
\item Let $C_1 = {\bf k} g \oplus {\bf k} \gamma$, $g$ as above, and
let $\Delta \gamma = g \otimes \gamma + \gamma \otimes g$, $\epsilon(\gamma) = 0$.
Then $\phi: C_1 \to {\rm Hom}_{{\bf k}}(A,B)$ measures if and only if
$\phi(g)$ is an algebra homomorhpism and $\phi(\gamma)$ is a
derivation with respect to $\phi(g)$. That is,
\begin{equation}
\phi(\gamma)(aa') = \phi(\gamma)(a)\phi(g)(a') + \phi(g)(a)\phi(\gamma)(a').
\end{equation}
\item More generally if $L$ is a Lie algebra over ${\bf k}$, then $L\oplus C_0$
can be
given the structure of coalgebra with comultiplication
$\Delta \gamma = \gamma \otimes g + g \otimes \gamma$ and
$\epsilon(\gamma) = 0$ for $\gamma$ in $L$. Suppose
\begin{equation}
\phi: L\oplus C_0 \to {\rm Hom}_{{\bf k}}(A,A)
\end{equation}
is a measuring map such that
\begin{equation}
\phi[\nu,\gamma] = [\phi\nu, \phi\gamma], \qquad \phi(g) = Id
\end{equation}
for $\nu, \gamma$ in $L$. By the universal property of $P(A,A)$
there is a map of measuring coalgebras
\begin{equation}
\rho: L \oplus C_0 \to P(A,A).
\end{equation}
However $P(A,A)$ is an algebra, and in fact the following is true.
\end{enumerate}
}
\end{Examples}
\begin{Proposition}\label{ULprop} {\rm If the map $\phi$ is injective on $L$ the subalgebra
of $P(A,A)$ generated by the image of $\rho$ is isomorphic to the
universal enveloping algebra $UL$.
The proof follows from the universal property of $P(A,A)$ and facts
about bialgebras [2]. In that paper I considered subalgebras of
$P(A,A)$ generated by measuring coalgebras $L \oplus \Bbc{C} K$,
where $K$ is a group, $\Bbc{C} K$ has the usual comultiplication
$\Delta k = k \otimes k$ for $k$ in $K$, and elements of $L$ have a slightly skew
version of the usual comultiplication for derivations,
\begin{equation}
\Delta E = E \otimes k + k^{-1} \otimes E
\end{equation}
which is characteristic of difference operators. These objects
resemble quantum groups. The construction in section \ref{GENsec}
of this paper uses the same procedure to construct subalgebras of
the universal measuring comodule (defined below) which are related
to central extensions of loop algebras.
}\end{Proposition}
\begin{Definition}{\rm {\em Measuring comodules.}
Let $M$ be an $A$ module and let $N$ be a $B$ module (all
modules and algebras are vector spaces over ${\bf k}$). When it is
necessary to emphasize the algebra over which $M$ and $N$ are
modules write ${}^AM$, ${}^BN$. Let $(C,\phi)$ be a measuring
coalgebra in $C(A,B)$. Recall that a {\em comodule} over $C$ is a
vector space with a comultiplication
\begin{equation}
\Delta: D \to C \otimes D, \quad \Delta(d) = \sum_{(d)} d_{(1)} \otimes d_{(0)}
\end{equation}
In addition I will assume that $(\epsilon \otimes1)\Delta = 1$.
When it is necessary to keep track of the coalgebra over which $D$
is a comodule, write ${}_CD$. A ${\bf k}$-linear map
\begin{equation}
\psi:D \to {\rm Hom}_{{\bf k}}(M,N)
\end{equation}
{\em measures} if
\begin{equation}
\psi(am) = \sum_{(d)}\phi d_{(1)}(a) \psi d_{(0)}(m).
\end{equation}
The pair $(D,\psi)$ is called a measuring comodule, and $\psi$ is
called a measuring map.
Equivalently $\psi$ measures if and only if the corresponding transpose
map
\begin{equation}
\psi: M \to {\rm Hom}_{{\bf k}}(D,N)
\end{equation}
is a map of $A$ modules, where the $A$ module structure on
${\rm Hom}_{{\bf k}}(D,N)$ is given by
\begin{equation}
a\bullet \beta(d) = \sum_{(d)} \phi d_{(1)}(a) \beta d_{(0)}.
\end{equation}
Again, results from [4] are summarized in the following proposition.
}\end{Definition}
\begin{Proposition} \hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Given a measuring coalgebra $C$ in $C(A,B)$
there is a category ${}_CD(M,N)$ whose objects are measuring
comodules $(D,\psi)$ and whose maps $\sigma:(D,\psi) \to
(D',\psi')$ are comodule maps
$\sigma: D \to D'$ such that the diagram
\par
\setlength{\unitlength}{0.7cm}
\begin{picture}(12,5)
\put(3,3.5){$D$}
\put(4,3.5){\vector(1,0){5}}
\put(10,3.5){${\rm Hom}_{{\bf k}}(M,N)$}
\put(4,3){\vector(1,-1){1.8}}
\put(8.0,1.0){\vector(1,1){1.8}}
\put(6.5,0.5){$D'$}
\end{picture}
\par
commutes.
\item The subcategory of ${}_CD(M,N)$ whose objects are the finite
dimensional measuring comodules is a dense subcateory of
${}_CD(M,N)$.
\item The category ${}_CD(M,N)$ has a final object, ${}_CQ(M,N)$. This has the
property that there is a correspondence
\begin{equation}
C-\mbox{comodule maps}(D, {}_CQ(M,N)) \longleftrightarrow A-\mbox{module maps}(M, {\rm Hom}(D,N)).
\end{equation}
\item If $M_i$ are modules over algebras $A_i, i = 1,2,3$, and if $C$, $C'$ are in
$C(A_1,A_2), C(A_2,A_3)$ respectively, then there is a map
\begin{equation}
{}_CD(M_2,M_3)\times {}_{C'}D(M_1,M_2) \stackrel{m}{\to} _{m(C \times C')}D(M_1,M_3).
\end{equation}
In particular, ${}_CQ(M,M)$ is a comodule algebra, for $C \to
{\rm Hom}(A,A)$ a measuring coalgebra, $M$ an $A$ module.
\item If $A = B$ and $M = N$, then ${}_CQ(M,N)$ is a comodule algebra.
\end{enumerate}
Proofs. ii. Again the important step is to show that if $D$ is a
$C$ comodule then each $d$ in $D$ is contained in a finite
dimensional subcomodule.
Define an action $\bullet$ of $C'$, the linear dual of $C$ on $D$ via
\begin{equation}
a\bullet d = \sum_{(d)} a(d_{(1)})d_{(0)}.
\end{equation}
Choose an element $d$ of $d_0$ and let $D_0 = C'\bullet d_0$.
Evidently $D_0$ is a finite dimensional $C'$ module and $d_0 =
1\bullet d_0$ is in $D_0$.
The full linear dual $D'$ (of $D$) is also a $C'$ module, with the action given
explicitly by
\begin{equation}
a*d(d) = \sum_{(d)} a(d_{(1)})d(d_{(0)}).
\end{equation}
The subset $D_0^{\perp} = \{d \epsilon D' : d(D_0) = 0\}$ is a
submodule, and $D_0' = D'/D_0{}^{\perp}$.
But $(D_0^{\perp} )^{\perp}$ is then a subcomodule of $D$, and $D$ is contained in
$(D_0{}^{\perp})^{\perp}$. Since $(D_0{}^{\perp})^{\perp}$ includes in $(D'/D_0^{\perp})'$,
$(D_0^{\perp})^{\perp}$ must be a finite dimensional comodule as
required.
The proofs of i and iii are identical in format to the corresponding
statements for measuring coalgebras.
}\end{Proposition}
\begin{Examples}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Let $C_0 = {\bf k} g$ as in example \ref{EXAMP}.i and suppose that
$\phi:C \to {\rm Hom}_{{\bf k}}(A,B)$ measures, so that $\phi(g)$ is an
algebra homomorphism. Let $D$ be the comodule with
$D = {\bf k} d$ and comultiplication $\Delta d = g \otimes d$. Let
$\psi:D \to {\rm Hom}_{{\bf k}}(M,N)$
be a linear map.
Recall that the pullback of $N$, $\phi(g)^*N$ is an $A$ module. Then $\psi$ measures
if and only if
\begin{equation}
\psi(d): M \to \phi(g)^{*}N
\end{equation}
is a map of $A$ modules.
\item If $A = B$ and if $C$ contains the measuring comodule $C_0$ with
$\phi(g) = 1$, the measuring comodule ${}_CQ(M,N)$ contains the vector space $H$ of
all genuine $A$ module maps from $M$ to $N$ as follows.
Any vector space, for example $H$, is trivially a $C$ comodule with
comultiplication $\Delta h = g \otimes h$. The inclusion
$\psi: H \to {\rm Hom}_{{\bf k}}(M,N)$ is then a measuring map. By the universal
property there is a unique map of measuring comodules
$\rho: H \to {}_CQ(M,N)$. Since $\psi$ is an inclusion, so must $\rho$ be.
\item Any algebra can be considered as a module over itself acting by
left multiplication. If $C \to {\rm Hom}(A,B)$ is a measuring
coalgebra, by considering $C$ as a comodule over itself,
$C \to {\rm Hom}(A,B)$ is also a measuring comodule.
\item For an element $a$ of an algebra $A$ let $\iota_a$ denote the inner derivation
\begin{equation}
\iota_a(b) = [a,b] = ab - ba
\end{equation}
Let $I_A$ denote the Lie algebra of inner derivations of $A$. As
in example 1.3.iii, let $C$ be the measuring coalgebra
$C = I_A \oplus C_0, C \to {\rm Hom}_{{\bf k}}(A,A)$. Now put a $C$ comodule structure on $A$,
\begin{equation}
\Delta(a) = g \otimes a + \iota_a \otimes 1.
\end{equation}
Now let $M$ be an $A$ module. With the comodule structure above
the inclusion $A \to {\rm Hom}_{{\bf k}}(M,M)$ sending $a$ to left
multiplication by $a$ gives $A$ the structure of a measuring
comodule. This construction generalizes the observation that for
modules over commutative rings, left multiplication is a module
map.
\end{enumerate}
}
\end{Examples}
\begin{Remarks}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $\tau:(C,f) \to (C',f')$ is a map of measuring coalgebras,
in particular $\tau$ is a comodule map, so that ${}_CQ(M,N)$ can be
considered as a $C'$ comodule. Since $\tau$ is a map of measuring
coalgebras ${}_CQ(M,N)$ is in fact in ${}_{C'}D (M,N)$, and hence
by the universal property there is a unique map
${}_CQ(M,N) \to {}_{C'}Q(M,N)$. All universal measuring comodules ${}_CQ(M,N)$ thus map to
$_{P(A,B)}Q(M,N)$, which will often be denoted $Q(M,N)$.
\item The construction $Q(M,N)$ serves as the set of ``module maps from
an $A$ module $M$ to a $B$ module $N$'' even when $A$ is not the
same as $B$. The paper [4] arose from the desire to put this
curiosity into a sound categorical context.
\end{enumerate}
}\end{Remarks}
\section{Connections}
Given an algebra $A$ of functions, and a set $V$ of derivations of $A$(vector
fields), a connection is that which is needed to define covariant
differentiation by elements of $V$ on a module $M$ (for example,
sections of a bundle) over $A$.
This is a completely algebraic statement and as such lends itself to
restatement in terms of measuring comodules.
\begin{Definition}{\rm
{\em Loose connections} Let $A$ be an algebra and let $M$ be a module over $A$. Let $C$
be a measuring coalgebra, and let D be a comodule over $C$ which is
also an $A$ module. A loose connection is a measuring map
\begin{equation}
\nabla: D \to {\rm Hom}_{{\bf k}}(M,M)
\end{equation}
which additionally satisfies the requirement that $\nabla$ be a map
of $A$ modules in the sense that
\begin{equation}
\nabla(a\xi)(m) = a\nabla\xi(m).
\end{equation}
}\end{Definition}
\begin{Examples}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Connections on a vector bundle. Let
$A = C^{\infty}(Y)$ where $Y$ is a smooth manifold and Let $V$ be the Lie algebra
of vector fields on $Y$ and let $C = V \oplus \Bbc{C} 1$. Let
$D = V \oplus A$ with the comultiplication
\begin{eqnarray}
\Delta&:& D \to C \otimes D, \nonumber\\
\Delta(\psi) &=& 1\otimes \psi + \psi\otimes 1, \psi \in V\nonumber\\
\Delta(a)& =& 1\otimes a + \iota_a \otimes 1 a \in A
\end{eqnarray}
Notice that $D$ is an $A$ module. Let $E$ be a vector bundle over
$Y$ and let $\Gamma(Y,E)$ denote the smooth sections of $E$ over
$Y$. Thus $M = \Gamma(Y,E)$ is a module for $A$. In this setting loose connections
are precisely Koszul connections (see [8]).
\item Connections on a principle bundle. (See [7].) Let $Y$ be a manifold
and let $P$ be a principle $G$ bundle over $Y$. Let
$M = C^{\infty}(P)$. Observe that $C^{\infty}(Y)$ includes in $M$ as those
functions which are constant on the fibres of $P$, hence $M$ is a
$C^{\infty}(Y)$ module. In addition the group algebra ${\bf R} G$ acts on $M$ via
right translation. The action of ${\bf R} G$ commutes with the action
of $C^{\infty}(Y)$.
Let $A = C^{\infty}(Y) \otimes{\bf R} G$. Let $V$ be the Lie algebra of vector fields on $Y$.
Observe that the coalgebra $C$ above becomes a measuring coalgebra
with measuring map
\begin{eqnarray}
\phi: C &\longrightarrow &{\rm Hom} \left(C^{\infty}(Y) \otimes{\bf R} G,C^{\infty}(Y) \otimes{\bf R} G \right), \nonumber\\
\phi(\psi)f \otimes g &=& \psi f \otimes g.
\end{eqnarray}
Let $D$ be the comodule of example i above. This is a
$C^{\infty}(Y)\otimes {\bf R} G$ module with the trivial action of
$G$ on $C^{\infty}(Y)$ and $V$. Also notice that $D$ contains $C$
as a subcomodule (with its usual coproduct). A loose connection
\begin{equation}
\nabla: D \to {\rm Hom}(C^{\infty}(P),C^{\infty}(P))
\end{equation}
in this setting corresponds to a connection on the principle bundle
if and only if additionally $\nabla$ restricted to the subspace $C$
defines a measuring coalgebra
\begin{equation}
\nabla: C \to {\rm Hom}(C^{\infty} (P),C^{\infty}(P)).
\end{equation}
\end{enumerate}
}
\end{Examples}
\subsection{Curvature} {\rm
Curvature can be defined for any measuring comodule equipped with
Lie bracket, in particular for loose connections. Recall that any
measuring comodule, $D$ in particular, comes with a map of
measuring comodules
\begin{equation}
\rho:D \to Q(M,M).
\end{equation}
Recall that Q(M,M) is a comodule-algebra. If $D$ contains a
subspace $V$ on which a Lie bracket is given, for $\xi,\psi$ in
$V$, write
\begin{equation}
\Omega(\xi,\psi) = \rho(\xi)\rho(\psi) - \rho(\psi)\rho(\xi) - \rho([\xi,\psi]).
\end{equation}
This map
\begin{equation}
\Omega: V\otimes V \to Q(M,M)
\end{equation}
is the {\em curvature} of the loose connection $\nabla$ on V.
\begin{Remark} {\rm In all classical cases, the coalgebra $C$ is always
$V \oplus \Bbc{C} 1$, the comodule $D$ is always $V \oplus A$, where $V$ is the Lie algebra of
derivations of $A$, and one is only interested in the restriction
of $\nabla$ to $V$. There is no harm, however, in allowing this
greater generality. In the next section a very different example
demonstrates the advantages of being broad minded.
This section concludes with a result which is well known for
conventional connections.
}\end{Remark}
\begin{Proposition} {\rm If V is a set of primitive elements, ie, with comultiplication
\begin{equation}
\Delta x = \xi \otimes 1 + 1 \otimes \xi
\end{equation}
then $\Omega(\xi,\psi)$ determines a module map
\begin{equation}
\Omega(\xi,\psi): M \to M.
\end{equation}
}\end{Proposition}
{\em Proof }. Direct calculation (observing that Q(M,M) is a comodule algebra, ie,
that multiplication preserves the comodule structure) shows that
\begin{equation}
\Delta \Omega(\xi,\psi) = 1\otimes \Omega(\xi,\psi).
\end{equation}
But this is exactly the statement that $\Omega(\xi,\psi)$ is a
module map.
\section{Generalizations of universal enveloping algebras using measuring comodules}
\label{GENsec}
In proposition \ref{ULprop} the universal enveloping algebra is constructed as a
subalgebra of the universal measuring coalgebra generated by a Lie algebra of
derivations. The original Lie algebra can be identified as the subspace of
primitive elements.
This construction can be generalized, replacing primitive elements
(derivations) with elements $E$ of a measuring coalgebra with the
assymetric comultiplication $\Delta E = E \otimes K + K^{-1}
\otimes E$, where $K$ is an (invertible) group like element. The
algebras generated by such $E$ and $K$ resemble quantum groups, and
were the subject of [2].
This construction is now generalized again, replacing the universal
measuring coalgebra with the universal measuring comodule. The
resulting algebra has, as the analogue of its Lie algebra of
primitive elements, a Lie algebra related to the central extensions
of loop algebras. The central term arises as the trace of the
curvature.
\begin{Construction}{\rm
Let $A$ be an algebra and let $M$ be an $A$ module. Let $P_0$ be a
subcoalgebra subalgebra of $P(A,A)$, and define
\begin{eqnarray}
V_0 &=& \{v \epsilon Q(M,M): \Delta v \epsilon P_0 \otimes Q(M,M)\}. \nonumber\\
V &=& \{v \epsilon Q(M,M), \Delta v \epsilon P(A,A)\otimes 1 + 1
\otimes Q(M,M) + P_0 \otimes Q(M,M)\}.\nonumber\\
\end{eqnarray}
Evidently $A$ is contained in $V$. The subcomodule $V$ is the
generalization of primitive elements referred to above.
Suppose a ``trace''
\begin{equation}
\tau: V_0 \to \Bbc{C}
\end{equation}
is given. Let $K$ be the kernel of $\tau$. Define
\begin{eqnarray}
V_{0\tau} &=& \{v \epsilon V_0: [K,v] \leq K\} \nonumber\\
V_{\tau} &=& \{v \epsilon V: [V_{0\tau},v] \leq K\}
\end{eqnarray}
Observe that $V_{\tau}$ is not an algebra, but the Jacobi identity
guarantees that it is closed under Lie bracket. $V_0$ is a
subalgebra of $Q(M,M)$ and hence $V_{0\tau}$ is a Lie subalgebra.
It is not hard to check that there is a short exact sequence of Lie
algebras
\begin{equation}
0 \to V_{0\tau}/K \to V_{\tau}/K \to V_{\tau}/V_{0\tau} \to 0.
\end{equation}
Moreover $V_{\tau}/K$ is a central extension of
$V_{\tau}/V_{0\tau}$.
Suppose now $\mu:V_{\tau}/V_{0\tau} \to V_{\tau}$ is any linear section of the
projection $V_{\tau} \to V_{\tau}/V_{0\tau}$. The image
$\mu(V_{\tau}/V_{0\tau})$ inherits a Lie bracket from
$V_{\tau}/V_{0\tau}$: hence the associated curvature $\Omega_{\mu}$
takes values in $V_{0\tau}$. While the $\Omega_{\mu}$ may depend on
the section $\mu$, the trace of the curvature does not. In fact
$V_{\tau}/K$ is the central extension of $V_{\tau}/V_{0\tau}$ with
cocycle $c$ defined by
\begin{equation}
c(v,w) = \tau(\Omega_{\mu}(\mu v,\mu w))
\end{equation}
for $v, w$ in $V_{\tau}/V_{0}$. Familiar examples arise from
looking at particular subspaces of $V$.
}
\end{Construction}
\begin{Examples}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Let $A = M = \Bbc{C}[x]$. Let
\begin{equation}
P_0 = \{p \epsilon P(\Bbc{C}[x],C): p(x^n) = 0 \mbox{ for almost all } n\}.
\end{equation}
Explicitly $P_0$ has basis $\{\beta_{j}\}$ with comultiplication
and measuring map given by
\begin{equation}
\Delta \beta_{j}
= \sum_{k=0} \beta_k \otimes\beta_{j-k},
\qquad \phi(\beta_{j}) = j! \frac{d}{dx}|_0 .
\end{equation}
Define a trace $\tau$ on $V_0$
\begin{equation}
\tau(v) = \sum_{j=0}^{\infty} \beta_{j}(v(x^j)).
\end{equation}
To see this is well defined, observe that for $v$ in $V_0$
\begin{equation}
v(x^n) = \sum_{(v)}v_{(1)}(x^n)v_{(0)}(1).
\end{equation}
Thus $\beta_{j}(v(x^n)) = 0 $ for greater than the greatest degree
of the $v_{(0)}(1)$, and the sum defining $\tau$ is always a finite
sum.
This example contains two very well known examples as Lie
subalgebras. First consider $\Bbc{C}[\alpha^{-1}]$. This is can be
given the structure of a coalgebra with
\begin{eqnarray}
\Delta(\alpha^{-i}) &=& \alpha^{-i}\otimes 1
+ \sum_{k=0}^{i-1} \beta_k \otimes \alpha^{-i+k}, i>0. \nonumber\\
\epsilon(\alpha^{i}) &=& \delta_{i,0}.
\end{eqnarray}
Define a map $\phi:\Bbc{C}[\alpha^{-1}] \to {\rm Hom}(\Bbc{C}[x],\Bbc{C}[x])$
via
\begin{equation}
\phi(\alpha^{i})x^n = \left\{ \begin{array}
{r@{\quad}l}
x^{i+n} & i + n \geq 0 \\
0 & \mbox{otherwise}.
\end{array} \right.
\end{equation}
It is routine, if surprising, to verify that this map measures. Now
$\Bbc{C}[\alpha,\alpha^{-1}]$ can be considered a comodule over
$\Bbc{C}[\alpha^{-1}] \oplus P_0$ with comultiplication given by
\begin{equation}
\Delta \alpha^{i} = 1 \otimes \alpha^{i}
\end{equation}
for $i>0$ otherwise as above. Clearly the measuring map $\phi$
above extends to all of $\Bbc{C}[\alpha,\alpha^{-1}]$. It is not
hard to check that the image of $\Bbc{C}[\alpha,\alpha^{-1}]$ lies in
$V_{\tau}$. The image of $\Bbc{C}[\alpha,\alpha^{-1}]$ in
$V_{\tau}/K$ is the familiar central extension of the abelian Lie
algebra $\Bbc{C}[\alpha,\alpha^{-1}]$ with cocycle
\begin{equation}
c(\alpha^k,\alpha^j) = k\delta_{k,-j}.
\end{equation}
\item With $P_0$ and $\tau$ as before let $T$ be the vector
space with basis $\{T_i, i \in \Bbc{Z}\}$, and put a comodule
structure on $T$ via
\begin{equation}
\Delta(T_i) = T_i \otimes 1 + 1 \otimes T_i +
\sum_{i+k<0} k \beta_k \otimes \alpha^{k+i} + \beta^k \otimes T_{k+i} .
\end{equation}
Observe that $T \oplus P_0\oplus \Bbc{C}[\alpha^{-1} ]$ is in fact a
coalgebra if the counit on $T$ is defined to be identically $0$.
Extending $\phi$ of the previous example via
\begin{equation}
\phi(T_i)(x^n) = \left\{ \begin{array}
{r@{\quad}l}
x^{i+n} & i + n \geq 0 \\
0 & \mbox{otherwise}.
\end{array} \right.
\end{equation}
gives $T \oplus P_0\oplus \Bbc{C}[\alpha^{-1} ]$ the structure of a
measuring coalgebra, and hence a measuring comodule. Again the
image lies in $V_{\tau}$. The image of $T$ in
$V_{\tau}/V_{0\tau}$ is isomorphic to the Lie algebra of
derivations of $\Bbc{C}[x,x^{-1}]$, and its image in $V_{\tau}/K$ is
the variant of the Virasoro algebra with cocycle
\begin{equation}
c(T_m,T_n) = \frac16 (m^3-m) \delta_{m,-n}.
\end{equation}
\item Return now to a general algebra $A$, and suppose
that $M = A$, and suppose also that $\tau$ is given so that $K$,
$V_{\tau}$ and $V_{0\tau}$ are as described in 3.1. Let ${L}$ be a
finite dimensional (semi-simple) Lie algebra, which is faithfully
represented by $\rho: {L} \to \mbox{End}(W)$. Let $M(W) = A
\otimes W$. Then the identification of
${\rm Hom}(M(W),M(W))$ with ${\rm Hom}(A,A) \otimes {\rm Hom}(W,W)$ provides a map
\begin{equation}
\phi\otimes \rho:V \otimes {L} \to {\rm Hom}(M(W),M(W))
\end{equation}
which measures. Moreover, $V\otimes {L}$ is closed under Lie
bracket, as is $V_0 \otimes {L}$.
If $\kappa$ is the Killing form on ${L}$ then $\tau \otimes \kappa$ is well defined on $V_0 \otimes {L}$, and
$\tau \otimes \kappa([V_{\tau}\otimes {L}, V_{0\tau}\otimes {L}]) =
0$. Let $K({L})$ be the kernel of $\tau \otimes \kappa$. There is
then a short exact sequence of Lie algebras
\begin{equation}
0 \to V_{0\tau} \otimes {L}/K({L}) \to V_{\tau} \otimes {L}/K({L}) \to V_{\tau}\otimes {L}/V_{0\tau}\otimes {L} \to 0
\end{equation}
The Lie algebra $V_{\tau} \otimes{L}/K({L})$ is the central
extension of the loop algebra
${L} \otimes \Bbc{C}[x,x^{-1}] \approx V_{\tau} \otimes {L}/V_{0\tau} \otimes{L}$.
It turns out that the cocycle $c$ of the central extension is
given by
\begin{equation}
c(v \otimes \xi,w \otimes \psi) = \tau\Omega (\mu v,\mu w)\kappa(\xi,\psi).
\end{equation}
In the case of example i, $V_{\tau} \otimes {L}/V_{0\tau} \otimes {L} = {L}[x,x^{-1}]$,
the loop algebra of ${L}$, and $c$ is the expected central
extension,
\begin{equation}
c[x^m\xi,x^n\psi] = m \delta_{-m,n}\kappa(\xi,\psi).
\end{equation}
\end{enumerate}
}
\end{Examples}
\section{Dual comodules, positive energy representations, and smooth
representations}
\subsection{Dual coalgebras and dual comodules}{\rm If $A$ is an algebra, and
$M$ an $A$ module, the the constructions $P(A,\Bbc{C}), Q(M,\Bbc{C})$,
have alternative descriptions which make hands on calculations
easy. }
\begin{Proposition}\hfil\break {\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $P(A,\Bbc{C}) =: A^{*} = \{\alpha: A\to \Bbc{C}: {\rm ker} \alpha \geq I,
I \mbox{ an ideal}, dim A/I< \infty\}$
\item $Q(M,\Bbc{C}) =: M^{*} = \{\mu: M \to \Bbc{C}: {\rm ker}\mu \geq W, AW \leq W, dimM/W< \infty\}.$
\end{enumerate}
}\end{Proposition}
{\em Proof }. i.Observe that since $A/I$ is finite dimensional
multiplication in $A/I$ gives the linear dual $(A/I)'$ the
structure of a coalgebra with the obvious measuring map into
${\rm Hom}(A,\Bbc{C})$. Then, since $(A/I)'$ maps to $P(A,\Bbc{C})$ by the
universal property, $A^{*} = \lim(A/I)' \leq {\rm Hom}(A,\Bbc{C})$ maps to
$P(A,\Bbc{C})$.
But now observe that the measuring map $\pi:P(A,\Bbc{C}) \to {\rm Hom}(A,\Bbc{C})$ has
its image in $A^{*}$. To see this consider $c$ in $P(A,\Bbc{C})$.
Let $C$ be a finite dimensional subcoalgebra of $P(A,\Bbc{C})$
containing $c$. Then the restriction of the measuring map $\pi:C\to
{\rm Hom}(A,\Bbc{C})$ corresponds to an algebra homomorphism $\pi:A\to{\rm Hom}
(C,\Bbc{C})$, this last being a finite dimensional algebra. Let $J$
be the kernel of
$\pi$. Since $\pi:A\to {\rm Hom}(C,\Bbc{C})$ factors through $A/J$, $\pi(c):A \to \Bbc{C}$ must factor through $A/J$,
and $\pi(c)$ is in $A^{*}$ as required.
ii. The argument is exactly parallel to that of (i).
\begin{Remarks}\hfil\break{\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Evidently $M^{*}$ becomes a module for the opposite algebra $A^{{\rm op}}$
under the action
\begin{equation}
am = \sum_{(m)}m_{(1)}(a)m_{(0)}.
\end{equation}
One can ask what representations arise as dual comodules. It is
evident that such a representation $V$ must have the property that
every element of $V$ lives in some finite dimensional submodule of
$V$. Representations which have this property will be called
locally finite.
\item More generally, given modules $M$, $N$ over $A$, $B$ respectively,
$Q(M,N)$ can be considered an $A^{{\rm op}}$ module.
\end{enumerate}
The ingredients for the applications of interest are an algebra $A$ and a
representation of $A$ on a vector space $V$, and a distinguished
subalgebra $B$. Considered as an $A$ module, $V^{*} = ({}^AV)^{*}$
is not very interesting, and may in fact be zero. However,
considered as a $B$ module, $({}^BV)^{*}$ is not only a $B$ module,
but an $A$ module. The property of the subalgebra $B$ which gives
$({}^BV)^{*}$ the structure of an $A$ module is as follows. }
\end{Remarks}
\begin{Definition} {\rm
Say $B \leq A$ is {\em quasi-normal} if and only if for every $a$
in $A$ there exists $a_1,..., a_n$ such that
\begin{equation}
BaB = \sum_1^{l} Ba_i = \sum_{1}^{l}a_iB.
\end{equation}
}\end{Definition}
\begin{Lemma}\label{QNAlem} {\rm Suppose $B$ is quasi normal in $A$, and let $s:A \to A$
be an anti-automorphism. Then if $M$ is a representation of $A$,
$Q({}^BM, \Bbc{C})$ is an $A$ module.
{\em Proof }. Notice that the action of $A^{{\rm op}}$ on ${\rm Hom}(M,\Bbc{C})$ given
by
\begin{equation}
a\mu(m) = \mu(am)
\end{equation}
coincides with the action of $B^{{\rm op}}$ on $({}^BM)^{*}$
whenever $\mu$ is in $({}^BM)^{*}$ and $a$ is in $B$. Prefacing
this action with the antiautomorphism $s$
\begin{equation}
a\bullet \mu(m) = \mu(s(a)m)
\end{equation}
defines an action of $A$ on {\rm Hom}(M,\Bbc{C}). The claim is that
$({}^BM)^{*}$ is fixed by this action.
Let $\alpha$ be in $({}^BM)^{*}$ and let $a$ be in $A$.
By 4.2 $\alpha: M \to \Bbc{C}$ vanishes on
$N$, a $B$ submodule with $M/N$ finite dimensional. The problem is
to show that there is a $B$ submodule $N_a$ such that $M/N_a$ is
finite dimensional and $(a\bullet \alpha)(N_a) = 0$.
Since $B$ is quasi normal write
\begin{equation}
Bs(a)B = \sum_{1}^l a_iB = \sum_1^l Ba_i.
\end{equation}
Then define linear maps
\begin{equation}
\alpha_{i}~: N \to M \to M/N, \quad \alpha_{i}(n) = a_in +N
\end{equation}
and let
\begin{equation}
N_i = {\rm ker} \alpha_{i}, \quad N_a = \cap N_i.
\end{equation}
Now observe that $N_a$ is a $B$ submodule of $M$: consider $a_ibn$
for $b$ in $B$, $n$ in $N_a$. We can write
\begin{equation}
a_ib = \sum_1^l b_j a_j
\end{equation}
so that
\begin{equation}
\alpha_i (bn) = a_i bn + N = \sum_1^l b_j a_j n +N.
\end{equation}
Since $n$ is in $N_a$, $a_jn$ is in $N$ for all $j$, and hence so
is $b_ja_jn$. Thus $bn$ is in the kernel of $a_i~$ for all $i$.
Finally check that $N_a$ is contained in $ker a \bullet \alpha$. For
$n$ in $N_a$\ $a \bullet\alpha(n) = \alpha(s(a)n)$. But $s(a)$ is in
$Bs(a)B$, so $s(a) = \sum_1^l b_ja_j$ and $s(a)n = \sum_1^l b_ja_j n$.
Since $n$ is in ${\rm ker} a_i~$ for each $i$, $a_in$ is in $N$ for
each $i$, hence $s(a)n$ is in $N$ and $\alpha(s(a)n) = 0$ as
required.
}\end{Lemma}
\subsection{Application to totally disconnected groups}
Let $G$ be a totally disconnected group (see [3] for a survey of the
representation theory of these objects) with a given compact open
subgroup $K$, and let $M$ be a complex representation of $G$, hence
a representation of $\Bbc{C} G (= A)$ and $\Bbc{C} K (=B)$.
\begin{Lemma}\label{QNlem} {\rm $\Bbc{C} K$ is quasi normal in $\Bbc{C} G$.
{\em Proof }. Let $g$ be in $G$. The double coset $KgK$ is a finite union of
either right or left cosets of $K$ and the left coset
representatives $\{g_i\}$ may be chosen to be the same as the right
coset representatives. Then
\begin{equation}
\Bbc{C} K g\Bbc{C} K = \sum_{1}^n\Bbc{C} Kg_j = \sum_1^n g_j\Bbc{C} K
\end{equation}
as required.
}\end{Lemma}
\begin{Corollary}\label{QREPCG} {\rm $Q({}^{\Bbc{C} K} M,\Bbc{C})$ is a representation of $\Bbc{C} G$ which is locally
finite as a representation of $\Bbc{C} K$.
{\em Proof }. All that is needed to meet the conditions of lemma \ref{QNAlem}
is the choice of an appropriate antiautomorphism. Clearly the
map $s(g) = g^{-1}$ is a suitable choice.
The representation $Q({}^{\Bbc{C} K} M,\Bbc{C})$ is almost, but not quite the smooth dual
of $M$. The relationship can be described in coalgebraic terms.
}\end{Corollary}
\begin{Definition} {\rm If $F$ is a subcoalgebra of $C$, and $D$ is a $C$
comodule, define the {\em restriction} of $D$ to $F$ to be
\begin{equation}
{}_F|D = \{d \in D: \Delta d \in F \otimes D\}.
\end{equation}
Thus $ {}_F|D$ is a $C$ sucomodule and an $F$ comodule.
In particular, the coalgebra $P(\Bbc{C} K, \Bbc{C}) = (\Bbc{C} K)^*$ contains as an important
subcoalgebra the vector space with basis $K^{\wedge}$, the set of
group homomorphisms $\rho:K \to \Bbc{C}$. The trivial homomorphism
$\tau:K \to \Bbc{C}$ in particular is in $K^{\wedge}$. Consider the subcomodule
\begin{equation}
{}_{\Bbc{C}\tau}|({}^{\Bbc{C} K} M).
\end{equation}
}\end{Definition}
\begin{Proposition}\hfil\break {\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $K'$ is another compact open subgroup of $G$ then $({}^{\Bbc{C} K} M)^{*} = ({}^{\Bbc{C} K'}M)^{*}$.
\item If $K'\leq K$, and if $\tau, \tau'$ are the corresponding trivial homomorphisms,
then
\begin{equation}
{}_{\Bbc{C}\tau}|({}^{\Bbc{C} K} M)^* \leq {}_{\Bbc{C}\tau'}|({}^{\Bbc{C} K'}M)^*.
\end{equation}
\item The union $\cup {}_{\Bbc{C}\tau}|({}^{\Bbc{C} K} M)^*$ over all compact open $K$ is the smooth
dual of $M$.
\end{enumerate}
{\em Proof }. The only statement which is not immediate is the first.
Suppose that $K'\leq K$. The inclusion induces a map
$(\Bbc{C} K)^{*} \to (\Bbc{C} K')^{*}$, and any $(\Bbc{C} K)^{*}$ comodule is automatically a
$(\Bbc{C} K')^{*}$ comodule. Moreover, any $(\Bbc{C} K)^{*}$ comodule $D$ equipped
with a measuring map
$\rho:D \to {\rm Hom}(M,C)$ is also a measuring comodule for $(\Bbc{C} K')^{*}$. Thus
$({}^{\Bbc{C} K} M)^{*} \to (\Bbc{C} K'M)^{*}$.
Less obviously $(\Bbc{C} K'M)^{*} \to ({}^{\Bbc{C} K} M)^{*}$. Let $\alpha:M \to C$ vanish on $N'$
which is a $\Bbc{C} K'$ submodule with $M/N'$ finite dimensional. The
aim is to show that there exists $N$, a $\Bbc{C} K$ submodule with
$\alpha(N) = 0$ and $M/N$ finite dimensional.
Write $K'KK' = \sqcup k_iK' = \sqcup K'k_i$. Since $K$ and $K'$ are compact open
this is a finite union. The argument now is the same as that which
established \ref{QNlem}. Define maps
\begin{eqnarray}
k_i: N' &\to& M \to M/N' \nonumber\\
k_in' &=& k_n'+N' \mbox{\ for\ } n' \in N'
\end{eqnarray}
and set
\begin{equation}
N= \cap {\rm ker} k_i.
\end{equation}
The arguments that (i) $N$ is a $CK$ module, (ii) $N$ is contained
in ${\rm ker} \alpha$ and (iii) $M/N$ is finite dimensional follow the
pattern of \ref{QNAlem}
%
}\end{Proposition}
\subsection{Application to loop algebras}
(See [5] for basic information on the subject.) Let ${L}$ be a
finite-dimensional simple Lie algebra over $\Bbc{C}$ and let
${L}[x,x^{-1}]$ denote the loop algebra of ${L}$ consisting of
Laurent polynomials in $x$ with coefficients in ${L}$. A
representation $M$ of ${L}$ is a projective representation with
cocycle $c$ if
\begin{equation}
(\xi x^i)(\psi x^j) m = \left( (\psi x^j)(\xi x^i) \right)
+ [\xi,\psi]x^{i+j}m + c(\xi x^i,\psi x^j)m
\end{equation}
for all $m$ in $M$. The representation is said to be of level $k$
if it is projective with cocycle $c$ given by
\begin{equation}
c(\xi x^i,\psi x^j) = ik\kappa(\xi,\psi)\delta_{i,-j}
\end{equation}
where $\kappa(\, ,\, )$ is the Killing form on ${L}$.
A projective representation of ${L}[x,x^{-1}]$ corresponds to an
ordinary representation of the central extension ${L}[x,x^{-1}]
\oplus \Bbc{C} c$ in the usual way. Thus level $k$ representations are
representations in which $c$ acts as multiplication by $k$. In
addition, there is an outer derivation $d$ of ${L}[x,x^{-1}]$ given
by
\begin{equation}
d\xi x^i = i \xi x^i.
\end{equation}
Form the Lie algebra ${L}[x,x^{-1}] \oplus \Bbc{C} c \oplus \Bbc{C} d$,
setting $[d,\xi x^i] = i \xi x^i$, $[d,c]=0$. The algebras of
interest are universal enveloping algebras of this Lie algebra and
certain subalgebras. Write
\begin{eqnarray}
U &=& U({L}[x,x^{-1}]\oplus \Bbc{C} c \oplus \Bbc{C} d),\nonumber\\
U_{\geq} &=& U({L}[x] \oplus \Bbc{C} c \oplus \Bbc{C} d) \nonumber\\
U_{\leq} &=& U({L}[x^{-1}] \oplus \Bbc{C} c \oplus \Bbc{C} d) \nonumber\\
U_> &=& U({L}[x]x) \nonumber\\
U_< &=& U({L}[x^{-1}]x^{-1})
\end{eqnarray}
The isomorphisms as vector spaces
\begin{equation}
{L}[x,x^{-1}] = {L}[x^{-1}]x^{-1} \oplus {L}[x]
= {L}[x^{-1}]x^{-1} \oplus {L} \oplus {L}[x]x
\end{equation}
induce isomorphisms of vector spaces
\begin{equation}
U = U_< \otimes U_{\geq}
= U_< \otimes U({L} \oplus \Bbc{C} c \oplus \Bbc{C} d) \otimes U_>.
\end{equation}
The bracket with $d$ provides a $\Bbc{Z}$ grading (as vector
spaces) of all the universal enveloping algebras described here.
With respect to this grading
\begin{equation}
U_< = \oplus_{n \leq 0} (U_<)_n
\end{equation}
where $(U_<)_n$ is the set of elements of degree $n$. Each
$(U_<)_n$ is finite dimensional. Hence the subspace
\begin{equation}
(U_<)_{(n)} = \oplus _{0 \leq j \leq n} (U_<)_j
\end{equation}
is also finite dimensional and
\begin{equation}
U_< = \oplus_{n\leq 0} (U_<)_{(n)}.
\end{equation}
\begin{Lemma}\label{UQNlem} {\rm $U_{\geq}$ is quasi-normal in $U$.
{\em Proof }. This is essentially a consequence of the analogue of the Poincare-
Birkhoff-Witt theorem for universal enveloping algebras. Observe that
\begin{eqnarray}
(U_<)_{(n)}\otimes U_{\geq} &=& U_{\geq} \otimes(U_<)_{(n)} \nonumber\\
U_{\geq} \otimes(U_<)_{(n)} &=& (U_<)_{(n)}\otimes U_{\geq}.
\end{eqnarray}
The result follows since $a$ in $U$ is in some $(U_<)_{(n)}
\otimes U_{\geq}$. If $\{a_{i}\}$ is a basis for $(U_<)_{(n)}$ then
\begin{equation}
U_{\geq}aU_{\geq} =\sum_{i}a_iU_{\geq} = \sum_{i} U_{\geq}a_i
\end{equation}
as required.
The antiautomorphism commonly used is that determined by the Lie
algebra antiautomorphism $s:{L}[x,x^{-1}] \oplus \Bbc{C} c \oplus
\Bbc{C} d
\to
{L}[x,x^{-1}]\oplus \Bbc{C} c \oplus \Bbc{C} d$,
\begin{equation}
s(\xi x^i) = -\xi x^{-i}, s(c) = c, s(d) = -d.
\end{equation}
}\end{Lemma}
\begin{Proposition}\hfil\break {\rm If $M$ is a $U$ module then
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $({}^{U_{\geq}}M)^*$ is a level $k$ representation if $M$ is.
\item $({}^{U_{\geq}}M)^*$ is locally finite as a $U_{\geq}$ module.
\end{enumerate}
{\em Proof }. i. This is more or less a direct corollary of \ref{UQNlem}. Calculate
\begin{eqnarray}
\lefteqn{ [(\xi x^i)_(\psi x^j)\alpha - (\psi x^j)_(\xi x^i)\alpha - [x,\psi]xi+j\alpha ](m)} \nonumber\\
&=& \alpha[(s(\psi x^j)(s(\xi x^i) - s(\xi x^i)s(\psi x^j) - s([x,\psi]x^{i+j}))m] \nonumber\\
&=& \alpha[((\psi x^{-j})(\xi x^{-i}) - (\xi x^{-i})(\psi x^{-j}) - ([\psi,x]x^{-i-j}))m] \nonumber\\
&=& \alpha[c(\psi x^{-j},\xi x^{-i})m] \nonumber\\
&=& c(s(\psi x^j),s(\xi x^i))\alpha(m) \nonumber\\
&=& -c(s(\xi x^i),s(\psi x^j))\alpha(m) \nonumber\\
&=& c(\xi x^i,\psi x^j)\alpha_(m)
\end{eqnarray}
since $c(\xi x^i,\psi x^j) = ik\delta_{i,{-j}}\kappa(x,\psi) =
{-j}k\delta_{{-j},i}\kappa(\psi,x)
= c(\psi x^{-j},\xi x^{{-i}})$. This establishes i.
For ii observe that $s(U_{\geq}) = U_{\geq}$. The result then
follows from 4.6.
}\end{Proposition}
\begin{Definition} {\rm Say a representation $M$ of $U$ is positive
energy if $d$ acts diagonally with real eigenvalues and the
eigenvalues of d are bounded above.
}\end{Definition}
As with the category of smooth representations of totally disconnected
groups, so the category of positive energy representations of a
loop algebra admits the existence of a dual. As the smooth dual of
a representation of a totally disconnected group can be identified
in terms of restricted comodules, so the dual positive energy
representation of a representation $M$ can be identified as an
appropriate restriction of $({}^{U_{\geq}}M)^{*}$. It remains to
identify the appropriate subcoalgebra of $(U_{\geq})^{*}$.
The universal enveloping algebra $U_{>}$ has an augmentation ideal
$U_{>}{}^{+} = \oplus_{n>1} (U_{>})_n$. This generates an ideal
$U_0$ of $U_{\geq}$
\begin{equation}
U_0 = U_{\geq} (U_{>}{}^{+}).
\end{equation}
A short calculation shows that $U_0$ is in fact a two sided ideal.
Define
\begin{equation}
P_0{}^N = im (U_{\geq}/(U_0)^N)^{*} \to (U_{\geq})^{*}.
\end{equation}
Since $P_0{}^{N+j} \geq P_0{}^N$, define
\begin{equation}
P_0 = \cup P_0{}^N.
\end{equation}
\vfil\eject
\begin{Proposition} \hfil\break {\rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item ${}_{P_0}|({}^{U_{\geq}}M)^*$ is a $U$ submodule of $({}^{U_{\geq}}M)^{*}$.
\item If ${}_{P_0}|({}^{U_{\geq}}M)^{*}$ is generated by a finite set of eigenvectors for $d$ then it
is positive energy.
\end{enumerate}
}\end{Proposition}
{\em Proof } i. Check that for $z$ in $U$, $q$ in ${}_{P_0}|({}^{U_{\geq}}M)^*$, $z\bullet q$ is
in $P_0 | ({}^{U_{\geq}}M)^*$, or equivalently, for some $N$, any
$u$ in $s(U_0{}^N)$, $u\bullet z\bullet q = 0$. Using $P_0 = \cup
P_0{}^N$ it can be shown that
\begin{equation}
{}_{P_0}|({}^{U_{\geq}}M)^* = {}_{\cup P_0^N}|({}^{U_{\geq}}M)^*.
\end{equation}
Suppose then that $q$ is in $P_0^{N'}|({}^{U_{\geq}}M)^*$ for some
$N'$, that is, $u\bullet q = 0$ for all $u$ in $s(U_0{}^{N'})$.
If $i \geq 0$ and $z$ in $(U_{\geq})i$, then $u\bullet z\bullet q = 0$ since $z$ is in
$ U_{\geq}$ and $s(U_0{}^{N'})$ is an
ideal of $U_{\geq}$. If $i< 0$, observe that
\begin{equation}
s(U_0{}^N)(U_{\geq})_i \leq (U_{\geq})_{(i)}s(U_0{}^{N+i}).
\end{equation}
Thus for $z$ in $(U_{\geq})_i$ , $q$ in
$P_){}^{N'}|({}^{U_{\geq}}M)^{*}$, $u\bullet z \bullet q = 0$ provided $N> N'{{-i}}$.
ii. Write $V = {}_{P_0}|({}^{U_{\geq}}M)^*$. Assume that $\{q_i\}$ is a finite generating set
for $V$ of $d$ eigenvectors. Since $V$ is locally finite as a
$U_{\geq}$ module, $U_{\geq}\{q_i\}$ is a finite dimensional
$U_{\geq}$ module, call it $D$. In particular, the element $d$ acts
on $D$, and is diagonalizable on $D$ with finitely many
eigenvalues. But then since
\begin{equation}
V = UD = U_{<}D
\end{equation}
$d$ acts diagonally on $V$ and the eigenvalues are bounded below.
\vskip 0.1cm
{\bf References}
\vskip 0.1cm
\begin{enumerate}
\item Batchelor, M. In search of the graded manifold of maps between
graded manifolds. In {\em Complex Differential Geometry and
Supermanifolds in Strings and Fields.} P.J.M. Bongaarts and R.
Martini, eds. LNP 311. Springer, 1988.
\item Batchelor, M. Measuring coalgebras quantum group-like objects,
and non-commutative geometry. In {\em Differential Geometric
Methods in Theoretical Physics.} C. Bartocci, U. Bruzzo and R.
Cianci, eds. LNP 375. Springer. 1990.
\item Cartier, P. Representations of p-adic groups: a survey. In {
\em Automorphic forms and L-functions. Proceedings of Symposia in Pure
Mathematics.} Volume XXXIII, part 1. p 111-155. AMS. Providence,
1979.
\item Hyland, M. and Batchelor, M. Enrichment in coalgebras and
comodules: an approach to fibrations in the erniched context. {\em
In preparation.}
\item Kac, V. G. {\em Infinite dimensional Lie algebras.}
Birkhauser, Boston, 1983.
\item Kelly, G. M. {\em Basic Concepts of Enriched Category Theory.}
London Mathematical Lecture Note Series 64. Cambridge University
Press, 1985.
\item Kobayashi, S. and Nomizu,K. {\em Foundations of Differential
Geometry.} Wiley. 1996.
\item Spivak, M. {\em A Comprehensive Introduction to Differential
Geometry,} vol. 2. 2nd ed. Publish or Perish, 1979.
\item Sweedler, M. {\em Hopf Algebras.} Benjamin. New York, 1969.
\end{enumerate}
\end{document}
|
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}
| 8,222
|
Bustul lui Consantin I. Nottara, amplasat în curtea Muzeului C. I. Nottara și C. C. Nottara, a fost realizat de sculptorița Milița Petrașcu (1892 - 1976). Bustul este turnat în bronz și este așezat pe un soclu de piatră.
Constantin I. Nottara (1859 – 1935) a fost un important actor al Teatrului Național din București, debutând pe scena acestuia în 1877. A jucat roluri precum: Shylock, Hamlet și Regele Lear din teatrul shakespearean, Don Salluste din "Ruy Blas" de Hugo, Ștefan cel Mare din "Apus de Soare" și Tudose din "Hagi Tudose" de Delavrancea. De asemenea, a lucrat ca director de scenă și profesor la Conservatorul dramatic din Bucuresti.
Lucrarea este înscrisă în Lista monumentelor istorice 2010 - Municipiul București - la nr. crt. 2294, .
Monumentul este situat în curtea Muzeului C. I. Nottara și C. C. Nottara din sectorul 2 de pe Bulevardul Dacia nr. 105.
Note
Statui din București
Monumente istorice de for public din București
|
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| 1,402
|
You will receive emails featuring helpful information and stories about living with symptoms of nOH, as well as notifications when we add new information to nOHMatters.com.
Learn how you can help others by sharing your experience with symptoms of nOH as an nOH Champions ambassador.
I found two programs that I think you might be interested in. The first is the nOH email program, which provides advice, education, and patient stories for people living with neurogenic orthostatic hypotension (nOH). The second is nOH Champions, an ambassador program for people who want to share their nOH story.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 795
|
(function ($) {
$.fn.craftyslide = function (options) {
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| 286
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Last Updated on Mon, 10 Sep 2018 | Minor Planets
young people. Fujii has fostered public awareness of astronomy through television broadcasts from his Chiro Observatory in Shirakawa, Fukushima prefecture, and he toured Japan in 1986, offering the public views of Halley's Comet with a trailer-mounted 60-cm reflector. Fujii is best known outside of Japan for his stunning celestial photographs, a hallmark of which is technical perfection. (M 22829)
Name proposed by D. di Cicco and R. W. Sinnott and endorsed by the discoverer.
Celestial, Photographs
Grafton Planet Images
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{
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| 3,595
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Dr. Faulkner is a plastic and reconstructive surgeon who specializes in surgery of the breast and body, including breast reconstruction, cosmetic breast surgery, breast reduction, and body contouring after massive weight loss, childbirth, and aging.
Dr. Faulkner was born and raised in NYC. She attended University of Southern California for college, medical school and general surgery residency, and is board certified by both the American Board of Surgery and the American Board of Plastic Surgery. During residency, Dr. Faulkner spent 2 years in Boston, and earned a Masters in Public Health (MPH) degree from Harvard School of Public Health, and was a Harvard Medical School plastic surgery research fellow at Boston Children's Hospital. Dr. Faulkner attended plastic surgery residency at Vanderbilt University Medical Center in Nashville, TN, followed by fellowship at Mass General Hospital, specializing in reconstructive and aesthetic breast surgery.
Dr. Faulkner is a member of the faculty of Harvard Medical School. She is Chief of Plastic Surgery at North Shore Medical Center. She is an active member of the American Society of Plastic Surgeons and the New England Society of Plastic and Reconstructive Surgeons, and a candidate member of the American Society for Aesthetic Plastic Surgery. She is a Fellow of the American College of Surgeons (FACS), and member of the Association of Women Surgeons and the Association of Academic Surgery, and other national professional organizations. She has authored publications (under her maiden name Rosen and married name Faulkner) on pediatric and adult plastic surgery. She has given research presentations at national and international plastic surgery meetings. Her clinical areas of interest include breast reconstruction, cosmetic breast surgery, body contouring after weight loss/childbirth/aging, labiaplasty, and the use of Botox and fillers for non-surgical facial rejuvenation. Dr. Faulkner is committed to excellence in patient care and attention to detail.
Faulkner HR, Colwell AS, Liao EC, Winograd JM, Austen WG Jr. Thermal Injury to Reconstructed Breasts from Commonly Used Warming Devices: A Risk for Reconstructive Failure.
Nuzzi LC, Cerrato FE, Webb ML, Faulkner HR, et al. Psychological impact of breast asymmetry on adolescents: a prospective case-control study.
Ganske I, Verma K, Rosen H, et al. Minimizing complications with the use of acellular dermal matrix for immediate implant-based breast reconstruction.
Chun YS, Verma K, Sinha I, Rosen H, et al. Impact of prior ipsilateral chest wall radiation on pedicled TRAM flap breast reconstruction.
Rosen H, Webb ML, et al. Adolescent gynecomastia: not only an obesity issue.
Rosen H, Saleh F, Lipsitz SR, Rogers SO Jr, Gawande AA. Downwardly mobile: the accidental cost of being uninsured.
Heather R. Faulkner, MD, MPH, is a plastic and reconstructive surgeon who specializes in surgery of the breast and body, including breast reconstruction and cosmetic breast surgery, and body contouring after massive weight loss, childbirth and aging. She discusses options for women at midlife in this talk from the 2018 Midlife Women's Health Community Conference.
Dr. Heather Faulkner speaks about breast reduction surgery and answers common questions about the surgery and recovery.
An introduction to Dr. Heather Faulkner, a plastic and reconstructive surgeon at Mass General.
|
{
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| 303
|
\section{Introduction}
A stochastic interaction of a quantum system with a bath
brings up the term $\ehat\ffrnforce$ in the relations for time-dependent expectation values of system momenta $\hat\pp{=}\{\hat p_1,\ldots,\hat p_N\}$ and positions $\hat\xx{=}\{\hat x_1,\ldots,\hat x_N\}$:%
\begin{subequations}\label{ODM-eqs}
\begin{gather}
\label{ODM-dp/dt}
\tder{}{t}\midop{\hat p_n}{=}{-}\midop{\tpder{}{\hat x_n}U(\hat \xx)}{+}\midop{\ehat\frnforce_
},\\
\label{ODM-dx/dt}
\tder{}{t}\midop{\hat x_n}{=}\tfrac{1}{m_n}\midop{\hat p_n},
\end{gather}
\end{subequations}
where $U(\hat \xx)$ is a potential energy operator and $m_k$ are effective masses. In this Letter, we study the case where $\ehat\ffrnforce{=}\ehat\ffrnforce(\hat{\pp})$ is position-independent. In this form, Eqs.~\eqref{ODM-eqs} apply to many quantum phenomena including the translational motion of an excited atom in vacuum \cite{2017-Sonnleitner}, Brownian motion in a dilute background gas \cite{2003-Accardi}, light-driven processes in semiconductor, nanoplasmonic and optomechanical systems \cite{BOOK-Auffeves,BOOK-Milburn,2015-Barchielli}, superconducting currents \cite{1996-OConnell}, quantum ratchets \cite{2002-Reimann}, energy transport in low-dimensional systems \cite{2008-Dhar}, dynamics of chemical reactions \cite{2000-Kuhn}, two-dimensional vibrational spectroscopy and NMR signals \cite{2000-Steffen,2006-Tanimura} as well as more exotic entirely quantum dissipative effects \cite{2010-Rezek,1999-Kardar}.
The term $\ehat\ffrnforce(\hat{\pp})$ in Eqs.~\eqref{ODM-eqs} admits a simple classical interpretation as friction acting on effective particles moving in a potential $U(\xx)$.
Such classical dynamics are described by the familiar Langevin, Drude and Fokker-Plank models when the system-bath interactions are treated as
\begin{enumerate*}[label=(\roman*)]
\item \label{A_i}
memoryless (Markovian) and
\item \label{A_ii}
translation invariant (position-independent).
\end{enumerate*}
However, we will show that these two assumptions are at odds with quantum thermodynamics.
Specifically, we will prove a long-standing no-go conjecture that completely positive\footnote{Positivity of quantum evolution guarantees satisfaction of the Heisenberg uncertainty principle at all times. It was shown that the requirements for positivity and complete positivity coincide for some quantum systems including a harmonic oscillator \cite{1997-Kohen}.} Markovian translation-invariant quantum dynamics obeying Eqs.~\eqref{ODM-eqs} cannot thermalize.
The no-go conjecture was demonstrated by Lindblad as early as in 1976 \cite{1976-Lindblad-a} for a quantum harmonic oscillator with a Gaussian damping%
\footnote{The Gaussian damping corresponds to $\Lrel{=}\Lbd_{\set{\mu}\hat\xx{+}\set{\eta}\hat\pp}$ ($\set{\mu},\set{\eta}\in \mathbb{C}^N$) in Eq.~\eqref{Quantum_Liouville_equation_nD(a)} and can be cast to form \eqref{theorem:trans_inv_Lbd}, as shown in \appref{@APP:theo:trans_inv_Lbd}). The original paper \cite{1976-Lindblad-a} deals with one-dimensional case. The multidimensional extension can be found e.g. in \cite{1985-Dodonov}.}.
Subsequently his particular result was extended to a general quantum system under the weight of mounting numerical evidence, however without proof. The no-go conjecture is de-facto incorporated in all popular models such as the Redfield theory \cite{1957-Redfield}, the Gaussian phase space ansatz of Yan and Mukamel \cite{1988-Yi_Jing_Yan}, the master equations of Agarwal \cite{1971-Agarwal}, Caldeira-Leggett \cite{1983-Caldeira}, Hu-Paz-Zhang \cite{2005-Ford}, and Louisell/Lax \cite{1965-Louisell}, and the semigroup theory of Lindblad \cite{1976-Lindblad} along with specialized extensions in different areas of physics and chemistry. These models break either one of assumptions \ref{A_i} and \ref{A_ii} or the complete positivity of quantum evolution (see \cite{1997-Kohen,2000-Yan,2016-Bondar} for detailed reviews, note errata \cite{2016-Bondar-a}). This circumstance is a persistent source of controversies (see e.g.~the discussions \cite{1998-Wiseman,2001-OConnell,2001-Vacchini} of original works \cite{1997-Gao,2000-Vacchini}).
The matters were further complicated by the discovery that the free Brownian motion $U(\hat\xx){=}0$ circumvents the conjecture \cite{2002-Vacchini} (we will identify the full scope of possible exceptions below).
The no-go result challenges studies of the long-time dynamics of open systems. On the one hand, model's thermodynamic consistency is undermined by assumptions \ref{A_i} and \ref{A_ii}. On other hand, the same assumptions open opportunities to simulate large systems that are otherwise beyond the reach. Specifically, the abandonment of Markovianity
entails a substantial overhead to store and process the evolution history. The value of assumption \ref{A_ii} can be clarified by the following example. Consider the re-thermalization of a harmonic oscillator coupled to a bath (represented by a collection of harmonic oscillators) after displacement from equilibrium by, e.g., an added external field, a varied system-bath coupling, or interactions between parts of a compound system.
To account for such a displacement without assumption \ref{A_ii}, one needs to self-consistently identify the equilibrium position for each bath oscillator, re-thermalize the bath and modify the system-bath couplings accordingly. In practice, this procedure is intractable
without gross approximations that lead to either numerical instabilities or physical inaccuracies. Choosing among a polaron-transformation-based method, Redfield, and F\"orster (hopping) models of quantum transfer epitomizes this dilemma \cite{2013-Jang}.
Remarkably, assumption \ref{A_ii} enables to model the displaced state equilibrium by simply adjusting the potential energy $\hat U$. Fig.~\ref{@FIG.01b'}a shows that without this assumption the potential adjustment yields steady state $\rhost$ significantly different from the canonical equilibrium $\rhoth{\theta}{\propto}e^{{-}\frac{\hat H}{\theta}}$, where $\theta{=}k_{\idx{B}}T$ and $\hat H$ is system Hamiltonian.
Motivated by these arguments, we propose in this Letter a general recipe to construct approximately thermalizable bath models under assumptions \ref{A_i} and \ref{A_ii}. Fig.~\ref{@FIG.01b'} illustrates this recipe in application to the above example.
The resulting mismatch between $\rhost$ and $\rhoth{\theta}$ is small, especially at high temperatures and in the weak system-bath coupling limit. (The calculations details will be explained below.)
It will be shown elsewhere that the proposed recipe is capable of accurately accounting for electronic and spin degrees of freedom. We found it helpful in reservoir engineering and optimal control problems. Moreover, the resulting bath models are realizable in the laboratory and can be used for coupling atoms and molecules nonreciprocally \cite{2016-Zhdanov}. However, the scope of our recipe is limited by the applicability of assumptions \ref{A_i} and \ref{A_ii} and, therefore, cannot encompass strongly correlated systems (as in the case of Anderson localization \cite{2015-Nandkishore}).
\begin{figure}[tbp]
\centering\includegraphics[width=0.99\columnwidth]
{{fig.01b}.eps}
\caption{
The errors (expressed in the terms of Bures distance $D_{\idx{B}}$ between the thermal state $\rhoth{\theta}$ and its approximation $\rhost$) in modeling thermal states of a 1D quantum harmonic oscillator in the displaced equilibrium configurations (due to a change $U(\hat x){\to}U(\hat x{-}\Delta x_0)$ in the potential energy) using the conventional
quantum optical master equation (dashed lines) and the proposed translation-invariant dissipation model defined by Eqs.~\eqref{Quantum_Liouville_equation_nD},\eqref{theorem:trans_inv_Lbd} and \eqref{_optimal_f_k} (solid lines).
(a) The error dependence on displacement $\Delta x_0$ for several temperatures $\theta$. (b) The error dependence on temperature $\theta$ for different values of $\kappa$ (in units of $\kappa_0{=}\hbar^{-1}\beta^{-\frac12}$).\label{@FIG.01b'}}
\end{figure}
\section{The key results}
Starting by formalizing the problem, we write the general master equation that accounts for memoryless system-bath interactions and ensures positivity of the system density matrix $\hat\rho$ at all times \cite{1976-Lindblad}:%
\begin{subequations}
\label{Quantum_Liouville_equation_nD}
\begin{gather}\label{Quantum_Liouville_equation_nD(a)}
\tpder{}{t}\hat\rho{=}{\cal L}[\hat\rho],~{\cal L}{=}\Lvn{+}\Lrel,\\%~\Lrel{=}\textstyle\sum_{k{=}1}^K\Lbd_{\hat L_k}\\
\Lvn[\odot]{=}\tfrac{i}{\hbar}[\odot,\hat H],~\hat H{=}H(\hat\pp,\hat\xx){=}\textstyle\sum_{n=1}^N\frac{\hat p_n^2}{2 m_n}{+}U(\hat\xx),\label{Quantum_Hamiltonian_nD}\\[-4pt]
\Lrel{=}\sum_{k{=}1}^K\Lbd_{\hat L_k},
\label{_Lindbladian_definition}
~~
\Lbd_{\hat L}[\hat\rho]{\defeq}\hat L{\hat\rho}\hat L^{\dagger}{-}\tfrac12(\hat L^{\dagger}\hat L{\hat\rho}{+}{\hat\rho}L^{\dagger}\hat L),
\end{gather}
\end{subequations}
where $\odot$ is the substitution symbol defined, e.g., in \cite{2000-Grishanin}. The superoperator $\cal{L}_{\idx{rel}}$ accounts for system-bath couplings responsible for the friction term $\ehat\ffrnforce$ in Eq.~\eqref{ODM-dp/dt} and depends on a set of generally non-Hermitian operators $\hat L_k$.
Based on theorems by A. Holevo \cite{1995-Holevo,1996-Holevo}, B. Vacchini \cite{2005-Petruccione,2005-Vacchini,2009-Vacchini} has identified the following criterion of translational invariance for the $\cal{L}_{\idx{rel}}$:
\begin{lemma}[The justification is in \appref{@APP:theo:trans_inv_Lbd}]\label{@theo:trans_inv_Lbd}
Any translationally invariant superoperator $\Lrel$ of the Lindblad form \eqref{_Lindbladian_definition} can be represented as%
\begin{subequations}\label{theorem:trans_inv_Lbd}
\begin{gather}
\displaybreak[0]\Lrel{=}\textstyle\sum_k\Lbd_{\hat A_k}+\Laux \mbox{ with } \displaybreak[0]\\
\hat A_k{\defeq}e^{{-}i\kkappa_k\hat\xx}\tilde f_k(\hat\pp),~~\Laux{=}{-}i[\kkappaaux\hat\xx{+}\faux(\hat \pp),\odot].
\end{gather}
\end{subequations}
where $\kkappa_k$ and $\kkappaaux$ are $N$-dimensional real vectors, $\tilde f_k$ are complex-valued functions and $\faux$ is real-valued
\footnote
{
The Gaussian dissipators $\Lbd_{\kkappaG_k\hat\xx{+}\ffG_k(\hat\pp)}$ $(\kkappaG_k{\in}\mathbb{R}^N)$ can be reduced to the form
Eq.~\eqref{theorem:trans_inv_Lbd} as a limiting case $\kkappa_k{\to}0$, as shown in \appref{@APP:theo:trans_inv_Lbd}. The
generalized unitary drift term $\Laux$ accounts for ambiguity of the separation of the quantum Liouvillian ${\cal L}$ in Eq.~\eqref{Quantum_Liouville_equation_nD(a)} into Hamiltonian and relaxation parts.
}
.The converse holds as well.
\end{lemma}
The primary findings of this work are summarized in the following two no-go theorems.
\begin{theorem} \label{@theo:no_go(T=0)
Let $\ket{\Psi_0}$ be the ground state (or any other eigenstate of $\hat H$), such that $\matel{\Psi_0}{\hat\pp}{\Psi_0}{=}0$, and which momentum-space wavefunction $\Psi_{0}(\pp){=}\scpr{\pp}{\Psi_0}$ is nonzero almost everywhere, except for some isolated points. Then, no translationally invariant Markovian process of form \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} can steer the system to $\ket{\Psi_0}$.
\end{theorem}
The idea of the proof, whose details are given in \appref{@APP:theo:no_go(T=0)}, is to show that the state $\hat\rho_{0}{=}\proj{\Psi_{0}}$ can be the fixed point of superoperator $e^{t\cal L}$ only if $\Lrel{\equiv}0$. First, note that the linearity and translation invariance of the dissipator \eqref{theorem:trans_inv_Lbd} imply that $\Lrel[\int g(\xx')e^{{-}\frac{i}{\hbar}\xx'\hat\pp}\hat\rho_{0}e^{\frac{i}{\hbar}\xx'\hat\pp}d^N\xx']{=}0$ for any function $g(\xx')$. This equation can be equivalently rewritten as
\begin{gather}\label{ngt1_sample_eq}
\Lrel[\Psi_{0}(\hat\pp)g(\hat\xx)\Psi_{0}(\hat\pp)^{\dagger}]{=}0
\end{gather}
using the identities $e^{{-}\frac{i}{\hbar}\xx'\hat\pp}\ket{\Psi_{0}}{=}\sqrt{2\pi\hbar}\Psi_{0}(\hat\pp)\ket{\xx'}$ and $\int g(\xx')\proj{\xx'}d^N\xx'{=}g(\hat\xx)$, where $\ket{\xx'}$ is the eigenstate of position operator: $\hat x_k\ket{\xx'}{=}x'_k\ket{\xx'}$. Let us choose $g(\xx){=}e^{-i\llambda\xx}$, where $\llambda$ is an arbitrary real vector, and move to the right the $\hat\xx$-dependent terms in the lhs of Eq.~\eqref{ngt1_sample_eq} using the commutation relations $e^{-i\tilde\llambda\hat\xx}\hat\pp = (\hat\pp{+}\hbar\tilde\lambda)e^{-i\tilde\llambda\hat\xx}$ with $\tilde\llambda{=}\llambda,\pm\kkappa_k$. This rearrangement brings Eq.~\eqref{ngt1_sample_eq} to the form $\tilde G_{\llambda}(\hat\pp)e^{-i \llambda\hat{\xx}}{=}0$ (note that all the operators of form $e^{\pm i\tilde\kkappa_k\hat\xx}$ expectedly cancel out owing to translation invariance of $\Lrel$). The last equality can be satisfied only if the function $\tilde G_{\llambda}(\pp)$ vanishes identically for all $\pp$ and $\llambda$.
However, careful inspection of \appref{@APP:theo:no_go(T=0)}~shows that the latter happens only if $\Lrel{=}0$.
The statement of the \ref{@theo:no_go(T=0)}-st no-go theorem can be strengthened for a special class of quantum systems. Let $\BFn(\pp,\llambda)$ be the Blokhintsev function \cite{1940-Blokhintzev}, which is related to Wigner quasiprobability distribution $W(\pp,\xx)$ as
\begin{gather}\label{_Blokhintsev_function}
\BFn(\pp,\llambda){=}\textstyle\int_{{-}\infty}^{\infty}\ldots\textstyle\int_{{-}\infty}^{\infty} e^{i \llambda\xx} W(\pp,\xx)\diff^N\xx.
\end{gather}
\begin{theorem}[]\label{@theo:no_go(T>0)}
Suppose that the Blokhintsev function $\BFn_{\theta}(\pp,\llambda)$ of the thermal state $ \rhoth{\theta}{\propto}e^{-\frac{\hat H}{\theta}}$ characterized by temperature $k_{\idx{B}}T{=}\theta$ is such that%
\begin{subequations}\label{_B(p,lambda)-features}
\begin{gather}\label{_B(p,lambda)-features-a}
\forall\pp,\llambda: \BFn_{\theta}(\pp,\llambda){>}0,~~\BFn_{\theta}(\pp,{-}\llambda){=}\BFn_{\theta}(\pp,\llambda),\\
\forall\pp{\ne}\zzero,\llambda{\ne}\zzero: \BFn_{\theta}(\pp,\llambda){<}\BFn_{\theta}(\zzero,\zzero).\label{_B(p,lambda)-features-b}
\end{gather}
\end{subequations}
Then, no translationally invariant Markovian process \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} can asymptotically steer the system to $\rhoth{\theta}$.
\end{theorem}
The proof of this theorem is given in \appref{@APP:theo:no_go(T>0)}~and generally follows the same logic as the outlined proof of the \ref{@theo:no_go(T=0)}-st no-go theorem.
Using Eq.~\eqref{_Blokhintsev_function} and the familiar formula for the thermal state Wigner function \cite{1949-Bartlett}, it is easy to check that the criteria \eqref{_B(p,lambda)-features} are satisfied for any $\theta$ in the case of a quadratic potential $U$. This means that the Lindblad's original conclusion on inability to thermalize the damped harmonic oscillator using the Gaussian friction term $\Lrel{=}\Lbd_{\set{\mu}\hat\xx{+}\set{\eta}\hat\pp}$ is equally valid for all Markovian translationally invariant dissipators.
\begin{corollary}\label{@cor:no_go_qho(T>0)}
No translationally invariant Markovian process of form \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} can steer the quantum harmonic oscillator into a thermal state of form $ \rhoth{\theta}{\propto}e^{{-}\frac{\hat H}{\theta}}$.
\end{corollary}
\section{Practical implications of the no-go theorems} In classical thermodynamics, the bath is understood as a constant-temperature heat tank ``unaware'' of a system of interest. However, the no-go theorems indicate that system-bath correlations of at least one kind -- spatial or temporal -- become obligatory for thermalization once quantum mechanical effects are taken into account. These correlations break the bath translation invariance or Markovianity assumptions, respectively.
Nevertheless, in the view of computational advantages outlined above, it is desirable to incorporate these assumptions into the master equations \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd}. Now we are going to introduce the recipe to construct such models with a minimal error in the thermal state.
In order to proceed, note that in the limit $(\hbar\kkappa_k)^2{\ll}\midop{\hat\pp^2}$ Eqs.~\eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} reduce to the familiar Fokker-Planck equation
\begin{gather}
\tpder{}{t}{\varpi(\pp)}{\stackrel{}{\simeq}}\Tr[\delta(\pp{-}\hat\pp)\Lvn[\hat\rho]]{+}\notag\\
\sum_{n,l}\pder{^2D_{n,l}(\pp)\varpi(\pp)}{p_n\partial p_l}{-}\sum_{n}\pder{\frnforce_n(\pp)\varpi(\pp)}{p_n}\label{_Fokker_Planck_equation}
\end{gather}
for the momentum probability distribution $\varpi(\pp){=}\Tr[\delta(\pp{-}\hat\pp)\hat\rho]$. The friction forces $\ffrnforce$ in Eq.~\eqref{_Fokker_Planck_equation} as well as Eq.~\eqref{ODM-dp/dt} have the form
\begin{gather}\label{_Bondarian_gen}
{\ffrnforce}(\hat\pp){=}{-}\textstyle\sum_k\hbar\kkappa_{k}|\fc_k(\hat\pp)|^2,
\end{gather}
whereas the momentum-dependent diffusion operator is
\begin{gather}\label{_Diffusion_operator}
D_{n,l}(\hat\pp){=}\textstyle\tfrac{\hbar^2}2\textstyle\sum_k|\fc_k(\hat\pp)|^2\kappa_{k,n}\kappa_{k,l}.
\end{gather}
Equations~\eqref{_Bondarian_gen} and \eqref{_Diffusion_operator} can be satisfied by different sets of $\kkappa_k$ and $\tilde f_k(p)$. We will exploit this non-uniqueness to reduce the system-bath correlation errors. Our strategy is reminiscent to the familiar way of making density functional calculations practical via error cancellation in approximated exchange-correlation functionals. We shall demonstrate the generic procedure
by considering a one-dimensional oscillator with the Hamiltonian $\hat H{=}\tfrac m2\hat p^2{+}\frac{m\omega^2}2\hat x^2$ (here the dimension subscript $n$ is omitted for brevity).
Corollary~\ref{@cor:no_go_qho(T>0)} implies that $\Lrel[\rhoth{\theta}]{\ne}0$ and $\rhost{\ne}\rhoth{\theta}$ for any $\theta$, where $\rhost{=}\left.\hat\rho\right|_{t\to\infty}$ is the actual fixed point of the evolution operator $e^{t{\cal L}}$. However, the net discrepancies can be reduced by imposing the following thermal population conserving constraint:
\begin{gather}\label{_energy_distribution_conservation}
\left.\tder{}{t}\midop{e^{-\alpha\hat H}}_{\theta}\right|_{t{=}0}{=}0;~\left|\tder{^2}{t^2}\midop{e^{-\alpha\hat H}}_{\theta}\right|_{t{=}0}{\to}\min \mbox{ for all }\alpha,
\end{gather}
where $\midop{\odot}_{\theta}(t){=}\Tr[\odot e^{t{\cal L}}[\rhoth{\theta}]]$. This constraint can be intuitively justified when the characteristic decay rates are much smaller than the typical transition frequencies, such that the dissipation can be treated perturbatively. Since the term $\Lrel[\rhoth{\theta}]$ generates only rapidly oscillating off-diagonal elements in the basis of $\hat H$, Eq.~\eqref{_energy_distribution_conservation} ensures that the first-order perturbation vanishes on average for the exact thermal state: $\lim_{t\to{\infty}}\tfrac{1}t\int_0^{t}e^{\tau\Lvn}\Lrel e^{(t-\tau)\Lvn}[\rhoth{\theta}]\diff\tau{=}0$.
In the case of the driftless dissipation $\Laux{=}0$, Eq.~\eqref{_energy_distribution_conservation} is satisfied by the following functions $\tilde f_{k}(p)$ in Eq.~\eqref{theorem:trans_inv_Lbd}:
\begin{gather}\label{_optimal_f_k}
\tilde f_{k}(p){=}c_k e^{p \beta \hbar \lambda_{k}},~~\lambda_k{=}\kappa_{k}\tanh(\tfrac{\hbar\omega}{4\theta}),
\end{gather}
where $\beta{=}(m\hbar\omega)^{-1}$ and the constants $c_k$ should be chosen to satisfy Eq.~\eqref{_Bondarian_gen}. The corresponding dissipator \eqref{theorem:trans_inv_Lbd} reproduces the familiar microphysical model of quantum Brownian motion (see e.g. Eq.~(16) in Ref.~\cite{2002-Vacchini}) in the limit $\kkappa{\to}0$, $\omega{\to}0$. Furthermore, the resulting dynamics tends to decrease (increase) the average system energy $\midop{\hat H}_{\theta}$ if its initial temperature $\theta'$ is higher (lower) than $\theta$:
\begin{gather}\label{_stability_check}
\tder{}{t}{\midop{\hat H}_{\theta'}}\big|_{t{=}0}{=}\tfrac{c_k^2}{\omega}\tilde\gamma^{\idx{en}}_k(\theta',\theta)(\midop{\hat H}_{\theta}{-}\midop{\hat H}_{\theta'})\big|_{t{=}0},
\end{gather}
where $\tilde\gamma^{\idx{en}}_k(\theta',\theta){=}2\omega{\beta\hbar^2\kappa_k \lambda_k \exp\left({{\beta\hbar^2\lambda_k^2}{\coth(\frac{\hbar\omega}{2 \theta' })}}\right)}{>}0$.
Equation~\eqref{_stability_check} suggests that $\rhost$ is close to $\rhoth{\theta}$. This conclusion is supported by the simulations presented in Fig.~\ref{@FIG.01'}a for the isotropic dissipator $\Lrel{=}\Bdn_{\kappa,\fciso}$,
\begin{gather}\label{theorem:trans_inv_Lbd-isotropic}
\Bdn_{\kappa,\fciso}{\defeq}\Lbd_{\hat A^+}{+}\Lbd_{\hat A^-},~~\hat A^{\pm}{=}e^{{{\mp}}i\kappa\hat x}\fciso({\pm}\hat p).
\end{gather}
One can see that the high-quality thermalization is readily achieved by tuning the free parameters $c_k$ and $\kappa_k$ even in the strong dissipation regime.
\begin{figure}[tbp]
\centering\includegraphics[width=\columnwidth]
{{fig.01-}.eps}
\caption{(a) The accuracy of thermalization of the harmonic oscillator at $\theta{=}0$ by the dissipator $\Lrel{=}\Gamma\Bdn_{\kappa,\fciso}$ as function of $\kappa$ and $\Gamma$. The solid curves show the Bures distance $D_{\idx{B}}$ between the thermal state $\rhoth{\theta}$ and its approximation $\rhost$ for the case $\fciso(p)$ defined by Eq.~\eqref{_optimal_f_k} with $c{=}\omega/\sqrt{\tilde\gamma^{\idx{en}}(0,0)}$. The dotted curves represent the clipped versions \eqref{__clipping_rule} of $\fciso(p)$.
The dashed curves correspond to the case of functions $\fciso(p)$ approximated by Eq.~\eqref{_coherent_Doppler_f(p)} with parameters $\tilde c_{i}$ chosen such that $\left.\tder{^l}{p^l}(\fciso(p){-}\fcisoD(p))\right|_{p=0}{=}0$ for $l{=}0,1,2$.
(b) The Doppler cooling setup to test the model \eqref{Quantum_Liouville_equation_nD}, \eqref{theorem:trans_inv_Lbd} in the laboratory.\label{@FIG.01'}}
\end{figure}
To understand the result \eqref{_optimal_f_k}, note that the terms $\Lbd_{\hat A_k}$ in Eq.~\eqref{theorem:trans_inv_Lbd} represent independent statistical forces $\midop{{-}\hbar\kkappa_{k}|\tilde f_k(\hat\pp)|^2}$ contributing to the net friction $\midop{\ehat\ffrnforce}$.
In classical mechanics, such forces at $\theta{=}0$ steer the system to the state of rest by acting against the particles' momenta, hence
\begin{gather}\label{__clipping_rule}
\tilde f_k(\hat\pp){=}0\mbox{ when }\pp\kkappa_k{<}0~~~\mbox{(classical mechanics)}.
\end{gather}
However, clipping the functions \eqref{_optimal_f_k} according to Eq.~\eqref{__clipping_rule} introduces significant errors, as displayed by dotted curves in Fig.~\ref{@FIG.01'}a. Thus, the ``endothermic'' tails of $\tilde f_k(\hat\pp)$ at $\pp\kkappa_k{>}0$ break the thermalization in the classical case, but reduce errors in the quantum mechanical treatment. To clarify this counterintuitive observation, note that the
physical requirement $\der{}{t}\midop{\hat O}_{\theta}{=}0$ for any observable $\hat O$ in the thermodynamic equilibrium $\rhost{=}\rhoth{\theta}$ is violated by the master equations \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} due to the no-go theorems, i.e.,
\begin{gather}\label{_2-nd_moments_x^2}
\tder{}{t}{\midop{\hat x_n^2}_{\theta}}\bigg|_{t{=}0}{=}\hbar^2\sum_k\midop{\bigl|\tpder{}{\hat p_n}{
\fc_k(\hat\pp)
}\bigr|^2}_{\theta}\bigg|_{t{=}0}{>}0
\end{gather}
in the driftless case $\Laux{=}0$. The inequality~\eqref{_2-nd_moments_x^2} provides further evidence for the no-go theorems
and is the hallmark of the ``position diffusion'', a known artifact in the quantum theory of Brownian motion \cite{2009-Vacchini}.
According to Eq.~\eqref{_2-nd_moments_x^2}, $\tder{}{t}\midop{\hat x^2}_{\theta}\big|_{t{=}0}$ is sensitive to smoothness of $\fc_k(\pp)$. Specifically, the rhs of Eq.~\eqref{_2-nd_moments_x^2} is exploded by any highly oscillatory components of $\fc_k(\pp)$ and diverges if $\fc_k(\pp)$ is discontinuous. This entirely quantum effect is the origin of poor performance of the clipped solutions \eqref{__clipping_rule} seen in Fig.~\ref{@FIG.01'}a. Equation~\eqref{_2-nd_moments_x^2} uncovers unavoidable errors in the potential energy. The optimal solutions \eqref{_optimal_f_k} enforce error cancellation $\tder{}{t}\midop{\tfrac{\hat p^2}{2m}}_{\theta}\big|_{t{=}0}{=}{-}\tder{}{t}\midop{U(\hat x)}_{\theta}\big|_{t{=}0}$ between kinetic and potential energies leaving the total energy intact $\tder{}{t}\midop{\hat H}_{\theta}\big|_{t{=}0}{=}0$. In fact, the error cancellation is achieved with a large class of physically feasible functions $\fc_k(\pp)$ that may substantially differ from the solutions \eqref{_optimal_f_k} everywhere but the region of high probability density $\varpi(p){=}\Tr[\delta{(\hat p{-}p)\rhoth{\theta}}]$ (however, note the remark in \appref{@APP:phys_meaning}). This is illustrated in Fig.~\ref{@FIG.01'}a by dashed curves overlapping with solid curves.
The master equations \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} provide accurate non-perturbative description of collisions with a background gas of atoms or photons \cite{1996-Poyatos,2001-Vacchini-a,2005-Vacchini,2008-Hornberger,
2015-Barchielli
. Hence, the above theoretical conclusions can be directly tested in the laboratory using well-developed techniques, e.g., the setup shown in Fig.~\ref{@FIG.01'}b. Here a two-level atom is subject to two orthogonally polarized counterpropagating monochromatic nonsaturating laser fields of the same amplitude $\cal E$ and frequency $\omlas$. We show in \appref{@APP:phys_meaning}~that the translational motion of the atom can be modeled using Eq.~\eqref{Quantum_Liouville_equation_nD} with an isotropic friction term of form $\Lrel{=}\Bdn_{\kappa,\fcisoD}$. Here
\begin{gather}\label{_coherent_Doppler_f(p)}
\kappa{=}\tfrac{\omlas}c,~~\fcisoD(p){=}\tilde c_{1}({\tilde c_{2}^2{+}(p{-}\tilde c_{3})^2})^{{-}\frac12},~~\tilde c_{k}{\in}\mathbb{R}
\end{gather}
and the parameters $\tilde c_{k}$ can be tuned by $\cal E$ and $\omlas$.
Now we are ready to clarify why the deviations from canonical equilibrium increase with $|\kappa|$ in Fig.~\ref{@FIG.01'}a. The parameters $\hbar|\kappa|$ and $\fcisoD(p)^2$ in Eq.~\eqref{_coherent_Doppler_f(p)} can be regarded as the change of atomic momentum after absorption of a photon and the absorption rate. The case of small $\hbar|\kappa|{\ll}\sqrt{\midop{\hat p^2}}$ and large $\fcisoD(p)^2$ implies tiny and frequent momentum exchanges subject to the central limit theorem. The net result is a velocity-dependent radiation pressure with vanishing fluctuations. The opposite case of large $\hbar|\kappa|{\gg}\sqrt{\midop{\hat p^2}}$ and small $\fcisoD(p)^2$ is the strong shot noise limit, where the stochastic character of light absorption is no longer averaged out, notably perturbing the thermal state. Note that a similar interpretation applies to quantum statistical forces in Ref.~\cite{2016-Vuglar}.
The dissipative model \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd} with optimized parameters \eqref{_optimal_f_k} is further analyzed in Fig.~\ref{@FIG.01b'} using the same parameters as in Fig.~\ref{@FIG.01'}a. Both Figs.~\ref{@FIG.01b'} and \ref{@FIG.01'}a indicate that thermalization can be modeled for a wide range of recoil momenta $\hbar\kappa\in\left({-}(\hbar\sqrt{\beta})^{-1},(\hbar\sqrt{\beta})^{-1}\right)$ and the higher the temperature, the better the accuracy. Thus, Eqs.~\eqref{_Bondarian_gen} and \eqref{_Diffusion_operator} enable to simulate a variety of velocity dependences of friction and diffusion.
Finally, Fig.~\ref{@FIG.01b'}a benchmarks such simulations against the commonly used quantum optical master equation (QOME) \cite{BOOK-Gardiner} defined by Eq.~\eqref{_Lindbladian_definition} with $K{=}2$, $\hat L_1{=}\sqrt{2\Gamma\omega}({1{-}e^{{-}\frac{\hbar\omega}{\theta}}})^{{-}\frac12}\hat a$, $\hat L_2{=}\sqrt{2\Gamma\omega}({e^{\frac{\hbar\omega}{\theta}}}{-}1)^{{-}\frac12}\hat a^{\dagger}$, where $\hat a$ is the harmonic oscillator annihilation operator. For a correct comparison, the parameters of both models are adjusted to ensure identical decay rates in Eq.~\eqref{_stability_check}. Systematic errors in our model and QOME are comparable for the equilibrium displacements $\Delta x_0{\sim}\hbar\beta^{{-}\frac12}$ at zero temperature and $\Delta x_0{\sim}0.1\hbar\beta^{{-}\frac12}$ for $\theta{\sim}\hbar\omega$. For low-frequency molecular vibrational modes ($m{\sim}10^4$\,atomic units, $\omega{\sim}200$\,cm$^{-1}$), these shifts are of order $0.4$\,\AA~and $0.04$\,\AA, respectively, which are in the range of typical molecular geometry changes due to optical excitations or liquid environments. We found the displacement-independent errors in the model \eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd}
to be very important for quantum control via reservoir engineering.
Furthermore, the same feature can also be exploited for engineering the mechanical analogs of nonreciprocal optical couplings \cite{2015-Metelmann} and energy-efficient molecular quantum heat machines \cite{2016-Zhdanov}. These subjects will be explored in a forthcoming publication.
\let\section\oldsection
\begin{acknowledgments}
We thank to Alexander Eisfeld for valuable discussions and drawing our attention to Refs.~\cite{1995-Holevo,1996-Holevo,2005-Petruccione,2005-Vacchini,2009-Vacchini}. We are grateful to the anonymous referees for important suggestions and insightful critic. T.~S{.} and D.~V.~Zh{.} thank the National Science Foundation (Award number CHEM-1012207 to T. S.) for support.
D.~I.~B{.} is supported by AFOSR Young Investigator Research Program (FA9550-16-1-0254).
\end{acknowledgments}
\SUPPLEMENTALPLACEMACRO
\section*{Introduction}
This supplemental material is organized as follows. In the Sections~\ref{@APP:theo:trans_inv_Lbd}, \ref{@APP:theo:no_go(T=0)} and \ref{@APP:theo:no_go(T>0)} we give the proofs of Lemma~\ref{@theo:trans_inv_Lbd}, first, and second no-go theorems, respectively. Finally, the supporting mathematical derivations for the Doppler cooling model briefly discussed in the main text are provided in Section~\ref{@APP:phys_meaning}.
The Roman numbers in parentheses refer everywhere to the equations in the main text of the letter.
\section{The proof of lemma~\ref{@theo:trans_inv_Lbd}\label{@APP:theo:trans_inv_Lbd}}
\newcommand{\SpSh}{{\cal R}
The property of the translational invariance can be formulated as
\begin{gather}\label{Quantum_Liouville_trans_inv_nD(reformulated)}
\forall \dxx: {\cal R}_{\dxx}\Lrel{\cal R}_{-\dxx}{=}\Lrel,
\end{gather}
where
\begin{gather} \label{__spatial_shift_superoperator}
\SpSh_{\dxx}{=}e^{{-}\frac{i}{\hbar}\dxx\hat\pp}\odot e^{\frac{i}{\hbar}\dxx\hat\pp}
\end{gather}
is the superoperator of translational shift: $\forall g(\hat\xx): \SpSh^{\intercal}_{\dxx}[g(\hat\xx)]{=}g(\hat\xx{+}\dxx)$.
With the help of the canonical commutation relations, any operator $\hat L_k{=}L_k(\hat\pp,\hat\xx)$ can expanded in the series
$\hat L_k{=}\sum_{l,m}c_{k,l,m}\hat B_{l,m}$, where $\hat B_{l,m}{=}e^{{-}i\set{\kappa}_l\hat\xx}g_m(\hat\pp)$ and the functions $g_m(\pp)$ constitute a set of (not necessarily orthogonal) basis functions. Using this expansion, any superoperator of form $\Lrel{=}\sum_k\Lbd_{\hat L_k}$ can be rewritten as
\begin{gather}
\Lrel{=}\sum_{k,l_1,m_1,l_2,m_2}c_{k,l_1,m_1}c_{k,l_2,m_2}^*\tilde{\cal L}^{\idx{lbd}}_{\hat B_{l_1,m_1},\hat B_{l_2,m_2}},
\end{gather}
where
\begin{gather}
\tilde{\cal L}^{\idx{lbd}}_{\hat A_1,\hat A_2}{\defeq}\hat A_1\odot\hat A_2^{\dagger}-\frac12(\hat A_1\hat A_2^{\dagger}\odot{+}\odot\hat A_1\hat A_2^{\dagger}).
\end{gather}
It follows from Eq.~\eqref{Quantum_Liouville_trans_inv_nD(reformulated)} that if $\Lrel$ is translationally invariant then it should satisfy the identity
\begin{gather}
\Lrel{=}\left.\frac{1}{(2L)^N}\int_{-L}^{L}...\int_{-L}^{L}\,{\cal R}_{\dxx}\Lrel{\cal R}_{-\dxx}\mathrm{d}^N\diff\dxx\right|_{L{\to}\infty}{=}\notag\\
\sum_{l}\sum_{m_1,m_2}\tilde c^{(l)}_{m_1,m_2}\tilde{\cal L}^{\idx{lbd}}_{\hat B_{l,m_1},\hat B_{l,m_2}},\label{Quantum_Liouville_trans_inv_nD_a}
\end{gather}
where the Hermitian matrices $\tilde c^{(l)}$ are defined as
\begin{gather}
\tilde c^{(l)}_{m_1,m_2}{=}\sum_{k}c_{k,l,m_1}c_{k,l,m_2}^*.
\end{gather}
Let us substitute in Eq.~\eqref{Quantum_Liouville_trans_inv_nD_a} the matrices $\tilde c^{(l)}$ with their Jordan decomposition $\tilde c^{(l)}{=}\tilde u^{(l)}\tilde\gamma^{(l)}{\tilde u^{(l)}}^{\dagger}$, where $\tilde u^{(l)}$ is unitary and $\tilde\gamma^{(l)}$ is diagonal. The result is
\begin{gather}\label{Quantum_Liouville_trans_inv_nD_b}
\Lrel{=}\sum_{l,m}{\cal L}^{\idx{lbd}}_{\hat A_{l,m}},
\end{gather}
where $A_{l,m}{=}\tilde f_{l,m}(\hat\pp)e^{{-}i\set{\kappa}_l\hat\xx}$ and $\tilde f_{l,m}(\hat\pp){=}\sqrt{\tilde\gamma^{(l)}_{m,m}}\times\sum_{m'}\tilde u^{(l)}_{m',m}g_{m'}(\hat\pp)$. Finally, note that Eq.~\eqref{Quantum_Liouville_trans_inv_nD_b} can be cast into the form \eqref{theorem:trans_inv_Lbd} by replacing the compound index $\{l,m\}$ with the single consecutive index $k$. The lemma is proven.
\begin{remark}[Remark 1.]
In this work, the Gaussian (continuous) translationally invariant dissipators of form
\begin{gather}\label{_Lbd_Gaussian}
{\LvnG}{=}\sum_k\Lbd_{\ehat\AG_k},~~\ehat\AG_k{=}\kkappaG_k\hat\xx{+}\ffG_k(\hat\pp)~~(\kkappaG_k{\in}\mathbb{R}^N)
\end{gather}
are treated as the limiting case of Eq.~\eqref{theorem:trans_inv_Lbd} with $\kkappa_k{=}\epsilon\kkappaG_k{\to}0$. Specifically, one can verify by direct calculation that
\begin{gather}\label{_Lbd_Gaussian[lemma-form]}
\Lbd_{\ehat\AG_k}{=}\left.\Lbd_{\ehat\AG_{k,+}}{+}\Lbd_{\ehat\AG_{k,-}}\right|_{\epsilon{\to}0}{-}i\frac{\hbar}2\kkappaG_k\left[\pder{\ffG_k(\hat\pp)}{\hat\pp},\odot\right],\\
\ehat\AG_{k,\pm}{=}\tfrac1{\sqrt 2}\left(\tfrac{i}{\epsilon}{\pm}\ffG_k(\hat\pp)\right)e^{{\mp}i\epsilon\kkappaG_k\hat\xx}.
\end{gather}
\end{remark}
\begin{remark}[Remark 2.]
The translation invariance criterion is generalized to non-Markovian dynamics in Ref.~\cite{2017-Gasbarri}.
\end{remark}
\section{The proof of no-go theorem~\ref{@theo:no_go(T=0)} (by contradiction)\label{@APP:theo:no_go(T=0)}}
Suppose that some eigenstate $\ket{\Psi_{0}}$ of Hamiltonian $\hat H$ is also the fixed point of the quantum Liouvillian $\cal L$ defined by Eqs.~\eqref{Quantum_Liouville_equation_nD} and \eqref{theorem:trans_inv_Lbd}. Since $\Lrel$ is assumed translation invariant, it should commute with spatial shift superoperator \eqref{__spatial_shift_superoperator} for any $\dxx$. Hence, $\Lrel[\SpSh_{\dxx}[\hat\rho_0]]=\SpSh_{\dxx}[\Lrel[\hat\rho_0]]{=}0$, where $\hat\rho_0{=}\proj{\Psi_{0}}$. Furthermore, the linearity of $\Lrel$ implies that
\begin{gather}\label{__pre_w_g}
\forall g(\xx'): \Lrel[\int g(\xx')\SpSh_{\dxx}[\hat\rho_0]d^N \xx']{=}0.
\end{gather}
Equation \eqref{__pre_w_g} can be further simplified using the identity
\begin{gather}\label{__state_shift_identity}
e^{{-}\frac{i}{\hbar}\xx'\hat\pp}\ket{\Psi_{0}}{=}\sqrt{2\pi\hbar}\Psi_{0}(\hat\pp)\ket{\xx'},
\end{gather}
where $\ket{\xx'}$ is the eigenstate of position operator: $\hat x_k\ket{\xx'}{=}x_k'\ket{\xx'}$, $\scpr{\xx''}{\xx'}{=}\delta(\xx''{-}\xx')$.
The validity of Eq.~\eqref{__state_shift_identity} can be verified by comparing the wavefunctions in momentum representation corresponding to its left and right sides. Identities \eqref{__state_shift_identity} and $\int g(\xx')\proj{\xx'}d^N\xx'{=}g(\hat\xx)$ allow to equivalently rewrite Eq.~\eqref{__pre_w_g} as
\begin{gather}\label{_w_g}
\forall g(\xx'): \Lrel[\hat w_g]{=}0,~~~\hat w_g{=}\Psi_{0}(\hat\pp)g(\hat\xx)\Psi_{0}(\hat\pp)^{\dagger}.
\end{gather}
Consider the case $g(\xx){=}g_{\llambda}(\xx){=}e^{-i\llambda\xx}$, where $\llambda$ is some real $N$-dimensional vector. Note that the operator $\Lrel[\hat w_g]$ then includes the explicit dependence on coordinate operators $\hat\xx$ only in forms of matrix exponentials $e^{-i\llambda\hat\xx}$, $e^{\pm i\kkappa_k\hat\xx}$ and commutators $[\hat x_k,\odot]$. Using the commutation relation $e^{-i\tilde\llambda\hat\xx}\hat\pp{=}(\hat\pp{+}\hbar\tilde\lambda)e^{-i\tilde\llambda\hat\xx}$ with $\tilde\llambda{=}\llambda,\pm\kkappa_k$, it is possible to group out the momentum and coordinate operators in $\Lrel[\hat w_g]$ and rewrite the condition \eqref{_w_g} as:
\begin{gather}
0{=}\Lrel[\hat w_{g_{\llambda}}]{=
\tilde G_{\llambda}(\hat\pp)e^{-i \llambda\hat{\xx}}\label{_Lrel(w_g)},
\end{gather}
where
\begin{gather}\label{__G_lambda}
\tilde G_{\llambda}(\pp){=}G(\pp,\pp{+}\hbar\llambda)\Psi_0 (\pp)\Psi_0(\pp{+}\hbar\llambda)^{*}
\end{gather}
and
\begin{gather}
G(\pp,\pp'){=}\sum_k\left(
F_k(\pp)F_k(\pp')^{*}{-}\tfrac{|\tilde f_k(\pp)|^2{+}|\tilde f_k(\pp')|^2}2
\right){+}\\\notag
\hbar\kkappaaux(\tpder{\ln(\Psi_0 (\pp))}{\pp}{+}\tpder{\ln(\Psi_0 (\pp')^{*})}{\pp'}){-}i(\faux(\pp){-}\faux(\pp')),
\\
F_k(\pp){=}\tilde f_k (\pp{+}\hbar\kkappa_k) \frac{\Psi_0(\pp{+}\hbar\kkappa_k)}{\Psi_0(\pp) }
\end{gather}
Note that the cancellation of all the matrix exponentials $e^{\pm i\kkappa_k\hat\xx}$ at the rhs of Eq~\eqref{_Lrel(w_g)} is the consequence of the translation invariance of $\Lrel$. Condition \eqref{_Lrel(w_g)} implies that
\begin{gather}\label{condition_in_Psi_0}
\forall \pp{\in}\mathbb{R}^N,\forall\lambda\in\mathbb{R}: \tilde G_{\lambda}(\pp){=}0,
\end{gather}
and $\forall \pp,\pp'{\in}\mathbb{R}^N: G(\pp,\pp'){=}0$ (except a possible zero measure subset of points $\{\pp,\pp'\}$ where $\Psi_0(\hat\pp)\Psi_0(\hat\pp')^{\dagger}{=}0$). In particular, this means that
\begin{gather}\label{dG/dp_1dp_2}
\begin{split}
\forall n,\forall \pp,\pp'{\in}\mathbb{R}^N:& \pder{^2}{p_{n}\partial p'_{n}} G(\pp,\pp'){=}\\
&\sum_k
\tpder{F_k^{(n)}(\pp)}{p_n}\left(\tpder{F_k^{(n)}(\pp')}{p_n'}\right)^{*}{=}0.
\end{split}
\end{gather}
Equality \eqref{dG/dp_1dp_2} can be satisfied only if $\forall k: F_k(\pp){\propto}$const, i.e., if $\tilde f_k (\pp){=}c_k\frac{\Psi_0(\pp{-}\hbar\kkappa_k)}{\Psi_0(\pp)}$, where $c_k$ is some real constant. Substitution of this expression and $\llambda{=}\zzero$ into Eq.~\eqref{__G_lambda} gives $\tilde G_{\zzero}(\pp){=}\sum_kc_k^2\left({|{\Psi_0(\pp)}|^2{-}|\Psi_0(\pp{-}\hbar\kkappa_k)}|^2\right){+}\hbar\kkappaaux\tpder{|{\Psi_0}(\pp)|^2}{\pp}$
\footnote{Using \eqref{_Lbd_Gaussian[lemma-form]}, it is straightforward to deduce that the corresponding summand for Gaussian dissipator \eqref{_Lbd_Gaussian} takes form $\frac12\hbar^2\kkappaG\pder{}{\pp}(\kkappaG\pder{}{\pp}|{\Psi_0(\pp)}|^2)$. The resulting contribution in the lhs of Eq.~\eqref{condition_in_Psi_0_integral} is $|\kkappaG|^2\hbar^2$.
}.
Multiplication of the both sides of Eq.~\eqref{condition_in_Psi_0} by $\pp^2$ and subsequent integration over $\pp$ gives:
\begin{align}\label{condition_in_Psi_0_integral}
\int&\pp^2\tilde G_{\zzero}(\pp)\diff^N\pp{=
\notag\\&\sum_kc_k^2\hbar^2\kkappa_k^2{-}
2\hbar(\kkappaaux{+}\sum_kc_k^2\kkappa_k)\matel{\Psi_0}{\hat\pp}{\Psi_0}{=}0.
\end{align}
According to our assumption, $\matel{\Psi_0}{\hat\pp}{\Psi_0}{=}\zzero$\footnote{The equality $\matel{\Psi_0}{\hat\pp}{\Psi_0}{=}\zzero$ holds for any non-degenerate eigenstate of the time-reversal invariant Hamiltonian \eqref{Quantum_Hamiltonian_nD}}.
Hence, Eq.~\eqref{condition_in_Psi_0_integral} implies that $\sum_kc_k^2|\kkappa_k|^2{=}0$. This equality holds only if $\forall k:\kkappa_k{=}\zzero$. However, in this case all functions $\tilde f_k (\pp){=}c_k$ reduce to constants, so that $\Lrel{=}0$.
\newcommand{\boldsymbol{\chi}}{\boldsymbol{\chi}}
This result completes the proof.
\section{The proof of no-go theorem~\ref{@theo:no_go(T>0)} (by contradiction)\label{@APP:theo:no_go(T>0)}}
Denote as $\Psi_{k}(\pp)$ and $E_k$ $(k{=}0,...,\infty)$ the momentum-space wavefunction and energy of the $k$-th eigenstate $\ket{\Psi_{k}}$ of the Hamiltonian $\hat H$. The thermal state $\rhoth{\theta}$ can be expressed in these notations as
\begin{gather}
\rhoth{\theta}{=}\rnorm\sum_ke^{{-}\frac{E_k}{\theta}}\proj{\Psi_{k}}
\end{gather}
Suppose that there exists such relaxation superoperator of form \eqref{theorem:trans_inv_Lbd} that $\Lrel[\rhoth{\theta}]{=}0$.
Owing to assumed linearity and translational invariance of $\Lrel$, the thermal state $\rhoth{\theta}$ should satisfy the relation similar to \eqref{__pre_w_g}:
\begin{gather}\label{__pre_w_(theta,g)}
\forall g(\xx'): \Lrel[\int g(\xx')\SpSh_{\dxx}[\rhoth{\theta}]d^N \xx']{=}0,
\end{gather}
where $\SpSh_{\dxx}$ is the spatial shift superoperator defined by Eq.~\eqref{__spatial_shift_superoperator}. With the help of relation \eqref{__state_shift_identity}, one can apply to Eq.~\eqref{__pre_w_(theta,g)} the same procedure as was used to derive the equality \eqref{_w_g} from Eq.~\eqref{__pre_w_g}. The result is
\begin{gather}\label{_Lrel(_w_{theta,g})=0}
\forall g(x): \Lrel[\hat w_{\theta ,g}]{=}0,
\end{gather}
where
\begin{gather}\label{_w_{theta,g}}
\hat w_{\theta,g}{=}\rnorm\sum_ke^{{-}\frac{E_k}{\theta}}\Psi_{k}(\hat\pp)g(\hat\xx)\Psi_{k}(\hat\pp)^{\dagger}.
\end{gather}
Consider the case $g(\xx){=}g_{\llambda}(\xx){=}e^{{-}i\llambda\xx}$, where $\llambda$ is some real $N$-dimensional vector. The result of application of $\Lrel$ to $\hat w_{\theta,g_{\llambda}}$ can be represented after some algebra as
\begin{gather}\label{__Lrel[w]}
\Lrel[\hat w_{\theta,g_{\llambda}}]{=}G_1\left(\hat\pp{+}\tfrac{\hbar\llambda}{2},\llambda\right)e^{{-}i\llambda\hat\xx},
\end{gather}
where
\begin{align}\label{__G_1(p,lambda)}
G_1(\pp,&\llambda)={-}i\BFn_{\theta}(\pp,\llambda)\left(\faux(\pp{-}\tfrac{\hbar\lambda}2){-}\faux(\pp{+}\tfrac{\hbar\lambda}2)\right){+}\notag\\
&
\hbar\kkappaaux\tpder{\BFn_{\theta}(\pp,\llambda)}{\pp}{+}\sum_k \biggl(Q_{k,n}(\pp{+}\hbar\kkappa_k,\llambda){-}\notag\\
&\tfrac12\BFn_{\theta}(\pp,\llambda)\biggl({\left|\tilde{f}_k\left(\pp{+}\tfrac{\hbar \llambda }{2}\right)\right|^2{+}\left|\tilde{f}_k\left(\pp{-}\tfrac{\hbar \llambda}{2}\right)\right|^2}
\biggr)\biggr)
,
\\
Q_k(\pp,&\llambda )=\BFn_{\theta}(\pp,\llambda)\tilde{f}_k(\pp{-}\tfrac{\hbar \llambda }{2}) \tilde{f}_k^*(\pp{+}\tfrac{\hbar \llambda }{2}).
\end{align}
In derivation of \eqref{__G_1(p,lambda)} the identity
\begin{gather}
\BFn(\pp,\llambda ){=}
\rnorm\sum_k e^{{-}\frac{E_k}{\theta}}\Psi_{k}(\pp{-}\tfrac{\hbar\llambda}{2})\Psi_{k}^{*}(\pp{+}\tfrac{\hbar\llambda}{2})
\end{gather}
was used which follows directly from the definition \eqref{_Blokhintsev_function} of the Blokhintsev function.
Eqs.~\eqref{_Lrel(_w_{theta,g})=0} and \eqref{__Lrel[w]} require that
\begin{gather}\label{__G_1(p,lambda){=}0}
\forall\pp,\llambda:G_1(\pp,\llambda){=}0,
\end{gather}
and hence $\forall \llambda: \bar G_2(\llambda){=}\int_{{-}\infty}^{\infty}\ldots\int_{{-}\infty}^{\infty}\diff^N \pp\,G_2(\pp,\llambda){=}0$,
where
\begin{gather}\label{__G_2(p,lambda)}
\begin{split}
G_2(&\pp,\llambda){=}G_1(\pp,\llambda){+}G_1(\pp,{-}\llambda){=}
\\
&\sum_k\biggl\{
{-}\left|\tilde{f}_k\left(\pp{+}\tfrac{\hbar\llambda }{2}\right){-}\tilde{f}_k\left(\pp{-}\tfrac{\hbar \llambda}{2}\right)\right|^2\BFn_{\theta}(\pp,\llambda){+}
\\
&\sum_{\alpha,\beta{=}\pm1}\beta Q_k\left(\pp{+}\tfrac{\beta{+}1}2\hbar\kkappa_k,\alpha\llambda\right)\biggr\}
{+}2\hbar\kkappaaux\tpder{\BFn_{\theta}(\pp,\llambda)}{\pp}
.
\end{split}
\end{gather}
The last equality in \eqref{__G_2(p,lambda)} is obtained assuming that $\BFn_{\theta}(\pp,{-}\llambda){=}\BFn_{\theta}(\pp,\llambda)$ (see Eq.~\eqref{_B(p,lambda)-features-a}).
It is easy to check that the integrations over all terms in the last line of \eqref{__G_2(p,lambda)} cancel out, so that
\begin{align}
\bar G_2(\llambda){=}&{-}\int_{{-}\infty}^{\infty}\ldots\int_{{-}\infty}^{\infty}\diff^N \pp{\times}\notag\\
&\sum_k\left|\tilde{f}_k\left(\pp{+}\tfrac{\hbar\llambda }{2}\right){-}\tilde{f}_k\left(\pp{-}\tfrac{\hbar\llambda}{2}\right)\right|^2\BFn(\pp,\llambda).\label{_bar_G_2(p,lambda)}
\end{align}
According to the assumption \eqref{_B(p,lambda)-features-a}, the integrand in \eqref{_bar_G_2(p,lambda)} is nonnegative. Moreover, $\bar G_2(\llambda){=}0$ iif $\forall k: \tilde f_k(\pp){=}c_k{=}$const. Hence, the expression \eqref{__G_1(p,lambda)} for $G_1(\pp,\llambda)$ can be simplified as
\begin{gather}\label{__G_1_simplified(p,lambda)}
G_1(\pp,\llambda )=\sum _k c_k^2 \left(\BFn\left(\pp{+}\hbar\kkappa_k,\llambda \right){-}\BFn(\pp,\llambda )\right).
\end{gather}
Note that the terms $\Lbd_{\hat A_k}$ in Eq.~\eqref{theorem:trans_inv_Lbd} with $\tilde f_k(\pp){=}$const will have non-trivial effect only if $\kkappa_k{\ne}0$%
\footnote
{In the case of Gaussian dissipator \eqref{_Lbd_Gaussian} Eq.~\eqref{__G_1_simplified(p,lambda)} reduces to
\begin{gather}\label{__G_1_simplified(p,lambda)[Gaussian]}
G_1(\pp,\llambda)=\frac12\hbar^2\sum_{k,m,n}\mu_{k,n}\mu_{k,m}\pder{^2}{p_np_m}\BFn(\pp,\llambda ).\tag{\ref*{__G_1_simplified(p,lambda)}*}
\end{gather}
By assumption \eqref{_B(p,lambda)-features-b}, the quadratic form $\pder{^2}{p_np_m}\BFn(\pp,\llambda )$ in \eqref{__G_1_simplified(p,lambda)[Gaussian]} is negative-definite at $\{\pp,\llambda \}{=}\{\zzero,\zzero\}$. Hence, $G_1(\zzero,\zzero){<}0$, which contradicts Eq.~\eqref{__G_1(p,lambda){=}0} and completes the proof for this case.
}%
. However, it follows from \eqref{_B(p,lambda)-features-b} that in this case $G_1(\zzero,\zzero){<}0$ which contradicts Eq.~\eqref{__G_1(p,lambda){=}0}. The theorem is proven.
\section{Testing the model \texorpdfstring{(\ref{Quantum_Liouville_equation_nD}) and (\ref{theorem:trans_inv_Lbd})}{} in the laboratory\label{@APP:phys_meaning}}
In this section, we provide the detailed analysis of the Doppler cooling example introduced in the main text (see Fig.~\ref{@FIG.01'}b in the main text) and prove that the cooling mechanism is the quantum friction of form \eqref{theorem:trans_inv_Lbd-isotropic}.
In the proposed setup an atom is subject to two orthogonally polarized counterpropagating beams of the same field amplitude $\cal E$ and carrier frequency $\omlas$ (hereafter in
this section we will omit the subscript l for shortness since it will not cause any ambiguity). We assume that $\omega$ is close to the frequency $\omega_{\idx{a}}$ of the transition $\es{g}{\LR}\es{e}$ between the ground $\es{g}$ and degenerate excited $\es{e}$ electron states of $s$- and $p$-symmetries, respectively. Let $d$ be the absolute value of the transition dipole moment and $\gamma$ be the excited state spontaneous decay rate.
For the spatial arrangement depicted in Fig.~\ref{@FIG.01'}b the translation motion of the atom along $x$-axis is coupled to the field-induced electron dynamics since each absorbed
or coherently emitted photon changes the $x$-component of atomic momentum hereafter denoted as $p$. Furthermore, we will assume that the spontaneous decay does not affect the $x$-component of atomic momentum. The latter condition can be achieved using, e.g., an arrangement shown in Fig.~\ref{@FIG.A01}.
\begin{figure}[tbp]
\centering\includegraphics[width=0.7\columnwidth]
{{fig.A01}.eps}
\caption{The possible Doppler cooling setup where stochastic recoil accompanying the spontaneous emission is damped along the $x$-axis. Here the atom of interest $A$ is put into intersected orthogonal optical cavities formed by pairs of mirrors $M_1$, $M_1'$ and $M_2$, $M_2'$. The cavities are tuned resonant to the atomic $\es{g}\LR\es{e}$ transition and force atom to spontaneously emit absorbed photons predominantly in the directions perpendicular to the $x$-axis via the Purcell effect. The decay rate $\gamma$ can be controlled by changing the cavities Q-factors. The collateral increase of the energy of motions along $y$- and $z$-axes is restricted by sympathetic cooling by two auxiliary atoms $B$ and $C$.\label{@FIG.A01}}
\end{figure}
The master equation which describes this coupled dynamics can be written within the rotating wave approximation in the form \eqref{Quantum_Liouville_equation_nD} with
\begin{gather}
\begin{split}
\hat H{=}&\frac{\hat p^2}{2m}{-}\hbar\omega_{\idx{a}}\proj{\es{g}}{+}
\biggl\{\xi_1(t)\proj[\es{e}_1]{\es{g}}e^{-i(\omega t{-}\kappa \hat x)}{+}\\&
\xi_2(t)\proj[\es{e}_2]{\es{g}}e^{-i(\omega t{+}\kappa \hat x)}{+}\mbox{h.c.}
\biggr\}
\end{split}
\end{gather}
and
\begin{gather}\label{doppler_L_rel}
\Lrel{=}{\gamma}\sum_{n{=}1}^2\Lbd_{\proj[\es{g}]{\es{e}_n}}.
\end{gather}
Here $\xi_k(t){=}{-}\frac12\vec d_k\vec{\cal E}_k(t)$, where $\vec d_1$ and $\vec d_2$ are the transition dipole moments associated with the $s{\to}p_z$ and $s{\to}p_y$ electronic transitions into degenerate electronically excited sublevels $\es{e}_1$ and $\es{e}_2$, respectively, and $\vec{\cal E}_k(t)$ is the slowly varying complex amplitude of the associated field component. The remaining notations are defined in the main text.
\newcommand{\evsop}[2]{\mathop{{{\cal U}^{#1}_{#2}}}}
\newcommand{\evsopt}[2]{\mathop{{{\cal U}^{#1}_{#2}}^{\intercal}}}
\newcommand{\stackrel{\Rightarrow}{{\cal T}}}{\stackrel{\Rightarrow}{{\cal T}}}
\newcommand{\stackrel{\Leftarrow}{{\cal T}}}{\stackrel{\Leftarrow}{{\cal T}}}
\newcommand{\hat P_{\es{g}}}{\hat P_{\es{g}}}
The mean value of any observable of form $\hat O{=}f(\hat p,\hat x)$ can be written in Heisenberg representation as:
\begin{gather}\label{_<O>}
\midop{\hat O(t)}{=}\Tr[\hat\rho_0\evsopt{\cal L}{t,t_0}[\hat O]],
\end{gather}
where we define:
\begin{gather}\label{_evsop}
\forall {\cal L}(t): \evsop{\cal L}{t,t_0}\stackrel{\idx{def}}{=}\stackrel{\Rightarrow}{{\cal T}} e^{\int_{t{=}t_0}^{t}{\cal L}\diff t}.
\end{gather}
The symbol $\stackrel{\Rightarrow}{{\cal T}}$ in \eqref{_evsop} denotes the chronological ordering superoperator which arranges operators in direct (inverse) time order for $t{>}t_0$ ($t{<}t_0$). Let us also define the following notations for the interaction representation generated by arbitrary splitting ${\cal L}(t){=}{\cal L}_0+{\cal L}_1(t)$:
\begin{gather}\label{superoperator_interaction_representation}
(\evsop{\cal L}{t,0})^{\intercal}{=}\evsop{(\cal L_0^{\intercal})}{t,0}\evsop{({\cal L}_{\idx{I}}^{\intercal})}{t,0},
\end{gather}
where the interaction Liouvillian reads
\begin{gather}\label{interaction_Liouvillian}
{\cal L}_{\idx{I}}^{\intercal}(\tau){=}{\evsop{({\cal L}^{\intercal}_0)}{{-}\tau,0}}{\cal L}_1^{\intercal}(t{-}\tau){\evsop{({\cal L}_0^{\intercal})}{\tau,0}}.
\end{gather}
In the case ${\cal L}_0'{=}\frac{-i}{\hbar}[\frac{\hat p^2}{2m}{-}\hbar\omega_{\idx{a}}\proj{\es{g}},\odot]$ the associated interaction liouvillian \eqref{interaction_Liouvillian} in the rotating wave approximation takes the form:
\begin{gather}\label{L_I'-}
{\cal L}_{\idx{I}}'{\simeq}\frac{-i}{\hbar}[\hat H',\odot]{+}\sum_{n{=}1}^2\Lbd_{\proj[\es{g}]{\es{e}_n}},
\end{gather}
where
\begin{gather}
\hat H'(\tau){=}\sum_{n{=}1}^2\hat\chi_n(\tau)\proj[\es{g}]{\es{e}_n}{+}\mbox{h.c.};\\
\hat\chi_1(\tau){=}\xi_1^*(t{-}\tau)e^{i(\omega t{-}\kappa \hat x{-}(\Delta{-}\frac{\kappa \hat{p}}{m})\tau) };\\
\hat\chi_2(\tau){=}\xi_2^*(t{-}\tau)e^{i(\omega t{+}\kappa \hat x{-}(\Delta+\frac{\kappa \hat{p}}{m})\tau)},
\end{gather}
and $\Delta{=}\omega{-}\omega_{\idx{a}}$ is detuning of carrier frequency of radiation from atomic resonance in the case of system at rest.
Repeated application of the transformation \eqref{superoperator_interaction_representation} to \eqref{L_I'-} with ${\cal L}_0''{=}\Lrel{=}\gamma\sum_{n{=}1}^2\Lbd_{\proj[\es{g}]{\es{e}_n}}$ leads to expression:
\begin{gather}
(\evsop{\cal L}{t,0})^{\intercal}{=}\evsop{{\cal L_0'}^{\intercal}{+}\Lrel^{\intercal}}{t,0}\evsop{({{\cal L}_{\idx{I}}''}^{\intercal})}{t,0},
\end{gather}
so that
\begin{gather}
\midop{\hat O(t)}{=}\Tr[(\evsop{{\cal L_0'}{+}\Lrel}{t,0}[\hat\rho_0])\evsop{({{\cal L}_{\idx{I}}''}^{\intercal})}{t,0}[\hat O]]\stackrel{t{\gg}\gamma^{-1}}{=}\\
\Tr[\hat P_{\es{g}}{(\evsop{{\cal L_0'}{+}\Lrel}{t,0}[\hat\rho_0])}\hat P_{\es{g}}({\evsop{({{\cal L}_{\idx{I}}''}^{\intercal})}{t,0}[\hat O]})\hat P_{\es{g}}]\label{<O(t)>-doppler(t->inf)},
\end{gather}
where $\hat P_{\es{g}}{=}\proj{\es{g}}$ and the last equality is due to the exponential damping of excited states populations induced by relaxation superoperator \eqref{doppler_L_rel}.
Let us consider the evolution $\hat O(t)$ generated by the superoperator $\evsop{{{\cal L}_{\idx{I}}''}^{\intercal}}{t+\delta t,t}$:
\begin{gather}\label{generator_2-order-expansion}
\begin{split}
\hat O(t{+}&\delta t){\simeq}\biggl(1{+}\int_t^{t{+}\delta t}{{\cal L}_{\idx{I}}''}^{\intercal}(\tau)\diff\tau{+}\\
&\int_t^{t{+}\delta t}\diff\tau_2\int_t^{\tau_2}d\tau_1{{\cal L}_{\idx{I}}''}^{\intercal}(\tau_2){{\cal L}_{\idx{I}}''}^{\intercal}(\tau_1)\biggr)\hat O(t).
\end{split}
\end{gather}
Integrands in Eq.~\eqref{generator_2-order-expansion} include the terms oscillating at frequencies $|\Delta{\pm}\frac{k \midop{\hat{p}}}m|$. In sequel we will consider the so-called weak-field limit when these oscillations are rapid relative to the characteristic timescales of the relevant processes, so that the contributions of the associated terms asymptotically vanish. In this limit, the second term in rhs of Eq.~\eqref{generator_2-order-expansion} disappears. The remaining terms constitute two decoupled evolution equations for the reduced density matrices $f_{\es{x}}(\hat p,\hat x,t{+}\delta t){=}\matel{\es{x}}{\hat O(t)}{\es{x}}$ ($\es{x}{=}\es{g},\es{e}$):
\begin{gather}\label{f_g(t)-func}
\begin{split}
f_{\es{g}}(\hat p,&\hat x,t{+}\delta t){=}\biggl({\odot}{+}\frac{1}{\hbar^2}\int_t^{t{+}\delta t}\diff\tau_2\int_t^{\tau_2}d\tau_1e^{\frac{1}{2} \gamma (\tau_1{-}\tau_2)}\times\\
&\sum_{n{=}1}^2\biggl\{\hat\chi_n(\tau_2){\odot}{\hat\chi_n^{\dagger}(\tau_1)}{+}{\hat\chi_n(\tau_1)}{\odot}\hat\chi_n^{\dagger}(\tau_2){-}\\
&{\odot}{\hat\chi_n(\tau_1)}{\hat\chi_n^{\dagger}(\tau_2)}{-}\hat\chi_n(\tau_2) {\hat\chi_n^{\dagger}(\tau_1)}{\odot}\biggr\}\biggr)[f_{\es{g}}(\hat p,\hat x,t)];
\end{split}\\
f_{\es{e}}(\hat p,\hat x,t{+}\delta t){=}{\cal G}[f_{\es{e}}(\hat p,\hat x,t)]
\end{gather}
The explicit form of $\cal G$ is irrelevant in view of Eq.~\eqref{<O(t)>-doppler(t->inf)}. The first two terms in the curly brackets in Eq.~\eqref{f_g(t)-func} can be transformed as
\begin{subequations}\label{f_g(t)-term}
\begin{gather}
\begin{split}\label{f_g(t)-1-st_term}
\hat\chi_1&(\tau_2){f_{\es{g}}(\hat p,\hat x,t)}{\hat\chi_1^{\dagger}(\tau_1)}{=}\\
&\xi_1^*(t{-}\tau_2)\xi_1(t{-}\tau_1)f_{\es{g}}(\hat{p}{+}\hbar\kappa,\hat{x}{+}\tfrac{\hbar\kappa}{m}\tau_2,t)e^{i \Delta_1(\hat p)(\tau_1{-}\tau_2)}{=}\\
&\xi_1^*(t{-}\tau_2)\xi_1(t{-}\tau_1)e^{i\hat\Delta_1(\hat p)(\tau_1{-}\tau_2) }f_{\es{g}}(\hat{p}{+}\hbar\kappa,\hat{x}{+}\tfrac{\hbar\kappa}{m}\tau_1,t),
\end{split}
\end{gather}
\begin{gather}
\begin{split}\label{f_g(t)-2-nd_term}
\hat\chi_1&(\tau_1){f_{\es{g}}(\hat p,\hat x,t)}{\hat\chi_1^{\dagger}(\tau_2)}{=}\\
&\xi_1(t{-}\tau_2)\xi_1^*(t{-}\tau_1)f_{\es{g}}(\hat{p}{+}\hbar\kappa,\hat{x}{+}\tfrac{\hbar\kappa}{m}\tau_1,t)e^{{-}i \Delta_1(\hat p)(\tau_1{-}\tau_2)}{=}\\
&\xi_1(t{-}\tau_2)\xi_1^*(t{-}\tau_1)e^{{-}i\hat\Delta_1(\hat p)(\tau_1{-}\tau_2) }f_{\es{g}}(\hat{p}{+}\hbar\kappa,\hat{x}{+}\tfrac{\hbar\kappa}{m}\tau_2,t),
\end{split}
\end{gather}
\end{subequations}
where
$
\Delta_1(p){=}\Delta{-}\frac{\kappa( p{+}\frac{\hbar\kappa}{2})}{m}.
$
The extra displacements $\frac{\hbar\kappa}{m}\tau_n$ in the $x$-dependencies of $f_{\es{g}}$ in Eqs.~\eqref{f_g(t)-term} account for the change of the velocity of atom after the photon absorption.
These displacements are typically very small compared to the characteristic scales of spatial change of the function $f_{\es{g}}$ and can be neglected. With this approximation, the exponentials and functions $f_{\es{g}}$ in Eqs.~\eqref{f_g(t)-term} commute, which allows to write:
\begin{subequations}\label{approximation_for_f_g(t)-term}
\begin{gather}
\begin{split}
\frac{1}{\hbar^2}\int_t^{t{+}\delta t}&\diff\tau_2\int_t^{\tau_2}d\tau_1e^{\frac{1}{2}\gamma(\tau_1{-}\tau_2)}{\times}\\
&\left(\hat\chi_1(\tau_2){\odot}{\hat\chi_1^{\dagger}(\tau_1)}{+}{\hat\chi_1(\tau_1)}{\odot}\hat\chi_1^{\dagger}(\tau_2)\right)[f_{\es{g}}(\hat{p},\hat{x},t)]{\simeq}\\
&2C_{+}(\hat p,t)f_{\es{g}}(\hat{p}{+}\hbar\kappa,\hat{x},t)C_{+}(\hat p,t)\delta t,
\end{split}
\end{gather}
\begin{gather}
\begin{split}
\frac{1}{\hbar^2}\int_t^{t{+}\delta t}&\diff\tau_2\int_t^{\tau_2}d\tau_1e^{\frac{1}{2}\gamma(\tau_1{-}\tau_2)}{\times}\\
&\left(\hat\chi_2(\tau_2){\odot}{\hat\chi_2^{\dagger}(\tau_1)}{+}{\hat\chi_2(\tau_1)}{\odot}\hat\chi_2^{\dagger}(\tau_2)\right)[f_{\es{g}}(\hat{p},\hat{x},t)]{\simeq}\\
&2C_{-}(\hat p,t)f_{\es{g}}(\hat{p}{-}\hbar\kappa,\hat{x},t)C_{-}(\hat p,t)\delta t,
\end{split}
\end{gather}
\end{subequations}
where
\begin{subequations}\label{formulas_for_C+-}
\begin{widetext}
\begin{align}\label{-C+-}
C_{+}(p,t)&{=}
\sqrt{s_+(p){+}s_+^{*}(p)}, &s_+(p)&{=}\frac{1}{2\hbar^2\delta t}\int_t^{t{+}\delta t}\diff\tau_2\int_t^{\tau_2}d\tau_1\xi_1^*(t{-}\tau_2)\xi_1(t{-}\tau_1)e^{(i\Delta_1(p){+}\frac{\gamma}2)(\tau_1{-}\tau_2)},
\\
C_{-}(p,t)&{=}\sqrt{s_-(p){+}s_-^{*}(p)}, & s_{-}(p)&{=}\frac{1}{2\hbar^2\delta t}\int_t^{t{+}\delta t}\diff\tau_2\int_t^{\tau_2}d\tau_1\xi_2^*(t{-}\tau_2)\xi_2(t{-}\tau_1)e^{(i\Delta_1({-}p){+}\frac{\gamma}2)(\tau_1{-}\tau_2)}.
\end{align}
\end{widetext}
\end{subequations}
Substitution of approximations \eqref{approximation_for_f_g(t)-term} into \eqref{f_g(t)-func} gives:
\begin{gather}\label{effective_Liouvillian}
f_{\es{g}}(\hat p,\hat x,t{+}\delta t){=}\evsop{{\cal L}_{\idx{eff}}^{\intercal}}{t{+}\delta t,t}[f_{\es{g}}(\hat p,\hat x,t)],
\end{gather}
where
\begin{gather}
{\cal L}_{\idx{eff}}(t){=}{-}\frac{i}{\hbar}[\hat H_{\idx{eff}},\odot]{+}\Lrel^{\idx{eff}},
\end{gather}
\begin{gather}
\label{effective_friction}
\Lrel^{\idx{eff}}{=}
\Lbd_{e^{i\kappa \hat x}C_{+}(\hat p,t)}{}+\Lbd_{e^{{-}i\kappa \hat x}C_{-}(\hat p,t)},
\end{gather}
\begin{gather}
\hat H_{\idx{eff}}{=}i\hbar\sum_{m=\pm}(s_m(\hat p)-s_m^{*}(\hat p)).
\end{gather}
Eq.~\eqref{effective_Liouvillian} allows to calculate the averaging in \eqref{<O(t)>-doppler(t->inf)} within the reduced Hilbert space which involves only the translational degree of freedom:
\begin{gather}
\midop{\hat O(t)}\stackrel{t{\gg}\gamma^{-1}}{=}\Tr[\hat\rho_0^{\idx{red}}{\evsop{\frac{i}{\hbar}[\frac{\hat p^2}{2m},\odot]}{t,0}\evsop{{\cal L}_{\idx{eff}}^{\intercal}}{t,0}[\hat O]}]_{\idx{spatial}}\label{<O(t)>-spatial_only}.
\end{gather}
Here $\hat\rho_0^{\idx{red}}{=}\Tr[\hat\rho]_{\idx{el}}$ whereas $\Tr[\odot]_{\idx{el}}$ and $\Tr[\odot]_{\idx{spatial}}$ denote the partial traces over the electronic and translational subsystems.
The dissipator \eqref{effective_friction} reduces to the isotropic friction of form \eqref{theorem:trans_inv_Lbd-isotropic} provided that
\begin{gather}
\forall p:C_{+}({-}p,t){=}C_{-}(p,t){=}\fciso(p).
\end{gather}
It is easy to verify that this condition is realized in two important cases.
\subsection{Weak coherent laser driving}
In this regime, $\xi_1(t){=}\xi_2(t){=}\xi{=}$const, and there exists such $\delta t$ in the range of applicability of the second-order expansion \eqref{generator_2-order-expansion} that $\delta t{\gg}\gamma^{-1}$. Thence, the integrals in \eqref{formulas_for_C+-} can be easily computed, which gives:
\begin{gather}\label{_coherent_C_{+-}}
\Lrel^{\idx{eff}}{=}\Bdn_{\kappa,\fciso},~
\fciso(p){=}{\frac{|\xi|}{\hbar}\frac{\sqrt{\gamma/2}}{\sqrt{(\frac{\gamma}2)^2{+} \Delta_1^2(-p)}}},\\
\hat H_{\idx{eff}}{=}{-}\frac{|\xi|^2}{\hbar}\sum_{\alpha{=}\pm1}\frac{\Delta_1(\alpha \hat p)}{(\frac{\gamma}2)^2+ \Delta_1^2(\alpha \hat p)}.
\end{gather}
Note what the Hamiltonian $\hat H_{\idx{eff}}$ describes the effect of the optical quadratic Stark shift which also can induce the effective potential forces on the system in the case of spatially non-uniform fields $\xi{=}\xi(x)$.
\subsection{Incoherent driving}
Suppose that the the atom is illuminated by the two classical light sources with the equal spectral densities $I(\omega)$ at the atomic site and having coherence times in the range $\Delta_1^{-1}(p){\ll}t_{\idx{coh}}{\ll}\gamma^{-1}$. In this case, $\xi_{1}(t)$ and $\xi_{2}(t)$ represent the uncorrelated stationary stochastic processes. This allows one to choose such $\delta t$, that $\gamma^{-1}{\gg}\delta t{\gg}t_{\idx{coh}}$, and calculate the integrals in Eqs.~\eqref{formulas_for_C+-} neglecting the terms $\frac{\gamma}2$ in the exponents, which gives
\begin{gather}
\Lrel^{\idx{eff}}{=}\Bdn_{\kappa,\fciso},~
\fciso(p){=}\frac{\pi d}{\hbar}\sqrt{\frac{1}{2 c}I(\omega{+}\Delta_1(-p))},
\end{gather}
where $I(\omega)$ is the spectral density of each beam. Also, here we assumed equal transition dipole momenta: $d{=}|\vec d_1|{=}|\vec d_2|$.
\begin{remark}[Remark.]
The setup sketched in Fig.~\ref{@FIG.A01} as well as in Fig.~\ref{@FIG.01'} of the main text in principle can be used to measure both the momenta and positions of the environmental photons by registering the scattered photons and the position of atom. This implies that there must exist the fundamental restrictions on the physically admissible shapes and smoothness of profiles $\fciso(p)$ and, more generally, on admissible forms of operators $\hat L_k$ in Eq.~\eqref{_Lindbladian_definition}, that would prevent these measurements from violating the Heisenberg uncertainty principle. The detailed analysis of implications of this important observation is way beyond the scope of this paper and will be the subject of future work.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}\label{sec:intro}
\IEEEPARstart{T}{he} optimal transport of delay-constrained multimedia services over WLANs requires adaptation to many aspects of \gls{OSI} model layers starting from delay constraints and bandwidth variations of the traffic at the application layer up to accommodation to wireless channel conditions and power constraints at the physical layer. The efficiency of 802.11e \gls{HCCA} function mainly depends on the accuracy of its scheduler in assigning network resources, such as channel bandwidth, to the traffic streams without jeopardizing the QoS constraints such as delay and throughput. Moreover, with the presence of delay-sensitive multimedia traffic with variable profile, the existing scheduling approaches become inefficient. Thus, the scheduler is required to consider the fluctuation of traffic in the scheduling process.
This article introduces an overview of the prime challenges for provisioning QoS for multimedia traffic with emphasize on \gls{VBR} traffic in IEEE~802.11e wireless networks. Then, it presents a taxonomy for the existing solutions, and describes the most representative properties, advantages, and design challenges. This taxonomy comprises the core approaches and techniques on IEEE802.11e protocol, with more emphasize on \gls{HCCA} enhancements. Additionally, a systematic summarization and comparison for research contributions in each field are used to clearly identify the current challenges for further research. Finally, the article discusses the most critical issues which hinder the provisioning of QoS in wireless networks with a special attention to polling and \gls{TXOP} allocation enhancements.
This paper is a survey of QoS provisioning for video transmission in IEEE802.11e, which is organized as follows: Section~\ref{sec:11eStandard} exhibits the background about IEEE802.11e standard and its functions. Section~\ref{sec:QoSChalng} presents the main challenges in IEEE802.11e WLANs. Section~\ref{sec:classification} classifies and reviews the core approaches in IEEE802.11e WLANs, which were proposed to enhance QoS provisioning for multimedia traffic. A number of leading approaches aiming at improving the QoS for multimedia traffic has been discussed in Section \ref{sec:HCCA_enhancement}. Section~\ref{sec:comparison} shows a general comparison of the IEEE802.11e approaches and their targeted features, and lists some of the strength and limitation criteria of these approaches. Section~\ref{sec:ORI} and Section~\ref{sec:futureWork} identifies research trends, challenges, and potential future areas related to the article's scope, and finally Section~\ref{sec:conclusion} concludes the article.
\section{IEEE802.11e standard}
\label{sec:11eStandard}
Several amendments have been made to the legacy IEEE802.11 WLAN standard \cite{IEEEStand1999}, as shown in Table~\ref{tab:IEEEFamily}. IEEE802.11e is one of the approved versions of IEEE802.11 standard, which defines a combination of \gls{QoS} improvements on the \gls{MAC} layer for WLAN applications, as shown in Fig. \ref{fig001}. The standard is critically important for applications that are very sensitive to delay, such as \gls{VoWLAN} and multimedia streaming.
\begin{table*}[!t]
\caption {The family of IEEE-802.11 versions}
\centering
\begin{tabular}{p{2.2cm}p{8cm}p{6cm}}
\hline
Standard & Objective & Frequency and Modulation\\ \hline
IEEE802.11 \cite{IEEE80211Stand} & To provide up to 2 Mbps bit rate. & 2.4 GHz by utilizing DSSS and FHSS. \\
IEEE802.11a \cite{IEEE80211aStandard} & To provide up to 54 Mbps bit rate. & 5 GHz by utilizing OFDM. \\
IEEE802.11b \cite{IEEE80211bStandard} & To provide up to 11 Mbps bit rate. & 2.4 GHz by utilizing HRDSSS. \\
IEEE802.11c \cite{IEEESTD11c} & To ensures proper bridging operations. & -\\
IEEE802.11d \cite{IEEE80211d} & To covers more regulatory domains. & -\\
IEEE802.11e \cite{IEEE80211e2005} & To define new QoS enhancements to 802.11a and 802.11b. & -\\
IEEE802.11f \cite{IEEE80211f} & To provide interoperability for roaming among different APs. & -\\
IEEE802.11g \cite{IEEE80211g} & To provide up to 54 Mbps bit rate. & 2.4 GHz by utilizing OFDM. \\
IEEE802.11n \cite{IEEE80211n} & To provide up to 600 Mbps bit rate. & 2.4 and 5 GHz by utilizing MIMO-OFDM.\\ \hline
\end{tabular}
\label{tab:IEEEFamily}
\end{table*}
In IEEE802.11e, the QoS feature includes an extra coordination function called \gls{HCF}. This function combines both functionalities of the well-known \gls{PCF} and \gls{DCF}. In order to permit the use of a uniform assortment of frame exchange sequences for QoS data transfers during the time of both \gls{CP} and \gls{CFP}, the HCF introduced some enhanced frame subtypes and QoS-specific mechanisms. As for contention-based transfer, HCF employs a contention-based channel access approach, namely \gls{EDCA}, while for contention-free transfer it uses a controlled-channel access method, so-called \gls{HCCA}. \glspl{STA} might obtain TXOPs using \gls{EDCA}, \gls{HCCA} or both schemes together. Thus, a \gls{TXOP} is defined as \gls{EDCA} \gls{TXOP} if it is obtained by the contention-based channel access, while it is defined as \gls{HCCA}-\gls{TXOP} if it is obtained by the controlled channel access.
\Figure[h!](topskip=0pt, botskip=0pt, midskip=0pt)[width=0.9\linewidth]{MACArchitechture}
{MAC architecture in IEEE802.11e.\label{fig001}}
\subsection{Enhanced distributed channel access (EDCA)}
\gls{EDCA} mechanism has been designed to provide sort of differentiated distributed access to \gls{WM} for \glspl{STA} by using eight uneven \glspl{UP}. It determines four \glspl{AC} to provide support for traffic delivery at the \glspl{STA} using \glspl{UP}, which produces the \gls{AC}, as shown in Table~\ref{tab:ACMap}. For every \gls{AC}, an enhanced variant of \gls{DCF}, called \gls{EDCF}, contends for TXOPs using a set of \gls{EDCA} parameters. For more details about the \gls{EDCF} refer to \cite{IEEEStandard2012}. Implementation of this mechanism is easy; however, the QoS requirement of a realtime traffic can not always be met, especially when the heavy load conditions occur. In heavy loaded scenarios, higher prioritized traffic QoS requirement may easily be broken even though it exhausts most of the available bandwidth. However, lower prioritized traffic may be starved and severely deteriorated in both efficiency and effectiveness.
\begin{table}
\centering
\caption {Mappings of user priority to access category}
\begin{tabular}{p{2cm}p{1.5cm}p{1.5cm}p{2cm}}
\hline
Priority & UP & AC & Designation \\ \hline
Lowest & 1 & AC\_BK & Background \\
& 2 & AC\_BK & Background \\
& 0 & AC\_BE & Best Effort \\
& 3 & AC\_BE & Best Effort \\
& 4 & AC\_VI & Video \\
& 5 & AC\_VI & Video \\
& 6 & AC\_VO & Voice \\
Highest & 7 & AC\_VO & Voice \\ \hline
\end{tabular}
\label{tab:ACMap}
\end{table}
\subsection{HCF controlled channel access (HCCA)}
As known in IEEE-802.11e, a synchronization signal is rhythmically sent to all of the connected stations in the \gls{BSS}. The time between two subsequent signals makes a super-frame, where a service can be delivered through this super-frame over two periods of time, \gls{CFP} and \gls{CP}. The data of any station has to be transmitted during a period of time, namely \gls{TXOP}, which is dedicated for a \gls{QSTA} to transfer its \glspl{MSDU}. Fundamentally, \gls{TXOP} is acquired through the contention-based access, which is known as EDCA-TXOP. As for the controlled medium access, the \gls{HC} grants the \gls{TXOP} to the \gls{QSTA} (known as polled \gls{TXOP}). Fig. \ref{fig:MACArch} shows a clear example of 802.11e super-frame which demonstrates the interchanging of one controlled medium access and one contention-based period, where the later includes one \gls{QAP} and three \glspl{QSTA}. In general, controlling medium access occurs either within the \gls{CP} or through the \gls{CFP} if the medium remains idle for at least one period of \gls{PIFS}. In order to support \gls{QoS} in \gls{HCCA}, many researchers have proposed to improve the existing \gls{PCF} by controlling the transmission only within the \gls{CFP}. Therefore, the data packets of any wireless station in \gls{HCCA} can be only transmitted during a declared period of time in the poll frame.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\linewidth]{HCCAMechanism}
\caption{An 802.11e super-frame example, CFP and CP. In the CFP, the frame exchange takes a place throughout the polling mechanism, while in CP the QSTAs have to listen to the medium transmitting data packets.}
\label{fig:MACArch}
\end{figure}
\subsubsection{Reference design of \gls{HCCA}}
At the point if a \gls{QSTA} wants to transmit its realtime \gls{TS} within the contention-free period, it has to send an ADDTS-Request to the \gls{QAP}. This ADDTS-Request declares the requirements of QoS for that specific \gls{TS} within the relevant \gls{TSPEC} domain. Consequently, the \gls{QAP} will try to fulfill the requirements while conserving the QoS of existing admitted flows. If the ADDTS-Request is accepted, the \gls{QAP} will reply an ADDTS-Response back to the relevant station, then, this station will be admitted to the \gls{QAP} polling list. Table~\ref{tab:TSPECsymbol} shows the compulsory \gls{TSPEC} parameters and their symbols.
\begin{table} [!h]
\caption {Symbols used for \gls{TSPEC} and scheduling parameters}
\centering
\begin{tabular}{p{2cm}p{6cm}}
\hline
Notation& Description\\ \hline
$\rho$ & Mean Data Rate\\
$L$ & Nominal \gls{MSDU} Size \\
$M$ & Maximum \gls{MSDU} Size \\
$D$ & Delay Bound\\
$SI$ & Service Interval\\
$mSI$ & Minimum Service Interval\\
$MSI$ & Maximum Service Interval\\
$R$ & Physical Transmission Rate\\
$BI$ & Beacon Interval \\
$O$ & \gls{PHY} and \gls{MAC} Overhead \\
$N$ & Number of packets \\
$T$ & super-frame duration \\
$T_{CP}$ & Contention-based duration \\ \hline
\end{tabular}
\label{tab:TSPECsymbol}
\end{table}
After accepting new ADDTS-Request, the \gls{HCCA} scheduler will go through the following steps:
\begin{enumerate}
\item \textit{Assigning service interval} \\
\label{SIassign}
\gls{HCCA} \cite{IEEEStandard2012} computes the \gls{SI} as a sub-multiple of the whole Beacon Interval $BI$, which is calculated as the minimum of the maximum \glspl{SI} of all priorly accepted traffic streams including the incoming data traffic. Equation (\ref{si}) is used to calculate the \gls{SI}:
\begin{equation}
\label{si}
SI = \frac{BI}{\left \lceil\frac{BI}{MSI_{min}} \right \rceil},
\end{equation}
where $MSI_{min}$ is computed as in Equation (\ref{minSI}):
\begin{equation}
MSI_{min} = min(MSI_{i}) , i \in [1, n],
\label{minSI}
\end{equation}
where $MSI_{i}$ denotes the maximum $SI$ of the $i^{th}$ stream and $n$ denotes the number of all previously admitted \glspl{QSTA}' traffic streams.
\item \textit{Allocating TXOP} \\
Variant \gls{TXOP} is allocated by \gls{HC} to every accepted \gls{QSTA} based on the declared QoS parameters in the \gls{TSPEC}, which allows the \gls{QSTA} to obtain the required QoS. The HC calculates \gls{TXOP} for the $i^{th}$ \gls{QSTA} based on the expected MSDUs, which may arrive at $\rho_{i}$, as calculated in Equation~\eqref{eq:N}:
\begin{equation}
N_{i}=\left \lceil \frac{SI\times\rho_{i}}{L_{i}} \right \rceil,
\label{eq:N}
\end{equation}
where $L_{i}$ denotes the \gls{MSDU} of the $i^{th}$ station.\\
Thereafter, the \gls{TXOP} of the $i^{th}$ station $(TXOP_{i})$ is calculated as the required time to transmit $N_{i}$ \gls{MSDU} or one maximum \gls{MSDU} at the relevant physical rate $R_{i}$, as in Equation~\eqref{eq:TXOP} below:
\begin{equation}
TXOP_{i}=max\left (\frac{N_{i} \times L_{i}}{R_{i}} + O, \frac{M}{R_{i}} + O \right)
\label{eq:TXOP}
\end{equation}
where $O$ represents the total overhead, including MAC and physical headers, poll frames overheads, inter-frame spaces (IFSs) and acknowledgments.
\item \textit{Admission control} \\
The \gls{ACU} regulates the admission of the \gls{TS} while maintaining the QoS of the previously admitted \glspl{TS}.
When the \gls{ACU} receives a request of admitting a new \gls{TS}, the \gls{ACU} calculates a new $SI$ using Equation \eqref{si} and estimates the number of MSDUs that may arrive at this new $SI$ based on Equation \eqref{eq:N}. Then, the \gls{ACU} calculates the $TXOP_{i}$ for the particular \gls{TS} using Equation~\eqref{eq:TXOP}. Finally, the \gls{ACU} would admit the relevant \gls{TS} only if the following inequality is satisfied:
\begin{equation}
\label{eq:ACU}
\frac{TXOP_{n+1}}{SI}+\sum_{i=1}^{n} \frac{TXOP_{i}}{SI}\leq \frac{T- T_{CP}}{T}
\end{equation}
Fig.~\ref{fig:ACU} shows an example of an admitted stream from $STA_{i}$. The beacon interval is 100ms and the maximum \gls{SI} for the stream is 60ms. The scheduler sets a scheduled $SI$ to 50ms with complying to Equation~\eqref{eq:ACU}, where $n$ represents the number of all admitted streams, $n+1$ denotes the index of incoming \gls{TS}, $T$ indicates the beacon interval and $T_{CP}$ is the time reserved for \gls{EDCA} contention-period.
The \gls{HC} sends an ADDTS-Response to the relevant \gls{QSTA} only if Equation \eqref{eq:ACU} is satisfied, and it sends a message of rejection otherwise. Then, the \gls{HC} will add the accepted \gls{TS} to its polling list.
\begin{figure}[!h]
\centering
\includegraphics[width=0.8\linewidth]{ACUeg}
\caption{Schedule for streams from STAs i to k. The streams are scheduled in Round-Robin fashion govern by the admission control unit}
\label{fig:ACU}
\end{figure}
\end{enumerate}
\begin{figure}[!th]
\centering
\includegraphics[width=.6\linewidth]{IPQoS-x}
\caption{QoS architecture of the IP Network. The QoS parameters are defined in the MAC layer}
\label{fig:IPQoS}
\end{figure}
\section{QoS challenges in IEEE802.11e WLANs}
\label{sec:QoSChalng}
QoS is the overall effect of the service performance, which defines the satisfaction degree of a service user and manifests itself in a number of subjective or objective parameters \cite{rec1994800}. There are two ways to investigate the QoS, subjective (perceptive) and objective (network) measurements. In the subjective measurement, the user involves to carry out a series of assessment tests, while in objective measurement, typical network performance throughput, packet loss, packet jitter and delay is evaluated. In order to meet the user satisfaction, the subjective QoS parameters shall be translated into a set of objective QoS parameters, e.g. throughput, delay and losses.
QoS could be supported in different ways at different protocol layers as illustrated in Fig.~\ref{fig:IPQoS}. Some applications have the capability to adapt the generated traffic to the conditions of the underlying network in order to meet user expectations. An example is the use of the \gls{RTP} and associated \gls{RTCP} \cite{jacobson2003rtp} to dynamically adapt the parameters of an audio and/or video streams, minimizing the losses due to congestion in the network \cite{busse1996dynamic}. Nevertheless, application layer mechanisms are usually not enough, since end-to-end QoS requires support in the lower layers of the protocol stack throughout the network nodes that the traffic must traverse from sender to receiver. However, this work mainly concerns with QoS provisioning at \gls{MAC} layers.
The QoS provisioning of diverse multimedia streams in a wireless environment imposes a chain of challenges due to many factors of \gls{OSI} model layers ~\cite{ISO7498_1994,Delsing2012,alani2014osi} ranging from traffic characteristics in application layer down to the wireless channels nature in physical layer. In this section, a review of the major challenges that may emerge when providing QoS for delay-sensitive applications in IEEE802.11e wireless networks.
\subsection{Adaptation to fluctuation of application profile}
\label{sec:Adpt2Aplica}
Generally, the application profile of a traffic is defined by the alternation of the traffic over the time. The QoS provision of a \gls{VBR} flow is substantially influenced by the variation of the application profile over the time. The accurate estimation of the traffic at the application layer can significantly enhance the performance of underlying functions of \gls{MAC} layer to adapt its parameters according to these changes.
The \gls{VBR} video source can be generally classified into three main categories~\cite{Inan2006,Huang2009}: I) variable packet size with constant Generation Interval (GI), e.g., MPEG-4 videos; II) constant packet size with variable GI, e.g., Voice over Internet Protocol (VoIP); and III) variable packet size with variable GI, e.g., H.263.
The transmission of video streams can be significantly affected by the compression techniques used, such as MPEG-4 and H.263. The nature of the frame structure and the compression algorithm used along with the variations within video scenes can significantly influence the burstiness level of the stream ~\cite{trathgeb1993,hattacharyya2014}. The burstiness of a \gls{VBR} stream traffic increases the complexity of network resources management to ensure QoS support for continuous stream playback. Although, the reference design of the \gls{HCCA} scheduler is simple and efficient in supporting constant application profile, yet it is not adequate since it cannot address the fast-changing imposed by the \gls{VBR} bursty traffic, which hinders the performance of \gls{HCCA} by causing packets to wait for a longer time in their transmission queues.
In case of downlink traffic, from \gls{QAP} to \glspl{QSTA}, the \gls{QAP} is aware about its data queues and shall use its highest priority to seize the channel if it remains idle for a duration of \gls{PIFS} without undergoing back-off procedure. However, due to the fact that \gls{QAP} suffers from the lack of information about the uplink transmission queue status, an adaptive scheme is required to allow the scheduler to adjust its behavior based on the current application characteristics. Generally, adaptation to the application can be categorized according to its variability level-based in the three well-known types mentioned in \ref{sec:Adpt2Aplica}.
In \gls{MAC} layer, the uplink traffic profile can be determined using different ways, such as estimating the data buffer of the flow, predicting the packet generation time and/or traffic load at a specific time, or obtaining actual information through cross-layer architecture design. By having the traffic profile, the \gls{HCF} can adjust one or more of its functions such as polling \cite{ramos2007}, \gls{SI} assignment \cite{Grilo2003}, \gls{TXOP} allocation mechanisms \cite{Jansang2011} which allows it to instantaneously adapt to QoS requirement of the flow.
The QoS of VBR video transmission is ungoverned due to the fact that those packets are queued for a duration equivalent to SI until already-queued packets in the buffer are delivered. Recall that during each SI, the reference HCCA scheduler allocates a fixed TXOP to each QSTA based on its mean rate requirements regardless the real VBR traffic changes. There are three QoS challenges relevant to Class I, II and III of VBR traffics.
\subsubsection{QoS Challenges of Class I video flows}
\label{sec:QoSClassIIChallanges}
HCCA scheduler fails to accommodate to variability Class I traffic which, in turn, leaves the wireless bandwidth in underutilization status. Assume, without loss of generality, that an identical TXOP duration is allocated for every QSTA, consequently, each QSTA will waste the same amount of unused TXOP ($T_u$). Thus, Equation \eqref{eq:ACU} can be rewritten as follows:
\begin{equation}
\label{eq:ACUa}
\frac{TXOP_{n+1}}{SI}+ \frac{TXOP-T_u}{SI} \leq \frac{T- T_{CP}}{T}
\end{equation}
According to Reference \cite{Qinglin2007}, using different SIs for different streams will improve the bandwidth utilization up to 50\%. In other words, the $T_u$ in Equation \eqref{eq:ACU} will be equal to $\frac{TXOP}{2}$. Therefore, the Equation \eqref{eq:ACU} can be again rewritten as follows:
\begin{equation}
\label{eq:Bim}
\frac{TXOP_{n+1}}{SI}+ \frac{TXOP}{2 \times SI} \leq \frac{T- T_{CP}}{T},
\end{equation}
which means that the number of admitted flows can be maximized to double the number of admitted flows when different SIs are used.
\subsubsection{QoS Challenges of Class II video flows}
\label{sec:QoSClassIChallanges}
In Class II, when $QSTA_{i}$, at any SI, exploits only portion of its allocated $TXOP_{i}$ at the traffic setup time, namely $T_{eff}^{i}$, leaving an unspent amount of $T_{u}^{i}$. Thus, the following relation can be held \cite{almaqri2016}:
\begin{equation}
\begin{split}
\sum_{i=1}^{N}{T}'_{i} & = T_{eff}^{1} + T_{eff}^{2} +\cdots + T_{eff}^{N}\\
& = TD_{1} - T^{1}_{u} + TD_{2} - T^{2}_{u} +\cdots +TD_{N} - T^{N}_{u} \\
& = \sum_{i=1}^{N} TD_{i} - \sum_{i=1}^{N}T^{i}_{u} \\
\end{split}
\end{equation}
where $T^{i}_{u} \geq 0$, $\sum_{i=1}^{N}{T}D_{i}$ and $\sum_{i=1}^{N}{T}'_{i}$ is the total TXOP scheduled in any SI used in HCCA and ATXOP, respectively. It is worth noting that $TD_{i}$ is the TXOP duration of the $QSTA_i$ including the poll overhead. Thus, the delay of $QSTA_{i}$ in an SI is computed as follows:
\begin{equation}
D_{SI}^{i}=\sum_{j=1}^{i-1} (TD_{i} - T^{i}_{u}) + T_{L}^{i} + T_{poll} + 2 \times SIFS
\label{eq:SIdlyAXTOP}
\end{equation}
Altogether, the real QoS challenge is to minimize packet delay by minimizing the surplus amount, namely $T^{i}_{u}$.
\subsubsection{QoS Challenges of Class III video flows}
\label{sec:QoSClassIIIChallanges}
In video streams like H.263, the deviation comprises not only packet size but also shows up to high variation in generation interval which makes the matter much worse. In any \gls{SI}, scheduling a \gls{QSTA} based on its \gls{TSPEC} likely imposes allocating surplus of \gls{TXOP} duration which leads to wasting of the resources. This waste of resources due to the variations in data rate influences the efficiency of the scheduler that does not implement any recovery policy. Besides, due to the variation in the packet generation interval, perhaps there are some QSTAs that are not ready to transmit which will be considered as over-polling state. This waste of resources, due to the variations in data rate, influences the efficiency of the scheduler that does not implement any recovery policy. Overall, it hinders the meet of delay bounds requirements, which leads to a degradation in QoS provisioning.
Consider the example illustrated in Figure \ref{fig:HCCAVideoTrans} where four QSTAs are polled for transmission in both \gls{CFP} and \gls{CAP}. In this example, $TXOP_{1}$, $TXOP_{2}$, $TXOP_{3}$ and $TXOP_{4}$ to $QSTA_{1}$, $QSTA_{2}$, $QSTA_{3}$ and $QSTA_{4}$, respectively. The wasted \gls{TXOP} and over-polling issues experienced using reference \gls{HCCA}, inspired from the example, are as illustrated in Figure \ref{fig:HCCAVideoTrans}.
\begin{figure*}
\centering
\includegraphics[width=.8\linewidth]{HCCAVideoTrans}
\caption{Wasting TXOP and poll issue with VBR traffic transmission}
\label{fig:HCCAVideoTrans}
\end{figure*}
\begin{itemize}
\item \textbf{Over-polling of \glspl{QSTA}}
As illustrated in this example, due to the lack of awareness about the change in the traffic profile, some QSTAs may receive unwanted poll messages as their transmission queues are empty. $QSTA_{2}$ and $QSTA_{3}$ in \gls{CP}, and $QSTA_{2}$ in \gls{CFP} will respond with a null-frame causing unwanted delay to all QSTAs that may come after them in the same \gls{SI}.
\item\textbf{Wasted \gls{TXOP} duration}
Since some $QSTAs$, such as $QSTA_{1}$, experience a high instant drops-down in data rate, only a short amount of the given \gls{TXOP} duration is utilized. In this case, the channel might remain idle for a period of time greater than the \gls{SIFS} and the control of the medium conveyed to \gls{AP} to poll the next station in the list. Even though the effect of wasted \gls{TXOP} duration in the packet delay is not as high as that caused by over-polling case, however, it is considerably can go high as the number of stations in the network increases.
\end{itemize}
\subsection{Adaptation to varying network conditions}
\label{sec:Adapt2Net}
Due to the phenomena of path loss, multipath fading, shadowing, and interference, wireless networks likely suffer from \gls{SINR} \cite{Grilo2003}. The fluctuation of the underlying channel capacity will hinder the QoS provisioning for time-sensitive applications. Consequently, two possible ways to be applied on the QoS algorithms in order to encounter this challenge and meet the required QoS needs. The first one is by computing the transmission time for the packets based on the minimum physical bit rate announced. By doing so, the QoS is guaranteed, however, this technique gives rise to degradation in bandwidth efficiency as the bandwidth might get higher anytime while only the minimum link rate is considered. The second one is to encourage the QoS algorithm to take into account the link adaptation mechanism of WLANs over the time.
Although, piggybacking feature of \gls{HCCA} is basically designed to improve the channel capacity, it may inversely behave when a station experience successive retransmission or channel noise. This issue has been referred to in \cite{Lee2007} as "the piggyback problem as the low physical transmission rate". If any \gls{QSTA} was transmitting at a low physical rate due to channel error, \gls{QAP} will accordingly decrease the transmission rate of the piggybacked \gls{CF-Poll} frame. This, in turn, will result in channel efficiency degradation and will increase the TSs' frame delay of other stations involving in the \gls{NAV} process.
In case of \gls{VBR} traffic transmission over WLANs and apart from the issues and challenges of \gls{HCCA} reported in \cite{Inan2006,madhar2012}, the major issue of the reference scheduler is the unawareness about the inherent wireless time-varying channel condition\cite{arora2010}. Keeping aware about the channel status has a major impact on the scheduler performance as it can potentially degrade the service differentiation process, even though, \gls{HCCA} has been observed to perform well in heavy loaded network \cite{cicconetti2005,Chen2011}, especially with the emergence of several physical layer technologies such as \gls{AMC} schemes.
\subsection{Bandwidth utilization}
\label{sec:BwUtil}
In \gls{HCCA}, after receiving an ADDTS-request from the station, the scheduler needs first to calculate the required \gls{TXOP} duration taking into account its \gls{TSPEC} parameters. Thereafter, the used admission control mechanism will check the ability to accept the new \gls{TS}. If the new \gls{TS} is accepted, the \gls{SI} will be computed as the minimum among all delay bounds of admitted streams, which is enough to meet the most urgent delay requirement to guarantee the required QoS service for the admitted streams. Finally, the round robin approach is used to allocate TXOPs to the involved station. Even though the use of this design is very simple and straightforward, it still suffers from some challenging issues related to the efficient use of the bandwidth. Indeed, the use of round robin approach in \gls{HCCA} scheduler to serve all TSs in one \gls{SI} might lead to over-allocating the bandwidth, which in turn leads to under-utilizing the channel bandwidth. Moreover, the waste of the wireless bandwidth may reach up to more than 50\% in some cases \cite{zhao2007}.
In fact, based on the minimum physical rate and the characteristics of the incoming TSs, the \gls{ACU} decides the number of admitted TSs to which the wireless resource will be allocated. This approach leads to allocating a constant amount of resource to every \gls{TS} using the mean of single physical transmission rate, which is not compatible with the condition of current wireless bandwidth, especially \gls{VBR} traffic. In other words, the \gls{ACU} should consider both the physical layer and the service specific QoS parameters in order to be able to achieve effective bandwidth utilization \cite{lee2010}.
With noticeably \gls{VBR} flows, one of two scenarios likely occurs at some specific SIs. In the first scenario, the data rate becomes lower than the average value determined in the \gls{TSPEC}, thus, the allocated \gls{TXOP} will not be completely consumed which is considered as a wasting of resources. As for the second scenario, the data rate becomes greater than the average value determined in the \gls{TSPEC}, thus, the assigned \gls{TXOP} will not be enough to transmit the relevant data which increases the end-to-end delay of the flow. The possible solutions to solve these two problems as explained in these references \cite{cecchettielAL2012,ruscelli2014} are: (1) By increasing the \gls{TXOP} duration to the average \gls{TXOP} of traffic for the first case, knowing that it will reduce the bandwidth utilization, especially if the data rate is dropped down. (2) By applying the bandwidth reclaiming approach \cite{Cecchetti2012,lo2007,rashid2008,ruscelli2011}.
\subsection{Network resources management}
\label{sec:NetResMan}
Indeed the \gls{HCF} of IEEE802.11e protocol is targeted to the provisioning of QoS throughout the service differentiation, yet the proper network resource management, such as coordinating between distributed (CP) and controlled (CAP) periods and link layer resources still in request \cite{ramos2005}. In addition, a feasible \gls{ACU} scheme is also required, which in such way can ensure that the QoS requirements are satisfied.
The \gls{HCCA} scheduler operates based on the static configuration of its traffic \gls{TSPEC} parameters where they are constantly served for their lifetime to enforce resource sharing with ensuring that the desired QoS constraints are met. To this aim, a good resource utilization is often left to the heuristic network administrator know-how. However, this constant resource sharing policy might highly cause a scarce bandwidth utilization since it cannot adapt to the transformation of the traffic profile and the lifetime due to dynamic \gls{VBR} traffic evolution.
As a resolution to this issue, a bandwidth sharing strategy is suggested to rely on a criteria which is driven by the performance \cite{navarro2010}, in which a common performance metric is recommended to be defined to differentiate between the traffic streams based on their performance requirements.
\section{Classification of QoS support for multimedia traffic approaches in IEEE802.11e WLAN}
\label{sec:classification}
In general, the enhancement approaches in IEEE802.11e protocol can be classified based on the access medium control fashion into distributed control and centralized control enhancements. In IEEE802.11e, \gls{EDCA} operates based on the distributed access control while the \gls{HCCA} represents the centralized access control. In \cite{ni2004}, many QoS enhancements for 802.11 WLAN have been proposed and classified along with their advantages and disadvantages. Another survey in \cite{luo2011} has focused on the QoS provisioning in both \gls{EDCA} and \gls{HCCA} over IEEE802.11e networks. The \gls{HCCA} enhancement approaches can be themselves classified into different categories according to several aspects such as the functional, structural, environmental and location aspects. In \cite{gao2005} and \cite{liu2009}, the authors presented a survey of various admission control in IEEE802.11e and they classify schemes based on several aspects such as Measurement-Based, Model-Based and Hybrid schemes. In \cite{piro2012}, the delay-EDD based scheduler has been compared to the feedback control based scheduler in order to provide a better comprehension about the so-called packet scheduling in 802.11 WLANs. Below is a short description of the different possible ways of IEEE802.11e approaches classification.
\begin{itemize}
\item \textbf{Traffic flow direction:} In infrastructure mode of IEEE802.11 WLANs, the traffic directions would be either downlink and uplink. "Downlink" refers to a traffic flow transmitted from \gls{AP} to a mobile device, while "uplink" refers to a flow with a reverse direction. IEEE802.11e enhancements can be tailored to enhance the performance for either downlink or uplink traffic or in some cases for both directions.
\item \textbf{Targeted environment:} Although IEEE802.11e \gls{MAC} was originally designed for wireless infrastructure networks and widely used in WLANs, there have been some enhancements for adapting IEEE802.11e to work with other networks such as the improvement of polling and scheduling scheme over IEEE802.11a/e \cite{Schaar2006}, IEEE802.11p networks \cite{zhang2013}, Ad-hoc Wireless Networks in \cite{Fiandrotti2008} or in Integrated model of IEEE802.11e and IEEE802.16 \cite{Naeini2012,Lee2009wimax}.
\item \textbf{Delay-EDD based and feedback control based:}
The pre-knowledge of packets arrival time is only possible for the downlink. While in the uplink, neither the delays of the head of line packets nor the quota of bandwidth needed by each flow are possible to be known by the access point. For this reason, IEEE802.11e schedulers have been categorized into the earliest due date and the feedback control class. A thorough comparison between these types has been presented in \cite{piro2012}.
\item \textbf{Layered vs. cross layer:} IEEE802.11e enhancements can be introduced in two structures, cross-layer and layered approaches. The cross-layer approaches rely on interactions between two layers of the \gls{OSI} architecture. These approaches were motivated by the fact that providing lower or higher layer information to \gls{MAC} layer to perform better. The layered approaches rely on adapting \gls{OSI} layers independently of the other layers. Cross-layer is a promising direction to improve the overall performance of WLAN since it takes into account the interactions among layers \cite{shankar2007}. Thus, several enhancements \cite{Cicconetti2007,noh2010,luo2011cross} prefer to use cross-layer design for obtaining accurate information for scheduling purposes.
\item \textbf{Technique or mechanism used:} The \gls{HCCA} scheduling approaches can be classified based on the techniques and/or mechanisms used in the design. In the literature, a diverse techniques were developed for \gls{HCCA} scheduling to boost its performance for multimedia transmission over error-prone WLANs such as estimation based approaches \cite{Grilo2003,Ansel2004}, predicting traffic profile \cite{Yoo2002,kuo2011}. Moreover, some of these approaches modified one or more of \gls{HCCA} mechanisms such as \gls{TXOP} assignments \cite{ju2013,lee2013,noh2011,Huang2008}, polling mechanism \cite{Chen2004,Ramos2012,Son2004} or \gls{ACU} \cite{noh2010,Kim2006,Zhu2007,lee2010}.
\item \textbf{Analysis method used:} The approach might be analyzed and/or evaluated using one of three methods, namely analytical model \cite{Harsha2006,Cecchetti2008,leonovich2013,Jansang2013}; simulation experiment \cite{ng2013,ruscelli2012,Kim2006TWOPoll} and test bed \cite{Ng2005}. It is worth noting that the analytical model usually is done to capture the characteristics and the shortcoming of the approach and prior to the proposal of a solution. Although, in simulation and test bed methods used for evaluating the proposed scheme, they might be carried out to provide a preliminary study to investigate a particular issue in the existing scheme for possible remedy.
\end{itemize}
\section{QoS Enhancements in IEEE802.11e controlled access mode}
\label{sec:HCCA_enhancement}
This section presents some of the leading approaches proposed in the literature to improve the QoS provisioning for multimedia traffic. More emphasis has been put on the transmission of \gls{VBR} video streams in IEEE802.11e WLAN. The approaches are classified into six sets based on the strategy used to improve \gls{HCCA} performance. However, some approaches can be matched to several types of strategies, but are only classified to their main strategy. Moreover, in the layered approaches the focus was only on the enhancements of \gls{HCCA} at the \gls{MAC} layer. The representative approaches are defined and their mechanisms are described along with a discussion concerning their strengths and weaknesses in improving QoS performance in IEEE802.11e WLAN. In addition, a comparison of the main characteristics of various \gls{HCCA} approaches is provided for each category. Besides some mathematical models that study and provide insights to improve the \gls{HCF} functions have been presented which can provide a promising avenue for further research and investigation.
\subsection{HCCA polling enhancements}
\label{sec:pollEnh}
The polling mechanism of the legacy \gls{HCCA} is responsible for the scheduling and the allocation of \gls{TS} based on their fixed reservations. Thus, the efficiency of this mechanism highly depends on the accuracy of the flow specification declared to the \gls{HC}. Yet, as the flow profile of \gls{VBR} might highly vary over the time, the allocation based on fixed reservations will cause degradation in quality of multimedia flows even when the channel resource is surplus. More particularly, several issues may affect the efficiency of the \gls{HCCA} such as the inefficient Round-Robin scheduling algorithm, the overhead induced by the poll frames, and the lack of coordination between the APs of the neighboring \gls{BSS}. The representative approaches that address these issues and even more are summarized as follows.
\textbf{CP-Multipoll} is a robust multipolling mechanism aims to increase the channel utilization and to minimize the corresponding implementation overhead, which can be robust in error-prone environments like WLANs \cite{Lo2003}. Moreover, the proposed scheme provides a polling schedule to ensure the bounded delay requirements of real-time traffic and it also provides an admission control mechanism. The main aim of this scheme is to design an efficient polling mechanism, due to its high impact on the performance of \gls{HCCA}, which is able to serve both CBR and VBR real-time traffic. Unlike SinglePoll schemes where every \gls{STA} receives one poll frame when polled, CP-Multipoll aggregates many polls in a single multipolling frame incorporating the \gls{DCF} into the polling mechanism. The frame format of CP-Multipoll scheme is as shown in Fig.~\ref{fig:CP-Multipolling}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.8\linewidth]{CPMultipolling}
\caption{CP-Multipoll frame format}
\label{fig:CP-Multipolling}
\end{figure}
Basically, CP-Multipoll conveys the polling order into the contending order. This can be achieved by assigning different back-off values to the streams in the polling list with accordance to their ascending order in the polling list and allow the back-off to execute as soon as they receive the CP-Multipolling frame. Besides minimizing the polling overhead by transmitting one polling message for all \glspl{QSTA} in the polling list instead of sending polls as many \glspl{STA}, the proposed scheme has other advantages over other multipoll schemes. The bursty traffic is better supported since the \gls{STA} holds the channel only for a period needed to transmit its local buffered data. Moreover, in \gls{DCF} access mode, if the \gls{STA} does not use the poll frame due to empty data buffer, the other STAs in this polling group will immediately detect that the channel is idle and it will advance the starting of channel contention.
However, the proposed mechanism is prone to hidden terminal problems since each \gls{STA} will decrement back-off counter when it senses that the channel is idle. Thus, if hidden terminals exist in the network, different STAs will complete their back-off simultaneously and collision will happen. Due to the inherent hidden node issue of infrastructure wireless networks, CP-Multipoll cannot guarantee that all STAs in the \gls{BSS} can sense the transmission of other STAs. In this case, the station will transmit its data immediately upon the expiration of its back-off timer leading to a collision.
\textbf{CF-Poll piggyback scheme} is presented by Lee and Kim \cite{lee2006} to optimize the usage rule of the CF-Poll piggyback scheme as defined in the IEEE802.11 standard \cite{IEEEStand1999} according to the \gls{TS} load and the minimum physical transmission rate of a \gls{QSTA} which suffer the deep channel fading. Consider the case of piggybacking, the CF-Poll in the QoS-ACK frame from \gls{QAP} to $QSTA_{3}$, illustrated in Fig.~\ref{fig:CFPoll}, must be listened by all \gls{QSTA} in the \gls{BSS}. If any of the QSTAs experience low physical rate, which implies that $QSTA_{3}$ requires more time to receive the frame, the delay for all other QSTAs will be increased and the channel efficiency will be decreased.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{CFPoll}
\caption{CF-Poll piggyback issue with an example of piggybacking CF-Poll on data frame}
\label{fig:CFPoll}
\end{figure}
Motivated from the aforementioned issue, the proposed work provides a guideline for the optimal usage of the CF-Poll piggyback scheme in IEEE802.11e and IEEE802.11n protocols. Simulation-based results reveal that the frame transmission delay is majorly affected by the minimum physical rate when CF-Poll is piggybacked in the QoS data frame while it is slightly influenced by the traffic load. The results show an inverse relationship between the \gls{CF-Poll} piggyback scheme and the traffic load. Despite the presented analysis and guidelines, the recommendations reckon on a number of assumptions that are: the traffic is \gls{CBR} and each \gls{QSTA} has only one \gls{TS} calculated based on the Equation~\eqref{eq:N} which cannot be suitable for supporting the transmission of multimedia applications with variable profile.\\
\textbf{\gls{DEB} method for \gls{HCCA}} is an enhancement of \gls{HCCA} which performs virtual polling through sensing the carrier of the wireless channel \cite{Huang2010}. This technique highlighted the issue of the collision incurred due to polling the nodes in the overlapping area of two adjacent BSSs at the same time. This actually occurs due to the lack of coordination in \gls{HCCA} between the adjacent APs. Consider the nodes 5, 6, 7, 8 in the overlapping area illustrated in Fig.~\ref{fig:DEBScheme}. Since \gls{AP} A cannot hear \gls{AP} B, therefore the collision occurs between the nodes in the overlapped area.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{DEBPolling}
\caption{Collision due to polling the STAs in the overlapping area}
\label{fig:DEBScheme}
\end{figure}
\gls{DEB} uses a similar idea of sensing the carrier of \gls{EDCA} since it manifests high robustness and flexibility controlling the medium at the overlapping BSSs. A virtual polling has been achieved in a distributed manner. The \gls{DEB} arranges the back-off timer of station to guarantee that the polled stations will have different back-off. When the back-off timer expires, the station can be polled without colliding with others. However, \gls{DEB} is only functioning in \gls{CFP} whereas \gls{HCCA} is supposed to work in both \gls{CFP} and \gls{CP}, for this reason one of the significant merits of \gls{HCCA} will be untapped. Moreover, there is no clear consideration of the readiness of the station, \gls{STA} with no data ready to send will be given a \gls{TXOP} which, in turn, be wasted.
\textbf{\gls{NPHCCA}} is presented in \cite{Chen2011} to provide an enhancement over \gls{HCCA} mechanism. Since the \gls{VBR} traffic exhibits variability in packet generation time, the station will not always have pending data to transmit, thus, it will waste time for the \gls{AP} to send polling messages to the stations that have no data to transmit. For this reason, the proposed solution modifies the \gls{HCCA} scheme in such way it allows stations that have pending frames to report their readiness status to \gls{AP} through exchanging messages. Then, the \gls{AP} schedules the only ready stations in appropriate transmission sequence.
The mechanism of the \gls{NPHCCA} is carried out throughout a sequence of messages exchanging. First, a station with data will send a transmission frame request to the \gls{AP} in order to update it about its transmission queue status, including information such as required Priority, Queue status, etc. A station only sends this frame after it receives the beacon message from the \gls{AP} and senses whether the medium is idle for \gls{SIFS}. Accordingly, the \gls{AP} maintains this information in its scheduling table. Finally, the \gls{AP} determines a transmission sequence and notifies stations to transmit data according to this transmission sequence broadcast in the beacon messages. Fig.~\ref{fig:NPHCCA} demonstrates The components of the \gls{NPHCCA}.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{NPHCCA}
\caption{\gls{NPHCCA} mechanism}
\label{fig:NPHCCA}
\end{figure}
Although, \gls{NPHCCA} has shown improvement in the transmission delay when the network is light-loaded, the performance was similar to that of \gls{HCCA} when the network is heavy-loaded. Besides, the messaging exchange of the beacon and transmission request frames added extra overhead to the network, especially when the number of the nodes increases.
\textbf{F-Poll} In Feasible Polling Scheme (F-Poll) \cite{maqri2015}, the application layer gives the accurate arrival-time of the upcoming data frame over the uplink connection to the MAC layer, where this approach is known as a cross-layering approach. F-Poll is suitable for both type II and III of video types categorized in Subsection~\ref{sec:Adpt2Aplica}, where the exact information of the next inter-arrival time is sent to the QAP in order to enhance the scheduling of the TSs. In order to avoid polling a station that have no ready data to transmit, a decision is made of whether to poll the relevant station in the upcoming SI or not directly after receiving a data frame. As a result, the packet access delay is minimized and a great amount of unused TXOP duration is conserved which efficiently enhances the channel utilization. Fig.~\ref{fig:FPollMechanism} elaborates the F-Poll Mechanism.
\begin{figure}[!h]
\centering
\includegraphics[width=0.8\linewidth]{FPollFramework}
\caption{F-Poll scheme mechanism}
\label{fig:FPollMechanism}
\end{figure}
\textbf{AMTXOP} \cite{almaqri2016} like D-TXOP \cite{Almaqri2013}, the \gls{AMTXOP} calculates the TXOP for a certain data stream based on the actual frame size. Since the polling messages can increase the overheard among all QSTAs, the BSS broadcasts one multi-polling message to the QSTAs in a single SI instead of sending one polling message for each. This approach minimizes the polling overhead and also reduces the packet delay which significantly improves the bandwidth utilization. Due to this integration, the AMTXOP outperforms both HCCA and its ancestor, D-TXOP, in terms of channel utilization and packet delay.
\subsection{\gls{TXOP} allocation enhancements}
\label{sec:TXOPEnh}
Usually, if a QSTA's buffer queue is empty during its \gls{TXOP} because of a non-uniform data flow from the upper layer, the media will be unutilized for the whole \gls{TXOP} of the station. However, according to the 802.11e standard, the \gls{QSTA} should send a QoS-NULL frame to the \gls{QAP} to enforce it to start polling other sessions immediately \cite{Jansang2013}. On the other hand, if the allocated \gls{TXOP} is not enough to send the backlogged packets, these data will be served in the next \gls{SI} causing more delay and might impair the designated QoS requirements \cite{rashid2008}. Several techniques have been presented in the literature to address the limitation of the \gls{TXOP} assignment mechanism of IEEE802.11e, we overview here some representative approaches.
\textbf{Scheduling Based on Estimated Transmission Times-Earliest Due Date (SETT-EDD)} \cite{Grilo2003} has proposed a novel scheduling technique for the so-called IEEE802.11e \gls{HCF}. A simple mechanism similar to the \gls{CAP} timer has been employed to limit the polling-based transmission in \gls{HCF} which so-called \gls{TXOP} timer. This \gls{TXOP} timer increases at a constant rate equal to the proportion of that \gls{TXOP} duration to the minimum service interval ($TD/mSI$), which reflects the fraction of time consumed by the station in polled TXOPs. The maximum value of this timer is equal to the maximum \gls{TXOP} duration ($MTD$). The consumed time by a station in a polled \gls{TXOP} is subtracted from the \gls{TXOP} timer by the end of the \gls{TXOP}. Thus, the station can be polled only if the \gls{TXOP} timer value is greater than or equal to the minimum \gls{TXOP} duration ($mTD$), which guarantees the transmission of at least one data frame at the minimum \gls{PHY} rate.
Since the \gls{TXOP} is allocated in SETT-EDD based on earliest deadlines, the transmission delay and data loss have been reduced. That is why SETT-EDD shows flexibility and considered a representative dynamic scheduler, as well as it provides compatibility to the link adaptation implemented in the commercial WLANs. However, it still lacks an efficient technique to be able to calculate the accurate required \gls{TXOP} for every \gls{QSTA} transmission instead of estimating \gls{TXOP} based on the average data rate of each \gls{TS} and the packet time interval between two consecutive transmissions.
\textbf{Adaptive Resource Reservation Over WLANs (ARROW)} is another algorithm where the \gls{TXOP} assignment is calculated dynamically based on the queued data size of the QSTAs \cite{Skyrianoglou2006,Passas2006}. In ARROW, the SETT-EDD \cite{Grilo2003} has been extended, where the available bandwidth is allocated based on the existing amount of data which is ready for transmission in every \gls{STA}. In contrast to SETT-EDD, which allocates the channel bandwidth based on the expected arriving data in every \gls{STA}. In this mechanism, \gls{QSTA} advertises the size of the total queued packets waiting for transmission with every poll. This information is piggybacked with the data frame prior the sending back to \gls{QAP}. So, the next \gls{TXOP} allocation for any particular stream will be calculated based on the advertised queue size.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{ARROW}
\caption{The ARROW Mechanism}
\label{fig:ARROW}
\end{figure}
In this algorithm, the channel is allocated based on the exact transmission requirements for each \gls{QSTA}, which is expressed by the \gls{QS} field indicated during the previous \gls{TXOP}. By doing so, the \gls{TXOP} is assigned to meet the transmission requirement at the time when the previous \gls{TXOP} assignment is made and consequently the data buffered in the \gls{QSTA} is taken into account at any \gls{SI} leading to efficient adaptation of bandwidth allocation to actual requirements. Specifically, as illustrated in Fig.~\ref{fig:ARROW}, data arrive during $[t_{i}(x), t_{i}(x + 1)]$ can only be transmitted after the elapsing of $t_{i}(x + 2)$, which results in a delay of packets for at least one \gls{SI}.
\textbf{Enhanced \gls{EDD} QoS scheduler:} presented by \cite{Lee2009} and it is an EDD-based algorithm mainly aims at addressing the above-mentioned weakness of the ARROW scheduler. Similar to ARROW scheduler, the Enhanced \gls{EDD} also uses the queue length information like ARROW. However, the Enhanced \gls{EDD} estimates the number of arriving packets immediately after the end of the previous transmission, as shown in Fig.~\ref{fig:EnhancedEDD}. Thereafter, it calculates just the enough \gls{TXOP} to clear up the buffer queue by the end of current transmission. To reduce the average delay, when the buffer is not empty after the current transmission completes, the next \gls{SI} begins earlier, which can be achieved by changing the value of mSI and \gls{MSI}.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{EnhancedEDD}
\caption{TXOP assignment in enhanced EDD}
\label{fig:EnhancedEDD}
\end{figure}
The \gls{TXOP} allocation in Enhanced \gls{EDD} is calculated for each station $STA_{i}$ as the summation \gls{TXOP} calculated exactly as in ARROW and a duration enough to transmit the packets generated during the current \gls{SI} as below
\begin{equation}
TXOP_{i}=TXOP_{avg}^i+TDr_{i}
\label{eq:EnhancedEDD1}
\end{equation}
where $TXOP_{avg}$ is calculated as follows
and $N_{curSI}$ is the expected number of packets generated from time $t_{pre}$ until $t'$.
\textbf{Dynamic \gls{TXOP} \gls{HCCA} \gls{DTH}} involves a bandwidth reclaiming mechanism into a centralized \gls{HCCA} scheduler in order to improve the transmission capacity and to provide additional resources to \gls{VBR} TSs \cite{cecchettielAL2012}. The main concept of \gls{DTH} is to prevent wasting the underutilized portion of transmission time in order to allocate it to the next polled station that needs longer transmission period. This approach relies on the unspent amount of the \gls{TXOP} from the previous poll time of a $QSTA_{i}$ as follows
\begin{equation}
TXOP_{i}= \left\{\begin{matrix}
TXOP_{AC(i)} & \text{if } T_{spare}\equiv 0 \\
t_{est(i)+T_{spare}}& \text{if } T_{spare} > 0
\end{matrix}\right.
\end{equation}
If there is no surplus \gls{TXOP} duration from previous poll time, which implies that the station exhausts the whole \gls{TXOP} duration. The next \gls{TXOP} duration will be the same as the one calculated in Equation~\eqref{eq:TXOP}. Otherwise, it will be calculated as the summation of the unused \gls{TXOP} duration and the estimated transmission time, computed through the \gls{SMA} of the effectively utilized duration in the previous polling intervals. Simulation results show that this approach can improve the performance, especially in terms of transmission queue size, data loss and delay, and the approach can absorb and follow the variation of VBR. Additionally, another analytical study confirms that the \gls{DTH} approach has no effect on the policy of the centralized scheduler. However, the estimation of transmission time using Moving Average needs more investigation as the \gls{VBR} traffic tends to high variability during the time, thus it might be not efficient to find the best setting of the mobile sampling windows.
\textbf{The Dynamic TXOP (D-TXOP) scheduling algorithm} \cite{Almaqri2013} analyzes the video of the prerecorded streams before the call setup, which has been previously highlighted in \cite{Haddad2012}. The D-TXOP is suitable for transmitting type (I) of VBR video source categorized in Subsection~\ref{sec:Adpt2Aplica}, which shows variability in packet size. Indeed, this approach assigns the \gls{TXOP} for a stream based on the real frame size rather than the estimated average of frame size. It uses the unused \gls{QS} field of IEEE802.11e \gls{MAC} header to send the actual size of the upcoming frame to the \gls{HC}. Thus, the wasted TXOPs have been minimized by this approach, which reflects lower delay compared to the previous solutions. Moreover, the EDCA benefits from the surplus TXOP duration from unused TXOP of the preceding STAs as illustrated in Fig.~\ref{dig:HCCAvsATAV}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{HCCAvsATAV}
\caption{Dynamic TXOP assignment scheduling algorithm.}
\label{dig:HCCAvsATAV}
\end{figure}
\subsection{\gls{HCCA} admission control enhancements}
\label{sec:ACUEnh}
The main purpose of \gls{HCF} admission control is to administer policy or regulate the available bandwidth resources which is used by the \gls{HC}. The admission control is used to limit the amount of traffic admitted under a certain service category in order to guarantee the highest possible QoS level, while maximizing the utilization of the medium resources. Fig.~\ref{fig:TSPECElements} depicts a common frame format for carrying \gls{TSPEC} parameters. Since the admission control relies on a fixed \gls{TSPEC} element, it cannot efficiently cope with the high variability of \gls{VBR} streams. To solve this problem, numerous enhancements and optimizations have been proposed to tackle this deficiency in the legacy \gls{ACU} mechanism.
\begin{figure}[!t]
\centering
\includegraphics[width=.9\linewidth]{TSPECElements}
\caption{\gls{TSPEC} element format}
\label{fig:TSPECElements}
\end{figure}
\textbf{Rate-Variance-envelop-based Admission Control (RVAC)} mechanism uses the \gls{DTB} shaper to guarantee the desired QoS specification \cite{gao2008}. The authors of these two references \cite{ knightly1997, knightly1998} have derived the delay probability based on the aggregate traffic statistics rather than considering each flow individually to accept a new flow for admission \cite{ fan2004, chou2005}. The effective \gls{TXOP} duration of a recently arrived \gls{VBR} stream can be inversely derived based on a given packet loss rate as in Equation~\eqref{eq:ACU}.
Indeed, the RVAC takes the multiplexing gain of \gls{VBR} traffic into account unlike the guarantee-rate-based scheme. More specifically, if the arrival time of data streams extends over a wide range, where the RVAC can fully utilize the multiplexing gain among the \gls{VBR} streams, the performance gain can be noticeable. Additionally, the RVAC considers both uplink and downlink traffic streams. Simulation results have shown that the admission capacity of the RVAC approach is more than the double of its equivalent in the GRAC approach. In addition, the RVAC scheme will not violate the 0.1 second delay requirement as long as the starting time of the streams are spread over a wide time range of not less than 2 seconds. However, the performance of the RVAC in the wireless channel errors environment is not studied.
\textbf{Equal-SP} \cite{zhao2008} has been designed of \gls{HCCA} scheduling, in which the spacing of a particular stream is determined as the period of time between two consecutively scheduled streams. It has been called equal-SP scheduling because a particular stream will always get an equal spacing for its scheduling slices in the schedule. Indeed, the equal-SP scheduling relies on the well-known \gls{RM} algorithm to achieve the QoS requirements. Despite that the equal-SP approach is similar to the SETT-EDD in terms of the general scheduling concept, however, the former assigns equal spacing for each particular stream, which is proven to violate the delay requirement in some cases.
In the example as shown in Fig.~\ref{fig:EqualSP}, the scheduler assigned 25 ms, 50 ms, and 150 ms time spicing for the flows 1, 2, and 3, respectively, which makes $T_{11}=T_{12}=T_{1}=$25 ms, $T_{2}$=50 ms, and $T_{3}=$150 ms. The equal-SP approach is easy to be implemented and it can guarantee the delay requirements and efficiently utilize the bandwidth while maintaining the compatibility to the standard since it uses the same \gls{TSPEC} parameters. However, the equal-SP approach encounters the same issues faced by the standard; if a newly admitted stream has a smaller delay bound than the current $T_{1}$, the current $T_{1}$ will be set to less than or equal to the new delay bound. Therefore, the \gls{TXOP} durations for the previously admitted flows are required to be recalculated with the $T_{i}$s. Additionally, the scheduler needs to reassign indexes to the admitted flows in order to maintain the condition of \mbox{$T_{1} \leq T_{2} \leq \cdots \leq T_n$}.
\begin{figure}[h!]
\centering
\includegraphics[width=.8\linewidth]{EqualSP}
\caption{An Example of the Equal-SP scheduling. The QoS is guaranteed by applying admission control}
\label{fig:EqualSP}
\end{figure}
\textbf{PHCCA}, as described in Fig.~\ref{fig:PHCCA}, is a priority based QoS and admission control used for queue management mechanism \cite{hantrakoon2010}. In this approach, a mechanism for borrowing and returning bandwidth among queues has been studied. The higher priority queue, called class, has permission to borrow bandwidth from lower priority queues with the awareness of starvation protection for each priority queue. PHCCA modifies the \gls{HCCA} by classifying the traffic into 3 classes, which has not been divided by the standard. Class 1 is the highest priority class suitable for voice and conference traffic implementation. Class 2 is the second highest priority class suitable for broadcast video traffic. Class 3 is the lowest priority traffic suitable for FTP and HTTP traffic.
\begin{figure}[h!]
\centering
\includegraphics[width=.8\linewidth]{PHCCA}
\caption{PHCCA admission control mechanism}
\label{fig:PHCCA}
\end{figure}
Experimental results reveal that the proposed PHCCA can accept more requests for Class 1 (70\% better) compared to the regular \gls{HCCA}. Meanwhile, the second highest priority (Class 2 in this case) is still served quite similar to the regular \gls{HCCA}. Despite that the PHCCA significantly outperforms the regular \gls{HCCA}, it is still not able to guarantee the required QoS for every flow; since it assumes that flows of similar types (e.g. VoIP) will have exact QoS requirements. Besides, the performance of NPHCCA could merely achieve the performance level of Best-Effort for VoIP and CBR-video applications. Moreover, parameters and environment factors for bandwidth borrowing mechanism, such as distance from the access point or mobility, should be investigated.
\textbf{AF-HCCA} \cite{almaqri2017a} enhances the experienced delay of video traffics by utilizing the surplus bandwidth and mitigates the over-polling issue. It computes the TXOP for a traffic stream based on the knowledge about the actual upcoming frame size instead of assigning TXOP according to the mean characteristics of the traffic which is unable to reflect the actual traffic. This scheduler exploits the queue size field of IEEE802.11e MAC header to transfer this information to the HC.
In AF-HCCA, the QSTAs will be prevented from receiving unnecessary large TXOP which produces a remarkable increase in the packet delay. Furthermore, the surplus time of the wireless channel conserved by reducing the number of poll frames throughout the feedback is another benefit of this research.The integrated scheme of AF-HCCA shows superior performance compared to IEEE802.11e HCCA, Enhanced EDD \cite{Lee2009} and F-Poll \cite{maqri2015} schedulers in terms of delay and channel utilization without affecting the system throughput. However, preserved TXOP time is not efficiently utilized to enhance the flow capacity.
\textbf{Feedback-based Admission Control Unit (FACU)} \cite{almaqri2017} aims at maximizing the utilization of the surplus bandwidth which has never been tested in previous schemes. The FACU exploits piggybacked information containing the size of the subsequent video frames to increase the number of admitted flows.
The FACU introduces an enhancement to admission control mechanism of Adaptive-TXOP. Analytical results reveal the efficiency of FACU over the examined schemes. The results show that the conserved channel bandwidth of Adaptive-TXOP can be utilized to increase the number of admitted QoS flows and enhance the overall QoS provisioning in IEEE802.11e WLANs.
\section{\gls{HCCA} scheduling approaches comparison}
\label{sec:comparison}
Table~\ref{tab:comparison} presents a summary and comparison for the \gls{HCCA} enhancements in IEEE802.11e along with their targeted features classified based on the place of the enhancement. The solution column briefly describes the used technique. The complexity of an approach can be high, medium or simple estimated based on Likert-type rating scale. The complexity here represents the volume of the operations of that particular approach. The method that involves more operations is considered high-complex and vice versa. The main targeted traffic of the enhancement is stated. The targeted flow direction, which is considered by the approach, is also presented.
\begin{table*}[h!] \scriptsize
\begin{center}
\caption{Comparison of The main characteristics of the \gls{HCCA} approaches}
\label{tab:comparison}
\begin{tabular}{m{2cm} m{2.5cm} m{3.5cm} m{1cm} m{1.5cm} m{1.6cm} m{2.5cm}} \hline
Strategy & Approach & Solution & Complexity Level$^*$ & Targeted & Flow Direction & Main QoS challenge\\\hline
& CP-Multipoll \cite{Lo2003} & Multipolling scheme & High & Voice/video & Uplink/downlink & Packet delay\\
HCCA polling & CF-Poll \cite{lee2006} & Piggyback & Medium & Voice/video & Uplink & Flows capacity\\
mechanisms & DEB method for HCCA \cite{Huang2010} & Deterministic polling & Simple & CBR/VBR video & Uplink & Flows capacity\\
& NPHCCA \cite{Chen2011} & Non polling feedback-based & Simple & Voice/video & Uplink & Packet delay\\
& F-Poll \cite{maqri2015} & Feasible Polling Scheme & Simple & Video & Uplink & Packet delay and flow capacity\\ \hline
& SETT-EDD \cite{Grilo2003} & Token bucket and Earliest Due Date based & Medium & Voice and video & Uplink/downlink & Packet delay\\
& AMTXOP \cite{almaqri2016} & Adaptive Multipolling TXOP Scheme & Simple & Video & Uplink & Packet delay, flow capacity\\
TXOP allocation & ARROW \cite{Skyrianoglou2006} & Feedback based & Simple & Voice and video & Uplink/downlink & Flow capacity\\
mechanism & Enhanced ED \cite{Lee2009} & Estimation and feedback based & Medium & Voice and video & Uplink & Packet delay\\
& Dynamic TXOP HCCA \cite{cecchettielAL2012} & Bandwidth reclaiming mechanism & High & Voice and video & Uplink & Packet delay, flow capacity\\
& D-TXOP \cite{Almaqri2013} & The Dynamic TXOP Scheduling algorithm & Simple & VBR traffic & Uplink & Packet delay\\
\hline
& RVAC \cite{gao2008} & Dual token bucket (DTB) shaper & Medium & VBR traffic & Uplink/downlink & Flows capacity\\
HCCA admission control & Equal-SP \cite{zhao2008} & Equal spacing scheduling & Simple & Voice and video & Uplink & Flows capacity\\
& PHCCA \cite{hantrakoon2010} & Priority based & Simple & Voice, video & Uplink/downlink & Flows capacity\\
& AF-HCCA \cite{almaqri2017a} & Adaptive Feedback-based HCCA & Simple & Video & Uplink & Packet delay \\
& FACU \cite{almaqri2017} & Feedback-based Admission Control Unit & Simple & Video & Uplink & Packet delay, flow capacity \\
\hline
\end{tabular}
\begin{flushleft}
\hspace{0.2cm} {\tiny $^*$Note that the complexity level reflects the volume of the operations as explained in Section \ref{sec:comparison} based on Likert-type rating scale}
\end{flushleft}
\end{center}
\end{table*}
\section{Open research issues}
\label{sec:ORI}
Although the existing approaches provide several possible solutions to alleviate the deficiency of scheduling for \gls{VBR} multimedia traffic in IEEE802.11e WLANs, many issues have been thoroughly discussed in the literature review section, which are potential research topics. This section highlights the most important issues in order to determine the directions for potential future research. One of the problems with \gls{HCCA} is the coexistence. Several mechanisms claim to be able to coordinate different HCs that operate on the same frequency channel. Since HCCA's QoS guarantee depends on the exclusive usage of the frequency channel, multiple \gls{HCCA} can hardly coexist. On the other hand, additional delay may occur by the polling STAs with scalable video that exhibits constant quality yet introduce high variation in the traffic profile. From the cross-layer perspective and to the best of our knowledge, there is no proactive scheme that provides a good solution to the adequate interaction between the fluctuation of the uplink \gls{VBR} traffic profile at the application layer and the flexible scheduling policy at \gls{MAC} layer which exhibit low-complexity design.
In summary, some issues are needed to be considered to provide optimal enhancement for the transmission of \gls{VBR} traffic in IEEE802.11e WLANs. We believe the following suggestions are desirable for designing a good \gls{HCCA} scheme in IEEE802.11e wireless networks.
\begin{itemize}
\item Providing efficient estimation of the bandwidth in order to achieve high connection throughput.
\item Designing a scheme coupled with link adaptation mechanisms in order to provide efficient adaptation to dynamic network behavior.
\item Exploiting the distributed feature of \gls{EDCA} in designing a hybrid \gls{HCCA} scheme in order to yield high integration and interoperability without jeopardizing the system simplicity.
\item Enabling the fragmentation and the block acknowledgment introduced in the standard \cite{IEEEStandard2012} with \gls{HCCA} scheme.
\item Achieving low algorithm complexity.
\end{itemize}
\section{Research trend on QoS support in IEEE802.11e}
Many researches have been conducted in the Literature since the first advent of the HCCA protocol draft in IEEE802.11e standard \cite{IEEE80211eDraft}. These researches can be classified into five research areas as in Table~\ref{tab:ResearchTrend} aims at demonstrating the trend of the research since the first presence of the HCF functions till 2009 and from 2010 to present. The collection includes over 89 journal and conference papers. These scientific documents have been collected using IEEE Explorer Digital Library, Springer Link, ScienceDirect and Google Scholar. One can notice that the polling and TXOP allocation mechanisms have greatly received the researchers' attention since the evolution of the HCCA till now, while admission control mechanisms have less interest. It is worth noting that the design of the hybrid EDCA-HCCA scheme has scarcely studied. The HCCA performance and mathematical analysis have been fairly covered. Yet, only few efforts have focused on designing a comprehensive analytical model for HCCA protocol. The aggregated number of papers published in three periods, namely 2004 to 2007; 2008 to 2011 and 2012 to present are depicted in Fig.~\ref{fig:ResearchTrends}. The figure shows that the polling and TXOP mechanism have received a great amount of attention compared to ACU and hybrid scheme. On the other hand, recently there has been a few analysis of HCCA protocol, in contrast to the period from 2004 up to 2011.
\begin{center}
\begin{table*}[!t]
\centering
\caption {Researches in QoS provisioning of Multimedia traffic in IEEE802.11e }
\centering
\begin{tabular}{l|l|l
\hline
Area & Published from 2004 to 2009 & Published since 2010\\ \hline
HCCA Polling &
\cite{kim2004,Son2004,Xiyan2004,Chen2004,ByungSeoKim2005,lee2006,Ramos2006,Park2007,Chen2008} & \cite{Huang2010,He2011,Chen2011,Chou2011,Viegas2012,zhang2013,Li2013,ZiTsan2014,maqri2015,Zhang2015} \\ \hline
TXOP Allocation & \cite{Ansel2004,Inan2006,Kim2006,Yamane2006,Skyrianoglou2006,Ansel2006,Choi2007,Qinglin2007,Rashid2007,Kyung2007}, & \cite{Byung2010,noh2010,arora2010,ruscelli2011,luo2011cross,Jansang2011,Yong2011,noh2011,cecchettielAL2012,Cecchetti2012} \\
& \cite{Floros2008,rashid2008,Hsieh2008,Huang2008,Chu2008} & \cite{ju2013,lee2013,almaqri2016} \cite{almaqri2017a}\\ \hline
Admission Control &
\cite{fan2004,Deyun2005,Deyun2005a,Kim2006,vander2006,Jang2006,Cecchetti2007,gao2008,Qinglin2008,Zeng2008} & \cite{lee2010,Didi2010,hantrakoon2010,palirts2010,Chie2011,lee2013,Huang2015}, \cite{almaqri2017} \\ \hline
Hybrid EDCA-HCCA &
\cite{xiao2004,Fallah2007,Fallah2008,Zhu2008109} &
\cite{Siddique2010,ruscelli2012,ng2013} \\ \hline
HCCA Analysis & \cite{Boggia2005325,Qiang2005per,Tresk2006,Harsha2006,Karanam2006,rashid2006,Siris2006,binMuhamad2007,rashid2008,Cecchetti2008}, & \cite{Perez2010,Minseok2010,Ghazizadeh2010,lyakhov2011,Pastrav2012,Lagkas2013,leonovich2013,Jansang2013}\\
& \cite{Ghazizadeh2008,Lin2008,Jansang2009}& \\ \hline
\end{tabular}
\label{tab:ResearchTrend}
\end{table*}
\end{center}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{ResearchTrends}
\caption{Number of publication of the investigated research areas}
\label{fig:ResearchTrends}
\end{figure}
\section{Future directions}
\label{sec:futureWork}
Although all proposed schemes in their current states improve the transmission of prerecorded video, there still some issues need to be addressed and investigated. Below, we highlight some future works that need to be carried out for further enhancement to:
\begin{itemize}
\item Enhance the HCCA to cope with more complicated wireless scenarios, where the hidden node problem exists and QSTAs communicate using RTS/CTS mechanism with \gls{MAC} level fragmentation and multi-rate support enabled.
\item Study the scalable \gls{HCCA} \gls{MAC} for video over wireless mesh networks that are also scalable to a wider range of \gls{MAC} settings to support more robust time-bounded media applications.
\item Design a new admission control algorithm to utilize the excess bandwidth and to manage the available resources among the \gls{HCCA}, \gls{HCF} and \gls{EDCA} in order to maximize the number of served streams or applications in the network.
\item Examine the performance of HCCA with the presence of collision occurred in the overlapping area when polling stations among two neighboring \glspl{BSS} simultaneously.
\end{itemize}
\section{Conclusion}
\label{sec:conclusion}
IEEE802.11e is aimed at providing stringent QoS support to multimedia applications such as video streaming. The controlled based function of IEEE802.11e standard, \gls{HCCA} scheduler, consider a fixed \gls{TSPEC} for scheduling the traffic while in fact the \gls{VBR} traffic tends to change their characteristics such as data rate and packet size over the time. The inability of the IEEE802.11e \gls{MAC} protocol to accommodate to the high fluctuation of VBR video profile motivates many researches to be conducted. Several enhancements have been made to alleviate these shortcomings. These enhancements tend to address particular issues or applications by improving, in most cases, one of the \gls{HCF} functions. In general, designing a robust \gls{HCF} solution that provides an integrated solution for all traffic classes is still a challenging task for future research.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,684
|
\section{ Introduction}
An on-shell quantization of general gauge theories which are reductible
and/or whose classical gauge algebra is not closed (for a review see e.g.
Ref. \cite{Henneaux}), can be successfully performed by the
Batalin-Vilkovisky (BV) formalism \cite{batalin22}. The BV formalism is a
very general covariant Lagrangian approach which overcomes the need of
closed classical algebra by a suitable construction of BRST operator. The
construction is realized by introducing a set of new antifields besides the
fields occurring in the theory. The elimination of these antifields at the
quantum level via a gauge-fixing procedure leads to the quantum theory in
which effective BRST transformations\ are nilpotent only on-shell.
Another possibility for quantizing reductible and/or open gauge theories
consists to introduce a set of auxiliary fields\ as in supersymmetric
theories, such as the Wess-Zumino model for which it is only with auxiliary
fields that one can obtain a tensor calculus \cite{Van-Nieuwen}, or in
topological antisymmetric tensor gauge theories, the so-called BF theories
\cite{tahiri2}. The introduction of auxiliary fields are needed to close the
gauge algebra, and then provides an efficient way to use the usual
Faddeev-Popov quantization method \cite{becchi}.
On the other hand, it is well known that in the superspace formalism one
can\ naturally introduce a set of auxiliary fields that gives rise to the
construction of the off-shell BRST invariant quantum action. Indeed, as
shown in Ref. \cite{tahiri2} for the case of $4D$ non-Abelian BF theory and
Ref. \cite{tahiri22} for the case of the simple supergravity where the
classical gauge algebra is open \cite{Van-Nieuwen}, the superspace formalism
has been used in order to realize the quantization of such theories. It
leads to introduce the minimal set of auxiliary fields ensuring the
off-shell invariance of the quantum action.
In the context of superspace formalism, the gauge field, the ghost and
anti-ghost fields in gauge theories can be incorporated into a natural gauge
superconnection by extending spacetime to a $(4,2)-$dimensional superspace
\cite{Bonora}. In this framework, the BRST and anti-BRST transformations can
be derived by imposing horizontality conditions on the supercurvature
associated to a superconnection on a superspace.
Let us mention that in the case of supersymmetric theories the corresponding
supersymmetric algebra close only modulo equations of motion \cite%
{Sohnius,Wess}. So, the appropriate framework to quantize such theories is
the BV\ one. Indeed, in Ref. \cite{Baulieu}\ it has been discussed how to
realize the quantization of supersymmetric systems in BV approach.
As shown in \cite{Lavrov1,Lavrov2}, a superfield description of the BV
quantization method can be realized by simply introducing superfields whose
lowest components coincide with the usual fields in the BV formalism.
Superfield method has provided a powerful tool for producing supersymmetric
field equations for any degree of supersymmetry. In \cite{Sohnius,WEinberg},
one has also established an off-shell superfield formulation of four
dimensional $\mathcal{N}=1$ supersymmetric Yang-Mills theory. Considerably
more involved off-shell superfield formulations are also available for $%
\mathcal{N}=2$ in terms of harmonic and analytic superspace \cite{Galperin},
while the off-shell formulation of $\mathcal{N}=4$ supersymmetric Yang-Mills
theory with non-Abelian gauge group $SU(N)$ is not available in terms of
unconstrained fields because it would require the introduction of an
infinite set of auxiliary fields \cite{Falk}.
The purpose of the present paper is to derive, in the framework of BRST
superspace\footnote{%
Here, we call the superspace obtained by enlarging spacetime with two
ordinary anticommuting coordinates BRST superspace, in order to distinguish
it from superspace of supersymmetric theories.}, the off-shell nilpotent
version of the BRST and anti-BRST transformations for $\mathcal{N}=1,$\ $D=4$
supersymmetric Yang-Mills theory $(SYM_{4})$, in analogy to what is realized
for the case of\ simple supergravity \cite{tahiri2} and the four-dimensional
non-Abelian BF theory \cite{tahiri22}. The Action for the $\mathcal{N}=1$\
supersymmetric Yang-Mills in four dimensions is given by
\begin{equation}
S_{0}=\int dx^{4}\left( \frac{-1}{2g^{2}}trF_{\mu \nu }F^{\mu \nu }+\frac{%
\theta }{8\pi ^{2}}trF_{\mu \nu }\widetilde{F}^{\mu \nu }-\frac{i}{2}tr%
\overline{\lambda }\sigma ^{\mu }D_{\mu }\lambda \right) ,
\label{Lagrangian}
\end{equation}%
where $(A_{\mu },\lambda )$ is the gauge multiplet, $g$ is the gauge
coupling, $\theta $ is the instanton angle, the field strength is $F_{\mu
\nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }+\left[ A_{\mu
},A_{\nu }\right] ,$ $\widetilde{F}_{\mu \nu }=\frac{1}{2}\epsilon _{\mu \nu
\rho \sigma }F^{\rho \sigma }$ is the dual of $F,$ and $D_{\mu }=\partial
_{\mu }+\left[ A_{\mu },\right] $. By construction, the Action (\ref%
{Lagrangian}) is invariant under the supersymmetry transformations
\begin{eqnarray}
\delta _{\xi }A_{\mu } &=&i\overline{\xi }\overline{\sigma }_{\mu }\lambda -i%
\overline{\lambda }\overline{\sigma }_{\mu }\xi , \notag \\
\delta _{\xi }\lambda &=&\sigma ^{\mu \nu }F_{\mu \nu }\xi ,
\label{Supersy transf}
\end{eqnarray}%
where $\xi $ is a spin $1/2$ valued infinitesimal supersymmetry parameter.
Our paper is organized as follows. In Section 2, the BRST superspace
approach and horizontality conditions for $\mathcal{N}=1,$\ $D=4$
supersymmetric Yang-Mills theory are discussed. In Section 3, we introduce\
the various fields and derive the off-shell nilpotent\ BRST and anti-BRST
transformations. The construction\ of the BRST-invariant quantum action for $%
\mathcal{N}=1$ super Yang-Mills theory in terms of this off-shell structure
is described in Section 4. The obtained quantum action permits us to see
that the extrafields are nondynamical, auxiliary fields. Section 5 is
devoted to concluding remarks.
\section{BRST superspace}
Let $\Phi $\ be a super Yang-Mills connection on a $(4,2)-$dimensional BRST
superspace with coordinates $z=(z^{M})=(x^{\mu },\theta ^{\alpha }),$\ where
$(x^{\mu })_{\mu =1,..,4}$\ are the coordinates of the spacetime manifolds
and $(\theta ^{\alpha })_{\alpha =1,2}$\ are ordinary anticommuting
variables. Acting the exterior covariant superdifferential $D$ on $\Phi $ we
obtain the supercurvature $\Omega $ satisfying the structure equation, $%
\Omega =d\Phi +\frac{1}{2}\left[ \Phi ,\Phi \right] ,$ and the Bianchi
identity, $d\Omega +\left[ \Phi ,\Omega \right] =0.$ The superconnection $%
\Phi $\ as 1-superform on the BRST\ superspace can be written as
\begin{equation}
\Phi =dz^{M}(\Phi _{M}^{i}I_{i}+\Phi _{M}^{\mu }P_{\mu }+\Phi _{M}^{\alpha
}Q_{\alpha }), \label{eq1a}
\end{equation}%
where $\{I_{i}\}_{i=1,...,d},$ and $\{P_{\mu },Q_{\alpha }\}_{\mu
=1,...,4;\alpha =1,...,4}$ are the generators of the internal symmetry group
$(G)$ and the $\mathcal{N=}1$ supersymmetric group $(SG)$ respectively. They
satisfy the following commutation relations
\begin{eqnarray}
\left[ I_{i},I_{j}\right] &=&f_{ij}^{k}I_{k}, \notag \\
\left[ I_{i},P_{\mu }\right] &=&\left[ Q_{\alpha },P_{\mu }\right] =\left[
P_{\nu },P_{\mu }\right] =0, \notag \\
\left[ Q_{\alpha },Q_{\beta }\right] &=&2(\gamma ^{\mu })_{\alpha \beta
}P_{\mu }, \notag \\
\left[ I_{i},Q_{\alpha }\right] &=&b_{i}^{\ast }Q_{\alpha }, \label{eq2}
\end{eqnarray}%
where $\{\gamma ^{\mu }\}_{\mu =1,..,4}$ are the Dirac matrices in the Weyl
basis, $b_{i}^{\ast }=b_{i}$ for $\alpha =1,2$ and $b_{i}^{\ast }=-b_{i}$
for $\alpha =3,4$ giving the representation of the internal symmetry of $%
Q_{\alpha }$ and $\left[ ,\right] $ the graded Lie bracket. Let us mention
that the supersymmetric generators $\{Q_{\alpha }\}$ are given in the
Majorana representations \cite{Wess}. Note that the Grassmann degrees of the
superfield components of $\Phi $ are\ given by $\mid \Phi _{M}^{i}\mid =\mid
\Phi _{M}^{\mu }\mid =m,$ $\mid \Phi _{M}^{\alpha }\mid =m+1$ $(\func{mod}2$
), where $m=\mid z^{M}\mid (m=0$ for $M$=$\mu $ and $m=1$ for $M=\alpha ),$
since $\Phi $ is an even 1-superform.
However, we assign to the anticommuting coordinates $\theta ^{1}$ and $%
\theta ^{2}$ the ghost numbers $(-1)$and $(+1)$ respectively, and ghost
number zero for an even quantity:\ either a coordinate, a superform or a
generator. These rules permit us to determine the ghost numbers of the
superfields $(\Phi _{\mu }^{i},$ $\Phi _{\mu }^{\nu },$ $\Phi _{\mu
}^{\alpha },$ $\Phi _{1}^{i},$ $\Phi _{2}^{i},$ $\Phi _{1}^{\nu },$ $\Phi
_{2}^{\nu },$ $\Phi _{1}^{\alpha },$ $\Phi _{2}^{\alpha })$ which are given
by $(0,$ $0,$ $0,$ $+1,$ $-1,$ $+1,$ $-1,$ $+1,$ $-1).$
Upon expressing the supercurvature $\Omega $ as
\begin{equation}
\Omega =\frac{1}{2}dz^{N}\wedge dz^{M}\Omega _{MN}=\frac{1}{2}dz^{N}\wedge
dz^{M}(\Omega _{MN}^{i}I_{i}+\Omega _{MN}^{\mu }P_{\mu }+\Omega
_{MN}^{\alpha }Q_{\alpha }), \label{eq4}
\end{equation}%
we find\ from the structure equation
\begin{subequations}
\begin{eqnarray}
\Omega _{\mu \nu } &=&\partial _{\mu }\Phi _{\nu }-\partial _{\nu }\Phi
_{\mu }+\left[ \Phi _{\mu },\Phi _{\nu }\right] , \label{Eq5a} \\
\Omega _{\mu \alpha } &=&\partial _{\mu }\Phi _{\alpha }-\partial _{\alpha
}\Phi _{\mu }+\left[ \Phi _{\mu },\Phi _{\alpha }\right] , \\
\Omega _{\alpha \beta } &=&\partial _{\alpha }\Phi _{\beta }+\partial
_{\beta }\Phi _{\alpha }+\left[ \Phi _{\alpha },\Phi _{\beta }\right] .
\end{eqnarray}%
Similarly, the Bianchi identity becomes
\end{subequations}
\begin{subequations}
\label{bianchi_4a}
\begin{gather}
D_{\mu }\Omega _{\nu \kappa }+D_{\kappa }\Omega _{\mu \nu }+D_{\nu }\Omega
_{\kappa \mu }=0, \label{Eq bian} \\
D_{\alpha }\Omega _{\mu \nu }-D_{\nu }\Omega _{\mu \alpha }+D_{\mu }\Omega
_{\nu \alpha }=0, \\
D_{\alpha }\Omega _{\beta \gamma }+D_{\beta }\Omega _{\alpha \gamma
}+D_{\gamma }\Omega _{\alpha \beta }=0, \\
D_{\mu }\Omega _{\alpha \beta }-D_{\alpha }\Omega _{\mu \beta }-D_{\beta
}\Omega _{\mu \alpha }=0,
\end{gather}%
where $D_{M}=\partial _{M}+\left[ \Phi _{M},.\right] $ is the $M$ covariant
superderivative. Now, we shall search for the constraints to the
supercurvature $\Omega $ in which the consistency with the Bianchi
identities $(7)$ is ensured. This requirement insures then the off-shell
nilpotency of the BRST and anti-BRST algebra. The full set\ of
supercurvature constraints turns out to be given by
\end{subequations}
\begin{equation}
\Omega _{\mu \alpha }=0,\text{ \ \ \ \ \ \ \ }\Omega _{\alpha \beta }=0.
\label{contraintes}
\end{equation}%
It is easy to check the consistency of this set of supercurvature
constraints through an analysis of the Bianchi identities. Indeed, we remark
that identities $(7c)$ and $(7d)$ are automatically satisfied because of the
constraints (\ref{contraintes}) while identities $(7a)$ and $(7b)$ yield a
further restriction on supercurvature $\Omega $
\begin{eqnarray}
\Omega _{\mu \nu \mid }^{i} &=&F_{\mu \nu }^{i}=\partial _{\mu }A_{\nu
}^{i}-\partial _{\nu }A_{\mu }^{i}+\left[ A_{\mu },A_{\nu }\right] ^{i},
\notag \\
\Omega _{\mu \nu }^{\kappa } &=&0,\text{ \ \ \ \ \ \ \ \ \ }\Omega _{\mu \nu
}^{\alpha }=0. \label{CONTRAINT}
\end{eqnarray}%
At this point, let us mention that the consistency of the horizontability
conditions (\ref{contraintes}) and (\ref{CONTRAINT}) with the Bianchi
identities $(7),$ as we will see later, guarantees automatically the
off-shell nilpotency of the BRST and anti-BRST transformations on all the
fields belonging to $\mathcal{N}=1$ super Yang-Mills theory.
\section{Auxiliary fields}
In order\ to derive the off-shell BRST structure of$\ \mathcal{N}=1$ super
Yang-Mills theory using the above BRST superspace formalism, it is necessary
to give the geometrical interpretation of the fields occurring in the
quantization of such theory. Besides the gauge potential $\Phi _{\mu \mid
}^{i}=A_{\mu }^{i},$ there exists the superpartener $\Phi _{\alpha \mid
}^{i}=\lambda _{\alpha }^{i}$ of $A_{\mu }$ which is introduced via the
field redefinition $\Phi ^{\rho i}=-(\gamma ^{\mu })^{\rho \sigma }/4\left[
Q_{\sigma },\Phi _{\mu }^{i}\right] ,$ $\Phi _{\mu \mid }^{\nu }=0$
representing the fact\ that the gauge potential associated to the
translation will not exist in the super Yang-Mills theory and an auxiliary%
\footnote{%
By definition an auxiliary field does not describe an independent degree of
freedom; its equation of motion is algebraic.} real field $\Phi _{\mid
}^{i}=\Lambda ^{i}$ which is required for the construction of off-shell
nilpotent BRST and anti-BRST transformations and introduced via the
supersymmetric transformations of the superpartener of the gauge potential
\begin{equation}
\Phi ^{i}=\delta _{\sigma }^{\rho }/4\left[ Q_{\rho },\Phi ^{\sigma i}\right]
. \label{auxiliary}
\end{equation}%
Furthermore, we introduce the following now: $\Phi _{1\mid }^{i}=c_{1}^{i}$
is the ghost for Yang-Mills symmetry, $\Phi _{2\mid }^{i}=\overline{c}%
_{2}^{i}$ is\ the antighost of $c_{1}^{i},$ $B^{i}=\partial _{1}\Phi _{2\mid
}^{i}$ is the associated auxiliary field, $\Phi _{1\mid }^{\alpha }=\chi
_{1}^{\alpha }$ is the supersymmetric ghost, $\Phi _{2\mid }^{\alpha }=%
\overline{\chi }_{2}^{\alpha },$ is\ the antighost of $\chi _{1}^{\alpha },$
$G^{\alpha }=\partial _{1}\Phi _{2\mid }^{\alpha }$ is the associated
auxiliary field$,$ $\Phi _{1\mid }^{\mu }=\xi _{1}^{\mu }$ is the
translation symmetry ghost, $\Phi _{2\mid }^{\mu }=\overline{\xi _{2}^{\mu }}%
,$ is\ the antighost of $\xi _{1}^{\mu }$ and $E^{\mu }=\partial _{1}\Phi
_{2\mid }^{\mu }$ is the associated auxiliary field$.$ Let us mention that
the symmetry ghosts antighosts $\chi _{\rho }^{\alpha }$ are commuting
fields while the others $c_{\rho }^{i}$ and $\xi _{\rho }^{\mu }$ are
anticommuting. Note that the symbol $``\mid "$\ indicates that the
superfield is evaluated at $\theta ^{\alpha }=0.$ We also realize the usual
identifications: $Q_{\alpha }(X_{\mid })=\partial _{\alpha }X_{\mid },$
where $X$ is any superfield $Q=Q_{1}$ and $(\overline{Q}=Q_{2})$ is the BRST
(anti-BRST) operator.
The action of the $\mathcal{N=}1$ supersymmetric generators $\{P_{\mu
},Q_{\alpha }\}_{\mu =1,...,4;\alpha =1,...,4}$ on these fields is given by
\begin{eqnarray*}
\left[ P_{\mu },X\right] &=&\partial _{\mu }X, \\
\left[ Q_{\alpha },A_{\mu }^{i}\right] &=&-(\gamma _{\mu })_{\alpha \beta
}\lambda ^{\beta i}, \\
\left[ Q_{\alpha },\lambda ^{\beta i}\right] &=&-\frac{1}{2}(\sigma ^{\mu
\nu })_{\alpha }^{\beta }F_{\mu \nu }^{i}+\delta _{\alpha }^{\beta }\Lambda
^{i},
\end{eqnarray*}
\begin{eqnarray}
\left[ Q_{\alpha },c_{\rho }^{i}\right] &=&2(\gamma ^{\mu })_{\alpha \beta
}\chi _{\rho }^{\beta }A_{\mu }^{i}, \notag \\
\left[ Q_{\alpha },\Lambda ^{i}\right] &=&\gamma ^{\mu }D_{\mu }\lambda
_{\alpha }^{i}, \notag \\
\left[ Q_{\alpha },F_{\mu \nu }^{i}\right] &=&(\gamma _{\mu })_{\alpha \beta
}(D_{\nu }\lambda )^{\beta i}-(\gamma _{\nu })_{\alpha \beta }(D_{\mu
}\lambda )^{\beta i}, \label{eqalgeb1}
\end{eqnarray}%
where $D_{\mu }=\partial _{\mu }+\left[ A_{\mu },.\right] $ and $X$ any
fields.
It is worthwhile to mention that we are interested in our present
investigation on the global supersymmetric transformations, so that the
parameters of the $\mathcal{N}=1$ supersymmetric and translation groups must
be space-time constant, i.e.
\begin{eqnarray}
\partial _{\mu }\chi _{\alpha }^{\rho } &=&0, \notag \\
\partial _{\mu }\xi _{\alpha }^{\nu } &=&0. \label{const1}
\end{eqnarray}
Using the above identifications with (\ref{const1}) and inserting the
constraints (\ref{contraintes}) and (\ref{CONTRAINT}) into the Eqs. $(6)$
and $(7b),$ we obtain
\begin{eqnarray}
\partial _{\alpha }\Phi _{\mu \mid }^{i} &=&(D_{\mu }c_{\alpha })^{i}-\xi
_{\alpha }^{\nu }\left[ P_{\nu },A_{\mu }^{i}\right] -\chi _{\alpha }^{\rho }%
\left[ Q_{\rho },A_{\mu }^{i}\right] , \notag \\
\partial _{\alpha }\Phi _{\beta \mid }^{i}+\partial _{\beta }\Phi _{\alpha
\mid }^{i} &=&-\left[ c_{\alpha },c_{\beta }\right] ^{i}-\xi _{\alpha }^{\nu
}\left[ P_{\nu },c_{\beta }^{i}\right] -\chi _{\alpha }^{\rho }\left[
Q_{\rho },c_{\beta }^{i}\right] \notag \\
&&-\xi _{\beta }^{\nu }\left[ P_{\nu },c_{\alpha }^{i}\right] -\chi _{\beta
}^{\rho }\left[ Q_{\rho },c_{\alpha }^{i}\right] , \notag \\
\partial _{\alpha }\Phi _{\beta \mid }^{\gamma }+\partial _{\beta }\Phi
_{\alpha \mid }^{\gamma } &=&0, \notag \\
\partial _{\alpha }\Phi _{\beta \mid }^{\nu }+\partial _{\beta }\Phi
_{\alpha \mid }^{\nu } &=&-2\chi _{\alpha }^{\rho }(\gamma ^{\nu })_{\rho
\sigma }\chi _{\beta }^{\sigma }, \notag \\
\partial _{\alpha }\Phi _{\mid }^{\kappa i} &=&-\left[ \lambda ^{\kappa
},c_{\alpha }\right] ^{i}-\xi _{\alpha }^{\nu }\left[ P_{\nu },\lambda
^{\kappa i}\right] -\chi _{\alpha }^{\rho }\left[ Q_{\rho },\lambda ^{\kappa
i}\right] +\chi _{\alpha }^{\kappa }\Lambda ^{i}, \notag \\
\partial _{\alpha }\Omega _{\mu \nu \mid }^{i} &=&-\left[ c_{\alpha ,}F_{\mu
\nu }^{i}\right] -\xi _{\alpha }^{\tau }\left[ P_{\tau },F_{\mu \nu }^{i}%
\right] -\chi _{\alpha }^{\rho }\left[ Q_{\rho },F_{\mu \nu }^{i}\right] ,
\notag \\
\chi _{\alpha }^{\rho }(\partial _{\beta }\varphi _{\mid }^{i})+\chi _{\beta
}^{\rho }(\partial _{\alpha }\varphi _{\mid }^{i}) &=&-\chi _{\alpha }^{\rho
}\left( \left[ c_{\beta },\Lambda \right] ^{i}+\xi _{\beta }^{\nu }\left[
P_{\nu },\Lambda ^{i}\right] +\chi _{\beta }^{\rho }\left[ Q_{\rho },\Lambda
^{i}\right] \right) \notag \\
&&-\chi _{\beta }^{\rho }\left( \left[ c_{\alpha },\Lambda ^{i}\right] +\xi
_{\alpha }^{\nu }\left[ P_{\nu },\Lambda ^{i}\right] +\chi _{\alpha }^{\rho }%
\left[ Q_{\rho },\Lambda ^{i}\right] \right) , \notag \\
\partial _{\alpha }\Phi _{\mid }^{\rho i} &=&(\gamma ^{\mu })^{\rho \sigma
}/4\left[ Q_{\sigma },\partial _{\alpha }\Phi _{\mu \mid }^{i}\right] ,
\notag \\
\partial _{\alpha }\Phi ^{i} &=&-\delta _{\sigma }^{\rho }/4\left[ Q_{\rho
},\Phi _{\sigma }^{i}\right] . \label{Inser}
\end{eqnarray}
Inserting Eq. (\ref{eqalgeb1}) into (\ref{Inser}), and evaluating these at $%
\theta ^{\alpha }=0$, we find the following BRST transformations
\begin{eqnarray}
QA_{\mu }^{i} &=&D_{\mu }c^{i}-\xi ^{\rho }\partial _{\rho }A_{\mu
}^{i}+\chi \gamma _{\mu }\lambda ^{i}, \notag \\
Q\lambda _{\alpha }^{i} &=&-f_{jk}^{i}c^{j}\lambda _{\alpha }^{k}-\xi ^{\mu
}\partial _{\mu }\lambda _{\alpha }^{i}+\frac{1}{2}(\chi \sigma ^{\mu \nu
})_{\alpha }F_{\mu \nu }^{i}+\chi _{\alpha }\Lambda ^{i}, \notag \\
Qc^{i} &=&-\frac{1}{2}f_{jk}^{i}c^{j}c^{k}-\xi ^{\mu }\partial _{\mu
}c^{i}+\chi \gamma ^{\mu }\overline{\chi }A_{\mu }^{i}, \notag \\
Q\xi ^{\rho } &=&-\chi \gamma ^{\rho }\overline{\chi }, \notag \\
Q\Lambda ^{i} &=&-f_{jk}^{i}c^{k}\Lambda ^{j}-\xi ^{\rho }\partial _{\rho
}\Lambda ^{i}-\chi \gamma ^{\mu }D_{\mu }\lambda ^{i}, \notag \\
QF_{\mu \nu }^{i} &=&-f_{jk}^{i}c^{k}F_{\mu \nu }^{j}-\xi ^{\rho }\partial
_{\rho }F_{\mu \nu }^{i}-\chi ^{\rho }\left\{ (\gamma _{\mu })_{\rho \sigma
}(D_{\nu }\lambda )^{\sigma i}-(\gamma _{\nu })_{\rho \sigma }(D_{\mu
}\lambda )^{\sigma i}\right\} \notag \\
Q\chi ^{\alpha } &=&0, \notag \\
Q\overline{c}^{i} &=&B^{i}, \notag \\
QB^{i} &=&0, \notag \\
Q\overline{\xi }^{\mu } &=&E^{\mu }, \notag \\
QE^{\mu } &=&0, \notag \\
Q\overline{\chi ^{\alpha }} &=&G^{\alpha }, \notag \\
QG^{\alpha } &=&0, \label{brst -trans}
\end{eqnarray}%
and also the anti-BRST transformations, which can be derived from (\ref{brst
-trans})\ by the following mirror symmetry of the ghost numbers given by: $%
X\rightarrow X$ if $X=A_{\mu }^{i},$ $\lambda _{\alpha }^{i},$ $\Lambda
^{i}; $ $X\rightarrow \overline{X}$ if $X=Q,$ $c^{i},$ $B^{i},\xi ^{\mu
},E^{\mu },\chi ^{\alpha },G^{\alpha },$and $X=\overline{\overline{X}}$ where
\begin{eqnarray}
B^{i}+\overline{B^{i}} &=&f_{jk}^{i}c^{k}\overline{c}^{j}-\xi ^{\rho
}\partial _{\rho }\overline{c^{i}}-\overline{\xi }^{\rho }\partial _{\rho
}c^{i}-\chi \gamma ^{\mu }\overline{\chi }A_{\mu }^{i}-\overline{\chi }%
\gamma ^{\mu }\chi A_{\mu }^{i}, \\
E^{\mu }+\overline{E^{\mu }} &=&2\chi \gamma ^{\mu }\overline{\chi }, \notag
\\
G^{\alpha }+\overline{G^{\alpha }} &=&0. \label{brst-trans2}
\end{eqnarray}
Let us note that the introduction of an auxiliary real field $\Lambda ^{i}$
besides the fields present in quantized $\mathcal{N}=1$ super Yang-Mills
theory in four-dimensions, guarantees\ automatically the off-shell
nilpotency of the$\left\{ Q,\overline{Q}\right\} $-algebra and make easier
then, as we will see in the next Section, the gauge-fixing process.
\section{Quantum action}
In the present section, we show how to construct in the context of our
procedure a BRST-invariant quantum action for $\mathcal{N}=1$ super
Yang-Mills theory as the lowest component of a quantum superaction. To this
purpose, let us recall that the gauge-fixing superaction similar to that
obtained in the case of Yang-Mills theories \cite{tah1,tah2} and
gauge-affine gravity \cite{meziane} is given by%
\begin{eqnarray}
S_{sgf} &=&\dint d^{4}xL_{sgf,} \notag \\
L_{sgf} &=&(\partial _{1}\Phi _{2})(\partial ^{\mu }\Phi _{\mu })+(\partial
^{\mu }\Phi _{2})(\partial _{1}\Phi _{\mu })+(\partial _{1}\Phi
_{2})(\partial _{1}\Phi _{2}). \label{fixing1}
\end{eqnarray}
We note first that in the case of Yang-Mills theory the superaction involve
a Lorentz gauge \cite{tah11} given by
\begin{equation}
\partial _{\mu }\Phi _{\mid }^{\mu }=0. \label{gauge-fix1}
\end{equation}
In the case of super Yang-Mills theory we shall choose a supersymmetric
gauge-fixing which is the extension of the Lorentz gauge. This gauge fixing
can be obtained from (\ref{gauge-fix1}) by using the following substitution
\begin{equation}
\Phi _{\mu }\longrightarrow \widetilde{\Phi _{\mu }}=\Phi _{\mu }+\left[
\partial _{\mu }\Phi ^{\alpha },Q_{\alpha }\right] . \label{gauge-fix2}
\end{equation}
Now, it is easy to see that the gauge-fixing superaction (\ref{fixing1}) can
be put in the following form
\begin{equation}
S_{sgf}=\dint d^{4}x(\partial _{1}\Phi _{2})(\partial ^{\mu }\widetilde{\Phi
_{\mu }})+(\partial ^{\mu }\Phi _{2})(\partial _{1}\widetilde{\Phi _{\mu }}).
\label{gauge-fix3}
\end{equation}
To determine the gauge-fixing action $S_{gf}$ as the lowest component of the
gauge-fixing superaction $S_{gf}=S_{sgf\mid }$, we impose the following rules%
\begin{eqnarray}
Tr(I^{m}I_{n}) &=&\delta _{n}^{m} \notag \\
Tr(\left[ Q_{\alpha },Q_{\beta }\right] ) &=&2\gamma _{\alpha \beta }^{\mu }%
\mathcal{\partial }_{\mu } \notag \\
Tr(P^{2}) &=&0. \label{eqfix5}
\end{eqnarray}
These rules permit us to compute the trace of each term in (\ref{gauge-fix3}%
) . Indeed, from (\ref{eqfix5}) it is easy to put the gauge-fixing action $%
S_{^{gf}}$ in the form
\begin{eqnarray}
S_{gf} &=&S_{sgf\mid }=\dint d^{4}x(B\partial ^{\mu }A_{\mu }+2b_{j}^{\ast
}G(\gamma ^{\mu }\mathcal{\partial }_{\mu }\square \lambda ^{j}) \notag \\
&&+(\partial ^{\mu }\overline{c})\left\{ D_{\mu }c+\xi ^{\nu }\partial _{\nu
}A_{\mu }+\chi \gamma _{\mu }\lambda \right\} \notag \\
&&-2b_{j}^{\ast }(\partial ^{\mu }\overline{\chi })\gamma ^{\nu }\mathcal{%
\partial }_{\nu }\partial _{\mu }\left\{ f_{ik}^{j}\lambda ^{i}c^{k}+\xi
^{\tau }\partial _{\tau }\lambda ^{j}+\frac{1}{2}\chi \sigma ^{\tau \upsilon
}F_{\tau \upsilon }^{i}+\chi D^{j}\right\} ). \label{gaugefixac}
\end{eqnarray}
On the other hand, the presence of the extrafields breaks the invariance of
the classical action (\ref{Lagrangian}). In fact, the only terms which may
contribute to the $Q$-variation of the classical action $S_{0}$ are those
containing the extrafield $\Lambda ^{i}.$ This follows from the fact that
the BRST transformations up to terms $\Lambda ^{i}$ represent the $\mathcal{N%
}=1$ super Yang-Mills transformations expressed \`{a} la BRST. A simple
calculation with the help of the BRST transformations (\ref{brst -trans})
leads to%
\begin{equation}
QS_{0}=\Lambda ^{i}\gamma ^{\mu }D_{\mu }\lambda _{i}. \label{action auxi}
\end{equation}%
Thus the classical action $S_{0}$ is note BRST-invariant, and in order to
find the BRST-invariant extension $S_{inv}$ of the classical action, we
shall add to $S_{0}$ a term $\widetilde{S_{0}}$ so that
\begin{equation}
Q(S_{0}+\widetilde{S_{0}})=0. \label{Q(inva)}
\end{equation}%
To this end, we propose to write $\widetilde{S_{0},}$\ which define the
extension action for the auxiliary field $\Lambda ^{i}$ as follows
\begin{equation}
\widetilde{S_{0}}=-\frac{1}{2}\Lambda ^{i}\Lambda _{i}. \label{aux actio22}
\end{equation}%
Then, it is quite easy to show that $Q(S_{0})=-Q(\widetilde{S_{0}})$ by a
direct calculation with the help\ of the transformations (\ref{brst -trans}).
Having found the BRST-invariant extension action $S_{inv}$ we now write the
full off-shell BRST- invariant quantum action $S_{q}$ by adding to the $Q$%
-invariant action $S_{inv}=S_{0}+\widetilde{S_{0}}$ the $Q$-gauge-fixing
action $S_{gf}$
\begin{equation}
S_{q}=S_{0}+\widetilde{S_{0}}+S_{gf}. \label{quan act}
\end{equation}
Let us mention that the quantum action (\ref{quan act}) and the off-shell
BRST transformations (\ref{brst -trans}) obtained in the previous section
are equivalent to those proposed in \cite{Baulieu},\ where the
Batalin-Vilkovisky formalism has been considered to close the supersymmetric
algebras without relying on the unusual auxiliary fields.
It is worth nothing that the quantum action (\ref{quan act}) allows us to
see that the auxiliary field $\Lambda ^{i}$ does not propagate, as its
equation of motion is a constraint
\begin{equation}
\frac{\delta S_{q}}{\delta \Lambda ^{i}}=-\Lambda ^{i}+2b_{i}^{\ast }(%
\overline{\chi }\gamma ^{\mu }\mathcal{\partial }_{\mu }\square \chi )=0.
\label{eq motion}
\end{equation}
Thus the essential role of the nondynamical auxiliary field $\Lambda ^{i}$
is to close the BRST and anti-BRST algebra off-shell.
The elimination of the auxiliary field $\Lambda ^{i}$ by means of its
equation of motion (\ref{eq motion}) leads to the same gauge-fixed theory
with on-shell nilpotent BRST transformations obtained in the context of BV
formalism \cite{Baulieu} as well as in the framework of the superfibre
bundle approach \cite{tah1}.
Moreover, in our formalism we have also introduced an anti-BRST operator $%
\overline{Q}$ and it is important to realize that both the BRST symmetry and
anti-BRST symmetry can be taken into account on an equal footing. To this
end, we simply use the fact that there is a complete duality, with respect
to the mirror symmetry of the ghost number, between the $Q$- and $\overline{Q%
}$-transformations. So, the $\overline{Q}$-variation of the classical action
$S_{0}$ is given by
\begin{equation*}
QS_{0}=\overline{\Lambda }^{i}\gamma ^{\mu }D_{\mu }\overline{\lambda }_{i}
\end{equation*}%
where $\overline{\Lambda }^{i}$ represents an auxiliary field in the context
of $\overline{Q}$-symmetry. Using however the $Q$-transformations of the
auxiliary field ( see Eqs. (\ref{brst -trans}) with the mirror symmetry), we
obtain that the $Q$-invariant action $S_{inv}$=$S_{0}+\widetilde{S_{0}}$ is
also $\overline{Q}$-invariant. Furthermore, the $Q$-gauge-fixing action can
be also written as in Yang-Mills theories in $\overline{Q}$-form. At this
point, we remark that the mirror symmetry allows to replace, in particular
the auxiliary field $\Lambda ^{i}$ in the context of $Q$-symmetry by $%
\overline{\Lambda }^{i}$. Therefore the full off-shell BRST-invariant
quantum action $S_{q}=S_{0}+\widetilde{S_{0}}+S_{gf}$ is also an off-shell
anti-BRST-invariant quantum action.
\section{\protect\bigskip Conclusion}
In this paper we have presented a BRST superspace approach in order to
perform the quantization of the four dimensional $\mathcal{N}=$1
supersymmetric Yang-Mills theory as model where the classical gauge algebra
is not closed. The construction is entirely based on the possibility of
introducing a set of auxiliary fields via the supersymmetric transformations
of the superpartener of the gauge potential associated to a super Yang-Mills
connection. The gauge fields and their associated ghost and antighost fields
occurring in\ quantized four dimensional $\mathcal{N}=1$ supersymmetric
Yang-Mills theory have been described through a super Yang-Mills connection,
whereas the extrafields coming from the supersymmetric transformations are
required to achieve the off-shell nilpotency of the BRST and anti-BRST
operators. The minimal set of extrafields is defined after having imposed
constraints on the supercurvature in which the consistency with the Bianchi
identities is guaranteed.
Furthermore, we have performed a direct construction of the BRST invariant
extension of the classical action for $\mathcal{N}=1,$ $4$ $D$
supersymmetric Yang-Mills theory in analogy with what it is realized in
simple supergravity \cite{tahiri22} and four-dimensional BF theories \cite%
{tahiri2}. The obtained quantum action allows us to see that the extrafields
enjoy the auxiliary freedom, i.e. their auxiliary fields do not propagate,
being vanishing on-shell. The elimination of these auxiliary fields\ using
the solution of their equations of motion permits us to recover the standard
quantum action with on-shell nilpotent BRST symmetry. The transformations of
this minimal set of auxiliary fields and the obtained BRST invariant action
agree with the standard results. By using the mirror symmetry between the
BRST and anti-BRST transformations, we can see that the BRST invariant
action is also anti-BRST invariant. Therefore the full quantum action is
BRST and anti-BRST invariant, since the gauge-fixing action can be written
as in the Yang-Mills case in BRST as well as anti-BRST exact form, due to
the off-shell nilpotency of the BRST-anti-BRST algebra.
Let us note, that the Batalin-Vilkovisky formalism can be used to obtain the
on-shell BRST invariant gauge fixed action for $\mathcal{N}=1$
supersymmetric Yang-Mills theory in four dimensions without requiring the
set of\ auxiliary fields but by extending the fields in the theory to
include the so-called antifields \cite{Baulieu}.
Finally, we should mention that the BRST superspace formalism presented in
this paper was successfully applied to several interesting theories: simple
supergravity \cite{tahiri22} and $4D$ non-Abelian BF theory where the
symmetry is reductible \cite{tahiri2}. Such formalism permit us to determine
the off-shell nilpotent BRST and anti-BRST algebra for gauge theories. In
particular, it gives another possibility leading to the minimal set of
auxiliary fields. Thus, it would be a very nice endeavour to use this basic
idea to study the structure of auxiliary fields in other gauge theories.
These are some of the issues that are under investigation at the moment.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 970
|
\section{INTRODUCTION} \label{sec-intro}
When attempting to evaluate the mass parameter $\OM$
and/or the cosmological constant $\Lambda$, observations of quantities such as magnitude, angular
separation, and redshift are made on objects distant enough for curvature effects to be detected.
As an example, for Type Ia supernovae (SNe Ia) corrected magnitudes and redshifts ($m$-$z$) are
measured, plotted, and compared with theoretical $m(\OM,\Lambda; z)$ curves computed for the FLRW
models (\cite{PS1}, \cite{PS2}, \cite{GP}). In spite of the fact that the FLRW models contain
only homogeneously and isotropically
distributed perfect fluid gravity sources, one of these models is assumed to represent the ``large
scale" geometry of the universe. Relations like $m(\OM,\Lambda;z)$ are also commonly assumed to
be valid, on average. This latter assumption may well be incorrect for some distant observations
including SNe Ia, but even if technically correct may not be useful in determining $\OM$ and
$\Lambda$. In particular if the underlying mass density approximately follows luminous matter
(\ie associated with bounded galaxies) then effects of inhomogeneities on relations like
$m(\OM,\Lambda;z)$ must be taken into account. The majority of currently observed SNe Ia are not
being seen through foreground galaxies and whether or not this is due entirely to selection
(rather than statistics) is not important. If the objects observed do not have the average FLRW
mass density $\rho_0$ in their foregrounds then the FLRW \mz\ relation does not apply to them
(see \cite{KR2}).
Ultimately some SNe Ia should exist behind foreground galaxies (\cite{RK}) and for these, \mz\ should be
computed using the lensing formulas. These formulas (\cite{BR} and \cite{CJ}) contain
source-observer,
deflector-observer, and source-deflector distances, respectively $D_s, D_d$, and $D_{ds}$, all of
which depend on the mass density in the observing beam, \underline{excluding} the deflector.
These distances will not be given by the standard FLRW result if the observing beam contains less
than the average FLRW mass density.
In \S \ref{sec-optics} the average area-redshift equation (\ref{Af}) for a light beam traveling
through a FLRW Swiss cheese universe is given and its solution is related to the luminosity
distance--redshift relation $D_{\ell}(z)$. In \S \ref{sec-B=0} the solution of this equation is
given for the case where gravitational lensing can be neglected. The new result of this paper,
$D_{\ell}(z)$ without lensing for FLRW Swiss cheese can be found in equations (\ref{Dell0}) and
(\ref{Dellinfty}), and for the special case $\Omega_0=1$ in equations (\ref{Dell01}) and
(\ref{Dellinfty1}) of Appendix A. In \S \ref{sec-mzplots}
numerous \mz plots are given to illustrate the
effects of inhomogeneities and some conclusions are drawn.
It is argued that if homogeneities are not taken into
account when attempting to determine $\Omega_m$ and $\Lambda$, errors as large as 50\% could be
made. Even though the $D_{\ell}(z)$ given here has been derived using the exact Swiss cheese
cosmologies, the result are valid for observations in essentially any perturbed
pressure-free FLRW
models in which lensing can be neglected.
Inhomogeneous models of the Swiss cheese type and their associated optical equations discussed
here are often mistakenly attributed to Dyer and Roeder (see Appendix B).
Appendix C contains some useful simplifications for evaluating the real-valued Heun functions
needed in the analytic \mz relations given here. Appendix C also contains six useful lines of
Mathematica code which numerically evaluates and plot these same \mz relations.
\section{SWISS CHEESE OPTICS} \label{sec-optics}
Some years ago the author (\cite{KR}), used the ``Swiss cheese"
cosmologies to study the effects of local inhomogeneities in the FRW mass density on the
propagation of light through an otherwise homogeneous and isotropic universe. That analytic work
was undertaken because prior results computed in perturbed FRW models were suspect.
In particular, results were inconsistent with weak lensing results, \eg on average, the
luminosity of a distant object was not given by the FRW result (\cite{BB}). Numerous sources of error were
suggested but particularly mistakes inherent in using perturbative gravity were suggested. For
example the FRW relations between radius, redshift, and affine parameters were
(and still are
for approximate GR solutions)
assumed valid in the presence of mass perturbations. At that time \cite{KR} put to rest any question of the possible
existence of an effect on the mean luminosity; theoretically it could exist! Because the Swiss
cheese models are exact solutions to the Einstein equations the accuracy of the FRW relations
between radius, redshift, and affine parameters could be directly established. Today Swiss
cheese itself is under attack as the source of the `erroneous' prediction (see \cite{FJ} and
\cite{WJ}, and additionally see the related work of \cite{PP}). The intent of this particular paper is not to defend Swiss cheese
\underbar{predictions} against these attacks, that can be done elsewhere. The reader should not
dismiss distance--redshift predictions made by Swiss cheese models because of their
non-physical distribution of matter, \eg cheese and holes. Because these models contain
the only two types of gravitational curvature (Ricci and Weyl) that affect optical
observations, and because they are fairly flexible in including density perturbations, they
\underline{should} adequately describe optical observations at least as far as $z=1$
(a distance to which galaxies and other inhomogeneities are thought to have undergone
only minor changes). The conjectured extension of the validity of
the optical equation (\ref{Af}) used here beyond the Swiss cheese models has been made frequently
since it was first derived and is argued by \cite{SEF} in their Sections 4.5.2 and 4.5.3.
The purpose of this paper is
to extend analytic results for distance--redshift relations in inhomogeneous FRW models to FLRW,
\ie to include the cosmological constant (see \cite{KVB} and
\cite{SS}). \cite{KR} derived a 2nd order intergal-differential equation [see (43a) of that paper or (1) of \cite{KVB}
as well as the equivalent 3rd order differential equation, (43b) of that paper or (6) of \cite{KVB}] for the
average cross-sectional area $A$ of a beam of light starting from a distant source and
propagating through a $\Lambda=0$ Swiss cheese universe (see Figure 1). The solution of this
equation, with appropriate boundary conditions, gives all average quantities relating to
distance--redshift. This equation and its derivation were easily extended to include a
cosmological constant by \cite{DC3}.
However, as will be seen below,
extension of the equation's analytic solutions of \cite{KVB} and \cite{SS} is somewhat involved and the special
functions required are much less familiar to the math/physics community. Weinberg sign
conventions will be again used (\cite{MTW}).
As a light beam from a distant SN Ia propagates through the universe
(Figure 1) the cheese of the model produces the same focusing effect
as does the transparent
material actually appearing within the beam. The holes in the cheese
with their condensed central masses
reproduce the optical effects of the remaining Friedmann matter that has been condensed
into clumps, \eg galaxies.
The extended equation for the average area $A$ traversing the universe, that is randomly focused by numerous
clouds of transparent matter and lensed by numerous clumps is:
\be
{\sqrt{A}^{''}\over \sqrt{A}} +
{\langle\xi^2\rangle\over A^2} = -{3\over 2}\ {\rho_D\over \rho_0}\ \Omega_m\ (1+z)^5\ ,
\label{Af}
\end{equation}
where prime ($'$) is differentiation with respect to an affine parameter,
\be
{'}\
\equiv -(1+z)^3 \sqrt{1+\OM z+\OL[(1+z)^{-2}-1]} \ {d \ \over d z}\ , \label{affine}
\end{equation}
and
$\langle\xi^2\rangle$ is the average of $(\sigma/A)^2$, the square of the wavefront's shear over
its area,\footnote[2]{The form of $\langle\xi^2\rangle$ depends on structure details of the clumps.
What is given in (3) is for objects completely condensed into opaque spherical masses.
This particular type of Swiss cheese clumping is expected to produce maximum lensing.
In the following sections we will be interested in observations where even maximum
lensing is negligible.}
\be
\langle\xi^2\rangle = {15\over 2}\ {\rho_I\over \rho_0}\ B_0\ \Omega_m \int \
{A^2(1+ z)^6\over z'} d z\,. \label{xi}
\end{equation}
In (\ref{Af}) $\rho_D$ ($D$ is for dust) is the
average mass density of all transparent material interior to light beams used to observe the given objects
and $\rho_I$ ($I$ is for inhomogeneous) is the average mass density of all types of clumpy
material systematically or statistically excluded from the light beams.
The average shear term in (\ref{Af}) comes from the Weyl (conformal) curvature tensor of inhomogeneous
material exterior to the beams. The ${\rho_D/\rho_0}$ term comes
from the Ricci tensor of transparent material within the beams.
For the SNe Ia observations, $\rho_D$ would certainly include those ubiquitous
low mass neutrinos (if they exist)
as well as other
transparent material not confined to galaxies, while $\rho_I$ would contain all
matter clumped with galaxies.
If there is no correlation of mass and light, deciding what goes in $\rho_D$ and what goes in
$\rho_I$ is problematic, and the relative value becomes another unknown parameter of the theory.
The current Friedmann
mass density is the total $\rho_0=\rho_D+\rho_I$ and the curvature parameter $\Omega_0\equiv \Omega_m
+\Omega_{\Lambda}$ consists of a mass part and a cosmological constant part:
\be
\Omega_m \equiv
{8\pi G \rho_0\over 3H_0^2} \hskip .25 in {\rm and } \hskip .25 in \Omega_{\Lambda} \equiv
{\Lambda c^2\over 3H_0^2}.
\end{equation}
Inclusion of the cosmological constant $\Lambda$ (using FLRW
rather than FRW) only modifies the functional relationship between redshift and affine parameter
(\ref{affine}), see \cite{DC3}. The unitless gravitational lensing parameter $B_0$ is defined in equation
(A2) of \cite{KVB} and its effects on the solution of equation (\ref{Af}) are described in \cite{DC3}.
In this paper analytic solutions to equation (\ref{Af}) will be given for $B_0=0$, \ie for distance--redshift
in any Swiss cheese model where $\langle\xi^2\rangle$ is neglected. In \cite{KVB} it was argued that even maximal lensing effects ($B_0\ne 0$) are not expected to be significant when
observing SNe Ia at $z\le 1$ and, as pointed out above, when lensing events do occur, the
observed magnitudes should be analyzed by using the lensing formula and not by incorporation into
the \mz\ relation.
The resulting $B_0=0$ equation (\ref{Ab0}) represents the equation for the average
area of a light beam only Ricci focused by part of the mass density, $\rho_D(\le\rho_0)$.
Such a light beam is not conformally lensed by inhomogeneities ($\rho_I=\rho_0-\rho_D$)
that remain exterior to the beam.
If Weyl lensing is infrequent,
a distribution of areas will occur for which the $B_0=0$
equation gives the maximum value for the area (\ie the lower bound on the distribution of
luminosities).
In \cite{KR2} the resulting \mz is appropriately dubbed the `intergalactic' magnitude-redshift
relation because it is \mz without galactic focusing. If significant galactic lensing is an unusual
event as apparently is the case with
SNe Ia beams passing exterior to galaxies, the `intergalactic' \mz approximates the `mode' value
(the most likely).\footnote[3]{This assertion is consistent with that part of
the numerical work of \cite{HD} which treated galaxies as condensed objects.
When galaxies were modeled by 200 kpc isothermal spheres the `mode' moved
towards the `mean' and away from
the minimum (intergalactic) value as expected of a more homogeneous model.}
If galaxies are compact (20 kpc) the `intergalactic' \mz relation
should be more useful in determining $\OM$ and $\OL$
from SNe Ia observations than is the mean \mz relation (standard FLRW relation).
If galaxies are more diffuse (200 kpc) exact modeling of the lensing galaxies will be important.
To relate the differential equation (\ref{Af}) to observations, consider a source at redshift
$z_s$ radiating power ${\delta\cal P}$ into solid angle $\delta\Omega$. The flux received by an
observer at $z=0$ in area $A|_0$ is given by ${\cal F} = {\delta\cal P}/ A|_0(1+z_s)^2$. The two
factors of $(1+z_s)$ can be thought of as coming separately from the redshift of the observed
photons and their decreased rates of reception. The definition of luminosity distance is
motivated by this result, \ie
\be
D_{\ell}^2\equiv {A\big|_0 \over \delta\Omega}(1+z_s)^2.
\label{Dl}
\end{equation}
The observed area $A|_0$ is evaluated by integrating equation (\ref{Af}) from the
source $z=z_s$ to the observer $z=0$ with initial data which makes the wave front satisfy
Euclidean geometry when leaving the source (area=radius$^2\times$ solid angle):
\bea
\sqrt{A}|_s&=&0,\nonumber\\ {d\sqrt{A\big|_s }\over dz}&=& -\sqrt{\delta\Omega} {c\over
H_s(1+z_s)}, \label{Aboundary}
\eea
where in FLRW the value of the Hubble parameter at $z_s$ is
related to the current value $H_0$ at $z=0$ by
\be
H_s=H_0(1+z_s)\ \sqrt{1+\OM
z_s+\OL[(1+z_s)^{-2}-1]}. \label{Hs}
\end{equation}
The series solution of equation (\ref{Af}), combined
with (\ref{Dl}) and (\ref{Aboundary}) is:
\bea D_{\ell}(\OM,\OL,\nu,B_0;z)&=& \sqrt{{A\over
\delta\Omega}}\Bigg\vert_0(1+z) ={c\over H_0} \Biggl\{ z + {1\over2}\left[1+\OL-{1\over
2}\OM\right] z^2 \nonumber\\ &+&{1\over2} \left[ {1\over2}\OM \left(
{1\over2}\OM+{\nu(\nu+1)\over 6}-1 \right) - \OL\biggl(1+\OM-\OL\biggr) \right] z^3\nonumber\\
&+&{1\over8} \Biggl[ \OM\left( {1\over 8}\OM \biggl[10 - 2\nu(\nu+1)-5\OM\biggr]
-B_0{\nu(\nu+1)\over 6} \right)+\nonumber\\ &&\hskip .5 truein\OL \left( 5+{1\over
2}\biggl[5+\nu(\nu+1) \biggr]\OM +{15\over 4}\OM^2 +5 \OL^2-{5\over 2}\OL\biggl[4+3\OM\biggr]
\right) \Biggr] z^4\nonumber\\ &+&O[z^5] \Biggr\}, \label{Dlseries} \eea where the source
redshift $z_s$ has been simplified to $z$ and ${\rho_I/ \rho_0}$ has been replaced for later
convenience by a clumping parameter $\nu$, $0\le\nu\le 2$,
\be
\nu\equiv
{\sqrt{1+24(\rho_I/\rho_0)}-1\over 2} \Rightarrow {\rho_I\over \rho_0}={\nu(\nu+1)\over 6}\end{equation}
This series is useful for understanding the low-redshift sensitivity of $D_{\ell}$ to the various
parameters; \eg $\OM$ and $\OL$ appear in the $z^2$ term, the clumping parameter $\nu$ first
appears in the $z^3$ term, whereas the lensing parameter $B_0$ doesn't appear until the $z^4$
term. Additionally, analytic results computed in the next section can be checked by comparison
with this series.
\section{THE ANALYTIC SOLUTION FOR \hbox{{\it D}$_{\ell}$}(${\lowercase {z}}$)
WHEN LENSING CAN BE NEGLECTED} \label{sec-B=0}
In this section the general $B_0=0$ solution of (\ref{Af}) will be given for boundary conditions
appropriate for $D_{\ell}(z)$. If apparent-size (angular) distances are desired the reader has
only to compute $D_<(z)=D_{\ell}(z)/(1+z)^2$. The new solution appears in (\ref{Dell0}),
(\ref{Dellinfty}), (\ref{Dell01}), and
(\ref{Dellinfty1}) expressed in terms of Heun functions $Hl$. All previously known special
solutions are limiting cases of the general solution (\ref{Dellinfty}) [see
(\ref{hypergeometric}) and Appendix B]. To solve equation (\ref{Af}) it is first rewritten as:
\bea &&(1+z)^3\sqrt{1+\OM z+\OL[(1+z)^{-2}-1]}\times\nonumber\\ &&\hskip 1 in {d\ \over
dz}(1+z)^3\sqrt{1+\OM z+ \OL[(1+z)^{-2}-1]}\,{d\ \over dz}\sqrt{A(z)}\nonumber\\ &&\hskip 2.0 in
+ {(3+\nu)(2-\nu)\over 4}\OM(1+z)^5\sqrt{A(z)}=0. \label{Ab0} \eea
This equation is often attributed to Dyer-Roeder (\cite{DC1}, \cite{DC2}) in the literature.
(see Appendix B for some history
of this equation).
To date only numerical solutions have been obtained when $\OL\ne 0$, \eg see \cite{AH},
\cite{SY}, \cite{KRa}.
It can be put
into a recognizable form by changing the independent variable from $z$ to $y$ and the dependent
variable from $\sqrt{A(z)}$ to $h$,
\bea
y&=&y_0(1+z)={\OM\over 1-\OM-\OL}(1+z),\nonumber\\
h&=&(1+z)\sqrt{{A\over\delta\Omega}}.
\label{newvariables}
\eea
The resulting equation is
\be
{d^2h\over dy^2} +{\left(1+{3\over 2}y\right)y\over y^3+y^2-b_{\Omega}}\
{dh\over dy} - { {1\over 4}\nu(\nu+1)y+1\over y^3+y^2-b_{\Omega}}\ h = 0\ , \label{h1}
\end{equation}
where
$b_{\Omega}\equiv -\OM^2\OL/(1-\OM-\OL)^3$. When the cubic $y^3+y^2-b_{\Omega}=(y-y_1)(y-y_2)(y-y_3)$ is
factored, (\ref{h1}) simplifies to a recognizable form of the Heun equation (see \cite{RA},
\cite{EA}, \cite{WE}, \cite{HK}):
\be
{d^2h\over dy^2} +\left({\gamma\over y-y_1}+{\delta\over
y-y_2}+{\epsilon\over y-y_3}\right) {dh\over dy} + { \alpha\beta\ y-q\over
(y-y_1)(y-y_2)(y-y_3)}\ h = 0\ , \label{h2}
\end{equation}
where (\ref{h1}) requires
\bea
&&\gamma=\delta=\epsilon={1\over 2},\nn &&\alpha = -{1\over 2}\nu,\nn &&\beta= {1\over
2}(\nu+1),\nn &&q=1,
\label{exponents}
\eea
and additionally the three roots to be constrained by: \bea
&&y_1y_2y_3=b_{\Omega}=-\OM^2\OL/(1-\OM-\OL)^3,\nonumber\\ &&y_1+y_2+y_3=-1,\nonumber\\
&&y_1y_2+y_1y_3+y_2y_3=0. \label{rootconstraints} \eea
The Heun equation is slightly more complicated than the hypergeometric equation; it possesses
four regular singular points in the entire complex plane rather than just three. In the form
given by (\ref{h2}) one of the two exponents of each finite singular point $(y_1,y_2,y_3)$
vanishes and the other exponent is given respectively by
$(1-\gamma,1-\delta,1-\epsilon)$.\footnote[4]{ Recall that an exponent gives the analytic
behavior of a solution within the neighborhood of a regular singular point, \eg
$h=(y-y_1)^{1-\gamma}(1+ c_1(y-y_1)+ \cdots)$.} The point at $\infty$ is the fourth singular
point and its exponents are $\alpha$ and $\beta$. For the point at $\infty$ to also be regular
(\ie for this to be a Heun equation) all exponents must sum to a value of 2, equivalently \be
\alpha+\beta+1=\gamma+\delta+\epsilon. \label{exponentsum}
\end{equation}
For (\ref{h1}) this necessary
constraint is satisfied. From (\ref{rootconstraints}) it follows that at least one root has to
be real and complex roots must come in conjugate pairs. For convenience $y_1$ will be
chosen as real throughout. This Heun equation (\ref{h2}) is conveniently expressed in terms of a
Riemann P-symbol as:
\be
P\left\{ \begin{array}{cccccc} y_1 & y_2 & y_3 & \infty \\ 0 & 0 & 0 &
\alpha & y & q \\ 1-\gamma & 1-\delta & 1-\epsilon & \beta\\ \end{array} \label{Py} \right\}.
\end{equation}
The first 4 columns of (\ref{Py}) are the 4 regular singular points and their 2 exponents,
and the 5th column is the independent variable, all analogous to the Riemann P-symbol for the
hypergeometric equation. The 6th column is the constant $q$ from the numerator of the
coefficient of $h$ in the Heun equation [when put into the standard form of (\ref{h2})].
The hypergeometric equation is uniquely specified by information about its regular singular
points but Heun requires the extra parameter $q$. Because the three finite singular points
of this Heun equation have
values of 1/2 for their nonvanishing exponents, (\ref{h1}) can be transformed
into the Lame$^{\prime}$ equation. In this paper, solutions of (\ref{h1}) will be given as local
Heun functions and in a following paper they will be expressed as Lame$^{\prime}$ functions.
When $\Lambda=0$ ($\Rightarrow b_{\Omega}=0$) equation (\ref{h1}) has only three regular singular points
\ie this Heun equation simplifies to an equation of the hypergeometric type. Additionally the
corresponding Lame$^{\prime}$ equation reduces to the associated Legendre equation. Solutions
for $\Lambda =0$ can be written either as combinations of hypergeometric functions or as
associated Legendre functions [see (\ref{hypergeometric}) and (\ref{Legendre}) below]. To
motivate the form of the $\OL\ne 0$ solution, the $\OL=0$ solution will be given first [\cite{KVB} and
\cite{SS}],
\bea
&&D_{\ell}(\OM=\Omega_0,\OL=0, \nu ; z)\nonumber\\
&&= {c\over H_0}{1\over
(\nu+{1\over2})}\times \nonumber\\
&& \Biggl[\!\Biggl[ (1+\Omega_0 z)^{1+\nu/2}\ \ {}_2F_1\left({\nu\over2}+2,
{\nu\over2}+{3\over 2}; \nu+{3\over2} ; 1-\Omega_0\right) {}_2F_1\left(-{\nu\over2}-1,
-{\nu\over2}-{1\over2}; {1\over2}-\nu; {1-\Omega_0\over 1+\Omega_0 z}\right) \nonumber\\
&-&{(1+z)^2\over (1+\Omega_0 z)^{3/2+\nu/2} } \ {}_2F_1\left(-{\nu\over2}-1, -{\nu\over2}-{1\over2}; {1\over2}-\nu;
1-\Omega_0\right) {}_2F_1\left({\nu\over2}+2, {\nu\over2}+{3\over 2}; \nu+{3\over2} ; {1-\Omega_0\over
1+\Omega_0 z}\right) \Biggr]\!\Biggr].\nonumber\\ \label{hypergeometric}
\eea
The expected form of the $\OL\ne
0$ solution follows the above where the hypergeometric functions ${}_2F_1$ are replaced by local
Heun functions $Hl$ [see (\ref{Dell0}) and (\ref{Dellinfty})].
From (\ref{rootconstraints}) the three singular points $(y_1,y_2,y_3)$ are chosen from the six
permutations of the three roots:
\bea
Y_1&=&-{1\over 3}\left[1-{1\over
v_+}-v_+\right],\nonumber\\
Y_2&=&-{1\over 3}\left[1+{1\over v_-}+v_-\right],\nonumber\\
Y_3&=&-{1\over 3}\left[1+{e^{-i{\pi\over 3}}\over v_+}+ e^{i{\pi\over 3}}v_+\right],
\label{yroots}
\eea
where
\bea
v_+&\equiv& \left[-1+b+\sqrt{b(b-2)}\right]^{1\over
3},\nonumber\\ v_-&\equiv& \left[1-b+\sqrt{b(b-2)}\right]^{1\over 3},
\label{yroots1}
\eea
with
\be
b\equiv {27\over 2}b_{\Omega}=-{27\over 2}\OM^2\OL/(1-\OM-\OL)^3.
\label{b}
\end{equation}
The
locations of the three finite singular points $(y_1,y_2,y_3)$ are determined by the value of the
single parameter $b,\ (-\infty<b<\infty)$. Some important values are shown as contours
in Figure 2.
The standard form for the Heun equation ordinarily has its singularities at $(0,1,a,\infty)$.
The simple linear transformation
\be
\zeta={y-y_1\over y_2-y_1}, \label{zeta}
\end{equation}
moves \bea y_1
&\rightarrow &0,\nonumber\\ y_2 &\rightarrow& 1,\nonumber\\ y_3 &\rightarrow& a={y_3-y_1\over
y_2-y_1}={y_1(2+3y_1)\over (y_2-y_1)^2} ={(y_3-y_1)^2\over y_1(2+3y_1)}. \label{zetaroots} \eea
The latter two equalities are consequences of the useful identity: \be
(y_2-y_1)(y_3-y_1)=y_1(2+3y_1), \label{rootidentity}
\end{equation}
which results from
(\ref{rootconstraints}). In terms of the new variable $\zeta$, (\ref{h1}) becomes
\be
{d^2h\over
d\zeta^2} +{1\over2}\left({1\over \zeta}+{1\over \zeta-1}+{1\over \zeta-a}\right) {dh\over
d\zeta} + { (-{1\over 2}\nu){1\over 2}(\nu+1)\zeta-q\over \zeta(\zeta-1)(\zeta-a)}\ h = 0\ ,
\label{h3}
\end{equation}
and the value of $q$ changes to:
\be
q={1+{1\over4}\nu(\nu+1)y_1\over y_2-y_1}.
\label{q}
\end{equation}
The new Riemann P-symbol is:
\be
P\left\{ \begin{array}{cccccc} 0 & 1 & a & \infty \\ 0 & 0 & 0 & -{\nu\over2} & \zeta &
{1+{1\over4}\nu(\nu+1)y_1\over y_2-y_1} \\ {1\over 2} & {1\over 2} & {1\over 2} & {\nu+1\over
2}\\ \end{array} \label{Pzeta} \right\}.
\end{equation}
See the figures in Figure 3 for locations of $a$ in
the complex plane and the trajectories of $\zeta(z)$ starting with $\zeta_0$ (the value of
$\zeta$ at zero redshift [see (\ref{newvariables}) and (\ref{zeta})]) for the following three cases:
\bea
&&b < 0 \longrightarrow y_1= {\rm real,\ } y_2= \bar{y}_3, {\rm \ and\ } |a|=1,\nonumber\\
&&0\le b \le 2 \longrightarrow y_1,y_2,y_3, a {\rm\ are\ all\ real},\nonumber\\
&&2< b
\longrightarrow y_1= {\rm real,\ } y_2= \bar{y}_3, {\rm \ and\ } |a|=1. \label{beta} \eea Figure 3
gives the proper choices for the three roots $(y_1,y_2,y_3)$ from the six possible orderings
of $(Y_1,Y_2,Y_3)$ in each of the three $b$ domains. It also contains values for $a, q$,
the new variables $\zeta$, and $\zeta_0$. Hyperbolic and trigonometric variables, $\xi$ and
$\phi$, can be used to parameterize the values of the three roots (rather than $b$) and they
are also given in Figure 3.
Boundary conditions on $h$ come directly from its definition (\ref{newvariables}) and the desired
boundary conditions on $\sqrt{A}$ [see (\ref{Aboundary})],
\bea \sqrt{A}|_s&=&0 \Longrightarrow
h(\zeta_s)=0,\nonumber\\
{d\sqrt{A}\over dz}\Bigg|_s &=& -\sqrt{\delta\Omega} {c\over H_s(1+z_s)}
\Longrightarrow {d h\over d z}\Bigg|_s =-{c\over H_s}. \eea
Equation (\ref{Dl}) then relates
$D_{\ell}$ to the value of $h$ at the observer,
\be
D_{\ell}(z_s)=(1+z_s)h(z=0). \label{Dlh0}
\end{equation}
Using these boundary conditions on two independent solutions $h_1\ \&\ h_2$ of (\ref{h3})
gives
\be
h(\zeta)=-{h_1(\zeta_s) h_2(\zeta)-h_2(\zeta_s) h_1(\zeta) \over h_1(\zeta_s) \dot
h_2(\zeta_s)-h_2(\zeta_s) \dot h_1(\zeta_s)} \left[ \left({c\over H_s}\right) \biggr/ {d\zeta\over
dz}\Bigg|_{z_s} \right], \label{h12}
\end{equation}
where $\dot h \equiv {d h\over d\zeta}$. From
(\ref{zeta}) and (\ref{newvariables})
\bea
&&\zeta={y_0(1+z)-y_1\over y_2-y_1},\nn
&&\zeta_0={y_0-y_1\over y_2-y_1},\nn
&&{d\zeta\over dz}={y_0\over y_2-y_1}.
\label{zeta0}
\eea
The denominator of $h(\zeta)$ in (\ref{h12}) can be evaluated using the Wronskian of (\ref{h3}),
\be
h_1(\zeta) \dot h_2(\zeta)-h_2(\zeta) \dot h_1(\zeta)= (C_W){1\over
\sqrt{\zeta(\zeta-1)(\zeta-a)}}, \label{Wronskian}
\end{equation}
where $C_W$ is a constant. The square
root in this term can be evaluated using \bea
\sqrt{\zeta(\zeta-1)(\zeta-a)}&=&{\sqrt{y^3+y^2-b_{\Omega}}\over (y_2-y_1)^{3/2}},\nn
\sqrt{y^3+y^2-b_{\Omega}}&=&(1+z){y_0^{3/2}\over \sqrt{\Omega_m}} \sqrt{1+\OM z+\OL[(1+z)^{-2}-1]}.
\label{}
\eea
With $D_{\ell}$ from (\ref{Dlh0}) and $H_s$ from (\ref{Hs}), equations (\ref{h12})
and (\ref{Wronskian}) give the desired result:
\be
D_{\ell}(z_s)=-{c(1+z_s)y_0^{1/2}\over
H_0\sqrt{\Omega_m}(y_2-y_1)^{1/2}(C_W)} \left[h_1(\zeta_s) h_2(\zeta_0)-h_2(\zeta_s)
h_1(\zeta_0)\right]. \label{Dell}
\end{equation}
Figure 2 shows domains in the $(\OM,\OL)$ plane separated
by $b =2$, $b =\infty$, and $ |\zeta_0|=1$. The $b =\infty$ contour is equivalent to
$\Omega_0=1$. These contours are important because they separate domains for which different choices
of the two independent solutions $h_1(\zeta)$ and $h_2(\zeta)$ must be taken. The $ |\zeta_0|=1$
contour divides the $ |\zeta_0|<1$ domain where solutions about the singular point $\zeta=0$ are
chosen from the $ |\zeta_0|>1$ domain where solutions about $\infty$ are chosen. These choices
are necessary for convergence of the local Heun functions. For the $\OL=0$ case, only
analytic expressions
about $\infty$ were required [see (\ref{hypergeometric})].
The Heun Function $Hl(a,q;\alpha,\beta,\gamma,\delta;\zeta)$ is the analytic solution of
(\ref{h2fn}) defined by the infinite series (\ref{Hseries}), see \cite{RA}. It converges in a
circle centered on $\zeta=0$ which extends to the nearest singular point $1$ or $a$. This
solution is analogous to the ${}_2F_1$ solution of the hypergeometric equation but unfortunately
does not appear in any of the common computer libraries. When $c_0=1$ is chosen (as will be done
here) the series is:
\be
Hl(a,q;\alpha,\beta,\gamma,\delta;\zeta) \equiv 1+ \sum_{r=1}^{\infty}
c_r\zeta^r, \label{Hseries}
\end{equation}
where the $c_r$ are constrained by a three term recursion
relation (take $c_{-1}=0$):
\be
P_rc_{r-1}-(Q_r+q)c_r+R_rc_{r+1}=0, \label{recursion}
\end{equation}
with
\bea &&P_r\equiv (r-1+\alpha)(r-1+\beta),\nn
&&Q_r\equiv
r[(r-1+\gamma)(1+a)+a\delta+\epsilon],\nn
&&R_r\equiv(r+1)(r+\gamma)a. \label{recursionCons}
\eea
The $\epsilon$ parameter is not included as an argument in
$Hl(a,q;\alpha,\beta,\gamma,\delta;\zeta)$ because of the constraint (\ref{exponentsum}). This
series corresponds to the zero exponent for the regular singular point $\zeta=0$ and will be
taken as $h_1(\zeta)$ in (\ref{Dell}) when $|\zeta_0|<1$. The second independent solution is \be
h_2(\zeta)= \zeta^{1-\gamma}Hl(a,q_{II};\alpha_{II},\beta_{II},\gamma_{II},\delta;\zeta),
\label{h2fn}
\end{equation}
where four parameters have changed, \bea &&q_{II}\equiv
(a\delta+\epsilon)(1-\gamma)+q,\nn &&\alpha_{II}\equiv \alpha+1-\gamma,\nn &&\beta_{II}\equiv
\beta+1-\gamma,\nn &&\gamma_{II}\equiv 2-\gamma. \eea
The constant $C_W$ in the Wronskian can be evaluated for the $\zeta\sim 0$ expansion using $h_1$
and $h_2$ above,
\be
C_W={1\over 2}\sqrt{a}.
\end{equation}
This gives an expression for the luminosity
distance appropriate for $|\zeta_0|< 1$,
\bea
&&D_{\ell}(\OM,\OL,\nu;z)= -{c(1+z)\over H_0{1\over
2}\sqrt{\Omega_m}}\sqrt{{y_0(y_0-y_1)\over y_1(2+3y_1)}}\times\nn
&&\Biggl[\!\Biggl[
Hl\left(a,{1+{1\over4}\nu(\nu+1)y_1\over\sqrt{y_1(2+3y_1)}}\sqrt{a}; -{\nu\over 2},{\nu+1\over
2},{1\over 2},{1\over 2};{y_0(1+z)-y_1\over \sqrt{y_1(2+3y_1)}}\sqrt{a}\right)\nn
&&\times Hl\left(a,{3+(\nu^2+\nu-3)y_1\over 4 \sqrt{y_1(2+3y_1)}}\sqrt{a}; -{\nu-1\over
2},{\nu+2\over 2},{3\over 2},{1\over 2};{y_0-y_1\over \sqrt{y_1(2+3y_1)}}\sqrt{a}\right) \nn
&&-\sqrt{{y_0(1+z)-y_1 \over y_0-y_1}} Hl\left(a,{3+(\nu^2+\nu-3)y_1\over 4
\sqrt{y_1(2+3y_1)}}\sqrt{a}; -{\nu-1\over 2},{\nu+2\over 2},{3\over 2},{1\over
2};{y_0(1+z)-y_1\over \sqrt{y_1(2+3y_1)}}\sqrt{a}\right)\nn
&&\times
Hl\left(a,{1+{1\over4}\nu(\nu+1)y_1\over \sqrt{y_1(2+3y_1)}}\sqrt{a}; -{\nu\over 2},{\nu+1\over
2},{1\over 2},{1\over 2};{y_0-y_1\over \sqrt{y_1(2+3y_1)}}\sqrt{a}\right)
\Biggr]\!\Biggr], \label{Dell0}
\eea
where the source redshift $z_s$ has again been replaced by $z$. The required values of $y_1$ and $a$ can be
found in Figure 3 for all three $b$ domains and $y_0=\OM/(1-\Omega_0)$ is the value of $y$ at $z=0$
[see (\ref{newvariables})]. Even though the above Heun functions contain complex arguments and
parameters, they are real valued functions of the real redshift varable $z$. As soon as these
functions become available in Mathematica, expressions (\ref{Dell0}) and (\ref{Dellinfty}) will
be immediately useful. Untill then, simpler expansions suitable for $|a|=1$ are indicated in
Appendix C.
The solution similar to (\ref{Dell0}) but suitable for $|\zeta_0|>1$ is given by choosing \be
h_1(\zeta)=\zeta^{-\alpha}Hl\left({1\over a},\underline{q};
\alpha,\underline{\beta},\underline{\gamma},\delta;{1\over \zeta}\right), \label{h1b}
\end{equation}
and
\be
h_2(\zeta)=\zeta^{-\beta}Hl\left({1\over a},\underline{q}_{II};
\underline{\alpha}_{II},\underline{\beta}_{II},\underline{\gamma}_{II},\delta;{1\over \zeta}\right),
\label{h2b}
\end{equation}
where seven parameters have now changed:
\bea
&&\underline{q}\equiv{q\over
a}-\alpha\left[\beta\left(1+{1\over a}\right) -{\delta\over a}-\epsilon\right]\nn
&&\underline{\beta}\equiv-\beta+\delta+\epsilon\nn &&\underline{\gamma}\equiv1+\alpha-\beta\nn
&&\underline{q}_{II}\equiv \underline{q}+\left({\delta\over a}+\epsilon\right)
\left(1-\underline{\gamma}\right),\nn &&\underline{\alpha}_{II}\equiv
\alpha+1-\underline{\gamma},\nn &&\underline{\beta}_{II}\equiv
\underline{\beta}+1-\underline{\gamma},\nn &&\underline{\gamma}_{II}\equiv 2 -\underline{\gamma}.
\label{bar}
\eea
For this choice of $h_1$ and $h_2$ the constant in the Wronskian
(\ref{Wronskian}) becomes
\be
C_W=-\beta+\alpha=-(\nu+{1\over 2}).
\end{equation}
From (\ref{Dell}) the
resulting expression for the luminosity distance is
\bea
&&D_{\ell}(\OM,\OL,\nu;z)={c(1+z)\over
H_0(\nu+{1\over 2})\sqrt{\Omega_m}\sqrt{(1+z)-y_1/y_0}}\times\nn
&&\Biggl[\!\Biggl[
\left({y_0(1+z)-y_1
\over y_0-y_1}\right)^{{\nu+1\over 2}}\nn
&&\hskip .25 in \times Hl\left({1\over
a},{4-\nu^2+\nu(1-2\nu)y_1\over 4\sqrt{y_1(2+3y_1)}}{1\over\sqrt{a}}; -{\nu\over 2},{1-\nu\over
2},{1-2\nu\over 2},{1\over 2}; {\sqrt{y_1(2+3y_1)}\over y_0(1+z)-y_1}{1\over\sqrt{a}}\right)\nn
&&\hskip .25 in \times Hl\left({1\over a}, {(3+\nu)(1-\nu)-(3+2\nu)(1+\nu)y_1\over 4
\sqrt{y_1(2+3y_1)}}{1\over\sqrt{a}}; {1+\nu\over 2},{\nu+2\over 2},{3+2\nu\over 2},{1\over 2};
{\sqrt{y_1(2+3y_1)}\over y_0-y_1}{1\over\sqrt{a}}\right) \nn
&&-\left({y_0-y_1 \over
y_0(1+z)-y_1}\right)^{{\nu\over 2}}\nn
&&\hskip .25 in \times Hl\left({1\over a},
{(3+\nu)(1-\nu)-(3+2\nu)(1+\nu)y_1\over 4 \sqrt{y_1(2+3y_1)}}{1\over\sqrt{a}}; {1+\nu\over
2},{\nu+2\over 2},{3+2\nu\over 2},{1\over 2}; {\sqrt{y_1(2+3y_1)}\over
y_0(1+z)-y_1}{1\over\sqrt{a}}\right)\nn
&&\hskip .25 in \times Hl\left({1\over
a},{4-\nu^2+\nu(1-2\nu)y_1\over 4\sqrt{y_1(2+3y_1)}}{1\over\sqrt{a}}; -{\nu\over 2},{1-\nu\over
2},{1-2\nu\over 2},{1\over 2}; {\sqrt{y_1(2+3y_1)}\over y_0-y_1}{1\over\sqrt{a}}\right) \Biggr]\!\Biggr].
\label{Dellinfty}
\eea
The special case of $\Omega_0\equiv\OM+\OL = 1$ can be obtained from (\ref{Dell0}) and (\ref{Dellinfty})
by taking the appropriate limits. Some details of this process along with the resulting
luminosity distance are given in Appendix A.
For those values of $\OM$ and $\OL$ where $|\zeta_0|< 1$
it is clear that for large enough values of $z$, $|\zeta|> 1$ and hence (\ref{Dell0}) is no longer
valid ($Hl$ no longer converges). For some values in the $(\OM,\OL)$ plane above the
$|\zeta_0|= 1$ contour in Figure 2, (\ref{Dell0}) will not converge for a SNe Ia range of $z\sim 0.5$,
but for most values it does.
In the next section several plots of magnitude vs. redshift are made to illustrate the importantce
of take clumping into account when attempting to
determine $\OM$ and $\OL$.
\section{\mz PLOTS FOR CLUMPY UNIVERSES \& CONCLUSIONS} \label{sec-mzplots} In this section several
magnitude-redshift plots are given to illustrate the effects that density clumps can have on the
\mz relation and consequently on a determination of $\OM$ and $\Lambda$ made by using this
relation. Because \mz depends differently on $\OM$ and $\OL$ as a function of redshift for the
FLRW models, both parameters could in principle be determined from a sufficient quantity of
accurate SNe Ia data. Clumping provides an additional parameter $\nu$ which complicates
any such determination.
As can be seen
from (\ref{Dlseries}) the dependence of \mz on this additional parameter could also be
determined
by enough data. However, such a triple determination is certainly more complicated.
What will be
done here to illustrate the effects of the $\nu$ parameter is to plot multiple \mz curves for
various values of all three parameters $\nu, \ \OM$, and $\OL$.
In all plots the unit of distance is taken to be $c/H_0$.
In these figures
$D_{\ell}$ is plotted on a magnitude scale, 5 Log $D_{\ell}$ (\ie the distance modulus plus
5\,Log $10pc\, H_0/c$).
In Figure 4, $\OL$ is held
fixed while $\nu$ and $\OM$ are varied and in Figure 5, $\OM$ is held fixed while $\nu$ and
$\OL$ are varied. In Figure 6, $\Omega_0=\OM+\OL=1$ is fixed while all three parameters
vary.
In Figure 7 the sensitivity of observed magnitudes to variations of $\OM$ is illustrated
by fixing $z=0.83$ and $\OL=0.1$.
In Figure 8 a similar plot is given showing the sensitivity to variations of $\OL$.
The importance of the clumping parameter is easily seen from these last two figures.
If the distance modulus of a source such as SN 1997ap at $z=0.83$ were precisely known
(\eg see the two sample horizontal lines in Figure 7) then a determination of $\OM$
could be made,
assuming $\OL$ were somehow known. Likewise, from Figure 8, a determination of $\OL$ could be made
if $\OM$ were somehow known. From Figure 7 the reader can easily see that
the determined value of $\OM$ depends on the clumping parameter $\nu$. The $\OM$ value
will be about 95\% larger for a $\nu=2$ completely clumpy universe than it will be
for a $\nu=0$ completely smooth FLRW universe. Equivalently, $\OM$ could be
underestimated by as much as 50\% if the FLRW is used. The maximum underestimate is reduced to 33\%
at the smaller redshift of $z=0.5$ (see a similar result for $\OL=0$ in \cite{KVB}).
These conclusions are not sensitive to the value of $\OL$.
Slightly different conclusions follow from Figure 8 about $\OL$.
The discrepancy in the determined value of $\OL$ is $\Delta \OL \sim -0.14$
for $\nu=2$ compared to $\nu=0$, and is not sensitive to the
distance modulus. The discrepancy is halved, $\Delta \OL \sim -0.07$, at
a smaller redshift of $z=0.5$.
A minimal estimate of the quantity of data required to begin distinguishing
between the various $\nu$ values can easily be made. At
$z=0.5$ the differences in observed magnitudes of a SN Ia in a $\nu=0$ (100\% smooth FLRW)
and a $\nu=2$ (100\% clumpy)
universe is about $\Delta m \sim 0.02$ if $\OM\sim 0.2$, and $\Delta m \sim 0.09$ if $\OM\sim 0.8$.
These differences are not sensitive to $\Lambda$.
With corrected-intrinsic and observed magnitude uncertainties of $\pm 0.2$,\ \cite{BD},
data on over 200 SNe Ia will be required if we live in a low density universe and over a dozen
if we live in a higher density one.
The results presented here (\ref{Dell0}),(\ref{Dellinfty}),(\ref{Dell01}), and (\ref{Dellinfty1})
for the `intergalactic' distance--redshift relation are quite general. They contain
corrections (for mass
inhomogeneities) to the standard FLRW result, applicable to observations where
gravitational lensing can be neglected, \ie observations where the conformal (Weyl)
curvature doesn't produce significant average shear in (\ref{Af}). Even though the original
area equation (\ref{Af}) was rigorously established for a particular type of Swiss cheese model,
the resulting equation which neglects lensing (\ref{Ab0}) is expected to be widely applicable
to observations at redshifts of $z=1$ and less.
Application of its solution to a given set of observations requires that
the average fraction of the mass density contained in the observing
beams (\ie the $\nu$ parameter) be determined. This fraction obviously depends on the
number as well as the type of object observed. Collecting CMB radiation at wide angles is
likely to produce a $\nu=0$ value but observing a few dozen SNe Ia might well result
in a value close to $\nu=2$ (\ie we might in fact live in a universe where mass, dark or otherwise, is primarily
associated with galaxies). If a significant fraction of
the universe's mass density is
clumped on galactic scales, then the effects of these clumps on SNe Ia observations should be
taken into account by using the lensing formulas rather than by decreasing $\nu$
to zero.
Recent numerical work by \cite{HD} confirms the assertion that, given galaxy clumping,
the cross-sectional area
of a \underbar{typical} light beam will not follow the FLRW area-redshift relation.
Instead the area will follow more closely one of the `intergalactic' \mz relations given here, until
a lensing event occurs.
The new luminosity distances presented here
represent the theoretical minimum of the observed magnitudes and are especially
applicable to situations where lensing is infrequent (\ie where the most probable value is closer
to the min than the mean). Because \cite{HD} did not include any diffuse
transparent matter, the applicable \mz relations given here are those with $\nu=2$.
For $\Lambda=0$ and $\Omega_0<1$ it is the Dyer-Roeder solution (\ref{DC1})
and for $\Lambda=0$ and $\Omega_0=1$
its the $\nu=2$ solution of Dashevskii \& Slysh (\ref{Dash}).
The $\nu=0$ (standard FLRW) result represents the theoretical `mean' for \mz for a universe
in which only weak-lensing events occur. For extremely non-symmetric probability distributions,
the ``mean"
is not likely the best estimator - in this case the ``most probable" is likely better, \cite{SD}.
\acknowledgements
The author would like to thank Tamkang University for their kind hospitality and support
during an extended visit to Taiwan in the Spring of '97 where this work was first presented.
The author would also like to thank D. Branch and E. Baron for suggesting changes in the final draft.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 23
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The transparency organization WikiLeaks has exposed acting President Michel Temer, who ascended to the Brazilian presidency after a soft coup deposed President Rousseff, as having been an intelligence informant for the United States.
Temer, who has served as Brazil's vice president since 2011, took power Thursday after Brazil's parliament suspended Rousseff pending the results of impeachment proceedings.
The cables — marked "sensitive but unclassified" — contained summaries of conversations Temer, a Brazilian federal lawmaker at the time, had with the U.S. intelligence officials.
The damning evidence was provided in a series of tweets by the WikiLeaks Twitter account that linked to diplomatic cables highlighting the information provided to the U.S. military and National Security Council by Temer.
Clearly, the evolution of Unconventional War (UW), spelled out in the 2010 Special Forces Unconventional Warfare manual, has come to full fruition with the events that have unfolded in Brazil.
This has unfolded in the form of Hybrid War, which is essentially a weaponization of chaos theory – widely embraced across the U.S. military/intelligence spectrum. The UW manual highlights the perceptions of a vast "uncommitted middle population" is essential in the road to success, and that these uncommitted eventually can be turned to oppose their political leaders.
The process encompasses everything from "supporting insurgency" (as in Syria) to "wider discontent through propaganda and political and psychological efforts to discredit the government" (as in Brazil). It explains that when an insurrection takes root and begins to escalate, the "intensification of propaganda; psychological preparation of the population for rebellion" should as well, as has been the case in Brazil.
BRICS is the acronym for an association of five major emerging national economies: Brazil, Russia, India, China and South Africa. The grouping was originally known as "BRIC" before the inclusion of South Africa in 2010.
This in large part explains the turmoil in Brazilian civic society. Although there are many structural problems inherent to the political apparatus in Brazil, these issues were exploited by external powers in an effort to create a more politically appealing environment for the U.S. and their neoliberal cronies to maneuver into positions of power as exhibited by U.S. intelligence informant, Temer ascending to the presidency of Brazil.
"The right wing is the driving force of the protests… Two of the principal groups responsible for organizing and mobilizing the demonstrations are the Free Brazil Movement (MBL) and Students for Liberty (EPL), both of which have direct ties to Charles and David Koch, the right-wing, neocon, US billionaires, as well as other leading figures of the far right, pro-business neoliberal establishment," Draitser wrote.
Make no mistake that there is a massive covert effort by the U.S. to maintain their hegemonic position in global affairs — undertaken by any means necessary — and aimed squarely at the BRICS. The events in Brazul are unquestionably yet another orchestrated coup organized by Washington. This new form of sophisticated Hybrid Warfare has shown itself to be an insidious weapon that can foment regime change without firing a single shot.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,604
|
package com.outofjungle.bluetoothspp.app;
import android.content.Context;
import android.view.LayoutInflater;
import android.view.View;
import android.view.ViewGroup;
import android.widget.ArrayAdapter;
import android.widget.TextView;
import com.outofjungle.bluetoothspp.app.models.Message;
import com.outofjungle.bluetoothspp.app.models.Writer;
import java.util.ArrayList;
public class MessageAdapter extends ArrayAdapter<Message> {
public MessageAdapter(Context context, ArrayList<Message> messages) {
super(context, R.layout.message_item, messages);
}
@Override
public View getView(int position, View convertView, ViewGroup parent) {
if (convertView == null) {
convertView = LayoutInflater.from(getContext()).inflate(R.layout.message_item, parent, false);
}
Message message = getItem(position);
TextView writerName = (TextView) convertView.findViewById(R.id.writerName);
TextView messageText = (TextView) convertView.findViewById(R.id.messageText);
Enum writer = message.getWriter();
if (Writer.ANDROID == writer) {
writerName.setTextColor(0xFF04B404);
} else if (Writer.ARDUINO == writer) {
writerName.setTextColor(0xFF0000FF);
}
writerName.setText(writer.toString());
messageText.setText(message.getText());
return convertView;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,818
|
De Crosstown Line is een metrolijntraject van de New York City Subway gelegen in de boroughs Brooklyn en Queens van New York. Het traject verbindt het noordwesten van Queens met het westen van Brooklyn.
De buurten Long Island City in Queens, en Greenpoint, Williamsburg, Bedford-Stuyvesant, Clinton Hill, Fort Greene en Downtown Brooklyn in Brooklyn worden met het metrolijntraject van de Crosstown Line bediend.
Het hele traject wordt met een lokale dienst verzorgd door de G-trein, die vervolgens in Brooklyn verder zuidwaarts rijdt langs de Culver Line.
Geschiedenis
De eerste plannen voor een verbinding van de verstedelijkte kernen van Queens en Brooklyn dateerden reeds uit 1912. In 1923 werden deze concreter onder aansturen van de Brooklyn-Manhattan Transit Corporation (BMT). Hun uitwerking van de plannen stootte op verzet van burgemeester John F. Hylan en inwoners van de betrokken wijken in Brooklyn omdat de BMT uit kostefficiëntie stond op een aanleg als verhoogde lijn, met viaducten boven de betrokken straten, wat veel extra lawaaihinder voor de betrokken gebieden ging meebrengen. Hierdoor werd een opening gecreëerd om het project door te schuiven naar het nieuwe stadsbedrijf van het Independent Subway System (IND). Op basis van nieuwe plannen werd het project IND gegund in 1928 en opende de eerste, noordelijke sectie als aftakking van de Queens Boulevard Line in 1933. De volledige lijn, tot aan de intersectie met de Culver Line werd afgewerkt in 1937.
Stations
Met het pictogram van een rolstoel zijn de stations aangeduid die ingericht zijn in overeenstemming met de Americans with Disabilities Act van 1990.
Metrotraject in New York
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,165
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essay we produce, we guarantee that it meets the grade you order. Order #451786, there is not much to say. Papers on different subjects can't be created with a single pattern. Hundreds of websites offer essay writer help online.
Business school essay writing service: need help to write my paper best dissertation writing companies project management assignment help people write papers for money write a mla essay essay writing on why you are joining collage homework help us history pay to do my paper. Get started with the best Essay Writing Service around. View samples, read our Fair Use Policy, what people are saying about UK Essays. Such a reputation can be hard to get rid. Introducing the best essay writing service. When we write custom papers, we pay much attention to essay requirements, no detail will get overlooked when an essay writer of ours gets down to work. Every student can order an academic writing piece from. We receive orders from thousands of people worldwide, from the Canada, United States, Australia, United Kingdom and OAE. Professionalism, attention to deadlines, and constant contact with customers are our trademark principles. All you do is register on our site, choose your writer, and send us all the necessary information, the topic of your essay, format, academic level, and the deadline.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,505
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\section{Introduction}
{For almost thirty years, the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence has been a bridge that allows us to relate gravity and strongly coupled conformal field theories \cite{Maldacena:1997re,Witten:1998qj}. Following this spirit, a new holographic dual of a CFT arises, which is defined on a manifold $\mathcal{M}$ with a boundary $\partial \mathcal{M}$, denoted as Boundary Conformal Field Theory (BCFT), proposed by Takayanagi \cite{Takayanagi:2011zk} and Takayanagi et al. \cite{Fujita:2011fp}, extending the AdS/CFT duality. This new holographic dual denoted as AdS/BCFT correspondence, is defined on a manifold
boundary in a $D$-dimensional manifold $\mathcal{M}$ to a $(D+1)$-dimensional asymptotically AdS space $\mathcal{N}$ in order to $\partial \mathcal{N}=\mathcal{M} \cup Q$. Here, $Q$ corresponds to a $D$-dimensional manifold that satisfies $\partial Q=\partial \mathcal{M}$ (see Figure \ref{p}).}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.22]{fig1.jpg}
\caption{Schematic representation of the AdS/BCFT correspondence. Here, $\mathcal{M}$ represents the manifold with boundary $\partial \mathcal{M}$ where the CFT is present. On the other hand, the gravity side is represented by $\mathcal{N}$, which is asymptotically AdS is $\mathcal{M}$. Together with the above, $\partial \mathcal{M}$ is extended into the bulk AdS, which constitutes the boundary of the $D-$dimensional manifold $Q$.}
\label{p}
\end{center}
\end{figure}
{At the moment to explore the AdS/CFT correspondence, we impose the Dirichlet boundary condition at the boundary of AdS,
and therefore we require the Dirichlet boundary condition on $\mathcal{M}$. Nevertheless, according to \cite{Takayanagi:2011zk,Fujita:2011fp}, for AdS/BCFT duality a Neumann
boundary condition (NBC) on $Q$ is required, given that this boundary should be
dynamical, from the viewpoint of holography, and there is no natural definite metric on $Q$ specified from the CFT side \cite{Nozaki:2012qd}.}
{On the other hand, the AdS/BCFT conjecture appears in many scenarios of the transport coefficients, where black holes take a providential role, such for example Hawking-Page phase transition, the Hall conductivity and the fluid/gravity correspondence \cite{Fujita:2011fp,Melnikov:2012tb,dosSantos:2022scy,Miao:2018qkc,Sokoliuk:2022llp,Santos:2021orr,Magan:2014dwa}. Together with the above, this duality finds its natural roots in the holographic derivation of entanglement entropy \cite{Ryu:2006bv} as well as in the Randall-Sundrum model \cite{Randall:1999vf}. In fact, this extension of the CFT's boundary inside the bulk of the AdS-space is considered a modification of a {\em thin} Randall-Sundrum brane, which intersects the AdS boundary. For this brane to be a dynamical object, we need to impose, as was shown before, NBC where the discontinuity in the bulk extrinsic curvature across the {\em defect}, is compensated by the tension from the brane. Furthermore, these boundaries are known as the Randall-Sundrum (RS) branes in the literature.}
{Following the above, Fujita et al. \cite{Fujita:2012fp} propose a model with gauge fields in the AdS$_{4}$ background with boundary RS branes. In this setup, the authors show that the additional boundary conditions impose relevant constraints on the gauge field parameters, deriving the Hall conductivity behavior in the dual field theory. Nevertheless, this approach does not consider the back reaction of the gauge fields on the geometry, constraining the geometry of the empty AdS space. A natural extension and generalization from the above work was constructed in \cite{Melnikov:2012tb}.}
{In the present paper, we are interested in constructing configurations describing a physical system at finite temperature and charge density. For this, we consider the most common playground, provided by the charged AdS$_{4}$ black holes. This background has} already been shown to encode many interesting condensed-matter-like phenomena such as superconductivity/superfluidity \cite{Gubser:2008px,Hartnoll:2008vx} and strange metallic behaviors \cite{Liu:2009dm}, {via an action characterized by the well-known Einstein-Hilbert structure together with a cosmological constant and Abelian gauge fields. It is interesting to note that the above toy model can be extended in the presence of boundaries within a special case of the Horndeski gravity \cite{Horndeski:1974wa}, (see for example \cite{Brito:2018pwe,Brito:2019ose,Santos:2019ljs,Santos:2020xox,Santos:2021guj,DosSantos:2022exb,Santos:2022lxj,Santos:2022uxm}). Here, the gravity theory is given through the Lagrangian
\begin{eqnarray}\label{eq:Lhorn}
{\cal L}_{\rm H}= \kappa \Big[ (R-2\Lambda)\label{L1} -\frac{1}{2}(\alpha g_{\mu\nu}-\gamma\, G_{\mu\nu})\nabla^{\mu}\phi\nabla^{\nu}\phi \Big],
\end{eqnarray}
where $R$, $G_{\mu \nu}$ and
$\Lambda$ are the scalar curvature, the Einstein tensor, and the cosmological constant respectively, $\phi=\phi(r)$ is a scalar field, $\alpha$ and $\gamma$ are coupling constants, while that $\kappa={1}/{(16 \pi G_N)}$, where $G_N$ is the Newton Gravitational constant. The Lagrangian (\ref{eq:Lhorn}) has been exhaustively explored from the perspective of hairy black hole configurations
\cite{Rinaldi:2012vy,Babichev:2013cya,Anabalon:2013oea,Bravo-Gaete:2014haa,Bravo-Gaete:2013dca}, boson and neutron stars \cite{Brihaye:2016lin,Cisterna:2015yla,Cisterna:2016vdx}, Hairy Taub-NUT/Bolt-AdS solutions \cite{Arratia:2020hoy}, as well as holographic applications such that quantum complexity and shear viscosity
\cite{Feng:2018sqm,Feng:2015oea,Bravo-Gaete:2022lno,Bravo-Gaete:2021hlc,Bravo-Gaete:2020lzs}.}
{On the other hand, through this work the physical system analyzed is based on the model proposed by \cite{Melnikov:2012tb,Fujita:2012fp}. Here, as we will see in the following lines, we start from the same Lagrangian for a Horndeski-Maxwell system, this is (\ref{eq:Lhorn}), together with the Maxwell Lagrangian
\begin{equation}
{\cal L}_{\rm M}=-\dfrac{\kappa}{4e^{2}} F^{\mu \nu} F_{\mu \nu}, \label{L3}
\end{equation}
where $e$ is a coupling constant and $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the Maxwell stress tensor, describing the gravity dual of a field theory on a half-plane.} In the simple plane-symmetric black hole ansatz, we have that only tensionless RS branes are allowed, and that the background solution must be not allowed to model the situation with external electric fields, as in \cite{Fujita:2012fp}. {Even more, as a result of the NBC for the gauge fields, and showing in \cite{Melnikov:2012tb}, the charge density $\rho$ in the dual field theory must be supported by an external magnetic field $B$, where the ratio $\rho/B$, which is equal to the Hall conductivity, is a constant inversely proportional to the coefficients.} In our prescription, this represents the topological terms present in the gravity action: namely, a $m^{2}$ in the bulk action, that is, {an} antisymmetric tensor field $M_{\mu\nu}$ {which} is the effective polarization tensor of the term in the boundary action on the RS branes \cite{Cai:2015bsa,Cai:2014oca,Ghotbabadi:2021mus}. Such behaviors are expected for a quantum Hall system tuned to a quantized value of the conductivity. Furthermore, we provided similar results in the AdS/BCFT holographic model, where, {for example}, we will see how accurately it can account for the physical behaviors expected in a quantum Hall system where, {as was showed before, through AdS/BCFT construction} the Hall conductivity is inversely proportional to the coefficients of the terms that appear in the gravity Lagrangian. {Additionally}, the ratio $\rho/B$ will indicate a localized condensate \cite{Hartnoll:2008kx,Hartnoll:2009sz}.
{Just for completeness, as discussed in \cite{Melnikov:2012tb}, for the classical Hall effect, the charge density and the external magnetic field are independent quantities, that is, the $\rho/B$ ratio depends on} the density of conductance electrons. On the other hand, in the quantum Hall Effect (QHE) the transverse conductivity given by $\sigma_{H}$, {has} plateaus that are independent of either $\rho$ or $B$. These plateaus are generally attributed to disorder \cite{Laughlin:1981jd,Moore:1991ks,Avron85}, {being responsible for the existence of localized electron states} \cite{Melnikov:2012tb}. {Here, the localized states fill the gaps between the Landau levels. Nevertheless, there is no active participation in the Hall conductivity.}
Finally, we {study} the properties of holographic paramagnetism-ferromagnetism phase transition in the presence of Horndeski gravity {(\ref{eq:Lhorn}). Here, from the matter field part,} we consider the effects of the Maxwell field {(\ref{L3})} on the phase transition of this system, {following \cite{Zhang:2016nvj,Wu:2016uyj}, introducing a massive 2-form coupled field, and neglect the effects of this 2-form field and gauge fields on the background geometry.} In our analysis, we observe that increasing the strength of parameter $\gamma$, {given in (\ref{eq:Lhorn}), decreases the temperature of the holographic model and leads to a harder formation of the magnetic moment in the black hole background.} {On the other hand, at low temperatures, spontaneous magnetization, and ferromagnetic phase transition happen, but when removes the external magnetic field, this magnetization disappears. As we know, ferromagnetic materials have coercivity, which is the ability to keep their elementary magnets stuck in a certain position. This position can be modified by placing the magnetized material in the presence of an external magnetic field. In this way, a material with high coercivity its elementary magnets resists the change of position. In the material science, experimental framework \cite{Muller2016}, there is a close relationship between the magnetic related to viscosity and coercivity, this relationship was predicted theoretically and observed experimentally. Thus, we have a fundamental role in both cases, that is, between viscosity and coercivity, where they play the so-called activation volume, which is the relevant volume where thermally activated and field-induced magnetization processes occur, respectively. In our work, we will study this way for the paramagnetic material to resist the external magnetic field, through the viscosity/entropy ratio. In our model, this relationship depends on the external magnetic field, the Horndeski parameters, and the boundary size $\Delta\,y_{Q}$ of the RS brane in a non-trivial way.}
{This work is organized as follows: In Section \ref{v1} we consider the gravitational setup, which contains all the information with respect to the AdS$_{4}$/BCFT$_{3}$ duality, showing the solution. Together with the above, in Section \ref{sec:charge} the charge density is obtained for then, in Section \ref{sec:Q} to present the boundary $Q$ profile. In Section \ref{sec:HR}, we perform a holographic renormalization, computing the Euclidean on-shell action, which is related to the free energy
of the corresponding thermodynamic system, where in particular we will focus on the black hole entropy, present in Section \ref{sec:ent}, and the holographic paramagnetism/ferromagnetism phase transition, given in Section \ref{sec:hol}. Finally, Section \ref{v4} is devoted to our conclusions and discussions.}
\section{{Black hole as a probe of AdS/BCFT}}\label{v1}
{As was shown in the introduction, we will present our setup starting} with the total action, which contains all information related to AdS$_{4}$/BCFT$_{3}$ correspondence with probe approximation, so that:
\begin{eqnarray}\label{açao}
S &=&{S^{\mathcal{N}}_{\rm H}+S^{\mathcal{N}}_{\rm M}+S^{\mathcal{N}}_{\rm 2-FF}+S^{\mathcal{N}}_{mat}+S^{Q}_{bdry}+S^{Q}_{mat}+S^{Q}_{ct}},\label{1}
\end{eqnarray}
where
\begin{equation}\label{eq:actionH-M}
S^{\mathcal{N}}_{\rm H}=\int_{\mathcal{N}}d^{4} x \sqrt{-g}\; {\cal L}_{\rm H},\qquad S^{\mathcal{N}}_{\rm M}=\int_{\mathcal{N}}d^{4} x \sqrt{-g}\; {\cal L}_{\rm M},
\end{equation}
with ${\cal L}_{\rm H}$ and ${\cal L}_{\rm M}$ given previously in (\ref{eq:Lhorn})-(\ref{L3}) respectively,
while that
$S^{\mathcal{N}}_{mat}$ is the action associated to matter sources and:
\begin{eqnarray}
S^{Q}_{bdry}&=&2\kappa\int_{Q}{d^{3}x\sqrt{-h}\mathcal{L}_{bdry}}\nonumber\\
S^{Q}_{mat}&=&2\int_{Q}{d^{3}x\sqrt{-h}\mathcal{L}_{mat}},\nonumber\\
S^{Q}_{ct}&=&2\kappa\int_{ct}{d^{3}x\sqrt{-h}\mathcal{L}_{ct}}\,,
\end{eqnarray}
with
{\begin{eqnarray}
&&\mathcal{L}_{bdry}=(K-\Sigma)-\frac{\gamma}{4}(\nabla_{\mu}\phi\nabla_{\nu}\phi n^{\mu}n^{\nu}-(\nabla \phi)^2)K-\frac{\gamma}{4}\nabla_{\mu}\phi\nabla_{\nu}\phi K^{\mu\nu}\,,\label{L5}\\
&&\mathcal{L}_{ct}=c_{0}+c_{1}R+c_{2}R^{ij}R_{ij}+c_{3}R^{2}+b_{1}(\partial_{i}\phi\partial^{i}\phi)^{2}+\cdots,\label{L6}
\end{eqnarray}}
{where in our notations $(\nabla \phi)^2=\nabla_{\mu}\phi\nabla^{\mu}\phi$. In Eq.(\ref{L5}), $\mathcal{L}_{bdry}$ corresponds to the Gibbons-Hawking $\gamma$-dependent terms associated with the Horndeski gravity (\ref{eq:Lhorn}), where $K_{\mu\nu}=h^{\phantom{\mu}\beta}_{\mu}\nabla_{\beta}n_{\nu}$} is the extrinsic curvature, $K=h^{\mu\nu}K_{\mu\nu}$ is the trace of the extrinsic curvature, $h_{\mu\nu}$ is the induced metric, $n^{\mu}$ is an outward pointing unit normal vector to the boundary of the hypersurface $Q$, $\Sigma$ is the boundary tension on $Q$. { $\mathcal{L}_{mat}$ is the matter Lagrangian on $Q$, while that in} Eq. (\ref{L6}) ${\cal L}_{ct}$ represents the boundary counterterms, which do not affect the bulk dynamics and will be neglected.
Following the procedures presented by \cite{Magan:2014dwa,Melnikov:2012tb,Fujita:2011fp,Takayanagi:2011zk,Santos:2021orr} we have imposed the {NBC}:
\begin{eqnarray}
K_{\alpha\beta}-h_{\alpha\beta}(K-\Sigma)-\frac{\gamma}{4}H_{\alpha\beta}=\kappa {\cal S}^{Q}_{\alpha\beta}\,,\label{L7}
\end{eqnarray}
where
\begin{eqnarray}
&&H_{\alpha\beta}\equiv(\nabla_{\sigma}\phi\nabla_{\rho}\phi\, n^{\sigma}n^{\rho}-(\nabla \phi)^2) (K_{\alpha\beta}-h_{\alpha\beta}K)-(\nabla_{\alpha}\phi\nabla_{\beta}\phi)K\,,\label{L8}\\
&&{\cal S}^{Q}_{\alpha\beta}=-\frac{2}{\sqrt{-h}}\frac{\delta S^{Q}_{mat}}{\delta h^{\alpha\beta}}\,.\label{L9}
\end{eqnarray}
Considering {the matter stress-energy tensor on $Q$ as a constant (this is ${\cal S}^{Q}_{\alpha\beta}=0$),} we can write
\begin{eqnarray}
K_{\alpha\beta}-h_{\alpha\beta}(K-\Sigma)-\frac{\gamma}{4}H_{\alpha\beta}=0\,.\label{L10}
\end{eqnarray}
{On the other hand, from the gravitational part, given by the Einstein-Horndeski theory and assuming that $S^{\mathcal{N}}_{mat}$ is constant, varying $S^{\mathcal{N}}_{\rm H}$ and $S^{Q}_{bdry}$ with respect to the dynamical fields, we have:
\begin{eqnarray}
{\cal E}_{\alpha\beta}=-\frac{2}{\sqrt{-g}}\frac{\delta S^{\mathcal{N}}}{\delta g^{\alpha\beta}}\,,\quad {\cal E}_{\phi}=-\frac{2}{\sqrt{-g}}\frac{\delta S^{\mathcal{N}}}{\delta\phi} \,,\quad {\cal F}_{\phi}=-\frac{2}{\sqrt{-h}}\frac{\delta S^{Q}_{bdry}}{\delta\phi} \,,\nonumber\\
\end{eqnarray}
%
where
\begin{eqnarray}
{\cal E}_{\mu\nu}&=&G_{\mu\nu}+\Lambda g_{\mu\nu}-\frac{\alpha}{2}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)\label{11}\nonumber\\
&+&\frac{\gamma}{2}\left(\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi R-2\nabla_{\lambda}\phi\nabla_{(\mu}\phi R^{\lambda}_{\nu)}-\nabla^{\lambda}\phi\nabla^{\rho}\phi R_{\mu\lambda\nu\rho}\right)\nonumber\\
&+&\frac{\gamma}{2}\left(-(\nabla_{\mu}\nabla^{\lambda}\phi)(\nabla_{\nu}\nabla_{\lambda}\phi)+(\nabla_{\mu}\nabla_{\nu}\phi)\Box\phi+\frac{1}{2}G_{\mu\nu}(\nabla\phi)^{2}\right)\nonumber\\
&-&\frac{\gamma g_{\mu\nu}}{2}\left(-\frac{1}{2}(\nabla^{\lambda}\nabla^{\rho}\phi)(\nabla_{\lambda}\nabla_{\rho}\phi)+\frac{1}{2}(\Box\phi)^{2}-(\nabla_{\lambda}\phi\nabla_{\rho}\phi)R^{\lambda\rho}\right),\\
{\cal E}_{\phi}&=&\nabla_{\mu}\left[\left(\alpha g^{\mu\nu}-\gamma G^{\mu\nu}\right)\nabla_{\nu}\phi\right]\,,\label{L11}\\
{\cal F}_{\phi}&=&-\frac{\gamma}{4}(\nabla_{\mu}\nabla_{\nu}\phi n^{\mu}n^{\nu}-(\nabla^{2}\phi))K-\frac{\gamma}{4}(\nabla_{\mu}\nabla_{\nu}\phi)K^{\mu\nu}\,,\label{L12}
\end{eqnarray}
and note that, ${\cal E}_{\phi}={\cal F}_{\phi}$, from the Euler-Lagrange equation.}
{Together with the above, and according to \cite{Rinaldi:2012vy,Babichev:2013cya,Anabalon:2013oea,Bravo-Gaete:2014haa,Bravo-Gaete:2013dca}
, we have a condition that deals to static black hole configurations, avoiding no-hair theorems \cite{Hui:2012qt}.
Here, we need to require that the square of the radial component of the conserved current must vanish identically without restricting the radial dependence of the scalar field. Such discussion implies that in Eq. (\ref{L11}):
\begin{equation}
\alpha g_{rr}-\gamma G_{rr}=0\label{L13}\,,
\end{equation}
and defining $\phi{'}(r)\equiv {\psi}(r)$, where $ (')$ denotes the derivative with respect to $r$, we can show that the equations ${\cal E}_{\phi}=0={\cal E}_{rr}$ are satisfied. In our setup, the four dimensional metric is defined via the following line element
\begin{equation}\label{ansatz}
ds^2= \frac{L^2}{r^2}\left(-f(r)\,dt^2+dx^2+dy^2+\frac{dr^2}{f(r)}\right),
\end{equation}
where $ x_1 \leq x \leq x_2$ and $ y_1 \leq y \leq y_2$, while that from Refs.\cite{Brito:2019ose,Santos:2021orr,Bravo-Gaete:2014haa}, $f(r)$ is the metric function which takes the form
\begin{eqnarray}
&&f(r)=\frac{\alpha L^{2}}{3\gamma}\left[1-\left(\frac{r}{r_{h}}\right)^{3}\right]\,,\label{L14}
\end{eqnarray}
while that $\psi(r)$ reads
\begin{eqnarray}
&&\psi^{2}(r)=(\phi'(r))^2=-\frac{2L^{2}(\alpha+\gamma\Lambda)}{\alpha\gamma r^{2}f(r)}\,,\label{L15}
\end{eqnarray}
where
\begin{equation}\label{eq:phi}
\phi(r)=\pm{\frac { 2\sqrt {-6(\alpha+\Lambda \gamma)}}{3 \alpha}}\, \tanh^{-1} \left(\sqrt{1-\frac{r^3}{r_h^3}} \right)+\phi_0. \end{equation}
Here, $\phi_0$ and $r_h$ are integration constants, where the last one is related to the location of the event horizon. Following the steps of \cite{Santos:2021orr,Brito:2019ose}, performing the transformations
\begin{eqnarray}
&&f(r) \rightarrow \frac{\alpha L^{2}}{3\gamma} f(r),\qquad t \rightarrow \frac{3\gamma}{\alpha L^{2}} t,\nonumber\\
&&x \rightarrow \sqrt{\frac{3\gamma}{\alpha L^{2}}} x, \qquad y \rightarrow \sqrt{\frac{3\gamma}{\alpha L^{2}}} y,\qquad L \rightarrow \sqrt{\frac{\alpha}{3\gamma}} L^2,\label{transfor}
\end{eqnarray}
we have that the line element (\ref{ansatz}) is invariant, but now the metric function $f(r)$ takes the form
\begin{eqnarray}
&&f(r)= 1-\left(\frac{r}{r_{h}}\right)^{3}\,\label{L16}
\end{eqnarray}
while the square of the derivative of the scalar field $\psi^2(r)$ takes the form given previously in (\ref{L15}). Here is important to note that from Eqs. (\ref{L15})-(\ref{eq:phi}) we can see that to have a real scalar field,
$$\alpha+\Lambda \gamma \leq 0,$$
where it vanishes when $\alpha=-\Lambda \gamma$.}
{It is important to note that, from the action (\ref{açao}), we can see that there is another contribution, denoted as $S^{\mathcal{N}}_{\rm 2-FF}$, which is responsible to construct the ferromagnetic/paramagnetic model. The above will be explained in the following section.}
\section{The finite charge density}\label{sec:charge}
{As was shown in the previous section, in the action (\ref{açao}) appears the additional contribution
$$S^{\mathcal{N}}_{\rm 2-FF}=\lambda^2 \int_{\mathcal{N}}d^{4} x \sqrt{-g}\; {\cal L}_{\rm 2-FF},$$
where
\begin{eqnarray}
&&{\cal L}_{\rm 2-FF}=-\frac{1}{12} (dM)^2 - \frac{m^2}{4}M^{\mu \nu} M_{\mu \nu}- \frac{1}{2}M^{\mu \nu}F_{\mu \nu}-\frac{J}{8}V(M).\label{L4}
\end{eqnarray}
Here, the above action defined from the seminal works \cite{Cai:2015bsa,Cai:2014oca}, is coupled through the constant $\lambda$ and constructed via the 2-form $M_{\mu\nu}$, $dM$ is the exterior differential of the 2-form field $M_{\mu\nu}$, this is $(dM)_{\tau \mu \nu} =3 \nabla_{[\tau} M_{\mu \nu]}$ and $(dM)^2=9 \nabla_{[\tau} M_{\mu \nu]} \nabla^{[\tau} M^{\mu \nu]}$, $m$ is a constant related to the mass, while that $V(M)$ describes the self-interaction of polarization tensor, with $J$ a constant, which reads
\begin{equation}
V(M)=(^{*}M_{\mu \nu} M^{\mu \nu})^2=[^{\ast} (M \wedge M)]^2,\label{L4.1}\\
\end{equation}
where $(^{*})$ is the Hodge star operator, this is $^{*}M_{\mu \nu}=\frac{1}{2!} \varepsilon^{\alpha \beta}_{\phantom{\alpha \beta} \mu \nu} M_{\alpha \beta}$ and $\varepsilon^{\alpha \beta}_{\phantom{\alpha \beta} \mu \nu}$ is the Levi-Civita Tensor. In the following lines, will restrict our analysis to the probe approximation, that is, from the action Eq. \eqref{açao}, one can derive the corresponding equations of motions for matter fields in the probe approximation, that is, $e^{2}\to\infty$ and $\lambda\to 0$, so that:
\begin{eqnarray}
\nabla^{\mu} \left(F_{\mu \nu}+\frac{\lambda^2}{4}\,M_{\mu\nu} \right) &=& 0,\label{eom1}\\
\nabla^{\tau} (dM)_{\tau \mu \nu} - m^2 M_{\mu \nu} - J(^{\ast}M_{\tau \sigma}M^{\tau \sigma})(^{\ast}M_{\mu \nu}) - F_{\mu \nu} &=&0\,. \label{eom2}
\end{eqnarray}
Given that we are focusing on the probe limit approximation, we are going to disregard any back reaction coming from the two-form field $M_{\mu \nu}$.
In order to analyze the holographic paramagnetism/ferromagnetism and paraelectric/ferroelectric phase transition, we consider the gauge fields $M_{\mu \nu}$ and $A_{\mu}$ we consider the following ansatz:
\begin{eqnarray}
M_{\mu \nu} &=& -p(r)\,dt\wedge dr + \rho(r)\,dx\wedge dy,\label{Mansatz} \\
A_{\mu} &=& A_{t}(r)\,dt + B x\,dy, \quad F = dA,\label{Fansatz}
\end{eqnarray}
where $B$ is the external magnetic field.} Using \eqref{ansatz}, \eqref{Mansatz}-\eqref{Fansatz} in the background (\ref{L16}), the field equations \eqref{eom1} and \eqref{eom2} are given by
\begin{eqnarray}
A_{t}'+\left(m^2-\frac{4\,J\,r^4\,\rho^2}{L^{4}}\right)\,p &=& 0,\label{L18} \\
\frac{\rho ''}{L^{2}} +\left(\frac{f'}{f}+\frac{2}{r}\right)\,\frac{\rho'}{L^{2}} - \left(\frac{4\,J\,r^2\,p^2}{fL^{4}}+\frac{m^2}{r^2\,f}\right)\,\rho - \frac{B}{r^2\,f} &=& 0,\label{L19} \\
A_t''+\frac{\lambda^2}{4}\,p' &=& 0\,,\label{L20}
\end{eqnarray}
As we work with probe approximation, the back reaction can be neglected. {Together with the above,} given that the behaviors are asymptotically AdS$_{4}$, we can solve the field equations (\ref{L18})-(\ref{L20}) near {to the boundary (this is $r\to 0$)}. Here, asymptotic solutions are given by
\begin{eqnarray}
&&A_{t}(r)\sim\mu-\sigma r,\\
&&p(r)\sim \frac{\sigma}{m^2},\\
&&\rho(r)\sim\rho_{+}r^{\Delta_{+}}+\rho_{-}r^{\Delta_{-}}-\frac{B}{m^2},\label{eq:rho}\\
&&\Delta_{\pm}=\frac{-1\pm\sqrt{1+4L^2 m^2}}{2}.\label{eq:m}
\end{eqnarray}
{Here,} $\rho_{+}$ and $\rho_{-}$ correspond to the source and vacuum expectation value of the dual operator in the boundary field theory (up to a normalization factor), respectively. {It is worth pointing out that one should take $\rho_{+}=0$, in order to obtain condensation spontaneously \cite{Cai:2015bsa}. From Eq. (\ref{eq:rho}), we can define $\rho_{+}$ and $\rho_{-}$ as
\begin{eqnarray}
&&\rho_{+}\equiv r^{-\Delta_{+}}_{h},\qquad \rho_{-}\equiv r^{-\Delta_{-}}_{h},
\end{eqnarray}
yielding to the asymptotic solution $\rho(r)$ the following structure
\begin{eqnarray}
\rho(r)\sim \left(\frac{r}{r_{h}}\right)^{\Delta_{+}}+\left(\frac{r}{r_{h}}\right)^{\Delta_{-}}-\frac{B}{m^2}.
\end{eqnarray}
Additionally, and according to \cite{Miao:2018qkc}, we can to analyze the electromagnetic field, extracted from the four dimensional {\em electromagnetic duality}, in a sense that the theory is invariant under
\begin{eqnarray}
{F}_{\mu\nu}\to ^{*}{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta} {F}^{\alpha\beta}\label{S3},
\end{eqnarray}
where, as before, $\varepsilon_{\alpha \beta \mu \nu}$ is the Levi-Civita Tensor, transforming the electric field into a magnetic field and vice versa.
Such duality gives that, from the action (\ref{L3}), ${F}_{\mu\nu} {F}^{\mu\nu}=(^{*}{F}_{\mu\nu})(^{*}{F}^{\mu\nu})$, showing that is invariant under (\ref{S3}). Besides, the transformation (\ref{S3}) shows that $\mathcal{F}_{rt}\to (^{*}\mathcal{F}_{rt})=\mathcal{F}_{xy}=\sigma=B$, where $\sigma$ ($B$) is the constant related to the electric (magnetic) field.}
\section{Q-boundary profile}\label{sec:Q}
In this section, we present the boundary $Q$ profile, we assume that $Q$ is parameterized by the equation {$y=y_{Q}(r)$, analyzing the influence of the Horndeski action (\ref{eq:Lhorn}), (\ref{eq:actionH-M}). For this, to find the extrinsic curvature, one has to consider the induced metric on this surface, which reads
\begin{eqnarray}
ds^{2}_{\rm ind}=\frac{L^{2}}{r^{2}}\left(-f(r)dt^{2}+dx^{2}+\frac{g^{2}(r)dr^{2}}{f(r)}\right)\,,\label{Q1}
\end{eqnarray}
where $g^{2}(r)=1+{y'}^{2}(r)f(r)$ {and $({'})$ denotes the derivative with respect to the coordinate $r$. Here, the normal vectors on Q are} represented by
\begin{eqnarray}
n^{\mu}=\frac{r}{Lg(r)}\, \left(0,0,\, 1, \, -{f(r)y{'}(r)}\right)\,.\label{Q2}
\end{eqnarray}
Considering the field equation ${\cal F}_{\phi}=0$ (\ref{L12}), one can solve the Eq. \eqref{L10}, yielding
\begin{eqnarray}
y{'}(r)&=&\frac{(\Sigma L)}{\sqrt{4+\dfrac{\gamma\psi^{2}(r)}{4}-(\Sigma L)^{2}\left(1-\left(\dfrac{r}{r_{h}}\right)^{3}\right)}}\,,
\end{eqnarray}
\noindent and, with $\psi^2(r)$ given {previously in} Eq. \eqref{L15}, we have
\begin{eqnarray}
y{'}(r)&=&\frac{(\Sigma L)}{\sqrt{4-\dfrac{\xi L^{2}}{2r^{2}\left(1-\left(\dfrac{r}{r_{h}}\right)^{3}\right)}-(\Sigma L)^{2}\left(1-\left(\dfrac{r}{r_{h}}\right)^{3}\right)}}\,,\label{prof}
\end{eqnarray}
where we define
\begin{equation}\label{eq:xi}
\xi=\dfrac{\alpha+\gamma\Lambda}{\alpha}.
\end{equation}
With all this information, we can plot the $y_{Q}$ profile from Eq. (\ref{prof}), representing the holographic description of BCFT considering the theory (\ref{eq:Lhorn}).
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.12]{fig01.jpg}
\caption{{The figure shows the numerical solution for Q boundary profile from Eq. (\ref{prof})} for the black hole within Horndeski gravity, considering the values for $\theta'=2\pi/3$, $\theta=\pi-\theta'$, $\Lambda=-1$, $\alpha=8/3$ with $\gamma=0$ ({\sl pink curve}), $\gamma=0.1$ ({\sl blue dashed curve }), $\gamma=0.2$ ({\sl red dot dashed} curve), and $\gamma=0.3$ ({\sl green thick curve}). The dashed parallel vertical lines represent the UV solution, Eq. \eqref{profuv}, that is, Randall-Sundrum branes. The region between the curves Q represents the bulk {$\mathcal{N}$.}}\label{p1}
\end{center}
\end{figure}}
{On the other hand, following the steps of \cite{Melnikov:2012tb,dosSantos:2022scy}, we have that the NBC on the gauge field is $n^{\mu}{F}_{\mu\nu}|_{Q}=0$, and $B=\sigma$}. The holographic model (AdS$_{4}$/BCFT$_{3}$) predicts that a constant boundary current in the bulk induces a constant current on the boundary $Q$. Such boundary current can be measured in materials graphene-like. Furthermore, $n^{\mu}M_{\mu\nu}|_{Q}=0$ provide
\begin{eqnarray}\label{eq:rho/B}
\frac{\rho(r)}{B}=\dfrac{f(r)y'(r)}{m^{2}}.
\end{eqnarray}
{Here, the density $\rho$ and the magnetic field $B$} are no longer two independent parameters. As {the ratio is the Hall conductivity,} this is very reminiscent of the quantum Hall effect (QHE), where this ratio is independent of both $\rho$ and $B$ and is inversely proportional to the topological coefficients, which in our case are the coupling constant $\gamma$ {presents in the Horndeski gravity, together with the parameter from the antisymmetric tensor field $M_{\mu\nu}$, this is $m^2$. In our case, the equation of $y'$ from (\ref{prof}) and then the $\rho/B$ ratio (\ref{eq:rho/B}) can be analyzed by numerical calculations, being represented in Fig. \ref{p0}. Here, we show the ratio $\rho/B$ with respect to external magnetic field $B$ for different values of the Horndeski gravity parameter $\gamma$, where we introduced $\Sigma L=\cos(\theta')$, where $\theta'$ represents the angle between the positive direction of the $y$ axis and $Q$.} At the boundary $Q$, the curves of solutions in the ($\rho,B$) plane will be a localized condensate \cite{Hartnoll:2008kx,Hartnoll:2009sz}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=1]{f.pdf}
\caption{{Graphic of the ratio $\rho/B$ with respect to external magnetic field $B$ versus $r$, for different values of the Horndeski parameter $\gamma$. Here, we consider the values $r_{h}=0.1$, $L=1$, $\theta{'}=2\pi/3$, $\Lambda=-1$, $\alpha=0.5$, $m=1$, and $\gamma=1$ (represented through the blue curve), $\gamma=4$ (represented through the red curve), and $\gamma=8$ (represented through the green curve).}}
\label{p0}
\label{ylinhaz}
\end{center}
\end{figure}
{Together with the above, in addition to the above numerical solution, we} can analyze some particular cases regarding the study of the UV and IR regimes. Thus, {for the first case,} performing an expansion at $r\to 0$ with, { as before}, $\Sigma L=\cos(\theta{'})$, the equation (\ref{prof}) becomes
{\begin{eqnarray}
y_{_{UV}}(r)=y_{0}+ \sqrt{\frac{2}{-\xi L^{2}}}\,{r\cos(\theta{'})},
\end{eqnarray}
where $y_0$ is an integration constant.} In the above equation, considering $\xi\to-\infty$, we have
\begin{eqnarray}
y_{_{UV}}(r)=y_{0}={\rm constant}.\label{profuv}
\end{eqnarray}
This is equivalent to keeping $\xi$ finite and a zero tension limit $\Sigma\to 0$, {considering the cases $\theta'=\pi/2$ and $\theta'=3\pi/2$.
Now, for this regime, we have that the $\rho/B$ ratio takes the form
\begin{eqnarray}
\frac{\rho}{B}=\sqrt{\frac{2}{-\xi L^{2}}}\dfrac{\cos(\theta{'})}{m^{2}}.\label{ratio}
\end{eqnarray}
Here, it turns out a straightforward generalization of a known AdS$_{4}$/CFT$_{3}$ solution, given by the plane-symmetric charged four-dimensional AdS black hole, where only allows for tensionless RS branes in the AdS$_{4}$/BCFT$_{3}$ construction \cite{Melnikov:2012tb}.} In this case, requires that the static uniform charge density is supported by a magnetic field. Specifically, we found that $\rho/B$ is a constant proportional to a ratio of the coefficients appearing in the Horndeski gravity. { These analyses indicate a generalization of the AdS$_{4}$ black hole can describe a quantum Hall system at a plateau of the transverse conductivity. Additionally,} the AdS/BCFT setup yields that the Hall conductivity is inversely proportional to a sum of the coefficients of the topological terms appearing in the gravity Lagrangian. That is, we obtain that $\sigma_{H}=\rho/B$, which from the equation (\ref{ratio})
{\begin{eqnarray}
\sigma_{H}=\sqrt{\frac{2}{-\xi L^{2}}} \dfrac{\cos(\theta{'})}{m^{2}},\label{Hall1}
\end{eqnarray}
where, as was shown in the introduction, in QHE the conductivity is related to the number of filled Landau levels (filling fraction), namely, by
\begin{eqnarray}
\dfrac{h}{e^{2}}\sigma_{H}=\sqrt{\frac{2}{-\xi L^{2}}} \dfrac{\cos(\theta{'})}{m^{2}},
\end{eqnarray}}
where $e^{2}/h$ is the magnetic flux quantum. In this way, the holographic description seems to provide results similar to the description of the QHE {obtained in } \cite{Moore:1991ks,Avron85}. In our case, we have an extension of the covariant form of the Hall relation $\rho=\sigma_{H}B$.
For the IR case, we take $r\to +\infty$ so that from Eq. \eqref{prof} implies that $\lim_{r\to +\infty} (\phi'(r))^2=0$, and then $\phi=$ constant, which ensures a genuine vacuum solution. Plugging this result in Eq. (\ref{prof}), in the limit $r\to\infty$, we have
{\begin{eqnarray}
y'_{_{IR}}(r)\sim \left(\frac{r_h}{r}\right)^{3/2}+O\left(\frac{1}{r^{2}}\right),
\end{eqnarray}
and $y^{'}_{_{IR}}(r) \to 0$ when $r \to +\infty$, which implies from (\ref{ratio}) that $\rho/B\to 0$}. Such value {becomes the on-shell} action finite.
{For the sake of completeness,} an approximate analytical solution for $y(r)$ can be obtained by performing an expansion for $\xi$ very small from Eq. (\ref{prof}), {this is
$$y'_{Q}=\frac{\cos(\theta')}{\sqrt{
4-\cos^2(\theta')f(r)}}+\frac{L^2
\cos(\theta) \xi}{4 r^2 f(r) (4-\cos^2(\theta')f(r))^{3/2}}+ O(\xi^2),$$
with $f$ given previously in (\ref{L16}), and considering} this expansion up to the first order, we obtain
{\begin{eqnarray}
&&y_{Q}(r)=y_{0}+\frac{r\cos(\theta{'})}{\sqrt{ (r^3-r_h^3)\cos^2(\theta{'})+4 r_h^3}}\,\sqrt{\frac{ 4 r_h^3-(r^3-r_h^3)\cos(2\theta{'})}{4-\cos^2(\theta{'})}}\nonumber\\
&&\times _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{r^3\cos^2(\theta{'})}{r^{3}_{h} (4-\cos^2(\theta{'}))}\right)+\xi \int \frac{L^2
\cos(\theta)}{4 r^2 f(r) (4-\cos^2(\theta')f(r))^{3/2}} \, dr+O(\xi^2),\label{prof1}
\end{eqnarray}}
{where $_2F_1(a,b;c;x)$ is the hypergeometric function.}
\section{Holographic renormalization}\label{sec:HR}
In our setup, we will compute the Euclidean on-shell action, which is related to the free energy of the corresponding thermodynamic system. Thus, our holographic scheme takes into account the contributions of AdS$_{4}$/BCFT$_{3}$ correspondence within Horndeski gravity. Let us start with the Euclidean action given by $I_{E}=I_{bulk}+2I_{bdry}$, i.e.,
\begin{eqnarray}
&&{I_{bulk}=-\frac{1}{16\pi G_N}\int_{\mathcal{N}}{\sqrt{g}d^{4}x\left(R-2\Lambda+\frac{\gamma}{2} G_{\mu \nu} \nabla^{\mu} \phi \nabla^{\nu} \phi\right)}-\frac{1}{8\pi G_N}\int_{\mathcal{M}}{d^{3}x\sqrt{\bar{\gamma}}\mathcal{L}_{\mathcal{M}}},}\\
&&{\mathcal{L}_{\mathcal{M}}=K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})}-\frac{\gamma}{4}(\nabla_{\mu}\phi\nabla_{\nu}\phi n^{\mu}n^{\nu}-(\nabla\phi)^{2})K^{(\bar{\gamma})}-\frac{\gamma}{4}\nabla^{\mu}\phi\nabla^{\nu}\phi K^{(\bar{\gamma})}_{\mu\nu}.}
\end{eqnarray}
Together with the above, $g$ is the determinant of the metric $g_{\mu\nu}$ on the bulk $\mathcal{N}$, {while that} $\bar{\gamma}$ is the induced metric, the surface tension on $\mathcal{M}$ {is represented} with $\Sigma^{(\bar{\gamma})}$, {and $K^{({\bar{\gamma}})}$ corresponds to the extrinsic curvature on $\mathcal{M}$}. On the other hand, for the boundary, {we have the following expressions}
\begin{eqnarray}
&&I_{bdry}=-\frac{1}{16\pi G_{N}}\int_{\mathcal{N}}{\sqrt{g}d^{4}x {\left(R-2\Lambda+\frac{\gamma}{2} G_{\mu \nu} \nabla^{\mu} \phi \nabla^{\nu} \phi\right)}}-\frac{1}{8\pi G_N}\int_{Q}{d^{3}x\sqrt{h}\mathcal{L}_{bdry}},\\
&&\mathcal{L}_{bdry}=(K-\Sigma)-\frac{\gamma}{4}(\nabla_{\mu}\phi\nabla_{\nu}\phi n^{\mu}n^{\nu}-(\nabla\phi)^{2})K-\frac{\gamma}{4}\nabla^{\mu}\phi\nabla^{\nu}\phi K_{\mu\nu}.
\end{eqnarray}
Thus, in order to compute the bulk action $I_{bulk}$, we consider the induced metric on { the bulk, which is obtained from (\ref{ansatz}) after the transformation $\tau=i t$, given by
\begin{eqnarray}
ds^{2}_{ind}=\bar{\gamma}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{L^{2}}{r^{2}}\left(f(r)d\tau^{2}+dx^{2}+dy^{2}+\frac{dr^{2}}{f(r)}\right).\label{mett1}
\end{eqnarray}
Here, we have that $0 \leq \tau \leq \beta$, where from Eq. (\ref{L16})
\begin{eqnarray}\label{eq:Th}
\beta=\frac{1}{T}=\left(\frac{|f'(r_h)|}{4 \pi}\right)^{-1}=\frac{4 \pi r_h}{3},
\end{eqnarray}
where $T$ is the Hawking Temperature, the above allows us to obtain the following quantities:
$$R=-\frac{12}{L^{2}},\qquad \Lambda=-\frac{3}{L^{2}},\qquad K^{({\bar{\gamma}})}=\frac{3}{L},\qquad \Sigma^{(\bar{\gamma})}=\frac{2}{L}.$$}
Thus, we have all elements needed to construct the bulk action $I_{bulk}$. {In the process of holographic renormalization, we need to introduce a cutoff $\epsilon$ to remove the IR divergence on the bulk side and we can provide that:}
{
\begin{eqnarray}
&&I_{bulk}=\frac{1}{16\pi G_{N}}\int{d^{2}x}\int^{\frac{4 \pi r_h}{3}}_{0}{d\tau}\int^{r_{h}}_{\epsilon}{dr\sqrt{g}\left(R-2\Lambda+\frac{\gamma}{2}G^{rr}\psi^{2}(r)\right)}\nonumber\\
&&+\frac{1}{16\pi G_{N}}\int{d^{2}x}\int^{\frac{4 \pi r_h}{3}}_{0}{d\tau}{\frac{L^{2}\sqrt{f(\epsilon)}}{\epsilon^{3}}},\\
&&I_{bulk}=-\frac{L^{2}V}{8r^{2}_{h}G}\left(1-\frac{\xi}{4}\right),\label{eq:bulk}
\end{eqnarray}
with $\xi$ given previously in (\ref{eq:bdry1}) and, in our notations, $V=\int{d^{2}x}=\Delta{x} \Delta{y}=(x_2-x_1)(y_2-y_1)$.}
Now, computing the $I_{bdry}$, {we introduce a cutoff $\epsilon$ to remove the UV divergence on the boundary side, and with this information, we have:}
\begin{eqnarray}
I_{bdry}=\frac{r_{h}L^{2}{\Delta y_{Q}}}{2G_{N}}\left(1-\frac{\xi}{4}\right)\int^{r_{h}}_{\epsilon}{\frac{\Delta y_{_Q}(r)}{r^{4}}dr}-\frac{r_{h}L^{2}\sec(\theta{'}){\Delta y_{Q}}}{2G_{N}}\int^{r_{h}}_{\epsilon}{\frac{\Delta y_{_Q}(r)}{r^{3}}dr}\label{Idry}
\end{eqnarray}
Here, $\Delta y_{Q}$ is a constant and {$\Delta y_{Q}(r):=y_{Q}(r)-y_0$ is obtained from the equation (\ref{prof1}).} As we know, from the point of view of AdS/CFT correspondence, IR divergences in AdS correspond to UV divergences in CFT. This relationship is known as the IR-UV connection. {Thus, based on this duality, we can reduce the above equation (\ref{Idry}) after some eliminations of terms that produce divergences to the following form:}
\begin{eqnarray}
&&I_{bdry}=-\frac{L^{2}\Delta\,y_{Q}}{2 G_{N}}\left(1-\frac{\xi}{4}\right)\left(\frac{\xi\,L^{2}b(\theta{'})}{5r^{4}_{h}}+\frac{q(\theta^{'})}{4r^{2}_{h}}\right)\nonumber\\
&&+\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}}{2G_{N}}\left(\frac{\xi\,L^{2}b(\theta{'})}{4r^{3}_{h}}+\frac{q(\theta^{'})}{2r_{h}}\right),\label{eq:bdry1}
\end{eqnarray}
\noindent where
\begin{eqnarray}
b(\theta{'})=\frac{\cos(\theta{'})}{4(4-\cos^{2}(\theta{'}))^{3/2}},\qquad
q(\theta{'})=\frac{\cos(\theta{'})}{\sqrt{4-\cos^{2}(\theta{'})}}\,\label{eq:bdry2}.
\end{eqnarray}
With all the above information, from Eqs. (\ref{eq:bulk}) and (\ref{eq:bdry1})-(\ref{eq:bdry2}), we can compute $I_{E}=I_{bulk}+2I_{bdry}$ as:
\begin{eqnarray}
&&I_{E}=-\frac{L^{2}V}{8r^{2}_{h}G_N}\left(1-\frac{\xi}{4}\right)-\frac{L^{2}\Delta\,y_{Q}}{G_{N}}\left(1-\frac{\xi}{4}\right)\left(\frac{\xi\,L^{2}b(\theta{'})}{5r^{4}_{h}}+\frac{q(\theta^{'})}{4r^{2}_{h}}\right)\nonumber\\
&&+\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}}{G_{N}}\left(\frac{\xi\,L^{2}b(\theta{'})}{4r^{3}_{h}}+\frac{q(\theta^{'})}{2r_{h}}\right)\label{freeEBH}.
\end{eqnarray}
{Here, $I_{E}$ is the approximated analytical expression for the Euclidean action. This equation is essential to construct the free energy and extract all thermodynamic quantities in our setup, as we show in the next section.}
\section{Black hole entropy}\label{sec:ent}
Now, we will compute the entropy related to the black hole considering the contributions of the AdS/BCFT correspondence in the Horndeski gravity. Free energy is defined as
\begin{equation}\label{FE}
\Omega=T I_E \,,
\end{equation}
one can obtain the corresponding entropy as:
\begin{eqnarray}
S=-\frac{\partial\,\Omega}{\partial T}\,\label{BT7}
\end{eqnarray}
{where $T$ is the Hawking Temperature}. By plugging the Euclidean {\em on-shell action} $I_E$ from Eq.\eqref{freeEBH}, {and replacing $T$ obtained previously in (\ref{eq:Th}), we have
\begin{eqnarray}\label{eq:ent-total}
S_{\rm total}&=&S_{\rm bulk}+S_{\rm bdry},
\end{eqnarray}
where
\begin{eqnarray}
S_{\rm bulk}&=&\frac{L^{2}V}{4r^{2}_{h}G_{N}}\left(1-\frac{\xi}{4}\right),\label{eq:Entbulk}\\
S_{\rm bdry}&=&\frac{L^{2}\Delta\,y_{Q}}{G_{N}}\left(1-\frac{\xi}{4}\right)\left(\frac{\xi\,L^{2}b(\theta{'})}{5r^{4}_{h}}+\frac{q(\theta^{'})}{4r^{2}_{h}}\right)\nonumber\\
&-&\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}}{G_{N}}\left(\frac{\xi\,L^{2}b(\theta{'})}{4r^{3}_{h}}+\frac{q(\theta^{'})}{2r_{h}}\right).\label{BT8}
\end{eqnarray}
The interpretation for this total entropy can be identified with the Bekenstein-Hawking formula for the black hole:
\begin{eqnarray}
S_{BH}=\frac{A}{4G_{N}}\label{BT9}\,,
\end{eqnarray}
where, in this case
\begin{eqnarray}
A&=&\frac{L^{2}V}{2r^{2}_{h}}\left(1-\frac{\xi}{4}\right)+4L^{2}\Delta\,y_{Q}\left(1-\frac{\xi}{4}\right)\left(\frac{\xi\,L^{2}b(\theta{'})}{5r^{4}_{h}}+\frac{q(\theta^{'})}{4r^{2}_{h}}\right)\nonumber\\
&&-4L^{2}\sec(\theta{'})\Delta\,y_{Q}\left(\frac{\xi\,L^{2}b(\theta{'})}{4r^{3}_{h}}+\frac{q(\theta^{'})}{2r_{h}}\right).\,\label{BT10}
\end{eqnarray}
Here, $A$ is the total area of the AdS black hole in the Horndeski contribution terms for the bulk and the boundary $Q$. We can see that the information is bounded by the black hole area. Then, the equation \eqref{BT10} suggests that the information storage increases with increasing $|\xi|$, as long as $\xi<0$.
Together with the above, with respect to the boundary contribution of (\ref{BT8}), we have that this expression is the entropy of the BCFT corrected by the Horndeski terms parametrized by $\xi$, given previously in (\ref{eq:xi}). In this case, the results presented in Refs. \cite{Melnikov:2012tb,Magan:2014dwa} are recovered in the limit $\xi\to 0$. Besides, still analyzing Eq. \eqref{BT8}, due to the effects of the Horndeski gravity, there is a non-zero boundary entropy even if we consider the zero temperature scenario, similar to an extreme black hole. This can be seen if one takes the limit $T\to 0$ (or $r_h \to \infty$) in Eq.\eqref{BT8}, then we do not get the denominated residual boundary entropy, as discussed in \cite{Santos:2021orr}.
{On the other hand, through Eq. (\ref{ratio}) we have
\begin{eqnarray}
S^{magnetic}_{bdry}&=&\frac{L^{2}\Delta\,y_{Q}}{G_{N}}\left(1-\frac{\xi}{4}\right)\left(-\frac{2B^{2}\cos^{2}(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{5r^{4}_{h}}+\frac{q(\theta^{'})}{4r^{2}_{h}}\right)\nonumber\\
&-&\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}}{G_{N}}\left(-\frac{2B^{2}\cos^{2}(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{4r^{3}_{h}}+\frac{q(\theta^{'})}{2r_{h}}\right),\label{BT11ext}
\end{eqnarray}
where $m^{2}>-1/(4L^2)$. For the entropy bound, the restriction on $m^{2}$ comes from Eq. (\ref{eq:m}). A well-defined probe limit demands that the charge density contributed by the polarization should be finite. At low temperatures, below the critical, in the ferromagnetic region, we can observe that our entropy is $S_{magnetic}^{bdry}\propto B^{2}$, that is, has a square dependence on the external magnetic field and this is a characteristic of ferromagnetic systems. Furthermore, we can observe that $S^{magnetic}_{bdry}$ is the magnetic entropy of the boundary $Q$, and we can observe that for ferromagnetic materials, the magnetic entropy is associated with the disorder of the magnetic moments. In addition, these materials have spontaneous magnetization. So when we remove the applied magnetic field, they still show magnetization.
\section{Holographic paramagnetism/ferromagnetism phase transition}\label{sec:hol}
In this section, we present the holographic paramagnetism/ferromagnetism phase transition through the boundary contribution from the entropy (\ref{BT11ext}). For this, we start considering the free energy $\Omega$ from (\ref{freeEBH}) -(\ref{FE}), where the first law of black holes thermodynamics, considering the canonical ensemble, takes the form
\begin{eqnarray}\label{eq:first-law}
d\Omega=-P dV-S dT,
\end{eqnarray}
where, in addition to the entropy $S$ as well as the Hawking temperature $T$, the pressure $P$ and the volume $V$ appear, yielding
$$\Omega=\epsilon-TS,$$
where $\epsilon$ takes the role of the energy density.
As a first thermodynamic quantity to study, we will consider the entropy $S$, from Eq. (\ref{eq:ent-total}), calculated in the previous section, and represented graphically in Fig. \ref{ST}, with respect to the Hawking temperature $T$ (\ref{eq:Th}). Here, in the right panel (left panel) there is (not) an external magnetic field $B$. Concretely, we see that the right panel exhibit similar behavior as analyzed in \cite{PhysRevB.72.024403}, as for example ferromagnetic materials with nearly zero coercivity and hysteresis. On the other hand, in the left panel, when the external magnetic field is removed (this is $B=0$), we still have a disorder of magnetic moments, this is a characteristic of paramagnetism.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.71]{f07.pdf}
\includegraphics[scale=0.71]{f071.pdf}
\caption{{\sl Right panel:} The behavior of the entropy $S$ with the temperature $T$ with different values for $\alpha=8/3$, $m=1/8$, $B=(4/5)T$, $\rho=1/4$, $\Lambda=-1$, $V=1$, $G_N=1$, $\theta{'}=2\pi/3$ with $\gamma=1$ ({\sl pink curve}), $\gamma=4$ ({\sl red dot dashed curve}), $\gamma=8$ ({\sl green thick curve}). {\sl Left panel:} The behavior of the entropy $S$ with respect the temperature $T$, with different values for $B=0$.}\label{p3}
\label{ST}
\end{center}
\end{figure}
The second parameter that we analyze is the heat capacity $C_{V}$, which allows us to analyze local thermodynamic stability, defined in the following form
\begin{equation}
\begin{gathered}
C_{V}=T\bigg(\frac{\partial S}{\partial T}\bigg)_{V}=-T\bigg(\frac{\partial^{2} \Omega}{\partial T^{2}}\bigg)_{V},\label{eq:QUANT.1}
\end{gathered}
\end{equation}
where the sub-index $V$ from Eq. (\ref{eq:QUANT.1}) represents at volume constant. From Fig. \ref{p4}, we can see that in the right panel, the black hole can switch between stable ($C_V>0$) and unstable ($C_V<0$) phases, depending on the sign of heat capacity $C_V$. This phase transition occurs, due to the spontaneous electric polarization, which was realized in our model from the application of the magnetic external field. Moreover, in the region $C_V>0$, we have structures built like magnetic domes on the boundary $Q$. Additionally, in Fig. \ref{p4}, one can see the influence of Horndeski gravity (represented via the constant $\gamma$) with respect to the temperature $T$, where the phase transition occurs for some ranges of values for $T$ when the external magnetic field is null, that is, $B=0$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.7]{f08.pdf}
\includegraphics[scale=0.7]{f081.pdf}
\caption{{\sl Right panel:} The behavior of the heat capacity $C_{V}$ with the temperature $T$ with different values for $\alpha=8/3$, $m=1/8$, $B=(4/5)T$, $\rho=1/4$, $\Lambda=-1$, $\theta{'}=2\pi/3$ with $\gamma=1$ ({\sl pink curve}), $\gamma=4$ ({\sl red dot dashed curve}), $\gamma=8$ ({\sl green thick curve}). {\sl Left panel:} The behavior of the heat capacity $C_{V}$ with respect the temperature $T$, with different values for $B=0$.}\label{p4}
\end{center}
\end{figure}
Additionally, we can obtain the heat capacity at constant pressure $C_P$, which reads
\begin{eqnarray}\label{eq:CP}
C_{P}=T\bigg(\frac{\partial S}{\partial T}\bigg)_{P},
\end{eqnarray}
and, from Fig. \ref{pp1}, we can see that in the right panel, the black hole can switch between stable ($C_P>0$), describing a ferromagnetic material, and unstable ($C_P<0$), describing a paramagnetic material, depending on the sign of heat capacity. This phase transition occurs, as in the previous case, due to spontaneous electric polarization. Moreover, in the region $C_P>0$, we have structures built like magnetic domes on the boundary $Q$, wherein the experimental specific frame, these heat curves without magnetic field can represent a material like $DyAl2$ \cite{PhysRevB.72.024403}. On the other hand, the left panel represents the heat capacity $C_P$ where $B=0$, where we can see, that is locally unstable ($C_P<0$).
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.71]{fr.pdf}
\includegraphics[scale=0.71]{fr1.pdf}
\caption{{\sl Right panel:} The behavior of the $C_{P}$ with respect to the temperature $T$ with different values for $\alpha=8/3$, $m=1/8$, $B=(4/5)T$, $\rho=1/4$, $\Lambda=-1$, $\theta{'}=2\pi/3$ with $\gamma=1$ ({\sl pink curve}), $\gamma=4$ ({\sl red dot dashed curve}), $\gamma=8$ ({\sl green thick curve}). {\sl Left panel:} The behavior of $C_{P}$ with respect $T$, with different values for $B=0$.}\label{pp1}
\end{center}
\end{figure}
{Additionally, we can derive other quantities, as for example the magnetization density $m$, and magnetic susceptibility $\chi$, following the steps of \cite{Hartnoll:2009sz}, given by}
{\begin{eqnarray}
m&=&-\bigg(\frac{\partial\,\Omega}{\partial B}\bigg)\nonumber\\
&=&\frac{L^{2}\Delta\,y_{Q}T}{G_{N}}\left(1-\frac{\xi}{4}\right)\left(\frac{4\cos^{2}(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{5r^{4}_{h}}\right)-\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}T}{G_{N}}\left(\frac{\cos(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{4r^{3}_{h}}\right),\label{mag2}\\
\chi&=&\bigg(\frac{\partial^{2} \Omega}{\partial B^{2}}\bigg)\nonumber\\
&=&-\frac{L^{2}\Delta\,y_{Q}T}{G_{N}}\left(1-\frac{\xi}{4}\right)\left(\frac{4B\cos^{2}(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{5r^{4}_{h}}\right)+\frac{L^{2}\sec(\theta{'})\Delta\,y_{Q}T}{G_{N}}\left(\frac{B\cos(\theta^{'})}{m^{2}\rho^{2}}\frac{b(\theta{'})}{4r^{3}_{h}}\right).\label{mag1}
\end{eqnarray}
As we can see from equations (\ref{mag2}) and (\ref{mag1}), the RS brane behaves like a paramagnetism material, that is, when we remove the external magnetic field, the equation (\ref{mag1}) disappears and the entropy linked disorder increases, as shown in Fig. \ref{p3}. On the other hand, from the equation (\ref{mag2}), the magnetization density is not null for zero magnetic fields (this is $B=0$). Thus, we can conclude that paramagnetic materials have a low coercivity, that is, their ability to remain magnetized is very low. Thus, one way to analyze coercivity is through viscosity $\eta$ in our model \cite{Muller2016}.
In order to be as clear as possible, the details about the computation of the shear viscosity and entropy density ratio are present in Appendix \ref{visc}. In particular, we will focus on the $\eta/S$ ratio, where from Eq. \ref{viscosity} and Fig. \ref{p10}, we can analyze the dependence of the viscosity on the magnetic field, characterizing a magnetic side effect, and describing the slow relaxation of the magnetization of paramagnetic materials when they acquire magnetization in the presence of an external magnetic field $B$ (left panel of Fig. \ref{p10}). In the right panel, we can observe that under an interval of the temperature $T$, the $\eta/S$ ratio is an increasing function when $B=0.$
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.71]{f10.pdf}
\includegraphics[scale=0.71]{f101.pdf}
\caption{{\sl Right panel:} The behavior of the $\eta/S$ ratio as a function of the temperature $T$ for different values for $\alpha=8/3$, $B=(4/5)T$, $\rho=1/4$, $\Lambda=-1$, $\gamma=1$ ({\sl pink curve}), $\gamma=2$ ({\sl red dot dashed curve}), $\gamma=2.5$ ({\sl green thick curve}). {\sl Left panel:} The behavior of $\eta/s$ for $B=0$.}\label{p10}
\end{center}
\end{figure}
}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.71]{f12.pdf}
\caption{The behavior of $\eta/S$ with respect to the magnetic field $B$, for different values for $\alpha=8/3$, $T=4/5$, $\rho=1/4$, $\Lambda=-1$, $\gamma=1$ ({\sl pink curve}), $\gamma=2$ ({\sl red dot dashed curve}), $\gamma=2.5$ ({\sl green thick curve}).}\label{p101}
\end{center}
\end{figure}
{On the other hand, and as we can see from Fig. \ref{p101} at a temperature $T$ fixed when we observe as the paramagnetic material, represented by the RS brane, we can obtain a relation between $\eta/S$ with respect to the magnetic field $B$, which is a decreasing function. Here, when $B$ becomes large, we have that $\eta/S\to 0$.}
We finalize this section showing the magnetic moment $N$ at a low temperature $T$, corresponding to order parameter $\rho$ in the absence of an external magnetic field, setting $B=0$, and then compute the value of $N$, defined as
\begin{eqnarray}
&&N=\dfrac{\lambda^{2}r_{h}}{2L}\int^{1}_{0}{\rho(r)dr}=-\frac{\lambda^{2}r_{h}}{2L}\left(-\dfrac{B}{m^{2}}+
\dfrac{1}{(\Delta_{+}+1)r^{\Delta_{+}}_{h}}+\dfrac{1}{(\Delta_{-}+1)r^{\Delta_{-}}_{h}}\right).\label{moment}
\end{eqnarray}
In Fig. \ref{p1}, it can be found that as the temperature decreases, the magnetization increases and the system is in the perfect order with the maximum magnetization at zero temperature. Thus, increasing the Horndeski parameters lowers the magnetization value and the critical temperature. Indeed, we have that the effect of a larger value of the parameters $\gamma$ and $m^{2}$ makes the magnetization harder and the ferromagnetic phase transition happen, which is in good agreement with previous works \cite{Zhang:2016nvj,Wu:2016uyj}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.7]{f03.pdf}
\caption{The behavior of magnetic moment $N$ with different values for $B=0$, $\alpha=8/3$ with $\gamma=1;m^{2}=2$ ({\sl blue curve}), $\gamma=4;m^{2}=4$ ({\sl red curve}), $\gamma=8;m^{2}=6$ ({\sl green curve}).
{\sl We consider in the Eq. \ref{moment} the transformations Eq.$\sim$(\ref{transfor}). }}\label{p1}
\end{center}
\end{figure}
Finally, we present the susceptibility density $\chi$ of the materials as a response to the magnetic moment. Thus, this behavior is an essential property of ferromagnetic materials. In order to study $\chi$ of the ferromagnetic materials in the Horndeski gravity and to consider the transformations Eq. (\ref{transfor}), we follow the definition
\begin{eqnarray}
&&\dfrac{\chi}{\lambda^{2}}=\lim_{B\to0}\dfrac{\partial N}{\partial B}=\left(\dfrac{3}{ 8 \pi m^{2}L^{2}}\right)\dfrac{1}{T}. \label{suscept}
\end{eqnarray}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6
]{f04.pdf}
\includegraphics[scale=0.6
]{f05.pdf}
\caption{The behavior of $1/\chi$ in the function of the temperature $T$ with different values for $\alpha=8/3$ with $\gamma=1;m^{2}=2$ ({\sl blue curve}), $\gamma=4;m^{2}=4$ ({\sl red curve}), $\gamma=8;m^{2}=6$ ({\sl green curve}).
{\sl We consider in the Eq. (\ref{suscept}) the transformations given in Eq.(\ref{transfor}).}}\label{p2}
\end{center}
\end{figure}
In Fig.\ref{p2}, we have the behavior of $1/\chi$ and $\chi$ as a function of the temperature $T$ for different choices of $m^2$ and $\gamma$. In our case, in the {\sl right panel}, we have that increasing each one of these parameters makes the susceptibility value decrease when the temperature increases. This fact agrees with our expectation of paramagnetic materials because when we remove the external magnetic field, the paramagnetic substance loses its magnetism. Its magnetic susceptibility is very small, but positive, and decreases with increasing temperature. In fact, this magnetic susceptibility is only part of the background black hole and the other part of the polarization field. For pure dionic Reissner-Nordström-AdS black hole, we have a diamagnetic material. In this sense, in the chemical reference, we have that a particle (atom, ion, or molecule) is paramagnetic or diamagnetic when the electrons in the particle are paired due to the external magnetic field \cite{Zhang:2016nvj,Wu:2016uyj}.
\section{Conclusions and discussions}\label{v4}
In four dimensions, we analyzed an AdS/BCFT model of a condensed matter system at finite temperature and charge density living on a 2+1-dimensional space with a boundary, showing an extension of the previous work presented in \cite{Santos:2021orr}, where in addition to the contributions of the theory together with the boundary terms, we include the components $A_{\mu}$ and $M_{\mu\nu}$, responsible to construct the ferromagnetic/paramagnetic model.
Via the resolution of the field equations, and using the no-hair theorem, we extend to the four-dimensional configuration obtained in
\cite{Santos:2021orr,Bravo-Gaete:2014haa}. From the above solution, we present the $Q$ profile, found a numerical solution, and present it in Fig. \ref{p1}, where the Horndeski parameter $\gamma$ takes an important role. Together with the above, we show that components of $M_{\mu\nu}$ can be viewed as dual fields of the order parameter in the paraelectric/ferroelectric phase transition in dielectric materials. Through the NBC over $n^{\mu}M|_{Q}$, we found the ratio $\rho/B$, where for some particular cases is a constant proportional to a ratio of the coefficients appearing in the gravity action. These properties resemble a quantum Hall system, which suggests at the boundary $Q$ in the ($\rho, B$) plane will be a localized condensate.
Additionally, via the solution we performed a holographic renormalization, calculating the Euclidean on-shell action, which is related to the free energy $\Omega$, and allowing us to obtain the entropy $S$ and the heat capacities $C_V, C_P$, thanks to the contribution to the bulk as well as the boundary. With respect to the entropy $S$, we show that when the magnetic field is present we see it exhibits similar behavior as for example ferromagnetic materials with nearly zero coercivity and hysteresis. Nevertheless, when $B=0$ the disorder entropy of the magnetic moments increases, being a characteristic of paramagnetism. Together with the above, with respect to $C_V$ and $C_P$, we obtained for both cases stable and unstable
phases, due to the spontaneous electric polarization, which was realized in our model from the application of the
magnetic external field $B$, being influence via the Horndeski gravity, represented through $\gamma$. We also show that the specific heat $C_P$ behaves like a material of the type $DyAl_{2}$, having a growth behavior similar to that expected from the experimental point of view, as presented by \cite{PhysRevB.72.024403}.
Currently, we can observe that the microscopic differences between real experimental systems, in relation to theories with gravitational dual suggest that, in the near future, we will have measurements of these values for experimental quantities obtained holographically. So many measurements can realistically aspire to more than useful benchmarks. Furthermore, it is important to highlight in this regard the need to take the big limit $N$ in holographic calculations \cite{Maldacena:1997re}. We now have a clarity of the value of the ratio between shear viscosity and entropy density, $\eta/S=1/4\pi$, which is universal in classical gravity to usual classical gravity \cite{Kovtun:2004de}. Furthermore, in the Horndeski gravity, these relations are modified by the parameter $\gamma$. However, there are controlled corrections $1/N$ for this result, which can be both positive and negative and which for realistic values of $N$ show significant changes in the numerical value of the ratio. As we show in our model, the violation of this universal bound in the Horndeski gravity with gauge fields changes the $\eta/S$ ratio (see Fig.\ref{p10} and Fig.\ref{p101}), where this behavior is similar to the results of \cite{Hartnoll:2016tri}. Furthermore, as $\gamma$ increases, we can observe a translational symmetry breaking that survives the lower energy scales. According to Fig. \ref{p101}, we have $\eta/S\to 0$ at low temperatures.
One of the strongest motivations for working with AdS/BCFT for condensed matter physics rests on two pillars. The first is that, although theories with holographic duals may exhibit specific exotic features, they also have features that are expected to be generic to tightly coupled theories, for example, the quantum critiques. In this sense, theories with gravitational duals are computationally tractable examples of generic tightly coupled field theories, and we can use them both to test our generic expectations and to guide us in refining those expectations. Thus, the examples discussed here are special cases of the fact that real-time finite temperature transport is much easier to calculate via AdS/BCFT than almost any other microscopic theory.
\acknowledgments
F.S. would like to thank the group of Instituto de Física da UFRJ for fruitful discussions about the paramagnetic systems. In special to the E. Capossoli, Diego M. Rodrigues and Henrique Boschi-Filho. S.O. performed the work in the frame of the "Mathematical modeling in interdisciplinary research of processes and systems based on intelligent supercomputer, grid and cloud technologies" program of the NAS of Ukraine. M.B. is supported by PROYECTO INTERNO UCM-IN-22204, L\'iNEA REGULAR.
\begin{appendix}
\section{Shear viscosity and entropy density ratio with magnetic field}\label{visc}
We will present the calculation of the ratio $\eta/S$ following the procedures \cite{Brito:2019ose,Feng:2015oea,Bravo-Gaete:2022lno,Kovtun:2004de,Hartnoll:2016tri}. For this, we consider a perturbation along the $xy$ direction in the metric Eq.\ref{ansatz} \cite{Brito:2019ose,Feng:2015oea}, in this sense, we have
\begin{eqnarray}
ds^{2}=\frac{L^{2}}{r^{2}}\left(-f(r)dt^{2}+dx^{2}+dy^{2}+2\Psi(r,t)dxdy+\frac{dr^{2}}{f(r)}\right).\label{pertur}
\end{eqnarray}
From the overview point of the holographic dictionary, this procedure maps the fluctuation of the diagonal in the bulk metric in the off-diagonal components of the dual energy-momentum tensor. In this sense, we have a linear regime where fluctuations are associated with shear waves in the boundary fluid. Thus, substituting this metric (\ref{pertur}) in the Horndeski equation (${\cal E}_{\mu\nu}=0$) for $\mu=x$ and $\nu=y$, one obtains:
\begin{eqnarray}
&&{\cal P}_{1}\Psi^{''}(r,t)+{\cal P}_{2}\Psi^{'}(r,t)+{\cal P}_{3}\ddot{\Psi}(r,t)=0\,,
\end{eqnarray}
where we defined
\begin{eqnarray}
&&{\cal P}_{1}=9\gamma^{2}(\alpha-\gamma\Lambda)f^{2}(r),\qquad {\cal P}_{2}=-3\gamma(\alpha-\gamma\Lambda)f(r)(2\alpha L^{2}-6\gamma r^{3}/r^{3}_{h}),\cr
&&{\cal P}_{3}=-9\gamma^{2}r(3\alpha+\gamma\Lambda).
\end{eqnarray}
Using the ansatz:
\begin{eqnarray}
&&\Psi(r,t)=e^{-i\omega t}\Phi(r),\\
&&\Phi(r)=\exp\left(-i\omega K\ln\left(\frac{6\gamma^{2}r^{3}f(r)}{{\cal G}}\right)\right),\quad\,{\cal G}=\frac{L^{2}V}{G_{N}}\left(1-\frac{\xi}{4}\right),
\end{eqnarray}
we obtain
\begin{eqnarray}
K=\frac{1}{4\pi T}\sqrt{\frac{3\alpha+\gamma\Lambda}{\alpha-\gamma\Lambda}},
\end{eqnarray}
with $T$ the Hawking temperature given previously in (\ref{eq:Th}). At this point, we must evaluate the Lagrangian (\ref{eq:Lhorn}), using the metric function from (\ref{L16}), and expand it up to quadratic terms in $\Psi$ and its derivatives \cite{Feng:2015oea}. In this way, we can study the boundary field theory using the AdS/CFT correspondence where the quadratic terms in the Lagrangian, after removing the second derivative contributions using the Gibbons-Hawking term, can be written as
\begin{eqnarray}
&&{\cal H}_{shear}=P_{1}\Psi^{2}(r,t)+P_{2}\dot{\Psi}(r,t)+P_{3}\Psi^{'2}(r,t)+P_{4}\Psi(r,t)\Psi^{'}(r,t),
\end{eqnarray}
where
\begin{eqnarray}
&&P_{1}=-\frac{48L^{2}}{9r^{7}f(r)},\qquad P_{2}=\frac{4\gamma\,L^{2}}{r^{7}},\qquad P_{3}=\frac{6\gamma^{2}}{r^{3}f(r)},\qquad
P_{4}=(\alpha+\gamma\Lambda)\frac{2\gamma^{2}L^{4}}{\alpha\,r^{7}f(r)}.
\end{eqnarray}
Here, $ (\dot{\phantom{a}} )$ denotes the derivative with respect $t$. Finally, viscosity $\eta$ is determined from the term $P_{3}\Psi(r,t)\Psi^{'}(r,t)$ which reads
\begin{eqnarray}
\eta=\frac{1}{4\pi}\frac{{\cal G}}{4r^{2}_{h}}\sqrt{\frac{3\alpha+\gamma\Lambda}{\alpha-\gamma\Lambda}},
\end{eqnarray}
where the entropy, from (\ref{eq:ent-total})-(\ref{BT8}), can be written as
\begin{eqnarray}
S=\frac{{\cal G}\mathcal{F}}{4r^{2}_{h}},
\end{eqnarray}
with
\begin{eqnarray*}
\mathcal{F}&=&1+\left(\frac{B^{2}\cos^{2}(\theta{'})b(\theta{'})}{5m^{2}\rho^{2}}\left(\frac{4\pi T}{3}\right)^{4}+\frac{q(\theta^{'})}{4}\left(\frac{4\pi T}{3}\right)^{2}\right)\nonumber\\
&-&\frac{\sec(\theta{'})}{\left(1-\frac{\xi}{4}\right)}\left(-\frac{B^{2}\cos^{2}(\theta{'})b(\theta{'})}{2m^{2}\rho^{2}}\left(\frac{4\pi T}{3}\right)^{3}+\frac{q(\theta^{'})}{2}\left(\frac{4\pi T}{3}\right)\right),
\end{eqnarray*}
and $T$ given in (\ref{eq:Th}). Thus, after algebraic manipulation and imposing $V=1$, we have:
\begin{eqnarray}
&&\frac{\eta}{S}=\frac{1}{4\pi\mathcal{F}}\sqrt{\frac{3\alpha+\gamma\Lambda}{\alpha-\gamma\Lambda}},\label{viscosity}
\end{eqnarray}
where $B=0$ and $\theta{'}=\pi/2$, we recover the result of \cite{Feng:2015oea}.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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##
##
"A good man leaves an inheritance . . ."
PROVERBS 13:22
#
##
##
DAVE RAMSEY
© 2014 Lampo Licensing, LLC
Published by Ramsey Press, The Lampo Group, Inc.
Brentwood, Tennessee 37027
All rights reserved. No portion of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, scanning, or other—except for brief quotations in critical reviews or articles, without the prior written permission of the publisher.
Dave Ramsey, The Dave Ramsey Show, Financial Peace, Financial Peace University, The Legacy Journey, Rachel Cruze, and The Total Money Makeover and are all registered trademarks of Lampo Licensing, LLC. All rights reserved.
This publication is designed to provide accurate and authoritative information with regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering financial, accounting, or other professional advice. If financial advice or other expert assistance is required, the services of a competent professional should be sought.
Unless otherwise noted, Scripture quotations marked NKJV are taken from the New King James Version®. Copyright © 1982 by Thomas Nelson. Used by permission. All rights reserved.
Scripture quotations marked NASB are taken from the New American Standard Bible®, Copyright © 1960, 1962, 1963, 1968, 1971, 1972, 1973, 1975, 1977, 1995 by The Lockman Foundation. Used by permission.
Scripture quotations marked NIV are taken from the Holy Bible, New International Version®, NIV®. Copyright © 1973, 1978, 1984, 2011 by Biblica, Inc.™ Used by permission of Zondervan. All rights reserved worldwide.
Scripture quotations marked TLB are taken from The Living Bible copyright © 1971. Used by permission of Tyndale House Publishers, Inc., Carol Stream, Illinois 60188. All rights reserved.
ISBN (ePub) 978-1-937077-74-7
ISBN (Kindle) 978-1-937077-72-3
Editors: Allen Harris, Jennifer Gingerich
Cover Design: Melissa McKenney
## Dedication
To the wonderful Christians I have met around the world who take on the task of managing wealth for the good of God's kingdom. These wealthy men and women follow Christ and serve Him and His kingdom, but they seldom get accolades. On the contrary, they are often the target of hate, criticism, jealousy, and envy. They do most of their investing in God's kingdom without the media or the general public ever knowing their levels of giving. They are kind, compassionate, smart, and full of integrity. There are thousands of you, and I for one salute you and thank you for your example. I'm a more generous man—a better man—for knowing you.
## Acknowledgments
This book is the result of years of thought, prayer, and study—and a lot of conversations. Special thanks to those who walked with me on this journey and helped make this book possible:
Sharon Ramsey, my ever-encouraging wife, girlfriend, and confidant.
Allen Harris, my editor, for helping me get this book out of my head and onto the page.
Preston Cannon, for coordinating and leading this project.
Jen Gingerich, for providing outstanding editorial and project management support.
Bob Bunn, for weeks' worth of faithful Bible scholarship and research.
Luke LeFevre, Brad Dennison, and Melissa McKenney, for overseeing all design elements and cover art.
Debbie LoCurto, Jen Sievertsen, Brent Spicer, Brian Williams, and Robbie Poe, for helping talk through early versions of this material.
Mike Glenn and Michael Easley, for their pastoral encouragement and careful review of early drafts of this book.
# Contents
Dedication
Acknowledgments
CHAPTER ONE
The Problem with God's Ways of Handling Money
CHAPTER TWO
The War on Success
CHAPTER THREE
Snares and Dares
CHAPTER FOUR
The Law of Great Gain
CHAPTER FIVE
Your Work Matters
CHAPTER SIX
Safeguarding Your Legacy
CHAPTER SEVEN
Generational Legacy
CHAPTER EIGHT
Called to Generosity
CHAPTER NINE
A Legacy Worth Leaving
The Pinnacle Point
The Road to Awesome
Notes
Praise for This Book
### CHAPTER ONE
# The Problem with God's Ways of Handling Money
The light was so blinding, I had to squint to see anything at all. It was a bright, sunny day, and the detailers had washed, waxed, and polished until the whole car was gleaming. The day before, an ugly, mostly unreliable junker sat in that same parking spot, but now it was gone. For the first time in ten years, I had a nice car again.
The past decade had been hard. By age twenty-six, I had become a millionaire. By age twenty-eight, I was bankrupt. I spent several years after that learning everything I could about how to handle money God's ways. I didn't just want to rebuild my former wealth; I wanted to honor God with that wealth. I had finally figured out that I was just a manager and God was the Owner, and I wanted a second chance at success. He had trusted me with a lot early on, and I had failed. I had shown Him—and myself—that I wasn't a good manager. But now things were different. I was passionate about doing things the right way the second time around. My wife, Sharon, and I vowed after bankruptcy to never borrow another dime. We worked like crazy to get out of debt and clear of the bankruptcy. We pinched pennies, cut coupons, and skipped vacations. I worked eighty hours a week while she was home taking care of three little kids. We clawed our way out of the mess using biblical principles for handling money, and God blessed us. Our new business began to grow. Life got a little better. We were able to breathe a little easier. And bit by bit we were building wealth again—but you wouldn't have known it by my old car.
During those years, I drove the cheapest cars I could. We had more important things to do than buy nice cars, and honestly, I had gotten used to driving junkers. Then one day, I was driving myself and one of my company's vice presidents to an event where I was supposed to speak, and the latest in my proud line of cheap cars broke down. There we were, standing in the parking lot of a gas station with steam pouring out from under the hood of the car. We looked through the trash, found an empty jug, and used it to pour water into the radiator to cool down the car so we could get to the event. By that time, my net worth was well over $1 million again, so this whole scene was ridiculous. My VP chided me, saying, "You seriously have to get a better car. This is crazy! You have the money!" And he was right. The moment had arrived. I was way overdue for a nicer car.
I shopped all over for a great buy, and I finally found a great deal on a two-year-old Jaguar. That was kind of cool because I drove a Jag right before I went broke. It felt like God was saying He was restoring what the locusts had eaten. So there I stood in the parking lot of my office just looking at it. You'd think everything about that moment would have been perfect, right? I mean, I had done everything right this time. We had zero debt. I _owned_ that car free and clear, and it was a perfectly reasonable purchase for my family. In a way, it represented a new way of life because it was a physical symbol that God's ways work—that I could build wealth again the _right way_. But as I stood there, I wasn't thinking of any of that. Instead, I suddenly questioned my decision. With the sunlight shining off this beautiful car, the only question that came to mind was, _Did I do something wrong?_
I started thinking about all the hungry children in the world, and I wondered what they'd think of my new car. How many kids could I have fed with the check I had just written? Then I started to worry about what my friends and customers and clients would say. Would they think I was a phony because I talked so much about generosity but I decided to spend this money on something nice for myself? Standing in the parking lot that day, I realized something. God had been teaching me about _money_ for the past ten years, but now it was time for Him to teach me about _wealth_. Now, in _The Legacy Journey_ , I want to share with you what I've learned.
But let's be clear right up front. I'll go ahead and ruin the end of the story for you. I didn't do anything wrong by buying that car. I think God smiled at me as I wrote that check. Depending on who you listen to or what you read about wealth, that may shock you. If so, then get ready to be shocked _a lot_ in this book. It's time to see what God's Word really says about wealth. And trust me: It's a much different message than you're hearing in our culture today.
# THE STORY SO FAR
You probably know my story. I've told it a zillion times on the radio, in books, and in my nine-week class, _Financial Peace_ _University_ (FPU). I hate repeating my story for those who already know it, but let me give you the short version just so you know where I'm coming from. I started out with nothing and became a millionaire by the time I was twenty-six. But my entire net worth was supported by a huge pile of debt. One bank called one loan early, and that started a domino effect of me losing everything I owned over the course of two and a half years. Then, with a toddler, a new baby, and a marriage hanging on by a thread, Sharon and I threw up the white flag and declared bankruptcy.
## Wake Up!
As we went through those hard times of losing everything, God started showing us the basic principles for how to handle money in His Word: Get out of debt. Live on a budget. Live on less than you make. Save. Invest. Never cosign. Always get the counsel of your spouse. I discovered these fundamental tenets—the core things you do to win with money—straight out of Scripture. But because I was a young hotshot with all these letters and licenses after my name that said I'm supposed to know something about money, all of this was new to me and kind of jolted my system. God had used my financial crash to say to me, "Wake up, stupid! You're going the wrong way! What you're doing isn't in line with Scripture. It isn't even in line with common sense!"
I've talked with tens of thousands of families who have gone through FPU who have had the same experience. When they learned these principles for the first time, it felt like God was grabbing them by the shoulders and saying, "Wake up!" That's what FPU and my books _Financial Peace_ and _The Total Money Makeover_ have been about. Their purpose is to get you to that wake-up, then moment—that point where God gets your attention and you start to turn your financial life around.
So if _Financial Peace_ and _The Total Money Makeover_ are about waking up, _The Legacy Journey_ is about growing up. At some point we all have to move beyond just drinking milk and learn to eat solid food. We must mature in our view of wealth. That means we have to emotionally and spiritually grasp wealth as a concept through the lens of Scripture. As we do that, our perspective changes. We really start to take hold of our role as managers—stewards—of the resources God has placed in our hands. What you do with His resources is what your legacy is all about.
## What's the Problem?
Here's the problem: God's ways of handling money actually work. If you do the things I teach from God's Word, if you are diligent and wise with your income, then over time you will become wealthy. No matter who you are or where you're starting from, you will at some point become one of those "rich people." I've seen it more times than I can count.
I was doing a book signing a while back and saw a guy in coveralls standing at the end of the line. Where I'm from, coveralls mean you do _real_ work. You do the kind of work that's hard and dirty, so you put on the coveralls to keep your real clothes from getting ruined. This man waited patiently, but I could tell he was the kind of guy who didn't really like to stand around waiting. He had a head full of bushy gray hair and a matching gray beard. He looked tired too; it was easy to tell that he'd been on his feet all day. When he finally got to the book-signing table, I noticed that he didn't have a book in his hand. He was just holding a little scrap of paper.
He said, "Dave, I'm not here to buy your book." I thought, _Well, why would you want to stand in line for an hour just to tell me you didn't want to buy my book?_ He continued, "I just want to tell you that I started listening to you on the radio ten years ago. I heard all those things you said about the Scriptures and I went and looked them up for myself, and I started living that way. And I gotta tell you, the stuff you teach on the radio works." Then he put that scrap of paper down on the table in front of me. There were two numbers written on it: -$66,283 and $847,623. I looked back up at him and saw this strong, burly workman with big tears in his eyes. He said, "The first number is where I started thirteen years ago, and the second number is where I am now. I started with a negative net worth and $66,283 in debt, and today I am debt-free with $847,623 in mutual funds. This stuff works, Dave."
That conversation meant so much to me that I kept that little scrap of paper. If you were to come visit me in my office, you'd see it in a frame on my bookshelf. I love it because it doesn't just represent a financial turnaround; it represents a total change of direction for an entire family. This guy's family going back several generations had never seen that much money before, but there it was. And because he taught his kids how to handle money, it would be there for them. And as it grew over time and his kids added to it, it would be there for his grandkids. This one man applying God's ways of handling money—even with a modest income—had completely changed his family legacy. And that's the "problem" with handling money God's ways: You end up wealthy. So let's talk about that.
# THE LEGACY JOURNEY FRAMEWORK
In my classes and other books, I explain in detail what I call the Baby Steps. This is the process for taking control of your money, getting out of debt, and starting your family on the road to financial security and long-term wealth building. I'm not going to reteach all of that here, but I will take just a second to review the seven Baby Steps:
> **Baby Step 1:** Place $1,000 in a beginner emergency fund ($500 if your income is under $20,000 per year).
> **Baby Step 2:** Pay off all debt except your home mortgage using the debt snowball.
> **Baby Step 3:** Put three to six months of expenses into savings as a full emergency fund.
> **Baby Step 4:** Invest 15 percent of your household income into Roth IRAs and pretax retirement plans.
> **Baby Step 5:** Start college funding for your kids.
> **Baby Step 6:** Pay off your home early.
> **Baby Step 7:** **** Build wealth and give.
The first three Baby Steps are all about taking control of your money. In those steps, you're correcting bad behaviors, getting out of debt, putting some money in the bank, and basically cleaning up a mess. Those months or years can be tense, but they are crucial to changing your legacy. Once you hit Baby Step 4, though, you start to feel a different kind of tension. For maybe the first time, you're able to relax a little bit, take a breath, and realize that you're in a position to build wealth. And that's when you start facing the questions that popped into my head that day as I was looking at my new car. For this, there's a different process in place that works alongside the Baby Steps. I call it **NOW–THEN–US–THEM** , and it's the framework for your legacy journey.
## **NOW** : Taking Control
When you first have that wake-up moment and start doing a budget and working through the Baby Steps, you are laser-focused on taking control of your money and cleaning up a financial mess. I call that stage the **NOW**. At that stage, you may be broke, behind on your bills, struggling to put food on the table, and always worried about making one paycheck last until the next one arrives. Picture it for a moment: You are at the kitchen table with your head down, slumped over a pile of bills. You can't look up, because there are about a dozen different crises right under your nose. All you're trying to do is take care of all the little fires going on _right now._ That's not a fun place to be, but that's where a lot of us start.
In the **NOW** , your job, first and foremost, is to take care of your family. The Bible says, "If anyone does not provide for his own, and especially for those of his household, he has denied the faith and is worse than an unbeliever" (1 Timothy 5:8). At this stage, if you are a believer, you should be tithing (giving a tenth of your income) to your church, but now is not the time to give extra beyond that. And you shouldn't feel guilty about that either, because you are obeying your biblical mandate to take care of your family. This is also not the time to invest or fund the kids' college accounts. The long-term plan is to completely change your family's legacy, but that may seem like a tiny dot on the horizon at this stage. In the **NOW** , the goal is to stop the bleeding and get your feet under you.
## **THEN** : Getting a Future Focus
Getting through **NOW** may seem like an eternity because you are working so hard and feeling completely stressed out, but over time, you start to take control of things. As you keep moving through the Baby Steps, you start to relax and are able to breathe again. Finally, you start to get a little wiggle room. If you've already been through the **NOW** , you know exactly what I'm talking about. You might even remember a specific moment when you realized that things were changing. That's a great place to be because as soon as the pressure eases, you are able to lift your head up a bit. When that happens, you can take your eyes off the **NOW** and start focusing on the **THEN**.
The **THEN** stage challenges you to adopt a future focus. By this point, you should be at Baby Step 4, investing 15 percent of your income into retirement, and Baby Step 5, funding your kids' college accounts. You can even start attacking the mortgage to pay off the house early, which is Baby Step 6. That's a much different place mentally, spiritually, and emotionally than the **NOW** , isn't it? With **NOW** , you're just trying to make it through the end of the week, but in **THEN** , you're able to look out at the future you're working so hard to create for your family. Scripture says, "Where there is no vision, the people perish" (Proverbs 29:18 KJV). At this stage, you're getting a vision for where you're going, which is crucial to your long-term success.
## **US** : Creating a Family Legacy
While you're working on **NOW** and **THEN** , you often have tunnel vision because you're so focused on getting your own finances and future under control. But once you start building some wealth in your mutual funds and 401(k), and as you get closer to finally paying off the mortgage, you start to realize that you really are going to retire with dignity. You're not ready to retire yet, but that tiny dot on the horizon has gotten bigger. It's coming into focus, and it doesn't feel like a daydream anymore. This is going to happen, and you're going to be fine.
Just breathe that in for a second. Imagine your family—your kids, your grandkids, and maybe even your future great-grandkids—taken care of for generations to come. I call this stage **US** because it's not just about you and your spouse anymore. The Bible says, "A good man leaves an inheritance to his children's children" (Proverbs 13:22). This isn't about getting out of a mess; this is about changing your family tree.
In the **US** stage, you're talking to your kids about money and making sure they have the emotional and spiritual maturity to manage the wealth you might leave them someday. You're also teaching them to build wealth themselves. And more than anything, you're making sure they're not confused about God's ownership and their role as manager of what He's provided. If you're not intentional about growing your kids' characters and ability to handle money, you can work your whole life to build wealth only to have it end up ruining them. As you work through **US** , you're making sure that will never happen.
## **THEM** : Leaving a Legacy for Others
As you get to the end of the legacy journey framework, you know that the bills are paid—that's **NOW** _._ You know that you're going to be able to retire with dignity and send your kids to college—that's **THEN**. You know that you are taking steps to prepare the next generation to carry your new family legacy forward—that's **US**. And then, finally, you see the big picture—the **THEM**.
The **THEM** stage represents all the needs in your community and around the world, and you begin to see how God can use you to meet some of those needs. Your vision expands and you start to see the world through Christ's eyes. Suddenly you see wells that need to be drilled in Haiti. You see the HIV prevention efforts in Africa. You see mosquito nets that can be used to prevent malaria. You see the hungry children right down the street. You see the broke, scared single mom at the other end of the pew who can't pay the light bill this week—and who might need a reliable car too. At this stage, you're able to open your eyes and see—maybe for the first time—how you can make a difference in the world.
Proverbs says, "He who gives to the poor will not lack, but he who hides his eyes will have many curses" (28:27). I don't think most of us ever actually hide our eyes; I just think the pressures we're under keep our heads down so we can't see. But as we work through these biblical principles about taking care of our family first, casting a vision for the future, and leaving an inheritance to our children's children, we're able to lift our heads and look clearly into the future. God can do amazing things through us in that future. But we have to take it one step at a time, first through **NOW** , then **THEN** , then **US** , and finally to **THEM**. And that's where we get to help change the world. That's a powerful legacy.
# YOU ARE HERE
No matter where you are in the Baby Steps, _The Legacy Journey_ will give you a clear, biblical view of wealth and generosity. You may not be financially ready to change the world today, but you need to see where God might take you one day. If you are just getting started, meaning you aren't living on a budget, you have some debt (including cars and student loans), and you have nothing in savings, then you really need to get those things under control. Use _Financial Peace University_ or _The Total Money Makeover_ to get started immediately. It's never too early—or too late! That's the foundation for everything in this book, and those are the resources and tools that will get you through **NOW** and **THEN**.
Most of all, at this point I want you to grasp that if you are diligent in handling God's money God's ways, you _will_ walk through these steps of building wealth. So don't be surprised when, as you move out of debt, learn to live on less than you make, give, and save, you become wealthy. If you are diligent with the steps, your eyes will look up from the **NOW**. You will begin to look toward the future with the **THEN**. You will become concerned that the next generations of your family are set up in the **US** stage. And of course, once that is done, it will feel natural and easy to meet the needs that God sets before you in the **THEM**. This is a biblical process, so you are not doing anything wrong by first taking care of your family, then your future, then future generations of your family, then finally to be outrageously generous with the wealth that God has allowed you to manage for His glory. So if the world or toxic "Christian" voices are trying to send you on a guilt trip, take heart that you are walking through a biblical process.
_The Legacy Journey_ is primarily designed for those of us in the later Baby Steps—Baby Step 4 and beyond—who are working on **US** and **THEM**. We're going to talk a lot about wealth as a concept, and we're going to address a lot of the toxic messages in our culture that tell us we should be ashamed of the success God's given us. We'll talk about how to figure out how much is "enough" for your family, and we'll identify key things that should be part of your estate plan. We'll talk about how to keep money from ruining your relationships and your kids. And of course, we'll talk about that incredible day when your hard work and faithfulness have paid off, when you've paid the price to win, when you've lived like no one else, and you're finally ready to live—and give—like no one else. This is going to be a wild ride, and it's not just going to change your life. It's not just going to change your family tree. It's going to show you how to live _and_ leave a lasting legacy of excellence.
### CHAPTER TWO
# The War on Success
A couple of years ago, I had an unbelievable opportunity to spend time with a group of ten incredibly successful, insanely wealthy business leaders. These were rich folks— _really,_ _really rich_. The minimum net worth represented in the room was half a billion dollars. They were all totally committed, sold-out Christians. In fact, these guys were meeting to discuss the different ways their families had given to kingdom work in the past year. It was kind of a counsel of Christian philanthropists, and they met to discuss different giving opportunities and to encourage each other. When they added it all up, these ten families had given more than $1 billion to different ministries and mission work in just that one year alone. And there I was, sitting in the middle of it. Talk about feeling like a wiener in a steakhouse!
After a day of hearing these amazing stories of radical generosity, the meeting wrapped and the group headed out to dinner. The guy who called the meeting—the wealthiest man in the room—said, "Hey Dave, would you ride with me?" Now this guy is worth $2.2 billion. He's a business genius, a great Christian man, a great husband and father, and exactly the kind of person I like hanging out with. I want to learn from these kind of people because I want to grow up to be one! So when he asked me to ride with him, I was _in_. "Oh, you bet. I'll ride with you. You drive, and I'll take notes the whole time. Let's go."
As we were walking out, he said, "Listen, before we get out there, I want to apologize for my car." Honestly, it caught me off guard. He didn't really strike me as the kind of guy who'd roll up to a meeting of billionaires driving a junker. I said, "Why apologize? Are you driving a beater? We can take my car if there's something wrong with yours."
He said, "No, it's not that. I actually just bought a new car. In fact, it's the nicest car I've ever owned."
I had an idea of what was going on, because, like I said in the previous chapter, I had been there myself. But I asked anyway, "Why would you be ashamed of that?"
He hung his head a little bit and replied, "Well, it's a _really_ nice car." About that time, we walked up to his parking spot. He wasn't kidding. I was looking at a brand new Mercedes. I looked the thing up when I got home that night. It was a $130,000 car. It was _sweet_. But instead of being excited about it, he felt like he had to apologize for it.
Let me tell you a little about this guy. He's worth $2.2 billion. That year alone, he had personally given away around $500 million to Christian ministries and Christian work around the world. But, of course, you'd never hear about that in the news or a press release because he didn't do it for you. He doesn't care what you think. He did it because he's a passionate giver, not because he wants to impress anyone or make headlines. So that year, after giving away $500 million to kingdom work, he bought himself a nice $130,000 car.
Guess what happened? He started getting hate mail—from Christians. Now, I think "Christian hate mail" should be an oxymoron, but I've gotten enough of it myself to know that it's real. He got notes and emails telling him how wasteful and selfish he was. They said things like, "You should have used that money to drill more wells in Africa" or "I wonder how many starving children you could have saved with that $130,000." These strangers—none of whom, I'm sure, had ever seen $1 million, let alone $2.2 billion—crawled out of the woodwork to condemn his decision and question his faith just because he bought himself a nice car.
I have to be honest: That bugs me. It bothers me because I know this guy. I know his heart. He's a strong believer. He's a follower of Christ. He's applied God's ways of handling money, he's faithfully provided tremendous value to his customers, he's given hundreds of millions of dollars to others, and he has been richly blessed by God. To see him criticized for enjoying the blessings God's given him hurts me, but it's something I'm seeing more and more of in today's culture. There's a war on success happening in America today, and it's something we as believers need to talk about.
# THE ROOT OF ALL EVIL?
Money is not the root of all evil. Let me say that again: Money is _not_ the root of all evil. The Bible does not say that, no matter how often you've heard it. It's one of those things that people are sure is in Scripture, but they're wrong. It's like the saying "God helps those who help themselves." A recent study found that 82 percent of Americans believe that is pulled straight from the Bible It's not. But wait; it gets worse. Writing for Christianity.com, Albert Mohler went through more findings:
> A Barna poll indicated that at least 12 percent of adults believe that Joan of Arc was Noah's wife. Another survey of graduating high school seniors revealed that over 50 percent thought that Sodom and Gomorrah were husband and wife. A considerable number of respondents to one poll indicated that the Sermon on the Mount was preached by Billy Graham. We are in big trouble.
I wonder what the response would have been if this poll had asked people if "Money is the root of all evil" is a verse from the Bible. People get all these crazy ideas about what the Bible says, but they never take time to slow down and see what it _really_ says. If you want to truly mature in your Christian walk, you've got to take the time to actually dig into Scripture and see what's true and what's not.
## Gnostics in the Land
First Timothy 6:10 does not say that money is the root of all evil. It says, "For the _love_ of money is _a_ root of _all kinds_ of evil" (emphasis added). It's not about money; it's about our attitude. God doesn't like it when we worship anything other than Him. He's a jealous God, and if our love of money (or family, or career, or hobbies, or . . . you fill in the blank) gets in the way of our relationship with Him, then we have a problem. He doesn't like that.
Nevertheless, there are loud voices in our culture today of people who are supposedly steeped in intelligence and research and who constantly beat the drum that money is evil. But this false teaching is nothing new. In fact, it dates back almost to the birth of the Christian church in the first century when a group of early believers strayed from the teachings of Jesus and became heretics. That is, they preached and taught a false doctrine that went against the Scriptures. Central to their beliefs was the notion that all physical things, including material possessions, were evil. This group was known as the Gnostics from the Greek word _gnosis_ , which means "knowledge." For this group, the only purity was that which was spiritual. If something could be physically seen or touched, then it was unholy. Flash forward two thousand years, and you can still see Gnostics in the land trying to convince us that material things are inherently evil. So a good income, savings, investments, a nice car, a big house—they're all evil. It doesn't matter how much you give or how much good you do with your resources. It doesn't matter how closely and faithfully you've followed God's ways of handling money. Your wealth is evil, period. That's a form of gnosticism, and it's a dangerous, heretical lie. It was heresy in the first century, and it's heresy today.
## Lessons from a Rabbi
If money itself—not the love of money—is evil, then logic would say that people who _have_ money must be evil as well. That means we as a culture get to vilify people who have become successful. I mean, only a filthy, nasty, self-serving jerk would pursue wealth if money were inherently evil, right?
I have a good friend who is an orthodox Jewish rabbi. Rabbi Daniel Lapin wrote a book several years ago called _Thou Shall_ _Prosper_ , and in it he explores how and why Jewish people—a stark minority in America—have a disproportionate amount of wealth. He told me that only about 2 percent of the US population is Jewish, but about 25–30 percent of the Forbes 400 (the four hundred wealthiest people in America) are Jewish. How is that possible? Rabbi Lapin's book outlines "Ten Commandments for Making Money" according to his Jewish tradition, and the first one is the most important. If there is one Jewish attribute more directly responsible for Jewish success in business than any other, it is this one: Jewish tradition views a person's quest for profit and wealth to be inherently moral.
I can't overstate that point. Believing that making money is a selfish activity will undermine anyone's chances of success. This is a dangerous socioeconomic trend that's running wild in our culture today. It's dangerous because any time you vilify success, nobody wants to become successful because they don't want to get any of that "evil" money.
## Guess What: You're Rich
There's a point where the "rich is wrong" folks run into a problem. You see, the average household income in America is right around $50,000. Let's say your household income is well below the average and you make $34,000 a year. If you're living in America and your household makes $34,000 a year, you are in the top 1 percent of income earners in the world. Not too long ago, the news was full of marches and demonstrations of people raging against what they called "one-percenters," meaning the top income-earners in America. Well, guess what? If your household makes $34,000 a year, you're rich. Welcome to the global 1 percent.
Let's go a step further. If you have an annual household income of just $11,000, you are in the top 14 percent of income earners in the world. Here in the US, you'd be well below the federal poverty line, but from a global perspective, you'd still be pretty well off. Don't get me wrong; I know how hard it is to get by on $11,000 a year. I don't want anybody to have to do that. But from a global perspective, America has some of the richest poor people on earth! If you've ever visited a third-world country where people live in huts or shanties, have no running water, and don't know where their next meal will come from, then you return home to running water, indoor plumbing, and cabinets with food in them, your perspective changes. You realize pretty quickly how wealthy you really are.
See how this whole "wealth is wrong" belief system starts to break down? Can $34,000 be a "humble" or "righteous" amount of money in the US _and_ an "evil" or "greedy" amount of wealth globally at the same time? It just doesn't make sense. Now if you're reading this and your household income is $34,000, I know you probably don't feel rich. That's because here in America, we compare ourselves to people who are richer than we are. But globally, you're still in the top 1 percent. So when we as a culture look down on "those greedy rich people," we're creating a massive hypocrisy because the people saying these things are usually, from a broader perspective, rich people.
But being "rich"—however you define it from a global perspective—is okay! The Gnostics were heretics, remember? The Bible never says that wealth is wrong in and of itself. In fact, the Bible encourages God's people to take control of money for kingdom work. Over and over again, we see stories and examples of godly men and women who did just that. Abraham, Job, David, Solomon, and Lydia were all portrayed as wealthy in Scripture, but their wealth is never held against them. In fact, Scripture shows us some incredible things they were able to do with their money. Being able to use hard-earned wealth for kingdom work is an amazing honor and blessing! We should be excited that God's given us that opportunity. My friend Craig Groeschel, who pastors one of the largest churches in America today, says, "Why is it that the only blessing of God that we apologize for is wealth?" It's a great question. I think part of the reason is that we often don't handle our money with the right spirit.
# THREE SPIRITS OF WEALTH
When it comes right down to it, there are only three main spirits —or attitudes—you can have with wealth and possessions. I'll go ahead and tell you up front that two of them are wrong and one of them is right. I'll also tell you that I've been guilty of all three of them throughout my life, which means I've gotten it wrong more often than I'd like to admit. I first came across these three views in a book called _The Blessed Life_ by my friend Robert Morris. Robert points out how Mary, Martha, and Judas react to Jesus in two specific encounters, Luke 10:38–42 and John 12:1–8. Let's look at those passages and examine what each of these three people said and did. If you're like me, you'll probably see yourself in at least one (but probably all) of these positions.
## The First Spirit: Pride
> Now it happened as they went that He entered a certain village; and a certain woman named Martha welcomed Him into her house. And she had a sister called Mary, who also sat at Jesus' feet and heard His word. But Martha was distracted with much serving, and she approached Him and said, "Lord, do You not care that my sister has left me to serve alone? Therefore tell her to help me." And Jesus answered and said to her, "Martha, Martha, you are worried and troubled about many things. But one thing is needed, and Mary has chosen that good part, which will not be taken away from her." (Luke 10:38–42)
The first spirit of wealth is pride, which says money comes from me. I did it. I worked hard and created the money all by myself. I went out, killed something, drug it home, and now I can do whatever I want with it. This is the spirit I fell into early on, before I lost everything in my late twenties. It's also one that's easy for me to fall back into from time to time, because I believe in hard work.
That actually goes back several generations. The Ramsey family has always been a hardworking bunch of people. We've all had varying degrees of success, but we could outwork just about anyone around. That's one of the biggest lessons I learned from my parents. I remember once asking my dad for some money so I could ride my bike down to the local gas station and buy a Coke. He looked down at me and said, "You're twelve years old. You don't need money; what you need is a job!" So instead of a dollar, he gave me a lawnmower! Our family is originally of Scottish descent, and several years ago I had the chance to visit Scotland where we found the Ramsey Castle. As we toured the place, I noticed this huge coat of arms hanging on the wall. Printed above the coat of arms were two words in Latin: _Ora et Labora_ , which means "Pray and Work." For hundreds of years, the Ramseys have been known for those two things: prayer and work.
People with the spirit of pride believe that money only comes from hard work and force of will. These are performance-based individuals who believe their wealth (and maybe even their spiritual growth) is 100 percent tied to their effort. And again, I've spent a lot of time in this zone. People struggling with a prideful spirit about wealth like to quote the parts of the Bible that elevate hard work. We say things like, "If you don't work, you don't eat" (2 Thessalonians 3:10); "The diligent prosper" (Proverbs 13:4); "When you're faithful with a little, more is given to you to manage" (Luke 16:10); or "You reap what you sow" (Galatians 6:7).
I love verses like that. I believe in hard work, and I believe those passages exalt the need to work hard and to be diligent with the work God's given you to do. But I do not believe a farmer who is mature in his faith has any question about where the rain comes from. He can work the land, line up his crops, and plant the seed, but he knows all of that work is meaningless if God doesn't provide the rain. Ignoring God's role in our success is a dangerous place to be because it puts the full burden of the outcome on our shoulders—and our shoulders just aren't big enough.
That's the trap Martha falls into in Luke 10:38–42. Read the passage again. Martha is amazing! She's doing all the right things. She's giving Jesus her very best work. She's cooking and cleaning. She's getting the house ready. She's keeping everyone comfortable and fed. You can picture her running around with a pitcher of iced tea, making sure all the tea glasses stay full. She's working her tail off! I love Martha in this story because _I've been_ Martha so many times. And I'll tell you something else: Our churches _need_ people who get excited about working hard. There's nothing wrong with working hard. But here's the issue with Martha in this passage: Jesus, the Son of God, is sitting in her living room . . . _and she's running a vacuum!_ She had the opportunity to spend time at Jesus' feet, and, instead, she's doing the dishes. I've done that! I've been so blinded by the work that I've totally missed the real blessing in a situation. That's what we do when we think that money or wealth or possessions or God's favor and blessings come only from our performance.
## The Second Spirit: Poverty
> Then, six days before the Passover, Jesus came to Bethany, where Lazarus was who had been dead, whom He had raised from the dead. There they made Him a supper; and Martha served, but Lazarus was one of those who sat at the table with Him. Then Mary took a pound of very costly oil of spikenard, anointed the feet of Jesus, and wiped His feet with her hair. And the house was filled with the fragrance of the oil. But one of His disciples, Judas Iscariot, Simon's son _,_ who would betray Him, said, "Why was this fragrant oil not sold for three hundred denarii and given to the poor?" This he said, not that he cared for the poor, but because he was a thief, and had the money box; and he used to take what was put in it. But Jesus said, "Let her alone; she has kept this for the day of My burial. For the poor you have with you always, but Me you do not have always." (John 12:1–8)
The second spirit of wealth is poverty, which says that wealth and possessions are evil. There's that Gnostic influence again, right? People with the spirit of poverty can tell you all about what other people should do with their money. These are the people who would have sent hate mail to my friend in the opening of this chapter for buying himself a nice car. They like to say things like, "He shouldn't have bought that" or "He could have helped so many people with that money." It's like their goal is to make successful people feel guilty about their success.
Judas represents the spirit of poverty in John 12:1–8. Even though his real motive was to steal the money, he echoed the condemnation that's become so common in our own culture. "She _shouldn't_ have . . ." "She _could_ have . . ." "Why _didn't_ she . . ." Poverty thinking is always focused on other people's money, and it often gets obsessively focused on an unbiblical notion of equality (which we'll discuss in more detail in the next chapter). If there are two people in a room, one rich and one broke, then a modern-day Robin Hood should come in, take the rich person's money, and give it to the broke person. The fact that Robin Hood was a thief isn't important.
But the spirit of poverty doesn't care how the rich person got rich and how the broke person got broke. The spirit of poverty shouts, "Equality! Spread the wealth! Give your fair share!" It wants to take from the rich and give to the poor, no questions asked. And if the rich person refuses, then he's branded a selfish jerk and comes under harsh public condemnation. I've watched that happen to a lot of my friends, and it's not pretty.
## The Third Spirit: Gratitude
The third spirit of wealth is the proper response to wealth. The spirit of gratitude says wealth is _from_ God and _belongs to_ God. This is where I want to be and where I want to stay. This is where I pray you'll aspire to get to, as well. In John 12:1–8, which we just looked at, we see the spirit of gratitude in the actions of Mary. The spirit of gratitude knows that "the earth is the LORD's, and the fulness thereof" (Psalm 24:1 KJV). Psalm 50:10 says that He owns the cattle on a thousand hills. And guess what? He owns _the hills_ too! He owns everything, and He asks you and me to manage it. It's not ours. We're just His managers.
The spirit of gratitude says _thank you._ Thank You for giving me the money to take care of my family and put food on the table. Thank You for the ability to buy a decent car and take my wife on a great vacation. Thank You for providing for my needs today and my retirement tomorrow. Thank You for Your principles on how to handle money because using them has allowed me to change my family tree and leave a legacy that will outlive me. At its core, the spirit of gratitude says, _God,_ _I'm going to manage this wealth and this stuff Your way—because it's Yours. Thank You for trusting me to manage it for You._
I love the John 12:1–8 passage because it shows how radically extravagant our gratitude can and should be. Let me help you with this. Judas is disgusted with Mary's sacrifice because she could have sold the perfume for three hundred denarii. Do you have any idea what that would mean in today's terms? Three hundred denarii would have been the equivalent of about one year's worth of wages. The average income in America today is right around $50,000. So, for some present-day perspective, it would have been like Mary pouring $50,000 worth of perfume on Jesus' feet! And this was the _good_ stuff. My wife informed me a long time ago that cheap perfume isn't nearly as strong per ounce as the expensive ones. With the drugstore brands, you have to pour half the bottle on your neck. But the good stuff—all it takes is one drop, and it fills the room.
That's the picture here: Mary poured $50,000 of fine perfume on Jesus' feet. That smell wouldn't have just filled the room; it would have filled the street. The whole neighborhood must have known something special was going on. And Jesus made it perfectly clear that Mary wasn't just making an offering; she was anointing Him for His burial. Days later, when Peter and John burst into Jesus' tomb, what did they find? Well, I can tell you that Jesus' body wasn't there—but the sweet smell of that perfume probably was. In her extravagant spirit of gratitude, Mary didn't hold anything back. She was so overcome with gratefulness that she gave her very best to the Lord. I want to note, though, that Scripture doesn't tell us that she gave _everything_ she had or that she walked away from that encounter completely broke. People like to make that assumption, but it's just not in the text. We'll actually deal with the "give it all away" mentality later in this book.
## The Source of Gratitude
My daughter Rachel Cruze and I recently wrote a book together called _Smart Money Smart Kids_. While working on the chapter on contentment, we had a great discussion about where gratitude comes from. As we talked through it, we started to see a progression where people move toward gratitude and into contentment. I believe the start of that process is humility. When I talk about humility, don't misunderstand me. I don't mean humiliation. Humility and humiliation are two totally different things. Humiliation is about shame or embarrassment. Humility, though, is a right understanding of myself and my place in the world. It's the opposite of entitlement. Humility recognizes that God is 100 percent responsible for every blessing, every success, every outcome, and every reward in my life. I only have these things because God gave them to me. Sure, I worked hard, but God provided the result. Like Proverbs says, "The horse is made ready for the day of battle, but victory rests with the LORD" (21:31 NIV).
Humility, then, leads into gratitude. That's an understanding not only of ownership and the source of the blessing, but also of the price that was paid to provide that gift. A good friend of mine once told me about the bicycle he got one Christmas when he was a boy. He was at that age when all his friends started riding bikes everywhere, and he wanted one more than anything. It's all he asked for that Christmas. The problem was, though, that his parents were broke. These were hard-working people who were doing what it took to put food on the table and keep a roof over their heads. A bike under the Christmas tree was a big deal to this family. Unless something changed, it meant they wouldn't _eat_ for a few weeks! But this guy's dad wanted to give his son a bike for Christmas, and so he worked. He worked like crazy. He worked nights, he took extra shifts, and he got a weekend job. He was completely focused on this goal. And so, as my friend told me the story of waking up to see that bicycle under the tree that Christmas morning, he teared up.
That was decades ago. My buddy is a grown man now with a family of his own. But remembering the price his father paid to buy that bike still chokes him up. And do you think that bike was ever left in the rain? Do you think it was ever left outside a store or school without a chain on it? Do you think it ever had half-empty tires or scratches on the paint? No way! Even as a boy, my friend recognized the true value of the gift, and in his gratitude, he treated that bike like his baby. That's the kind of gift that brings tears to a grown man's eyes, and that's the kind of bike that you find in your grandfather's attic because he never could dream of throwing it away. There's just too much sentimental value attached to it.
That's gratitude, and it's evident in the way we manage the gifts we've been given. When I have a spirit of gratitude—when I am faithful with the little things because I know I don't really own any of it—the strangest thing happens. God gives me more to manage. And the more He gives, the more convinced I am that I don't own it. Looking to the future, I've instructed my kids that they don't own it either. When I die, they'll take over the _management_ , not the _ownership_ —because it's not mine to leave them. Who's the true heir of my estate? It's the current owner: Jesus Christ. He owns it now, and He'll own it then. If my kids or grandkids ever get confused about that, then there are clearly defined corrections written into my estate plan. We'll cut them out of the will if necessary because I refuse to let anyone try to take ownership of what God's given me to manage.
This is how I show my gratitude. This is how I try to say _thank You_. It's my best shot at being like Mary because Lord knows I've been like Martha and Judas plenty of times. I've been the guy who got puffed up about what _I_ accomplished through _my own_ hard work. I've put my faith in my accomplishments. I've judged other people because they gave too much or didn't give enough. I've looked down on people for buying "too nice" a car or house. I've operated under the spirit of pride and the spirit of poverty more times than I'd like to admit, but with God's help, I'm moving out of those spirits for good. My goal now as I get older is to operate most often in the spirit of gratitude, where how I act and how I do business and what I say are all saying the same thing: _thank You_. That's what I want for you too.
### CHAPTER THREE
# Snares and Dares
Over the past few years, I've really gotten into the whole social media thing. When Twitter first started, the "cool" people on my team tried to get me to create an account, but I laughed it off. Honestly, I just didn't get it. I've got more important things to do than to stop what I'm doing to tell the world what I ate for breakfast! Because back then, that's all you saw. Nobody really knew how to use the thing yet, so it was kind of a waste. Today is a totally different story, though. Once Twitter really got its legs under itself, it (and other social sites) completely changed how people interact with each other and with companies. For the first time in my career, I was able to have direct, one-on-one, back-and-forth, 140-character conversations with tens of thousands of people. It was awesome! I learned things about my audience that I never knew, and that part of it has been a blast! Twitter, Facebook, and the myriad of other social media sources give us instant polling of what is going on in the culture—what people are thinking—and that can be inspiring and/or really scary.
There's another side to social media, though, that's been equally eye-opening. Having a Twitter account meant that I was wide open to all kinds of people who just wanted to take shots at me or, worse, take shots at the biblical principles I teach. After a few years and a half-million followers (and counting), I think I've heard just about every anti-Christian, anti-Dave, and antiwealth comment you could imagine. A lot of it comes from people outside the church, and that's fine. I love hearing from people with different backgrounds and different perspectives. But a lot of it has come from fellow Christians—men and women who have honest disagreements with my take on things or who are coming at the whole idea of biblical wealth from a totally different angle. And honestly, I enjoy that too.
The dialogue I've been able to have with such a diverse audience is amazing, and it's shown me several key areas where people get confused when it comes to a biblical view of wealth. Some are honest disagreements and others are outright attacks on biblical principles, but the themes are important enough that I want to take a good look at the big ones. These are snares that keep tripping many of us up as we try to understand God's view of wealth and success. In fact, some of them are things that tripped me up too when I first started thinking through all of this. So let's take a look at the key snares and see how we can avoid them. Then, I'll follow that up by making some dares—laying out the practical steps that will serve as a roadmap for your legacy journey.
# SPOTTING THE SNARES
Before we get into the big snares, we need to look at a couple of nonnegotiables. These are two things that we might take for granted if we're not careful, but that can derail our whole legacy. The first thing is your commitment to a regular monthly budget. I've always said that your number one wealth-building tool is your income. Get-rich-quick is a fairy tale 99.9 percent of the time. The real key to building wealth is diligently managing your income over a long period of time. That's why the budget is so important, and that's why I stress the need to do a zero-based budget so much in my other books and all through our _Financial Peace University_ class. The budget is your blueprint; without it, the whole thing falls apart. Jesus said, "For which of you, intending to build a tower, does not sit down first and count the cost, whether he has enough to finish it—lest, after he has laid the foundation, and is not able to finish, all who see it begin to mock him, saying, 'This man began to build and was not able to finish'?" (Luke 14:28–30). The monthly budget is how you "count the cost" with your money. I know from experience and from walking with millions of people through the process that the fastest way to fail with your money is to skip the monthly budget.
That's why it's so weird to me when I hear people say things like, "I can't wait to be financially secure enough that I don't have to do a budget anymore." What? Let me clear something up for you: The moment you stop doing a monthly budget is the moment your wealth building will start to fall apart. I don't care how much wealth you have or how much money you make, you never outgrow the need to do a monthly budget. Even after all these years, you'd better believe that Dave and Sharon Ramsey sit down and do the budget every single month. This is the simplest but most important part of everything that we're trying to build here, so don't skip it. Ever.
Second, and this seems like a no-brainer, debt can never again be an option. The Bible says absolutely nothing positive about debt. Every time debt is mentioned, it comes with a warning or condemnation. Debt is a trap, and it will steal your legacy. The Bible says that the borrower is slave to the lender (Proverbs 22:7). You will never be so "sophisticated" that you can safely play around with debt. I've talked about that plenty in my other books and classes, so I won't go back into all of it here. Just don't do it, okay?
## A Word of Warning
Wealth is powerful and, therefore, dangerous. It is not for the spiritually immature. In fact, wealth is so powerful that a significant portion of the Bible's teaching on wealth focuses on the dangers. As we look at some of the warnings and snares, we should learn to take caution—but this does not mean we are called to avoid wealth as if it were evil. Most of the toxic, antiwealth messages in today's culture come from people who see a warning to remain cautious and mistakenly turn it into a prohibition. When wealth is viewed incorrectly or handled wrong, it can be a tremendous spiritual problem, so beware. But avoiding it altogether isn't the solution. Money is a fact of life, and it's one of God's blessings, so let's learn how to handle it well.
## Worship the Provider, Not the Provision
Sometimes people get confused about things when they finally have a little money. When you're broke and struggling, it may be easy to trust God to meet your needs. You don't have any other choice! But once you start to build some wealth, it's easy to become distracted. If you're spiritually immature, a pile of money can turn into an idol, and you may try to draw security from it instead of from the One who gave you that money in the first place. Scripture says, "The name of the LORD is a fortified tower; the righteous run to it and are safe. The wealth of the rich is their fortified city; they imagine it a wall too high to scale" (Proverbs 18:10–11 NIV). That is, the righteous run to God while the foolish trust their money. Proverbs 11:28 makes it even clearer: "He who trusts in his riches will fall, but the righteous will flourish like foliage."
Spiritually mature people, no matter how much wealth they do or don't have, understand that God is the Provider and money is the provision. You never want to fall into the trap of worshipping money. That's like getting an amazing gift from your parents on Christmas morning, but instead of thanking your parents, you thank the gift. It just doesn't make sense. The gift is an inanimate object; it doesn't have feelings or morals or intelligence. It's just a thing. Money is the same way. So throughout our whole legacy journey, we must always worship the Provider, not the method He uses to provide for us.
## Wealth Comes from Work
I believe without a doubt that God is the Provider, and He's the source of every dollar I've ever made. But that doesn't mean I've just sat around waiting for God to knock on my door with a handful of hundreds. That's not how it works. Remember the motto on the Ramsey family crest? Pray and Work. I think that's what God expects from us. That's the kind of attitude and activity He can bless. God promises to feed the birds, but He doesn't throw worms into their nests. Colossians tells us to do our work "heartily, as to the Lord and not to men, knowing that from the Lord you will receive the reward of the inheritance; for you serve the Lord Christ" (3:23–24). Proverbs says, "Wealth gained by dishonesty will be diminished, but he who gathers by labor will increase" (13:11). Do you see the connection here? "He who gathers by labor" (the guy who works hard) "will increase" (will build wealth). So wealth is a result of faithful, diligent, hard work along with biblical management of that income. If you're not interested in working hard, you aren't interested in building wealth. More importantly, you are missing that work is part of the mature believer's spiritual walk.
## Wealth Requires Maturity
Wealth can be deceptive. If you're not careful, it can change how you see yourself, other people, and everything around you. It can make you think that you're a bigger deal than you are. Wealth has a way of blinding people and making them forget what it took to get there. We've already talked about the passage that says the love of money is a root of all kinds of evil, but let's take another look at those two verses in a different light. First Timothy 6:9–10 says, "Those who want to get rich fall into temptation and a trap and into many foolish and harmful desires that plunge people into ruin and destruction. For the love of money is a root of all kinds of evil. Some people, eager for money, have wandered from the faith and pierced themselves with many griefs" (NIV).
Have you ever watched a cat chase a laser pointer? It's hysterical. If you put a little red dot of light in front of a cat, he will chase that dot around the floor, over furniture, and up the wall like he's lost his mind. It's like the whole world fades away and this insane animal only sees that one tiny target. That's how the Bible describes someone whose only goal is to get rich. You'll be blind to other people, you'll ignore needs around you, you'll neglect your family, and you'll do really stupid things with the money you have on the off chance of getting rich quick. Don't get me wrong; if you follow God's ways of handling money, you will build wealth. That's great, but the goal should never be wealth for wealth's sake. That's a petty, immature, self-centered attitude, and it's a trap for many people. Wealth demands maturity. Don't let wealth lead to laziness in your walk with God and indifference in your relationships with others. If you don't hold yourself to an ultrahigh standard, money will cause you to walk with a spiritual limp your whole life.
## Wealth Is Uncertain
No matter how careful you are or how detailed your financial plan may be, you can never, ever put your faith in wealth. Even with the most careful planning, wealth is uncertain—but God's provision is constant. You can count on God; you can't count on the stock market. Amen?
You can be wise with your budgeting, saving, real estate, and investments, but you just can't count on the economy. When you put all your faith in your stuff, your stuff becomes an idol. Like we just talked about, you start worshipping the provision, not the Provider. Money is not designed to be worshipped; it's just a tool. If a fine craftsman hand-built you a beautiful piece of furniture, would you thank the craftsman or the tools he used to build it? That sounds silly, but when we put our faith in money, it's the same as thanking a screwdriver for our new cabinets. It just doesn't make sense!
In 1 Timothy 6:17, Paul tells Timothy, his young apprentice, "Command those who are rich in this present age not to be haughty, nor to trust in uncertain riches but in the living God, who gives us richly all things to enjoy." I love that verse for several reasons. First, despite what anyone says about the "evils" of wealth, this verse doesn't give any hint that wealth is wrong. It simply states that there are wealthy people in the world, and then it instructs them not to be stuck-up, greedy, arrogant jerks. Second, it makes it clear that you can't put your faith in "uncertain riches." If you don't think riches are uncertain, think about someone who loses millions in the stock market just because he bet his whole fortune on one single company's stock. He was a millionaire one day and broke the next. Can you say Enron? Or what about saving up to buy a brand-new, shiny car that you probably love a little too much, then parking at the mall and watching some kid bounce his mom's van door off your car? That dent's a good reminder of how "uncertain" riches can be.
Riches are fine, but you will not be taking them with you. I have never seen a Ryder truck following a hearse. You can't trust in "uncertain riches," but you can completely trust the living God. Third—and this is awesome—where does Paul say wealth comes from? It's right there: "God, who gives us richly all things." How about that? If wealth is so evil, why does God give it so richly? And for what purpose does He give us richly all things? So we can _enjoy_ them. Isn't that interesting?
What's even more interesting is where this passage falls in Scripture—just seven verses away from the "root of all kinds of evil" verse that so many people use to prove that wealth is morally wrong. This is part of the _same discussion_! When you take one little step back and look at the context of the passage, it doesn't make sense to say that riches are evil in and of themselves, because if you read just seven more verses, you see that God is the One who gives us riches, and He gives them to us so that we can _enjoy_ them! And if you read one more verse ahead to 1 Timothy 6:18, you see how much good wealth can do in the world when the wealthy are "rich in good works, ready to give, willing to share."
Yes, wealth is uncertain, but God is not. God is a rock. He's steady and secure. You can trust Him to hold your wealth because He's the One who gave it to you in the first place. Trying to nitpick and micromanage every single economic variable is a recipe for disappointment. We have to be faithful and wise, but we can't _control_ the things that are _out of our control_. I like how Martin Luther said it: "I have held many things in my hands, and I have lost them all; but whatever I have placed in God's hands, that I still possess."
## Wealth Equality Is a Myth
There is a rising tide of popular thought in our culture today that says everyone's wealth should be equal. We hear all about the one-percenters and the ninety-nine-percenters; the haves and the have-nots; Wall Street and Main Street; the rich and the poor. We even hear this in many churches, with well-meaning Christians trying to figure out how to get everyone on equal ground. The problem is, Scripture doesn't teach that. Even Jesus said that we'd always have the poor with us (Matthew 26:11). So why do some argue that everyone's wealth should be equal across the board if even Jesus said it wouldn't be the case? Now, don't misunderstand me. I don't think Jesus meant that it's a _good thing_ that the poor will always be with us. I don't want anyone to struggle. But at the same time, I don't for one second think that striving for some kind of forced wealth equality is at all biblical or even moral.
Probably the most telling passage to me in the whole discussion of wealth equality is The Parable of the Talents in Matthew 25:14–30. Jesus tells the story of a rich man who went out of town for an extended trip, and he left three servants in charge of different portions of his wealth while he was gone. In the parable, the master gave one servant five talents, another servant two talents, and a third servant one talent. In today's terms, a talent is the equivalent of about $500,000, so we're talking about some serious wealth here.
When the master returned, he found that the servants with two and five talents each doubled their money by making wise investments. But the servant with one talent had a different story. He was afraid of his master, so he just dug a hole and hid the money. He didn't lose any, but he also didn't make any. It was a total wash, and the master lost any opportunity for growth on that money while he was away.
What did the master say to this guy? He called him a "wicked and lazy servant." This servant was a horrible manager. The master knew from then on that he couldn't trust this servant with anything, because the servant was too lazy or scared to act. He just dug a hole in the dirt and hid the money. And the master's response was completely counter to what we're seeing in our culture today. He didn't feel sorry for the servant. He didn't take money from the two other servants who had doubled their wealth so he could give a little to the third. No, he actually did the opposite. He took the one talent—half a million dollars—away from the third servant and gave it to the guy with ten times more! It's almost backward from what our culture thinks as just these days. He took it from the poor guy and gave it to the rich one. This is a story that Jesus is telling? And the hero of the story is the one who was more faithful in his management of God's stuff and built more wealth? That's kind of weird in our culture, isn't it?
Now I know this parable isn't just about money. You don't need to email me telling me I've missed the point of the whole thing. But one of the major points in this story for me is that fair doesn't always mean equal in the Bible. I believe in a fair playing field and equal opportunity (which all three of these servants had), but I don't believe there will ever be equal results. What matters is being grateful for what we're given, taking responsibility for it, and then managing it well. Talent, looks, initiative, and intelligence are not distributed equally and neither is wealth. I can't play a guitar like Brad Paisley, I can't play golf like Tiger Woods, I am not a computer genius like Bill Gates, and almost all of you have better hair than I do, so there will be different economic results. We all have equal value before the Lord and are equal as humans, but we do not all bring the same level of economic service to the marketplace. Service to the marketplace generates wealth, not your inherent value as a human. I often hear Christians say things like "no one should be paid $100 million to throw a football." Why not? If Peyton Manning generates ticket sales, TV ratings, and fans buying NFL apparel more than you do, he should be paid more than you get paid. That is biblical. If you disagree, go read The Parable of the Talents again.
## Can the Rich Go to Heaven?
It seems like every time I talk about the subject of biblical wealth, at least one person uses the story of the rich young ruler to prove me wrong. "Have you not heard, Dave, that rich people are not going to heaven? It says it right here. It's easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God! Your wealth is going to send you straight to hell!" I have heard that message preached at me a thousand times, usually by people who don't know any other verse in the whole Bible. They throw this verse out there as a gotcha—like they've found the key to undermine the idea of biblical wealth building. Sadly, they have no idea what that passage is all about. In fact, I think Luke 18:18–27 may be one of the most-often-misinterpreted passages in the whole Bible, so let's take a look at what it _really_ says.
This is how you usually see it quoted by the antiwealth crowd:
> Now a certain ruler asked Him, saying, "Good Teacher, what shall I do to inherit eternal life?" So Jesus said to him, "Why do you call Me good? No one is good but One, that is _,_ God. You know the commandments: 'Do not commit adultery,' 'Do not murder,' 'Do not steal,' 'Do not bear false witness,' 'Honor your father and your mother.' " And he said, "All these things I have kept from my youth." So when Jesus heard these things, He said to him, "You still lack one thing. Sell all that you have and distribute to the poor, and you will have treasure in heaven; and come, follow Me." But when he heard this, he became very sorrowful, for he was very rich. And when Jesus saw that he became very sorrowful, He said, "How hard it is for those who have riches to enter the kingdom of God! For it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God." (Luke 18:18–25)
That sounds pretty clear, doesn't it? Here's a powerful, wealthy guy who asks Jesus how he can have eternal life, and Jesus tells him to go follow all the commandments. That alone is impossible. No one can keep all the commandments perfectly. But that doesn't faze this guy. He says, "All these things I have kept from my youth," meaning he is a conscientious, rule-following kind of person. He basically said, "Yeah, yeah, Jesus. I got that. No problem there. What else?"
Jesus is so wise. He sees straight through to the heart of this person. He knows this young man is looking for a checklist of things to do to make it into heaven, but Jesus won't play that game. Instead, He immediately calls out the one thing He knows will be a stumbling block for this guy: his wealth. And when the rich guy goes away sad, Jesus gives what many take as a condemnation against wealth in general: "How hard it is for those who have riches to enter the kingdom of God! For it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God" (Luke 18:24–25).
Here's the problem: Most people stop reading the passage right there. They say, "Well there it is, Dave. It couldn't be clearer!" But if you stop at verse 25, you're missing the whole point of the passage. If you take it just two verses further, you see the true heart of the passage. When Jesus makes that bold statement in verse 25, saying how hard it is for the rich to enter the kingdom of God, the crowd is shocked. They reply in verse 26, "Who then can be saved?" You see, in those days, wealth was actually seen as a way to enter into the kingdom. Some thought you could buy your way in, and others thought that wealth was a sure sign of God's favor, so the wealthy were _obviously_ bound for heaven, right? So when Jesus says how hard it is for the rich to enter the kingdom, the crowd is dumbfounded. They basically say, "Wait a minute. If even the rich—who obviously have God's favor and who can afford whatever sacrifice it takes to get right with God—can't get into heaven, what chance do any of us have?" It's not a condemnation of wealth; it's a desperate plea for help.
How does Jesus answer their question: If even the wealthy can't be saved, what chance do any of us have? He says, "The things which are impossible with men are possible with God" (Luke 18:27). Surprise! This passage isn't about wealth at all. Jesus wasn't proclaiming a doctrine on money; He was illustrating a powerful teaching about grace. Money doesn't do anything to impact your salvation. It doesn't get you into heaven, and it doesn't keep you out of heaven. There's only one way to heaven, and that's through Jesus Christ. That's what Jesus is saying: Salvation would be impossible if it was left up to us, but it isn't. Our salvation is entirely in God's hands, and He alone makes the impossible possible.
The point of this passage is that no one can go to heaven without Jesus. Murderers can't get there. Drug addicts can't get there. Prostitutes can't get there. Gossips can't get there. The rich can't get there, and the poor can't get there. We are all sinners, and none of us can get there except through Jesus. It would be, as Jesus Himself said, impossible. So to say that the rich can't get into heaven because of their wealth is actually a form of heresy. When someone says that, they are essentially saying that the sacrifice of Christ on the cross is not enough to provide salvation to someone who has too many zeroes on their balance sheet.
Let's push it just a little bit further. Remember, when Luke sat down to write his gospel, he didn't put in all the chapter breaks and neat little subheads that we see in our Bibles today. It was just a continuous narrative. And only sixteen verses after Jesus' encounter with the rich young ruler, Luke introduces us to another __ rich guy, Zacchaeus, at the start of chapter 19. Remember him? He was a wee little man, and a wee little man was he. What do we know about Zacchaeus? Well, I can think of three things: He was a tax collector, he was rich, and he was short. This is a perfect counterstory to the Rich Young Ruler, because it tells us flat out that Zacchaeus had ripped people off by breaking the commandments (Luke 19:8). Compare that to the rich young ruler who said he'd never broken a commandment since he was a boy! So we have two rich guys, one who kept the commandments and one who didn't. Guess which one came to Christ? It was Zacchaeus, the crook! How is that possible? Because "the things which are impossible with men are possible with God" (Luke 18:27). The point of these stories is that salvation comes through Jesus, not through your money—and certainly not _in spite_ of your money.
## Giving Until It Hurts
Luke is on a roll in his gospel, showing Jesus interacting with all sorts of wealthy people, teaching about money, and using money as a way to teach deeper spiritual truths. We've already talked about the rich young ruler in Luke 18, then we saw Zacchaeus in Luke 19, and that brings us to Luke 20–21. Here we find a story that's traditionally known as the Widow's Mite. The story as we usually hear it picks up in Luke 21:1–4:
> And He looked up and saw the rich putting their gifts into the treasury, and He saw also a certain poor widow putting in two mites. So He said, "Truly I say to you that this poor widow has put in more than all; for all these out of their abundance have put in offerings for God, but she out of her poverty put in all the livelihood that she had."
I have heard dozens of sermons preached on that passage, and most of them usually give the same message: _You've got to give sacrificially. You've got to give until it hurts. You've got to give like the widow, who put every dime in the offering. When the rich give out of their abundance, it's no big deal. But when you give it all, just like the widow, you really put your faith in God and trust_ _Him for the blessing. Your heart should be just like the widow's who gave all she had._ If you've been in church for a while, I bet you've heard that sermon too.
This may be exactly what God meant when He inspired Luke to put this in his gospel. It's a fantastic point, and it's a great picture of giving. However, something about that interpretation has always bothered me. All through the Bible, we see examples of sincere, generous, godly men and women who have wealth and are never condemned or criticized for it. We are instructed that we don't actually own anything; we are just managers. We've already seen that Jesus' call to the rich young ruler wasn't a teaching on giving away everything; it was a spotlight on that one person's idolatry and an opportunity to explain the doctrine of grace. Instead, the Bible says the diligent prosper (Proverbs 21:5); that in the house of the wise are stores of choice food and oil (Proverbs 21:20); that if you don't take care of your own household, you're worse than an unbeliever (1 Timothy 5:8); that a good man leaves an inheritance for his children's children (Proverbs 13:22). That's the consistent message of Scripture: We are called to faithfully manage His resources and take care of our family. Sure, that means we should be constant, consistent givers. Giving should always be part of your financial plan, no matter where you are in the process. I've talked a lot about that in my other books and classes, and we'll talk more about it later in this book. But in this passage, we don't see someone giving _a lot_ ; we see someone giving _everything_ she had. The NIV translates it like this: "She out of her poverty put in all she had to live on" (Luke 21:4 NIV). Does that mean God wants us to stay in poverty and endanger our families? I don't think so. We have to ask what else may be going on here.
The first words out of Jesus' mouth in the passage may give us a clue. Like I said, most of the time when I've heard the Widow's Mite preached, the Scripture reading starts with Luke 21:1, "And He looked up and saw the rich putting their gifts into the treasury." Something about that feels weird to me. It feels like it's picking up in the middle of something, not starting a new story. Remember, Luke didn't break his gospel up by chapters and verses; that happened later. So if we ignore the chapter break between Luke 20 and 21 and look at the greater context of what Jesus was doing when "He looked up," the whole thing starts to look a little different. Let's back up just three verses to get a better sense of the context, and we'll run right through the Widow's Mite passage we looked at above:
> Then, in the hearing of all the people, He said to His disciples, "Beware of the scribes, who desire to go around in long robes, love greetings in the marketplaces, the best seats in the synagogues, and the best places at feasts, who devour widows' houses, and for a pretense make long prayers. These will receive greater condemnation." And He looked up and saw the rich putting their gifts into the treasury, and He saw also a certain poor widow putting in two mites. So He said, "Truly I say to you that this poor widow has put in more than all; for all these out of their abundance have put in offerings for God, but she out of her poverty put in all the livelihood that she had." (Luke 20:45–21:4)
Did you see it? Jesus condemns the religious leaders of that community. He calls them out for their hypocrisy, and He makes it clear to everyone around that those particular leaders just want to put on a show so people will think they're powerful and important. They make long, elaborate prayers so others will see how "spiritual" they are, and they always sit up front in the synagogue. They're _really_ special. At least that's what they want everyone to think. You know what else Jesus condemns them for? Devouring widows' houses. Jesus is commenting on a religious system that these leaders have put in place that is actively harming the poor and the weak. He says they not only take the best seats at the feast, but they go a step further to _devour_ widows' houses. They completely consume everything and everyone around them, including the weakest, most-struggling members of the community.
So if you ignore the chapter break, you see He _just_ made that comment about devouring widows' houses, and _immediately_ the passage continues, "And He looked up." What did He see when He looked up? He saw a poor widow putting _everything she had to live on_ into the scribes' treasury. We always talk about the Widow's Mite as a story of giving sacrificially out of a good heart, but isn't it interesting that Jesus never says anything about the woman's heart? It doesn't say that she gave "out of love" or "out of a generous spirit." The passage says that she gave "out of her poverty." We can make all kinds of assumptions about her spirit, but this story takes place in the context of Jesus' condemning a religious system for stealing from the poor—and particularly a widow.
Those who think wealth is evil and believe the only moral thing to do with money is to give it all away love the story of the Widow's Mite. And honestly, the traditional take on this story may be right. I'm not a biblical scholar; I'm just a guy who's studied the financial teachings in the Bible to try to understand God's ways of handling money. But when I look at this passage in this context, it's hard for me to believe that God's using this one, probably victimized, lady as a model for giving. Instead, I see it as a challenge for how the church should always strive to serve—not harm—its weakest members.
The stories of the Rich Young Ruler, Zacchaeus, and the Widow's Mite have been taught the same way for so long that it may be really uncomfortable to step back and look at them with fresh eyes. I get that. And I'm not saying that everything you've ever heard is wrong or that everything I'm saying here is right. But what I hope you'll do with these passages—and the whole Bible, for that matter—is to not take these amazing stories for granted. Dig in for yourself, read the passages, read what happened immediately before and immediately after each passage, and study commentaries. Don't get caught up in the chapter breaks and verse numbers that were added centuries later. And be very careful not to let your politics set you up to misinterpret Scripture. Use the power of the Holy Spirit and the brain God gave you to examine for yourself what message we're supposed to get from these passages. Then use that knowledge, guidance, and insight to avoid all the snares the Enemy has put in our way regarding wealth.
# TAKING THE DARE
My friend and former pastor Dan Scott says, "Adults are asked to manage dangerous things well for the glory of God. Children are not." And guess what? Money is dangerous. It's not evil; remember, it doesn't have morals. But it _is_ and _can be_ dangerous. I've used the brick analogy in the past: Money is like a brick. You can use it to build a church, or you can throw it through a church window. The brick doesn't care what you do with it. It can be a helpful tool, or it can be a weapon. To some degree, money is the same way. It is a responsibility that God's entrusted to us. With that in mind, I want to talk about what it means to accept that responsibility—or daring to win with money.
## It's Okay to Enjoy Wealth
Much of the toxic teaching about wealth is the result of spiritual immaturity. Some people are hit with the "money is evil" message so hard that they honestly feel guilty if they start to win. That's a trap! If money were evil, then why would God's Word contain so many examples of incredible, faithful men and women who have massive wealth and yet whose devotion to God is never questioned? Abraham, Isaac, Jacob, Joseph, Job, David, Solomon, Joseph of Arimathea, and Lydia are just a few examples of biblical heroes who honored God with the wealth He gave them.
The fifth and sixth chapters of Ecclesiastes contain some of the hardest, most sobering teachings about wealth in all of Scripture. Those passages make it clear that wealth is a responsibility, and it's easy for that responsibility to lead some people off a spiritual (and financial) cliff. Like I said, money is dangerous, and if you see yourself as the owner instead of the manager, wealth will lead you into trouble every time. However, if you keep your perspective straight—if you are always aware of the fact that you're just a steward of what God owns—then you have every right to enjoy the blessings and benefits of that gift.
Now, because I told you to always keep the broader biblical context in mind, let's look at Ecclesiastes for a minute. Tradition says that it was written by King Solomon in his old age. What do we know about Solomon? I can think of two things right off the bat. First, God blessed him with wisdom beyond what had ever been known before. Second, Solomon was probably the wealthiest person in history up to that point. So, God gave him wisdom, and God gave him wealth. Wisdom _and_ wealth. That's a pretty powerful combination. Of course, Solomon made mistakes. His wisdom and obedience to God were imperfect at times—after all, he was human. By the time he wrote Ecclesiastes, he had been through major ups and downs in his spiritual, personal, emotional, and financial life. He had seen it all, and at this point, he's ready to talk about it. So what does he say?
He spends most of chapter 5 warning us about the dangers of wealth and greed, which some use to support the "wealth is evil" belief. But, at the end of a rant against the misuse of wealth, Solomon pulls back and makes this observation:
> Here is what I have seen: It is good and fitting for one to eat and drink, and to enjoy the good of all his labor in which he toils under the sun all the days of his life which God gives him; for it is his heritage. As for every man to whom God has given riches and wealth, and given him power to eat of it, to receive his heritage and rejoice in his labor—this is the gift of God. (Ecclesiastes 5:18–19)
These two verses blow my mind! What is the focus of this message? It isn't me; it's not saying, "Look what I've done! I'm so awesome!" It isn't work; it's not saying, "My job is my provider." It isn't wealth; it's not saying, "Money is the goal, so go get you some." It isn't even the _enjoyment_ of wealth or the different ways you can bless others with it. There is one and only one focus of these two verses: God.
This passage makes it perfectly clear. God is the One who gives us "all the days of [our] life." God gives us work. He gives us the energy and power to do the work. He gives us the "riches and wealth" that come from our work. And—don't miss this—He gives us the "power to eat of it" and "rejoice in [our] labor." It's all from God! The days, the work, the power to work, the reward from work, and even the _enjoyment_ of the reward—the whole thing is His from start to finish! This is His gift to us, and it's not our _option_ to enjoy it. Scripture says that it is our "heritage" to "enjoy the good of all [our] labor"!
God gives us these blessings to faithfully manage, and that means we should always be wise stewards. It means we should always be giving. It means we should always be taking care of our families. And yes, it means that we should actually enjoy the unbelievable blessings He has put in our hands! Spiritually mature people with a right view of God's ownership can do all of that. We don't have to be scared of the wealth or ashamed of the fact that wealth enables us to do some fun things. That's what my friend Dan Scott meant when he said, "Adults are called to manage dangerous things well for the glory of God." We're adults, and we're _managing_ these things for God. According to Solomon, the wisest man who ever lived, part of managing our God-given wealth is honoring God with our enjoyment of that blessing.
## I Dare You
The first purpose of this chapter is to give you some biblical warnings about the dangers of wealth, which we've called the snares. The second purpose is to show, based on God's Word, that good stewardship of God's resources includes building wealth for kingdom purposes and managing and growing that wealth. That's the dare.
Building wealth isn't for wimps. Being diligent in managing your income and actually building some wealth is hard work. Managing that wealth for the kingdom is harder. And so I dare you. I dare you to build wealth and use it as a tool to serve your family. I dare you to build wealth so you can retire with dignity. I dare you to build wealth so you can truly leave an inheritance to your children's children. I dare you to build wealth so you can change not only your family tree but also your whole community. I dare you to be a good steward because Scripture is clear that good stewards are given more to manage. The diligent are going to prosper, and that means if you really follow God's ways of handling money, you are going to build wealth. But that's not the goal. This is not about you; it's not about some hocus-pocus, pseudo-Christian formula so you can get rich. I'm not about you getting rich; I'm about God being rich and you managing it well for Him! So go out there and be successful—I dare you!
### CHAPTER FOUR
# The Law of Great Gain
After more than twenty years of doing financial coaching, working one-on-one with families, talking about money on the radio for three hours every day, writing books, doing research, and leading classes about money, I think I may have stumbled upon the most powerful financial principle that there is. I truly believe if you get this principle right, you can get out of debt. If you get it right, you can live on less than you make. If you get it right, you can save, invest, and change your family tree. If you get it right, you can build wealth, give, and make an enormous impact on the world for the kingdom of God.
But if you get this principle wrong, those same things will be affected negatively. You might stay in debt your whole life, always living paycheck to paycheck. You'd probably never have any savings for emergencies, and you would most likely never be able to build any wealth. You may never be able to retire with dignity—or retire _at all_ , for that matter. You wouldn't have much to give, and you probably wouldn't leave much of an inheritance to your children's children.
Are you getting the idea that this is important? Does it sound like something that everyone on the planet, regardless of their level of wealth, income, or faith should pay attention to? Taking hold of this principle changes the shape, the speed, and the power of your legacy journey, but most people completely leave it out of their financial plan. But the truth is: I believe it is almost impossible to be successful without it. What's this big secret, this critically important financial principle? I'm talking about contentment.
# UNDERSTANDING CONTENTMENT
People don't understand contentment today. In our materialistic, stuff-driven society, where bigger is better and faster is master, it can seem impossible to actually find contentment—to slow down and say, "I'm content with the car I have, the house I own, and the job I love." That doesn't mean we should stop setting goals and working toward them. We'll talk more about that in a minute. Instead, it's a spiritual exercise to stop in the middle of our go-go, frenetic culture and just . . . breathe. The Bible says it this way: "Now godliness with contentment is great gain" (1 Timothy 6:6). Whenever I see a phrase in Scripture like "great gain," I not only view it through a spiritual lens but I also view it through a financial lens. I think this is a financial principle. It's not about piling up stuff for the sake of stuff. It's not a prosperity message; it's a responsibility message. Our ability to build wealth, use wealth for the kingdom, and enjoy the wealth God gives us all boils down to whether or not we can keep that wealth in perspective. And that's a matter of contentment.
## A Winning Attitude
What makes a bigger impact on your ability to win with money: your income or your attitude? Many people over the years have told me all about their debt and their inability to save, and they often say something like, "Dave, if I only made more money, then I could . . ." Well, sometimes that's true. Sometimes you really do just need to make a little more money. But more often than not, the problem isn't income; it's attitude. I can't tell you how many times I've talked to couples who say to me, "Dave, thank you for what you teach. We're doing so good. We have no debt and a full emergency fund. We've got $200,000 in our 401(k), and our kids' college is underway. Everything is awesome!" When I ask them what they make a year, they say $40,000. That's a great job.
But then I take two steps and someone else stops me. They'll say, "Dave, we need your help. We've got $50,000 in credit card debt. We owe $80,000 in student loans. We owe $30,000 on our car. We have no money in retirement and no savings for an emergency. Things are terrible!" Then I'll ask this couple how much they make, and they'll say something like $80,000. How is that possible? How can one couple be winning at $40,000 and another couple be losing—big time—at $80,000? The answer isn't math. If it were just about math, then the couple with twice the income should be doing twice as good, right? No, the problem is contentment. It really is the most powerful financial principle there is.
The Bible says, "Now godliness with contentment is great gain. For we brought nothing into this world, and it is __ certain we can carry nothing out. And having food and clothing, with these we shall be content" (1 Timothy 6:6–8). Contrary to some teaching, this passage is _not_ saying that we should never have anything more than food, clothing, and basic necessities. The focus of this Scripture is not on the _stuff_ but on the state of our spirit while we're in achievement mode. Contentment is not a lack of ambition or action or intensity. It's the condition of our hearts while we're pursuing those things. The question shouldn't be, "Is it okay for me to have some nice stuff?" The real question is, "If I _don't_ have nice stuff, will my spirit be okay with that? Do I have to have this stuff in order to have peace?" If the answer to that last question is yes, then you may have a problem with discontentment.
See, content people don't always _have_ the best of everything, but they always _make_ the best of everything. Someone will always have a nicer car or a bigger house. Keeping up with the Joneses is a fool's game because someone always has something better. Besides, I've done detailed research into the matter, and guess what? The Joneses are broke! They're up to their eyeballs in debt just because they want to impress you. They don't have "great gain," because they're trying to fill a void in their hearts with _stuff_. Listen, don't get me wrong. I want you to have some nice stuff; I just don't want your nice stuff to have _you_. That's the problem we fall into when discontentment guides our actions. Keeping this spirit in check is an exercise in spiritual maturity that is critical to having a quality legacy journey. And it's an exercise you'll work through over and over and over again. You'll never "get there" when it comes to contentment. That's because contentment is not a destination; it is a manner of traveling. It's not a place you're leaving from, and it's not a destination you're heading toward. It's how you're going about getting there. More than anything, it's how you feel in your spirit while you're making the trip.
## Spiritual Jellyfish
Sometimes I hear people in the church say, "Well, if you were a content Christian, you'd be happy to just sit at home and pray all day. You wouldn't worry so much about working hard and making money. You'd just trust God to show up and take care of all your needs." That would make you a spiritual jellyfish, a blob just floating around with no backbone and no direction. That attitude comes from a total misunderstanding of what contentment really is. Nowhere in Scripture do we get the idea that contentment means apathy or lack of ambition. I see that "godliness with contentment is great gain" (1 Timothy 6:6), but I also see that I'm called to "do [my] work heartily, as for the Lord" (Colossians 3:23 NASB).
A contented spirit is never an excuse for idleness. Just look at the apostle Paul. This guy was super motivated and ambitious before he met Jesus, and he certainly didn't slow down after he met Jesus. This man had things to do, places to go, and people to see! He was _moving_. He never took his foot off the pedal no matter what he went through or what he was heading into. Instead, he said, "forgetting those things which are behind and reaching forward to those things which are ahead, I press toward the goal for the prize of the upward call of God in Christ Jesus" (Philippians 3:13–14).
Paul was always pressing. He was always moving forward. He was always going somewhere and always had huge goals ahead of him. But isn't this the same guy who said, "I have learned to be content whatever the circumstances" (Philippians 4:11 NIV)? Sure it is. Paul didn't see a conflict between being content with what he had _and still_ pressing forward. His contentment was not an excuse to sit around and do nothing. He knew God had big plans for his life, and those plans involved a whole lot of action. But Paul didn't _need_ those plans to work out in order to maintain a peaceful spirit. In his heart, he was grateful for what he had, and that gratitude and contentment made him _more_ excited—not less—to keep pushing forward into whatever God had in store for him.
Contentment doesn't mean I'm not going to set goals and try to reach them; it doesn't mean I'm going to sit back and relax and stop working at my career, relationships, and wealth building. But it does mean that I'm not going to be torn up inside with a lust for _stuff_ while I'm working. I can be content, but I can still move forward.
# UNDERSTANDING ENOUGH
The key question when it comes to contentment is "How much is enough for you?" That's a huge question, and it's a question that people will crawl out of the woodwork to answer for you. When you have a little wealth, it's amazing how many people will suddenly appear to tell you what you should do with it. More than that, though, they'll tell you what you _shouldn't_ do with it. And too often, what they say you shouldn't do with it is enjoy it. Thank goodness it's none of their business.
## How Big Is Your Chalice?
Back in Chapter 2, I mentioned my friend Rabbi Daniel Lapin and his book _Thou Shall Prosper_. Like I said before, that book had a huge impact on me because it makes such a clear, compelling case for the morality of building wealth—as long as you do it from an unselfish, nongreedy, biblical perspective. In his book, Rabbi Lapin lays out specific principles that Jewish people have used to win with money for thousands of years. Using those Old Testament principles as a lens, I've been able to look at the New Testament teachings with fresh eyes and see things I've never seen before. I've also discovered a whole new way to view the contentment issue—in an image that illustrates this more beautifully and clearly than anything I've ever seen. It's an ancient Jewish ceremony called the _Havdalah_ , and Rabbi Lapin explains it this way:
> Jewish tradition strongly establishes the principle that each person makes his or her own needs the primary concern, although not the only concern. One could say that Judaism declares it necessary but insufficient to focus on one's own needs first. As the Sabbath ebbs away each Saturday night, Jewish families prepare for the productive work week ahead by singing the joyful _Havdalah_ service. This observance divides the Sabbath from the upcoming work week and asks God to increase both the families' offspring and their wealth. It also highlights their hands, as if to beseech blessing on the work of those very hands. The _Havdalah_ service is recited over a cup of wine that runs over into the saucer beneath.
> This overflowing cup symbolizes the intention to produce during the week ahead not only sufficient to fill one's own cup, but also an excess that will allow overflow for the benefit of others. In other words, I am obliged to first fill my cup and then continue pouring as it were, so that I will have sufficient to give away to others. So you fill up your own cup first, which symbolizes taking care of your own household first (1 Timothy 5:8). That's what "enough" is for your family—whatever it takes to fill that cup. And then you keep pouring so you have plenty to give to others. I love this image. A pastor friend gave me a chalice and saucer years ago, so in some of my live events and video classes, I use that to demonstrate what it looks like to fill your own cup first. Once the wine reaches the lip of the cup, it flows over the top and down the sides of the chalice and fills the bowl. It's a beautiful thing to watch, especially if you see it as a metaphor for taking care of your own family and then the needs of other people.
But here's what I want you to understand about the _Havdalah_ in light of this discussion on contentment: No one else can tell you how big your chalice should be. The size of your cup in this metaphor is 100 percent between you and God. There is no concrete, cookie-cutter answer for what "enough" looks like for every family. The top of the _Havdalah_ cup doesn't have a dollar amount written on it that says you can have _this much_ but no more. It's different for every family. A couple of hints though: First, if your family is lacking because of your extreme giving, then your cup might be too small. It isn't a thimble; it's a cup. Second, if there is never any overflow to help others, then your cup might be too big. It isn't a swimming pool; it's a cup.
Only you and God can decide how much is enough for you. After all, He gave you this wealth in the first place. He is your loving Father who loves to give His children good gifts. He adores you, and He knows what's best for you. He can see where you'll be in twenty years, and He can see the impact of every decision you've ever made or will ever make. He holds your life in His nail-scarred hands. He gets to decide the size of your chalice—no one else.
## The $100 Jeans Hypocrisy
Too often in our culture—yes, even our church culture—people around us try to play the role of the Holy Spirit by telling us how much is enough for us. The Pharisees and Gnostics are loose in the land, right? People pop up and say things like, "Her car is too expensive. No Christian should drive a car that nice." Or maybe, "Their house is entirely too big. They don't need that much space. What a waste of God's resources." Translation: "I have decided that your chalice is too big, and so I think you're a selfish jerk." Have you heard these messages? I've heard them my whole life, and I never knew what the measure was. But I think I've finally figured it out. Here's how it works for most people who spread these toxic messages: How nice a car is too nice? "A little nicer than the one I drive." How big a home is too big? "A little bigger than my home." If they can't even fathom ever owning something as nice as someone else, then that suddenly becomes the dividing line between holy and unholy. It's not based on Scripture, income, wealth, or anything like that. It's based on each critic's twisted, limited, self-centered point of view. And that's just sad.
That's how you end up with billionaires, like my friend I told you about a couple of chapters ago, who can give away a half-billion dollars in a single year and yet still be criticized for buying a $130,000 car. Let me put that in perspective for you. This fine, successful Christian businessman is worth $2.2 billion. When you look at $130,000 in the context of $2.2 billion, it doesn't even show up. It doesn't move the needle at all. Him buying a $130,000 car is the financial equivalent of you and me buying a biscuit! It just doesn't matter to his financial situation at all. I guarantee you that most of the people who criticized my friend were spending a much greater percentage of their income on luxuries than he spent on that car.
But the modern Pharisees don't see it that way. Instead, these critics put on their $100 blue jeans, leave their nice homes, drive their SUVs to the trendy coffee shop down the street, pull out their $1,000 MacBooks, sip their $8 coffees, and type blog posts about how wealth is evil. And somehow, the way they do it is holy, but the way my friend (who gave away $500 million in a year) does it is unholy. It just doesn't make sense! It's like Judas complaining about Mary pouring out the expensive perfume (John 12:1–8). Besides, we've already seen that anyone making $34,000 in America today is in the top 1 percent of income earners worldwide. The size of your cup is relative to the size of your income, your family, and a million other variables. Ultimately, though, it's between you and God, and it is nobody else's business. Don't let anyone steal your enjoyment of the success God's given you. Remember, it's your heritage to enjoy it (Ecclesiastes 5:19).
# DISCONTENTMENT DANGER SIGNS
What does it look like when we get confused about our role as manager and start thinking we're the owner? It gets ugly. Discontentment starts to grow in our hearts. It starts with just a little pinprick, but over time, that wound festers and gets infected until it's all about me. It's about what I want, what I can get, and what I think about what you have. We're going to look at a few signs that point to a problem with discontentment. I covered some of these briefly in my previous book, _Smart Money Smart Kids_ , which I wrote with my daughter Rachel Cruze. That book looked at contentment from a parenting point of view. Here, I want to go a little deeper to see how discontentment impacts our character as we build wealth.
## Attempting to Get Rich Quick
One sign of discontentment is attempting to get rich quick. You've seen these people, haven't you? They are always working some scheme, joining some weird investment group, and looking for a BBD—a bigger, better deal. They're basically dreaming of a lottery jackpot, and they'll throw thousands (or hundreds of thousands) of dollars into crackpot schemes thinking they've found a shortcut to wealth. Maybe they even want to share that "opportunity" with you—for a price. Let me burst that bubble: For 99.9999 percent of us, there is no shortcut. Sure, you could become wealthy very quickly if you invent the next Facebook (which took them years to build), but outside of that, get-rich-quick schemes are a fast track to losing all your money. Proverbs 28:20 says, "A faithful man will abound with blessings, but he who hastens to be rich will not go unpunished." Personally, I kind of like The Message translation of that verse a little better: "Committed and persistent work pays off; get-rich-quick schemes are ripoffs." Amen.
A bigger problem with trying to get rich quick, though, is that it represents a serious spiritual issue. God is the Owner, and He entrusts His wealth to us to manage. When you point all your guns at some crazy, overnight "wealth-building opportunity" that you saw on late-night cable TV or that your broke brother-in-law told you about, what are you really trying to do? Trick God into handing you more ahead of His schedule? Here's a thought: Maybe God knows how much you can handle right now and when you may be able to handle a little more. Most people don't have the emotional and spiritual maturity to go from broke to billions overnight. That's why you see so many horror stories of people whose lives are ruined within a couple of years of winning the lottery. When you drop a lot of money into a broken system, it just widens the cracks.
## Trying to Appear Wealthy
The next sign of discontentment is trying to appear wealthy. There's a saying for that in Texas: big hat, no cattle. We all know people like this. These are the people who appear to have it all. They have a large, beautiful home with a perfectly manicured lawn. There are two brand-new luxury cars in the driveway. Their kids are in private school. They're always talking about some incredible vacation they just took. They have the nicest clothes, and their fingers and necks are dripping with diamonds. They look perfect, don't they? Well, more often than not, all of that is a façade that is hiding a mountain of debt, stress, and fear. That perfect couple may spend every night yelling at each other over the debt. They might be buying their groceries with their credit cards. They could be (and probably are) one missed paycheck away from losing everything. And it's all because they want to put on a show for the world, dressing their lives up in what looks like success but is really a financial and emotional nightmare.
By now you know I'm not saying you shouldn't have nice things. I believe God wants you to enjoy His blessings, and that includes buying some luxuries when you can afford them and when they make sense in your plan. Then, those things can be a blessing for you and your family, and you can have them because you want them, not because you want to impress other people. And I can tell you with 100 percent certainty that anything you buy with debt—no matter how much you enjoy it and no matter what it makes other people think of you—is _not_ a blessing. "Well, Dave, how can you say that? Aren't you judging other people's purchases?" Yes, this is one time when I am! And I can back it up with Scripture. Proverbs 10:22 says, "The blessing of the LORD makes one rich, and He adds no sorrow with it." That last part is the key: "He adds no sorrow with it." Debt and payments are sorrowful. No one is excited or feels blessed because they _get_ to make a ridiculous car payment every month. It's a burden, it is stealing their legacy, and it is absolutely _not_ a blessing!
Even if you pay cash for these luxuries, they can still be ridiculously out of place if you go there too soon. For example, I don't think anyone should ever buy a brand-new car unless they have $1 million in net worth—even if they pay cash. The goal of your legacy journey is to make wise decisions over time, always taking care of the **NOW** , **THEN** , **US** , and **THEM** , and making sure that your giving, spending, and investing are proportional to your wealth. So if you're jumping the gun a little bit and spending too much money on stuff just to impress other people, stop it! You'll get there in time, but if you have a burning need to show __ other people your success, then you most likely have a discontentment problem.
## Anxiety about What You Don't Have
"My name is Dave, and I like stuff."
I have to admit that I'm a recovering stuffaholic. As a kid and young adult, I was driven by the need to have and acquire more and more stuff. It was a huge problem that got me into big trouble when I was young. I don't struggle with that anymore, though, because it was burnt out of me the hard way—through bankruptcy court. Now that I use God's ways of handling money (and have for more than twenty-five years) and now that I understand my role as manager, freaking out about what I don't have isn't a big deal to me anymore.
When you struggle with discontentment, it's easy to get obsessed about what you don't have. This isn't about wealth or possessions. This is a spiritual problem where you're trying to fill a void in your life with some _thing_. The problem is, there's always _another_ thing that you think will make you happy. And so you go through your whole life in an endless pursuit of stuff, always thinking that the _next_ thing will be _the_ thing __ that makes you happy. And when you buy it, that happiness lasts for about a second until you see something else you want. Then it starts all over again. I've been there. I've heard that voice in my head telling me that I _need_ this or that. I've felt that pressure and anxiety about what I think is missing in my life. I can't think of many other things that will steal the joy of your success more than always being fixated on what you don't have.
If this is a problem for you, here's a tip: Cut yourself off from the source of the problem. For example, if you're a shopaholic, stop going to the mall. If you can't walk past a store without suddenly being overcome by the need to buy everything in the display window, then for heaven's sake just stay away from there! There's no such thing as window shopping when you struggle with a burning need for more stuff. Or if you're not ready to buy a new house, then don't go looking at houses. At best, you'll just come home frustrated and depressed. Or at worst, you'll make a stupid buying decision that could put your whole financial legacy in danger. Looking at more stuff, reading about more stuff, and fantasizing about more stuff isn't going to help you get past a sense of anxiety and panic over what you think is missing. It just feeds the beast, so stay away from it.
# JEALOUSY AND ENVY
Probably the ugliest sign of discontentment is the one that's most common: jealousy and envy. I used to think these two words meant the same thing, but I was wrong. Now I know that there's a distinct difference between the two. Jealousy says, _I want what you have._ Envy, however, goes a step further. It says, _I want what you have, but I can't have what you have, and so I don't want you to have it either_. These two spirits are running wild in this culture today, and they're at the heart of a lot of the "wealth is evil" movement. If you go online for more than five minutes, I bet you'll read a rant about some evil rich guy wasting his money and how he should give it all to the poor.
I've already mentioned my friend Robert Morris, who pastors a huge church in Dallas. Robert is a great writer on the subject of generosity. It's a passion for him, and it's definitely a focus of his ministry. I've spent time at his church, and the whole congregation has an amazing spirit of generosity that obviously flows from the leadership down. Robert is one of those rare people who God has called to give everything away—twice. But again, this is rare. Even Robert will tell you that God doesn't call most people to give everything away. This was a specific calling on Robert's life at two specific times in his ministry. Please don't be confused about that. I'm just telling you that to give you a sense for Robert's heart for giving so you'll be able to understand what's coming next.
Robert's book _The Blessed Life_ gives a fantastic picture of jealousy and envy at work:
> I once remember riding in the car with someone and passing by the large, beautiful home of a person who I knew to be a committed Christian and who had prospered by following biblical principles and giving generously.
> I pointed out the house to my driving companion and mentioned the owner's faith. His response was, "Well, he ought to sell that thing and give the money to the poor." Of course, the person making that comment was living in a house that was nicer than nine-tenths of the world's population could ever dream of owning. And guess what? He had no intention of selling his home and giving the money to [the poor].
> The ugly truth is that he didn't care about the poor. He just resented the fact that someone had a nicer house than he did. A spirit of compassion didn't prompt the comment—a spirit of envy did.
> This false spirituality manifests itself in several different but similar comments. See if any of these sound familiar: "How could anyone in good conscience drive a car that expensive?" "She sure could have helped a lot of people for what she spent on that coat." Or my personal favorite: "I could sure do a lot of good with the money they spent on that [insert name of luxury item here]." Remarks such as these are pure selfishness and jealousy dressed up as religious superiority—and it's ugly.
The bottom line is that no amount of wealth can be called spiritual or unspiritual, and no one except you and God will truly know if you're being a wise or wasteful steward. And remember that you have a full-time job taking care of yourself, so don't bother trying to judge what anyone else should be doing with what God's given them to manage. He gives to each one according to His plan. Translation: Mind your own business!
# IT'S ALL ABOUT RATIOS
One big reason why people criticize what others do with their wealth is that they are only looking at a dollar amount. The problem is, dollar amounts are relative. The amount doesn't change, but its relative worth changes big time based on the person's financial situation. For some families, giving $1,000 is a monumental, once-in-a-lifetime opportunity. For others, $1,000 is the amount they tithe to their church each week. What really matters here isn't the dollar amount, but the amount as a percentage of their income. For that, it all comes down to ratios.
## Reality Check
Let's slow down for a minute to do a quick reality check. At this point in the Baby Steps, we'll assume that you have some wealth. I know that's not true for everyone, but for the sake of this discussion, I'm going to assume that you're out of debt, have a full emergency fund, are actively contributing 15 percent of your income into retirement, have your kids' college funding underway or taken care of, and you've paid off the house. Whew! Great job! If you're not there yet, don't worry about it. You'll get there. God's ways work, and if you keep plugging away, you'll be at this point a lot faster than you think. So with all that behind you, you're now into Baby Step 7 territory, which is building wealth and giving a bunch of it away.
What about the **NOW–THEN–** **US–THEM** model? Well, if you're at this point, it means that you've done a great job of taking care of the **NOW** , because everything is under control. The **THEN** is pretty much taken care of because you've been investing into retirement for a while and you've knocked out the mortgage. You're doing great with the **US** , because you're building wealth that will one day be passed generationally. And now it's time to kick the **THEM** stage into high gear. That's where you get serious about doing acts of mercy and evangelism for the kingdom with God's money. Sure, you've been tithing at every stage of the process, and once you got out of debt and got your emergency fund in place, you started doing some extra giving here and there. That's great! But when you hit the **THEM** stage, your giving comes totally unglued. At this stage, you've lived like no one else, so now you get to give like no one else. Again, I know you may not be there yet. You may still be getting out of debt or building up your emergency fund. That's okay, but I need you to go here emotionally with me for this discussion. Imagine that you've just sent the last mortgage payment to the bank. You're sitting at the kitchen table looking at your checkbook, and you realize that you are completely free. The house is paid for, and you have $500,000 or $1 million in your retirement account. Then comes the question _What do I do now?_ Let's talk about that.
## Cups, Swimming Pools, and Thimbles
The size of your chalice—the amount needed to take care of your own household—is completely up to you and God. Never forget that. But that means you actually do need to figure out how big your cup should be. It's different for everyone; there is no magic number that fits every family or every income level. That's why no one—not even me—can look at your situation and tell you what that number should be. I will encourage you, though, to be balanced as you think through this. Like I said before, if you set it too high, you'll end up with a swimming pool, not a cup. That's where greed creeps in, and there won't be any overflow to give to others. And if you set it too low, you'll have a thimble. That means you'll be giving everything away and you won't have a realistic amount for your family to spend and enjoy.
## Understanding Ratios
The best way I know to keep the size of your cup in balance is to use budget ratios. A ratio shows us a dollar amount based on a percentage of income, and I'm going to show you how to apply ratios to the overflow from your cup. You see, a ratio looks at the whole picture. Saying, "I'm going to give X percent of my overflow" is a lot different than saying, "I think I should give X dollars away." The dollar amounts are relative, remember? Setting specific dollar amounts for your giving breaks down over time. At some point in your wealth building, that dollar amount may be way too low, so your cup starts to look like a swimming pool. Or you might set it too high early on, and you'd be feeding your family out of a thimble. Ratios keep things in balance, and they keep you from going out of bounds.
There are a few things that we should always be doing with our money, regardless of our level of wealth. One of those things, of course, is giving. When we understand that God is the Owner, we remember that He's called us to be generous, joyful givers. If we're managing things well for the Owner, then we are definitely setting aside a portion of our income for giving—and at this stage, this is well beyond the tithe. If you're a Christian, giving a tithe to your church should be part of your budget no matter what Baby Step you're on. But when you're at Baby Step 6 or 7, you can (and should) kick it up a notch—or two or three.
The second thing you should do with the overflow is invest in the future. The Bible says, "In the house of the wise are stores of choice food and oil" (Proverbs 21:20 NIV). That means wise people save and invest money. At this stage, you're still investing into retirement funds, but you're also actively investing for wealth building. That creates more wealth to pass down generationally (Proverbs 13:22) and more wealth to give to others. In a sense, you can think of your wealth building as a way to grow the golden goose. The bigger the goose, the bigger the golden eggs you (and future generations) can generously give to others.
The third and final thing you should do with the overflow is enjoy a reasonable lifestyle based on your level of wealth. It's perfectly biblical (therefore right) to not only take care of your family, but also to actively enjoy a portion of what God's given you to manage. One of the biggest blessings that comes from using ratios is that it helps you maintain balance while keeping your heart in the right place. Luke's gospel says, "For where your treasure is, there your heart will be also" (Luke 12:34). I've heard so many people get that verse backward. People seem to think it's saying that you put your money where your heart is. Read it again: "For where your treasure is, there your heart will be also." To me, that's saying _our hearts follow our treasure_ , not the other way around. You may not be a selfish, greedy jerk at the start, but if all you do is spend money on "me, me, me," then your heart will follow your dollars. Guess what that makes you over time? A selfish, greedy jerk. Planning out your legacy journey using ratios protects you from this because it sets guardrails on your spending. It's fine to spend money on yourself as your wealth grows; ratios simply help make sure you're not spending _all_ of it on Jet Skis and Jacuzzis.
## Planning Ratios
The goal is to look at the overflow and set ratios for each of those three areas. You'll decide what percentage to give, what percentage to invest, and what percentage to use on lifestyle. The first thing you have to do, though, is decide what size cup you need to take care of your family. If you are at Baby Step 7, you'll set your zero-based budget at whatever income level you choose. That will serve as the "Income" blank on your monthly budget, as though that were all you brought home. We're going to put ratios in place for anything over that amount, so be sure it isn't so high that there's no overflow and isn't so low that you and your family can't enjoy the blessings God's given you. You have to find the balance, which definitely means you need to talk to the Owner. Pray about it, and ask God to show you what income is appropriate for your family. Ask Him to bless this exercise too because this is your commitment to be a good manager of His resources.
Let me give you some examples, but keep in mind that these are just examples. Nowhere in this book will I tell you how big your cup should be or what percentages you should set for your ratios. Maybe you're on Baby Step 7 and you have an annual income of $80,000. You could go through this exercise and decide that your family could live well on $50,000, so you'd apply ratios to the remaining $30,000. Or maybe you're a doctor and you make $250,000 a year. You could pray about it and, together with your spouse, decide that your family would live well on $150,000, meaning you'd apply ratios to the remaining $100,000. See how this works? First you set the baseline, then you apply ratios to anything over that amount.
Before we start looking at how the math works on this, let me go ahead and give you a word of warning. I've said a dozen times now that only you and God can decide the size of your cup and assign ratios. Nevertheless, even after you spend time in prayer and feel good about the decisions you make, you should expect criticism. People will always think you should have done more, given more, and kept less. They'll question your spirituality if (and when) your numbers don't measure up to _their_ standard of what is holy. This is honestly one of the hardest things about winning with money. Having people who have no idea what it's like to bear this responsibility question what you're doing with God's money is frustrating and heartbreaking, but you can't bend to their pressure. God's given you this money to manage, not them. Let them worry about what He's given them; you're responsible to the Owner, not the critics.
## Ratios in Action
Okay, let's flesh out how all this works. I'm going to give you an example with real numbers to work with, but remember that this is just an _example_. The ratios and percentages we'll use are what worked for one family that I counseled years ago. I'm not saying this is what you should do yourself. That's between you and God. Also, the income we're going to work with is astronomical. I get that. I chose it on purpose because I want you to see that this system works no matter what your income is. But this is definitely not the average American income, so don't write me any crazy letters telling me my example is too high!
Over the years, I've had the opportunity to work with a lot of wealthy and famous people. I'm going to use one of those families for this example, but obviously I won't use the guy's name. I'll just tell you that it was an NFL player at the top of his game, and he had an income of around $10 million a year. That's a lot. That kind of salary may seem like a jackpot to most people, but for this family, it was a huge responsibility. He and his wife love Jesus; I prayed with him several times, and it was powerful. He has an amazing spiritual walk, and he is extremely concerned with managing God's money wisely. Are you getting that this is a good man who took his responsibility as a manager seriously? Good.
The first time we talked, he said something like, "Dave, I don't know what to do with all this money. All my buddies just spend everything. They give it all to their family members who think they won the lottery. They're supporting friends who are engaged in toxic behaviors. They lose half of it in bad business deals because they didn't think things through. It just slips right through their fingers. But I feel a huge responsibility to manage this money for God, and I don't know what I'm supposed to do. Can you help?"
Isn't it weird to hear a guy making that much money be so insecure? The truth is, most of them are like that. All that money comes with a huge burden, and people don't often talk about managing wealth at a high level like we are in this book. That's why it scares so many people, and that's why we need to figure these things out.
I walked this couple through the same things we've talked about in this chapter. I told them about setting a baseline income and about ratios for extra giving, investing, and lifestyle. Their first obstacle was setting a reasonable income. They talked about it for a few minutes, and then he said, "Dave, we've talked about it, and I think we could live really well on $100,000 a year." Now this guy was making $10 million, and he said he wanted to live on $100,000. It was all I could do to not start laughing.
I said something like, "Nuh uh. Listen, you're doing really, really well. You've worked hard at your craft. God has blessed you tremendously. I think God would smile if you spent some of that money on yourself and your family. He's your Father, and He's excited that you're doing well. Let's try $400,000, okay? You've got $10 million; you're not really going to miss $400,000. Besides, you can always adjust things later if this isn't the right fit." He and his wife went home and prayed about it, and the next week we set his monthly budget income based on an annual income of $400,000. Then we had to run ratios on everything he made over that $400,000 mark.
The first thing we factored in was the tithe. They were already tithing on the $400,000 they set for their baseline income, so of course they were going to tithe on anything above that too. So that's 10 percent. The second thing to factor in was taxes. The government is definitely going to take their share of your wealth, so the couple figured on about 40 percent of their overflow going to taxes. That meant between tithe and taxes, they had 50 percent of their overflow left. For that, we applied ratios for extra giving, investing, and lifestyle.
The couple agreed that 10 percent felt like a good number for extra giving, so that's what they used. Again, this is above the tithe. By the way, for this guy, 10 percent is $1 million. That means he was tithing $1 million and then giving _another_ $1 million away—every year. That's actually not too uncommon among people at this level, but that's not what you hear about from the media. This level of giving is so far beyond what most people can even fathom, so do you understand how completely crazy it sounds when people say things like, "He should give more"? Come on, people.
Next, this couple settled on 35 percent for extra investing for wealth building. Keep in mind that we're not talking about wealth building just for the sake of growing a big pile of money. The investments are building wealth for kingdom purposes, so in a sense, a big part of the extra investing is actually delayed giving. You're going to invest that money now so that you'll have more to give later. Kind of sounds like the Parable of the Talents, doesn't it?
If you followed the math in this example, you see that this left 5 percent for extra lifestyle spending. The couple hesitated on this one and almost threw that money back into extra giving, but the truth is, God wants you to enjoy your success. As long as you keep it in balance, there's absolutely nothing wrong with that. Remember, it's your heritage to enjoy what God's blessed you with (Ecclesiastes 5:19). So they put the remaining 5 percent toward extra lifestyle.
Let's review how this turned out in this particular over-the-top, but real-life example. Remember, they were making $10 million a year and had set $400,000 aside as their baseline income. Everything over that was considered overflow for ratios. In this example, that means this couple applied ratios to $9,600,000 of overflow. Based on what they decided, these were their actual numbers:
> **Tithe for Overflow Ratio:** 10% = $960,000
> **Taxes for Overflow Ratio:** 40% = $3,840,000
> **Extra Giving for Overflow Ratio:** 10% = $960,000
> **Investing for Overflow Ratio:** 35% = $3,360,000
> **Extra Lifestyle for Overflow Ratio:** 5% = $480,000
> **Overflow Ratio Total:** 100% = $9,600,000
Interesting to note that their overall plan (baseline income plus overflow) had 40% going to taxes, 10% to tithe, 9.6% to extra giving, and their spending on themselves was only 6.8% of their income. Remember, you apply whatever ratios or percentages you would like to your overflow above your baseline budget. If you're a Christian and you're deciding on your ratios, it is easy to start by assuming 50 percent of your extra will go to tithe and taxes. That leaves you 50 percent of your extra income for investing, giving, and lifestyle.
This exercise was a huge blessing for this family, and it has enabled them to not only grow their own wealth over the years but to also give generously, enjoy their success, and ensure that their wealth never leads them off the proverbial cliff emotionally or spiritually. It's done the same thing for my family, and I think it'll do the same for yours. And because it's based on a percentage of income, not on fixed dollar amounts, this works whether your overflow is $10,000 or $10 million.
# A JOURNEY, NOT A DESTINATION
The apostle Paul wrote, "I have learned to be content whatever the circumstances. I know what it is to be in need, and I know what it is to have plenty. I have learned the secret of being content in any and every situation, whether well fed or hungry, whether living in plenty or in want" (Philippians 4:11–12 NIV). I love that last part: "whether well fed or hungry, whether living in plenty or in want." By the way, Paul wrote that while he was sitting in prison! Even in prison, Paul's contentment wasn't based on his circumstances. He knew that God is not necessarily concerned with shaping your circumstances; He's concerned with shaping your character. And God does that throughout the journey, in the good times and the bad, when you're doing well financially and when you're flat broke.
The world is addicted to "bigger and better," so the idea of actually slowing down and enjoying what you have before moving on to the next big thing is kind of a joke these days. But without contentment, your whole life will just be jumping from one thing to another, always hoping that the _next_ thing will be the thing that will make you happy. That's just not going to happen. Remember: Contentment isn't a destination; it's not a place you get to. It's a manner of traveling. It's an attitude that influences everything you do with money. If you leave it out of your plan, you'll never feel like you have "enough" of anything. And trust me, that's no way to live.
### CHAPTER FIVE
# Your Work Matters
Did you ever imagine you'd have all this?"
The reporter's question hung in the air for several seconds. I've been interviewed for different things literally thousands of times over the years. Most of these people ask the same set of questions, so I can usually breeze through the answers kind of on autopilot. But this time there was something about that question that hit me funny. She was doing a story on the twentieth anniversary of our radio show, and I have to admit, I was feeling a bit nostalgic. I'd been thinking about all the ups and downs I'd experienced since God started me on this professional journey, and my head was full of memories. So when she asked me that question, for the first time in a long time, I was stuck. It took me a few seconds to get my thoughts together enough to give her a good answer.
I thought about my prayer journal from 1993. Back then, "all this" consisted of a brand-new local radio show heard only in Nashville and a little blue self-published book called _Financial Peace_ that had sold a few thousand copies. I was fresh out of bankruptcy, and I was fighting and clawing to make a living as I sketched out my dream career. I had a strong sense that God was calling me into something beyond real estate, which is what I'd been doing before, but I didn't know much about publishing or radio back then. I was making it up as I went along.
If you were to look in my actual prayer journal from 1993, you'd find a page where I listed some initial goals. The second goal on that list says, "high-touch support group concept; combination seminar and counseling." From those few words, I started teaching a twenty-six-week-long class called _Life After Debt_. I taught it live a few nights a week in my community. The first night I taught it, we set up 135 chairs. Only four people showed up. Today, more than twenty years later, that class is now _Financial Peace University_. It's taught on video in basically every city in the country, and millions of families have been through it.
A couple of years later, I had another written goal: Sell 50,000 copies of _Financial Peace_. At that point, I had only sold 7,000 copies, and most of those were sold out of the back of my car. Selling 50,000 in a year was a huge goal—and I didn't hit it. But over time, I ended up selling 50,000, and then 100,000, and today, _Financial Peace_ has sold more than 2.5 million copies. That same year, I set a goal for our radio show. We were heard in only one city back then, and we started thinking about syndicating. I wrote a goal to be in twenty-five cities within a year. We didn't come close. A couple of years later, we had only gotten up to twelve cities. But today, we're up to more than five hundred stations with more than 8 million listeners and, as of this writing, we're the number three talk radio show in the country and the largest privately owned syndicated show.
So when that reporter asked, "Did you ever imagine you'd have all this?" my head was swimming with all this history. I replied, "Well . . . yes and no." See, I knew that there was a huge need in the country. I knew that people were hurting in the area of personal finance and that someone needed to stand in that gap to serve them and help them out of their mess. But at the same time, I never in a million years expected to be the guy God would use to do that on this scale. I'm just a boy from Antioch, Tennessee. I wasn't a great writer. I wasn't a trained speaker. I wasn't an old pro in radio. I wasn't wealthy. At the time, I was still paying off old debts and getting my own act together. I was only a couple of years past the mess that most of my clients and listeners were in. So when God opened my eyes to this huge need, my thought was, _Yeah, God, You're right. Somebody's got to do something to help these people. But until You find the right guy, I'll do what I can to help out._ That response probably made God laugh. I can picture Him reaching down, tapping me on the shoulder, and saying, "Nuh uh. I'm going to use _you_. People will see that if I can use a broke, broken, inexperienced hothead like you to do this, then it really is from Me!"
When we started out, I had no idea how hard this journey would be. I didn't realize how many hours, years, or air miles it would take. I didn't know how difficult it would be to watch my team grow from one to five to fifty to five hundred. And through it all, I still never got over the fact that God used me to do this work. That is the most humbling part. I remember early on, some older, wiser man told me, "Dave, success is a journey, not a destination." I had no idea what he meant. Now that I've been doing this for a while and I'm reaching old-man status myself, I think I finally get it. There has never been a single point in my career where I became "successful." God's been writing success in and out of my story the whole time.
Looking back, the whole journey has been satisfying to my soul, but I've never been confused about _whose_ work it is. This was all God's work, and for some reason, He chose me to do it. That's been one of the biggest blessings of my life. The truth is, though, He does this for all of us. Jeremiah 29:11 says, "'For I know the plans I have for you,' declares the LORD, 'plans to prosper you and not to harm you, plans to give you hope and a future'" (NIV). I believe that God has a plan for each and every one of us, and that plan includes a job, a career, a calling to do a specific work while we're on this earth—whether it's a nine-to-five job, a dream that starts in your basement, or being a stay-at-home parent. That means our work matters to God, and what we do with our days has a huge impact on the legacy we leave behind.
# WORK MATTERS
As we talk about work, I want to start by telling you what this chapter is _not_. I'm not here to give you a detailed, foolproof plan for finding the perfect job. We're not going to talk about what to wear to a job interview, how to figure out what you want to do with your life, or how to target and win over potential employers. If that's what you're looking for, I recommend you check out _48 Days to the Work You Love_ by my friend Dan Miller. If you're looking for a more philosophical discussion about how to find the perfect job, you can read _Start_ by Jon Acuff. You'll find a helpful summary of the _Start_ concepts in the appendix of this book.
But as far as this chapter goes, I don't want this to be a _how-to_ __ guide at all. Instead, I want it to be a _why-to_ chapter. Why does work matter to God? Why should the right kind of work matter to us? How does the work we do impact the legacy we leave? Those are huge questions, and we're not going to answer them fully. However, as you work on your legacy journey, these are all things you need to carefully examine.
## Nine Years of Your Life
Most people would consider the average American's working lifetime to be age twenty-five to age sixty-five. Of course, plenty of people start earlier than that, and plenty of people work well into their seventies, eighties, and even nineties. But for the sake of simple math, let's say that your working lifetime is right around forty years. If you work the average workday of eight hours and take three weeks off per year for vacation and personal time, that means you'll spend roughly 78,400 hours—almost nine years of your life—at work. That's nine years' worth of time you're not with your family, you're not at church, you're not doing mission work, and you're not doing any of the "important" things we think we're supposed to be doing with our lives. I meet so many people who struggle with this. They say things like, "Dave, I just feel so guilty. I feel like God's put me on earth to do something important, but I can't do it from my cubicle. What do I do?" I'll ask if they hate their job, and often they'll say, "Oh no! I love my job! We provide a valuable service that helps a lot of people, and we do it better than anyone else. I just wish I could spend more time doing what God wants me to do."
Why is it that we sometimes think God only blesses the work we do _outside_ the office? If God's put us in a system where we'll spend forty years of our lives working, doesn't it make sense that _that work_ matters to God too? We get in the habit of compartmentalizing our lives, thinking that we have to be one way at church, one way at home, and yet another way at work. Where I come from, we call that hypocrisy. I feel pretty strongly that God's called me to be _me_ , and that doesn't change based on where I am or what I'm doing. If I matter to God, and if God's given me something to do for forty hours a week, then that must mean my work matters to Him.
## Work Is Important to God
Work is a major theme all throughout the Bible. It seems like you can't make it through a single chapter in Proverbs without reading about the rewards of diligence and the dangers of laziness. Proverbs 13:4 says, "The soul of a lazy man desires, and has nothing; but the soul of the diligent shall be made rich." Proverbs 10:4 says, "He who has a slack hand becomes poor, but the hand of the diligent makes rich." Probably my favorite of these is Proverbs 21:5: "The plans of the diligent lead surely to plenty, but those of everyone who is hasty, surely to poverty." Tracking this theme through Proverbs is fascinating. If you've never read the whole book, I'd encourage you to read one chapter of Proverbs a day for a month (it's easy since there are thirty-one chapters). As you read, make a note in your Bible every time you see a reference to work or laziness. That exercise could be a game changer for you.
Isn't it interesting that over and over again, Scripture says the result of laziness is poverty, but the result of hard work (diligence) is wealth? That's why I've said for years that your greatest wealth-building tool is your income. Think about that: What does your income represent? Getting a paycheck regularly is the reward for diligence. If you keep showing up at work, they keep paying you. Big shock, I know. But too many people get distracted with get-rich-quick gimmicks or high-risk investing because they want to get rich overnight. Instead of building wealth one day at a time over the course of many years, they want a shortcut. They act like there's a big, mysterious secret to becoming very wealthy, and if they sit on the sofa and think about it long enough, they'll figure it out. That's just not going to happen, but careful, diligent management of your income over a long period of time allows practically anyone to become a millionaire.
When you're relaxing in a rocking chair in twenty or forty years, you can never forget Who gave you the work that enabled you to build wealth. In Chapter 4, we saw how our ability to work, the days to work, the reward of work, and the enjoyment of that reward all come from God (Ecclesiastes 5:18–19). Everything about our work is His. If you ever get confused about that, remember what the Bible says in Deuteronomy:
> You may say to yourself, "My power and the strength of my hands have produced this wealth for me." But remember the LORD your God, for it is he who gives you the ability to produce wealth, and so confirms his covenant, which he swore to your ancestors, as it is today. (8:17–18 NIV)
If God gives us the ability to produce wealth through hard work, then we have to treat that like a special gift from our Father. He could make it rain gold coins if He wanted to, but instead, He gave us work. That's a biblical truth that goes all the way back to Adam in the Garden of Eden.
## Warning against Idleness
As I read the New Testament, I see that the apostle Paul didn't have much patience for idle hands. The church at Thessalonica was a particular sore spot in this area. Certain members of that church had become so convinced that Jesus was returning any day that they stopped working. They thought, _Why should I go to work today if Jesus is coming back tomorrow?_ Instead, they sat around not doing anything except eating other families' food and becoming gossipy busybodies! Paul hit this bunch pretty hard in 2 Thessalonians 3:6–12:
> In the name of the Lord Jesus Christ, we command you, brothers and sisters, to keep away from every believer who is idle and disruptive and does not live according to the teaching you received from us. For you yourselves know how you ought to follow our example. We were not idle when we were with you, nor did we eat anyone's food without paying for it. On the contrary, we worked night and day, laboring and toiling so that we would not be a burden to any of you. We did this, not because we do not have the right to such help, but in order to offer ourselves as a model for you to imitate. For even when we were with you, we gave you this rule: "The one who is unwilling to work shall not eat." We hear that some among you are idle and disruptive. They are not busy; they are busybodies. Such people we command and urge in the Lord Jesus Christ to settle down and earn the food they eat. (NIV)
The model Paul says we should imitate is one of hard work, where we strive to take care of our own needs and not become a burden to others. That doesn't mean there won't be times when we need help; we've all been in a bad spot from time to time when the kindness of other people or the church was the only thing that got us through the storm.
What Paul is talking about here is an overriding attitude of laziness and idleness, acting as though our needs were someone else's responsibility. That's simply not a biblical position. When we discussed the meaning of **NOW** in the **NOW–THEN–US–THEM** framework, we saw that our first duty is to do what it takes to provide for our families: "If anyone does not provide for his own, and especially for those of his household, he has denied the faith and is worse than an unbeliever" (1 Timothy 5:8). Our work matters to God, if for no other reason than it is the means by which we take care of our families—and that's our top financial priority.
## Work Is Creative
My friend Rabbi Lapin, who I've already mentioned a couple of times, once told me, "Man is most in God's image when he is creating things." I love that line because I'm a creative guy. I love to come up with new ways to communicate information, and I surround myself with people who are better at it than I am. That's why my company is involved in so many different forms of media, from web to video to books to live events. Our company's mission statement sums it all up: "We provide biblically based, common-sense education and empowerment that give HOPE to everyone in every walk of life." Everything we do is focused on that one thing, and we fire all our creative cannons at making that happen.
For example, I remember the first time I sat down with some of my team members to discuss the topics that would ultimately become _The Legacy Journey_ class, which preceded this book. We knew people wanted a follow-up to our _Financial Peace University_ class, but we never make a new product just because we need something new to sell. Everything we do has to have a purpose, so some of us sat around my conference table one night and talked about whether or not God was calling us to create a new high-level discussion on wealth, contentment, and extreme generosity. As we talked that night, every single one of us lit up as we worked through new ideas and concepts. We felt an undeniable energy in the conversation, and there wasn't a person in the room who didn't think God was calling us to do something new. That started a months-long process of planning, meeting, writing, editing, and designing. One team was focused on capturing just the right message. Another team was responsible for the video shoot. Another team went to work on all the design elements. Another team planned the live event where we'd shoot the video. And I got to sit in and over all of it, watching God knit this new thing together. It was a blast! Those are always some of the most exciting, most rewarding days I ever have because that's when we're in our zone, creating brand-new things out of nothing. The times I've done that the best are the times in my life when I have felt closest to God, when it seems like we're riding the wave of the Holy Spirit. It's amazing!
So when you are creating, dreaming something, then helping it to be born, you are truly acting like you were made in God's image. Find some things to do that give you that childlike excitement periodically in your career. Your work, your life, everything about you—it all matters to God.
## Finding Purpose in Activity
In today's culture, so many people are wandering around in what I call _professional limbo_. They spend most of the day sitting around doing nothing. If you ask them if they are looking for a job, they'll say they want to figure out their purpose first. It's as though the very idea of taking an entry-level position at a good company is incomprehensible. Sure, they'll stay on government unemployment for two years, but filing papers, cleaning houses, or waiting tables is somehow beneath them. It doesn't make sense. Whether you're just out of school or newly out of work, get out there and do something! There are always pizzas to be delivered, houses to be cleaned, and lawns to be cut. I have a friend who says, "Residential landscaping is a gold mine. Rich people are scared of leaves!" One leaf blower may be all it takes to feed your family for months!
The key is activity. If your family is struggling or even if you're trying to launch your career from scratch, sitting around thinking about your "purpose" can be a form of procrastination. You'll end up like Cousin Eddie in _Christmas Vacation_ , who was out of work for seven years because he was "holding out for a management position." Let me tell you what you _should_ be managing: Your life. Your family. Your legacy. You can't find your purpose on your couch, unless your purpose is spare change. Get out there and move some things around, and your purpose will find you. It's called the Butterfly Effect and it means: A little bit of activity today can lead to enormous results later. You never know what the final result of any action will be. Doing something—anything—today may be the thing that leads you to the thing that leads you to the thing that leads you to the _perfect_ thing for you. It's not going to be a clear, easy-to-follow journey, and you're not going to step from the starting line immediately to the finish line. There's a whole race to be run between those points, so don't worry if you can't see the finish line yet. You're not supposed to. You can't control the finish line. As Jon Acuff says, "The only line you control is the starting line." You've got to get started. There are no results without activity.
I'll give you one warning, though. Some people take their professional activity way too far—to unhealthy extremes. Now, I'm a huge fan of work. I _love_ to work and to work hard. I surround myself with people who do the same. Everyone at our office is passionate about what they do. Most of them would probably stay there all night half the week if we let them. But here's the thing: We don't let them. We work hard, but at 5:30 every evening, it's time to shut down the computers and leave. We take the passion and enthusiasm home with us and pour it into our families. We work hard, sure, but we also play hard. We don't get confused about what's really important in our lives.
Sometimes we'll bring a new web programmer or publishing person on board who came from a business where eighty-hour workweeks were the norm. They can have a hard time adjusting to the way we do things, and so we have to help them by kicking them out the door at 5:30 p.m. The work we do and the work you do is important, but it's not more important than our marriages and children. I often tell my team that I refuse to become a success with a trail of divorces and broken homes behind me. Don't let that happen to your family either. Be passionate at work, but be more passionate at home.
# SUCCESS STARTS IN YOUR HEART
I talk to dozens of people about their money every day on my radio show. A lot of these men and women are calling because they want to get out of debt and they want to know how to do it. I can always tell which ones are serious and which aren't. There's something in their voices that communicates passion and conviction when they're really excited about getting out of debt. But if they're just playing around with the idea, if they're simply curious about it, then their voices are flat. If I don't hear any passion behind what they're saying, I know they aren't ready to cut up the credit cards and dump their debt for good. That's because getting out of debt isn't about solving a math problem; it's about changing your life—and that requires a change of heart. Proverbs says, "Above all else, guard your heart, for everything you do flows from it" (4:23 NIV). That means everything of value in my life—from getting out of debt to my career to my relationships to my wealth building—starts in my heart.
## The Strangest Secret
I'm a big fan of goal setting and positive thinking. I guess I can thank my parents for that. Mom and Dad were real estate agents, and they were good at it. They knew how to sell, and any good salesperson knows that a big part of being successful in sales is keeping a steady stream of positive thoughts running through your head. They taught me early on that if I want to win, I have to imagine myself winning. If I want to build wealth, I have to imagine myself building wealth. If I want a good marriage . . . If I want to raise great kids . . . If I want to grow in my faith . . . It all starts in the mind and heart, "for everything you do flows from it" (Proverbs 4:23 NIV).
Because of that attitude, my parents were always listening to sales-training and personal-development audiotapes every time they got in the car. That means little Dave Ramsey, riding in the backseat during long road trips, got a master's degree in positive thinking by age ten. Some of my favorite memories are of my family driving to a vacation spot while listening to the greats like Zig Ziglar, Charlie "Tremendous" Jones, and, of course, Earl Nightingale.
Nightingale's powerful message _The Strangest Secret_ got stuck in my brain from an early age. I remember hanging on every word of that audio presentation while sitting in the back of my parents' car. It's a message I've probably come back to a hundred times in the forty-five years since. What is "the strangest secret"? It's the simplest thing in the world: We become what we think about. Our thoughts are directly tied to our results. It sounds almost biblical, doesn't it? Nightingale explains:
> I want to tell you about a situation that parallels the human mind. Suppose a farmer has some land. And it is good fertile land. Now, the land gives the farmer a choice. He may plant in that land whatever he chooses. The land doesn't care. It's up to the farmer to make the decision.
> . . . Now let's say that the farmer has two seeds in his hand—one is a seed of corn, the other is nightshade, a deadly poison. He digs two little holes in the earth and he plants both seeds, one corn, the other nightshade. He covers up the holes, waters, and takes care of the land. And what will happen?
> Invariably, the land will return what's planted. As it's written in the Bible, "As ye sow, so shall ye reap." Now remember, the land doesn't care. It will return poison in just as wonderful abundance as it will corn. So up come the two plants: one corn, one poison.
> The human mind is far more fertile, far more incredible and mysterious than the land, but it works the same way. It doesn't care what we plant—success or failure, a concrete, worthwhile goal or confusion, misunderstanding, fear, anxiety, and so on. But what we plant must return to us.
> The human mind is the last great, unexplored continent on earth. It contains riches beyond our wildest dreams. It will return anything we want to plant.
But the key is, you have to plant something. You have to put something in your mind and work toward it. That's called a goal, and more than anything else in the world, setting and striving toward goals will determine whether or not you are a success in life. Earl Nightingale defined success as "the progressive realization of a worthy ideal," or goal. Notice that he didn't say it's the _accomplishment_ of a goal; it's the progressive realization—the act of working toward the goal. So when are you a success? You're a success when you get off the couch, set a good goal, and get to work! The minute you take that first step toward a goal, you become successful. Success isn't really about _reaching_ the goal—although it's best if you do; it's about setting your mind on something and doing what it takes to make that goal a reality. That's when you change your life.
My friend Zig Ziglar once said, "By altering our attitudes, we can alter our lives." That's true. It starts in our hearts and minds. Author Robin Sharma took that idea a step further: "Everything is created twice: first in the mind, and then in reality." Think about that. When an artist gets ready to paint a picture, she has to create it in her mind before the brush ever hits the canvas. Before a builder builds a house, someone has to dream up what that house will look like. Before you write a book, you have to have an idea of what you want to say. You get the idea. The bottom line is that you can't do a single thing in your job, family, or finances that doesn't start in your mind. The human mind is one of the most powerful God-given forces in all of Creation, and we each get the opportunity to steer and direct that force every day of our lives! So here's an idea: Take control of what's in your mind!
How many times did Jesus talk about the power of faith, of simply _believing_? He said, "According to your faith let it be done to you" (Matthew 9:29). "As you have believed, so let it be done for you" (Matthew 8:13). "Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you" (Matthew 7:7). "If you have faith and do not doubt, . . . you [can] say to this mountain, 'Be removed and be cast into the sea,' [and] it will be done" (Matthew 21:21). Are you seeing a theme here? Belief matters. Setting your mind on something and believing it's not only _possible_ but _probable_ is the key to success. I want to be clear here, though. I'm not talking about simply dreaming. Dreaming doesn't get you anywhere if you don't take the next step and tie it to a goal. We're striving for a progressive realization of a goal, not the endless repetition of a dream. Dreams are a great start, but a dream that's ready to go to work becomes a goal.
## Finding the Sweet Spot
A goal takes what starts in your mind and turns it into reality. It puts action to the creative force we've talked about. I told you that when we first started syndicating our radio show, I wrote a goal to go from one to twenty-five cities within a year. Once that idea became a goal, I had to get to work. It wasn't going to happen accidentally. Now, like I said, we didn't hit that goal for several years, but it gave us a target. By writing it down, my team and I were able to start working toward that goal. Whether we hit it within the time frame we set or not, according to Earl Nightingale, we were _already_ successful just because we were moving toward a worthy goal.
If you look at my early goals for the radio show, the _Financial_ _Peace_ book, and our _Financial Peace University_ class, one thing really stands out: The goals were just out of reach. We didn't hit any of them in the time frame we set. Does that mean we failed? No way! Remember, you're a success when you start working toward a goal. We had a target, and we got to work. It took several years to hit those first few goals, but we got there. And by the time we reached them, we were emotionally mature enough to handle the responsibility that went along with that level of success. God has a way of preparing you for the success that comes with reaching your goals while you're in the process of working toward them. That might be why some people seem to win overnight while it takes others a while to get there. Maybe God's working some things out in those people's lives along the journey. I know He did with me.
When you set goals for your life, you need to set them slightly out of your reach. Picture yourself reaching as far as you can, and then set your goal a few inches past that, where your fingertips can barely touch it. Jim Collins and Jerry Porras, authors of _Built to Last_ , call that a BHAG: a Big Hairy Audacious Goal. You know what that does? It leaves room for the power of God. There's a delicate balance between setting realistic, achievable goals and leaving room for the power of God to work. You don't want to set easy goals that you could almost stumble into just to make sure you reach them. And you also don't want to set goals that are so crazy and impossible that you'll never be motivated to work toward them. You want to leave room for the power of God, but that doesn't mean you can set a completely unrealistic goal and then simply sit back and wait for God to make it happen. That's not a goal; that's a dream. Remember, goals put you to _work_!
Have you ever held some food or a chew toy up in the air to encourage a dog to jump? If you hold it too high, the dog will know there's no way he can reach it, so he won't even try. If you hold it too low, the dog will get the prize once or twice and then get bored. But there's a sweet spot, where if you hold the toy in just the right spot—high enough to make it a challenge but low enough to make it seem achievable—that will keep the dog jumping all day. That's where you want to set your goals.
## The Wheel of Life
I've studied the practice of goal setting for a long time, and one, distinct paradigm continues to rise to the surface. Zig Ziglar called it the Wheel of Life. Picture a big wheel with spokes dividing the wheel into seven segments. The wheel is your life, and the seven sections are the key parts that make up your life. These are all specific areas that we have to be intentional about in our goal setting. The seven areas are Career, Intellectual, Financial, Social, Physical, Spiritual, and Family.
For an overall healthy life, you need to set goals in each of these seven areas. As you look at the list, you'll no doubt see a couple that come naturally to you. For example, the career and financial spokes have always come easy for me. I've always been able to make money. I used to have trouble _keeping_ it, but I was always able to _make_ it! Career-wise, I come up with a dozen new business ideas every day before breakfast. Most of them stink, but a few of them turn out to be pretty good. I'm just wired that way, so setting goals in those two areas is a breeze.
Social, on the other hand, is definitely my weakest area. I love people, and I love big crowds of people, but forced and concocted social events make me want to scream. The idea of standing over a plate of finger foods and going through the whole "What do you do?" exchange sends me running for the hills. I'm convinced that if I wasn't married, I wouldn't have any friends outside of work!
So every year, I set goals for myself in each of these seven areas. I don't give myself a pass on social just because I stink at it. In fact, I'm _more_ intentional about my social goals because if I don't write them down and hold myself accountable to them, they'll never get done. The point here is that you want to keep growing in every area of life. You'll always have areas where you excel and some where you struggle, but you want the wheel to keep turning. Zig Ziglar used to say that if you ignore one area, your wheel gets a flat spot. That leads to a flat tire, which makes it harder to move forward in life. You'll waste a lot of time, effort, and energy trying to push a flat tire, so you have to make sure every part of the wheel gets your attention.
I want to give you one word of warning, though. There are some who say you should always strive for balance in your life. That is, you always have to maintain a certain tension between all of these different areas to make sure you're not focusing too much on one area to the exclusion of the others. They may call it a "work-life balance" or "balanced life." That's a trap. The truth is, there will never be a time in your life when everything is perfectly balanced. If you're trying to launch a new business, you're going to spend more time in the career area. If you're trying to lose weight or run a marathon, you're going to spend more time in the physical area. If you're up to your eyeballs in debt and set a goal to be debt-free, you're going to spend more time in the financial area. Trying to give equal time to every area and every goal will practically guarantee failure. I believe in a balanced life, but that's a big-picture view. My goal is to look back over my life over a long period of time and feel good about the whole thing, not about where I spent my time in any specific day, week, or month. You never want to ignore any area—especially your family—but trying to evenly divide twenty-four hours a day across all seven areas will drive you insane and leave you exhausted.
## Five Guidelines for Good Goals
Many people I talk to are terrified of writing goals. They act as though they're carving these things into marble instead of writing them down on a legal pad. Then, if something goes wrong or they fall short, they end up with a cloud of guilt over their heads that keeps them from moving forward. If that's you, you need to relax. Goal setting isn't rocket science, and you get to control every part of the goal. It's like doing a budget: I don't care _what_ you spend your money on; I just want you to write it down and spend your money on purpose. With the Wheel of Life, I don't really care what your goals are; I just want you to write them down and do them on purpose too! So take a breath, let go of any anxiety you feel, and write down a few _good_ goals. You can do that by making sure your goals follow five guidelines:
1. Your goals must be specific.
2. Your goals must be measurable.
3. Your goals must have a time limit.
4. Your goals must be your own.
5. Your goals must be in writing.
The starting point for every goal is to make sure it's specific and measurable. "I want to lose weight" is not a goal; it's a wish. A goal has to be specific, and you have to know what winning looks like. So instead, you could say, "I want to lose thirty pounds." That gives you a definite target to aim for, and there will be no ambiguity about whether or not you reach it.
Next, your goals must have time limits. By what date do you want to lose thirty pounds? Sometime in the next decade? That won't work, because you'll never be motivated to get started. Instead, you could say, "I want to lose thirty pounds in three months." Now we're cooking. From there, you can break the goal down into bite-sized pieces. Thirty pounds in three months is ten pounds per month, or two-and-a-half pounds a week. At this point, specific, measurable, and timed are all working together to keep you on track.
But _why_ do you want to lose thirty pounds? If it's because your spouse wants you to, it'll never happen. Your goals must be your own. You'll never stay motivated and make the sacrifices needed to win if you're chasing _someone else's_ goals. But if it's a goal that you're passionate about, one that you've fully embraced and made your own, then you'll do whatever it takes to make it happen.
The last step is the one that most people overlook: Your goals must be in writing. Habakkuk 2:2 says, "Write the vision and make it plain." There is spiritual power in writing down your goals. Something happens in your heart when you take an idea out of your mind, carefully articulate it into a goal, and then write it down. Having it in writing keeps it in front of you. It holds you accountable. Again, you set your own goals, but once you write them down, you become accountable to that piece of paper. It's your measuring stick. That's what shows you if and how you're winning.
# SUCCESS IS A PROGRESSION
I have a confession to make. I wasn't a great leader twenty-five years ago. I also didn't know much about being a great husband or father back then. I was terrible at managing my money, and I was heading into bankruptcy because of my own financial stupidity. I was an immature, baby Christian in those days too. By the grace of God, I'm better at all of those things now. It didn't happen overnight, but over the course of years and decades, by setting goals and striving to reach them, by working hard "as to the Lord" (Colossians 3:23), God grew me in every one of those areas and more.
The truth is, every successful person I know is a failure. They have a long line of failures behind them—goals they fell short on or mistakes that threatened to derail their plans. Some of them, like me, actually _did_ have their plans derailed because they weren't on the paths God had set for them. God had to step in and do some course corrections. This has definitely been true in my life. I absolutely love what I get to do for a living, and it's truly my life's calling. However, if you were to ask that young, hotshot real-estate wiz in his twenties what he wanted to do with his life, he wouldn't have had a clue. He would have thought the next forty years would be an unending string of bigger and better house flips and investment properties. Spending decades helping people with their money wouldn't have even been an option. Fortunately for me, God stepped in and set my life on the journey He mapped out for me. He'll do the same for you.
So let's review: Work creates wealth. Work done as a career is more satisfying and usually pays better. But the best possible scenario is to begin working (no couches) so that you can run into opportunity, then polish that opportunity into a career. Then if you continue to polish and pray, while finding your passions, setting goals, and directing your thoughts toward success, your career will fully blossom into a calling. When you work in your calling, you are creative, talented, passionate, and intentional as unto the Lord, which will likely cause you to build wealth over time from an ever-increasing income. This progression is quite the opposite of people who never change, never grow, never set goals, and then stagnate in the marketplace. These people don't have twenty years of experience; they have one year's experience repeated twenty times. See the difference?
If you are going to be intentional about your wealth building, estate planning, and legacy journey, then you must be intentional about developing your calling into a wonderfully fulfilling, productive, and prosperous part of your life and your spiritual walk.
We said at the beginning of this chapter that success is a journey, not a destination—and I believe God's got big plans for every one of us. Scripture says, "'For I know the plans I have for you,' declares the LORD, 'plans to prosper you and not to harm you, plans to give you hope and a future'" (Jeremiah 29:11 NIV). He's got the whole journey already mapped out! That doesn't mean we can just sit back and wait for Him to carry us every step of the way, though. We've got to get off the couch, throw a brick through the television, and get to work.
### CHAPTER SIX
# Safeguarding Your Legacy
I have a good friend whose father was a two-term governor of our state—the youngest governor in the state's history, in fact. So my buddy spent eight years as a little boy in the Governor's Mansion. That family spent their time and energy invested in politics. Their dinner table conversations were about public service. Just by sitting at the table, he learned everything his dad knew about how to govern wisely. Is it any wonder that my friend grew up to become a US congressman? I have another friend whose father was a master mechanic. Growing up, he and his dad talked shop all the time. They spent several hours a week lying under or leaning over cars, rebuilding engines, and talking about carburetors. My friend soaked it up like a sponge. Is it any wonder that my friend grew up to be a serious gearhead? In both of these cases, the sons received an inheritance—a legacy —of love, time, attention, and resources that shaped their futures. They weren't little robots with no choice in what they would do with their lives; they were boys whose dads had shared their passions with their sons, and that passion became a family tradition. It was woven into the environments they grew up in.
The cool thing is that you get to choose the environment your kids grow up in. You can set the tone, the conversation, the shared passions—all of it. Of course, as your kids develop their own personalities, they're going to have interests of their own, and they may not like what you like. Believe me, I've got three kids, and they're all totally different! But the one thing they have in common is that they all grew up under my roof, and they all shared the experience of what it meant to grow up as a Ramsey. My wife, Sharon, and I were intentional about what we did and what we talked about. We taught them about God's ways of handling money and how to live for other people. Now that they're all grown up, they have different personalities and interests, but their core values are tied directly to the inheritance Sharon and I passed on to them.
As I've talked to literally hundreds of millionaires—and even a few billionaires—over the years, I've noticed that practically every super-wealthy family of faith is intentional about preparing their children to shoulder the financial responsibility of generational wealth. All you see on TV are the drama queens and train-wreck stories, but in real life, godly wealthy people take this responsibility very, very seriously. That's because they love their children, and they want this generational wealth to be a blessing for them. Nobody wants to ruin their kids' lives! But that's exactly what wealth does if it's passed down to adult children who don't have the emotional and spiritual backbone to carry it. So just like political families talk about politics, wealthy families talk about wealth. It shapes the conversations around the dinner table.
But kids are just one part of the equation. If you really want to build a legacy that lasts, you have to be intentional about safeguarding that legacy against anyone and anything that could tear it apart. That's what we're going to cover in this chapter.
# THE SAFEGUARD PARADIGM
As I've had the opportunity to talk to a lot of wealthy people (and people working to become wealthy), one of the most common questions I get is this: "Dave, as you and Sharon went broke and then rebuilt your wealth, how did you keep the new wealth from ruining your life, your marriage, and your kids?" That surprised me the first few times I heard it, but then I started to understand the heart of the question. How many times have we seen wealth poison a family? How many spoiled, crazy rich kids make headlines for running their lives off the rails? How many friendships have been destroyed because one guy got wealthy and his buddies were hurt or confused by it? We see it everywhere, and if we don't take steps to protect our relationships from it, it could happen to any of us. That's what I mean when I talk about "safeguards." We want to put guardrails in place to keep our wealth from running our lives and relationships into a ditch. This chapter is all about the **US** of the **NOW–THEN–** **US–THEM** framework. This is where we'll talk about protecting our closest relationships—the **US** —from the dangers of wealth.
## Blessings or Curses?
Deuteronomy 30:19 says, "I call heaven and earth as witnesses today against you, that I have set before you life and death, blessing and cursing; therefore choose life, that both you and your descendants may live." Obviously, that passage is talking about the decision to obey the Lord, but I think we can see this as a financial principle as well. You see, as your wealth expands, so do your opportunities to choose either blessings or curses, life or death, for future generations.
The decisions you make at each step of your legacy journey dictate the type of legacy you'll leave. If you're intentional about protecting your marriage, teaching your kids about money, giving generously, managing your wealth using ratios, and maintaining healthy relationships, then you'll leave not only a powerful, enduring financial legacy, but also a legacy of love and respect that will last several generations. However, if you aren't serious about these things and you just try to wing it, your wealth will be a curse to future generations. Your financial success could be responsible for a broken marriage, spoiled-rotten kids, and a legacy of greed and destruction. No one wants to be remembered that way. So, in order to protect your future legacy, you have to safeguard your life today.
## Four Areas through Three Lenses
We can safeguard our legacy by being vigilant in four key areas. If you're lazy in any one of these areas, you could put your entire legacy in jeopardy. The first area is your personal life—your legacy as an individual. This is crucial whether you are single or married, and whether you have kids or not. Who you are as an individual influences every other relationship in your life, so we have to be careful here. The second area is your marriage—your legacy as a couple. What are the traps that could endanger your marriage? The third area is your children—your legacy as a family. What are you doing to proactively protect your kids from the dangers of wealth? The fourth and last area is other relationships in your life—your legacy as a friend and extended family member. There are all kinds of problems that could pop up in any one of those relationships. So if you want your wealth to be a blessing instead of a curse, you have to make sure you're actively protecting each area.
How do we do that? I believe it comes down to viewing each of these four areas through three lenses: ownership (whose is it?), magnification (what does it do to each area?), and community (how does it affect those around me?). And I'll say up front that the Ramsey family is not perfect. We've done a lot of these things really well, but we've also done some of it poorly in the past. That's why we call this a legacy _journey_ —we're not there yet. However, I think we've figured out some things along our journey as a family. We've gotten the bumps and bruises, and we've gotten a lot of them under the microscope of public scrutiny. If you host a national radio show telling people how they can build wealth, you better believe there are plenty of people out there watching and waiting for you to mess up. But so far, with God's help and by being intentional in these key areas, we haven't had any major problems.
That's the safeguard paradigm: four relationship areas through three lenses. We'll view our personal lives, marriages, kids, and other relationships through the lens of ownership, magnification, and community. It sounds like a lot, but it breaks down pretty simply. Stick with me!
# PERSONAL LIFE: MY LEGACY AS AN INDIVIDUAL
In the previous chapter, we talked about Proverbs 4:23: "Above all else, guard your heart, for everything you do flows from it" (NIV). Think about that for a minute: "everything you do." That means your walk with Christ, your relationship with your spouse, how you interact with your children, the way you operate your business, how you manage your money, how you act around your friends . . . it all flows out of your heart. I learned a long time ago that if I wanted to serve the Lord, have quality relationships, and win with money, I had to take control of the guy in the mirror. He was the only thing holding me back. He was the only one who could ruin everything God wanted to do in and through me. As you work on safeguarding your legacy, the first and most important area you need to manage is yourself.
## The Ownership Lens
"Hey, dummy. You don't own anything."
Sometimes that's the first thing I tell myself when I wake up in the morning. I've already told you that I'm a recovering stuffaholic. I like stuff. God's blessed my family, and we have some nice stuff, but I'm not confused about Who really owns it. Psalm 24:1 says, "The earth is the LORD's, and the fulness thereof" (KJV). It's His. It's _all_ His. I'm a car guy, and love my car. But every time I get behind the wheel, I have to emotionally remind myself that God gave me that car, and if He tells me to give it away, I will. That's His right, because He's the owner. He could do the same with my house, my business, my money, or anything else because it's not mine. "The earth is the LORD's"—the whole thing.
But you know what's even scarier? That also means that _my family_ belongs to the Lord. I love my wife and kids more than anything on earth, and as a loving husband and father, it's hard for me to put those lives and relationships in His hands. I'm a doer; I'm always doing something or working on something. Finding the spiritual discipline to sit back and trust the most precious thing in my life—my family—to God is probably the hardest thing for me to do when it comes to recognizing God's ownership. However, as I read the Scriptures, I don't see any exclusions to the "God owns it all" principle.
If you ever get confused about this, just look at Job. God blessed Job more than possibly anyone else on earth at the time, but then God took it all back—even Job's children. I can't even imagine that. And as Job is sitting there grieving, and as his friends are giving him bad advice, what does God say? "Who has a claim against me that I must pay? Everything under heaven belongs to me" (Job 41:11 NIV). God had to step in and remind Job that He owns _everything under heaven_. But that's good news! God's hands are big enough to hold it all; mine aren't. If I want what's best for my family (or my wealth or my legacy), then I'm going to entrust it to God's care. I'm grateful to be a steward of all these blessings, but I have to constantly remind myself to hold them all with an open hand.
Not owning, but managing, changes not only our view of things, but also our process of handling them. For some reason, we do a better job taking care of things we manage than things we own. Making decisions about giving, saving, spending, and managing money like you were in charge of someone else's money changes your decisions and behaviors. The big thing about releasing ownership is that, when you do, you don't feel puffed up or entitled. Releasing ownership brings a type of financial humility.
## The Magnification Lens
When it comes to our legacy, the magnification principle boils down to this: More money makes you more of what you already are. Whatever you are—good or bad—is going to get bigger when you add wealth to the mix. If you're a jerk with nothing, you'll be a colossal jerk with money. If you have a little temper problem when you're broke, you'll be an out-of-control rageaholic with wealth. That's the downside. But the reverse is true too. If you're compassionate and generous with a little, then wealth will turn you into a world-changing philanthropist. If you do acts of mercy with no money, then more money will enable you to bless people beyond their (or your) imaginations—and they'll probably never even know it was you doing it, because wealthy givers tend to stay anonymous. They'd rather God get the credit because they're not confused about ownership.
I think this is a big part of what Scripture means when it talks about being a faithful steward. The Bible says "a person who is put in charge as a manager must be faithful" (1 Corinthians 4:2 NLT). Stewardship isn't only about what you _do_ with God's resources; it's also about your attitude while doing it. We see over and over in Proverbs that wise people prosper and foolish people don't. Why? Is it just because wise people know how to save and invest? That's part of it, but maybe—just maybe—the simple fact that they are wise puts them in a position to win. It's magnification, remember? If you're wise with a little, you'll be a genius with a lot; if you're foolish with a little, you'll be an unstoppable idiot with a lot! Who you are, what you do, and what you believe matter.
## The Community Lens
I've heard several leadership and relationship coaches say that everyone needs a teacher, a student, and a friend in their lives. As Christians, we may say that we all need a Paul, a Timothy, and a Barnabas. We each need to be learning, teaching, and loving at all times. That's how we grow. Now, I'm a natural teacher, so that part is no problem for me. And I love to read and sit with people who are further along in their journey than I am, so that part is easy. But I've already told you that I struggle with the social part of the Wheel of Life. If I'm not intentional about it, I'll end up only spending time with my family and the people I work with.
Every Wednesday morning for the past fourteen years, unless I'm out of town, I sit at a conference room table at 7:00 a.m. with my twelve closest friends. We call it our Eagle's Group. We spend an hour and a half talking, checking in with each other, studying Scripture, or reading a book together. Over the past fourteen years, I've shared everything with these men. They know me almost as well as my wife does. I get a lot of criticism and hate mail from strangers, and that doesn't faze me a bit. But if one of those guys says something or corrects me, then it's serious. That nearly carries the same weight as if Sharon said it. That's accountability. They've invested in me and I've invested in them; they've earned the right to speak into my life. God works through that kind of community to guide, encourage, and correct us. If you want to grow as an individual, you need people like that in your life.
# MARRIAGE: YOUR LEGACY AS A COUPLE
Wealth building in marriage is a team sport. Starting with the very first man and woman in history, the Bible says that married couples become "one flesh" (Genesis 2:24). That means if you're trying to take control of your money, get out of debt, build wealth, teach your kids about money, or do any other part of your legacy journey without your spouse, you're only operating with half your brain! This isn't about forcing your spouse to do things _your way_ or using _The Legacy Journey_ principles as a club to beat the other person over the head. If you're married, the goal here is to do this whole thing side by side, arm in arm, walking out God's legacy for your family as a married couple. So in a practical way, one of the best ways to safeguard your legacy is to safeguard your marriage.
## A Virtuous Wife (or Husband)
No passage of Scripture has had as big an impact on my marriage or my finances as Proverbs 31:10–11: "Who can find a virtuous wife? For her worth is far above rubies. The heart of her husband safely trusts her; so he will have no lack of gain." And for you ladies, I don't think God would mind if we also read that as, "Who can find a virtuous _husband_." This passage is all about teamwork. My wife, Sharon, is wired a lot differently than I am. She sees things that I don't. God speaks to her in a different way than He speaks to me. I learned a long time ago that every time I made a decision that went against one of Sharon's _feeeeelings_ , it never turned out well. God put her in my life to help keep me inside the boundary lines.
That's true in every area of our lives, and that certainly includes financial. Now, I'm definitely the Nerd of the family. I love spreadsheets and budgets; I get excited about calculators and playing around with numbers. Sharon? Not so much. But here's what I figured out after a few years of making stupid money decisions on my own early in our marriage: Proverbs 31:10–11 is a financial principle. Read it again: "So he will have no lack of . . ." What? Gain. No lack of _gain_. That means if I want to have some of that "gain," then I need to engage my whole brain—meaning Sharon and I have to work together on these things.
## The Ownership Lens
From the moment the preacher pronounces you man and wife, Scripture says that you are to "submit to one another out of reverence for Christ" (Ephesians 5:21 NIV). To me, that's a powerful call for teamwork within marriage. The message here isn't for one partner to submit to the other in the sense of giving up all of his or her dreams and blindly following whatever the other spouse wants to do. If I ever got confused about that, it would take about three seconds for Sharon to knock my head back on straight—possibly with a frying pan. It's not about making one partner superior to the other; it's about bowing down together before the Lord. It's about recognizing as a team that we don't own anything and that God's made us—our family—responsible for a portion of _His_ wealth.
When you and your spouse believe together, as a couple, that God owns it all, that changes the whole tone of the conversation about money. You can trust each other's motives because you're both asking the same question: "What is God telling us to do with these resources?" That takes the potential for selfishness out of the equation. You're not always wondering if he's angling for a new car or if she's secretly planning an unbudgeted vacation. I've talked to thousands of couples over the years who seem to bring a four-year-old's behaviors into a thirty-year marriage. That kind of selfishness steals the nobility in a marriage. It will ruin the relationship and rob you of your legacy. But when you lay your "rights" down together, all of a sudden, all of that stuff starts to go away, because money isn't all about "me, me, me" anymore. Money isn't even about "us" anymore; it's about becoming faithful managers.
## The Magnification Lens
We've already said that more money simply magnifies who you already are. That definitely applies to marriage as well. So if you start with a loving, supportive marriage, the wealth you'll build over time will be a huge blessing to you and your family. A long time ago, when I was struggling to get back on my feet after the bankruptcy, I talked to a wealthy older man who had just gotten back from vacation. He was so excited because he had just spent two weeks with the people who meant the most to him. This guy took his whole family—kids, grandkids, and even his in-laws—on a two-week, all-expenses-paid, luxury ski vacation. He paid for the airplane tickets, the chalet, the ski lift tickets, the food—everything. He told me it was the most fun he and his wife had ever had. Hearing him talk about his wife of more than four decades touched me. I could tell that this man was more in love with his wife at this point in their lives than he ever had been. It was obvious that this had been a wonderful, loving couple when they were young and broke, and the wealth they'd built over the years magnified the quality of their marriage. As a young man with a wife and three little kids at the time, this gave me something to shoot for. From that point on, I dreamed of what Sharon and I could do together if we worked as a team to build our legacy.
Sadly, the opposite is true too. If your marriage is struggling and you've got cracks in your foundation, then more money will simply widen the gap. I talk to so many men whose marriages need work. They say things like, "Dave, if I just made more money, we'd be so much closer." No, they wouldn't. People like to blame a lack of money for the problems in their marriages, but that's almost never the reason for the underlying issues. Sure, money fights and financial stress are the leading causes of divorce in North America, but I think that has more to do with how couples _handle_ their money than how much money they do or don't have. That's a measure of their _teamwork_ , not their assets.
The bottom line is that you can't heal a wounded marriage with money. In fact, money will destroy a wounded marriage. The wealth won't create new joy; it will only magnify the problems. Have you ever known a married couple who seems to become less happy and more critical of each other the wealthier they become? I have. In the end, it turns into a nasty divorce with a team of lawyers on each side fighting for the biggest share of the wealth that ultimately drove them apart. The wealth didn't cause the divorce; their broken marriage was just revealed as the wealth grew.
No matter where you are in your wealth building, if you're married, I want you to stop right now and seriously examine your marriage. Are there cracks in the foundation? Does it feel like you and your spouse are growing further apart each year, despite your financial progress? If so, you need help. The most important piece of financial advice I can give you is to get yourselves to a counselor or your pastor's office immediately. Don't let money drive you further apart. Remember, your wealth comes from and ultimately belongs to God. Don't let His financial blessings drive a wedge through your marriage. Faithful stewardship means taking care of your family—especially your marriage—first.
## The Community Lens
One way to safeguard your marriage and your legacy is to maintain quality relationships with other couples. The key word here is _quality_. What we tell our children is true: You really do become like the people you hang around with. That's actually a biblical principle. Proverbs 13:20 says, "He who walks with the wise grows wise, but a companion of fools suffers harm" (NIV). When it comes to walking with the wise, I recommend you and your spouse nurture two different types of couple relationships.
First, you should nurture spiritually healthy relationships at your financial level. That absolutely does not mean you should force some kind of wealth or income requirement on your circle of friends! I'm simply saying that we need people in our lives who understand the unique challenges and opportunities we face as we hit certain wealth milestones. When your net worth tops $1 or $2 million, for example, you'll have different estate planning issues than you used to have. It helps to have a peer group to bounce some ideas off of. Or if you're just starting to figure out your ratios for lifestyle, extra giving, and extra investing, it would be good to talk to a couple who has already been through that. These kinds of relationships can keep you from feeling isolated or, even worse, ashamed of your success. You always want to have friends around you who don't make you feel embarrassed about the fact that you're winning with money.
Second, you need to maintain your relationships with your old friends, regardless of where they are in their legacy journey. Old friends keep us grounded. These are the people who knew you when you were young and stupid. Sharon and I have friends at all different levels of wealth. Some of these couples walked with us through the bankruptcy and were instrumental in my faith walk as a new Christian. I'd never dream of leaving these friends behind! They mean the world to me, and their friendship has been a blessing to me and my family for decades. They loved us when we were broke, and they aren't impressed by our wealth now. You need these friends in your life to maintain a healthy, balanced view of your success.
# CHILDREN: YOUR LEGACY AS A FAMILY
Whenever I think about what it looks like to pass down a family legacy to my children, I always think about a conversation Jesus had in Luke 9. You know the story: Jesus says to someone, "Follow Me." And the man replies, "Lord, let me first go and bury my father" (9:59). That seems like a reasonable request to me. I mean, we're called to honor our mother and father, so giving this young man time to pay his final respects to his dying father is a good and proper thing to do, right? That's why I was confused for years about Jesus' response to this basic, perfectly understandable request. Jesus replies, "Let the dead bury their own dead" (Luke 9:60). He basically tells the guy no. Ouch. That always seemed harsh to me. Why didn't Jesus let this guy bury his dad?
That's how I looked at that passage for years, until I heard Larry Burkett teach it. According to Larry, the Jewish culture had a much different view of these things than we do. When we hear, "Let me bury my father," we naturally assume that the man's father is either dead or dying. We empathize with his grief and want to comfort and console him. But, according to Burkett, that's not what was happening here. Based on Jewish culture at that time, this young man was probably saying his father was retiring—he most likely wasn't sick at all. In the Jewish tradition, an old man would turn his wealth over to his eldest son before his death, and then the son would be responsible for managing the money and taking care of his parents and any unmarried siblings. This wasn't about actually burying his father; it was about a transfer of wealth. The father's death could still have been years away at this point. That's why Jesus said no.
This passage makes me wonder, though, what it would look like if we had that kind of tradition today. If you knew that after decades of hard work, after all those years of careful wealth building, after everything you've done to build your personal and financial legacy, if you'd have to hand it all over to your kids _while you were still alive_ and trust them to take care of you, how would it change the way you teach them about money? I don't know about you, but if I knew my golden years were completely dependent on my kids, I'm going to make sure they're financial geniuses! I'm going to teach them how to handle money as though my life depended on it, because guess what? Your legacy actually _does_ depend on it.
## The Ownership Lens
Without a proper view of God's ownership, your kids could feel like they've won the lottery if you leave them a pile of money someday. If that's the case, then that wealth will be a curse on them, and whatever legacy you've tried to build throughout your life will come to a screeching halt. That's because preparing your children to carry that legacy forward is a huge part of your legacy journey. This is so important that we'll spend most of the next chapter talking about it. For now, though, I want to impress on you that more is caught than taught. Your kids are watching you. They're going to do what you do. If they see you budgeting, saving, working, and giving, then that's what they'll most likely do. If they see you stress out about money, buy big things on impulse, go into debt for purchases, and never give a dime to anyone, then that's probably what they'll do. The best way to teach them a healthy view of God's ownership is to _show_ them. You do that by the way you live your life.
But you can't just _do_ these things; you have to talk about them too. The most powerful way to teach your children anything is to tell them _and_ show them. They have to see that your actions and your instructions line up. Scripture says:
> These commandments that I give you today are to be on your hearts. Impress them on your children. Talk about them when you sit at home and when you walk along the road, when you lie down and when you get up. Tie them as symbols on your hands and bind them on your foreheads. Write them on the doorframes of your houses and on your gates. (Deuteronomy 6:6–9 NIV)
That's a call to consistency and integrity in your parenting. These things have to be in your heart, in your actions, and in your words. Your home has to represent all the things you want your children to learn. It's up to you. God calls parents to be the primary mentors and teachers for their children. It's not the school's job or the church's job; teaching your kids a sense of God's ownership is your job.
When our kids were growing up, we tried our best to model the lessons God was teaching us, but we didn't do it perfectly. In the early days, we were more concerned with paying the light bill than teaching the kids anything. But we always looked for teachable moments. For example, one day I was paying bills at the kitchen table, and my oldest—eight years old at the time—asked me what I was doing. I told her, and then I took it a step further and got her to write out the check for the utility bill. She started to write the amount for the power bill, and she looked up at me with these huge, shocked eyes and said, "Dad! Electricity costs $200?" From then on, that child was always running from room to room, turning light switches off! Even though it wasn't costing her any of her own money, she felt a sense of responsibility for managing the cost of electricity in our home. She became a steward of someone else's resources; that's what a healthy view of God's ownership is all about!
## The Magnification Lens
We've said that money magnifies who and what you already are. That's definitely true for your children too. Proverbs tells us to "Train up a child in the way he should go, and when he is old he will not depart from it" (Proverbs 22:6). The challenge for parents is not only to teach your kids God's ways of handling money, but to also do it in a way that resonates with their natural bent. Every child is wired a certain way. As a new dad, I was surprised to see how much unique personality my kids displayed at such a young age. Sharon and I discovered pretty quickly that we had one who was extremely organized, one who was a little wild (like her dad), and one who was super sensitive. It's hard to see the uniqueness when you're looking at someone else's kids, but you can't miss it in your own.
Those individual traits are part of who they are; that doesn't change much over time. For example, our youngest child, Daniel, has always had a servant's heart. Even as a little boy, he was always looking for ways to serve other people. Sharon and I knew that was part of his wiring, so we did our best to "train him up" with that in mind. Like all our kids, Daniel had to save up money to buy his own car. All three kids worked hard, saved as much as they could, and then we matched their savings. We called it our 401DAVE program. We told all three kids about this plan at the same time, and since Daniel is the youngest, he had the longest time to save. Because he's a hard worker and isn't a natural spender, he managed to save up a whopping $12,000 on his own. That means, after the match we promised, he was sitting on $24,000 cash at sixteen years old! I sat down with him when I gave him the match and told him that I was proud of him, but there was no way I could let a sixteen-year-old kid buy a $24,000 car. He had already found a great $14,000 Jeep he liked, so he bought the Jeep and put the other $10,000 in the bank.
A while after that, there was a horrible earthquake in Peru, close to where Daniel had been on a mission trip years earlier. He loved his time there, and he felt a special connection to the people he had met. So, when he heard about the earthquake and the great need in that community, Daniel told me he wanted to send his $10,000 to the relief efforts. Can you believe that? A sixteen-year-old young man chose to send $10,000 of his own, hard-earned money to earthquake relief. I said something like, "Daniel, that's a lot of money. You'll be heading to college soon. Are you sure you want to give your money away?"
He replied matter-of-factly, "But Dad, it's not my money. It's God's money. That's what you taught us, right?" Sharon and I were blown away (and more than a little humbled), but as it sank in, his response made perfect sense. He was wired to be a servant and giver, and Sharon and I had done our best to teach him about working, saving, and God's ownership. That put him in a position to do something at sixteen that most adults can never do: give $10,000 away—and with a smile. That's the magnification principle at work: Because of his hard work, he had wealth to share. That wealth simply magnified what was already there.
## The Community Lens
Getting the community lens right with your children is absolutely crucial. You can spend all day every day with them for the first fifteen years of their lives, teaching them the Bible, walking with them through each stage of life, showing them what it takes to be a mature, valuable member of society. But if you don't step back and look at the bigger community around them, all of your hard work can be undone in the blink of an eye. The apostle Paul put it this way: "Evil company corrupts good habits" (1 Corinthians 15:33). You and I might say, "You become who you hang around with." We know this is true for adults, but it is so much more powerful for children—especially teenagers. I cannot stress this enough: You've got to know your kids' friends. That means you have to be involved and know what their friends are into.
This used to drive my kids crazy. Rachel—the more dramatic one—always used to argue with me when I told her to stay away from certain friends. She'd say, "Dad, I'm sixteen years old. I know what I stand for. I can hang around with them and be okay." But then, slowly but surely, she'd start coming home with new bad habits—nothing major, just things like being a little disrespectful to her mom or trying to bend the rules a bit. As soon as we noticed it, we cut her off from the source. I didn't have anything against her friends, but as her parent, I was not going to allow outside influences to turn her into something I knew she wasn't—whether or not she understood it at the time. Some of you may need to grow a backbone about this. No matter how much they start to look like adults, your teenagers are still growing up. It's up to you to steer them in the right direction, and that includes helping them identify bad influences that could ultimately wreck their lives.
And while we're at it, I want to encourage you to examine your own friends. Don't tell your children to stay away from bad influences if you aren't doing it too. The psalmist says, "Blessed is the one who does not walk in step with the wicked or stand in the way that sinners take or sit in the company of mockers" (Psalm 1:1 NIV). Even as an adult, you can get knocked off course by your messed-up friends just as easily as your kids can. Even worse, you could bring those bad influences into your children's lives. Don't let your own choices in friends destroy your family legacy.
Getting these messages to your children at a young age is so important that my daughter Rachel and I have written a book that talks about all of these things in more detail. If you still have kids in the house, you should check out _Smart Money Smart Kids_. Teaching your kids how and why to work, spend, save, and give throughout their lives is one of the most important things you could ever do to guarantee a legacy that lasts. Like the Proverb says, "The righteous man walks in his integrity; his children are blessed after him" (Proverbs 20:7).
## No Kids? No Problem.
If you don't have kids, you just need to redefine what **US** means for you in the legacy journey framework. Maybe it's investing in nieces and nephews. Maybe it's getting involved in a children's ministry, working with an orphanage, or financially supporting a family seeking to adopt. No matter what it is, I encourage you to talk to God about how He wants to use you to impact the next generation. You want a legacy that goes somewhere, so be intentional about finding ways the good work you're doing today can bless others later.
# OTHER RELATIONSHIPS: YOUR LEGACY EXTENDED
We've talked about how to safeguard ourselves as individuals, our marriages, and our children from the challenges that wealth can bring into our lives. That's a great start, but we're still left with how the legacy we're building can affect other relationships. The truth is, your personal wealth can have an unexpected impact on your extended family and friends. I've seen so many people caught off guard by this. They spend years or decades focusing on their legacy, only to have it torn apart by toxic relationships they weren't prepared for. To truly safeguard your legacy, you've got to take a good, hard look at the people around you and set some boundaries to protect yourself—and those relationships.
## The Ownership and Magnification Lenses
I guarantee you, no matter how great your family and friends are, no matter how much you think I'm exaggerating, you will have people in your circle who think you owe them money just because they're in your life. That's where the magnification lens comes in because your success may reveal the true characters of your friends and family members. As your wealth grows, some of these people will get bolder and bolder in making demands about how you should "help" them. We've said a few times that grown children sometimes feel like they've hit the jackpot when they receive their inheritance from wealthy parents, but sometimes the roles get reversed. I talk to successful young adults all the time whose parents have started making demands on them just because they "can afford it" or because "I'm your mother." It's heartbreaking because it puts a burden of guilt over someone's success, and it puts an adult child in the position of enabling a parent's bad behavior. That's a totally toxic situation.
This is an entitlement issue, whether it's coming from your children, your parents, your best friend, or your fourth cousin twice removed. Yes, we're called to honor our parents. Yes, we're called to take care of our families before we do anything else with money. But that doesn't mean we _owe_ anyone anything. If your mother is a cocaine addict, giving her money is the last thing you should do. That's not helping her. If your child refuses to grow up and get a job, supporting him financially is a complete waste of money because it's only keeping him from getting out there and doing what God's called him to do. If your best friend is up to her eyeballs in debt because she's an out-of-control shopaholic, paying off her credit card bills won't be a blessing. It will just wipe out the symptom while ignoring the behaviors that got her into trouble. This means you're going to have to get comfortable setting—and enforcing—clear boundaries in some of these relationships. _Boundaries_ is a scary word for many people, but they are a crucial part of safeguarding your legacy. You must be intentional about setting boundaries in your relationships, and you, your friends, and your family members need to know when something falls out-of-bounds.
How you respond in all of these situations is an ownership issue. Remember, this is God's money, so you have to keep asking yourself, _Is this how God would want me to spend His resources?_ Scripture says, "The blessing of the LORD makes one rich, and He adds no sorrow with it" (Proverbs 10:22). That tells me that God doesn't want us putting His wealth into a sorrowful situation. That won't be a blessing to anyone. When you find yourself caught in someone's twisted web of entitlement, just take yourself out of the equation. Then ask yourself this one question: Is this a situation that God would pour His blessings into? If the answer is yes, then maybe it's something He wants you to do with His money. If the answer is obviously no, then run!
## The Community Lens
The point here is not to be scared and freaked out about the people in your life. Yes, you're going to have some people around you who get twisted up and confused by your success. That hurts when it happens, but it's not your fault. You can't be responsible for how other people behave. What you can do, though, is make sure you surround yourself with quality friends who help you navigate through this mess. When we talked about marriage, I suggested you find some friends at your wealth level. That's your new financial peer group because you're all going through these unique challenges together. That comes into play with your extended family and friends too. You need people in your life who are going through the same things with their own crazy relatives! You can lean on each other and learn from each other. You don't have to go through this alone!
# A LEGACY IN RELATIONSHIPS
I talk to so many people who are only concerned about the math, the practical and tactical side of wealth building. That part is easy! There's no big secret to building wealth. It just takes time, patience, and wisdom. Sure, there's plenty to learn about investments (which we'll cover in "The Pinnacle Point," the bonus chapter in the back of this book). But what most people overlook is the fact that our relationships have as much or more impact on our wealth and legacy than our investment strategy does. You can do smart things with money your whole life and leave millions of dollars when you die, but if you have sick, weak, toxic relationships all around you, that wealth won't make it to your children's children. It will not only disappear, but it will also destroy your family along the way.
Years ago, a guy called into my radio show to talk about his mother's nursing home selection. She was a godly woman in her late eighties who had been widowed for twenty years at that point. She and her late husband never had a big income. Their family never had much, and honestly, they never really even thought about teaching their children about money. They never had those conversations around the dinner table. They were simple country folks who did the best they could with what they had, and they had done okay. Now, as she entered the season of life that required full-time care, she was almost ninety years old with $200,000 in the bank—enough to at least have a choice in the quality of care she could receive through her final years. But this guy's call made my jaw drop. He was calling to ask me how he could talk his mother into choosing a cheaper nursing home because he was afraid the one she picked would eat up his inheritance! Can you believe that? This middle -aged man was more concerned about how much he'd get than he was about how well his mother would be treated in the few years she had left. He was acting like that money was already his, and it bugged him that she'd _dare_ to spend _his money_ on her healthcare. That kind of attitude makes me sick. It's a toxic form of entitlement that, sadly, characterizes too many families.
Now, contrast that to another story. I have a good friend who came from a family that talked about money. His parents weren't wealthy, but they were intentional about teaching their kids the kinds of things we've been talking about in this book. They worked hard, saved, planned, and gave generously, and they made sure they lived these principles out in front of their children. My buddy lost his mother several years ago, and he and his dad have grown even closer in the years since then. As my friend grew his business, he started doing pretty well financially. He lived like his parents taught him to, always saving, always giving. Then, a few years ago, my friend did something he'd dreamed of doing for years: He wrote a check and paid off his dad's house! How cool is that? That's the kind of stuff you can do—and want to do—when you come from a family that is intentional about protecting their relationships when it comes to money.
Remember what we read earlier from Deuteronomy: "I call heaven and earth as witnesses today against you, that I have set before you life and death, blessing and cursing; therefore choose life, that both you and your descendants may live" (30:19). Having these conversations with your family and friends, working together to protect your marriage, making sure you're taking care of yourself mentally and spiritually . . . these are life and death decisions when it comes to money. Your wealth has the power to bless your family or completely tear it apart. The guy who was more concerned with money than he was his own mother's well-being? That family chose cursing. But my friend who used his wealth to show his love and appreciation for his dad? Clearly, that family chose blessing a long time ago. That's my hope for you too.
### CHAPTER SEVEN
# Generational Legacy
Bologna and tomato sandwiches.
Crazy as it sounds, that's one of the most powerful, meaningful memories I have from my college years: sitting at my grandparents' kitchen table eating bologna and tomato sandwiches with Grandpaw. When I was growing up, my grandparents lived only a couple of hours away, so I certainly knew them and had spent time with them. But it wasn't until I went to college and moved near them that I really got to know them as a young adult. They lived in Alcoa, Tennessee, just outside of Knoxville where I went to college. I'd often run over to their house for lunch between classes or for dinner some nights when I just needed a home-cooked meal. And hey, I was a broke college student, and they had free food!
Grandpaw worked for Alcoa Aluminum for thirty-eight years as their head cost accountant. He had a brilliant mind for numbers; maybe I got just a little bit of that from him. I guess when you're a kid, you kind of have a _sense_ that your grandparents actually have a life outside of their role as grandparents, but most of us don't stop to think about that. However, during those great conversations I had with Grandpaw at his kitchen table, I realized what an amazing man he really was. We talked about family and relationships. We talked about money. We talked about business. He had this great way of tying it all together, and it always felt like one conversation flowed perfectly into the next. He was a great, great man.
Scripture says, "A good man leaves an inheritance to his children's children" (Proverbs 13:22). Clearly, the Bible teaches us to leave an inheritance for future generations. My grandfather left me an inheritance of character and of wonderful memories. Sure, a big part of that inheritance is a legacy of love—the memories I have of my grandfather are precious to me. They are a big part of the man I've become, and I wouldn't trade them for anything in the world! But I also think Scripture makes it clear that we're called to manage God's resources well so that we can leave a financial legacy too. There's a biblical model for passing wealth generationally, and there are practical things we can do to make sure our legacy not only survives but also increases as it moves down to our kids, grandkids, and great-grandkids. We're going to take a serious look at that in this chapter, as we get our hands dirty with some practical and tactical things we should be doing today to create a financial legacy that lasts.
# A GOOD MAN . . .
We've already talked a lot about Proverbs 13:22. That verse sets the tone for _The Legacy Journey_. That's the goal, right? We want to focus on doing things today that impact our children, our grandchildren, our great-grandchildren, and beyond. What would it look like if we actually lived our lives like we believe this verse?
## Anyone Can Do It
Anyone in America can retire a millionaire. That's a bold statement, I know. If you've never read any of my other books or taken our _Financial Peace University_ class, then that might even be a shocking statement. Let me tell you why it sounds so surprising: It's because we get hit nonstop with all kinds of messages from our culture that tell us we can't do it. I've heard them all: "I just can't find a job." "The little man can't get ahead." "The system is rigged." I even have a relative who just walks around yelling, "The corporations! It's the corporations!" I have no idea what he means, but in his mind, "the corporations" are the reason why he can't win with money. Personally, I think most of these excuses are garbage.
I know that it can be hard finding a job, and I know the economy has some wild ups and downs. I get that. I didn't say it's _easy_ to become a millionaire; I just said anyone can do it. If you learned God's ways of handling money and used them throughout your life, if you stayed out of debt, if you saved for emergencies, if you planned and invested for the future, and if you did that consistently over time, where would you end up? You'd be wealthy. Did you know that practically any twenty-five-year-old person could end up with more than $1 million by retirement without breaking a sweat? Do the math. If you invest just $100 a month in a decent growth stock mutual fund from age twenty-five to age sixty-five, you'd have more than $1 million! And that's just with a hundred bucks a month! If a million is what you had at retirement, you could live pretty well just on the growth without ever touching the principal. That means you'd have $1 million to pass down generationally.
Let's not stop there. Let's say you did that, and more importantly, you taught your children to do that. If you have two kids, imagine them retiring at age sixty-five with $1 million each. That $2 million plus twenty years' worth of growth on the $1 million you left would come to almost $12 million combined! Take this same thing down one more generation—to your grandchildren—and we're looking at a financial legacy in the neighborhood of $100–200 million! What could your family do for the kingdom of God generationally if you started the ball rolling? What if you gave them not just money, but also the character and wisdom to actually do this stuff? And again, that's with only $100 a month. Just imagine what this would look like if you actually did what we teach and invested 15 percent of your income into retirement!
## So Why Don't Most People Do This?
If this is possible for practically every single family in America, why on earth don't people actually do it? I think we see the answer in Proverbs 21:20: "In the house of the wise are stores of choice food and oil, but a foolish man devours all he has" (NIV). Wise people save money, delay pleasure, and make sacrifices for their families. Foolish people blow all their money on fancy cars, exotic vacations, and expensive jewelry they can't really afford. They're more concerned about themselves—what they want and what they can do to impress other people. For too many, it's all about me, me, me. What I want. What I can consume. What I can buy. Sometimes you meet these people and it's like they think the axis of the world runs right through the top of their head. They talk and act as though the whole world revolves around them. And in our culture today, we sometimes even celebrate that kind of selfishness. We make jokes about it, and we slap bumper stickers on our luxury cars that say, "I'm spending my kids' inheritance." That's not funny to me; it's sad, and I don't think it's biblically wise. I think that's just another way of telling the world, "I don't care about future generations, because it's all about me." It's a sad statement about the maturity level of many Americans.
That's definitely not who I want to be. I want to be the grandfather or great-grandfather who changed his family tree forever. I want to be old man Rockefeller, the first generation to draw a line in the sand and say, "I can do better. My family can do better. And from this point forward, we will." There's nothing wrong with nice things and enjoying your success; we've covered that at length already. But you can never put nice things above the legacy you're leaving for your family.
# THE BIBLICAL MODEL FOR GENERATIONAL WEALTH
All through the Bible I see wealthy families passing down their wealth to future generations. I believe Scripture shows us that godly families are called to manage wealth for God's glory. The responsibility for managing God's resources doesn't end when the head of the household dies; that responsibility is passed along with the wealth. It's not a surprise, and it's not a Hail Mary pass at the deathbed. It should be the result of a lifetime of conversations with your children about how to manage that wealth for the kingdom of God. That's the calling God's put on the family, and that's the responsibility your children take on when they accept their role as heirs.
## Your Children's Children
Generational wealth is not about materialism or consumption. It's about taking responsibility—as a family over several generations—for managing the wealth that God's put in your care. Building up wealth just so your kids never have to work is not biblical. Warren Buffet says, "A very rich person should leave his kids enough to do anything but not enough to do nothing." We're _all_ called to godly work and productivity. Proverbs 10:4 says, "Lazy hands make for poverty, but diligent hands bring wealth" (NIV). The Bible never says that one generation should work hard so the following generations can take it easy! Instead, we see every generation building on the hard work and success of the previous one.
That's the story of Abraham and his son Isaac. Abraham was an extremely wealthy man. Genesis 24:1 says that God blessed Abraham "in every way," and that certainly included wealth. In verse 35, Abraham's servant explains, "The LORD has blessed my master abundantly, and he has become wealthy. He has given him sheep and cattle, silver and gold, male and female servants, and camels and donkeys" (NIV). Translation: Abraham was loaded! And what happened to that wealth after Abraham died? Genesis 25:5 says, "Abraham left everything he owned to Isaac" (NIV). Scripture shows that Abraham was generous throughout his life, but he still maintained his wealth so that he could pass it on to his son.
What was the result of that generational transfer? Genesis 25:11 says, "After Abraham's death, God blessed his son Isaac" (NIV). So Abraham saved and built wealth throughout his life, and then he passed it to his son. Isaac received that wealth and added to it through God's blessing. In Genesis 26:3, God says to Isaac, "I will be with you and will bless you. For to you and your descendants I will give all these lands and will confirm the oath I swore to your father Abraham" (NIV). Verses 12 through 13 of that same chapter show us how big that blessing really was to the second generation: "Isaac planted crops in that land and the same year reaped a hundredfold, because the LORD blessed him. The man became rich, and his wealth continued to grow until he became very wealthy" (NIV).
During Israel's long journey to the Promised Land, the call for families to pass down their wealth to family heirs was outright mandated. Speaking for the Lord, Moses commands the Israelites, "No inheritance shall change hands from one tribe to another, but every tribe of the children of Israel shall keep its own inheritance" (Numbers 36:9). Isn't that interesting? Each tribe's inheritance had to be passed down the family line, and they were prevented from sharing it with the other tribes. No wealth redistribution going on here.
And let's not forget about Job. We always talk about Job as being the guy who went through unbelievable hardships—and he did. I never want to go through _any_ of what he did. But what we often lose in that story is how richly God _blessed_ Job too. There are some inferences in Scripture that make me think Job may have been the wealthiest man on the planet at that time. But then Job lost everything, and I mean everything. He lost his wealth, his home, his family, his kids, his friends . . . everything. There was nothing left except a pile of garbage to sit on and a few clueless buddies ready to give him some bad advice.
What happened to Job at the end of the story? God showed up, and He restored everything several times over. Job 42:12 says, "Now the LORD blessed the latter days of Job more than his beginning; for he had fourteen thousand sheep, six thousand camels, one thousand yoke of oxen, and one thousand female donkeys." That's a _lot_. And here's what I like: "In all the land were found no women so __ beautiful as the daughters of Job; and their father gave them an inheritance among their brothers" (Job 42:15). Now this was in a time when the daughters got nothing; everything was passed to the men. But Job's wealth was so big, and he loved his family so much, he went against tradition and passed his wealth on to _all_ his children. Speaking as the father of two girls myself, I can't imagine doing anything else either.
Probably one of the greatest examples of generational wealth transfer is King David. Most of us picture David as a little shepherd kid in the story of David and Goliath, with old King Saul's armor hanging off his skinny frame. It's easy to forget that David went on to some pretty big things after that. He was a great king, and he had plans to build a massive temple for God. But then he got himself into a bit of trouble, and God removed David's right to build the temple. Instead, the assignment was given to David's son, Solomon.
I have a friend who is a brilliant economist, and he dug into the description of the temple in 2 Chronicles. Once he factored in all the gold, woodwork, and incredible craftsmanship and décor, my friend estimated that Solomon's temple would have cost about $21 billion in today's dollars. That's _billion_ with a "b." Where do you think that money came from? Solomon used _David's money_ to build the temple, so that would mean that David and Solomon were billionaires! And the part I really like is that Solomon didn't just receive David's _money_ ; he received David's _responsibility_ as well. That's what we've been saying throughout this book: Godly stewards don't just pass along wealth to future generations; they also pass along the responsibility for managing it for God's glory. I believe that's the biblical model for generational wealth.
## A 501(c)(3) Is Not Holier Than Your Children
Sometimes wealthy people fall into the trap of thinking the "holy" thing to do with their money is to leave it all to a nonprofit organization, often categorized as a 501(c)(3) for tax purposes. We hear those messages all the time, don't we? We look at the amazing things that people like Bill Gates are doing with their money, and we applaud them for their generosity. I want to be crystal clear here: I think Bill Gates is doing some incredible things, and there's no question that he is going to go down in history as one of the most generous, most giving, most world-changing figures in modern history. But I still disagree with part of his plan. He's gone on record saying that his plan is for most of his wealth to be completely spent within twenty years of his and his wife's death. On the surface, that may sound like a good or generous thing—and it is. The problem is, though, one of the biggest personal fortunes in the history of the world will totally vanish within one generation. Even though the money will be spent in wonderful ways, it will still be gone forever.
There's even a growing amount of political pressure to make sure family wealth never makes it to the second generation. One prominent billionaire and political activist recently went on record saying he thinks the government should continue to increase the estate tax to give people a greater incentive to leave more of their wealth to nonprofits. It's kind of funny that this is coming from a guy who inherited his billions from previous generations. If he really believes that, maybe he could prove it by giving all he has away to nonprofits today. He won't, of course, which makes him a hypocrite, or at the very least, someone who just likes to hear himself talk. Again, though, well-run nonprofits are great if they line up with your values and you choose to contribute to their causes. But there should never be pressure—social, political, or spiritual—to leave everything to one, and there definitely shouldn't be tax laws put in place to force people to do it.
I just don't see that kind of "use it or lose it" attitude about wealth and giving in the Bible. Instead, I see example after example of godly families passing down their wealth—and the management (stewardship) of that wealth—to future generations. As that happens, the wealth grows and more and more people are blessed over several generations. Why do we think it's better to give everything to a nonprofit than to entrust it to the children we've spent decades training? I've seen some nonprofits that are _terrible_ at managing money; why should a 501(c)(3) classification make that a better choice at managing my wealth for the kingdom than my own children?
If your kids are functional, know how and why to manage money God's ways, love Jesus, and aren't confused about the responsibility of wealth, then they should be first in line to carry this financial blessing into the next generation—even above a well-run nonprofit. If you're passionate about a particular charity or ministry and trust the way they manage their money, then by all means make that a part of your plan; just don't make it the entire plan. But a broke nonprofit? That shouldn't even be an option. Your money will be wasted faster and potentially do more damage in a poorly run 501(c)(3) than it would in almost any other situation.
## Nonprofits Are Not Anointed
Since I will be misunderstood, let me say it one more time: I am not against leaving your wealth to a nonprofit or your church (unless they are poorly run). I just want you to be freed from the toxic and false message that you should give your money to a "holy" organization rather than leave it to a functional, godly family. Nonprofits are not by definition "holy." And leaving your money to your family is neither selfish nor unholy. That is simply wrong.
Keep in mind, God did not anoint nonprofits; the IRS did. Prior to the tax code being put in place in the last century, there was no such thing as a nonprofit organization. Sure, there were churches and charities, but the word _nonprofit_ didn't really exist. This is an IRS designation, not a biblical designation.
# PRACTICAL ESTATE PLANNING
Okay, we've talked about the biblical model for passing wealth generationally, and it's clear that I believe it's both moral and biblical to build wealth with a goal of leaving it to responsible, godly heirs to further bless future generations. So let's switch gears and look at some practical things that need to be part of your estate plan. Whenever I bring this up, I always get someone who says, "But, Dave, if I talk about doing a will . . . I'll die!" Well, guess what? My team has done extensive research on the subject, and we've found with 100 percent certainty that you are, in fact, going to die. We all are. Humanity has a 100 percent mortality rate. With that in mind, let's make sure we're doing the right things now so our families won't have to deal with a huge estate crisis later in the middle of their grief.
## Raise Great Adults
I recently heard my friend, author Andy Andrews, talk about parenting. He said something that really stuck in my head because it lined up with how Sharon and I raised our three kids. He said, "I'm not that interested in raising great kids. I'm more interested in raising pretty good kids who grow up to be great adults." If you're focused only on having great kids, your parenting style will be all about basic rule following and stiff punishment for going out of bounds. Your kids will be well-mannered and polite, and you may have the picture of the perfect family. At least until they go to college. Then, these little robots you controlled for eighteen years will go crazy with rebellion, getting a degree in beer pong, and that will do more to shape who they become as adults than anything you did.
Now, there were definitely rules in place in the Ramsey house, and my kids knew what it meant to get a swat on the behind or spend a week or two grounded when they messed up. But that can't be all you do. If your goal is to raise kids who become great adults, you have to give them room to grow —and that means giving them the responsibility to manage some stuff and the freedom to fail. You want your kids to make their biggest mistakes while they're still under your roof. If your goal is to leave a huge financial legacy to the next generation, you have to make sure your kids have the maturity and character to manage that wealth wisely for the Lord. If you get this right, the wealth you leave them will truly be a blessing; if you get it wrong, your wealth will completely destroy their lives. Make sure you're doing all you can to prepare the next generation to shoulder the responsibility you'll leave them one day.
## You Need a Will
Sometimes it seems like I can't write a book or go one day on the radio without talking about the need for everyone over the age of eighteen to have a will. I talk about this over and over and over again, and I will until every adult in America gets the message: You need a will—today!
"But Dave, I don't have any wealth or big possessions. Why do I need a will?" Because a will does a lot more than just say who gets what. Besides, even if you only have one big thing like a home, it could take months or even years for a court to figure out what to do with it. There's just no point in that! And if you have young kids, don't even try to rationalize not having a will. A will outlines your plans for your children in detail, so there's no question where they go or what happens to them. Do you want the state to determine that? For all you know, the state could put your children in the care of your craziest relative. Don't put your children in the court's hands. Love them enough to spell out your wishes for them in a will.
Keep in mind that wills are state-specific. You can't get one boilerplate will that works the same all over the US Probate and estate-planning laws are governed by the state, not the federal government, so it changes from state to state. That means if you move to another state, you need to update your will. Even if you don't move out of state, you should still carefully review your will every couple of years to make sure it still reflects your current wealth level, possessions, and wishes.
One big objection people have to doing a will is the basic hassle factor. I'm not going to lie to you: Doing a will is not fun. The legal part of it is dry and boring, and then there's the emotional part. To do your will right, you have to seriously think about dying. Who wants to do that? Well, your kids want you to think about it and plan for it, because if you don't think about it now, you'll leave them in a mess later. One thing that makes this easier for married couples is what's called a mirror-image will. Assuming you're married and you both do the estate planning together (a necessity, by the way), then you can do a full draft of the will in one person's name and then mirror it for the other spouse. When Sharon and I did ours, we wrote everything up in my name for my will, and when we were done, the lawyer ran another copy of it but put her name in place of mine everywhere. That way, we have matching wills that reflect our wishes. It saves money and keeps you from having to figure everything out twice. Easy.
Other things that absolutely must be in your will include healthcare power of attorney, living will instructions, and child guardianship. Healthcare power of attorney states clearly who is authorized to make your healthcare decisions if you are incapacitated. So, if you're in an accident, who do you trust to make those life-saving decisions on your behalf? Put it in the will.
The living will outlines your life-ending wishes, which takes a huge amount of pressure off of your spouse or relatives. Say you're in a coma from a car wreck, and you aren't doing well. The doctors declare you brain-dead and say you'll never wake up, but they can keep you breathing on a machine for years. Is that what you want? Your living will is the place to make your wishes known. Don't force your spouse to make the decision to turn off the machines. You can make that decision yourself; you just have to make it now and put it in writing.
Child guardianship is an absolute must if you have minor kids. Like we already said, this outlines your exact instructions for who will take care of your children if you and the other parent are gone. You can name whomever you want; just get their permission first. It does not have to be a family member, but if you die without these instructions in place, the court is likely to put your children in the care of a family member, whether or not that's really the best place for them. Don't wimp out on this. Your kids are counting on you to look out for them—and that includes looking out for their needs after you're gone.
A will doesn't have to be too complicated unless you have built up a great deal of wealth. If your assets are under $1 million, you can probably get by just fine with a basic will through an attorney or online service. If you have more than $1 million, or if there are some unusual circumstances in your family or estate, then I recommend working with an estate planner to make sure you've accounted for everything. It may cost you $1000 or more, but that's a small price to pay for making sure your legacy passes to the next generation intact.
## Avoid the Drama
Picture it: The lawyer's conference room. The trophy wife. The goofy stepson. The grieving family. The letter opener slicing through the thick yellow envelope. The gasps as one family member discovers he's been cut out of the inheritance. Welcome to the reading of the will—if you're watching a movie, that is.
In real life, there should be no drama involved in "the reading of the will" after you're gone. That's because grown-ups don't leave surprises and disappointments in the will that the family doesn't already know about. If you don't want your family to find out your final wishes until after you're dead, you're what's known as a coward. Grow up and get over it. In more than two decades of working with clients and people who call in to my radio show, I've heard thousands of horror stories about ticking time bombs in someone's will that blew a family apart. Think about it: There is no worse time for a person to be surprised with disappointing news than when their whole family is going through a painful loss. Emotions are already running so high that one little spark could keep siblings from talking to each other for the next thirty years. You do a will to provide for and protect your family; don't let it be the thing that tears the family apart.
Avoid this by doing the reading of the will while you're still alive. No, you don't need to make this a big dramatic event, and you don't even need to read the will word for word. But you should make sure everyone knows your wishes. Tell them who gets what. Tell them what happens with the kids. Tell them how much you want to leave to which charities. Tell them everything, and tell them yourself! Let them hear it straight from you. That one simple act can save your family years of heartache and legal battles contesting the will.
I know one sweet old guy who's kind of a fanatic about this stuff. He watched his siblings get torn apart by a surprise in their parents' will, and he was determined that his own kids and grandkids would never go through that because of him. So, if you were to walk through this guy's house today, you'd see little sticky notes with people's names on almost everything. Walking through his living room is like taking a tour of his will! I don't recommend you go quite that far, but at least love your family enough to be honest with them today. If you have kids or grandkids who are addicted to drugs and who you can't trust with money, then tell them today that they won't receive anything in the condition they're in. Who knows? That alone could be the motivation they need to get help and clean up their lives. And if they do, then by all means update your will!
## A Matter of Trust
One thing that's becoming more popular in the world of estate planning lately is the living trust. Estate planners are pitching this pretty heavily inside the church too, but I'm not a fan of these arrangements at all. Here's the deal. I never recommend something to someone that I wouldn't use myself. And since a living trust makes no sense in almost any situation, I can't recommend it to you either. I'm not mad at the living trust people, and I don't think they're ripping anyone off; I just don't see any benefit to the hassle that a living trust creates. Unlike a standard will, living trusts require you to move your assets out from under your own control to a trust before your death. As a result, even basic financial decisions—like adjusting investments, managing bank accounts, giving to charities, and buying or selling real estate—have to go through a trustee because the trust legally owns everything. While it's true that larger estates require a higher level of sophistication, detail, and privacy in estate planning, a living trust probably isn't the answer in the vast majority of cases. It definitely wasn't in mine. More times than not, people in Baby Step 7 who are living and generously giving like no one else can protect their legacies through an estate plan built around a carefully designed will. And if it's a large estate, there will most likely be several trusts put in place upon death.
Despite how often the financial world pitches living trusts, there are three main problems that are hard to ignore. First, living trusts are the primary upsell in the estate-planning world. Rather than simplifying matters, living trusts create an expensive and usually unnecessary add-on to estate planning. They also require an ongoing relationship with an estate planner because you need a representative every time you want to adjust your assets.
Second, most living trusts are never fully funded. Setting up a trust is usually an expensive process, but the estate planners I've talked to say most of their clients never actually fully fund the living trust after setting it up. The hassle of retitling and redeeding everything into the name of the trust is so complicated and frustrating that only the nerdiest of the Nerds tend to follow through. But an unfunded trust isn't really a trust at all; it's just a waste of money that provides no benefit to you or your loved ones.
Third, living trusts create a cumbersome problem every time you want to do something with your investments or property. In a living trust, you don't personally own anything—the trust does. That means you can't make a move without getting the legal approval of the trustee. For example, I buy and sell real estate, review my mutual funds, and set up new investments all the time. If I had everything in a living trust, I'd have to move my estate planner and the trustee into an office next door just to handle the day-to-day operations of my estate! I just haven't seen any benefit to outweigh these issues, so I stay away from living trusts.
That said, I'm not saying that _all_ trusts are bad. For example, if you have a child with special needs, you should look into a special needs trust. I have a friend whose son has Down syndrome, so they set up a special needs trust as part of their estate plan. The trust kicks in when my friend and his wife die, and a portion of his wealth will go to fund the trust. Those funds will be managed by a trustee they've named and used exclusively to care for the ongoing needs of his son for the rest of his life. That's a good thing. There are other kinds of trusts that you can put in your estate plan that don't start until after you're gone, and Sharon and I have some of this as part of our plan. The point is that all trusts are not created equally. Just move slowly, get trusted counsel, and avoid the trusts that mess with your wealth while you're still alive.
## Avoiding Mr. Taxman
Federal estate tax planning may be your biggest challenge because the government puts massive taxes on estates once they reach a certain size. I want to be crystal clear here: There is absolutely nothing wrong or immoral about using every legal means available to avoid taxes. In fact, I'll take it a step further. I believe that taking advantage of every legal method of avoiding taxes is actually good stewardship. If you think that giving the government money that you don't _have_ to give them is good stewardship, I question your intellect—and your sanity. I've never seen any individual, corporation, or organization waste money as quickly and effectively as the federal government. If you want to bring the kingdom-shaping power of your money to a screeching halt, then by all means, give it to the government.
I am absolutely not suggesting you do anything immoral or illegal to hide your money from taxes. You need to give the government every penny you are obligated to give them—but not one penny more. I don't need the government to redistribute the wealth God asked me to manage; God has some pretty good ideas about how to distribute it Himself. And I'll start with giving to the causes that God directs me to give to and leaving my wealth to the three children I've raised to be wise stewards of God's resources.
## Till Death Do Us Part?
For years, as I've talked about these things, people have asked me, "Dave, if I'm supposed to be intentional about protecting my wealth, should I get a prenuptial agreement before I get married?" My answer has always been pretty clear. If you're getting married, you need to be able to put _everything_ into the marriage. If you're not willing to do that, then you might love your stuff more than you love your future spouse. That's a problem. One day, a lady called the radio show to talk to me. She was about to get married, but her future husband wanted her to sign a prenup because he had a '67 Ford Mustang he didn't want to lose if the marriage didn't last. I told that caller exactly what I would have said to my daughters if either of their husbands had ever tried something like that. I said, "Run! This guy loves his car more than he loves you!"
The old wedding vows said, "For richer and for poorer, in sickness and in health, 'til death do us part, _unto thee all_ _my worldly goods I pledge_." I think that's both beautiful and right. If you're not willing to commit all that you have, you shouldn't get married. That's what I've always believed, and it's what I still believe to this day. However, after all these years of working with people at all levels of wealth, I've just recently started to make a very slight exception to that rule. If one of the people coming into the marriage has extreme assets and the other person has little to no assets, then you might consider a prenup in that situation. I'm not saying you _have_ to; I'm just saying that's the only time I'd be willing to budge on my "no prenup" position.
Here's the reason: Wealth can make you a target. I've seen so much weirdness in this area. And it may not even be your future spouse who's weird; they may have weird family members or other influencers who twist the broke marriage partner into something you wouldn't have expected. But again, let's clarify what I'm talking about. This isn't about protecting a '67 Mustang; this is about someone with extreme wealth marrying someone with nothing. So if one partner is broke and the other has $2 million or more, then a prenup may make sense. Pray about it, talk about it, and get wise counsel before you make any decisions on this issue.
## Don't Deed the House Before You Die
The next practical piece of estate-planning advice I want to give you is something I've seen hundreds of times, but I've never understood. Do not deed your home to your family members prior to your death. If your parents are getting on in years, do not let them do this either. This is a nightmare situation that solves no problems and ends up costing the heirs a ton of money. This happens when someone or a couple gets the idea of transferring their home into their child's name before their death. For some reason, they think it's a wise move to go ahead and take care of the house before death so it doesn't get caught up in the will or anything. Bad, bad move.
Here's what happens: Let's say Mom and Dad bought their home in 1963 and paid $13,000 for it. If you weren't around or buying houses in 1963, you'll just have to trust me when I tell you that these are real numbers. Now, all these years later, that house they bought for $13,000 is worth $400,000. That wouldn't be unusual at all in a good neighborhood over fifty years. So Mom and Dad are sitting in a $400,000 home and decide to deed the home to their kids because someone told them it was a good idea. If Mom and Dad deed the home before death, then the kids' tax basis is the same as their parents' tax basis, which was $13,000—their original purchase price. That means the kids will have to pay capital gains tax on the difference between the $400,000 they sell it for and the $13,000 Mom and Dad originally paid for it. Basically, they have to pay capital gains tax on $387,000, which would be more than $50,000 in taxes.
However, if Mom and Dad had waited and simply left the house to the kids in their will after their death, their tax basis would be the current market value at the time of death, which is $400,000. So if the kids sell the house for $400,000, they pay zero taxes. That's how big a mistake this one move can be. You want to leave your kids what's called the stepped-up basis of current market value, not the basis on what you originally paid for the home. That one thing can save your children tens of thousands of dollars in unnecessary taxes.
## The Family Constitution
One last thing I'd encourage you to do as you work on your legacy journey is to create a family constitution. This is a document that defines your family's values, recognizes your family's dysfunctions, and considers your family's relationships. At its heart, the family constitution clearly spells out who and what your family is—and what it isn't. This is an exercise my family did several years ago, and it opened the door to all kinds of conversations for all of us. We found that there were several things we had all just assumed but had never really talked about. And there were other things that we discovered were incredibly important but had never even considered before. I have to admit, it was surprising. I've always been a big goal-setter, and, as we've talked about, one of the most powerful things you can do with goals is to simply write them down. As we worked on our family constitution, I discovered that the act of writing our values and beliefs down was just as important as writing goals. Once that piece fell into place for me, the whole thing clicked.
Because we're a family of faith, it was important for us to put the Bible at the center of our family constitution. So, at the very top of the first page is Joshua 24:15: "Choose for yourselves this day whom you will serve . . . But as for me and my house, we will serve the LORD." That sets the tone for everything that follows. That's who we are as a family: We're a group of men and women who are always striving to serve the Lord in everything we do. By putting this verse at the beginning of our family constitution, we're showing the most important boundary line of all. We're saying to each other, "If you're not interested in serving the Lord, then you're out of bounds." And since it's in writing and we've all agreed to it, choosing to go outside the boundaries means you're choosing to walk away from the blessings we share as a family. That may seem harsh, and it may not work for your family, but after we talked it through, this made perfect sense for us, so that's how our family constitution starts.
There are four keys to a successful family constitution. The first key is to create a family mission statement. My friend Dan Miller, one of the leading career coaches in the nation, says that a good personal mission statement takes into account your skills and abilities, your personality traits, and your values, dreams, and passions. I think the same is true when you're writing a family mission statement. Your family has skills and abilities. Your family has a unique personality. Your family has values, dreams, and passions. Take the time to examine all these things that you share collectively and use that insight to figure out what purpose your family is here to accomplish. Then, write it down.
The second key to a good family constitution is to tailor this to your specific family. There's no single, one-size-fits-all constitution that works for every family, so don't try to rubber-stamp it. You can find all sorts of examples online to help steer you in the right direction, but this has to be custom-made for your own weird, quirky, unique family.
Third, and this may be the most important key, you have to live out your family constitution every day. If this document really does represent who you are as a family, then your lives need to reflect what it says. Your constitution can't say that your family is generous and mission-minded if you're not giving—not to mention tithing—and you haven't been on a mission trip in the last decade. You're writing down the reality, not some aspirational, fairy-tale vision of what you _wish_ your family was like. This is crucial for parents, because the kids in the family are watching you. My daughter Rachel Cruze constantly reminds parents, "More is caught than taught." Your kids learn a lot more from what you're _doing_ than what you're _saying_. If you want to teach them how to live according to your family constitution, then you need to live that way yourself.
This doesn't have to be a long, overly detailed list of rules and regulations, though. Our family constitution is simply a one-page document, but it's enough. Just as an example, I'll share a couple of things that we chose to spell out in our constitution. First, like we've talked about several times already, our family believes in good, honest work. The Bible says, "If anyone will not work, neither shall he eat" (2 Thessalonians 3:10). Our family takes that pretty seriously, so if one of the future grandkids ever decides that he wants to sit around and take it easy on the family's dime, he'll have a rude awakening. That kind of behavior will result in him no longer being able to participate in the family's wealth until he decides to get off his tail and get back to work! That's not because we're workaholics or slave drivers; it's because we believe in the value of work. We think God put each one of us on this earth to do something, and if all you want to do is sit around all day watching daytime television, you'll never accomplish your God-given mission. So, I'm not going to support that kind of behavior. And, no, I'm not talking about stay-at-home moms here. Moms in the home may have the hardest—and certainly most important—job of all! Sharon was a stay-at-home mom, and she worked incredibly hard to raise our three kids.
Also, as a family of evangelical believers, we believe that a man and woman who aren't married shouldn't live together. That sounds pretty old-fashioned these days in some circles. You can call me a dinosaur; I'm okay with that. It doesn't change the fact that our family values don't line up with that behavior, so the family wealth won't support that decision. It's the same with drugs. If Junior is on drugs, we can't pour our family's wealth into Junior's drug habit. That doesn't mean we won't do everything we can to help him, but giving an addict a big pile of money isn't really helping him, is it? Of course, we outline a path of restoration for the prodigal child too. We spell out how he or she can come back home to full participation in the family and the family's wealth. We love our kids, and we will always want them to come back home if or when they stray. We just can't financially support whatever bad behavior they may fall into. That's one way we actually show our love for them, even though it's hard.
The last key to a good family constitution is forming a family council to update and enforce the constitution with regular meetings. Your family should meet regularly to go through all of this and discuss anything that needs to be added, removed, or updated. As the kids get older or married, this circle may start to grow. Since mine is the first Ramsey generation to start this, the meetings are basically made up of me, my wife, my three kids, and their spouses. As my kids have kids of their own, this group will get bigger. That's normal, but it doesn't necessarily mean every family member several cousins deep has to show up. You just want to make sure that the whole family (at least the part governed by the constitution) has a voice and a vote.
# THE LEGACY BOX: YOUR LAST, BEST GIFT
We have covered a lot of things in this chapter that you need to do as you work toward your legacy journey. As you pull all of these things together, you're going to end up with dozens of documents, records, account statements, estate-planning paperwork, advisor contact information, and a hundred other things. This stuff doesn't sound all that exciting, but your family's ability to quickly put their hands on any one of these things could be a major ordeal if or when something happens to you. As I wrote in _Dave Ramsey's Complete Guide to Money_ :
> If you were to die today, would your spouse or other family members know where all your important papers, life insurance policies, bank account information, computer passwords, and final instructions are? In my years doing the radio show, I have talked to countless mourning spouses who have no idea how to put their hands on this information. On top of the crushing loss of their husband or wife, they are immediately thrown into a financial nightmare because they don't know what to do or how to access the accounts or where the final instructions are. It's a total disaster. After talking to those spouses, I decided I was absolutely not going to leave my wife in that position if something happens to me.
That's when I decided to put together what I now call the Legacy Box.
## Your Legacy, in a Box
The idea behind the Legacy Box is simple: Put all of your important paperwork and information in a single place so your spouse and loved ones can find it when they need it. This is important regardless of what Baby Step you're on. There is simply no better way to say "I love you" than to take this step to minimize the stress and panic your family will feel in an already exhausting situation.
Your Legacy Box should have dividers, folders, or envelopes for at least each of the following items:
> **Executive Summary:** a one-page document that gives an overview of the box's contents
> **Legacy Letters:** personal letters from you to each of your family members to be distributed upon your death
> **Birth Certificates**
> **Social Security Cards**
> **Passports**
> **Marriage Certificate**
> **Auto Insurance**
> **Homeowner's/Renter's Insurance**
> **Health Insurance**
> **Long-Term Disability Insurance**
> **Long-Term Care Insurance**
> **ID Theft Protection**
> **Life Insurance**
> **Umbrella Liability Insurance**
> **Estate Plan**
> **Power of Attorneys**
> **Wills**
> **Funeral Instructions**
> **Financial Account Log**
> **Safe Deposit Log**
> **Monthly Budget**
> **Tax Returns**
> **Money Market Statements**
> **Mutual Fund Statements**
> **College Funds**
> **Retirement Accounts**
> **Rental Property Summary**
> **Car Titles**
> **Home Ownership Records**
> **Passwords/Combinations**
If you take our class _The_ _Legacy Journey_ , we even give you a nice wooden Legacy Box, complete with divider tabs for each of these categories.
When you finally get everything together, you need to do a couple more things. First, make a copy of _everything._ Basically, you need to create a second Legacy Box with copies and put it in a different location. That way, if your house burns down, you'll have a backup of the entire set of papers. You can also make a digital copy by scanning everything and using digital folders for filing. Then you can easily put a copy of your digital Legacy Box in your safe deposit box on a flash drive or CD-ROM for safekeeping. I just don't recommend people keep their _only_ copy of everything in the safe deposit box. Emergencies happen every day at all hours; don't make your family wait until the bank opens to get to the information they need in an emergency.
Last, you need to make sure your family members know where to find your Legacy Box and how to use it. You might even want to do a dry run, just like a fire drill. I have one friend who likes to do this to his wife randomly. They'll just be sitting in the living room with their kids, and my buddy will say, "Okay, honey. I just dropped dead, and you need my life insurance policy. Go!" Then she jumps up and runs to the Legacy Box and pulls the form. Okay, that's kind of ridiculous, but, hey, at least she'll know exactly what to do and where to go if or when something ever happens.
The bottom line with everything in this chapter is that you will always be able to find reasons to procrastinate. Unless someone has a terminal illness, the thought of doing a will always seems to stay in the "important but not urgent" category. I want to encourage you to think of everything we discussed in this chapter as both important _and_ urgent. Remember, none of us is going to make it out of here alive. We're all going to die one day. The only question is: What kind of legacy do you want to leave when you do?
As for me and my house, we will serve the Lord.
### CHAPTER EIGHT
# Called to Generosity
Back in my college business and finance classes, I was introduced to a fantastic, must-have, do-it-all, fictional product called the widget. The widget is used in case studies as an example of a generic product from a generic company. If a company is doing really well, they're manufacturing and selling a bunch of widgets. They're keeping their supply chains fully stocked, and they're keeping a close eye on inventory to make sure they never run out of widgets. I remember one case study in particular that has really stuck with me over the years:
There was this guy who started a business that grew bigger and faster than he could have imagined. He developed a cutting-edge widget that was on its way to becoming a household name. Everyone loved his product—including him. He liked to spend his free time wandering the aisles of his widget warehouse, gawking at the thousands of cases of widgets that lined the shelves. Every now and then, he'd pull a widget off the shelf and open it up like he'd never seen one before. He felt like a kid on Christmas morning! This guy _really_ loved his widgets.
As business picked up, he had to stuff more and more widgets into the warehouse, which was beginning to get cramped. He was just about out of room before he realized he needed to tear down the old warehouse and build a bigger, better, cutting-edge warehouse. He called in the best engineers from across the country and meticulously designed every square inch of the place. His new facility would store all the widgets he'd ever need! It was the world's first multistoried widget warehouse. Before long, other business executives were calling him and asking him how he'd managed to get so many widgets into one space.
Filling that giant warehouse took a while, but one day, the delivery guy came and put the last box on the last shelf. The place was completely full. The young businessman stood in the middle of the warehouse, beaming with pride and joy at how successful he'd become. For the first time in several years, he relaxed a little and began dreaming of all the fun things he could do. He was on top of the world! But then, just as he was heading to his car, God showed up and said, "You fool! This very night your life will be demanded from you. Then who will get what you have prepared for yourself?" (Luke 12:20 NIV).
Oh, wait a minute. Maybe that wasn't an example from business class after all. Maybe I heard it in Sunday school instead. It's The Parable of The Rich Fool from Luke's gospel, but it doesn't feel thousands of years old, does it? It shows the same kind of attitude we deal with today when it comes to wealth. It's the story of a farmer whose ground produced an enormous amount of crops—what we'd consider to be wealth. The farmer grew so enamored with his surplus that he filled a barn. When that barn was full, he tore it down and built another one. Then, after all that effort, God took him home.
Now, there was nothing wrong with the farmer's crops, and there's nothing wrong with the wealth we build today. The problem comes when, like the farmer, we store up wealth because we worship it and the wealth is our only source of security. God calls this idol worship, not management or stewardship. We take our eyes off of God and off of those around us, and instead we focus on how rich we can get and how much stuff we can accumulate. That's a totally self-centered, self-absorbed attitude, and it's what I've been warning against throughout this book. Just look at what the farmer says in Luke 12:18–19: "I will do this: I will pull down my barns and build greater, and there I will store all my crops and my goods. And I will say to my soul, 'Soul, you have many goods laid up for many years; take your ease; eat, drink, and be merry."' In the space of only two verses, the farmer says "I" or "my" eight times! From a legacy journey perspective, there's no stewardship, no **US** or **THEM** in this guy's mind. He's totally and completely wrapped up in himself and in his own twisted version of **NOW** and **THEN**. The problem is: It is so easy for us to fall into this same trap today. After decades studying God's Word on this, and after hundreds of conversations with godly millionaires and billionaires, I think I've figured out how to keep this attitude from creeping into our mind-set.
The answer is generosity. Generous giving is the antidote for selfishness.
# PRINCIPLES FOR GENEROUS GIVING
Before we dive into some nuts and bolts of generous giving, let's take one last look at the legacy journey framework. First, there's what we call the **NOW** stage, which is taking care of our immediate family before we do anything else. That includes getting out of debt, getting on a budget, and clearing away any financial emergencies that may be keeping us from moving forward. Next is the **THEN** stage, which is when our heads come up and we're able to see down the road a bit. That's when we start investing for retirement, saving for the kids' college, and setting a vision for our future. That leads to the **US** stage, which is when we focus on our family. We start to look at our children, grandchildren, and even our great-grandchildren, and we realize that the decisions we're making today set the tone for our generational legacy. Then, once we know our family is taken care of and we've set a fantastic legacy in motion, we look around and start to see the whole world through God's eyes. We see needs we never saw before. We see ways we can help people next door and around the world. We see people who are starving, orphaned, sick, or hurting, and our response is, "I can help!" That's a great place to be. That's what we call the **THEM** stage because at this stage, you're not focused on yourself, your future, or your family; those things are already taken care of. At this stage, you're focused on how God can use you to impact the world.
## We're Made to Be Givers
The key to having an enduring legacy is to practice generous giving throughout your life—at every stage of the **NOW–THEN–US–THEM** framework. Giving is a hallmark character quality of those who win with money. In fact, it's almost impossible to find someone you'd consider to be Christlike who is not a giver. That makes perfect sense, doesn't it? The Bible says that we are created in the image of God. The truth of our being—the innermost part of who we were created to be—is to be a reflection of God. Just breathe that in for a second. We get confused about our purpose all the time. So many times we ask someone _who they_ _are_ , and they answer by telling us _what they_ _do_. We confuse our jobs with our purpose. I am a teacher, an author, and a radio host. More important than that, I'm a husband and father. But even more important than that _,_ I am a follower of Christ, which means Christ is _in_ me.
When you peel back all the layers, what you find at the very center of me should be the image of my heavenly Father. Scripture says, "For you created my inmost being; you knit me together in my mother's womb" (Psalm 139:13 NIV). I once heard Ray Bakke talk about that passage. He described this beautiful image of God sitting in heaven with knitting needles actually _knitting_ your DNA double helix. That's such an amazing picture. God specifically, meticulously handcrafted every single one of us. And as He did, He infused us with His likeness. We are made in the image of God—and He is a giver. He gave the ultimate gift in His only Son. He gave us every single blessing we've ever had. We've said throughout this book that God is the Owner. That means everything we will ever have comes straight from His hand. Psalm 84:11 says, "The LORD will give grace and glory; no good thing will He withhold from those who walk uprightly." If He gives so freely and so richly, and if we're made in His likeness, then that must mean we're made to be givers too.
## Giving Is an Expectation
I believe that giving is an expectation for God's people. If we truly want to be like Christ, then we need to follow His example in giving—and He gave _a lot._ Scripture consistently shows the connection between working, earning an income, building wealth, and generous giving. The apostle Paul told the church in Ephesus, "Let him who stole steal no longer, but rather let him labor, working with his hands what is good, that he may have something to give him who has need" (Ephesians 4:28). That one verse ties it all together pretty well. You work, you earn, you give. That's what the _Havdalah_ service is all about, remember? We work to first fill our own cup, and then we keep pouring until there's an overflow to be shared with others. That overflow is for the good of other people.
But, like we saw in Chapter 4, we have to keep that cup in proper perspective. You and God set the size of your cup (the lifestyle and investing ratios), and what's left flows out to serve others (the extra giving ratio). Like I said before, you have to remember that this is a _cup_ ; it's not a thimble or a swimming pool. You should always size your lifestyle cup so that it is small enough to overflow, but not so small that your family will starve. I'll add one thing to that: It's also not a recirculating fountain. Once the overflow leaves the cup and spills into the bowl—meaning it's out of your hands—you can't take it back. You have to keep an emotional distance from what you've set aside for giving. That's why it is so important for you to spend plenty of time in prayer as you set your ratios. It keeps you from trying to reclaim the money you and God already decided to give away.
## Giving Should Be Fun and Private
If we're properly managing God's resources, then we're going to be giving—and we're going to be excited about the opportunity. Scripture says, "God loves a cheerful giver" (2 Corinthians 9:7 NIV). The word _cheerful_ comes from the Greek word _hilaros_ , which is where we also get the word _hilarious_. When you read it that way, you could say that God loves a _hilarious_ giver. Wouldn't it be cool if your church broke out in wild laughter and applause as the offering plate passed by? That's the spirit we're supposed to have when we get the opportunity to tithe and share our overflow.
The problem is, people get confused about why they should give. Some use their giving as a way to promote themselves or to help their reputations. Honestly, that makes me a little sick. That's no different than the Pharisees Jesus talked about who liked to say long, loud prayers on street corners just so people would see how "holy" they were (Matthew 6:5–6). So with rare exceptions, your giving should be done in private, and it should never be a spectacle. Jesus was perfectly clear about this:
> Be careful not to practice your righteousness in front of others to be seen by them. If you do, you will have no reward from your Father in heaven. So when you give to the needy, do not announce it with trumpets, as the hypocrites do in the synagogues and on the streets, to be honored by others. Truly I tell you, they have received their reward in full. But when you give to the needy, do not let your left hand know what your right hand is doing, so that your giving may be in secret. Then your Father, who sees what is done in secret, will reward you. (Matthew 6:1–4 NIV)
Please don't get too caught up in the "reward" this passage mentions. That's not why we give. We give to help other people, not to help ourselves. This isn't a transactional thing where we're always trying to get a return on our giving. We should give because we're called to be givers, because we are passionate about using God's resources to serve other people. We've been blessed not only for our good, but also for the good of others. We are blessed _to be a blessing_ (Genesis 12:2–3), and that's an exciting, win-win deal!
Charles Dickens said, "Do all the good you can, and make as little fuss about it as possible." Giving anonymously keeps our motives pure. It keeps us from making our giving all about me, me, me. And for those of us who are believers, that attitude leaves a wide-open door for people to experience God in a whole new way. When we lean into a big need, and we do it completely anonymously, who gets the credit? God does! If a struggling single mom finds a nice used car sitting in her driveway with the keys and title on the dashboard, she's going to tell the world that God provided big-time—if you're not standing in the spotlight. This all goes back to ownership and stewardship. The Owner, not the manager, should get the praise. If it's God's money (and it is), and if He is directing how you use it (which He should be), then He's the one who met that need—not you. Don't let your ego get in the way of a powerful testimony about how God showed up to change someone's life.
## Giving Is about More than Money
I had a lady call my radio show recently with an interesting take on tithing. She said, "I've heard people say that you should give more than money, that your giving should also include your time and talents. Well, I work at my church, so I'm just going to count the time I spend there as my tithe."
I replied, "Well, you can do whatever you want, but that's not what the tithe is. Giving should include your time, your talents, _and_ your treasure. So, biblically speaking, you should give your tithe—and why don't you give your time too? You're obviously good at drama; why don't you join the church's drama team?"
I was poking at her a little bit, but that call made me realize some people are confused about what giving is all about. Recently, my team and I have been discussing what biblical stewardship really means. We keep coming back to the idea that stewardship is handling all of God's blessings God's way for God's glory. And when we say "all of God's blessings," that's what we mean: _all_ of His blessings. That includes more than our money. It includes the time He gives us on earth. It includes the skills, passions, interests, and opportunities He gives us. It includes _everything_ we have to offer. Money is just one small part of that. That doesn't mean we can give our time and talents _instead of_ our money. God's given it all; as good stewards, we should be giving some of _all_ of those blessings.
## Giving Unlocks Your Full Potential
You will never reach your potential level of creativity, talent, passion, success, or excellence until you learn to give. I know that's a bold statement, but I believe all of Scripture backs me up on this. Luke 6:38 says, "Give, and it will be given to you: good measure, pressed down, shaken together, and running over will be put into your bosom. For with the same measure that you use, it will be measured back to you." Please don't hear this as a cause-and-effect, give-to-get statement. I don't think that's what the passage means at all. Instead, I think it's talking about what happens in your spirit when you give. Read it again: "Give, and it will be given to you: good measure, pressed down, shaken together, _and running over_ . . ." That kind of sounds like the overflow we've been talking about, doesn't it?
I think the inverse is true too. People who aren't givers, who are greedy and hold their money in clinched fists, never have an overflow. That's because they're not _flowing_ at all; they're stopped up! When I teach budgeting, I always tell people that money is active; it is always moving from somewhere and to somewhere. From a legacy journey perspective, we might say that money should always be _flowing to us_ and _flowing through us_. If we interrupt that flow by hoarding everything we get, then all we're doing is building bigger barns and stockpiling our wealth. That's a sick, twisted view of money. That's a picture of the manager trying to steal the Owner's resources, and that's not an attitude that God can bless.
Generous givers, however, are attractive. They have a spring in their step and an energy about them that draws people to them. They live their lives with open hands, and that changes one's whole perspective on life. Billy Graham once said, "God gives us two hands: one to receive with and one to give with." It takes both hands to create and leave a legacy, and you can't receive or give if your hands aren't open. An open hand is there to give money, sure, but it's also there for so much more. An open hand can lift someone out of a mess. An open hand can hug someone going through a difficult time. An open hand can applaud someone else's success without a trace of jealousy or envy. An open hand can serve the needs in the community. An open hand is ready to receive more and more of God's blessings because the person with an open hand can be trusted with more. Giving changes so much more than your money; it changes your life and sets you free to unbelievable levels.
# WISDOM FOR RADICAL GIVING
I remember the first time someone explained the concept of the tithe to me. I thought it was a scam. I came to a relationship with Christ as an adult, so I was already working and doing pretty well at the time—before our bankruptcy. So here I am making a good income, enjoying all the fun things that money can buy, and suddenly I'm supposed to cut 10 percent of my income right off the top? And I'm supposed to give it to whom? To God? It didn't look like He needed my money. To the church? We were in a beautiful church with a multimillion-dollar building. It looked to me like they were doing better than I was. If God loved me and blessed me financially, why on earth would He make me give Him a huge slice of my money? I didn't get it . . . at first.
## Giving for Dummies
I'll be honest: I don't have a full, detailed, expertly researched exegetical study on the tithe. I know people get into that sort of thing, and I think it's great if that's your passion. For me, though, the tithe is pretty simple. Once I understood that God is a giver, and that I'm also supposed to be a giver because I'm made in His image, the reason for the tithe came into focus for me. I don't think God tells us to tithe because He needs our money or because His church would fall apart without it. I think He tells us to tithe because He wants to teach us how to flex our giving muscles. In my book _Dave Ramsey's Complete Guide to Money_ , I explain it this way:
> I believe that God puts us through the mechanical act of giving—even when we don't fully understand the reasons why—because the act of giving changes us. It crushes our hearts and reforms us into something that looks and acts a little bit more like Christ. You can't say you're a follower of Christ when you're not giving. You can't walk around with the clenched fist and tell people about how amazing Jesus is. There's a disconnect. They won't believe you because your whole attitude is one of selfishness, fear, and greed. Remember, the clenched fist is the sign of anger. Jesus never talked to people about the love and grace of God with His hands balled up into fists!
> Every time I open my hand to put money in the offering plate at church, or to support a missionary, or to leave a huge tip for a struggling single mom waiting tables, it shifts how I see things. Every time I spend a weekend serving other people instead of skiing on the lake, it changes my heart a little bit. Over time, all those changes add up as we become more and more like Christ. Because we are designed in God's image, we are happiest and most fulfilled when we are serving and giving. Giving is hands down the most fun I have with money. I get such a thrill when God uses me to help families or ministries do things they thought were out of reach. One of the highlights of my year is our company's annual Christmas party, which has become somewhat of a local legend. Our team works insanely hard all year. These are families at every point on the wealth spectrum. Some are working on their debt snowball with gazelle intensity, doing whatever is necessary to take care of their **NOW**. Others are doing really, really well in building wealth and are fully in the **THEM** stage of their legacy journey. A lot of them are somewhere in between. We might have the most random assortment of cars of any parking lot in Nashville, with twenty-year-old junkers parked next to high-end Mercedes up and down the rows!
Once a year, a couple of weeks before Christmas, every one of these families gets dressed up to attend the Christmas party. We rent out the nicest banquet room we can find, and a committee plans every detail, down to the minute. We treat these families to gourmet food and world-class entertainment, and we (literally) roll out the red carpet to make them feel like the superstars they are. Then, at the end of the night, it is my honor to stand on stage and give each of them their Christmas gift. We've done laptops, iPhones and iPads, televisions, game systems, vacations, cash—you name it. The single goal of the whole event is to make everyone feel loved and appreciated.
This one party is a huge line item on the company's budget. I won't tell you how much we spend on that event and the gifts, but I'll just say no one ever feels shortchanged. So, why do we do it? That party represents a ton of money that would otherwise go into my pocket and my leaders' pockets. Why don't we just give each team member a ham and keep the rest of the money? It's because almost thirty years ago God told me to start tithing. I wasn't a natural giver back then, but I was committed to following God's ways of handling money. That meant giving. When I started, it was just the tithe; that was the minimum, which is a good thing, because I don't think I could have afforded any more. But learning how to give, even only 10 percent, opened the doors to the most powerful, most exciting, most enjoyable part of building wealth: the opportunity to share it with others.
## Giving above the Tithe
The tithe gets you started and helps you keep an emotional distance from your money. It teaches you how to give. That's why there's a 10-percent giving blank at the top of all of my budget forms. I don't care what Baby Step you're on, you need to be giving a tithe. It's not only important in your **NOW** , but it will gradually shape you into the kind of person you'll be later in your legacy journey. If you go through the process of building wealth with the purpose of leaving a legacy, your journey won't be **NOW–THEN–US–** **ME** ; it will be **NOW–THEN–US–** **THEM**. That means giving well above the tithe—when you are ready.
But when are you ready? Extra giving above the tithe enters your plan after Baby Step 3, which means you're out of debt and you have three-to-six months' worth of expenses saved in an emergency fund. At that point, you haven't started building wealth yet, but you can start dipping your toes in the water of extra giving. Up until this point, you haven't had any overflow to share, but now your cup is about to start overflowing for the first time, so it's a great time to start sharing some of that overflow. You just can't go crazy with extra giving at this point, because you're still taking care of your family first. You have to make sure your **THEN** and **US** stages are taken care of not just for today, but for the future.
When you're pretty much finished with the **US** stage, here's what life looks like: Your retirement savings are maxed out. You and your spouse are going to be just fine heading into your retirement years. Your kids are going to have the money to go to college debt free. You have investments above your retirement accounts for the purpose of building wealth, which creates overflow. You're using budget ratios to keep everything balanced, including extra lifestyle, investing, and—that's right—extra giving. When you're at this point, it's time for your giving to come unglued. When you get to Baby Step 7, you have zero debt (including the house), and you're a wealth-building machine. That's when you get the joy and honor of practicing radical, extravagant generosity regularly. God may specifically call you to do some big giving before this point, and that's fine. Just consider those to be exceptions. But otherwise, save your mind-blowing level of giving for this stage, where your golden goose (your wealth) is laying enormous eggs that you can give and share freely!
## Your Biggest Investment
"Dave, you're a _horrible_ giver!"
That comment caught me off guard. You might think this was said in the early days when I was still trying to figure out the tithe, but you'd be wrong. That comment came much, much later. In fact, it was just a few years ago. The problem wasn't that I wasn't giving; the problem was that I wasn't being a wise giver. Sharon and I were budgeting and living on budget ratios, but we weren't doing a good job directing all those giving dollars. In fact, too often that money would collect month after month, and I'd look up in December and get in a big rush to give it all away in time to get the tax write-off for the year. At that point, I'd just start writing checks to ministries without taking any time to get to know who they were, what they did, or how they handled their money.
I was talking about this with a friend who was further along in his wealth building than I was, and he said, "Dave, you're a horrible giver. You spend a lot of time on your business investigating every little thing. You spend a lot of time on your personal investments looking at charts and spreadsheets and lifetime performance. In those areas, you do due diligence because you're making an investment. But Dave, when you give, you're investing in the kingdom of God. That's the biggest investment you'll ever make! Why aren't you doing due diligence with that?"
As a math and business nerd, that question hit me like a ton of bricks. It was so obvious when he said it, but it's something I hadn't taken seriously before. From that day on, though, I started treating all my giving as investments. That meant I had to slow down, do my research, and pray. I had to really get in there and see what this money would do inside a ministry. If you are giving to a ministry that has bad operational and money-management practices, you may be harming them. You are in essence an enabler, just like if you gave money to a family who had poor spending habits. If your giving is a large percentage of the ministry's budget, are you artificially playing God in their planning? After facing these questions, we started to work really hard at investing God's money in ministries as if they were a business investment—because they are.
We ultimately created a family foundation to manage all of our giving. My oldest daughter, Denise, who had nonprofit experience, is the director of the Ramsey Family Foundation. Her full-time job is to explore every single organization our family donates to. Here's what I found out through this process: As a family, we had a good understanding of God's ownership, but we weren't being good stewards of His resources—at least in the area of giving—because we weren't doing a good job of managing His wealth for the good of others. Hiring Denise as the foundation director put that into perspective. We were paying her a salary to manage our giving budget. If she did a sloppy, lazy, haphazard job, then we'd probably fire her and find someone to do a better job. But here's the thing: That's exactly how God was viewing me! He's the Owner, and He gave me this wealth to manage, but I was doing a sloppy, lazy, haphazard job in this area! Making that connection flipped a switch in my brain, and since we started the foundation and began viewing these gifts as investments in the kingdom, we've been able to do more with those dollars than we'd done in all the years before. We were finally doing our giving like I teach people to do their budgets: on paper and on purpose—before we gave a dime.
## Give to Your Passions and Values
There will always be causes, ministries, and charities that need your help. That is an enormous responsibility to us as we start to give outrageously. How do you decide which ones to help? How do you choose the right ones out of a million good choices? I've known a lot of wealthy people who get so worked up about this that they try to give to _everything._ They spread their giving a mile wide, which means they don't make a big impact in any one organization. That's not the strategy Sharon and I took when we started giving above the tithe. Instead, we decided early on that we wanted to go a mile _deep_ with our giving dollars. We felt like God had given us the opportunity to make a huge difference in a small handful of ministries. Would you rather give $1 to a million charities or give $1 million to one? That's a personal question I can't answer for you; I can only tell you what we chose.
Learning to say no to a need we observe or a giving request is really hard for most people—including us. There are so many wonderful works going on around the world that are inspired and directed by God. So how can you possibly say no to them? We all realize we have a limited giving budget; the money is not infinite. We had to come to grips with the fact that, while we are managing God's money, we are not God. Once we got over our messiah complex, thinking we were God, we were released from the guilt of not giving to every valid, wonderful cause. Steven Curtis Chapman's song "God Is God (and I Am Not)" helps me remember this.
But we still had to figure out where to do our giving. As we prayed about it, we realized that God had given each of us specific passions and values. Sharon grew up in a small town with a fabulous local church, so she has a heart for supporting local church ministries. I came to Christ as an adult, so I'm big on evangelism. Those are big passions for us, so we let them guide our giving.
The same is true for your value system. Don't put money into something that goes against your values. For example, I don't believe people should go into debt. Big shock, right? That means I'm not going to give money to a ministry that is deeply in debt and about to go millions deeper into debt on a building campaign. That doesn't align with my value system. It is completely opposed to everything I believe about God's ways of handling money, so clearly, I'm not going to pour money into it. God gave me these passions and values for a reason, so I'm going to use them as a filter for my giving.
# DISPLACERS IN A DIRTY WORLD
There are voices in our culture today who believe we should give everything we have away. We've talked about that more than once in this book. Maybe we've talked about it too much, but I know so many godly, generous men and women of faith who have been beaten up over this issue. They've heard it so often and from such well-respected people that this toxic message has crept into their own hearts. They look around at their success and start asking themselves if they're doing something wrong by having nice things. Even if they give away millions or even billions of dollars over the course of their lives, like my friend I told you about earlier, they still question whether or not they're being good stewards of God's resources.
## Don't Give Away the Golden Goose
Let's go back to basics again. God is the Owner. I am the manager. He's given me a portion of His resources, and He's given me and my family the responsibility of managing it for Him in the way He directs me. If I bow to cultural pressure and give away all of my wealth, what am I really doing? I'm just surrendering my role as manager. I'm saying to God, "Thanks, but no thanks. I don't think I should be trusted with what You've entrusted to me, so I'm going to pass that responsibility on to someone else." But here's the problem: God didn't entrust that wealth to them; He entrusted it to me. It wasn't their responsibility; it was mine. He gave me a job to do, and I told Him no because I was scared or ashamed or guilty or whatever those toxic voices wanted me to be. It may not be a popular opinion these days, but I think that's dumb. That makes me a terrible steward, and if I care more about what other people think than I do about the job God's given me to do, maybe He really can't trust me to manage more.
It is practically impossible for your legacy to outlive you if you kill the golden goose. I talk about this in my book _More Than Enough_ :
> If you want to be a powerful giver you should view your wealth as the goose and give the golden eggs. If you give away the goose, the golden eggs are gone and so is your ability to help others. Those of you who think "those nasty rich people should be made to give up the wealth they have earned" are not only stupid; your short-sightedness kills the goose, and the poor are not really helped.
If you kill the goose, you get one good meal. If you keep the goose, you get egg after egg and meal after meal. What if you considered it your job to manage the goose and give away the eggs? Couldn't you do more?
Of course, there are times when God specifically calls certain people to give away everything. He's put that call on my friend Robert Morris, who I told you about before. But those calls are for specific people for a specific purpose at specific times. It's not the typical biblical model. If God puts that call on your life, and if that's your conclusion after spending days and weeks in prayer, then that's fine. I would never stand in the way of what God is calling anyone to do. I'm just saying that would be a very, very unusual calling, so I'd encourage you to talk to your pastor, friends, peer group, and maybe a few wealthy people of faith before taking that big step. And never do it just because you think it's a way to "trick" God into giving you more. He may, and He may not. If you give everything away, you need to be emotionally and mentally prepared to live in poverty the rest of your life—or at least to completely start over with nothing. You have no idea what God may have in store for you next.
## The Displacement Method
I love Dallas Willard's perspective on this issue. In his book _The Spirit of the Disciplines_ , Willard argues that if Christians view money as evil and filthy, then godly people should never have money. And if godly people never have money, we are effectively surrendering all the wealth of the world to the enemy. If the people of God pull out of the marketplace, we leave a hole in the world that other forces will rush in to fill. Somehow I don't think that's what God had in mind.
Instead, we should practice the displacement method. Here's what I mean by that: If you push a physical object into a crowded space, it pushes other things out of the way. It _displaces_ whatever's in its way. My wife, Sharon, is a health nut. She runs, works out, does yoga, and eats a near-perfect diet. It drives me nuts. You'd think after all these years, I would have gotten healthier just by osmosis, but no such luck. She recently got a juicer, and that's when things got really weird. She's juicing _everything_! She's juicing things I'm sure God never intended to be juiced. I'm talking about broccoli and Brussels sprouts here. I love her, but some of this stuff is just gross. The worst part is, sometimes when she's done with her health shake, the glass it was in looks like it's covered in mold. It has green crud in the bottom and a thick, nasty film all over the inside of the glass. Just looking at it, you'd think you'd have to throw out the glass, but here's the thing: When she takes one of those grimy glasses and puts it in the sink with the water running and she leaves it there, at first, the green grime holds on tight while the clean water runs over it. After a minute, though, all that junk starts coming loose and flowing over the top and out of the glass. A few seconds later, you can't see any trace of her shake. Simply pouring clean water continuously into the glass displaces all the junk that was there a minute before. That's what we're called to do with the money God gives us to manage for His kingdom. We're supposed to go into a dark world with our time, talents, and treasure and engage the culture in such a way that we create displacement. And the more of that we do, the cleaner the society becomes. The more we surrender and pull back from that, the more the filth takes over—because one side is going to displace the other.
That's true spiritually too. You can put something good in and push something bad out. It works both ways. Good and bad things can knock each other out of the way. For example, I know a musician who is crazy about Jesus and loves making music. Even though he's a strong Christian, most of his songs are what most people would call "secular" hits, meaning they don't talk explicitly about his faith. However, this is a man of God serving with integrity in a business that's often filled with filth. When he has a hit song on the radio, even if he doesn't mention the name of Jesus, he is displacing other trashy songs that could have been taking up that airtime. The same thing happens when a Christian teacher steps into a classroom, or when a Christian businessman opens a business, or when a Christian politician runs for office. They don't have to put the name of Jesus on a campaign poster in order to make an impact for the kingdom of God. All they need to do is serve with excellence, keep their integrity, and maintain their commitment to God, and that alone makes a huge impact. Just by _being there_ , they are displacing the forces of the enemy.
My goal is to equip an entire army of displacers—godly men and women who have been spiritually, emotionally, and financially set free—to be so radically generous that, together, we can move the needle in some huge area of need in the world. What if the church was known as the organization that ended the AIDS epidemic or that wiped out malaria? How cool would it be if God's people, using His resources His ways, changed the world like that? That's a powerful vision, and it's entirely possible— _if_ we move past all the toxic messages the world is throwing at us right now.
# EATING CAKE OR LIGHTING CANDLES
Starting today, there's a phrase I want you to watch out for in the media. It's something I hear at least weekly, and it always seems to get thrown out there whenever there's a story about a wealthy person giving away a lot of money. Are you ready? The phrase is "giving back." You've heard it before. A wealthy athlete makes a big donation to charity, and the talking heads on TV say something like, "He's been so successful. It's great to see him giving back!" They say it with a smile, but think about what they're saying. My friend Rabbi Daniel Lapin says "giving back" implies that someone _took_ something. If I rob a bank and return some of the money, then yes, I'm giving some of the wealth _back_. But if I apply God's ways of handling money and therefore build wealth because I've proved to be a wise steward, then I can't _give back_. I can only _give_.
## Enough for Everyone
Phrases like _giving back_ in this context reveal a fundamental misunderstanding of the economy. Like I said, _giving back_ implies that I took something. If I'm wealthy, then I got that wealth by _taking it out_ of the economy. So, _giving back_ must imply that I'm _putting wealth back into_ the economy. That's just not how the system works. This is what some people call a "fixed pie" view of economics. That is, there is only so much wealth in the world, and the only way I can get wealthy is by taking money away from someone else. So, the wealthier I become, the poorer someone else becomes because there's only a fixed pie of wealth to go around. But that's a false view of economics. There's not a set amount of money in the world, and the reason the poor are poor is _not_ because the rich are rich. There's more at work here.
Rabbi Lapin explains it as the difference between cake and candles. If we have a cake, there is a fixed, limited amount of cake for all of us to share. We might get a ruler and T-square to carefully measure each slice to make sure everyone gets a totally equal distribution. In that case, everyone would get exactly the same amount of cake, whether they were invited to the party or not. It doesn't matter if you're starving or stuffed, skinny or fat, diabetic or healthy—everyone gets the same size piece of cake. Period. That's what I think of when I hear people talk about "wealth equality," which we discussed earlier in this book.
That's one way of doing it. Here's another: When the cake comes out, it could be a free-for-all. Everyone can get as much as they want and leave little or nothing for the next person. Now I like cake, so I might cut off a huge hunk. If you're a few spots down the line from me, you're probably going to get a little less because I'll be in the corner licking frosting off my fingers. This is the view many in our culture today have of wealthy people. They see them as gorging themselves while everyone else at the party goes hungry. And honestly, that view makes sense if you think there's a set amount of cake—or wealth—to go around.
But Rabbi Lapin shows a problem with that way of thinking: There is no cake. Wealth is more like candles than cake. Have you ever been to a Christmas Eve service at your church? These are sometimes called candlelight services, because at some point, all the lights in the worship center are turned off except for one simple, small candle. I've been in some services where the room goes pitch black. I couldn't see two feet in front of my face. The only thing I could see was the one little flame all the way across the room. And then, that guy lights the candle of the person next to him. And that person lights another candle, and so on. Before long, you see candlelight moving down each aisle and up and down each row. By the time you get to the second verse of "Silent Night," the whole room is lit up by the soft, bright glow of hundreds of candles—and little kids are dripping hot wax on everything.
With that image in your mind, let me ask you a question: What happened to the first guy's candle? Did he lose anything by lighting another candle? The whole room is now burning bright because he shared his light, but he still has what he started with. He could light a million more candles, and it wouldn't hurt his light at all. The same is true with money. When I use money to serve you or when you use money to serve me, neither of us have lost anything. We can both get a lift. And if I take my candle outside and use it to start a bonfire, I'll have a great big flame—but that doesn't take anything away from anyone else. That's how money works. I can keep it, grow it, spend it, or give it all away, and it doesn't impact your money at all. But if we all work together, we can combine our efforts and light up the whole world. As believers who have been given the responsibility of managing God's resources, I think that's our calling.
### CHAPTER NINE
# A Legacy Worth Leaving
My favorite hero from the Civil War era is someone you've never heard of. Clyde Eckles West was seventeen years old when he fought in the war. I won't tell you which side he fought for, because it doesn't matter. There were great men in blue, and there were great men in gray. And Clyde Eckles West was definitely a great man. By the time the war was over, Clyde had had enough of the fighting. A man of faith, he felt a strong call on his life to do something of great value for God's kingdom. As soon as he was released from the army, Clyde put everything he owned, including his precious leather-bound Bible, into two saddlebags and threw them on a mule. Leaving the war behind, he headed south—preaching all the way. For years, Clyde Eckles West took that mule from one town to another, proclaiming the name of Jesus as he worked his way through the South.
Over time, Clyde's preaching turned to teaching, and he started a couple of colleges—one of which is still open to this day. He finally settled down in Maryville, Tennessee. He got married. He had kids. And years later, with a legacy of preaching the name of Jesus, educating young minds, loving his wife, and raising some pretty incredible children, Clyde Eckles West passed away. Now, that story may not sound all that remarkable to you. I'm sure there were thousands of fine young men like Clyde back in those days, and any one of their stories could be as powerful or more powerful than the one I just told. There's one difference, though. There's something that always brings me back to this particular itinerate preacher. There's a reason the story of Clyde Eckles West is so important to me. You see, he was my great-great-grandfather.
If, like me, you have trouble backtracking through all the "great-greats," I'll just say that Clyde was my grandmother's grandfather. Several years ago, not long after my grandmother passed away, some of us kids and grandkids were at her home visiting. While we were there, someone suggested we take a few things to remember her by. Some of my cousins flipped through the photo albums, and others looked through her old jewelry for keepsakes. I'm not much for jewelry, but I am a book guy. I wandered into her home library and looked over the shelves. I've always thought that a person's bookshelves can tell you a lot about who they are and what they believe. Just about the only mementos I have from deceased friends and relatives are some of their old, beloved books.
As I looked over her books, I came across her old Bible. It was beautiful. I flipped through it and saw her handwritten notes and the passages she had underlined. They all pointed to a life well lived. I felt like I was holding on to the most important part of my grandmother. As I sat there with that precious book in my hand, I looked down and saw another leather-bound book that had sat on the shelf beside her Bible. I reached down and pulled it off the shelf, and my mouth fell open. I was holding the Bible of my great-great-grandfather, Clyde Eckles West. By that time in my life, I knew all about Clyde. I had heard the stories and had read his memoirs. I had actually read about this old Bible in his memoirs, how his church had given it to him and how much he treasured it. This was the Bible that he carried in his Civil War saddlebags as he rode from town to town preaching the Good News. I gently flipped through it, and I saw Clyde's handwritten notes and a couple of sermons held together by nineteenth-century paperclips. I felt like Indiana Jones when he found the Holy Grail! I just stood there speechless for a while, and when I left my grandmother's house that day, those two well-worn Bibles were under my arm. Clyde's Bible will always be one of my most treasured, precious possessions.
Flash forward a few years. My son and youngest child, Daniel, was in school at the University of Tennessee in Knoxville. Sharon and I were in town for the football game, and we had some time to kill. Maryville is pretty close to Knoxville, so Sharon suggested we drive over to visit my grandparents' gravesite. I knew where the cemetery was, but I had no idea where their graves were. I called my aunt once we got there, and she directed me to the right spot. While I stood there in front of my grandmother's grave, my aunt told me over the phone, "If you look up and to the right, you'll see some old, huge pine trees. A bunch of the family is buried up there too. You'll recognize some of the names."
By that point, I was curious, so I wandered up to those pine trees and looked around. I spotted a few names I recognized, and then my eyes fell on one old tombstone that looked about 150 years old. I had a feeling in my gut about it, and sure enough, there lay the grave marker of Clyde Eckles West, my great-great-grandfather—the man who rode all over the South with a Bible in his saddlebags, a Bible that now sat in my bookcase. I just soaked it in for a while. This man took God's call seriously. This good man left an inheritance of faith to his children's children. Four generations later, his Bible is still preaching. Clyde never could have imagined that. He never could have known that a century and a half later, he'd have an heir who held his trusty old Bible and considered it a legacy. Clyde wasn't focused on that. He was just focused on leaving a legacy in the best way he knew how. He followed the Lord's directions, but he left the outcome up to God. He put his legacy in God's hands.
As I stood there looking at his headstone, thoughts kept running through my mind about the legacies we leave. Then I remembered the old story that motivational speakers used to tell about the numbers on a headstone. I've heard it all my life. They'd say something like, "When we die, we'll have two numbers on our headstone: the year we were born and the year we die. Those numbers are separated by a little dash. You don't get to decide either of those numbers, but you do get to decide what to do with the dash." The dash represents your life. Everything you have ever done or will do is represented by that little dash on your headstone. And all the great speakers ask the same question: "What are you going to do with your dash?"
As we bring _The Legacy Journey_ to a close, I want to broaden the horizon a little bit. Your dash isn't your legacy. Your real legacy is what happens in your children's dashes and in your children's children's dashes. That's the inheritance Proverbs 13:22 talks about—the impact you have not just on your life, but also on the lives that come after you. The good news is that you get to choose today what your dash will be, and if you've made some mistakes, you get to correct them. Even if you think you've run your life into a brick wall, guess what? Your children have a clean slate. Your mistakes don't have to transfer to them. Your legacy might be setting your kids up to be the first generation to be debt-free, or to be free of several generations' worth of emotional baggage, or to be free of whatever you think has gotten in your own way. You can choose today to put an end to wrong views and wrong behaviors that might have derailed past generations in your family. You can choose to do things differently and safeguard future generations to do things even better. You can do "all things through Christ who strengthens [you]" (Philippians 4:13). You can choose today what kind of legacy you want to leave.
Make it a great one.
# The Pinnacle Point
# Chapter Excerpt from _Dave Ramsey's Complete Guide to Money_
Special Note
The following is a full chapter excerpt from my previous book _Dave Ramsey's Complete Guide to Money_. It provides a detailed explanation of the different kinds of investments I recommend for building wealth. I have inserted a couple of updates to the excerpt below to fill in some additional investing information. You'll see those additions in italics.
For more on the basic how-to issues around taking control of your money and building wealth, be sure to check out my books _The Total Money Makeover_ and _Dave Ramsey's Complete Guide to Money_ , as well as our classes _Financial Peace University_ and _The Legacy Journey_. Find out more at daveramsey.com.
When I was a kid growing up in Tennessee, one of my favorite things to do was to go out with my buddies and ride bicycles. Now, my bike didn't have all the fancy gears and options that you see on bikes today. My bike had one gear. And it didn't have a little engine to help it get moving. All it had was two little stumpy legs to get it moving. And those little legs had to work hard to keep that thing going, because in Tennessee, there aren't many flat stretches of land. You pretty much have two options: up or down.
I remember a million times as a kid, I'd be riding along and all of a sudden the road in front of me would just start going straight up. Sometimes it was too steep to pedal straight up, so I'd start steering right and left, swooping side to side to keep my momentum going. You've been there, right? I'd fight and fight to get up that hill and then . . . there it was. After all that struggling to get to the top of the hill, there was a moment where the hill leveled off for a second just before the downhill ride of my life began.
That's a great place to be, and I'm not just talking about bicycles here. I'm talking about that point in life where all the hard work and struggle is behind you, and all the fun is in front of you. That's the Pinnacle Point, and as fun as it was on my bike as a kid, it's even more fun in my financial life as an adult. In your money, the Pinnacle Point is the place when your savings and investments—after years and years of dedication and hard work—make more money for you in a year than you make for yourself. It's when your investments produce a higher return than your work. That's the best downhill ride you'll ever have.
# THE TIME IS RIGHT NOW
This is one of my favorite topics, and it's one of my favorite lessons in our _Financial Peace University_ class. But every time I start talking about investments or we get to this lesson in FPU, we always hear the same few objections, so let's just go ahead and get them out of the way.
## "Oh, Dave, Investing Is So BORING!"
Every time I teach on investing, I can immediately spot the Nerds and the Free Spirits in the room. The Nerds perk up, pull out their pencils, and start to run the numbers in the margins of their notebooks. The Free Spirits? Well, they go to their happy places. I can see all the Free Spirits in the room start to float out of their bodies. Their eyes glaze over and, all of a sudden, they are running through a wheat field or singing in the rain in their minds. They just totally check out.
So here's my suggestion for those of you who find this stuff boring. When I use the word _investing_ , picture a vacation home in the French countryside. Or picture a ski trip with your whole family in a beautiful chalet that you've rented for a month. Or picture your spouse being home with you to hang out, laugh together, and just enjoy life. That's what investing is all about. It's not about the dollars; it's about the kind of life you want to live later on. What you do today will determine that.
## "I'm on Baby Step 1! I Can't Even Think about Investing Right Now!"
I know you may just be starting this whole process, and that's okay. Wherever you are in the Baby Steps, I'm on your team. But even if investing is a few years off, you need to learn some basics so you'll know what to do next as you knock out the Baby Steps.
I love to ski. In the snow, on the water, I don't care where. I just like moving fast and having two long planks strapped to my feet. If you've ever been on skis, you know the first thing they tell you is that your whole body will go wherever you're looking. If you're looking straight ahead, you'll go straight ahead. If you look to the right, you'll drift right. If you look down, don't forget to tuck and roll, because you're about to hit the ground or water. That's true with your money too. I promise, if you do the things we teach, you're going to get out of debt and save up a full emergency fund faster than you ever thought possible. And when you do, it'll be time to invest for wealth building. So let's get your eyes on that goal, okay?
## "Long-Term Investing Is Too Slow! I Want a Fast Return on My Money!"
The only people who get rich from get-rich-quick schemes are the people selling them. They play on your emotions, set you up for a quick return, take your money, and then leave you high and dry. And the truth is, risky investments have become a playground for people with gambling problems. Hoping to turn $100 into $1,000 overnight isn't investing; it's gambling. You've heard me say before that the stuff I teach isn't always easy and it isn't a quick fix, but it absolutely works every time.
# KISS YOUR INVESTING
Back when I was just starting to sell houses, one of my biggest problems was that I talked too much. Hard to believe, right? More times than I'd like to remember, I just talked and talked and talked, and I ended up talking myself out of a sale. The reason is that I was flooding buyers with information—more information than they wanted or even needed! I went on and on about all these features, contract issues, upgrades, neighborhood stats, and everything else I could think to say. Fortunately, I finally figured out how to shut up, but not before a lot of people missed out on some good houses just because I was overcomplicating the whole process.
Investing can be like that. Sometimes "financial people" come in and start talking about all the options, tricks, and strategies, and it makes our eyes glaze over. As a result, we either sign whatever they put in front of us, letting them make all our financial decisions, or we just decide it's not worth it and we walk away. Either way, we lose.
That's why I always recommend the KISS strategy for investing: "Keep It Simple, Stupid." No, this does not mean that you are stupid if you make simple investments! Just the opposite. I'm saying that people get in trouble when they overcomplicate things. I've met with a lot of really, really rich people over the years—multimillionaires and even several multi _billionaires_ —and most of them have a simple, even _boring_ , investment plan. They do the same few, simple things over and over again, over a long period of time. Why? Because it works.
But a lot of people truly believe that investing has to be complicated, or that there's some trick to it—as if there's one big secret to investing and those people who figure it out get to be rich. But nothing will send you to the poorhouse faster than stupid, long-shot, high-risk investments. It's like that old joke: What's the most common last words for a redneck? "Hey, y'all. Watch this!" In investing, I think the most famous last words would be, "Don't worry. I know what I'm doing!"
## "Financial People"
Too often, we play these games because "financial people" sit across the table and talk down to us like we're children. That drives me crazy! A financial advisor is usually an invaluable part of your team, as long as he remembers what his primary job is: to teach you how to make your own decisions. You need someone with the heart of a teacher who will sit down with you and teach you this stuff so that you can then make your own decisions about how, where, and how much to invest. You should never buy any financial product or service if you can't explain to someone else how it works. That level of education is what you're paying your advisor for!
There are two words you should say to financial and insurance people who talk down to you or won't (or can't) teach you how their products work: _YOU'RE FIRED!_ Remember, these people work for you. If they aren't doing the job you're paying them for, cut them loose. And if you need help, be sure to check out our list of Endorsed Local Providers (ELPs) in your area. We've handpicked excellent men and women all over the country to help you make your own investing decisions. You can learn more about that at daveramsey.com or, if you're in an FPU class, in the online resources for this lesson.
# FANCY TERMS: $10 WORDS FOR $3 CONCEPTS
Whenever I write, one of my guiding principles is that I don't use $10 words or words that sound too highbrow or stuffy. There are a few investing terms, though, that might fall into that category, so I'm going to take a minute to lay them out for you.
## Diversification: Spreading the Love
_Diversification_ is one of those terms that financial people throw around just to sound impressive. Let me take the wind out of their sails by clearing this up. Diversification just means to spread around. It's a really simple idea. Basically, don't bet the family farm on a one-horse race. This is a financial principle you may have learned in Sunday school, even if you didn't realize it at the time. Ecclesiastes 11:2 says, "Give portions to seven, yes to eight, for you do not know what disaster may come upon the land" (NIV).
Grandma said it too, didn't she? She always said, "Don't put all your eggs in one basket." The problem is, if you put all your eggs in one basket, something bad might happen to the basket. If it does, then you lose all your eggs. In recent years, we've had a lot of bad things happen to a lot of baskets. We've seen 9/11, Katrina, political drama, massive unemployment, and even a full-blown recession slam into a lot of people's single baskets, breaking a lot of eggs. But those who spread out their investments over several different options were better protected. If you spread them out over a wide area, you won't lose the whole thing when something goes wrong in one part of it.
The bottom line is that diversification lowers risk. So basically, what we're saying is that investments are like manure. Left in one pile, it starts to stink. But when you spread it around, it grows things. I bet your financial guy never laid it out like that!
## Risk-Return Ratio
With virtually all investments, as the risk goes up, so does the hopeful return. That is, if I don't take much of a risk, I'm not going to make as much money. All investing requires some degree of risk; there really is no sure thing. In my book _Financial_ _Peace Revisited_ , I explain it this way:
> The lion at the zoo is a pitiful sight—the king of beasts is eating processed food. You can see deep down in his soulful eyes that he misses the thrill of the hunt. Any of you who want a guarantee on your money need to understand that you are paying the same price as the lion.
The only real method to totally guarantee you won't lose your money is to put it in a cookie jar, but your return will be equal to your risk: zero. Actually, the cookie jar can't even offer a foolproof guarantee if your house is robbed or burns down. Besides, earning zero return is the same as moving backward once you factor in inflation, which we'll cover in a minute.
When you lay out investment options, you start to see a progression of risk. You start with the cookie jar—no risk, no return. One step up from that is a savings account, which is fantastic for your emergency fund but a joke for your investing. Your money will be fairly safe, but you'll be lucky to make 2 percent on it. A step up from that would be a Certificate of Deposit (CD), which is not much better than a savings account. A few more steps leads you to a mutual fund. A little more risk gets into single stocks. A few steps later and you are into day trading, where, risk-wise, you completely jump off the cliff.
I don't recommend single stocks or day trading, or Vegas for that matter, because the risk is just too high. But I also don't recommend cookie jars or CDs for long-term (more than five years) investing. The sweet spot, which we'll talk about later, is mutual funds. That's a great balance of reasonable risk and excellent returns.
## Inflation
Something that should be handled alongside risk-return ratio is inflation. This is something that is too often left out of risk calculations. When we talked about the cookie jar, we said that it was basically no risk, no return. But that's not really true. With inflation working against us, if we left $100 in a cookie jar for a year, we'd still have $100 a year later, but that $100 would be worth less.
Inflation has averaged around 4.2 percent over the last seventy years, according to the Consumer Price Index (CPI). So if your money is not earning at least a 4.2 percent return, you're actually losing money every year. In fact, once you factor in taxes on your growth, you really need your investments to make around 6 percent just to stay ahead of inflation. So if you stick with cookie jars, savings accounts, and CDs as your long-term investing strategy, your money will be relatively safe but inflation will tackle you from behind. You've got to see 6 percent as your break-even point, so a little risk is going to be vital to your long-term plan.
## Liquidity
_Liquidity_ is a funny word, but, again, the concept is simple. It just means availability. If you have a liquid investment, then you have quick and easy access to your money. The cookie jar is totally liquid because you can walk over and get your cash out whenever you want. A savings account is about the same, and a CD is fairly liquid even though there's a short time frame attached. Pretty much the least liquid investment is one that a lot of Americans own, which is real estate. If you're a homeowner, your house is likely your greatest investment, but it is not liquid at all. If an emergency came up, you'd have a tough time liquidating your house by tomorrow.
# TYPES OF INVESTING
We can already see the ground rules for investing. We're going to keep it simple. We're going to find investments that show a good, reliable balance between risk and return. We're going to listen to advisors but make our own decisions. We're going to stay away from gimmicks and get-rich-quick schemes. We're going to keep our investments diversified. And, of course, we're going to stay ahead of inflation. Before we do any of that, though, we're going to make sure it's time for us to start investing. Investing is Baby Step 4, so before you start saving up for retirement, you are debt-free except for your house, and you have three to six months of expenses saved up in an emergency fund. If you've done that, then it's time to start investing. So let's look at some of the most common types of investing, or investing "vehicles."
## CD: The Certificate of Depression
A CD is a certificate of deposit, typically at a bank. This is just a savings account. That's it. I always crack up when people tell me they have solid investment strategy, and then tell me all their money is in CDs. It's a _savings account_. The word _certificate_ does not make it sophisticated; the certificate is basically a receipt showing that you made a deposit at a bank. Whoopee! My kids had savings accounts when they were six years old! That's barely a step up from a piggy bank!
A CD will give you a higher rate of return than a standard savings account because you'll be required to leave your money alone until the CD matures. You may get a five-year CD, which means you deposit the money for a guaranteed rate of return—and you can't take the money back out until it matures at the end of the five years. If you do, you pay all sorts of fees and penalties. Even without the fees, though, the CD gives you a lousy rate of return. At the end of the day, a CD requires you to tie up your money for a few years, but then it gives you practically nothing in exchange. I don't own a single CD I just don't see a need for them.
## Money Markets
So what if you want to get _some_ return on a pile of money you don't need _today_ , but you know you'll need in a few years? This is like when you're saving up for a house or a car over three to five years. I use money market accounts for that. This is essentially a checking account that you open up with a mutual fund company. These are low-risk, and they offer about the same rates as you'd get with a six-month CD. However, your money isn't tied up and you have check-writing privileges (with no penalty) in case you need to access the account sooner than you thought. Just keep in mind that money market accounts are for _savings_ ; this is not an investment. That's why money markets with a mutual fund company make a great place to put your emergency fund. It keeps the money liquid while still giving you at least a little return.
## Single Stocks
Single stock purchases give you a tiny piece of a company. The company issues a number of shares to sell to shareholders, and those shareholders jointly "own" the company. That's what it means when a company "goes public." They go from being privately held to being publicly owned through the issuance of publicly traded stock. The value of the shares is tied to the value of the company. If the company's value skyrockets, the value of each share—each individual piece of ownership—also goes up. That's good if you get in and out at the right time.
For example, let's say you were an Apple® fan in the early 1990s. The technology company was having some trouble back then, and the stock hit the low twenties per share around 1993. But for some reason, you had a good feeling about where the company was going, so you bought one thousand shares at $23 each. By 2011, that $23,000 investment would have been worth somewhere around $350,000. Not bad, right?
But here's the reality: For every Apple, IBM®, or Google®, there are hundreds of publicly traded companies in bankruptcy, and it is impossible to know what the future holds for individual businesses. Remember Enron®? The collapse of that one company completely obliterated $74 billion of wealth in the four years leading up to Enron's eventual collapse. More than twenty thousand former employees were thrown into years and years of legal battles and lawsuits just to get back a piece of the money they lost in company stock.
Why? Because they had put all their eggs in one basket, and that basket fell apart. The same could be true for any business, at any point in time. That's why diversification is so important, and single stocks are one of the most UN-diversified investments you can make. Stay away!
## Bonds
A bond is a debt instrument by which a company owes you money. Instead of buying a piece of ownership, as with a stock, you're pretty much loaning a company (or the government, in the case of government bonds) some money. That means instead of becoming an owner, you become a creditor. Like I said in _Financial Peace Revisited_ :
> When you purchase a bond, the company that issued it becomes your debtor. The income is usually fixed, but again, the value or price of the bond will go up or down according to the performance of the company and prevailing interest rates.
I personally do not like bonds for several reasons. First, it's based on debt, and it's no secret what I think about debt—borrowing _or_ lending. Second, bonds are high risk because the company's ability to repay your investment is tied to their performance. So in that sense, it's like a single stock and has no diversification. And last, the performance of bond-based portfolios is generally pretty weak. This is another one I just stay away from.
## Mutual Funds: The Alphabet Soup of Investing
Now we come to one of my favorite vehicles for long-term investing, the mutual fund. I _love_ mutual funds. They have excellent returns, and they have built-in diversification that keeps me from having all my eggs in one basket. The problem is, a lot of people are scared off because they don't understand what a mutual fund is. Heck, I graduated college with a finance degree and I still had trouble understanding it! It's really not that complicated, though, once you strip out all the highbrow financial lingo. Let's take a look.
Picture a big bowl in the center of a table. You and your ten best friends are sitting around the table. Everyone puts a dollar in the bowl. That bowl is a mutual fund. You and your friends have all contributed, so you have _mutually funded_ the bowl. It is a _mutual_ fund. Get it?
So, what's in the bowl? It contains little pieces of stock in a whole bunch of companies. Picture it like a bowl of alphabet soup. If you look in there, you might see an "I" floating around, which could be IBM. You could see a "W," which might be Walmart®. There's an "A" and an "M," so that could be Apple and Microsoft®. Just imagine several companies floating around together in the bowl you and your friends funded.
A professional portfolio manager manages the fund, making sure that only the best investments are in the bowl. And this guy isn't flying solo. He's got a huge team of Nerds working for him—the best, brightest, nerdiest Nerds in the world! If the fund includes tech stocks, he'll have a Tech Nerd. If it includes restaurants, he'll have a Restaurant Nerd. These specialists spend all day, every day, learning every detail about these companies and industries. They pass that information to the fund manager, and the manager uses it to keep the fund filled with the best of the best investments. That crack team of Nerds can do a lot better job than any average bubba sitting at home picking stocks using a dartboard and dumb luck!
What the fund does depends on the goal, or the _fund objective_. If our fund is a growth stock fund, the fund manager will buy growth stocks. If it's a bond fund, the manager will buy bonds. If it's an international stock fund, the manager will buy—can you guess?—international stocks. You're getting it!
Diversification is fantastic in mutual funds. Let's say I wanted to invest in some good ol' American companies. If I were using single stocks, I could put $20,000 in Ford®, for example. But what happens if Ford implodes? I lose all my money! We talked about that with single stocks. That's just too much risk. But instead of going all-in with Ford, I could get a mutual fund that has a little Ford in it, along with up to two hundred other great American companies. So if Ford's value goes way up, I still benefit because I have a little Ford in my mutual fund. But if Ford goes bankrupt, I won't feel that much of a loss because it's just one small part of my fund. The other two hundred companies in the fund can protect me from a loss because they're all in there together.
Now, is there some risk involved with mutual funds? Sure, there are no guarantees. But remember what we said about inflation? If you just park your money in a "safe" CD, you're already behind the curve because inflation will take your legs out from under you. Besides, if you are invested in mutual funds containing the best and brightest two hundred companies in the country and they _all_ fail at the same time, you've got bigger problems than your mutual fund. That would mean the entire US economy has fallen apart, the stock market would be worth zero, and the FDIC would have collapsed—so your "safe" bank savings would be worthless too!
## Mutual Fund Diversification
Like I said, there are different kinds of mutual funds, and they all have built-in diversification. But I still recommend you diversify a little further by spreading your investments out over four different kinds of mutual funds. I tell people to put 25 percent in each of these four types: growth, growth and income, aggressive growth, and international.
Growth stock mutual funds are sometimes called mid-cap or equity funds. _Mid-cap_ refers to the fund's capitalization, or money. So, a mid-cap fund is a medium-sized company. These are companies that are still in the growth stage; that's why it's called a _growth_ stock fund.
Growth and income mutual funds are the calmest funds of the bunch. These are sometimes called large-cap funds because they include large, well-established companies. These funds usually don't have wildly fluctuating values. That's good and bad; they won't shoot up as much when the market's up, but they also won't fall as much when the market's down. These are basically slow-moving, lumbering dinosaurs.
Aggressive growth mutual funds are the exciting wild child of mutual funds. They represent small companies (so they're often called small-cap funds), and these are active, emerging, exciting companies. This is the roller coaster of mutual funds. There will be really high highs, and probably some really low lows. I had one back in the 1990s that had a 105 percent rate of return one year, then lost it all the next year. You absolutely don't want to put all your money here, but you need some aggressive funds in your plan.
International mutual funds are sometimes called overseas funds, and they represent companies outside the United States. I recommend putting a fourth of your investments in international funds for two reasons. First, you get to participate in the growth of some foreign products that you probably already enjoy. And second, it adds another layer of diversification just in case something weird and unexpected happens to the US stock market.
## Picking Mutual Funds
Always look at the track record of mutual funds before you buy one. And make sure that it has a good track record over at least five years, preferably ten or more. My favorites are the ones that have been around more than twenty years and have proved themselves to be quality, reliable investments. I have some mutual funds that are more than fifty years old! Those may be harder to find, but they give you an excellent track record. Whatever you do, don't buy a fund that's less than five years old. These are babies! If you can't see a track record of at least five years, keep looking. If it's a fund that really interests you for some reason, just make a note of it and check it a few years later.
I personally like to find funds with a good track record averaging at least 12 percent. But every time I say that, I get a million emails from people saying, "BUT DAVE, you can't get 12 percent on your investments! Are you crazy?" No, I'm not crazy. I use 12 percent because that's the historical average annual return of the S&P 500®, which gauges the performance of the five hundred largest, most stable companies in the Stock Exchange. The average annual return from 1926, the year of the S&P's inception, through 2010 is 11.84 percent. Just keep in mind that's the eighty-year _average_.
Sure, within that time frame there are up years and some down years. I'm not that interested in the performance of any individual years or even any short-term timespans, but just for fun, let's take a look at a few. From 1991–2010, the S&P's average was 10.66 percent. From 1986–2010, it was 11.28 percent. In 2009, the market's annual return was 26.46 percent. In 2010, it was 8 percent. See, this thing is up and down all the time, so 12 percent isn't really a magic number. But based on the history of the market, it's a reasonable expectation for your long-term investments.
Bottom line: Mutual funds make excellent long-term investments, but don't bother with them unless you can leave that money alone for _at least_ five years. This is where you park your money for the long haul, looking toward retirement.
## Update: Betas, Loads, and Low-Turnover Mutual Funds
_Since we're getting our hands dirty with serious investing at this point in your legacy journey, let's look at some of the nerdier parts of mutual fund investing that I don't often dive into. We'll quickly cover the beta, load versus no-load, and low-turnover mutual funds._
_The beta is the statistical measure of risk for a mutual fund. A beta of 1.0 mirrors the market. That means a fund with a 1.0 beta is exactly in line with the ups and downs of the stock market. Anything over a 1.0 is a little wilder and less predictable than the_ _market average, and anything under a 1.0 is calmer than the market average. So, something like a growth and income fund, which is the slow-moving, more predictable fund, might have a beta of 0.8,_ _meaning it's fairly safe. But an aggressive growth stock fund with a beta of 2.0 is really wild; it has twice as much risk as the market average. Those are fun and they can generate great returns, but you don't want your whole portfolio to be a roller-coaster ride. Remember the risk-return ratio and keep your funds balanced among the four types we've discussed._
_Now let's deal with load versus no-load mutual funds. A load simply means a commission is charged when you purchase a fund. A loaded fund has a commission built in, and a no-load fund does_ _not. That's the difference. Now, you may think it's better to avoid the commission, but that's not always true. All funds—both load_ _and no-load—have maintenance fees and other expenses attached. Sometimes, you can buy a loaded fund—which is more expensive on the front end—that has really low fees. So, over the course of many years you end up saving money because you aren't getting hit as hard with fees every year. That's not always true, but it's been the case with several of my investments._
_So which is better: load or no-load? There's no clear answer. The_ _truth is, I own both. The trick is to dig into the details and explore each fund on its own without making a blanket assumption that no-load is the better deal. That's where a quality broker comes in. Just be sure to get someone with the heart of a teacher who can give you all the information you need to make your own decision._
_The last type of fund I want to talk about is the low-turnover_ _mutual fund. That's a fund that holds almost all the stock it purchases. It almost never sells the stocks inside the fund, which is why it's called "low turnover." Let me explain why this is helpful. Let's say you paid $200,000 for a rental house, and the value of_ _that property increases to $300,000 over time. As long as you hold_ _the property, you don't have any taxes on that increase. Taxes aren't due until you sell it and realize that gain, which results in_ _capital gains tax. So in a sense, you have tax-deferred growth on that_ _property, right?_
_The same is true if you buy a single stock that goes up in value. If you buy a stock for $100 that goes up to $300 over twenty years, you don't pay any taxes on that gain until you sell. I don't recommend you buy single stocks, but remember that mutual funds are made up of a whole bunch of stocks. If it's a low-turnover fund, the stocks inside the fund are basically sitting there for a long time. They aren't being sold out of the fund, which means I'm not paying taxes on the gains as the stocks leave the mutual fund. So that's another way to help defer capital gains taxes on my investments. I love these kinds of funds and use them a lot._
## Rental Real Estate
Real estate can be a lot of fun, especially for old real estate guys like me. However, this is the least liquid investment you can make. You know what they call houses that sell fast? Cheap. Don't mess around with real estate until you are out of debt, have a full emergency fund, have maxed out your 401(k) and Roth IRA options, have paid off your own house, and have some wealth built up. Only _then_ are you ready.
And, of course, don't even think about real estate as an investment until you can pay cash for the property. Never, never, _never_ borrow money for an "investment." The risk is enormous. I've seen literally thousands of so-called investors lose their shirts in real estate because they bought houses when they were broke. If you start playing with rental real estate without any money, you will crash. I promise. That is _exactly_ how I went broke and ended up in bankruptcy court myself. I know what I'm talking about here.
_Update: Let me jump in with a few more details here. Since_ _rental real estate is one of only two types of investments that I do (the other one is mutual funds), people often ask me for tips as they consider getting into real estate themselves. So, I came up with my Four Rules for Buying Rental Real Estate:_
1. Buy Slowly and Pay Cash. _Like I already said, never use debt to finance an investment. That's a surefire way to lose your shirt in real estate. Move slowly, make wise decisions, and never get rushed into a surprise "deal of a lifetime." Do due diligence with every property, and always have a lot of cash on hand after the purchase. You'll need it for whatever rehab you plan to do to get the property ready to rent or sell._
2. You Make Your Money at the Buy. _Don't talk to me about how this neighborhood is going to skyrocket in value, so you're going to overpay for a property today. If you're going to buy a piece of property, you need to make money on it the minute you close. You want to buy it so cheap that it's an instant win._
3. Pay No More than 70–80 Percent Market Value. _This ensures you get a bargain and you make money the instant you buy it. Never even consider anything close to retail when buying investment properties. If I find something I like that we estimate to be worth $200,000 market value, I'll offer $140,000—and I'll pay in cash, and we can close this week. Do a lot of people say no? Yes, but I don't care. There are plenty of others who say yes._
4. Only Buy Properties in Your Area. _Don't get into the long-distance landlord game. It's a nightmare. I own a lot of properties, and every one of them is an easy drive from my house. If you own it, you need to be able to drive by it whenever you want. Otherwise, you could have someone changing the oil of their Harley in the living room of a house you own five hundred miles away._
## Annuities
I personally don't use annuities very much, but they are options and they do have a place in some people's investing strategy, so let's take a quick look. There are two types of annuities: fixed and variable. Fixed annuities are terrible. They're basically a savings account with an insurance company, and they pay somewhere around CD rates. They're really not that different from a CD you'd get from your local bank.
Variable annuities are the only ones I like. These are essentially mutual funds inside an annuity. The annuity provides some protection against taxes for the mutual funds inside, so if you've already maxed out your other tax-favored plans like a 401(k) and Roth IRA, then a variable annuity might make sense. There are fees involved, but in exchange for the fees, you don't have to worry about taxes on the investment. Plus, some variable annuities offer a guarantee on your principal. So if you put $100,000 in the investment and the value drops below that level, you'll still be able to get your $100,000 back out of it. They're not for everyone, but if you're further along in your investing, you may want to look into some quality variable annuities. Just don't buy an annuity with an investment that is _already_ tax-protected. If the investment is already safe from taxes, the added fees of an annuity just don't make sense.
## All That Glitters Is NOT Gold
I just might be the only talk radio host in the country that does not endorse gold! The reason is simple: Gold is a horrible investment! Here's the deal: Gold value has been rising since September 11, 2001. People started buying it more not because it's a good investment, but because of some false belief that if the economy totally collapses, gold will hold its value. If that were true, then gold coins would have become the dominant currency in New Orleans during the Katrina disaster! That's a great picture of a mini-economic breakdown. If you had been there at the time, I bet you would have gotten a lot more for a bottle of water than for a gold coin!
But let's look at the numbers. Gold was worth $21 an ounce in 1833, up to $275 per ounce by 2001, and then shot up to about $1,345 by 2010. Even including the crazy growth gold experienced in the first decade of this century, the 177-year track record for gold is just 2.38 percent. Would you look at a mutual fund with a 2 percent rate of return over the past 177 years? No way! Besides, if gold was _ever_ a good investment, it doesn't make much sense to buy it at its 177-year high! You always want to buy low and sell high. Gold gives you the chance to do the opposite. That's a bad plan.
# THE TORTOISE WINS
I have been blessed throughout my career to have had the chance to sit down and talk with a lot of wildly successful, super-wealthy men and women. I love those opportunities, and I always pull out a pen and some paper because I want to hear from them how they got where they are. I believe you'll stop growing the instant you stop learning, so I'm always looking for some new insight.
One day, I was talking to a really rich guy—I mean _billionaire_ rich. We were having lunch, and I gave him my standard billionaire question: "What can I do today that will get me closer to where you are in your business and in your wealth building?"
He leaned back and said, "Okay, here are two things. First, I've never met anyone who wins at money who doesn't give generously. You've got to keep a giving spirit if you want to win long term." Hey, that's no problem. Giving is one of my favorite things in the world! Check.
"Second," he said, "I want you to read a book. This is my favorite book. I read it several times a year. I read it to my children, and now I read it to my grandchildren over and over. It will change your life, your money, and your business forever." Now I'm a huge reader, so I'm pretty excited at this point. This megabillionaire is about to tell me the book that changed his life. Let's go!
"Dave, have you ever read _The Tortoise and the Hare_?" Huh? A children's book? An old fable? What does this have to do with building wealth? I just sat there for a second trying to decide if he was kidding or not. He wasn't.
He leaned in and said, "Dave, we live in a world full of hares. Everyone's racing around doing all kinds of crazy stuff. They're running ahead and falling back, running ahead and falling back. They're going back and forth, side to side, and all in circles. But the tortoise just keeps moving forward, slow and steady. And you know what? Every time I read the book, the tortoise wins."
That phrase has been stuck in my head for years: "Every time I read the book, the tortoise wins." It's not about fancy options. It's not about jumping on every new and exciting investment that comes down the pike. True, reliable, long-term wealth building is surprisingly simple—and even a little boring. It's a matter of doing just a couple of things over and over and over again, over a long period of time. Eventually, over time, when it counts . . . the tortoise wins.
# The Road to Awesome
Special Note
_Like we discussed in Chapter 5, "Your Work Matters," how you spend your time at work over the course of your life plays a huge part in the legacy you leave behind. To help you learn how to grow in your passions and turn your career into a calling, here's a brief overview of the great book_ Start _by Jon Acuff._
At some point, you might have stopped dreaming. If so, fear pressured you into a dull job with a steady paycheck, and before you knew it, you were stuck in average.
Get unstuck.
How? The best way to eat an elephant is one bite at a time, and the best way to move toward your life's calling is one steady step at a time. So focus on finding work that matters, and then shift into living out your dream with passion and purpose. This is called your Road to Awesome.
It won't be an easy trek, and you'll have to learn, edit, and master as you go, but pretty soon you'll be enjoying a successful harvest and guiding others along their own awesome roads. Changing the world starts with changing you. So get ready—your legacy awaits.
## Age Doesn't Matter Anymore
We're going to talk about five stages that every successful life goes through, but before I lay those out for you, I want to give you a word of warning: It doesn't matter how old you are today. As recently as a generation or two ago, your age was incredibly important. People got out of school, took a job at a big company, worked there for forty years, and then retired. Back then, your career path followed your age almost in lockstep. In your twenties, you were a rookie still trying to figure things out. In your thirties, you were getting better and making choices about what you liked about your job and what you didn't. In your forties, you relaxed a little bit because you knew how things worked and you might have even gotten a promotion. In your fifties, all the hard work of the past couple of decades started paying off and you probably had the best income of your life. Then in your sixties, your career started winding down, you spent more time helping other people do their jobs, and ultimately you left the building with a pension and a gold watch. The end.
I've got good and bad news. The bad news is, those days are long gone. And the good news is, those days are long gone! The problem back then was that you had to figure things out in your twenties. If you didn't figure out your calling until you were fifty, the cards were stacked against you. It was hard to work your way into a job that late in the game. Not anymore! Nowadays, age is out the window and you can jump on the train at any point. Wherever you are in life, right now is the perfect time to start pursuing your passion and turn your career into a calling.
I can give you three reasons why age doesn't matter anymore. First, traditional retirement is dead. There's a whole generation of Baby Boomers who are realizing the finish lines they were promised simply don't exist anymore—and they never will again. People in their forties and fifties are getting laid off unexpectedly and having to start over again in the middle of their career. Others are a decade into one career and figuring out that they hate what they do all day. Twenty more years of that feels like a death sentence! So they're starting over too. The old finish lines are gone!
Second—and this is the really cool part—anyone can play now. The technological advancements of the past twenty years or so have done away with all the old gatekeepers that used to stand in the way of people chasing their dreams. If you want to write a book today, you don't need a publisher or professional marketing company. You can do it all yourself and promote it to millions of potential readers through social media. If you're interested in mobile app development, you can get all the training you need (mostly for free) online, build an app, and sell it directly to customers through Apple's and Google's app stores. Think about your favorite app or website. Chances are, you have no idea if the person who created it is sixteen or sixty. Even better, you don't care! You're just looking for a quality product, and you're more than happy to give your time, attention, and dollars to someone who can meet that need. That's a perfect environment for an entrepreneur with passion!
Third, hope is the new currency for people chasing their passions today. That sounds a little weird, so let me explain. In the old system, someone may work forty years while nursing the dream to give their time and money to a good cause _someday_. They thought, _Once I retire, I'll be able to_ . . . But here's the thing: We don't have to wait until _someday_ anymore. The current generation doesn't want to change the world _eventually_ ; they want to change it _right now_. So they're leveraging their skills and talents along with today's resources to add their desire to serve into the work they're already doing. That's how you end up with a national shoe manufacturer who gives a free pair of shoes away for every pair they sell or a local coffee shop that gives a certain percentage of every sale to charity. It's amazing! But that's the kind of stuff you can do when you realize you're no longer part of an outdated system that says you have to go to school, then go to work, then retire, and then die. That's just not who we are anymore.
## You Can't Skip Stages
The five stages are not based on age. You can follow this path at any time, no matter where you're starting from. But there is one catch: You can't skip the stages. It doesn't matter what you've done in the past; once you decide to start (or start over), you're going to walk through each of these five stages one at a time. Remember when Michael Jordan decided to be a baseball player? He was the best basketball player on earth; he was a master of that game. Did that mean he could jump right to the mastery stage of another sport? Uh . . . no. He spent some time with a minor-league team, performed horribly, and never got called up to the majors. The dues he had paid in basketball didn't transfer over to baseball. He couldn't skip ahead. Neither can you. The good news, though, is that these aren't decade-long stages anymore. You can't skip them, but you can shorten them.
# NAVIGATING THE FIVE LANDS
So let's look at the five stages every awesome life goes through. Remember, age doesn't matter, but you can't skip the stages. You have to start where you are today. None of this is going to happen by accident either. Use these five stages as a map as you plan out the career side of your legacy journey.
## Stage 1: The Land of Learning
Learning is where you begin again. Regardless of your age, your work experience, or the overwhelming odds against you, the Road to Awesome starts with rediscovering your God-given passions.
What do you love doing _?_ Building decks? Caring for kids? Writing a blog? No book can spell out your life's passion. But that's okay, because it's already inside you. Just play around with the dreams you already have, and see what sticks. Experiment like crazy. Don't just try to _find_ your purpose; _live_ with purpose. Make your passions count whether you ever turn them into a career or not. God put those things inside you for a reason!
Finding your new beginning will also take time. Time, like purpose, doesn't just magically appear. It will take getting up early and going to bed late. It will take working harder than you've ever worked before. And it will take a ton of follow-through and a lot of frustrating firsts. So give yourself some grace. Too often, we think we need to have the finish line in place before we ever take the first step, but that's a trap. Stephen Covey says we should "begin with the end in _mind_ ," but sometimes we misinterpret that to mean "begin with the end in _stone_." You have no control over the finish line. The only line you can control is the starting line; that one is entirely up to you.
Start experimenting in this stage. Throw some stuff on the wall and see what sticks. Set some goals and start working toward them. But if you can't finish one thousand words per day on your sci-fi novel or attend five baking workshops a month, don't sweat it. This is just a time of exploration. Pat yourself on the back for what you got done and let the undone stuff slide. Pick it back up tomorrow, and most importantly—keep learning!
## Stage 2: The Land of Editing
Editing is all about subtracting distractions. You're not adding new things to your dream here; you're removing things that are almost-but-not-quite the right things. This is where you start to separate the good from the great. This freaks a lot of people out, so let's make it simple. If you died today, what would you regret _not_ doing? Spending more time with your family? Running a marathon? Opening your own coffee shop? These are your passions.
Next, ask yourself what might be the most frightening question of all: _Are these the things I'm doing now?_ You see, we have a way of putting our dreams on a shelf and looking at them like museum exhibits, but we never actually do what it takes to make those dreams a reality. If you'd regret not spending more time with your kids, then here's an idea: Start spending more time with your kids _today_. You can't flip a switch and suddenly find yourself living in your passions, but you can (and should) start making the daily decisions that will ultimately lead you into your calling. That means you must make choices every day about whether to spend your time doing _this thing_ that takes you further from your passion or _that thing_ that brings you closer to your passion. That's what you do in the Land of Editing.
## Stage 3: The Land of Mastering
You've found your passion. That's great, but don't quit your day job. First, you need to gain loads of experience—preferably without bleeding your bank accounts dry or cashing in your 401(k). It's hard to live with purpose if you can't pay your bills.
In his book _Outliers_ , Malcolm Gladwell asserts that it takes ten thousand hours of hands-on experience to become a master of anything. That's a lot of hours, but it goes by fast. At a forty-hour-per-week pace, you'd hit the ten-thousand-hour mark in less than five years. Think about where you were five years ago. It probably doesn't seem that long ago. And the older you get, the faster it goes by. Why not spend that time gaining experience at what you're passionate about? You could be a full-fledged, world-class expert at it in just five years!
I'm not saying you should quit your day job tomorrow and spend the next three months sitting on your sofa dreaming. That's a mistake way too many people make. There are safer and easier ways to log some experience hours. Can you say "part-time job"? If your dream is to run a local coffee shop, maybe you should spend a few hours a week behind the counter at Starbucks just to make sure you like it. If you're going to hate it, let's hate it at that level, not after you've walked away from the safety net of a full-time job.
You could also volunteer somewhere that gets you closer to your passion, and you should spend time with amazing mentors. In all of this, your goal is to really get to know your dream and then get wildly, ridiculously good at it. This takes time. The fastest way to become an expert at your new calling is to do so slowly—by acquiring insane amounts of experience.
## Stage 4: The Land of Harvesting
You're killing it. And you're making money! Sink your toes in the sand and sip from a coconut.
Not so fast. Farmers don't go on vacation during the harvest. Neither should you. This is the time to build upon your success and surround yourself with a circle of support to keep you grounded. Your spouse (or a close friend or family member if you're not married) is your inner circle of support. This person keeps your big head deflated and your bruised ego encouraged. Don't be afraid to lean on your rock.
Rely on your outer circle of support—friends and family—for celebrating milestones and learning from failures. And don't forget to support their dreams too. Fear fears community. So scare the tar out of it.
## Stage 5: The Land of Guiding
Welcome to your legacy. It's time to guide.
You may not feel ready to guide others yet. That's okay; do it anyway. Guiding someone is as easy as starting a conversation and following through with your time. Mentoring doesn't have to be hard; it just has to be intentional. So mold the relationship to your comfort level, and tweak as you go. The goal here is to help someone else on their own road to awesome. You don't have to have all the answers, and you don't have to be the wise old man on the hill. Just be you—but be the version of you who's done what it takes to win and now has some wisdom to share. Remember how much it meant to you when someone a few steps ahead of you slowed down to walk with you for a while? Now it's your turn.
# THE END OF THE ROAD?
Congratulations! Now you're officially awesome, and that's exactly what God wants for you. We shouldn't settle for "ordinary" because we don't serve an ordinary God. He didn't create you to be average; He created you to be awesome! But remember that the end is not really the end. It's easy to get stuck in your own success and accidentally stop growing in the process. Find something new to spark your interest, go back to the beginning if necessary, and keep on learning!
# Notes
Chapter One
1. Reference to Joel 2:25: "So I will restore to you the years that the swarming locust has eaten."
Chapter Two
1. Albert Mohler, "The scandal of biblical illiteracy: it's our problem," Christianity.com accessed April 17, 2014, <http://www.christianity.com/1270946/.>
2. Ibid.
3. Morgan Housel, "Attention, protestors: you're probably part of the 1%," _Yahoo! News_ , October 28, 2011, <http://news.yahoo.com/attention-protestors-youre-probably-part-1-153806044.html>.
4. Courtney Blair, "Compared to the rest of the world Americans are all the 1%," _Policy.Mic_ , December 6, 2011, <http://policymic.com/articles/2636/compared-to-the-rest-of-the-world-american-are-all-the-1>.
Chapter Four
1. Daniel Lapin, _Thou Shall Prosper_ (Hoboken, NJ: John Wiley & Sons, 2002) 150.
2. Robert Morris, _The Blessed Life_ (Ventura, CA: Regal Books, 2002) 179.
Chapter Five
1. Earl Nightingale, _The Strangest Secret_ , Keys Publishing, Inc., 1956, compact disc.
2. Robin S. Sharma, _The Monk Who Sold His Ferrari: A Fable about Fulfilling Your Dreams and Reaching Your Destiny_ (Fort, Mumbai: Jaico Publishing House, 2003).
Chapter Seven
1. Dave Ramsey, _Dave Ramsey's_ _Complete Guide to Money_ (Brentwood, TN: Lampo Press, 2011) 168–169.
Chapter Eight
1. Dave Ramsey, _Dave Ramsey's Complete Guide to Money_ (Brentwood, TN: Lampo Press, 2011) 312.
2. Dave Ramsey, _More than Enough_ (New York: Penguin Books, 1999) 269–270.
The Pinnacle Point
1. Dave Ramsey, _Dave Ramsey's_ _Complete Guide to Money_ (Brentwood, TN: Lampo Press, 2011) 195–216.
2. Dave Ramsey, _Financial Peace Revisited_ (New York: Viking Penguin, 2003) 146.
3. Kathy Rebello, Peter Burrows, and Ira Sager, "The fall of an American icon," _Business Week_ , February 5, 1996, <http://www.businessweek.com/1996/06/b34611.htm>.
4. Kris Axtman, "How Enron awards do, or don't, trickle down," _The Christian Science Monitor_ , June 20, 2005, http://www.csmonitor.com/2005/0620/p02s01-usju.html.
5. Dave Ramsey, _Financial Peace Revisited_ (New York: Viking Penguin, 2003) 129.
_Notes_
# Praise for This Book
I love this book. I am making it required reading in the Lucado family. It is powerful, practical, and most of all, biblical! These pages will go a long way toward keeping people out of guilt, debt, and bondage. Great work, Dave!
MAX LUCADO
Best-selling author of _Before Amen: The Power of a Simple Prayer_
_The Legacy Journey_ is truly one of the most important books for our culture today. Get it. Read it. And live it. When you do, you'll be equipped, inspired, and compelled to live beyond today and leave a legacy that will bless generations to come.
CRAIG GROESCHEL
Pastor of LifeChurch.tv and author of _From This Day Forward: Five Commitments to Fail-Proof Your Marriage_
In _The Legacy Journey_ , Dave clears up the popular myth that wealth is evil and explains that as we take the journey toward wealth, it is important not to feel guilty about success, but to rejoice in all that God can do through a lifestyle of generosity.
ROBERT MORRIS
Founding senior pastor of Gateway Church and best-selling author of _The Blessed Life_ , _From Dream to Destiny_ , and _The God I Never Knew_
"It's about time!" That was my first thought when I heard that Dave Ramsey was writing this book. He handles every "yeah, but what about" with both common sense and biblical insight—and, of course, a good dose of unapologetic boldness! _The Legacy Journey_ is one worth embarking on, and this book is your GPS.
STEVEN FURTICK
Founding lead pastor of Elevation Church and _New York Times_ best-selling author of _Crash the Chatterbox_ , _Greater_ , and _Sun Stand Still_
As someone who was once incredibly overwhelmed with debt, I cannot begin to tell you how learning from Dave has helped me and my family. His plan is simple and straightforward. I highly recommend everyone to apply the wisdom Dave teaches in all of his literature, but _The Legacy Journey_ is a new favorite!
PERRY NOBLE
Founding and senior pastor of NewSpring Church
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\section{Introduction}
Structure formation in the Universe governed by the $\Lambda$ cold dark matter ($\Lambda$CDM) paradigm pursues a hierarchical model. Smaller DM (sub)haloes merge in order to build up larger objects (e.g. \citealt{white-rees,moore}). Such mergers occur on different scales, ranging from minor (1:3 - 1:50 of total mass) to major (mass ratios $\gtrsim$ 1:3 of total galaxy mass) mergers. These events are believed to impact the dynamics of galaxies, and even change their morphology (e.g. \citealt*{baugh,borne,hernandez,lotz10,kannan}). Current hydrodynamical cosmological simulations, which account for different small- and large-scale physical processes such as radiative cooling, feedbacks, star formation, magnetic fields and so on, are able to reproduce a spectrum of realistic galaxies with different morphologies, including ellipticals, barred, and unbarred spirals \citep{vogelsberger}.
Observations of galaxies in the local and the high redshift Universe, have shown a high fraction of barred spirals (e.g. \citealt{bergh,abraham}). The properties of the host disc galaxy is affected by the presence of a bar, which possesses a strong non-axisymmetric gravitational potential. For instance, the redistribution of stars/gas within the bar's corotation radius, together with the evolution of the bulge, are strongly correlated with the bar (e.g. \citealt{hohl,kormendy82,athanassoula05}). Early N-body simulations were successful in generating spiral galaxies hosting bars (e.g. \citealt*{miller,ostriker,athanassoula86}).
According to observations, a higher fraction of barred spirals are found in denser environments, such as galaxy groups and clusters, than in isolation, suggesting the importance of tidal interactions for bar formation \citep*{elmegreen}. \citet{noguchi87} pioneered numerical studies of bar formation in the presence of tidal interactions for initially stable galaxy models. In his runs, parabolic prograde planar very massive encounters with $M_\textrm{sat}/M_\textrm{galaxy} = 1 $ or 3 were able to only produce transient bars. Also, \citet{walker} showed that a satellite with $M_\textrm{sat}/M_\textrm{disc}=0.1$ on a circular prograde orbit with a pericentre distance of 6$R_\textrm{d}$ and inclined by 30$^\circ$ with respect to the disc plane is capable of destroying the bar in the initially bar-unstable models (their galaxies transformed into Sa Hubble morphological type). In \citet{moetazedian} we investigated the disc heating due to satellites infall in discs that were initially bar stable. In seven runs with different distributions of satellites extracted from zoom-in cosmological simulations of Milky Way-like hosts, no bar was induced, despite the mass ratios $0.003 < M_\textrm{sat}/M_\textrm{disc} < 4$. All these numerical experiments disfavour the tidally induced bar formation scenario in originally stable discs.
\citet{DBS08} discussed that massive satellite galaxies crossing the inner region of stable host galaxies may excite a bar or a spiral structure in the disc through the process of Toomre's swing amplification \citep[][hereafter, T81]{T81}, where the disc's inner part becomes vulnerable to such amplifications.
\citet{kazan08} analysed the impact of massive encounters on the disc galaxy evolution in the cosmological context. For $z=1$ epoch, the disc mass is expected to be lower, and the satellites distribution contains more massive satellites with smaller pericentre distances compared to the present day epoch $z=0$. The adopted disc mass and the exponential scale length are $M_{\textrm d} = 3.53 \times 10^{10} M_\odot$ and $R_{\textrm d} = 2.82$~kpc, respectively. Also, the satellites were on eccentric orbits and possessed a mass range $0.21-0.57 M_{\textrm d}$ with pericentre distances of $0.5 -6.2 R_{\textrm d}$. Unfortunately, the authors do not provide information concerning the stability of the isolated run. The models with satellites show growth of strong central bars, and the disc axisymmetry is not restored at late times. We note that the Toomre stability parameter $Q=2.2$ adopted for the isolated model is insufficient to avoid local non-axisymmetric \citep{PPS97} and global instabilities such as bar instability as proved by our calculation of the isolated model with exactly the same Toomre parameter.
For the host galaxy in the present analysis we employ a stellar dynamical model with a cuspy bulge generated using the GalactICS code~\citep{WPD08}, which was already studied in detail in our previous papers~\citep{PBJ16a, PBJ16b}. It turns out that the process of bar formation is affected by randomly incoming waves towards the disc centre, delaying it for some time, and then facilitating the bar growth \citep{P16}.
The analysis with different disc mass favours a young bar hypothesis, according to which the bar instability is saturated only recently~\citep{PBJ16b}. Thus, the present paper is aimed at investigating the importance of satellite galaxy encounters on the time of bar formation in a Milky Way-like galaxy using high-resolution N-Body simulations, with disc parameters and satellite distribution closer to the present day epoch, in contrast to \citet{kazan08}. In order to have realistic initial conditions (ICs), we use the distribution of satellites with masses exceeding $10^8 M_\odot$ extracted from cosmological simulations likely to host Milky Way-like galaxies \citep{aquarius,moetazedian}.
In section~\ref{sec:nbody} we discuss the cosmological ICs employed together with the characteristics of our desired host galaxy and the satellite galaxies. Section~\ref{sec:inst-host} gives a short overview of bar instability in the host galaxy. The spectral analysis employed here follows that described by \citet{EP05}. The details of our simulations and the results obtained from our bar formation analysis are discussed in section~\ref{sec:results}. The relevance of the azimuthal phase between a weak bar and incoming satellites towards the bar formation epoch is investigated in section~\ref{sec:phase}. Also, the results from the run with an early satellite encounter are presented in section~\ref{sec:early}. Section~\ref{sec:observations} contains comparisons with observations. We finalise the paper with a summary and discussion of the results in section~\ref{sec:summary}.
\section{N-body models}
\label{sec:nbody}
This section includes a description of the models recruited for setting up our desired Milky Way model, which consists of a disc, a bulge, and a DM halo together with the DM-only satellite galaxies.
\subsection{Cosmological initial conditions}
In order to account for a realistic image of satellite galaxies' infall onto the Milky Way's disc, the distribution of satellites was withdrawn from the Aquarius cosmological suite \citep{aquarius}. This consists of a set of six realisations of DM haloes likely to host Milky Way-like galaxies and which have not experienced a recent major merger. For the purpose of this study the level 2 of the Aquarius-D simulation, hereafter Aq-D2, was employed. We have shown in~\citet{moetazedian} that Aq-D2 is a fair representative of a typical Milky Way-like system with contribution towards the vertical heating of the Galactic disc regarded as average compared to the rest of Aquarius simulations.
In our study we use the $z=0$ snapshot which corresponds to the present day distribution of DM substructures. The snapshot contains information such as position, velocity, maximum circular velocity $V_\textrm{max}$, the radial distance of this velocity $r_\textrm{vmax}$ and the tidal mass $M_\textrm{tid}$ for every substructure (satellite) within the simulation box. The parent host halo has an enclosed mass, $M_\textrm{200}=1.774\times10^{12} M_{\odot}$, which corresponds to the mass within a sphere with radius $r_\textrm{200}=242.8$~kpc; this represents the radius at which the mean density of the DM halo is 200 times the critical density of the Universe ($\rho_\textrm{crit}$). Also the radius, where the halo encloses the mass with mean density 50 times $\rho_\textrm{crit}$ is denoted as $r_\textrm{50}$ and has a value of 425.7~kpc.
As mentioned earlier we are interested in satellites with a chance of passing close to the disc. The detailed procedure of identifying these candidates, is described in a previous paper~\citep{moetazedian}. The statistics of the satellite distribution is shown in Tab.~\ref{tab:2}; $N_\textrm{sub}$ is the total number of satellites with $M_\textrm{tid}\geq10^{6} M_{\odot}$ and $f_\textrm{sub}$ ($< r_\textrm{50}$) is the fraction of these objects within $r_\textrm{50}$. The percentage fraction of crossed satellites (i.e. objects that come closer than 25~kpc to the host halo's centre during their 2~Gyr orbit) is represented by $f_\textrm{cross}$ ($< r_\textrm{50}$). The last four rows show the number of crossed satellites with masses $\geqslant$ 1, 3, 5 and $10\times10^{8} M_{\odot}$. We argue in~\citet{moetazedian} that only satellites with $M_\textrm{tid}\geqslant10^{8} M_{\odot}$ potentially contribute towards the heating of the Galactic disc. The Aq-D2 simulation has 23 satellites with $M_\textrm{tid}\geqslant10^{8} M_{\odot}$ out of which four have masses larger than $10^{9} M_{\odot}$.
\begin{table}
\caption{Number statistics of Aq-D2 satellites.}
\label{tab:2}
\begin{tabularx}{\linewidth}{XX}
\hline
Quantity & Value \\ \hline
$N_\textrm{sub}$ ($M_\textrm{tid}\ge10^{6} M_{\odot}$) & 72,380\\
$f_\textrm{sub}$ ($<$ $r_\textrm{50}$) & 24.01 \% \\
$f_\textrm{cross}$ ($<$ $r_\textrm{50}$) & 8.85 \% \\
$N_\textrm{cross}$ ($>$ 10$^{8}$ $M_{\odot}$) & 23 \\
$N_\textrm{cross}$ ($>$ 3 $\times$ 10$^{8}$ $M_{\odot}$) & 8 \\
$N_\textrm{cross}$ ($>$ 5 $\times$ 10$^{8}$ $M_{\odot}$) & 5 \\
$N_\textrm{cross}$ ($>$ 10$^{9}$ $M_{\odot}$) & 4 \\ \hline
\end{tabularx}
\tablefoot{$f_\textrm{sub}$ ($< r_\textrm{50}$) is the percentage of the satellites originally within $r_\textrm{50}$ of the total number of satellites $N_\textrm{sub}$ with $M_\textrm{tid}\geq10^{6} M_{\odot}$, while $f_\textrm{cross}$ shows the percentage of satellites inside $r_\textrm{50}$ which cross the disc in 2~Gyr. $N_\textrm{cross}$ give the numbers of crossed satellites above the noted threshold.}
\end{table}
The host halo, in which these satellites would be inserted, has a slightly different enclosed mass ($M_{200}$). Therefore, we needed to rescale the phase-space properties of the satellites, together with their masses, following a recipe mentioned by~\citet{kannan}, in order to have a physically sensible analysis. The scaling factor is defined as \textit{f}=$M_\textrm{200,Aq-D2}/M_\textrm{200}$ (=1.37 in our case),
\begin{equation}
M=M_\textrm{orig}/f,
\label{eq:scl1}
\end{equation}
\begin{equation}
v_\textrm{\textit{x,y,z}}=\frac{v_{\textrm{orig},x,y,z}}{f^{1/3}} \qquad \textrm{and}
\label{eq:scl2}
\end{equation}
\begin{equation}
x=\frac{x_\textrm{orig}}{f^{1/3}}
\qquad
y=\frac{y_\textrm{orig}}{f^{1/3}}
\qquad
z=\frac{z_\textrm{orig}}{f^{1/3}}.
\label{eq:scl3}
\end{equation}
The subscript ``orig'' represents the original non-scaled values.
Early CDM N-body simulations have shown that the density profile of DM (sub)haloes could be well fitted via the known Navarro-Frank-White (NFW) profile~\citep{nfw}
\begin{equation}
\rho_\textrm{NFW}(r)=\frac{\rho_\textrm{s}}{(r/r_\textrm{s})(1+r/r_\textrm{s})^2}
\label{eq:nfw}
\end{equation}
and
\begin{equation}
\delta_c=\frac{\rho_\textrm{s}}{\rho_\textrm{crit}} = \frac{200}{3}\frac{c^{3}}{\ln(1+c)-c/(1+c)}\ .
\label{eq:deltac}
\end{equation}
Here $\rho_\textrm{s}$ and $r_\textrm{s}$ are the scale density and the scale radius of the halo, respectively. The density contrast $\delta_c$ with respect to $\rho_\textrm{crit}$ is calculated with the concentration $c$ of the halo being the ratio of $r_\textrm{200}/r_\textrm{s}$.
As discussed in section~\ref{sec:sat}, all the DM satellites also follow a NFW profile, which is tidally truncated at $r_\textrm{tid}$ according to the satellite mass $M_\textrm{tid}$.
\subsection{The host galaxy}
Our three-component model is adopted from~\citet{WPD08, KD95}, and consists of a stellar disc, a bulge, and a DM halo. The complete list of characteristics for the disc, the bulge and the halo components are listed in Tab.\,\ref{tab:4}. For a recent discussion of global parameters of the Milky Way see \citet{BG16}.
The disc is exponential, with radial scale length $R_\textrm{d}=2.9$~kpc and truncation radius 15~kpc. The radial velocity dispersion $\sigma_R$ is approximately exponential with central value $\sigma_{R0}=140$~km\,s$^{-1}$ and radial scale length $R_{\sigma}=2R_\textrm{d}$. The solar neighbourhood location is at $R=8$~kpc (e.g. \citealt{gillessen,reid}).
\begin{table}
\caption{The characteristics of disk+bulge+halo for our host galaxy model.}
\label{tab:4}
\begin{tabularx}{\linewidth}{XX}
\hline
Quantity & Value \\ \hline
$\Sigma_\textrm{sol}$ & 50 $M_{\odot}$pc$^{-2}$\\
$R_\textrm{d}$ & 2.9 kpc \\
z$_\textrm{d}$ & 300 pc \\
$M_\textrm{d}$ & 4.2 $\times$ 10$^{10}$ $M_{\odot}$ \\
$\sigma_{R}$ & 35 km\,s$^{-1}$ \\
$\sigma_{R0}$ & 140 km\,s$^{-1}$ \\
$N_\textrm{d}$ & 6 million \\ [2mm]
$R_\textrm{e}$ & 0.64 kpc \\
$\sigma_\textrm{b}$ & 272 km\,s$^{-1}$ \\
$M_\textrm{b}$ & 1.02 $\times$ 10$^{10}$ $M_{\odot}$ \\
$N_\textrm{b}$ & 1.5 million \\[2mm]
$M_{200}$ & 1.29 $\times$ 10$^{12}$ $M_{\odot}$ \\
$r_\textrm{s}$ & 17.25 kpc \\
$r_\textrm{200}$ & 229.3 kpc \\
$N_\textrm{h}$ & 9.25 million \\ \hline
\end{tabularx}
\tablefoot{$N$ represents the particle number for the component and $M$ the mass, while the subscripts d, b, and h correspond to disc, bulge, and halo. The solar neighbourhood surface density $\Sigma_\textrm{sol}$ together with the thin disc scale height and scale length, z$_\textrm{d}$ and $R_\textrm{d}$ characterize the disc density distribution. $\sigma_{R}$ and $\sigma_{R0}$ correspond to the solar and central radial velocity dispersions of the disc. In the case of the bulge, $R_\textrm{b}$ and $\sigma_\textrm{b}$ are the effective radius and the characteristic velocity scale.}
\end{table}
The bulge component takes a density profile of the following form
\begin{equation}
\rho_\textrm{b}(r) = \rho_\textrm{b} \left( \frac r{R_e} \right)^{-p} \textrm{e}^{-b(r/R_e)^{1/n} }\ ,
\label{eq:bulge_dens}
\end{equation}
where $r$ is the spherical radius, corresponding to a S\'{e}rsic surface brightness profile with effective radius $R_e$.
The scale density $\rho_\textrm{b}$, can be replaced by the bulge velocity scale
\begin{equation}
\sigma_\textrm{b} \equiv \left\{ 4\pi G n b^{n(p-3)} \Gamma[n(3-p)] R^2_e \rho_\textrm{b} \right\} ^{1/2}\ .
\label{eq:bulge_veld}
\end{equation}
Here, $\sigma^2_\textrm{b}$ corresponds to the depth of the gravitational potential associated with the bulge. In addition, we require $n=1.11788$ leading to $p\simeq0.5$ parameter, in order to define the bulge density profile.
We have employed a truncated NFW profile for the halo, with the truncation radius at $r_\textrm{200}$. The bulge component dominates the inner rotation curve at $R \gtrsim 0.01$~kpc, despite the halo having a more cuspy profile.
The total circular velocity profile (solid curve) and the contribution from each component (dashed/dotted) are shown in the top panel (a) of Fig.~\ref{fig:1}. The bulge dominates at radii $R\lesssim 1.5$~kpc, while the halo takes over at $R > 10$~kpc. At $R\approx 6$~kpc, where the contribution of the disc component reaches the maximum, the force from the halo is approximately two thirds of that from the disc in the Galactic plane.
\begin{figure}
\centering
\centerline{\includegraphics[width = \linewidth]{vaq_w09.eps}}
\caption{Initial profiles for the basic model: (a) the total circular velocity and its components due to disc, bulge, and halo; (b) angular velocity $\Omega(R)$ and curves $\Omega(R) \pm \kappa(R)/2$; (c) Toomre $Q$ initially set by GalactICS and the one actually obtained in the simulations.}
\label{fig:1}
\end{figure}
Panel (b) presents the angular velocity profile $\Omega(R)$ in the equatorial plane. In addition, positions of the inner (ILR) and outer Lindblad resonances (OLR) are shown as $\Omega \pm \kappa/2$. For a given pattern speed $\Omega_\textrm{p}$, the ILR is calculated using
\begin{equation}
\Omega_\textrm{p} = \Omega_\textrm{pr}(R) \equiv \Omega(R) - \frac12 \kappa(R) \ .
\label{eq:res}
\end{equation}
The quantity, $\Omega_\textrm{pr}(R)$ is responsible for determining a precession rate of nearly circular orbits. Hence, it can be referred to as `precession' curve, which diverges weakly as $R \to 0$ with $R^{-\alpha/2}$ and $\alpha \approx 0.5$.
The lower panel (c) of Fig.\,\ref{fig:1} demonstrates the Toomre $Q$ profile
\begin{equation}
Q = \frac{\kappa \sigma_R}{3.36 G\Sigma_\textrm{d}}\ ,
\label{eq:Qs}
\end{equation}
where $\Sigma_\textrm{d}$ represents the disc surface density. The solid line shows the profile which was initially set by the GalactICS code, while the dashed curve corresponds to the actual measured profile in our Galaxy run. The deviation for $2 <R<12$ kpc is explained by larger actual radial dispersion. The initial minimum $Q_\textrm{min}=1.8$ resides at $R=5.9$~kpc. This minimum rises to $Q_\textrm{min}\approx 2.1$ in case of the actual run.
\subsection{Satellites}
\label{sec:sat}
In order to insert the satellites into our N-body simulations, each satellite needs to be generated as a distribution of particles following their corresponding NFW profile.
The parameters $\rho_\textrm{s}$ and $r_\textrm{s}$ in Eq.~\ref{eq:nfw} are determined by $r_\textrm{vmax}$ using $r_\textrm{s}=0.4623\,\,r_\textrm{vmax}$ and $V_\textrm{max}$ using the enclosed mass inside $r_\textrm{vmax}$. Next we derive the tidal radius $r_\textrm{tid}$ as cut-off radius from the cumulative mass profile reaching the satellite mass $M_\textrm{tid}$.
The distribution function introduced by~\citet{WPD08} and implemented in~\citet{lora} was used for generating cuspy NFW profiles. Such a profile has an infinite cumulative mass at $r \rightarrow \infty$; therefore, we use the satellites' $r_\textrm{tid}$ as the cut-off radius. In this work, all satellites are represented using 50,000 particles.
The satellites can be ranged according to their ability to excite density waves in the disc of the host galaxy. In order to quantify it, we shall introduce a `tidal impact' parameter, which is a normalised tidal velocity perturbation. For a crude estimate, we use the formulae for tidal shocks \citep[][sect. 8.2.1]{BT08}, calculating the ratio of the perturbation to the circular velocity at some typical radius, for example, $v_{\textrm c}(R_{\textrm d}) $:
\begin{equation}
\left( \frac{\Delta v}{v_{\textrm c}}\right)_{R_{\textrm d}} \sim \frac{2 G M_{\textrm{tid}}}{b^2 V} \frac{R_{\textrm d}}{v_{\textrm c}} \sim \frac{G M_{\textrm{tid}} R_{\textrm d} }{b^2 v^2_{\textrm c}} \ .
\label{eq:dV}
\end{equation}
Here $b$ is the pericentre distance and we assumed that velocities at the time of encounters $V$ are similar and approximated as $V \simeq 2 v_{\textrm c}$. Fig.\,\ref{fig:sat_stat} shows the reverse cumulative histogram of the number of encounters with the tidal impact parameter (\ref{eq:dV}) greater than a given one, for all encounters with $M_{\textrm{tid}} > 10^8 M_\odot$ among seven runs discussed in \citet{moetazedian}. The Aq-D2 run can be regarded as a natural choice, since it contains at least a couple of satellites with tidal impact parameters at the high end. The marked red arrow corresponds to the Aq-D2 satellite with the highest parameter (referred to hereafter as `primary'), coming as close as $\sim4$~kpc to the centre, and having the second highest mass in Aq-D2. The primary satellite is expected to have a larger tidal impact on the disc, by a factor of $\sim2$, compared to the most massive satellite.
\begin{figure}
\centering
\centerline{\includegraphics[width = \linewidth]{mtid_peri_hist_all.eps}}
\caption{The reverse cumulative histograms of the number of encounters with the tidal impact parameter (\ref{eq:dV}) greater than the given one, for all seven runs of \citet{moetazedian}. We consider the encounters with $M_{\textrm{tid}} > 10^8 M_\odot$ (black) and $M_{\textrm{tid}} > 10^9 M_\odot$ (blue) before applying the rescaling. The red arrow marks the position of the Aq-D2 encounter used in simulations with one satellite and it is the 5th encounter from the right hand-side of the upper cumulative line. The satellite with the highest impact ($\sim0.4$) has a mass $1.1\times10^8 M_\odot$ and comes as close as 0.25~kpc to the centre.}
\label{fig:sat_stat}
\end{figure}
The host halo has a different enclosed mass and concentration compared to the Aq-D2 main halo. This means the orbits of Aq-D2 satellites would differ if inserted into this model. In order to gain fair and physically sensible results, we decided to scale $M_\textrm{tid}$, positions and velocities of Aq-D2 satellites using the factor $f=1.37,$ which corresponds to the ratio of Aq-D2 and the targeted host halo mass. Simulations with these initial conditions are called `fully-rescaled'. For comparison we also performed simulations where only the masses of the satellites are rescaled, but not their initial orbital positions and velocities, which we call `mass rescaled' (see Tab.\,\ref{tab:runs}).
\section{Bar instability in the host galaxy}
\label{sec:inst-host}
The isolated galaxy is prone to instability leading to the formation of a bar. Bar properties on low amplitudes, such as a pattern speed and an amplitude (exponential) growth rate, can be obtained as global solutions from matrix equations \citep[e.g.][]{K71, K77}, describing a disc with stellar orbits of different eccentricities. It opposes tightly wound local WKB solutions of Lin--Shu--Kalnajs obtained in the epicyclic approximation.
The applicability of the razor-thin disc model for the description of real discs with finite thickness and a density cusp in the centre has been studied in detail by \citet{PBJ16a}. The main idea is to consider an effective rotation curve, which takes into account the $z$-dependence of the radial force on the stars that elevate above the equatorial plane. The effective rotation curve lacks cuspy features and possesses a maximum, which allows for a bar in a wide range of pattern speeds.
Here we use a matrix method by \citet{EP05} that has a form of the linear matrix equation,
\begin{equation}
{\mathbf A} {\mathbf x} = \omega {\mathbf x}\ ,
\label{eq:me}
\end{equation}
which allows us to find unstable modes effectively without prior information on the localisation of modes.
A serious flaw in all matrix methods is the inability to calculate unstable global modes in a live halo and bulge. Although the substitution of a live halo with its rigid counterpart gives nearly the same pattern speeds as in the fully live models, the growth rates are smaller by a factor of two or even more. The theoretical bar pattern speed estimate for our host galaxy is $\Omega_\textrm{b} \approx 48.34$ km\,s$^{-1}$\,kpc$^{-1}$ and for the growth rate we obtained $\omega_\textrm{I} \approx 1.23$~Gyr$^{-1}$, so the expected value for the fully live models is 2...3~Gyr$^{-1}$.
\section{N-body simulations}
\label{sec:results}
The initial conditions of stars for single mass simulations have been generated by the `GalactICS' code provided by~\citet{WPD08}. In our previous N-body simulations~\citep{PBJ16a} we used three different codes ({\tt SUPERBOX--10}, {\tt bonsai2}, {\tt ber-gal0}) and demonstrated the practical independence of the results on the chosen code.
The simulations in this work are restricted to the {\tt bonsai2} tree-code and {\tt ber-gal0}\footnote{\tt ftp://ftp.mao.kiev.ua/pub/users/berczik/ber-gal0/}~\citep{ZBGPJ2015} routines as an auxiliary tool to analyse the snapshot data. The modified version of the recently developed N-body Tree--GPU code implementation {\tt bonsai2}~\citep{BGPZ2012a, BGPZ2012b} includes expansion for force computation up to quadrupole order. The opening angle used had a value of $\theta=0.5$, accompanied by individual gravitational softening of 10~pc. {\tt bonsai2} employs the leap-frog integration scheme with a fixed time--step $\Delta t=0.2$~Myr.
The current set of simulations were carried out with the GPU version of the code using the ARI GPU cluster {\tt kepler} and also the GPU cluster {\tt MilkyWay} specially dedicated to the SFB 881 (``The Milky Way System''), located in the J\"ulich Supercomputing Centre in Germany.
Tab.\,\ref{tab:runs} contains the summary of our runs. The capital letter `B' denotes the numerical \textsc{bonsai2} Tree code. It is worthwhile to mention, the Milky Way halo components have multi-mass halo particles in order to achieve a better resolution in the bar region. For this, we modified the GalactICS code utilising the same strategy as described in \citet{DBS09}. In the region between 0.1 and 1~kpc, the number density ratio of our multi-mass and single mass runs varies from 10 to 100, thus the effective numerical resolution there is enhanced by this factor.
\begin{table}
\begin{center}
\caption{Summary of the runs}
\label{tab:runs}
\begin{tabular}{l l l l l l l l l}
\hline
Model & $n_\textrm{S}$ & $\Delta_\textrm{S}$ & $t_1$ & $t_2$ & $t_3$ & $\Delta_B$ & $\Omega_\textrm{p}$ & $\omega_\textrm{I}$ \\
\hline
B-0 & -- & -- & 0.75 & 1.4 & 2.2 & -- & 49 & 2.9 \\[2mm]
B-1 & 1 & -- & 0.75 & 1.2 & 1.8 & -200 & 48 & 3.0 \\
B-1m & 1 & -- & 0.75 & 2 & 2.4 & 300 & 49 & 2.9 \\
B-4 & 4 & -- & 0.75 & 1.3 & 2.0 & -300 & 49 & 2.9 \\
B-4m & 4 & -- & 0.75 & 1.7 & 2.5 & 470 & 48 & 3.0 \\
B-23 & 23 & -- & 0.75 & 1.1 & 1.7 & -310 & 48 & 2.9 \\ [2mm
B-1-110 & 1 & +110 & 0.75 & 1.6 & 2.5 & 600 & 48 & 2.9 \\
B-1-500 & 1 & --\,500 & 0.40 & 1.0 & 2.2 & -20 & 48 & 2.7\\
\hline
\end{tabular}
\end{center}
\vspace{-2mm}
\tablefoot{$n_\textrm{S}$ is the number of satellites; `m' denotes only mass-rescaled satellites while the remaining simulations are fully -- mass, positions and velocities -- rescaled. $t_1$, $t_2$, $t_3$ denote the times (in Gyr) of the jump, the onset and the end of the exponential growth of the bar, respectively. $\Delta_B$ is the delay of the bar growth in Myr with respect to the isolated case B-0. Pattern speed $\Omega_\textrm{p}$ in km\,s$^{-1}$ and exponential growth rate $\omega_\textrm{I}$ in Gyr$^{-1}$ are given in the last columns.
}
\end{table}
The subscript `m' corresponds to the simulations, which are only mass-rescaled using Eq.~\ref{eq:scl1}; while the remaining simulations are fully-rescaled. In the case of simulations marked using `B-1', we use the primary satellite from Aq-D2 simulation with rescaled mass $M_\textrm{tid}=2.45\times10^{9} M_{\odot}$. For runs marked as `B-4', the four most massive Aq-D2 crossed satellites with $M_\textrm{tid} > 10^{9} M_{\odot}$ were employed.
\begin{figure}
\centering
\centerline{\includegraphics[width = 85mm]{sat_orbit_bonsai_sep.eps}}
\caption{The orbit of the primary satellite from our five runs as a function of time.}
\label{fig:sat_orbit_bs}
\end{figure}
The orbital positions and velocities of the satellites are determined by their density centres. Fig.~\ref{fig:sat_orbit_bs} shows the galactocentric orbital distance for Aq-D2's primary satellite from our five runs. There exists a 110~Myr difference between the first pericentre passage of B-1 and B-1m runs. The pericentre distances have the values of 2.6 and 3.5~kpc, occurring at 0.85 and 0.96~Gyr, respectively. Also, the orbits from the two simulations with $+110$ and $-500$~Myr time shift are shown.
\subsection{Isolated simulation B-0 of the host galaxy}
In the absence of the satellites, the host galaxy appears unstable as manifested in the formation of a bar. In Fig.\,\ref{fig:gS1} one can see snapshots at times 2, 2.5, 3, and 4~Gyr of the bar oriented along the $x$-axis. At earlier times, the asymmetry of the density distribution is weak, because of a delay in bar formation (e.g.~\citealt{PBJ16a,P16}). Moreover, there are no noticeable spirals throughout the simulation.
\begin{figure}
\centering
\centerline{\includegraphics [width = \linewidth]{spirals_live.eps}}
\caption{Bar patterns in the B-0 run oriented along the $x$-axis at different stages of bar evolution. The curves are isolines of the surface density evenly spaced in log scale (ten levels for every factor of 10). Each frame size is $32\times32$~kpc.}
\label{fig:gS1}
\end{figure}
The bar is a wave, and its growth can be investigated using relative amplitudes of perturbed quantities, such as the surface density $A_2/A_0$,
\begin{equation}
A_m(t) = \sum\limits_{j} \mu_j \textrm{e}^{-im\theta_j}
\label{eq:A2A0}
,\end{equation}
(here $\mu_j$ and $\theta_j$ are mass and polar angle of star $j$; $j$ spans particles within some fixed radius, for example, the radial scale length $R_\textrm{d}$), or velocity components $U_{k,m}$. The latter are defined as the perturbed velocity amplitudes,
\begin{equation}
V_{k,m}(t) = \frac{1}{A_0} \sum\limits_{j} \mu_j \tilde v_{k,j} \textrm{e}^{-im\theta_j}\ ,
\label{eq:Vkm}
\end{equation}
normalized to the average velocity dispersion $\sigma_k(t)$ calculated by
\begin{equation}
\sigma^2_k(t) = \frac{1}{A_0} \sum\limits_{j} \mu_j \tilde v^2_{k,j}\ .
\label{eq:sigma}
\end{equation}
The cylindrical components $R$, $\theta,$ and $z$ correspond to the quantity $k,$ and $\tilde v_{k,j}$ is the residual velocity component of the particle $j$.
Panel (a) in Fig.~\ref{fig:bs_live} shows the relative bisymmetric amplitudes $A_2/A_0$, and $U_{k,2}$ for the B-0 run, representing the isolated galaxy. All the curves, except $U_{z,2}$,
show a typical phase of low amplitude fluctuations, followed by a jump. The evolution then enters a phase characterised by a typical exponential growth, and finally reaching a plateau. In Tab.~\ref{tab:runs}, $t_1$ denotes the jump, $t_2$ corresponds to the start of the exponential growth and $t_3$ marks the end of the exponential growth. The simultaneous growth of the perturbed density and velocity components in the plane is expected for unstable density waves, in particular for bars. However, the vertical velocity component $U_{z,2}$ does not respond to the bar formation, and we exclude it from our analysis.
The calculation of components of the inertia ellipsoid can also be used for the analysis of the bar growth, and to obtain the pattern speed of the bar. Slopes of the bar amplitudes and the bar strength,
\begin{equation}
B(t) = 1 - I_{yy}/I_{xx}\ ,
\label{eq:barstr}
\end{equation}
give estimates for the growth rates, which appear to be very close to one another (and $\simeq 3$ Gyr$^{-1}$ as predicted from the rigid halo analysis).
The pattern speed is obtained from an angle of the rotation of the ellipsoid's main axes (e.g. \citealt{WPT16}). In panel (b) of Fig.~\ref{fig:bs_live} we plotted the pattern speed of the bar $\Omega_\textrm{p}$ determined by an average centred finite difference over two time-intervals of $\pm 10$~Myr before and after the evaluated time (dots) and the pattern speed after filtering out the highest frequencies. No definite structure for the pattern speed value can be found before the jump, $t<t_1$. Just after the jump, the centred finite difference shows much less scatter, and the filtered $\Omega_\textrm{p}$ varies only slowly until $t=1.9$~Gyr. Due to a bar slowdown, the pattern speed gradually decreases to 28.6~km\,s$^{-1}$kpc$^{-1}$ at $t=4$~Gyr.
\begin{figure}
\centering
\centerline{\includegraphics [width = \linewidth]{tsd_V_W09.eps}}
\vspace{-7mm}
\centerline{\includegraphics [width = \linewidth]{t_Omega_W09.eps}}
\caption{Bar formation in the isolated model. (a) bar amplitudes $A_2/A_0$ and $U_{R\theta,2}$, and the bar strength $B(t)$ demonstrate lags, jumps, and exponential growths with a rate $\omega_\textrm{I} \approx 3$~Gyr$^{-1}$ (the slope is shown by the black dashed curve). (b) the pattern speed of the bar $\Omega_\textrm{p}$ determined as an average centred finite difference of the bar phase (dots), and the spline smoothed $\Omega_\textrm{p}$ (red curve).}
\label{fig:bs_live}
\end{figure}
Fitting ellipses to the isophotes is one of the possibilities to quantify the bar shape~\citep{Inma}. We define a bar radius, $R_\textrm{b}$, as a radius at which the ellipticity $\varepsilon (r)\equiv 1-b_\textrm{e}/a_\textrm{e}$ ($a_\textrm{e}$ and $b_\textrm{e}$ are the major and minor semi-axes of ellipses) declines by $\sim15$\% from its maximal value. The obtained radii and ellipticities are given in Tab.~\ref{tab_rb_live}. For $t=1.0$ Gyr, the bar radius is approximately 1.1~kpc, and $\varepsilon \simeq 0.13$. At the end of the exponential growth, $t\approx 2.5$, the bar radius is 4.1~kpc, with an ellipticity of $\varepsilon \simeq 0.65$.
\begin{table}
\begin{center}
\begin{tabular}{l l l l l l l}
\hline
Time [Gyr] & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 4.0 \\
\hline
$R_\textrm{b} $ [kpc] & 1.1 & 1.1 & 2.4 & 4.1 & 5.0 & 5.7 \\
$\Omega_\textrm{p} $ [km/s/kpc] & 48.8 & 48.0 & 45.3 & 37.8 & 33.3 & 28.6 \\
$R_\textrm{C} $ [kpc] & 4.4 & 4.5 & 4.8 & 5.9 & 6.7 & 7.8 \\
$\varepsilon $ & 0.13 & 0.42 & 0.69 & 0.65 & 0.63 & 0.62 \\
\hline
\end{tabular}
\end{center}
\vspace{-2mm} \caption{Parameters of the bar (bar radius, pattern speed, corotation radius, ellipticity) in B-0 run at different moments in time.}
\label{tab_rb_live}
\end{table}
With the determined $\Omega_\textrm{p} \approx 49$~km\,s$^{-1}$\,kpc$^{-1}$ at $T=1$\,Gyr, the corotation resonance occurs at $R_\textrm{C} \approx 4.4$~kpc. The ILR and OLR are located at 0.7 and 7.9~kpc, respectively. As the bar grows between $t_2$ and $t_3$, the ratio ${\cal R} \equiv R_\textrm{C}/R_\textrm{b}$ attains the value of 1.4 and remains roughly constant until the end of the simulation despite the significant slow-down of the bar in the saturation phase.
As can be seen from Fig.\,\ref{fig:bs_live}a, the bar instability in the real N-body finite thickness disc does not manifest itself as an exactly exponential amplitude growth from the early beginning of the simulation. For a long time, the amplitude is on the noise level. The start of a mild growth is possible only after a sufficiently strong wave appears in the disc at some moment $t=t_1$, and provokes a delayed perturbation in the centre (see also top panels in Fig.\,\ref{fig:SD}). The emergence of such a wave perturbation is a random event, the probability of which is smaller for less unstable discs \citep{P16}.
The delayed wave is a bar-like perturbation which rotates with a well-defined pattern speed (Fig.\,\ref{fig:bs_live}b), suggesting that a seed bar is already formed at $t_1$. However, the amplitude growth now is irregular and slower than exponential, until the wave amplitude $A_2$ reaches 1 or 2 per cent of the axisymmetric background $A_0$ ($t=t_2$). The exponential growth with a growth rate $\omega_\textrm{I} \approx 3$~Gyr$^{-1}$ lasts until $t=t_3$, when the instability saturates.
\subsection{Simulations with satellites}
Possible effects from satellite interactions were studied through five runs, B-1, B-1m, B-4, B-4m and B-23. The numbers in the run codes reflect number of the satellites, while `m' reflects the runs in which only the mass of the satellites was rescaled.
In Fig.~\ref{fig:ba_mass} we compare the bar amplitudes $A_2/A_0$ of these models with the one calculated for B-0 (solid red). The jumps in all curves occur at the same time, $t_1 = 0.75$~Gyr, meaning that it is independent of the satellites, but rather intrinsic to the disc itself. The subsequent behaviour is different: mass rescaled runs B-1m and B-4m show a delay in bar formation, while in the fully-rescaled runs, B-1, B-4, and B-23, the bar forms earlier compared to B-0. The time difference in the bar formation $\Delta_B$ is given in Tab.~\ref{tab:runs}.
\begin{figure}
\centering
\centerline{\includegraphics [width = \linewidth]{tsd_6.eps}}
\vspace{-7mm}
\centerline{\includegraphics [width = \linewidth]{Omega_t_smooth.eps}}
\caption{Bar formation in models with satellites: (a) bar amplitudes $A_2/A_0$ are given for B-0, B-1, B-1m, B-4, B-4m and B-23 run. The black dashed line shows the slope corresponding to a growth rate $\omega_\textrm{I} = 3$~Gyr$^{-1}$; (b) the spline smoothed pattern speeds for the same models.}
\label{fig:ba_mass}
\end{figure}
In the mass-rescaled runs, the initial positions of the satellites are further away from the host galaxy than in the fully-rescaled runs, so the interaction with the disc peaks later in the mass rescaled runs. In particular, the minimum distance in the B-1m run is at 0.96~Gyr, while in the B-1 run it occurs at 0.85~Gyr. The delay $\Delta_B$ is larger in the B-4m run where four satellites are used.
The fully-rescaled models show contrary effect on the bar formation, compared to the mass-rescaled simulations. In the B-1 run the satellite advances the bar formation by 200~Myr (negative delay), and the higher the number of satellites, the larger the impact. In the B-4 run where the four largest satellites are used, the time difference is 300~Myr. A comparison with the B-23 run shows that the most massive satellites determine mainly the time difference -- it is only 10~Myr larger than in the B-4 run.
In section~\ref{sec:phase_relation} we investigate in more details the true character of the observed delay/advancement.
\section{Phase relation between bar and satellite perturbations}
\label{sec:phase_relation}
A conjecture we wish to explore is focused on the measured mutual phase of an initial bar and perturbations induced by the massive satellites, and the importance towards advancement or delay in bar formation. Within the course of this section we inspect phases calculated using the surface density and velocity perturbations, assuming a simplified description of cold fluid discs.
\subsection{Perturbations from a satellite}
\label{sec:phase}
In this subsection we concentrate on the velocity perturbations in a cold fluid disc as a result of a satellite passage. In our runs, the typical encounter speed is $V \approx 400$~km\,s$^{-1}$; for a galaxy of size $d\sim20$~kpc the encounter lasts $d/V \approx 50$~Myr, which is a fraction of the rotation period of the bar, $\approx 130$ Myr, measured at the beginning of its formation.
To obtain velocity perturbations, one can integrate the acceleration of fluid particles moving on exactly circular orbits:
\begin{equation}
{\rm d} {\mathbf v}(R, \theta) = -G \frac{M_\textrm{s}(t) ({\mathbf R}-{\mathbf r}_\textrm{s} )}{|{\mathbf r}_\textrm{s} - {\mathbf R}|^3} {\rm d} t\ , \quad {\rm d}\theta = \Omega(R) {\rm d} t\ ,
\label{eq:acl_sat}
\end{equation}
where $M_\textrm{s}(t)$ and ${\mathbf r}_\textrm{s}(t)$ are known functions for the satellite mass and its distance from the disc centre, ${\mathbf R}$ is the radius vector of a fluid particle. We take only the in-plane acceleration into account and disregard vertical perturbations here.
The phase patterns, which are irregular at the beginning of the integration, become steady and regular as the satellite approaches the disc. In Fig.~\ref{fig:sat_v}, we show amplitudes and phases of the bisymmetric $m=2$ radial and azimuthal velocity perturbations during the encounter of the primary satellite for the period between 0.82 and 0.87~Gyr. Loci of the maxima of $V_{R,2}$ and $V_{\theta,2}$ are practically unchanged from $t=0.7$ to 0.84~Gyr. After the fly-by, the maxima lines begin to wind up, and eventually turn into a tightly wound spiral. The amplitudes of velocity components grow up to $t=0.86$~Gyr, then remain practically unchanged.
\begin{figure*}
\centering
\centerline{\includegraphics [width = 150mm]{vr_vp_m2_0000.eps} }
\vspace{-1mm}
\centerline{\includegraphics [width = 150mm]{vr_vp_m2_0001.eps} }
\caption{Time sequence of phase (upper panels) and amplitudes (lower panels) of the $m=2$ perturbations of radial (red triangles) and azimuthal (blue circles) velocities induced by the primary satellite (pink diamonds, visible for $T=0.84 - 0.86$\,Gyr) in the cold fluid disc.}
\label{fig:sat_v}
\end{figure*}
\subsection{Phase relations in the cold fluid disc}
Equations of fluid dynamics are employed frequently to analyse phenomena taking place in stellar discs \citep[e.g.][]{Ber14}. Indeed, when the stellar disc is cold enough, mean radial and azimuthal velocities can be obtained from linearised Euler equations for zero-pressure discs, because a cold stellar disc is dynamically equivalent to a fluid disc with zero pressure. If the perturbation has $m$-fold rotation symmetry and $\propto \textrm{e}^{i(m\theta-\omega t)}$, the perturbed surface density $\Sigma_{da}$ and average velocity components ($v_{\theta a}, v_{R a}$) obey the following relations \citep{BT08}:
\begin{equation}
v_{Ra}(R) = \frac{i}{\Delta} \Big[ (\omega-m\Omega) \frac{{\rm d}\Phi_a}{{\rm d} R} - \frac{2m\Omega\Phi_a}{R} \Big]\ ,
\label{eq:vr}
\end{equation}
\begin{equation}
v_{\theta a}(R) = -\frac{1}{\Delta} \Big[ 2B \frac{{\rm d}\Phi_a}{{\rm d} R} + \frac{m(\omega-m\Omega)\Phi_a}{R} \Big]\ ,
\label{eq:vp}
\end{equation}
\begin{equation}
-i(\omega-m\Omega) \Sigma_{da} + \frac{1}R \frac{d}{dR}(Rv_{Ra}\Sigma_0) + \frac{im\Sigma_0}{R} v_{\theta a} = 0\ ,
\label{eq:sa}
\end{equation}
where $B \equiv -\kappa^2/4\Omega$ is the Oort constant; $\Sigma_0$ is the disc surface density and $\Phi_a$ is the potential perturbation;
\begin{equation}
\Delta \equiv \kappa^2 - (\omega-m\Omega)^2
\label{eq:delta}
\end{equation}
is positive between the Lindblad resonances.
In this analysis, we shall distinguish two extreme cases: tightly wound spirals and the bar. In the {\it tightly wound} approximation, a perturbed quantity (e.g. the potential) can be written in the form
\begin{equation}
\Phi_a(R) = |\Phi_a| \textrm{e}^{iF(R)} = |\Phi_a| \textrm{e}^{i\int^R k\, dR} \ ,
\label{eq:pot}
\end{equation}
with phase function $F(R)$, where $k=dF(R)/dR$ and $|kR| \gg 1$. From eq. \ref{eq:sa} one has
\begin{equation}
kv_{Ra}\Sigma_0 = (\omega-m\Omega) \Sigma_{da}\ ,
\label{eq:sa1}
\end{equation}
that is, surface density of the wave is in phase with the radial velocity outside corotation, and in anti-phase inside corotation:
\begin{equation}
\begin{array}{rcl}
F_{R} - F_{\Sigma} = \pi & : & R<R_C \ ,\\
F_{R} - F_{\Sigma} = 0 & : & R>R_C \ .
\end{array}
\label{eq:F1}
\end{equation}
The equation for the azimuthal velocity (eq. \ref{eq:vp}) simplifies to
\begin{equation}
v_{\theta a}\Sigma_0 = \frac{-2iB}{\Delta} k \Phi_a \ ,
\label{eq:vp1}
\end{equation}
and taking into account that $\Phi_a = -2\pi G\Sigma_{da}/|k|$,
\begin{equation}
F_{\theta} - F_{\Sigma} = -\pi/2\ ,
\label{eq:F2}
\end{equation}
meaning that the surface density lags behind the azimuthal velocity by $\pi/4$ for an $m=2$ perturbation.
In the {\it bar region} we assume that the radial derivative is negligible compared to the other term in square brackets in (\ref{eq:vr}, \ref{eq:vp}). For the radial velocity,
\begin{equation}
v_{Ra}(R) = -\frac{2im}{\Delta} \frac{\Omega}{R} \Phi_a \ ,
\label{eq:v2}
\end{equation}
that is,
\begin{equation}
F_{R} - F_{\Sigma} = \pi/2\ ,
\label{eq:F3}
\end{equation}
meaning that radial velocity lags behind the surface density by $\pi/4$ ($m=2$). For the azimuthal velocity,
\begin{equation}
v_{\theta a}(R) = -\frac{m}{\Delta} \frac{(\omega-m\Omega)}{R} \Phi_a \ ,
\label{eq:vp2}
\end{equation}
that is,
\begin{equation}
\begin{array}{rcl}
F_{\theta} - F_{\Sigma} = \pi & : & R<R_C \ ,\\
F_{\theta} - F_{\Sigma} = 0 & : & R>R_C \ .
\end{array}
\label{eq:F4}
\end{equation}
\subsection{Phase relations in the stellar discs}
To compare the phases of surface density and velocity perturbations in the stellar discs, we calculate a relative surface density, $A_2(R,t)/A_0(R,t)$,
\begin{equation}
A_m(R,t) = {\sum\limits_{j}}' \mu_j \textrm{e}^{-im\theta_j}\ ,
\label{eq:ART}
\end{equation}
and velocity components, $V_{k,2}(R,t)/A_0(R,t)$,
\begin{equation}
V_{k,m}(R,t) = {\sum\limits_{j}}' \mu_j \tilde v_{k,j} \textrm{e}^{-im\theta_j}.
\label{eq:VRT}
\end{equation}
In both expressions, the summations are taken over disc particles within a ring $\Delta R$ near $R$.
Fig.~\ref{fig:b0_v} shows the loci of maxima $\theta(R) = -F/m$ for the B-0 run without satellites. Note that the maxima of $V_{\theta,2}$ are shifted by $\pi/2$ from the surface density maxima, and the maxima of $V_{R,2}$ lag by $\pi/4$ from the surface density maxima, in accordance with the phase relations for a bar in the cold fluid disc derived above.
\begin{figure*}
\centering
\centerline{\includegraphics [width = 150mm]{vr_vp_sd_m2_B0_0000.eps}}
\vspace{-1mm}
\centerline{\includegraphics [width = 150mm]{vr_vp_sd_m2_B0_0001.eps}}
\caption{Loci of maxima of $m=2$ perturbations (velocity and surface density) in the B-0 run (no satellites) at the time of the encounter in B-1 (the pink diamond shows the position of the primary satellite in B-1 for visualisation).}
\label{fig:b0_v}
\end{figure*}
The comparison with Fig.\,\ref{fig:sat_v} shows that the wave velocity components are approximately in phase with disc perturbations caused by a satellite when the satellite approaches the disc centre, but become out of phase after $t=0.85$~Gyr.
Fig.~\ref{fig:b1m_v} is similar to Fig.~\ref{fig:b0_v}, but for the primary satellite in the mass scaled models. The orientation of the bar is roughly perpendicular to that of the bar in Fig.~\ref{fig:b0_v} at the corresponding moments of time (time difference is $\sim 110$~Myr). So in this case the bar is out of phase when the satellite approaches the pericentre.
\begin{figure*}
\centering
\centerline{\includegraphics [width = 150mm]{vr_vp_sd_m2_B1m_0000.eps} }
\vspace{-1mm}
\centerline{\includegraphics [width = 150mm]{vr_vp_sd_m2_B1m_0001.eps} }
\caption{Same as in Fig.\,\ref{fig:b0_v}, but at the time of the encounter in B-1m.}
\label{fig:b1m_v}
\end{figure*}
The derived difference in phases during the satellite approach is essential, but seems to be only part of a more complex story. It follows from Fig.~\ref{fig:ba_mass}, where one can see a rather complicated bar amplitudes' behaviour. In Fig.~\ref{fig:tsd_3} we show a zoom-in of the early growth phase for the B-0, B-1, and B-1m runs. In particular, we stress the following features.
In the fully-rescaled B-1 run, the maximum occurs at $t=0.97$ Gyr, that is, 120~Myr (almost one period of bar rotation) after the pericentre passage; the 0.97~Gyr maximum is followed by minima at 1.06~Gyr and 1.16~Gyr. In the mass-rescaled model B-1m, a deep minimum occurs at $t=1.01$~Gyr, that is, 50~Myr after the pericentre passage; the minimum is followed by a large maximum exceeding the amplitude in the B-0 run at 1.11~Gyr followed by a nearly constant behaviour until 1.5~Gyr. These features cannot be understood from simple phase arguments as discussed above.
\begin{figure}
\centering
\includegraphics [width = \linewidth]{tsd_3.eps}
\caption{Bar amplitudes $|A_2/A_0|$ for the B-0, B-1, B-1m runs (selected zoom of Fig.~\ref{fig:ba_mass}).}
\label{fig:tsd_3}
\end{figure}
\subsection{Interference of bar and satellite perturbations}
The interaction between the central bar-mode and a perturbation from the satellite can be studied using density surface maps showing the total bisymmetric ($m=2$) amplitude in the $(R, t)$-plane. Left panels in Fig.\,\ref{fig:SD} show $\log|A_2/A_0|$ for the isolated run B-0, as well as for one satellite runs B-1 (fully-rescaled) and B-1m (mass-rescaled). The lower left panel contains the map for not-yet-mentioned B-1-110 run, which differs from B-1 by a 110~Myr time delay of the satellite encounter. The lag is calculated to mimic a passage of the satellite near the pericentre of the mass-rescaled B-1m run (see Fig.\,\ref{fig:sat_orbit_bs}).
\begin{figure*}
\includegraphics [width = 85mm]{SD_abs_4.eps} \hspace{1mm} \includegraphics [width = 85mm]{SD_real_4.eps}
\caption{Surface density maps $\log|A_2/A_0|$ (left panels, red = high, blue = low amplitude); $\Re A_2/A_0$ (right panels, red = positive, blue = negative amplitude at the $x$-axis) for B-0, B-1, B-1m, and B-1-110 runs (top to bottom), reflecting the interference between the bar-like perturbation in the centre, random perturbations, and tidally induced waves. The time period is $0.6<t<1.8$ Gyr.}
\label{fig:SD}
\end{figure*}
The descending `tongues' observed in the panels are nothing but interference patterns of two waves. Even in the upper figure for the isolated B-0 run, it is the interference between a small bar-like perturbation mostly residing in the central part, and random perturbations of the disc at $R \sim 1...2\,R_\textrm{d}$. These perturbations may act in accordance and accelerate bar formation, or, on the contrary, destroy one another.
The random peripheral perturbations are repeated in the lower panels. In addition, they contain imprints from the satellite for the different cases B-1, B-1m, B-1-110.
In Fig.\,\ref{fig:b1} one can see for case B-1 the onset of the disc perturbation after the encounter (0.87 and 0.89 Gyr), which clearly has a bar-like global shape and obeys phase relations obtained for bars in Section 5.2. This disc perturbation evolves so that its shape winds up gradually to the tightly wound spiral (last row of Fig.\,\ref{fig:b1}; the subsequent evolution is not shown here), but also it causes a disc response in the central region similar to those presented in T81; we refer to Figs. 1, 2 therein. Recall that Toomre analysed responses from weak corotating (Fig.1) and immobile (Fig.2) transient imposed dipole forces in a {stable} Mestel disc. The disc shows strong spiral responses developing from the centre outward, and sheared with a speed comparable with material arm shear, $\Omega(R)$. This is a manifestation of `a strong cooperative effect' called {\it swing amplification}. For this mechanism to be efficient, the parameter
\begin{equation}
X\equiv \frac{\kappa^2 R}{2m \pi G \Sigma_0}
\label{eq:Xt}
\end{equation}
should not exceed 3, which is the case in our model in the range $0.6<R<5.5$~kpc, with a minimum $X\simeq 2$ at $R\simeq 2.3$~kpc.
Much like in Toomre simulations, we expect similar responses to the disc external perturbations imposed by the satellite. Such responses grow on the time-scale of one period in the centre and interact with a small bar, strengthening or weakening the latter.
Right panels of Fig.~\ref{fig:SD} show $\Re A_2/A_0$, that is, a joint $m=2$-amplitude along the $x$-axis, for the same runs B-0, B-1, B-1m, B-1-110 from top to bottom. In the upper panel, one can see red and blue spots standing for maxima and minima and showing emergence of the bar, starting from $t\sim 0.75$~Gyr ($R<2$~kpc). These spots extend vertically in time as the bar length grows. Also one can note smaller slightly shifted spots above the bars. These are interference patterns between the shearing external disc response to the encounter and the spiral arm adjacent to the rotating bar in the centre.
The interference spots are localised roughly along the straight lines with inclination growing in time. Using these lines one can infer periods of the interference pattern as a function of radius, which roughly obey the linear law, $T_\textrm{i} = aR+b$. Since frequency difference between the bar and the spiral response $\Omega_\textrm{b}-\Omega_\textrm{s}(R) = 2\pi/T_\textrm{i}$, one can calculate the local pattern speed $\Omega_\textrm{s}(R)$ for wave patterns due to the encounter. This is shown by red circles in Fig.~\ref{fig:wkbw}, along with $\Omega$ and $\Omega-\kappa/2$ curves.
\begin{figure}
\centering
\includegraphics [width = \linewidth]{WKB_winding.eps}
\caption{Local pattern speeds $\Omega_s$ of the wave patterns (red circles) compared to main frequencies of the disc ($\Omega$ is the angular velocity, $\kappa$ is the epicyclic frequency, $\Omega_*$ is given by (\ref{eq:Oms})).}
\label{fig:wkbw}
\end{figure}
Local pattern speeds $\Omega_\textrm{s}$ of the wave patterns can be explained using the simplest fluid WKB dispersion relation. Each wave pattern evolves according to the relation,
\begin{equation}
\omega = \omega(k,R),
\label{eq:WKB_g}
\end{equation}
meaning that $R = R(k,\omega)$ is a function of $k$; $\omega \equiv m\Omega_\textrm{s}$ is a parameter. For very open spiral\footnote{Application of WKB theory should be treated here as extrapolation only.}, $k$ is close to zero, so
$\Omega_\textrm{s} = \Omega(R) - \kappa(R)/m$, and $R$ is close to the ILR, $R_\textrm{ILR}$. As the local wave pattern evolves, $(\omega-m\Omega)^2$ attains its minimum at some $k=k_*$, so $R$ attains its maximum \citep{T69}. Then $k_*$ can be estimated from the dispersion relation for fluid discs:
\begin{equation}
(\omega - m\Omega(R))^2 = \kappa^2(R) - 2\pi G \Sigma_0 |k| + k^2 c^2\ ,
\label{eq:WKB}
\end{equation}
where $k$ is the wavenumber and $c$ is a sound speed, which we use interchangeably with the radial velocity dispersion. Thus
\begin{equation}
k_* = \frac{\pi G \Sigma_0}{c^2}\ ,
\label{eq:k_s}
\end{equation}
and substitution to the WKB relation gives
\begin{equation}
\Omega_* (R) = \Omega(R) - \frac{\kappa(R)}m (1-Q(R)^{-2})^{1/2}\ .
\label{eq:Oms}
\end{equation}
In the last expression, $Q$ denotes Toomre stability parameter for fluid discs,
\begin{equation}
Q(R) = \frac{\kappa(R)c(R)}{\pi G \Sigma_0(R)}\ .
\label{eq:Qt}
\end{equation}
As is seen from Fig.~\ref{fig:wkbw}, $\Omega_*$ and $\Omega_\textrm{s}$ agree relatively well, which proves that the picture presented here is an interference pattern between the bar and the density wave. We note that due to large $Q$ ($>2$), radial excursions of the wave patterns are small, and are localised close to their ILR.
The interference of the bar and the delayed response to the disc perturbation in the B-1 run leads to a bar enhancement. This is seen from comparison of the right panels in the first and second rows of Fig.~\ref{fig:SD}. In particular, a red bar spot immediately after $t=1$~Gyr in the B-1 run is definitely stronger than its counterpart in the B-0 run.
Contrary to simulations performed in T81, our external perturbation does not appear for only a short time. Thus it provides a number of delayed responses in the centre which can also either enhance or destroy the bar, unless the external perturbation becomes tightly wound. In case of B-1, it also works for enhancement.
In the last two rows, we see the opposite behaviour. In B-1m, the red bar spot immediately after $t=1$~Gyr is definitely weaker than in B-0 run, and it is seen that the delayed response is not in phase with the original bar. Moreover, it seems that the established perturbation also suffers from the subsequent delayed response waves.
We also note powerful random waves seen in B-0 panels between 1.05 and 1.3~Gyr, and 1.15 and 1.4~Gyr. They also came into play and decrease amplitudes in B-1m and B-1-110 runs.
\section{The early encounter}
\label{sec:early}
In the previous section we argue that satellite encounters may cause a delay or advancement in bar formation after the small bar is already formed in the centre ($t>t_1$), but its amplitude is still insignificant ($t<t_2$). Here we determine what occurs if the encounter takes place well before $t_1$. For this purpose, we performed B-1-500 run, in which the fully scaled satellite approaches the host galaxy 500~Myr earlier than in the B-1 run.
In the surface density maps (Fig.\,\ref{fig:SD500}), we see disc perturbations due to the encounter immediately after $t=0.35$~Gyr. While there is no visible random perturbation in the disc (see upper left panel), the only perturbation that exists is the one due to the satellite and subsequent delayed response in the central region ($R<2$ kpc), visible only after $t=0.4$~Gyr. Then it disappears at 0.6~Gyr, and a second perturbation appears after a short gap. Once the first random wave has reached the centre at $t \simeq 0.75$~Gyr, the large scale spiral becomes tightly wound and ineffective in providing a delayed response in the centre. This follows from comparison of the amplitudes in the left panels for $R<2$~kpc.
The resulting bar amplitude calculated for disc particles within characteristic scale length $R_\textrm{d}$ is shown in Fig.~\ref{fig:rtmaps} together with the bar amplitude of the B-0 run. It is seen that after a jump at $t\simeq 0.4$~Gyr due to the delayed response waves, the central perturbation gradually fades away until the random wave forms a bar perturbation at $t\simeq 0.75$ Gyr -- seen already in the previous runs. Afterwards, the B-1-500 curve continues almost in pace with the B-1 curve.
\begin{figure}
\centering
\includegraphics [width = \linewidth]{tsd_delay-new.eps}
\caption{Bar amplitudes for B-0 and B-1-500 runs.}
\label{fig:rtmaps}
\end{figure}
\section{Comparison with observations}
\label{sec:observations}
In this section we would like to discuss how our results compare with observations. Throughout this paper we have argued that the Aq-D2 is believed to be a fair representative of a Milky Way-like host. Fig.~3 in \citet{moetazedian} is very helpful for comparing the distribution of satellites from the Aq-D2 run (red data points) with observations of our Milky Way. According to this figure, there resides a handful of massive satellites with $M_\textrm{tid} > 10^9 M_\odot$ among the seven simulations with pericentre distances within 25~kpc from the centre. The Milky Way's Sagittarius dwarf galaxy (Sgr) has an estimated current total mass $M_\textrm{Sgr} \sim 5 \times 10^8 M_\odot$ and a pericentre of $\sim20$~kpc \citep*{law}. If we assume Sgr to be on its second passage, then it has probably lost $\sim$50\% of its mass during the initial crossing. In the Aq-D2, an analogue to Sgr can be found at $\sim20$~kpc. In addition, more massive Milky Way satellites such as SMC with a mass $\sim 6.5 \times 10^9 M_\odot$ \citep{bekki} and galactocentric distance of 60~kpc \citep{smc} have a corresponding analogue in the Aq-D2 realisation.
As previously shown by Eq.~\ref{eq:dV}, the impact of a satellite with a given tidal mass and a pericentric distance $b$ scales as $\sim M_\textrm{tid}/b^2$. For a Sgr or SMC-like satellite we expect smaller impact as their passage occurs at a further distance to the centre (50-60~kpc) and hence, we fall below $10^{-4}$ (Fig.~\ref{fig:sat_stat}).
The distribution of satellites from both N-body cosmological simulations and observational studies are dominated by lower mass objects. In the Aquarius satellites distribution there exists a fair number of such low-mass satellites with $M_\textrm{tid} < 10^9 M_\odot$ and pericentres within the inner 10~kpc of the disc. Observations of the Local Group have shown the existence of so-called ultra-faint dwarf galaxies with velocity dispersions of the order of 10~kms$^{-1}$ \citep{simon}. Recent observations as part of the Dark Energy Survey (DES) have increased the total number of detected ultra-faint dwarfs to 17 \citep{des}. This acts in favour of narrowing down the gap between cosmological model predictions and observations at the lower mass end of the galaxy mass function, that is, ``the missing satellite problem''. The DES year-two quick release (Y2Q1) predicts a total of $\sim$100 ultra-faint dwarfs in the full sky cover. These objects are highly dark matter dominated with a very low stellar budget. In order to estimate the DM mass content of the satellites, their velocity dispersion needs to be measured using spectroscopy; this has only been done for a few satellites (e.g. \citealt{martin16,simon16}). These galaxies would be subject to strong tidal stripping in case of very close encounters with the disc, meaning it is very difficult to observe any remnants. Therefore, our current observational sample of ultra-faint dwarfs is limited and fainter magnitudes need to be reached.
There exists an inconsistency in current models of the Milky Way bar, where a broad range of values are discussed for bar properties such as age, pattern speed etc. (e.g.~\citealt*{wegg,monari}). Therefore we are unable to constrain our models, making it impossible to determine the age of the Milky Way's bar.
\section{Conclusions}
\label{sec:summary}
In this paper we analyse how the bar formation time can be affected by dark matter satellite encounters. Our Milky Way-like galaxy model is studied mainly using N-body simulations. All components, including an exponential stellar disc, a NFW halo, and a S\'{e}rsic cuspy bulge are tailored to the Milky Way and are live, that is, represented by particles that interact according to Newton's law of gravity.
The adopted parameters of the host galaxy provide a disc mildly sensitive to bar instability with a characteristic e-folding time (inverse growth rate) $\tau_\textrm{e} = 340$~Myr. This time, as well as the bar pattern speed, $\Omega_\textrm{p} = 48...49$~km\,s$^{-1}$kpc$^{-1}$ were obtained from our simulations and agree well with global mode calculations in the framework of linear perturbation theory. Such a large e-folding time is primarily due to the relatively high radial velocity dispersion (35 km\,s$^{-1}$ in the Solar neighbourhood), and a weak central cusp with $\rho \propto r^{-0.5}$. The cusp does not prohibit bar formation, as long as a disc with finite thickness rather than a razor-thin disc is considered \citep{PBJ16a}.
Bar formation in the isolated galaxy occurs in the presence of random swing amplified wave packets appearing in a range of radii with Toomre parameters $X<3$ and $Q<3$ (between 2 and 4 kpc in our case). These packets interfere with a bar-like perturbation being a density wave in the inner part, which grows on its own due to bar instability. The interference leads randomly to demolition or enhancement of the bar. The latter is seen as a jump of the bar amplitude at some arbitrary moment in time ($t_1$ in Fig.\,\ref{fig:bs_live}). However, the exponential growth occurs later after additional random wave packets reinforce the inner bar, thus the moment $t_2$ at which the bar starts to grow exponentially is also a random quantity \citep[see also][]{P16}.
The properties and orbital parameters of the satellite galaxies are derived from the Aquarius D-2 cosmological simulation. All satellites with tidal mass $M_\textrm{tid}$ above $10^8 M_\odot$ and pericentric distances to the host galaxy below 25~kpc were selected. Our set of runs with satellites consists of four runs with one satellite, two runs with four satellites (with tidal mass M$_\textrm{tid} \geq 10^9 M_{\odot}$) and one run with all 23 satellites. Each satellite was represented by 50,000 particles. In all cases, the pattern speed and the e-folding time of the bar in the host galaxy remain unchanged. However, we noted that from run to run, the bar amplitudes grow asynchronously. In the earliest case, the bar forms 300~Myr earlier than in the reference isolated run without satellites. In the latest run, the bar appeared with a 600~Myr delay. Most of the advancement (or delay in the other two cases) is caused by the primary satellite whose pericentre is the closest to the galactic centre. The effect from other massive satellites is appreciable, yet turns out to be less significant. The lighter satellites play practically no role.
Satellites add complexity to the physical picture of bar formation. One satellite on the same orbit can give both advancement and delay, simply by passing in a slightly different time. Thus, orientation of the bar is important. However, it is difficult to specify `a rule of thumb' for the sign of the effect, since the satellite angular speed is at least two times higher than the bar pattern speed, so a range of various bar phases is seen in all cases.
The main features of the satellite impact on the bar formation can be understood again through the wave interactions. After the satellite passage, a tidal wave is induced across the disc (Fig.\,\ref{fig:b1}), which then evolves in accordance with the WKB relation (\ref{eq:Oms}). It winds up and propagates to the centre where it interferes with the bar, as follows from Fig.\,\ref{fig:SD}. While it remains relatively open, it produces a delayed response in the inner disc. This mechanism is similar to swing amplification observed in simulations of the stable Mestel disc that turns out to be surprisingly vital under the influence of minor dipole perturbations (T81). This interference, in some cases, leads to bar formation delay (B-1m, B-1-110), or advancement (B-1), depending on enhancement or attenuation of the central structure. Keeping the phase of the tidal wave
fixed, the interference pattern depends on the bar phase, which is illustrated by comparison of B-1 and B-1-110 runs. This basic mechanism is contaminated by additional random waves, which appear throughout the disc, also producing delayed responses in the inner disc, regardless of any encounter.
The effect is possible only if the encounter occurs after a seed bar is formed (after the amplitude jump at $t=t_1$) and before the exponential growth phase, $t\gtrsim t_2$. If the encounter occurs well before $t_1$, the tidal wave decays with no effect. This is proved by our B-1-500 run, in which the tidal wave occurs at $t \approx 400$ Myr, as follows from Fig.\,\ref{fig:SD500}. The bar formation time then coincides with one in the isolated run B-0. During the period of vulnerability, $t_1 \lesssim t \lesssim t_2$, the tidal wave amplitude is comparable to the amplitude of the seed bar, so the consequences of the interference are most evident. After $t_2$, the bar becomes strong so that it cannot be affected by relatively weak tidal waves.
It was known that the bar formation time depends on the instability properties of the galactic disc, which is determined by many parameters including the disc mass, velocity dispersion, dark matter halo, and a central matter concentration. This leads to uncertainties in predicting the bar age from simulations. The discussed phenomenon adds a scatter in the age of the bars. \citet{P16} showed that less unstable models have longer time between the first jump and the start of the exponential growth. This implies that at higher redshifts $z > 0.5$ satellites have more chances to interfere with the process of bar formation an thus the effect should be stronger. This analysis supports the hypothesis that all observed barred galaxies should be bar unstable in order to develop long-lasting bars.
\begin{figure*}
\centerline{\includegraphics [width = 150mm]{B1_0083.eps} }
\vspace{-6mm}
\centerline{\includegraphics [width = 150mm]{B1_0085.eps} }
\vspace{-6mm}
\centerline{\includegraphics [width = 150mm]{B1_0087.eps} }
\vspace{-6mm}
\centerline{\includegraphics [width = 150mm]{B1_0089.eps} }
\vspace{-6mm}
\centerline{\includegraphics [width = 150mm]{B1_0091.eps} }
\caption{Onset of the large-scale bar-like perturbation from the satellite in the B-1 run (time from top to bottom, $\Sigma_\textrm{d}$, $V_\textrm{R}$ and $V_\theta$ from left to right).}
\label{fig:b1}
\end{figure*}
\begin{figure*}
\centering
\includegraphics [width = 85mm]{SD_abs_2.eps} \hspace{1mm} \includegraphics [width = 85mm]{SD_real_2.eps}
\caption{As in Fig.\,\ref{fig:SD} for B-0 and B-1-500 runs for the period $0.2<t<1.4$ Gyr.}
\label{fig:SD500}
\end{figure*}
\begin{acknowledgements}
We would like to thank the anonymous referee for their constructive comments and remarks which have greatly improved this paper. The main production runs were done on the {\tt MilkyWay} supercomputer, funded by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre (SFB 881) ``The Milky Way System'' (subproject Z2), hosted and co-funded by the J\"ulich Supercomputing Center (JSC).
The special GPU accelerated supercomputer {\tt laohu} funded by NAOC/CAS and through the ``Qianren'' special foreign experts program of China (Silk Road Project), has been used for some of code development. We also used a smaller GPU cluster {\tt kepler} for data analysis, funded under the grants I/80 041-043 and I/81 396 of the Volkswagen Foundation. This work was supported by the Sonderforschungsbereich SFB 881 ``The Milky Way System'' (subproject A2 and A6) of the German Research Foundation (DFG). E.P. acknowledges financial support by the Russian Basic Research Foundation, grants 15-52-12387, 16-02-00649, and by the Basic Research Program OFN-15 `The active processes in galactic and extragalactic objects' of Department of Physical Sciences of RAS. P.B. acknowledges the special support by the NASU under the Main Astronomical Observatory GRID/GPU ``golowood'' computing cluster project.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,446
|
\section{Introduction}
\label{intro_sect}
An interest to the exotic systems, which consist of antikaons and nucleons, rose recently
after the statement \cite{AY1,AY2}, that deep and narrow quasi-bound states can exist
in $\bar{K}NN$ and $\bar{K}NNN$ systems. Due to this, several calculations of
the quasi-bound state in the lightest $\bar{K}NN$ system with $J^P = 0^-$ quantum numbers,
that is $K^- pp$, were performed. Among all calculations those using Faddeev-type equations
are the most accurate ones. The first results of the two accurate calculations
\cite{myKpp_PRL,myKpp_PRC,ikedasato_first} confirmed the existence of
the quasi-bound state in the $K^- pp$, but the evaluated binding energies and widths
are far different from those predicted in \cite{AY1,AY2}. The results of the two groups
\cite{myKpp_PRL,myKpp_PRC,ikedasato_first} also differ one from the other, and the main reason
for this is a choice of the antikaon-nucleon interaction, being an input for the three-body
calculations.
The question of the quasi-bound state in the $K^- pp$ system is far from being settled from
experimental point of view as well. The first experimental evidence of the $K^- pp$ quasi-bound state
existence occurred in the FINUDA experiment~\cite{FINUDA} at the DA$\Phi$NE $e^+ e^−$ collider.
Recently performed new analyses of old experiments, such as OBELIX~\cite{OBELIX}
at CERN and DISTO~\cite{DISTO} at SATURNE also claimed the observation of the state.
However, there are some doubts, whether the observed structure corresponds to the
quasi-bound state. The experimental results also differ from each other, moreover, their
binding energies and widths are far from all theoretical predictions. Since the question
of the possible existence of the quasi-bound state in the $K^- pp$ system is still highly
uncertain, new experiments are being planned and performed by HADES~\cite{HADES} and LEPS~\cite{LEPS}
Collaborations, and in J-PARC E15~\cite{J-PARC_E15} and E27~\cite{J-PARC_E27} experiments.
It was demonstrated \cite{myKpp_PRC} that the $\bar{K}N$ interaction plays a crucial role
in the $\bar{K}NN$ calculations. It is much more important than the nucleon-nucleon
one, but is far less known. Therefore, a model of the antikaon-nucleon interaction,
which is more accurate in reproducing experimental data than that from \cite{myKpp_PRL,myKpp_PRC},
was necessary to construct. The experimental data on $\bar{K}N$ interaction,
which can be used for fitting parameters of the potential, are: near-threshold cross-sections
of $K^- p$ scattering, their threshold branching ratios, and shift and width
of $1s$ level of kaonic hydrogen (which should be more accurately called ``antikaonic
hydrogen''). The last observable has quite interesting experimental history and finally was
measured quite accurately. As for the theoretical description of kaonic hydrogen, many
authors used approximate Deser-type formulas, which connect $1s$ level shift
of an hadronic atom with the scattering length, given by strong interaction
in the pair. The question was, how accurate are the approximate formulas, derived for
the pion-nucleon interaction, for the antikaon-nucleon system. It was demonstrated in
\cite{revai_deser,our_KN} and later in \cite{cieply} that Deser-type formulas has low accuracy
for the antikaon-nucleon system.
Another question of antikaon-nucleon interaction is a structure of the $\Lambda(1405)$
resonance, which couples the $\bar{K}N$ system to the lower $\pi \Sigma$ channel.
$\Lambda(1405)$ is usually assumed as a quasi-bound state in the higher $\bar{K}N$ channel
and a resonance state in the lower $\pi \Sigma$ channel. But it was found in \cite{2L1405}
and later in other papers that a chirally motivated model of the interaction lead to
a two-pole structure of the resonance. Keeping these two points of view in mind,
two phenomenological $\bar{K}N$ potentials with one- and two-pole structure of the $\Lambda(1405)$
resonance were constructed in \cite{our_KN}. Their parameters were fitted to the experimental data,
and $1s$ level shift and width of kaonic hydrogen were calculated directly, without any
approximate formulas.
It turned out \cite{our_KN} that it is possible to construct phenomenological potentials with
one- and two-pole $\Lambda(1405)$ resonance which describe existing low-energy
experimental data with the same level of accuracy. Due to this, the two $\bar{K}N$ potentials
were used as an input in calculations of the low-energy elastic $K^- d$ scattering in \cite{my_Kd}.
But after the publication of the $K^- d$ results, SIDDHARTA collaboration
reported results of their measurement of kaonic hydrogen characteristics \cite{SIDDHARTA}.
The results turned out to be quite different from the previously measured results of
DEAR experiment \cite{DEAR} and compatible with older KEK data \cite{KEK}. Due to this,
the $\bar{K}N$ potentials were refitted in such a way, that they reproduce the most recent
experimental data on the $1s$ level shift and width of kaonic hydrogen. The calculations of
the low-energy elastic kaon-deuteron scattering were repeated in \cite{my_Kd_sdvig} with
the new potentials. In addition, an approximate calculation of the $1s$ level shift and width
of kaonic deuterium was performed. It was done approximately using a complex $K^- - d$ potential,
reproducing the elastic three-body $K^- d$ amplitudes.
It was found that the three-body $K^- d$ system also does not allow to make preference to one
of the two phenomenological $\bar{K}N$ potentials and by this to solve the question
of the number of $\Lambda(1405)$ poles. In order to support the statement, one more,
a chirally motivated $\bar{K}N$ potential was constructed. As other chiral models, it has two poles
forming the $\Lambda(1405)$ resonance. Parameters of this potential were also fitted
to the low-energy experimental data on $K^- p$ scattering and kaonic hydrogen,
the chirally motivated $\bar{K}N$ potential reproduces all antikaon-nucleon data with
the same accuracy as the two phenomenological models.
Another way of investigation of the $\Lambda(1405)$ resonance was suggested and
realised in \cite{revai_1405}, were low-energy breakup of the $K^- d$ system was considered.
The idea was that the resonance should be seen as a bump in so called deviation spectrum of
neutrons in the final state of the reaction. However, $\Lambda(1405)$ is so broad that it
was seen as a bump in some cases only.
Finally, the calculations of the three-body $\bar{K}NN$ system with different quantum
numbers were repeated in \cite{ourKNN_I,ourKNN_II} using all three models of the $\bar{K}N$ interaction.
In particular, the binding energy and width of the $K^- pp$ quasi-bound state were evaluated,
the low-energy $K^- d$ amplitudes were calculated and the $1s$ level shift and width of kaonic
deuterium were predicted. A search of the quasi-bound state in the $K^- d$ system was also performed,
but the results are negative.
After the approximate calculations of the characteristics of deuterium, the exact
calculations were performed in \cite{our_Kdexact}. Namely, Faddeev-type equations with strong
plus Coulomb interactions, suggested in \cite{Papp1}, were solved.
It was the first time, when the equations \cite{Papp1}, initially written and used for a system
with Coulomb interaction being a correction to a strong potential, were used
for investigations of an hadronic atom, where Coulomb potential plays the main role.
Since the equations are much more complicated than ``usual'' AGS ones (containing short-range
potentials only), the calculations were performed with simple complex $\bar{K}N$ potentials,
reproducing only some of the experimental $K^- p$ data. Comparison of the dynamically exact
three-body results with the previous approximate ones shown that the approximation of the kaonic
deuterium as a two-body system is quite accurate for this task.
Another three-body exotic system, consisting of two antikaons and one nucleon, was studied
in \cite{my_KKN}. It was expected that a quasi-bound state can exist in the $\bar{K}\bar{K}N$ system too.
The three $\bar{K}N$ potentials were used, and a quasi-bound state was found with smaller binding
energy than in the $K^- pp$ and larger width. It is interesting, that the parameters
of the state allow to associate it with a $\Xi$ state mentioned in the Particle Data Group \cite{PDG}.
The paper summarises results of the series of exact or accurate calculations
\cite{myKpp_PRL,myKpp_PRC,our_KN,my_Kd,my_Kd_sdvig,ourKNN_I,ourKNN_II,our_Kdexact,my_KKN}.
The next section contains information about the two-body interactions, necessary for the three-body
calculations. Faddeev-type Alt-Grassberger-Sandhas equations with coupled channels, which were used
for three-body calculations with strong interactions, are described in Section \ref{AGS_sect}.
Section \ref{qbs_sect} is devoted to the quasi-bound states in the $K^- pp$, $K^- d$, and $K^- K^- p$
systems. The near-threshold $K^-d$ scattering is considered in Section \ref{Kd_elastic_sect},
the kaonic deuterium -- in Section \ref{kaonic_deu_sect}. The last section summarises the results.
\section{Two-body interactions}
\label{Interactions_sect}
In order to investigate some three-body system it is necessary to know
the interactions of all the pairs of the particles. The interactions,
necessary for investigations of the $\bar{K}NN- \pi \Sigma N$ and
$\bar{K}\bar{K}N - \bar{K} \pi \Sigma$
systems, are $\bar{K}N$ and $\Sigma N$ with other channels coupled to them,
and the one-channel $NN$ and $\bar{K}\bar{K}$ interactions (the rest of them were
omitted in the three-body calculations). All potentials, except one of the $NN$ potentials,
were specially constructed for the calculations. They have a separable form and
$\mathbb N$-term structure
\begin{equation}
\label{V_Nterm}
V_{i,II'}^{\alpha \beta} = \sum_{m=1}^{{\mathbb N}_i^{\alpha}}
\lambda_{i(m),II'}^{\alpha \beta} \,
|g_{i(m),I}^{\alpha} \rangle \langle g_{i(m),I'}^{\beta} |,
\end{equation}
which leads to a separable $T$-matrix
\begin{equation}
\label{T_Nterm}
T_{i,II'}^{\alpha \beta} = \sum_{m,n = 1}^{{\mathbb N}_i^{\alpha}}
|g_{i(m),I}^{\alpha} \rangle
\tau_{i(mn),II'}^{\alpha \beta} \langle g_{i(n),I'}^{\beta} | \,.
\end{equation}
${\mathbb N}_i^{\alpha}$ in Eq.~(\ref{V_Nterm}) and Eq.~(\ref{T_Nterm}) is a number of terms
of the separable potential, $\lambda$ is a strength constant, while $g$ is a form-factor.
The two-body isospin $I$ in general is not conserved. In particular, the two-body
isospin is conserved in the phenomenological $\bar{K}N-\pi \Sigma$ potentials, but
not in the corresponding $T$-matrices due to Coulomb interaction and the physical masses,
taken into account. The chirally motivated $\bar{K}N - \pi \Sigma - \pi \Lambda$ potential
does not conserve the isospin due to its energy and mass dependence.
The separable potentials are simpler than other models of interactions. However,
the potentials, entering the equations for the antikaon-nucleon systems, were constructed
in such a way that they reproduce the low-energy experimental data for every subsystem
very accurately. From this point of view they are not worse than other models of
the antikaon-nucleon or the $\Sigma$-nucleon interaction (in fact, they are even better
than some chiral models). The one-term NN potential does not have a repulsive part,
but the two-term model is repulsive at short distances. Finally, all three-body observables
described in the present paper turned out to be dependent on the $NN$ and $\Sigma N$
interactions very weakly, therefore the most important is the accuracy of the $\bar{K}N$
potential.
The antikaon-nucleon interaction is the most important one for the three-body systems
under consideration. There are several models of the $\bar{K}N$ interaction, some of them
are ``stand-alone'' ones having the only aim to reproduce experimental data, others were used
in few- of many-body calculations. The problem is that the first ones are too
complicated to be used in few-body calculations, while the models from the second group are
too simple to reproduce all the experimental data properly. Due to this, the $\bar{K}N$ potentials,
which are simple enough for using in Faddeev-type three-body equations and at the
same time reproduce all low-energy antikaon-nucleon experimental data, were constructed.
\subsection{Antikaon-nucleon interaction, experimental data}
\label{KN_exp_sect}
\underline{$\Lambda(1405)$ resonance}
The $\Lambda(1405)$ resonance is a manifestation of the attractive nature of the antikaon-nucleon
interaction in isospin zero state, it couples $\bar{K}N$ to the lower $\pi \Sigma$ channel.
Not only position and width, but the nature of the resonance itself are opened questions.
A usual assumption is that $\Lambda(1405)$ is a resonance in the $\pi \Sigma$ channel and
a quasi-bound state in the $\bar{K}N$ channel. According to the most recent Particle
Data Group issue~\cite{PDG}, the resonance has mass $1405.1^{+ 1.3}_{- 1.0}$ MeV
and width $50.5 \pm 2.0$ MeV. There is also an assumption suggested
in \cite{2L1405} and supported by other chiral models, that the bump, which
is usually understood as the $\Lambda(1405)$ resonance, is an effect of
two poles. Due to this, the two different phenomenological models of the antikaon-nucleon
interaction with one- or two-pole structure of the $\Lambda(1405)$ were constructed.
The third model is a chirally motivated potential, which has two poles by construction.
Extraction of the resonance parameters from experimental data is complicated for
two reasons. First, it cannot be studied in a two-body reaction and can be seen in
a final state of some few- or many-body process. Second, its width is large,
so the corresponding peak could be blurred.
A theoretical paper~\cite{revai_1405} was devoted to the possibility of tracing
the $\Lambda(1405)$ resonance in the neutron spectrum of a $K^- d$ breakup reaction.
The neutron spectra of the $K^- d \to \pi \Sigma n$ reaction were
calculated in center of mass energy range $0 - 50$ MeV. The three-body system with coupled
$\bar{K}NN$ and $\pi \Sigma N$ channels was studied using the Faddeev-type AGS equations,
described in Section~{\ref{AGS_sect}}, with four phenomenological $\bar{K}N$
potentials with one- or two-pole structure of $\Lambda(1405)$.
It was found that kinematic effects completely mask the peak corresponding
to the $\Lambda(1405)$ resonance. Therefore, comparison of eventual experimental
data on the low-energy $K^- d \to \pi \Sigma n$ reaction with theoretical results
hardly can give an answer to the question of the number of $\Lambda(1405)$ poles.
Later, similar calculations of the same process were performed for initial kaon
momentum $1$~GeV in \cite{Kdbreak_repeat1,Kdbreak_repeat2}. Coupled-channel AGS equations
were solved as well with energy-dependent and -independent $\bar{K}N$ potentials.
The authors predict a pronounced maximum in the double-differential cross section
with a forward emitted neutron at $\pi \Sigma$ invariant mass $1.45$ GeV. However,
applicability of the $\bar{K}N$ potentials, fitted to the near-threshold data,
and of the nonrelativistic Faddeev equations for such high energies is quite doubtful.
Several arguments, suggested in support to the idea of the two-pole structure
of the $\Lambda(1405)$ resonance, were checked in~\cite{my_Kd} using the one- and two-pole
phenomenological models of the antikaon-nucleon interaction. One of the arguments is
the difference between the $\pi \Sigma$ cross-sections with different charge combinations,
which is seen in experiments, e.g. in CLAS~\cite{CLAS}. The elastic $\pi^+ \Sigma^-$, $\pi^- \Sigma^+$,
and $\pi^0 \Sigma^0$ cross-sections were plotted to check the assumption, that
the difference is caused by the two-pole structure. However, it turned out that
the cross sections are different and their maxima are shifted one from each other
for both one- and two-pole versions of the $\bar{K}N$ potential (see Fig. 5 of ~\cite{my_Kd}).
Therefore, the effect is not a proof of the two-pole structure, but a manifestation of
the isospin non-conservation and differences in the background.
Another argument for the two-pole structure comes from the fact, that the poles in
a two-pole model are coupled to different channels. Indeed, a gradual switching
off of the coupling between the $\bar{K}N$ and $\pi \Sigma$ channels turns
the upper pole into a real bound state in $\bar{K}N$, while the lower one becomes
a resonance in the uncoupled $\pi \Sigma$ channel (see e.g. Fig.2 of \cite{ourKNN_I}).
Consequently, it was suggested, that the poles of a two-body model
manifest themselves in different reactions. In particular, the $\bar{K}N - \bar{K}N$,
$\bar{K}N - \pi \Sigma$, and $\pi \Sigma - \pi \Sigma$ amplitudes should
``feel'' only one of the two poles. The hypothesis was also checked in~\cite{my_Kd},
and indeed, the real parts of the $\bar{K}N - \bar{K}N$, $\bar{K}N - \pi \Sigma$, and
$\pi \Sigma - \pi \Sigma$ amplitudes in $I=0$ state cross the real axis at different energies.
But it is true for the both: the one- and the two-pole versions of the potential
(see Fig. 6 of \cite{my_Kd}). This effect must be caused by different background
contributions in the reactions. Therefore, a proof of the two-pole structure of
the $\bar{K}N$ interaction does not exist.
\vspace*{2mm}
\noindent
\underline{Cross-sections and threshold branching ratios}
Three threshold branching ratios of the $K^- p$ scattering
\begin{eqnarray}
\label{gammaKp}
\gamma &=& \frac{\Gamma(K^- p \to \pi^+ \Sigma^-)}{\Gamma(K^- p \to
\pi^- \Sigma^+)} = 2.36 \pm 0.04, \\
\label{Rc}
R_c &=& \frac{\Gamma(K^- p \to \pi^+ \Sigma^-, \pi^- \Sigma^+)}{\Gamma(K^- p \to
\mbox{all inelastic channels} )} = 0.664 \pm 0.011, \\
\label{Rn}
R_n &=& \frac{\Gamma(K^- p \to \pi^0 \Lambda)}{\Gamma(K^- p \to
\mbox{neutral states} )} = 0.189 \pm 0.015
\end{eqnarray}
were measured rather accurately in~\cite{gammaKp1, gammaKp2}. Since the phenomenological
$\bar{K}N$ potentials, used in the calculations, take the lowest
$\pi \Lambda$ channel into account indirectly, a new ratio
\begin{equation}
\label{RpiSigma}
R_{\pi \Sigma} =
\frac{\Gamma(K^- p \to \pi^+ \Sigma^-)+\Gamma(K^- p \to \pi^- \Sigma^+)}{
\Gamma(K^- p \to \pi^+ \Sigma^-) + \Gamma(K^- p \to \pi^- \Sigma^+) +
\Gamma(K^- p \to \pi^0 \Sigma^0) } \,,
\end{equation}
which contains the measured $R_c$ and $R_n$ and has an ``experimental'' value
\begin{equation}
\label{RpiSigma_exp}
R_{\pi \Sigma} = \frac{R_c}{1-R_n \, (1 - R_c)} \, = \, 0.709 \pm 0.011
\end{equation}
was constructed and used.
In contrast to the branching ratios, the elastic and inelastic total cross sections
with $K^- p$ in the initial state~\cite{Kp2exp,Kp3exp_1,Kp3exp_2,Kp4exp,Kp5exp,Kp6exp}
were measured not so accurately, see Figure~\ref{CrossKp.fig}.
\vspace*{2mm}
\noindent
\underline{Kaonic hydrogen}
The most promising source of knowledge about the $\bar{K}N$ interaction is kaonic hydrogen
atom (which correctly should be called ''antikaonic hydrogen''). The atom has rich
experimental history, several experiments measured its $1s$ level shift
\begin{equation}
\Delta E_{1s} = E_{1s}^{Coul} - {\rm Re }(E_{1s}^{Coul+Strong})
\end{equation}
and width $\Gamma_{1s}$, caused by the strong $\bar{K}N$ interaction in comparison to pure
Coulomb case, with quite different results. The most recent measurement was performed by
SIDDHARTA collaboration~\cite{SIDDHARTA}, their results are:
\begin{equation}
\label{SIDDHARTA}
\Delta E_{1s}^{\rm SIDD} = -283 \pm 36 \pm 6 \; {\rm eV}, \quad
\Gamma_{1s}^{\rm SIDD} = 541 \pm 89 \pm 22 \; {\rm eV}.
\end{equation}
Paradoxically, the directly measurable observables are not reproduced in the same
way in the most of the theoretical works devoted to the antikaon-nucleon interaction.
Some approximate formula are usually used for reproducing the $1s$ level shift.
The most popular is a ``corrected Deser'' formula~\cite{corDeser}, which connects
the shift with the scattering length $a_{K^- p}$ of the $K^- p$ system:
\begin{eqnarray}
\label{corDes}
\Delta E^{cD} - i \, \frac{\Gamma^{cD}}{2}
&=& - 2 \alpha^3 \mu_{K^-d}^2 \, a_{K^-p} \\
\nonumber
&{}& \times [1 - 2 \alpha \mu_{K^-p} \, a_{K^- p} \, (\ln \alpha - 1)].
\end{eqnarray}
The formula is one of quite a few versions of the original formula, derived
by Deser for the pion-nucleon system. It differs from the original one by the second term
in the brackets. However, it was shown (e.g. in \cite{our_KN} and other papers) that
for the antikaon-nucleon system the formula is not accurate, it gives $\sim 10\%$ error.
\subsection{Phenomenological and chirally motivated $\bar{K}N$ potentials}
\label{V_KN_sect}
The constructed phenomenological models of antikaon-nucleon interaction with one- or
two-pole structure of the $\Lambda(1405)$ resonance together with the chirally motivated
model reproduce the $1s$ level shift and width of kaonic hydrogen, measured
by SIDDHARTA collaboration, directly, without using approximate formulas. The potentials
also reproduce the experimental data on the $K^- p$ scattering and the threshold branching
ratios, described in the previous subsection. All three potentials are suitable for using
in accurate few-body equations.
The problem of two particles interacting by the strong and Coulomb potentials,
considered on the equal basis, was solved. The method of solution of Lippmann-Schwinger
equation for a system with Coulomb plus a separable strong potential is based on the fact
that the full $T$-matrix of the problem can be written as a sum $T = T^c + T^{sc}$.
Here $T_c$ is the pure Coulomb transition matrix and $T^{sc}$ is the Coulomb-modified
strong $T$-matrix. It was necessary to extend the formalism to describe the system
of the coupled $\bar{K}N$, $\pi \Sigma$ (and $\pi \Lambda$ for the chirally
motivated potential) channels. The physical masses of the particles were used in the equation,
therefore, the two-body isospin of the system is not conserved. More details on the formalism
can be found in~\cite{our_KN}.
\begin{center}
\begin{table}
\caption{Physical characteristics of the three antikaon nucleon potentials:
phenomenological $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$ and
$V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ with one- and two-pole structure of
the $\Lambda(1405)$ resonance respectively, and the chirally
motivated $V^{Chiral}_{\bar{K}N - \pi \Sigma-\pi \Lambda}$ potential:
$1s$ level shift $\Delta E_{1s}^{K^- p}$ (eV) and
width $\Gamma_{1s}^{K^- p}$ (eV) of kaonic hydrogen,
threshold branching ratios $\gamma$, $R_c$ and $R_n$ together
with the experimental data. The additional $R_{\pi \Sigma}$ ratio, see Eq.(\ref{RpiSigma}),
with its ``experimental'' value is shown as well. Scattering length of the $K^- p$
system $a_{K^- p}$ (fm) and pole(s) $z_1$, $z_2$ (MeV) forming the $\Lambda(1405)$ resonance
are also demonstrated.}
\label{phys_char.tab}
\begin{tabular}{ccccc}
\hline \noalign{\smallskip}
& $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$ & $V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ &
$V^{Chiral}_{\bar{K}N - \pi \Sigma-\pi \Lambda}$ & Experiment \\
\noalign{\smallskip} \hline \noalign{\smallskip}
$\Delta E_{1s}^{K^- p}$ & -313 & -308 & $-313$ & $-283 \pm 36 \pm 6$~\cite{SIDDHARTA} \\
$\Gamma_{1s}^{K^- p}$ & 597 & 602 & $561$ & $541 \pm 89 \pm 22$~\cite{SIDDHARTA} \\
$\gamma$ & 2.36 & 2.36 & 2.35 & $2.36 \pm 0.04$~\cite{gammaKp1,gammaKp2} \\
$R_c$ & - & - & 0.663 & $0.664 \pm 0.011$~\cite{gammaKp1,gammaKp2} \\
$R_n$ & - & - & 0.191 & $0.189 \pm 0.015$~\cite{gammaKp1,gammaKp2} \\
\noalign{\smallskip} \hline \noalign{\smallskip}
$R_{\pi \Sigma}$ & 0.709 & 0.709 & - & $0.709 \pm 0.011$~Eq.(\ref{RpiSigma_exp}) \\
$a_{K^- p}$ & -0.76 + i 0.89 & -0.74 + i 0.90 & -0.77 + i 0.84 & \\
$z_1$ & 1426 - i 48 & 1414 - i 58 & 1417 - i 33 & \\
$z_2$ & - & 1386 - i 104 & 1406 - i 89 & \\
\noalign{\smallskip} \hline
\end{tabular}
\end{table}
\end{center}
The phenomenological potentials describing the $\bar{K}N$ system with coupled
$\pi \Sigma$ channel are the one-term separable ones defined by~Eq.(\ref{V_Nterm}).
In momentum representation they have a form
\begin{equation}
\label{Vseprb}
V_{I}^{\bar{\alpha} \bar{\beta}}(k^{\bar{\alpha}},k'^{\bar{\beta}}) =
\lambda_{I}^{\bar{\alpha} \bar{\beta}} \;
g^{\bar{\alpha}}(k^{\bar{\alpha}}) \, g^{\bar{\beta}}(k'^{\bar{\beta}}),
\end{equation}
where indices $\bar{\alpha}, \bar{\beta} = 1, 2$ denote the $\bar{K}N$
or $\pi \Sigma$ channel respectively, and $I$ is a two-body isospin.
Different form-factors were used for the one- and two-pole versions
of the phenomenological potential. While for the one-pole version
Yamaguchi form-factors
\begin{equation}
\label{1res_ff}
g^{\bar{\alpha}}(k^{\bar{\alpha}}) = \frac{1}{(k^{\bar{\alpha}})^2 +
(\beta^{\bar{\alpha}})^2}
\end{equation}
were used, the two-pole version has slightly more complicated form-factors
in the $\pi \Sigma$ channel
\begin{equation}
\label{2res_ffpi}
g^{\bar{\alpha}}(k^{\bar{\alpha}}) =
\frac{1}{(k^{\bar{\alpha}})^2
+ (\beta^{\bar{\alpha}})^2} \,+\,
\frac{s \, (\beta^{\bar{\alpha}})^2}{[(k^{\bar{\alpha}})^2 +
(\beta^{\bar{\alpha}})^2]^2} \,.
\end{equation}
In the $\bar{K}N$ channel the form-factor of the two-pole version is also
of Yamaguchi form Eq.(\ref{1res_ff}).
Range parameters $\beta^{\bar{\alpha}}$,
strength parameters $\lambda_{I}^{\bar{\alpha} \bar{\beta}}$
and an additional parameter $s$ of the two-pole version were obtained by
fitting to the experimental data described in the previous subsection.
They are: the elastic and inelastic $K^- p$ cross-sections, the threshold branching
ratios and the $1s$ level shift and width of kaonic hydrogen. The first versions
of the potentials, presented in~\cite{our_KN} and used in \cite{my_Kd}, were
fitted to the KEK data \cite{KEK} on kaonic hydrogen. The actual versions of the
phenomenological potentials were fitted to the most recent experimental data of
SIDDHARTA collaboration \cite{SIDDHARTA}. The parameters of the one- and
two-pole versions of the phenomenological potentials fitted to SIDDHARTA data
can be found in~\cite{my_Kd_sdvig}.
All fits were performed directly to the experimental values except the threshold
branching ratios $R_c$ and $R_n$. The reason is that the ratios contain data on scattering
of $K^- p$ into all inelastic channels including $\pi \Lambda$, which is taken
by the phenomenological potentials into account
only indirectly through imaginary part of one of the $\lambda$ parameters. Due
to this the phenomenological potentials were fitted to the new ratio $R_{\pi \Sigma}$,
defined in Eq.(\ref{RpiSigma_exp}).
The third model of the antikaon-nucleon interaction is the chirally motivated potential.
It connects all three open channels: $\bar{K}N$, $\pi \Sigma$ and $\pi \Lambda$,
and has a form
\begin{equation}
\label{VchiralIso}
V_{I I'}^{\alpha \beta}(k^{\alpha},k'^{\beta};\sqrt{s} ) =
g_I^{\alpha} (k^{\alpha}) \, \bar{V}_{I I'}^{\alpha \beta}(\sqrt{s}) \,
g_{I'}^{\beta} (k'^{\beta}) \,,
\end{equation}
where $V_{I I'}^{\alpha \beta}(\sqrt{s})$ is the energy dependent part of the
potential in isospin basis. In particle basis the energy dependent part
has a form
\begin{equation}
\label{VchiralPart}
\bar{V}^{ab}(\sqrt{s} ) =
\sqrt{ \frac{M_{a}}{2 \omega_{a} E_{a} }} \,
\frac{C^{ab}(\sqrt{s})}{(2 \pi)^3 f_{a} f_{b}} \,
\sqrt{ \frac{M_{b}}{2 \omega_{b} E_{b} }} \,.
\end{equation}
Indices $a, b$ here denote the particle channels
$a, b = K^- p, \bar{K}^0n, \pi^+ \Sigma^-, \pi^0 \Sigma^0, \pi^- \Sigma^+$
and $\pi^0 \Lambda$. the square roots with baryon mass $M_{a}$, baryon energy
$E_{a}$ and meson energy $w_{a}$ of the channel $a$ ensure proper normalization
of the corresponding amplitude. SU(3) Clebsh-Gordan coefficients $C^{WT}_I$
enter the non-relativistic form of the leading order Weinberg-Tomozawa interaction
\begin{equation}
C^{ab}(\sqrt{s}) =
- C^{WT} \, (2\sqrt{s} - M_{a} - M_{b}).
\end{equation}
Since, as in the case of the phenomenological potentials, the physical
masses of the particles were used, the two-body isospin $I$ in Eq.(\ref{VchiralIso})
is not conserved. It is different from the phenomenological potentials situation since
in that case the potentials conserve the two-body isospin (but the corresponding
$T$-matrices do not). Another feature, which distinguish the chirally motivated
potential from the phenomenological ones, is the isospin dependence of its
form-factors:
\begin{equation}
g_I^{\alpha} (k^{\alpha}) =
\frac{(\beta_I^{\alpha})^2}{(k^{\alpha})^2 + (\beta_I^{\alpha})^2}.
\end{equation}
Besides, they are dimensionless due to the additional factor $(\beta_I^{\alpha})^2$ in
the numerator.
The pseudo-scalar meson decay constants $f_{\pi}$, $f_K$ and the isospin dependent
range parameters $\beta_I^{\alpha}$ are free parameters of the chirally motivated potential.
They also were found by fitting the potential to the experimental data in the same way as
in the phenomenological potentials case. The chirally-motivated
$\bar{K}N - \pi \Sigma - \pi \Lambda$ potential reproduces the elastic and inelastic
$K^- p$ cross-sections, SIDDHARTA $1s$ level shift and width of kaonic hydrogen. In contrast
to the phenomenological potentials, the chirally-motivated one directly reproduces all three
$K^- p$ branching ratios: $\gamma$, $R_c$ and $R_n$. The parameters of the potential
can be found in~\cite{ourKNN_I}.
The $\Lambda(1405)$ resonance can manifest itself as a bump in elastic $\pi \Sigma$
cross-sections or in $K^- p$ amplitudes. In the last case, the real part of the
amplitude crosses zero, while the imaginary part has a maximum near the resonance
position. It is demonstrated in \cite{my_Kd_sdvig} and \cite{ourKNN_I} that
the elastic $\pi \Sigma$ cross-sections, provided by the three potentials, have a bump
near the PDG value \cite{PDG} for the mass of the $\Lambda(1405)$ resonance
with appropriate width.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Six_plots_ChiralSIDD12.pdf}
\caption{Elastic and inelastic $K^- p$ cross-sections, obtained with
the one-pole $V^{\rm 1, SIDD}_{\bar{K}N - \pi \Sigma}$ (dash-dotted line),
the two-pole $V^{\rm 2, SIDD}_{\bar{K}N - \pi \Sigma}$ (dotted line)
phenomenological potentials,
and the chirally motivated $V^{Chiral}_{\bar{K}N - \pi \Sigma-\pi \Lambda}$
(solid line) potential. The experimental data are taken
from~\cite{Kp2exp,Kp3exp_1,Kp3exp_2,Kp4exp,Kp5exp,Kp6exp}.
\label{CrossKp.fig}}
\end{figure}
The physical characteristics of the three antikaon-nucleon potentials are shown in
Table~\ref{phys_char.tab}. In addition, the $K^- p$ scattering length $a_{K^- p}$
and the pole(s) forming the $\Lambda(1405)$ resonance, given by the potentials,
are demonstrated. The elastic and inelastic $K^- p$ cross-sections, provided by
the three potentials, are plotted in Figure \ref{CrossKp.fig} together with the
experimental data. It is seen form the Table~\ref{phys_char.tab} and
Figure \ref{CrossKp.fig} that the one- and two-pole phenomenological potentials
and the chirally motivated potential describe the all experimental
data with equal high accuracy. Therefore, it is not possible to choose one of
the three models of the $\bar{K}N$ interaction looking at the two-body system.
\vspace*{2mm}
\noindent \underline{Approximate versions of the coupled-channel potential}
In order to check approximations used in other theoretical works, two approximate
versions of the coupled-channel potentials~Eq.(\ref{Vseprb}), which have only one
$\bar{K}N$ channel, were also used. They are: an exact optical potential and
a simple complex one.
The exact optical one-channel potential, corresponding to a two-channel one,
is given by Eq.~(\ref{Vseprb}) with $\bar{\alpha}, \bar{\beta} = 1$ and
the strength parameter defined as
\begin{equation}
\label{lambdaOpt}
\lambda^{11,{\rm Opt}}_{I} = \lambda^{11}_{I} +
\frac{(\lambda^{12}_{I})^2 \,
\langle g^{2}_{I} |\, G_0^{(2)}(z^{(2)}) |\, g^{2}_{I} \rangle}{
1 - \lambda^{22}_{I} \, \langle g^{2}_{I} |\, G_0^{(2)}(z^{(2)}) |\, g^{2}_{I} \rangle
} \,.
\end{equation}
Here $\lambda^{\bar{\alpha},\bar{\beta}}_I$ are the strength parameters of the two-channel
potential, and $| g^{2}_{I} \rangle$ is the form-factor of the second channel.
Since a two-body free Green's function $G_0^{(2)}$ depends on the corresponding
two-body energy, the parameter $\lambda^{11,{\rm Opt}}_{I}$ of the exact optical
potential is an energy-dependent complex function. The exact optical potential has
exactly the same elastic amplitudes of the $\bar{K}N$ scattering as the elastic part
of the full potential with coupled channels.
A simple complex potential is quite often miscalled ``an optical'' one, however, it is
principally different. The strength parameter $\lambda^{11,{\rm Complex}}_{I}$
of a simple complex potential is a complex constant, therefore, the simple complex
potential is energy independent. The strength parameter of a simple complex potential
is usually chosen in such a way, that the potential reproduces only some characteristics
of the interaction. The simple complex potential as well as the exact optical one take
into account flux losses into inelastic channels through imaginary parts of the strength
parameters.
\subsection{Nucleon-nucleon and $\Sigma N (-\Lambda N)$ potentials}
\label{NN_SigN_sect}
\noindent \underline{$NN$ interaction}
Different $NN$ potentials, in particular, TSA-A, TSA-B and PEST, were used in order
to investigate dependence of the three-body results on the nucleon-nucleon interaction models.
A two-term separable $NN$ potential~\cite{Doles_NN}, called TSA, reproduces
Argonne $V18$~\cite{ArgonneV18} phase shifts and, therefore, is repulsive at
short distances. Two versions of the potential (TSA-A and TSA-B) with
slightly different form-factors
\begin{eqnarray}
&{}& g_{(m)}^{A,NN}(k) = \sum_{n=1}^2 \frac{\gamma_{(m)n}^A}{(\beta_{(m)n}^A)^2 + k^2},
\quad {\rm for \;} (m)=1,2 \\
\nonumber
&{}& g_{(1)}^{B,NN}(k) = \sum_{n=1}^3 \frac{\gamma_{(1)n}^B}{(\beta_{(1)n}^B)^2 + k^2},
\quad
g_{(2)}^{B,NN}(k) = \sum_{n=1}^2 \frac{\gamma_{(2)n}^B}{(\beta_{(2)n}^B)^2 + k^2}
\end{eqnarray}
were used. TSA-A and TSA-B potentials properly reproduce the $NN$ scattering lengths and
effective radii, they also give correct binding energy of the deuteron in the ${}^3S_1$ state.
For more details see Ref. \cite{my_Kd}.
A separabelization of the Paris model of the $NN$ interaction, called PEST potential~\cite{NNpot},
was also used. The strength parameter of the one-term PEST is equal to $-1$,
while the form-factor is defined by
\begin{equation}
g_{I}^{NN}(k) = \frac{1}{2 \sqrt{\pi}} \, \sum_{n=1}^6
\frac{c_{n,I}^{NN}}{k^2 + (\beta_{n,I}^{NN})^2}
\end{equation}
with $c_{n,I}^{NN}$ and $\beta_{n,I}^{NN}$ being the parameters.
PEST is equivalent to the Paris potential on and off energy shell
up to $E_{\,\rm lab} \sim 50$ MeV. It also reproduces the deuteron binding
energy in the ${}^3S_1$ state, as well as the triplet and singlet $NN$
scattering lengths.
The quality of reproducing the ${}^3S_1$ phase shifts by the three $NN$ potentials
is shown in Fig.8 of \cite{my_Kd}, were they are compared with those given by the Argonne V18
model. The two-term TSA-A and TSA-B potentials are very good at reproducing of
the Argonne V18 phase shifts. They cross the real axis, which is a consequence of the $NN$
repulsion at short distances. The one-term PEST potential does not have such a property,
but its phase shifts are also close to the ``standard'' ones at lower energies.
Only isospin-singlet $NN$ potential enters the AGS equations for the $K^- d$ system and
only isospin-triplet one enters the equations describing the $K^- pp$ system after
antisymmetrization.
\vspace*{2mm}
\noindent \underline{$\Sigma N$ interaction}
A spin dependent $V^{\rm Sdep}$ and an independent of spin $V^{\rm Sind}$ versions
of the $\Sigma N$ potential were constructed in \cite{my_Kd} in such a way, that
they reproduce the experimental $\Sigma N$ and $\Lambda N$
cross-sections~\cite{SigmaN1,SigmaN2,SigmaN3,SigmaN4,SigmaN5}.
The one-term separable potentials with Yamaguchi form-factors were used for the
two possible isospin states, but with different number of the channels.
Parameters of the one-channel $I=\frac{3}{2}$ state were fitted to the
$\Sigma^+ p \to \Sigma^+ p$ cross-sections. The $\Sigma N$ system in isospin one half
state is connected to the $\Lambda N$ channel, therefore, a coupled-channel potential
of the $I=\frac{1}{2}$ $\Sigma N - \Lambda N$ interaction was constructed first.
The coupled-channel $I=\frac{1}{2}$ potential together with the one-channel $I=\frac{3}{2}$
potential reproduce the $\Sigma^- p \to \Sigma^- p$, $\Sigma^- p \to \Sigma^0 n$,
$\Sigma^- p \to \Lambda n$, and $\Lambda p \to \Lambda p$ cross-sections.
It is seen in Fig.9 of \cite{my_Kd} that both $V^{\rm Sdep}$ and $V^{\rm Sind}$
versions of the $I=3/2$ $\Sigma N$ and $I=1/2$ $\Sigma N - \Lambda N$ potentials
reproduce the experimental data perfectly. Parameters of the potentials and
the scattering lengths $a^{\Sigma N}_{\frac{1}{2}}$, $a^{\Sigma N}_{\frac{3}{2}}$,
and $a^{\Lambda N}_{\frac{1}{2}}$, given by them, are shown in Table 5 of \cite{my_Kd}.
For the three-body $\bar{K}NN$ calculations, where a channel containing $\Lambda$ is not
included directly, not a coupled-channel, but one-channel $\Sigma N$ models of
the interaction in the $I=\frac{1}{2}$ state were used. They are an exact
optical $V^{\Sigma N, \rm{Opt}}$ potential and a simple complex $V^{\Sigma N, \rm{Complex}}$
one, corresponding to the $I=\frac{1}{2}$ $\Sigma N - \Lambda N$ model
with coupled channels. The exact optical potential has an energy dependent strength
parameter defined by Eq.~(\ref{lambdaOpt}), it reproduces the elastic $\Sigma N$ amplitude
of the corresponding two-channel potential exactly. The simple complex potential gives
the same scattering lengths, as the two-channel potential.
\subsection{Antikaon-antikaon interaction}
\label{V_KK_sect}
Lack of an experimental information on the $\bar{K}\bar{K}$ interaction
means that it is not possible to construct the $\bar{K}\bar{K}$ potential in the same
way as the $\bar{K}N$ or $\Sigma N$ ones. Due to this, theoretical results
of a modified model describing the $\pi\pi - K\bar{K}$ system developed by the J\"ulich
group \cite{Lohse90,Janssen95} were used. The original model yields a good description of
the $\pi\pi$ phase shifts up to partial waves with total angular momentum $J=2$ and for
energies up to $z_{\pi\pi}\approx 1.4$ GeV. In addition, the $f_0(980)$ and $a_0(980)$
mesons result as dynamically generated states.
Based on the underlying SU(3) flavor symmetry, the interaction in
the $\bar{K}\bar{K}$ system was directly deduced from the $K\bar{K}$ interaction
without any further assumptions. The $\bar{K}\bar{K}$ scattering length
predicted by the modified J\"ulich model is $a_{\bar{K}\bar{K},I=1}= -0.186$ fm,
therefore, it is a repulsive interaction. This version of the $\bar{K}\bar{K}$
interaction was called ''Original''.
Recent results for the $KK$ scattering length from lattice QCD simulations
suggest values of $a_{\bar{K}\bar{K},I=1}= (-0.141\pm 0.006)$ fm \cite{Beane:2007uh}
and $a_{\bar{K}\bar{K},I=1}= (-0.124\pm 0.006\pm 0.013)$ fm \cite{Sasaki:2013vxa}.
Those absolute values are noticeably smaller than the one predicted by the Original
J\"ulich meson-exchange model and, accordingly, imply a somewhat less repulsive
$\bar{K}\bar{K}$ interaction. Due to this, another version of the interaction that is
in line with the lattice QCD results was also constructed. It yields scattering length
$a_{\bar{K}\bar{K},I=1}= -0.142$ fm. This version of the $\bar{K}\bar{K}$ interaction
was called ''Lattice motivated''.
However, the models of the $\bar{K}\bar{K}$ interaction described above cannot be
directly used in the AGS equations. Due to this, the $\bar{K}\bar{K}$ interaction
was represented in a form of the one-term separable potential with form factors given by
\begin{equation}
g(k) = \frac{1}{\beta_1^2 + k^2} +
\frac{\gamma}{\beta_2^2 + k^2}.
\end{equation}
The strength parameters $\lambda$, $\gamma$ and range parameters
$\beta$ were fixed by fits to the $\bar{K}\bar{K}$ phase shifts and
scattering lengths of the ''Original'' and the `'Lattice motivated''
models of the antikaon-antikaon interaction.
\section{AGS equations for coupled $\bar{K}NN - \pi \Sigma N$ and
$\bar{K}\bar{K}N - \bar{K}\pi\Sigma$ channels}
\label{AGS_sect}
The three-body Faddeev equations in the Alt-Grassberger-Sandhas (AGS)
form~\cite{AGS} were used for the most of the three-body calculations.
The equations were extended in order to take the $\pi \Sigma$ channel, coupled to
the $\bar{K}N$ subsystem, directly. In practice it means that all operators
entering the system
\begin{equation}
\label{U_coupled}
U_{ij}^{\alpha \beta} = {\delta}_{\alpha \beta} \,(1-\delta_{ij})
\, \left(G_0^{\alpha} \right)^{-1} + \sum_{k, \gamma=1}^3 (1-\delta_{ik})
\, T_k^{\alpha \gamma} \, G_0^{\gamma} \, U_{kj}^{\gamma \beta} \, ,
\end{equation}
namely, transition operators $U_{ij}$, two-body $T$-matrices $T_i$
and the free Green function $G_0$, - have additional channel indices $\alpha, \beta = 1,2,3$
in addition to the Faddeev partition indices $i,j = 1,2,3$.
The additional $\pi \Sigma N$ ($\alpha=2$) and $\pi N \Sigma$ ($\alpha=3$) channels
were added to the $\bar{K}NN$ system , while the $\bar{K}\bar{K}N$ system
was extended to the $\bar{K} \pi \Sigma$ ($\alpha=2$) and $\pi \bar{K} \Sigma$
($\alpha=3$) channels. A Faddeev index $i$, as usual, defines a particle and
the remained pair, now in the particular particle channel $\alpha$. The combinations
of the $(i,\alpha)$ indices with possible two-body isospin values can be found
in~\cite{myKpp_PRC} for the $\bar{K}NN$ system and in~\cite{my_KKN} for
the $\bar{K}\bar{K}N$ systems, respectively.
Since the separable potentials Eq.(\ref{V_Nterm}), leading to the separable $T$-matrices
Eq.(\ref{T_Nterm}), were used as an input, the system Eq.(\ref{U_coupled})
turned into the new system of operator equations
\begin{equation}
\label{full_oper_eq}
X_{ij, I_i I_j}^{\alpha \beta} = \delta_{\alpha \beta} \,
Z_{ij, I_i I_j}^{\alpha} +
\sum_{k=1}^3 \sum_{\gamma=1}^3 \sum_{I_k}
Z_{ik, I_i I_k}^{\alpha} \, \tau_{k, I_k}^{\alpha \gamma} \,
X_{kj, I_k I_j}^{\gamma \beta}
\end{equation}
with $X_{ij, I_i I_j}^{\alpha \beta}$ and $Z_{ij, I_i I_j}^{\alpha \beta}$
being new transition and kernel operators respectively
\begin{eqnarray}
\label{X_definition}
X_{ij, I_i I_j}^{\alpha \beta} &=& \langle
g_{i,I_i}^{\alpha} | G_0^{\alpha} \, U_{ij, I_i I_j}^{\alpha
\beta} G_0^{\beta} | g_{j,I_j}^{\beta} \rangle \,, \\
\label{Z_definition}
Z_{ij, I_i I_j}^{\alpha \beta} &=&
\delta_{\alpha \beta} \, Z_{ij, I_i I_j}^{\alpha} =
\delta_{\alpha \beta} \, (1-\delta_{ij}) \,
\langle g_{i,I_i}^{\alpha} | G_0^{\alpha} | g_{j,I_j}^{\alpha}
\rangle \, .
\end{eqnarray}
\vspace*{2mm}
\noindent
\underline{$\bar{K}NN - \pi \Sigma N$ system}
The two states of the strangeness $S=-1$ $\bar{K}NN$ system were considered.
The $K^- pp$ and $K^- d$ systems differ from each another by the total spin value,
which leads to different symmetry of the operators describing the system containing
two identical baryons, $NN$. This fact is taken into account when the three-body
coupled-channel equations are antisymmetrized.
All calculations were performed under or slightly above the $\bar{K}NN$ threshold,
so that orbital angular momentum of all two-body interactions was set to zero
and, therefore, the total orbital angular momentum is also $L=0$. In particular,
the main $\bar{K}N$ potential was constructed with orbital angular momentum $l=0$ since
the interaction is dominated by the $s$-wave $\Lambda(1405)$ resonance. The interaction
of $\pi$-meson with a nucleon is weaker than
the other interactions, therefore, it was omitted in the equations.
An experimental information about the $\Sigma N$ interaction is very poor, and there
is no reason to assume significant effect of higher partial waves. Finally, the $NN$ interaction
was also taken in $l=0$ state only, since physical reasons for
sufficient effect of higher partial waves in the present calculation are not seen.
Spin of the $\bar{K}NN$ system is given by spin of the two baryons, which also defines
the $NN$ isospin due to the symmetry properties. Looking for the quasi-bound state
in $\bar{K}NN$, the isospin $I=1/2$ and spin zero state, usually denoted as $K^- pp$,
was chosen due to its connection to experiment. Another possible configuration
with the same isospin and spin one, which is $K^- d$, was also studied. As for
the $\bar{K}NN$ state with isospin $I=3/2$, it is governed by the isospin $I_i=1$
$\bar{K}N$ interaction, which is much weaker attractive than the one in the $I_i=0$ state
or even repulsive. Therefore, no quasi-bound state is expected there.
The nucleons, entering the highest $\bar{K}NN$ channel, require
antisymmetrization of the operators entering the system of equations~Eq.(\ref{full_oper_eq}).
Two identical baryons with symmetric spatial components ($L_i=0$) has
antisymmetric ($S_i=0$) spin components for the $pp$ state of the $NN$ subsystem
or symmetric ($S_i=1$) ones for the $d$ state.
The operator $X_{1,1}^1$ has symmetric $NN$ isospin components,
therefore, it has the correct symmetry properties for the $K^- pp$
system (here and in what follows the right-hand indices of
$X$ are omitted: $X_{ij,I_i I_j}^{\alpha \beta} \to X_{i,I_i}^{\alpha}$).
Another operator, $X_{1,0}^1$, has antisymmetric $NN$ isospin components,
so it drops out the equations for the $K^- pp$ system, but remains
in the equations describing the $K^- d$ system. All the remaining
operators form symmetric and antisymmetric pairs.
At the end there is a system of 9 (with PEST $NN$ potential) or 10
(with TSA nucleon-nucleon model) coupled operator equations, which has
the required symmetry properties.
The system of operator equations Eq. (\ref{full_oper_eq}) written in momentum space
turns into a system of integral equations. To search for a quasi-bound state
in a three-body system means to look for a solution of the homogeneous system
corresponding to Eq.~(\ref{full_oper_eq}). Calculation of three-body scattering amplitudes
require solution of the inhomogeneous system. In the both cases the integral equations
are transformed into algebraic ones. The methods of solution are different for
the quasi-bound state and scattering problems, so they are discussed in
the corresponding sections.
More details on the three-body equations with coupled $\bar{K}NN - \pi \Sigma N$
channels can be found in~\cite{myKpp_PRC} for the $K^- pp$ and in~\cite{my_Kd}
for the $K^- d$ systems.
\vspace*{2mm}
\noindent
\underline{$\bar{K}\bar{K}N - \bar{K} \pi \Sigma$ system}
As for the strangeness $S=-2$ $\bar{K}\bar{K}N$ system,
its total spin is equal to one half since an antikaon is a pseudoscalar meson. Since the
two-body interactions, namely the $\bar{K}N - \pi \Sigma$ and $\bar{K} \bar{K}$ potentials,
were chosen to have zero orbital angular momentum, the total angular momentum is also
equal to $1/2$. As in the case of the $\bar{K}NN$ system, the state of
the $\bar{K}\bar{K}N$ system with the lowest possible value of the isospin $I=1/2$
was considered.
Two identical antikaons should have a symmetric way function, therefore,
the $\bar{K}\bar{K}$ pair in $s$-wave can be in isospin one state only.
Accordingly, the three-body operators entering the AGS system for the $\bar{K}\bar{K}N$
system were symmetrized. Similarly to the $\bar{K}NN$ case, the transition operator
$X_{3,1}^1$, entering the equations describing $\bar{K}\bar{K}N$, already has the proper
symmetry properties. The remaining operators form pairs with proper symmetry properties.
\vspace*{2mm}
It is necessary to note that while Coulomb potential was directly included in
the two-body equations, used for fitting the antikaon-nucleon potentials,
the three-body calculations were performed without it (except the case of kaonic
deuterium calculations, of cause). The reason is that the expected effect of its inclusion
is small. In addition, the isospin averaged masses were used in all three-body calculations
in contrast to the two-body $\bar{K}N$ case. Accuracy of this approximation was checked
in \cite{revai_1405}, and it turned out to be quite high.
\section{Quasi-bound states}
\label{qbs_sect}
It was shown in Section \ref{V_KN_sect} that the phenomenological $\bar{K}N$ potentials
with one- and two-pole structure of the $\Lambda(1405)$ resonance and
the chirally motivated antikaon-nucleon potential can reproduce near-threshold
experimental data on $K^- p$ scattering and kaonic hydrogen with equal accuracy.
Therefore, it is not possible to choose one of these models looking at
the two-body system only. Due to this, the three-body calculations were performed
using all three models of the antikaon-nucleon interaction.
The quasi-bound state in the $K^- pp$ system was the phenomenon, which
attracted the present interest to the antikaon-nucleus systems. Additionally
to being an interesting exotic object, the state could clarify
still unanswered questions on the antikaon-nucleon interaction,
in particular, the nature of the $\Lambda(1405)$ resonance.
$\bar{K}\bar{K}N$ is one more possible candidate for a strange three-body system
with the quasi-bound state in it. However, the strangeness $S=-2$ system contains
$\bar{K}\bar{K}$ interaction, which is repulsive. The question was,
whether the repulsion is strong enough to overtake $\bar{K}N$ attraction and
by this exclude the possibility of the quasi bound state formation.
\subsection{Two ways of a quasi-bound state evaluation}
\label{qbs_details_sect}
The quasi-bound state, which is a bound state with a non-zero width, for the
higher $\bar{K}NN$ (or $\bar{K}\bar{K}N$) channel, is at the same time
a resonance for the lower $\pi \Sigma N$ ($\bar{K} \pi \Sigma$) channel.
Therefore, the corresponding pole should be situated between the
$\bar{K}NN$ ($\bar{K}\bar{K}N$) and $\pi \Sigma N$ ($\bar{K} \pi \Sigma$))
thresholds on the physical energy sheet of the higher channel and on
an unphysical sheet of the lower channel. Two methods of searching of the complex
pole position by solving the homogeneous system of equations were used.
The first one is the direct pole search with contour rotation. The correct
analytical continuation of the equations from the physical energy sheet
to the proper unphysical one is achieved by moving the momentum integration
into the complex plane. Namely, the integration was performed along a ray in
the fourth quadrant of the complex plane with some condition on the momentum
variable. After that the position $z_0$ of a quasi-bound state was found by solving
the equation ${\rm Det}(z_0) = 0$, where ${\rm Det}(z)$ is the determinant of
the linear system, obtained after discretization of the integral equations,
corresponding to Eq.~(\ref{full_oper_eq}).
Another way of a quasi-bound state searching, which avoids integration in
the complex plane, was suggested and used in~\cite{ourKNN_II}.
The idea is that every isolated and quite narrow resonance should manifest
itself at real energies. Namely, resonances are usually seen in cross-sections
of some reactions. The function $1/{\rm Det}(z)$ enters all possible amplitudes,
described by a system of three-body integral equations. Therefore,
the function $1/|{\rm Det}(z)|^2$, entering all possible cross-sections, can be
calculated instead of some cross-sections. The function is universal, it does not
contain additional information about the particular processes in the three-body
system. The corresponding bump of the $1/|{\rm Det}(z)|^2$ function, calculated
at real energies, can be fitted by a Breit-Wigner curve with arbitrary background.
In this way an information on the resonance position and width can be obtained.
It is clear that the second method can work only if the resonance bump is
isolated and not too wide. The bump corresponding to the $K^- pp$ quasi-bound
state satisfies these conditions \cite{ourKNN_II}, as is seen in Fig. \ref{detsBW.fig}.
The calculated $1/|{\rm Det}(z)|^2$ functions of the AGS system of equations
are shown there as symbols while the corresponding Breit-Wigner fitting curves
are drown in lines. The results obtained with the three $\bar{K}N$ potentials,
described in Section \ref{V_KN_sect}, are shown in the figure.
Since direct search of the complex root is a non-trivial task, the Breit-Wigner values
of the $1/|{\rm Det}(z)|^2$ function can give a good starting point for it. On the
other hand, the $1/|{\rm Det}(z)|^2$ method can be used as a test of the directly
found pole position, which is free of the possible uncertainty of the proper choice
of the Riemann sheet. However, the $1/|{\rm Det}(z)|^2$ method is not easier than
the direct search, since the calculation of the determinant, which is almost equal
to the solving of a scattering problem, should be performed.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{dets_new.pdf}
\caption{Calculated $1/|{\rm Det}(z)|^2$ functions of the AGS system of equations
for the $K^- pp$ system (symbols) and the corresponding Breit-Wigner fits of
the obtained curves (lines) \cite{ourKNN_II} for
the one-pole $V^{\rm 1, SIDD}_{\bar{K}N - \pi \Sigma}$ (triangles and dashed line),
the two-pole $V^{\rm 2, SIDD}_{\bar{K}N - \pi \Sigma}$ (circles and solid line)
phenomenological potentials,
and the chirally motivated $V^{Chiral}_{\bar{K}N - \pi \Sigma-\pi \Lambda}$
(squares and dash-dotted line) potential.}.
\label{detsBW.fig}
\end{figure}
\subsection{$K^- pp$ quasi-bound state: results}
\label{qbs_results_sect}
The first dynamically exact calculation of the quasi-bound state in the $K^- pp$
system was published in~\cite{myKpp_PRL},
while the extended version of the results appeared in~\cite{myKpp_PRC}.
Existence of the $I=1/2, J^{\pi}=0^-$ three-body quasi-bound state
in the $\bar{K}NN$ system, predicted in \cite{AY1,AY2}, was confirmed there,
but the evaluated binding energy and width were strongly different.
However, the $\bar{K}N - \pi \Sigma$ potentials used in \cite{myKpp_PRL,myKpp_PRC}
do not reproduce the experimental data on the $K^- p$ system as accurately, as those
described in Section \ref{V_KN_sect}. Due to this, the calculations
devoted to the $K^- pp$ system were repeated in \cite{ourKNN_II}. The one-pole
$V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$, two-pole $V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$
phenomenological potentials from~\cite{my_Kd_sdvig} and the chirally motivated
$V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$ potential from~\cite{ourKNN_I},
described in Section \ref{V_KN_sect}, were used as an input. The other two potentials
were the two-term TSA-B NN potential \cite{Doles_NN}
together with the spin-independent exact optical $\Sigma N$ potential in isospin
$I=1/2$ state and the one-channel $V_{\Sigma N}$ in $I=3/2$,
see Section \ref{NN_SigN_sect}.
The results of the last calculations \cite{ourKNN_II} of the $K^- pp$ quasi-bound state
are shown in Table~\ref{KNN_poles.tab}. First of all, comparison of
the results obtained using the direct pole search and the $1/|{\rm Det}(z)|^2$ method
demonstrates that they are very close each to other for the every given $\bar{K}N$ potential.
Therefore, the suggested $1/|{\rm Det}(z)|^2$ method of finding mass and width of
a subthreshold resonance is efficient for the $K^- pp$ system, and the two methods
supplement each another.
\begin{center}
\begin{table}
\caption{Binding energy $B_{K^-pp}$ (MeV) and width
$\Gamma_{K^-pp}$ (MeV) of the quasi-bound state in the $K^- pp$ system \cite{ourKNN_II}:
the results of the direct pole search and of the Breit-Wigner fit of the
$1/|{\rm Det(z)}|^2$ function at real energy axis. The AGS calculations
were performed with the one-pole $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$,
two-pole $V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ phenomenological
potentials from~\cite{my_Kd_sdvig} and the chirally motivated
$V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$ potential
from~\cite{ourKNN_I}.}
\label{KNN_poles.tab}
\begin{center}
\begin{tabular}{ccccc}
\hline \noalign{\smallskip}
& \multicolumn{2}{c}{Direct pole search} & \multicolumn{2}{c}{BW fit of $1/|{\rm Det(z)}|^2$} \\
\hline \noalign{\smallskip}
& $B_{K^- pp}$ & $\Gamma_{K^- pp}$ & $B_{K^- pp}$ & $\Gamma_{K^- pp}$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
$V^{1,SIDD}_{\bar{K}N - \pi \Sigma}$
& $53.3$ & $64.8$ & $54.0$ & $66.6$ \\ \noalign{\smallskip}
$V^{2,SIDD}_{\bar{K}N - \pi \Sigma}$
& $47.4$ & $49.8$ & $46.2$ & $51.8$ \\ \noalign{\smallskip}
$V_{\bar{K}N - \pi \Sigma - \pi \Lambda}^{\rm Chiral}$
& $32.2$ & $48.6$ & $30.3$ & $46.6$ \\
\noalign{\smallskip} \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
Another fact, seen from the results in Table~\ref{KNN_poles.tab}, is strong dependence
of the binding energy $B_{K^- pp}$ of the quasi-bound state and its width
$\Gamma_{K^- pp}$ on the $\bar{K}N$ interaction models. It was already observed in
\cite{myKpp_PRL,myKpp_PRC}, when older phenomenological antikaon-nucleon potentials
were used. In particular, it is seen from Table~\ref{KNN_poles.tab} that the quasi-bound
states resulting from the phenomenological potentials lie about $15-20$ MeV deeper than
those of the chirally motivated one. This probably is due to the energy dependence of the
chirally motivated model of the interaction. Really, all three potentials were fitted to
the experimental data near the $\bar{K}N$ threshold. When the $K^- pp$ quasi-bound state
is calculated at lower energies, the strengths of the phenomenological models of
the $\bar{K}N$ interaction are unchanged. As for the chirally motivated potential,
its energy dependence reduces the attraction at the lower energies in the $\bar{K}NN$
quasi-bound state region, thus producing the states with less binding.
The widths of the three quasi-bound states are also different: those of the two-pole models
of the $\bar{K}N$ interaction are almost coinciding, while the width evaluated using
the one-pole $V^{1,SIDD}_{\bar{K}N - \pi \Sigma}$ potential is much larger.
It is seen from Table~\ref{phys_char.tab} that the potentials with the two-pole
$\Lambda(1405)$ structure have very close positions of the higher poles, while the pole of
the one-pole potential is different. Therefore, the difference in widths might be
connected with the different pole structure of the corresponding $\bar{K}N$ interaction
models.
Importance of the proper inclusion of the second $\pi \Sigma N$ channel in the
calculations was first demonstrated in~\cite{myKpp_PRC}. A simple complex version
of the $\bar{K}N - \pi \Sigma$ potentials, described in Section \ref{V_KN_sect},
was used in~\cite{myKpp_PRC} together with the full version with coupled channels.
This allowed to check importance of the proper inclusion of the second channel.
Comparison of the result of the one-channel complex calculation
($B^{1 \,\rm complex}_{K^- pp},\Gamma^{1 \,\rm complex}_{K^- pp}$)
with the coupled-channel one ($B^{2 \,\rm coupled}_{K^- pp},\Gamma^{2 \,\rm coupled}_{K^- pp}$)
\begin{eqnarray}
\label{2chOLDKpp}
B^{2 \,\rm coupled}_{K^- pp} &=& 55.1 \; {\rm MeV}
\qquad \Gamma^{2 \,\rm coupled}_{K^- pp} = 101.8\; {\rm MeV} \\
\label{1chOLDKpp}
B^{1 \,\rm complex}_{K^- pp} &=& 40.2 \; {\rm MeV}
\qquad \Gamma^{1 \,\rm complex}_{K^- pp} = 77.4\; {\rm MeV}
\end{eqnarray}
shows that the quasi-bound state obtained in the full calculation with coupled channels is much
deeper and broader than the approximate one-channel one. (The values for the binding energy and
width in Eq.(\ref{2chOLDKpp}) differ from those in Table~\ref{KNN_poles.tab} since another
$\bar{K}N$ potential was used in \cite{myKpp_PRC}.) This means that the $\pi \Sigma$
channel plays an important dynamical role in forming the three-body quasi-bound state, over
its obvious role of absorbing flux from the $\bar{K}N$ channel. Thus, proper inclusion of
the second $\pi \Sigma$ channel is crucial for the $\bar{K}NN$ system.
It was found later in~\cite{ourKNN_II} that use of the exact optical
$\bar{K}N$ potential can serve an alternate way of direct inclusion
of the $\pi \Sigma$ channel. An accuracy of use of the exact
optical $\bar{K}N$ potential, which gives exactly the same on- and off-shell elastic
$\bar{K}N$ amplitude as the original potential with coupled channels, was checked
in one-channel AGS calculations for the three actual $\bar{K}N$ potentials.
The ``exact optical'' binding energies differ only slightly from the full coupled-channel results
from Table~\ref{KNN_poles.tab}, while the widths gain more visible error:
\begin{eqnarray}
B_{K^- pp}^{1,SIDD,Opt} &=& 54.2\; {\rm MeV}
\qquad \Gamma_{K^- pp}^{1,SIDD,Opt} = 61.0 \; {\rm MeV} \\
B_{K^- pp}^{2,SIDD,Opt} &=& 47.4\; {\rm MeV}
\qquad \Gamma_{K^- pp}^{2,SIDD,Opt} = 46.0 \; {\rm MeV} \\
B_{K^- pp}^{Chiral,Opt} &=& 32.9; {\rm MeV}
\qquad \Gamma_{K^- pp}^{Chiral,Opt} = 48.8 \; {\rm MeV.}
\end{eqnarray}
However, the difference in widths is not dramatic, so the one-channel
Faddeev calculation with the exact optical $\bar{K}N$ potential could be quite satisfactory
approximation to the full calculation with coupled channels.
\subsection{$K^- pp$ quasi-bound state: comparison to other results}
\label{qbs_comparison_sect}
The three binding energy $B_{K^- pp}$ and width $\Gamma_{K^- pp}$ values of
the $K^- pp$ quasi-bound state, shown in Table \ref{KNN_poles.tab}, can be compared
with worth mentioning other theoretical results. Those are: the original prediction
of the deep and narrow quasi-bound $K^- pp$ state~\cite{AY1,AY2}, the results obtained
in the earlier Faddeev calculation~\cite{myKpp_PRC}, the most recent results of alternative
calculation using the same equations with different input~\cite{ikedasato} and
two variational results~\cite{vaizeyap,evrei}. Only calculations presented
in~\cite{ikedasato} together with the earlier ones~\cite{myKpp_PRL,myKpp_PRC}
were performed with directly included $\pi \Sigma N$ channel. All others
take it into account approximately. The second problem is that none of the $\bar{K}N$
potentials, used in all other $K^- pp$ calculations, reproduce data on near-threshold
$K^- p$ scattering with the same level of accuracy as those described in Section~\ref{V_KN_sect}.
In addition, none of them reproduce the $1s$ level shift of kaonic hydrogen directly.
The binding energy of the quasi-bound $K^- pp$ state and its width were obtained in~\cite{AY1,AY2}
from a G-matrix calculation, which is a many-body technics. The one-channel simple
complex $\bar{K}N$ potential used in those calculations does not reproduce the actual
$K^- p$ experimental data. Finally, the authors of~\cite{AY1,AY2} take into
account only the $\bar{K}NN$ channel. In a funny way all the defects of the calculation
presented in~\cite{AY1,AY2} led to the binding energy ($48$ MeV) and width ($61$ MeV),
which are quite close to the exact results from Table~\ref{KNN_poles.tab}
obtained with the two-pole and the one-pole phenomenological potentials, respectively.
The earlier result for the binding energy $B_{K^- pp} = 55.1$ MeV~\cite{myKpp_PRC}
is very close to the actual one from Table~\ref{KNN_poles.tab} calculated with the one-pole
phenomenological $\bar{K}N - \pi \Sigma$ potential. In fact, in both cases the same
three-body equations with coupled $\bar{K}NN$ and $\pi \Sigma N$ channels
were solved. In addition, the same model of the antikaon-nucleon interaction was used,
but with different sets of parameters. This difference influenced the width of
the quasi-bound state: the older $\Gamma_{K^- pp} = 100.2$ MeV is much larger.
Coupled-channel AGS equations were also solved in~\cite{ikedasato}
with chirally motivated energy dependent and independent $\bar{K}N$
potentials. Therefore, in principle, those results obtained with the energy
dependent version of the $\bar{K}N$ potential $V^{E-dep}_{\bar{K}N}$
should give a result, which is close to those from Table~\ref{KNN_poles.tab}
with chirally motivated model of interaction
$V_{\bar{K}N - \pi \Sigma - \pi \Lambda}^{\rm Chiral}$. However, only
those width ($34-46$ MeV) is comparable to the $\Gamma_{K^- pp}^{\rm Chiral}$,
while the binding energy obtained in~\cite{ikedasato} ($9-16$ MeV) is much smaller
than the actual one from Table~\ref{KNN_poles.tab}. The situation is opposite
when the actual results are compared with those obtained
in \cite{ikedasato} using the energy independent antikaon-nucleon potential. Namely,
binding energies reported in that paper ($44-58$ MeV) are comparable with
those from Table~\ref{KNN_poles.tab} evaluated using phenomenological
$V^{1,SIDD}_{\bar{K}N}$ and $V^{2,SIDD}_{\bar{K}N}$
potentials, however, their widths $34-40$ MeV are much smaller.
The authors of \cite{ikedasato} neglect the $\Sigma N$ interaction in their
calculations. It was shown in \cite{myKpp_PRL,myKpp_PRC} that dependence of
the three-body $K^- pp$ pole position on $\Sigma N$ is weak. However,
when the interaction is switched off completely, like in the case of \cite{ikedasato},
some visible effect manifests itself.
An approximation used in the chirally motivated models used in \cite{ikedasato}
is more serious reason of the difference. Namely, the energy-dependent square root
factors, responsible for the correct normalization of the $\bar{K}N$ amplitudes,
are replaced by constant masses. This can be reasonable for the highest $\bar{K}N$ channel,
however, it is certainly a poor approximation for the lower $\pi \Sigma$ and
$\pi \Lambda$ channels. The role of this approximation in the AGS calculations
was checked in \cite{ourKNN_II}, the obtained approximate binding energy
$25$ MeV is really much smaller than the original one $32$ MeV, presented in
Table~\ref{KNN_poles.tab}. The remaining difference between the results
from Table \ref{KNN_poles.tab} and those from \cite{ikedasato} could be explained
by the lower accuracy of reproducing experimental $K^- p$ data by the $\bar{K}N$
potentials from \cite{ikedasato}.
Finally, no second pole in the $K^- pp$ system reported in~\cite{ikedasato} was found
in \cite{ourKNN_II}. The search was performed with all three $\bar{K}N$ potentials
in the corresponding energy region (binding energy $67-89$ MeV and width $244-320$ MeV).
Variational calculations, performed by two groups, reported the results which
are very close to those obtained in~\cite{ikedasato} with the energy-dependent
potential:
$B_{K^- pp} = 17-23$ MeV, $\Gamma_{K^- pp} = 40-70$ MeV in~\cite{vaizeyap}
and
$B_{K^- pp} = 15.7$ MeV, $\Gamma_{K^- pp} = 41.2$ MeV in~\cite{evrei}.
However, there are a few problematic points in~\cite{vaizeyap,evrei}.
First of all, the variational calculations were performed solely in the
$\bar{K}NN$ channel. The authors of the variational calculations
used a one-channel $\bar{K}N$ potential, derived from a chirally motivated model
of interaction with many couped channels. However, the potential is not
the ``exact optical'' one. In fact, it is not clear, how this one-channel potential
is connected to the original one and whether it still reproduces some experimental
$K^- p$ data.
Moreover, the position of the $K^- pp$ quasi-bound state was determined
in~\cite{vaizeyap,evrei} using only the real part of this complex $\bar{K}N$ potential,
as a real bound state. The width was estimated as the expectation value of
the imaginary part of the potential. This, essentially perturbative, treatment of
the inelasticity might be justified for quite narrow resonances, but the
$K^- pp$ quasi-bound state is certainly not of this type.
Another serious problem of the variational calculations is their method of
treatment of the energy dependence of the $\bar{K}N$ potential in the
few-body calculations. It was already shown in the previous subsection
that the energy dependence of the chirally motivated model of the $\bar{K} N$
interaction is very important for the $K^- pp$ quasi-bound state position.
While momentum space Faddeev integral equations allow the exact
treatment of this energy dependence, variational calculations in coordinate space
can use only energy independent interactions. Due to this the energy of the $\bar{K}N$
potential was fixed in ~\cite{vaizeyap,evrei} at a ``self-consistent'' value $z_{\bar{K}N}$.
A series of calculations using the exact AGS equations was performed in \cite{ourKNN_II}
with differently fixed two-particle $\bar{K}N$ energies $z_{\bar{K}N}$ in the couplings
of the chirally motivated interaction. The conclusion was, that it is not possible
to define an ``averaged'' $z_{\bar{K}N}$, for which the fixed-energy
chirally motivated interaction, even in the correct three-body calculation, can
yield a correct $K^- pp$ quasi-bound state position.
First, the calculations of \cite{ourKNN_II} show, that a real $z_{\bar{K}N}$ has
absolutely no chance to reproduce or reasonably approximate the exact quasi-bound
state position, even with correct treatment of the imaginary part of the interaction, unlike
in~\cite{vaizeyap,evrei}. Second, the way, how the ``self-consistent''
value of (generally complex) $z_{\bar{K}N}$ is defined in the papers does not seem
to guarantee, that the correct value will be reached or at least approximated.
In view of the above considerations, the results of~\cite{vaizeyap,evrei}
can be considered as rough estimates of what a really energy dependent $\bar{K}N$
interaction will produce in the $K^- pp$ system.
After publication of the exact results~\cite{ourKNN_II} one more paper on
the $K^-pp$ system appeared \cite{gruziny}. Hyperspherical harmonics in
the momentum representation and Faddeev equations in configuration space
were used there. However, the authors collected all defects of other
approximate calculations. In particular, they used simple complex antikaon-nucleon
potentials and, therefore, neglected proper inclusion of the $\pi \Sigma N$
channel, which is crucial for the system. In addition, the $\bar{K}N$ potentials are
those from \cite{AY2, vaizeyap}, which have problems with reproducing of
the experimental $K^- p$ data. Keeping all this in mind, the results of \cite{gruziny}
hardly can be reliable.
\subsection{$K^- d$ quasi-bound state}
\label{qbsKd_results_sect}
The strongly attractive isospin-zero part of the $\bar{K}N$ potential plays less
important role in the spin-one $K^- d$ state of the $\bar{K}NN$ system than
in the spin-zero $K^- pp$. Therefore, if a quasi-bound state exists in $K^-d$,
it should have smaller binding energy than in $K^- pp$. The Faddeev calculations
of the $K^- d$ scattering length $a_{K^- d}$, described in Section~\ref{Kd_elastic_sect},
gave some evidence that such a state could exists~\cite{ourKNN_I}.
A simple analytical continuation of the effective range formula below the $K^- d$
threshold suggests a $K^- d$ quasi-bound state with binding energy $14.6 - 19.6$ MeV
(the energy is measured from the $K^- d$ threshold) and width $15.6 - 22.0$ MeV
for the three antikaon-nucleon potentials from Section~\ref{V_KN_sect}.
However, a systematic search for these states, performed in \cite{ourKNN_I} with the
same two-body input as for the $K^- pp$ system, did not find the corresponding poles
in the complex energy plane between the $\pi \Sigma N$ and $K^- d$ thresholds.
The reason of discrepancy between the effective range estimations and the direct calculations
must be the validity of the effective range formula, which is limited to the vicinity
of the corresponding threshold. Since the $K^- d$ state is expected to have, similarly
to $K^- pp$, rather large width, it is definitely out of this region.
It was demonstrated in~\cite{ourKNN_I} that increasing of the attraction in
isospin-zero $\bar{K}N$ subsystem by hands (in the phenomenological antikaon-nucleon
potentials only) leads to appearing of $K^- d$ quasi-bound states. Therefore,
the isospin-zero attraction in the $\bar{K}N$ system is not strong enough to
bind antikaon to deuteron. It is necessary to note that the $K^- d$ system with
strong two-body interactions only is considered here. An atomic state caused by Coulomb
interaction, kaonic deuterium, exists and will be considered later.
\subsection{$\bar{K}\bar{K}N$ system: results}
\label{KKN_res_sect}
The calculations of the quasi-bound state in the $\bar{K}\bar{K}N$ system
were performed with the two $\bar{K}\bar{K}$ interactions
described in Section~\ref{V_KK_sect} (Original $V_{\bar{K}\bar{K}}^{Orig}$ and
Lattice-motivated $V_{\bar{K}\bar{K}}^{Latt}$)
and three $\bar{K}N$ potentials from Section~\ref{V_KN_sect}: the phenomenological
one-pole $V^{1,SIDD}_{\bar{K}N - \pi \Sigma}$ and
two-pole $V^{2,SIDD}_{\bar{K}N - \pi \Sigma}$ phenomenological potentials together
with the chirally-motivated potential $V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$.
The results are presented in Table \ref{KKN_poles.tab}.
It turned out that all combinations of the two-body interactions
lead to a quasi-bound state in the three-body $\bar{K}\bar{K}N$ system.
The quasi-bound state exists in the strangeness $S=-2$ system
in spite of the repulsive character of the $\bar{K}\bar{K}$ interaction
Comparison with the $K^- pp$ characteristics from Table~\ref{KNN_poles.tab}
shows that the quasi-bound state in the strangeness $S=-2$
$\bar{K}\bar{K}N$ system is much shallower and broader than the one in the
$S=-1$ $K^- pp$ system for the given $\bar{K}N$ potential.
\begin{center}
\begin{table}
\caption{Binding energy $B_{\bar{K}\bar{K}N}$ (MeV) and width
$\Gamma_{\bar{K}\bar{K}N}$ (MeV) of the quasi-bound state in the $\bar{K}\bar{K}N$
system \cite{my_KKN}: the results of the direct pole search and of the Breit-Wigner
fit of the $1/|{\rm Det(z)}|^2$ function at real energy axis. The AGS calculations
were performed with the one-pole $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$,
two-pole $V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ phenomenological
potentials from~\cite{my_Kd_sdvig} and the chirally motivated
$V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$ potential
from~\cite{ourKNN_I}. $V^{Orig}_{\bar{K}\bar{K}}$ and
$V^{Lattice}_{\bar{K}\bar{K}}$ models of the antikaon-antikaon interaction were used.}
\label{KKN_poles.tab}
\begin{center}
\begin{tabular}{cccccc}
\hline \noalign{\smallskip}
& & \multicolumn{2}{c}{Direct pole search} & \multicolumn{2}{c}{BW fit of $1/|{\rm Det(z)}|^2$} \\
\hline \noalign{\smallskip}
& & $B_{\bar{K}\bar{K}N}$ & $\Gamma_{\bar{K}\bar{K}N}$
& $B_{\bar{K}\bar{K}N}$ & $\Gamma_{\bar{K}\bar{K}N}$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
& $V^{1,SIDD}_{\bar{K}N - \pi \Sigma}$
& $11.9$ & $102.2$ & $17.1$ & $110.8$ \\ \noalign{\smallskip}
$V^{Orig}_{\bar{K}\bar{K}}$ and &
$V^{2,SIDD}_{\bar{K}N - \pi \Sigma}$
& $23.1$ & $91.4$ & $23.7$ & $77.6$ \\ \noalign{\smallskip}
& $V_{\bar{K}N - \pi \Sigma - \pi \Lambda}^{\rm Chiral}$
& $15.5$ & $63.5$ & $15.9$ & $57.4$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
& $V^{1,SIDD}_{\bar{K}N - \pi \Sigma}$
& $19.5$ & $102.0$ & $23.7$ & $103.7$ \\ \noalign{\smallskip}
$V^{Lattice}_{\bar{K}\bar{K}}$ and &
$V^{2,SIDD}_{\bar{K}N - \pi \Sigma}$
& $25.9$ & $84.6$ & $26.4$ & $76.8$ \\ \noalign{\smallskip}
& $V_{\bar{K}N - \pi \Sigma - \pi \Lambda}^{\rm Chiral}$
& $16.1$ & $61.3$ & $15.9$ & $60.0$ \\
\noalign{\smallskip} \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
Two methods of the quasi-bound state evaluation were used: the direct
search method and the Breit-Wigner fit of the inverse determinant.
It is seen from the Table \ref{KKN_poles.tab} that the accuracy of the inverse
determinant method is much lower for the phenomenological $\bar{K}N$ interactions
than for the chirally motivated one (and for the $K^- pp$ system too). The reason
is the larger widths of the ''phenomenological'' $\bar{K}\bar{K}N$ states, which
means that the corresponding bumps are less pronounced, so they hardly can be
fitted reliably by Breit-Wigner curves.
The found $\bar{K}\bar{K}N$ quasi-bound state has the same quantum numbers as
a $\Xi$ baryon with $J^P=(1/2)^+$. The available experimental information on
the $\Xi$ spectrum is rather limited, see PDG \cite{PDG}. There is a $\Xi$(1950)
listed by the PDG, but its quantum numbers $J^P$ are not determined, and it is
unclear whether it should be identified with the quark-model state. It is possible
that there are more than one resonance in this region. However,
in spite of the fact, that the $\Xi(1950)$ state is situated above
the $\bar{K}\bar{K}N$ threshold, four of the experimental values would be roughly
consistent with the quasi-bound state found in the calculation~\cite{my_KKN}.
Specifically, the experiment reported in Ref.~\cite{Dauber:1969} yielded a mass
$1894 \pm 18$ MeV and a width $98 \pm 23$ MeV that is compatible with the range
of values for the evaluated pole position.
An investigation on the $\bar{K}\bar{K}N$ system was also performed
in \cite{KanadaEn'yo:2008wm}, but several uncontrolled approximations
were done there. In particular, energy-independent as well as
energy-dependent potentials were used, but the two-body energy of the latter was fixed
arbitrarily. Moreover, the imaginary parts of all complex potentials
were completely ignored in the variational calculations in \cite{KanadaEn'yo:2008wm},
the widths of the state were estimated separately. As a result, the binding energies
are compared to the exact ones from \cite{my_KKN}, but the widths of
the $\bar{K}\bar{K}N$ state are strongly underestimated.
\section{Near-threshold $K^- d$ scattering}
\label{Kd_elastic_sect}
\subsection{Methods and exact results}
\label{Kd_scatt_exact_sect}
The $K^- pp$ quasi-bound state is a very interesting exotic object. However,
it is not clear whether the accuracy of experimental results will be enough
to draw some conclusions from comparison of the data with theoretical predictions.
No strong quasi-bound state was found in the $\bar{K}NN$ system with other quantum
numbers $K^- d$ \cite{ourKNN_I}, but an atomic state, kaonic deuterium, exists,
and its energy levels can be accurately measured. In addition, scattering of an antikaon
on a deuteron can be studied.
Exact calculations of the near-threshold $K^- d$ scattering were performed
in \cite{my_Kd_sdvig,ourKNN_I} using the three antikaon-nucleon potentials,
described in Section \ref{V_KN_sect}, and different versions of the $\Sigma N$ and $NN$
potentials, described in Section~\ref{NN_SigN_sect}. Namely, the exact optical and
a simple complex versions of the spin-dependent and spin-independent
$\Sigma N -\Lambda N$ potentials were used. The calculations were performed with TSA-A,
TSA-B, and PEST models of the $NN$ interaction.
The inhomogeneous system of the integral AGS equations, corresponding to
Eq.(\ref{full_oper_eq}) and describing the $K^- d$ scattering, was transformed into
the system of algebraic equations. It is known, that the original, one-channel, integral
Faddeev equations have moving logarithmic singularities in the kernels
when scattering above a three-body breakup threshold is considered. The $K^- d$
amplitudes were calculated from zero up to the three-body breakup $\bar{K}NN$ threshold,
so, in principle, the equations could be free of the singularities. However, the lower
$\pi \Sigma N$ channel is opened when the $K^- d$ scattering is considered, which
causes appearance of the logarithmic singularities even below the three-body
breakup $\bar{K}NN$ threshold. The problem was solved by interpolating of the unknown solutions
in the singular region by certain polynomials and subsequent analytical integrating
of the singular part of the kernels.
The $K^- d$ scattering lengths $a_{K^- d}$ obtained with the
one- $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$ and two-pole
$V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ versions of the phenomenological $\bar{K}N$ potential
in \cite{my_Kd_sdvig} together with the chirally-motivated potential
$V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$ in \cite{ourKNN_I}
are shown in Table~\ref{Kd_aReff_EGam.tab}. It is seen, that the chirally motivated
potential leads to slightly larger absolute value of the real and the imaginary
part of the scattering length than the phenomenological ones. However, the difference
is small, so the three different models of the $\bar{K}N$ interaction, which reproduce
the low-energy data on the $K^- p$ scattering and kaonic hydrogen with the same level
of accuracy, give quite similar results for low-energy $K^- d$ scattering.
It means that it is not possible to solve the question of the number
of the poles forming the $\Lambda(1405)$ resonance from the results
on the near-threshold elastic $K^- d$ scattering.
The small difference between the ``phenomenological'' and ``chiral'' results
of the $a_{K^- d}$ calculations
is opposite to the results obtained for the $K^- pp$ system (see Section~\ref{qbs_results_sect}),
where the binding energy and width of the $K^- pp$ quasi-bound state
were calculated using the same equations (the homogeneous ones with properly changed
quantum numbers, of cause) and input. In that case the three-body observables
obtained with the three $\bar{K}N$ potentials turned out to be very different each
from the other. The reason of this difference between the results for the near-threshold
scattering and the quasi-bound state calculations could be the fact, that while
the $a_{K^- d}$ values were calculated near the $\bar{K}NN$ threshold, the $K^- pp$ pole
positions are situated far below it.
The amplitudes of the elastic $K^- d$ scattering for kinetic energy from $0$ to $E_{\rm deu}$,
calculated using the three versions of the $\bar{K}N$ potential, are shown in Fig.3
of \cite{my_Kd_sdvig} and in Fig 5 of \cite{ourKNN_I} in a form of $k \cot \delta(k)$ function.
The chosen representation demonstrates that the elastic near-threshold $K^- d$
amplitudes can be approximated by the effective range expansion rather accurately
since the lines are almost straight. The calculated effective ranges $r^{\rm eff}_{K^- d}$
of the $K^- d$ scattering, evaluated using the obtained $K^- d$ amplitudes, are shown
in Table~\ref{Kd_aReff_EGam.tab}.
The dependence of the full coupled-channel results on the $NN$ and $\Sigma N (-\Lambda N)$
interaction models was investigated in \cite{my_Kd}. The antikaon-nucleon
phenomenological potentials, used there, reproduce the earlier KEK data on kaonic hydrogen
and not the actual ones by SIDDHARTA. However, the results, obtained
with those phenomenological $\bar{K}N$ potentials, are relative, so they must be valid
for the actual potentials as well. In order to investigate dependence of the three-body
results on the $NN$ model of interaction, TSA-A, TSA-B, and PEST nucleon-nucleon
potentials were used. It turned out that the difference for the $K^- d$ scattering length
is very small even for the potentials with and without repulsion at short distances
(TSA and PEST, respectively). Therefore, the $s$-wave $NN$ interaction
plays a minor role in the calculations. Most likely, it is caused by the relative weakness
of the $NN$ interaction as compared to the $\bar{K}N$ one. Indeed, the quasi bound state
in the latter system (which is the $\Lambda(1405)$ resonance with $E_{\bar{K}N} \approx -23$ MeV)
is much deeper than the deuteron bound state ($E_{\rm deu} \approx 2$ MeV). Due to this,
some visible effect from higher partial waves in $NN$ is also not expected.
The dependence of $a_{K^- d}$ on the $\Sigma N (-\Lambda N)$ interaction was also investigated
in \cite{my_Kd}. The $K^- d$ scattering lengths were calculated with the exact optical
and the simple complex versions of the spin dependent $V^{\rm Sdep}$ and spin independent
$V^{\rm Sind}$ potentials. The results obtained with the two versions of the $\Sigma N (-\Lambda N)$
potential $V^{\rm Sdep}$ and $V^{\rm Sind}$ in exact optical
form are very close, while their simple complex versions are slightly different.
However, the largest error does not exceed $3 \%$, therefore, the dependence of
the $K^- d$ scattering length $a_{K^- p}$ on the $\Sigma N - (\Lambda N)$ interaction is also weak.
\subsection{Approximate calculations and comparison to other results}
\label{Kd_approx_compare_sect}
It is hard to make a comparison with other theoretical results due to different
methods and inputs used there. Due to this, several approximate calculations,
in particular, one-channel $\bar{K}NN$ calculations with a complex
and the exact optical $\bar{K}N$ potentials, were performed in \cite{my_Kd}.
In addition, a so-called FCA method was tested there.
In order to investigate the importance of the direct inclusion of the $\pi \Sigma N$
channel, the one-channel AGS calculations were performed in \cite{my_Kd}. It means that
Eq.~(\ref{full_oper_eq}) with $\alpha = \beta = 1$ were solved, therefore,
only the $\bar{K}N$ and $NN$ $T$-matrices enter the equations. The exact optical
and two simple complex one-channel $\bar{K}N (- \pi \Sigma)$ potentials approximating
the full coupled-channel one- and two-pole phenomenological models of the interaction were used.
As written in Section \ref{V_KN_sect}, the exact optical potential $V^{\rm{Opt}}$ provides
exactly the same elastic $\bar{K}N$ amplitude as the coupled-channel
model of the interaction. Its energy-dependent strength parameters are defined by
Eq.~(\ref{lambdaOpt}) with $\bar{\alpha},\bar{\beta} = 1,2$ stands for the $\bar{K}N$
and $\pi \Sigma$ channels, respectively.
The complex constants of the simple complex potentials were obtained in two ways.
The first version of the simple complex $\bar{K}N$ potential reproduces the $K^- p$
scattering length $a_{K^- p}$ and the pole position $z_{1}$ of the corresponding
coupled-channel version of the potentials. The second one gives the same isospin
$I_i=0$ and $I_i=1$ $\bar{K}N$ scattering lengths as the full $\bar{K}N - \pi \Sigma$ potential.
It was found in \cite{my_Kd} that the one-channel AGS calculation with
the exact optical $\bar{K}N$ potential, giving exactly the same elastic $\bar{K}N$
amplitude as the corresponding coupled-channel phenomenological potential,
is the best approximation. Its error does not exceed $2$ percents.
(The same is true for the results obtained with the chirally motivated $\bar{K}N$ potential,
see \cite{ourKNN_I}.)
On the contrary, the both simple complex $\bar{K}N$ potentials led to very inaccurate
three-body results. Therefore, the one-channel Faddeev-type calculation with a simple
complex antikaon-nucleon potential is not a good approximation for the low-energy
elastic $K^- d$ scattering.
One more approximate method, used for the $a_{K^- d}$ calculations, is a
so-called ``Fixed center approximation to Faddeev equations'' (FCA), introduced
in~\cite{Kd_KOR}. In fact, it is a variant of FSA or a two-center formula.
The fixed-scatterer approximation (FSA) or a two-center problem assumes,
that the scattering of a projectile particle takes place on two much heavier target
particles, separated by a fixed distance. The motion of the heavy particles is
subsequently taken into account by averaging of the obtained projectile-target amplitude
over the bound state wave function of the target. The approximation is well known and
works properly in atomic physics, where an electron is really
much lighter than a nucleon or an ion. Since the antikaon mass is
just a half of the mass of a nucleon, it was expected, that FSA hardly can be a
good approximation for the $K^- d$ scattering length calculation.
The derivations of the FCA formula from Faddeev equations presented in~\cite{Kd_KOR}
already rises questions, while the proper derivations of the FSA formula was done
in~\cite{peresypkin}. The accuracy of the FCA was checked in \cite{my_Kd} using the same
input as in the AGS equations in order to make the comparison as adequate as possible.
First of all, the $\bar{K}N$ scattering lengths provided by the coupled-channel
$\bar{K}N - \pi \Sigma$ potentials together with the deuteron wave function, corresponding
to the TSA-B $NN$ potential, were used in the FCA formula. Second, all $\bar{K}^0 n$
parts were removed from the formula because they drop off the AGS system of equations
after the antisymmetrization. Finally, the fact, that the FCA formula was obtained for a local
$\bar{K}N$ potential, while the separable $\bar{K}N - \pi \Sigma$ potentials were used in
the Faddeev equations, was took into account, and the corresponding changes in the FCA
formula were made.
The results of using of the FCA formula without ''isospin-breaking effects'' stay far away
from the full calculation. While the errors for the imaginary part are not so large,
the absolute value of the real part is underestimated by about $30 \%$. Therefore,
the calculations performed in \cite{my_Kd} show that FCA is a poor approximation
for the $K^- d$ scattering length calculation. It is also seen from the figure that
the accuracy is lower for the two-pole model of the $\bar{K}N$ interaction.
Therefore, among the approximate results the FCA was demonstrated to be
the least accurate approximation, especially in reproducing of the real part
of the $K^- d$ scattering length. On the contrary, the one-channel AGS calculation
with the exact optical $\bar{K}N (-\pi \Sigma)$ potential gives the best approximation
to the full coupled-channel result. All approximations are less accurate for
the two-pole phenomenological model of the $\bar{K}N - \pi \Sigma$ interaction.
Calculations of the $K^- d$ scattering length were performed by other authors
using Faddeev equations in~\cite{Kd_BFMS_new,Kd_TGE,Kd_TDD,Kd_Deloff}, while
the FCA method was used in~\cite{Kd_KOR,ruzecky}.
The result of the very recent calculation with coupled channels~\cite{Kd_BFMS_new}
has real part of $a_{K^- d}$, which almost coincides with the result for chirally
motivated potential shown in Table \ref{Kd_aReff_EGam.tab}.
The imaginary part of the $K^- d$ scattering length from~\cite{Kd_BFMS_new} is
slightly larger. It might be caused by the fact that the model of the $\bar{K}N$
interaction, used there, was not fitted to the kaonic hydrogen data directly, but
through the $K^- p$ scattering length and the Deser-type approximate formula, which
has larger error for the imaginary part of the level shift.
The two old $a_{K^- d}$ values~\cite{Kd_TDD,Kd_TGE}, obtained within
coupled-channel Faddeev approach, significantly underestimate the imaginary
part of the $K^- d$ scattering length, while their real parts are rather close to those
in Table~\ref{Kd_aReff_EGam.tab}.
One more result of a Faddeev calculation~\cite{Kd_Deloff} lies far away
from all the others with very small absolute value of the real part of $a_{K^- d}$.
One of the reasons is that the $K^- d$ scattering length
was obtained in~\cite{Kd_Deloff} from one-channel Faddeev equations with
a complex $\bar{K}N$ potential. However, the underestimation of the absolute value
of its real part in comparison to other Faddeev calculations is so large, that it
cannot be explained by the method only. The additional reason of the difference
must be the $\bar{K}N$ potential, used in the paper. It gives so high position of
the $K^- p$ quasi bound state ($1439$ MeV), that it is situated above the $K^- p$ threshold.
The $a_{K^- d}$ values of~\cite{Kd_KOR} obtained using FCA method differ significantly
from all other results. The absolute value of the real part of $a_{K^- d}$ from~\cite{Kd_KOR}
and its imaginary part are too large, which is caused by two factors. The first one is the
FCA formula itself, which was shown to be inaccurate for the present system. The second reason
are too large $\bar{K}N$ scattering lengths, used as the inputs.
The result of \cite{ruzecky} was obtained by simple applying of two approximate
formulas: FCA and the corrected Deser formula, used for calculation of the $\bar{K}N$
scattering lengths, entering the FCA. The values of~\cite{ruzecky} suffer not only from
the cumulative errors from the two approximations, but from using of the DEAR results
on kaonic hydrogen $1s$ level shift and width as well. Indeed, it was already written
that the error of the corrected Deser formula makes about $10 \%$ for two-body case,
the accuracy of the FCA was shown to be poor. As for the problems with DEAR experimental
data, they were demonstrated in \cite{our_KN} and in other theoretical works.
\section{$1s$ level shift of kaonic deuterium}
\label{kaonic_deu_sect}
The shift of the $1s$ level in the kaonic deuterium (which, strictly speaking, is the
``antikaonic'' deuterium) and its width are caused by the presence of the strong
interactions in addition to the Coulomb one. It is a directly measurable value, which
is free of a few uncertainties connected with an experiment on the $K^- pp$ quasi-bound
state. However, from theoretical point of view it is harder task due to necessity to take
Coulomb potential into account directly together with the strong ones.
There are two ways to solve three-body problems accurately:
solution of Faddeev equations or use of variational methods.
However, for the case of an hadronic atom both methods face serious difficulties.
The problem of the long range Coulomb force exists in the Faddeev
approach, while variational methods suffer from the presence of two very
different distance scales, which both are relevant for the calculations.
Due to this, at the first step the $1s$ level energy of the kaonic deuterium was
calculated approximately using a two-body model of the atom.
At the next step a method for simultaneous treatment of a short range plus Coulomb forces
in three-body problems based on Faddeev equations~\cite{Papp1} was used,
and the lowest level of kaonic deuterium was calculated dynamically exactly.
\subsection{Approximate calculation of kaonic deuterium $1s$ level}
\label{Kdshift_approx_sect}
The approximate calculation of the kaonic deuterium was performed assuming
that the atom can be considered as a two-body system consisting of a point-like deuteron,
interacting with an antikaon through a complex strong $K^- - d$ potential and Coulomb.
By this the size of a deuteron was taken into account only effectively through
the strong potential, which reproduces the elastic three-body $K^- d$ amplitudes,
evaluated before. Keeping in mind the relative values of a deuteron and Bohr radius
of the kaonic deuterium, the approximation seemed well grounded.
The complex two-body $K^- - d$ potential, constructed and used for investigation
of the kaonic deuterium by Lippmann-Schwinger equation, is a two-term
separable potential
\begin{equation}
\label{VKd}
V_{K^- d}(\vec{k},\vec{k}') = \lambda_{1,K^- d} \, g_1(\vec{k}) g_1(\vec{k}')
+ \lambda_{2,K^- d} \, g_2(\vec{k}) g_2(\vec{k}')
\end{equation}
with Yamaguchi form-factors
\begin{equation}
\label{gKd}
g_i(k) = \frac{1}{\beta_{i,K^- d}^2 + k^2}, \qquad i=1,2.
\end{equation}
The complex strength parameters $\lambda_{1,K^- d}$ and $\lambda_{2,K^- d}$
were fixed by the conditions, that the $V_{K^- d}$ potential reproduces
the $K^- d$ scattering length $a_{K^- d}$ and the effective range
$r^{\rm eff}_{K^- d}$, obtained with one of the $\bar{K}N - \pi \Sigma$
potentials and presented in Table \ref{Kd_aReff_EGam.tab}. A variation of the real
$\beta_{1,K^- d}$ and $\beta_{2,K^- d}$ parameters allowed to reproduce the full
near-threshold $K^- d$ amplitudes from \cite{my_Kd_sdvig,ourKNN_I} more accurately.
As a result, the near-threshold amplitudes obtained from the three-body
calculations $f^{(3)}_{K^- d}$ are reproduced by the two-body
$K^- - d$ potentials through the interval $[0,E_{\rm deu}]$ with such
accuracy, that the two-body functions $k \cot \delta^{(2)}(k)$
are indistinguishable from the three-body $k \cot \delta^{(3)}(k)$.
The parameters of the potentials are shown in Table 3 of~\cite{my_Kd_sdvig}
and in Eqs.(18,19) of~\cite{ourKNN_I}. Both $\beta_{1,K^- d}$ and
$\beta_{2,K^- d}$ parameters for every $K^- - d$ potential are much
smaller than the corresponding $\beta^{\bar{K}N}$ parameter of the $\bar{K}N$
potential.
The constructed two-body complex potentials $V_{K^- d}$ were used
in the Lippmann-Schwinger equation. The calculations of the binding energy of
a two-body system, described by the Hamiltonian with the strong and
Coulomb interactions were performed in the same way as those of the $K^- p$
system, see Section~\ref{V_KN_sect}.
\begin{center}
\begin{table}
\caption{Scattering length $a_{K^- d}$ (fm) and effective range
$r^{eff}_{K^- d}$ (fm) of $K^- d$ system obtained from AGS calculations
with the one-pole $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$, two-pole
$V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$ phenomenological potentials and
the chirally-motivated $V^{Chiral}_{\bar{K}N - \pi \Sigma - \pi \Lambda}$
potential. Approximate results for the $1s$ level shift $\Delta E^{K^- d}_{1s}$
(eV) and width $\Gamma^{K^- d}_{1s}$ (eV) of kaonic deuterium, corresponding
to the AGS results on the near-threshold elastic amplitudes, are also shown.}
\label{Kd_aReff_EGam.tab}
\begin{center}
\begin{tabular}{ccccc}
\hline \noalign{\smallskip}
$\bar{K}N$ potential used & $a_{K^- d}$ & $r^{eff}_{K^- d}$
& $\Delta E^{K^- d}_{1s}$ & \quad $\Gamma^{K^- d}_{1s}$\\
\hline \noalign{\smallskip}
$V^{1,SIDD}_{\bar{K}N}$ & $-1.49 + i \, 1.24$
\qquad & $0.69 - i \, 1.31$ & -785 & 1018 \\ \noalign{\smallskip}
$V^{2,SIDD}_{\bar{K}N}$ & $-1.51 + i \, 1.25$
\qquad & $0.69 - i \, 1.34$ & -797 & 1025 \\ \noalign{\smallskip}
$V_{\bar{K}N}^{\rm Chiral}$ & $-1.59 + i \, 1.32$
\qquad & $0.50 - i \, 1.17$ & -828 & 1055 \\
\noalign{\smallskip} \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
The shifts $\Delta E^{K^- d}_{1s}$ and widths $\Gamma^{K^- d}_{1s}$
of the $1s$ level of kaonic deuterium, corresponding to the three models
of the $\bar{K}N$ interaction, described in Section~\ref{V_KN_sect}, are shown
in Table \ref{Kd_aReff_EGam.tab}. It is seen that the ``chirally motivated''
absolute values of the level shift $\Delta E_{1s}^{K^- d}$ and the width
$\Gamma_{1s}^{K^- d}$ are both slightly larger than those obtained using
the phenomenological $\bar{K}N - \pi \Sigma$ potentials.
However, all three results do not differ one from the other more than several
percents. It is similar to the case of the $K^- d$ scattering length calculations,
which turned out to be very close for the three $\bar{K}N$ potentials.
The important point here is the fact that all three $\bar{K}N$ potentials
reproduce the low-energy experimental data on $K^- p$ scattering and kaonic hydrogen
with the same level of accuracy. It is also important that the $1s$ level
of kaonic deuterium is situated not far from the $\bar{K}NN$ threshold.
The closeness of the results for kaonic deuterium means that comparison
of the theoretical predictions with eventual experimental results hardly could
choose one of the models
of the $\bar{K}N$ interaction, especially taking into account the large widths
$\Delta E^{K^- d}_{1s}$. Therefore, it could not be possible to say, whether
the potential of the antikaon-nucleon interaction should have one- or two-pole
structure of the $\Lambda(1405)$ resonance and whether the potential should
be energy dependent or not. It is seen from Tables~\ref{phys_char.tab}
and \ref{Kd_aReff_EGam.tab} that there is no correlation between the pole
or poles of the $\Lambda(1405)$ resonance given by a $\bar{K}N $
potential and the three-body $K^- d$ elastic scattering or kaonic deuterium
characteristics obtained using the potential.
Inaccuracy of the corrected Deser formula Eq.(\ref{corDes}) was already shown
for the two-body $K^- p$ system, but some authors use it for the kaonic deuterium
as well. Due to this, an accuracy of the formula was checked for this three-body system.
The results were obtained using the $a_{K^- d}$ values from Table~\ref{Kd_aReff_EGam.tab}.
Being compared to the $\Delta E_{K^- d}$ and $\Gamma_{K^- d}$ from the same
table, the "corrected Deser" results show large error for all three versions
of the antikaon-nucleon interaction. While difference for the shift is not so
drastic, the width of the $1s$ level of the kaonic deuterium is underestimated
by the corrected Deser formula by $\sim 30\%$.
The $1s$ level shift and width presented in Table \ref{Kd_aReff_EGam.tab}
are not exact, they were evaluated using the two-body approximation,
which, however, is well-grounded. Information on the three-body strong part
is taken into account indirectly through the $K^- - d$
potential, reproducing the exact elastic $K^- d$ amplitudes. On the contrary,
the corrected Deser formula contains no three-body information at all since
the only input is a $K^- d$ scattering length, which is a complex number. Moreover,
the formula relies on further approximations, which are absent in the accurate
approximate calculations.
\subsection{Exact calculation of kaonic deuterium:
Faddeev equations with Coulomb interaction}
\label{papp_eq_sect}
Exact calculations of the kaonic deuterium were performed using
a method~\cite{Papp1} for simultaneous treatment of short range plus Coulomb forces
in three-body problems, based on Faddeev equations.
The method was successfully applied for purely Coulomb systems with
attraction and repulsion and for the short range plus repulsive Coulomb
forces. The case of an hadronic atom with three strongly interacting particles and
Coulomb attraction between certain pairs was not considered before.
The basic idea of the method is to transform the Faddeev integral equations into a matrix
form using a special discrete and complete set of Coulomb Sturmian functions
as a basis. Written in coordinate space the Coulomb Sturmian functions are orthogonal with
the weight function $1/r$. So that they form a bi-orthogonal and complete set
with their counter-parts. The most remarkable feature of this particular set is, that
in this representation the matrix of the two-body $(z-h^c)$ operator, where $z$ is
an energy and $h^c$ is the pure two-body Coulomb Hamiltonian, is tridiagonal.
When this property is used for evaluation of the matrix elements of the two-body
Coulomb Green's function $g^c$, an infinite tridiagonal set of equations, which can be
solved exactly, is obtained. The same holds for the matrix elements of the free two-body
Green's function $g^0$.
The system of equations with the Coulomb and strong interactions was solved in \cite{our_Kdexact}
for kaonic deuterium. This calculation is different from all other three-body calculations,
described before. Already the initial form of the Faddeev equations for the kaonic
deuterium differs from those for the pure strong interactions,
described in Section~\ref{AGS_sect}. First, the equations should be written
in coordinate space, while the AGS equations were written in momentum
space. Second, since the Coulomb interaction acts between $K^-$ and the proton,
the particle basis was used and not the isospin one. Finally, the Faddeev equations with
Coulomb do not define the transition operators, as e.g. those in Eq.(\ref{U_coupled}),
but the wave functions.
The equations are written in the Noble form~\cite{Noble}, when the Coulomb interaction
appears in the Green's functions. As usual for Faddeev-type equations, there are three
partition channels $\alpha = (pn,K^-)$, $(pK^-,n)$, $(nK^-,p)$ and three sets of Jacobi
coordinates. The system of homogeneous equations to be solved contains the matrix elements of
the overlap between the basis functions from different Jacobi coordinate sets
and of the strong potentials. They all can be calculated directly. The remaining
parts of the kernel are matrix elements of the three-body partition Green's
functions $G_{\alpha}$. They are the basic quantities of the method, and their
calculation depends on the partition channel.
The partition Green function $G_{(pK^-,n)}$ of the $(pK^-,n)$ channel
contains Coulomb interaction in its ``natural'' coordinate. It describes
the $(pK^-)$ subsystem and the neutron, which
do not interact between themselves. Due to this, $G_{(pK^-,n)}$ can be
calculated taking a convolution integral along a suitable contour in the complex
energy plane over two two-body Green functions. As for the matrix elements
of the two-body Green functions, they can be calculated using the properties
of the Coulomb Sturmian basis and solving a resolvent equation.
The situation with the remaining $G_{\alpha}$ functions is more complicated.
In the case of the $\alpha = (pn,K^-)$ and $(nK^-,p)$ channels
the Coulomb interaction is written not in its ``natural'' coordinates.
Due to this, it should be rewritten as a sum of the Coulomb potential
in the natural coordinates plus a short range potential $U_{\alpha}$,
which is a ``polarization potential''. The three-body Green function
$G_{\alpha}^{ch}$ containing Coulomb potential in natural for the channel
coordinates is called the ''channel Green function'', and it is evaluated similarly
to the previous $\alpha = (pK^-,n)$ case. Namely, since the function describes
a two-body subsystem and the non-interacting with it third particle,
the $G_{(pn,K^-)}^{ch}$ and $G_{(nK^-,p)}^{ch}$ functions can be found
by taking a convolution integral with two two-body Green's functions.
At the last step the $G_{\alpha}$ function is found from the equation,
containing the obtained channel Green function $G_{\alpha}^{ch}$ and
the polarization potential $U_{\alpha}$
\begin{equation}
\label{Geq}
G_{\alpha}(z) = G^{ch}_{\alpha}(z) + G^{ch}_{\alpha}(z) U_{\alpha}
G_{\alpha}(z).
\end{equation}
For the kaonic deuterium calculations it was necessary to take the isospin
dependence of the $\bar{K}N$ interaction into account. In particle representation
it means that the strong $V_{pK^-}^s$ potential is a $2 \times 2$ matrix,
containing $V^s_{pK^-,pK^-}$, $V^s_{pK^-,n\bar{K}^0}$ and $V^s_{n\bar{K}^0,n\bar{K}^0}$
elements. Due to this, the final equations for the kaonic deuterium have
four Faddeev components, including the additional one in the $(n\bar{K}^0,n)$
channel.
The solution of the Faddeev-type equations with Coulomb gave the full energy of
the $1s$ level. Since the aim of \cite{our_Kdexact} was evaluation of the $1s$ level
shift of kaonic deuterium caused by the strong interactions between the antikaon
and the nucleons, it was necessary to define the energy, from which the real part
of the shift is measured. It can be the lowest eigenvalue of the channel Green function
or of the ``original'' Green function of the $(pn,K^-)$ channel. The first one corresponds
to a deuteron and an antikaon feeling a Coulomb force from the center of mass
of the deuteron. The second reflects the fact that the antikaon interacts via Coulomb
force not with the center of the deuteron, but with the proton. In principle,
the correct one should be the second variant, however, all approximate
approaches use an analogy of the first one as the basic point, due to this
it was used in \cite{our_Kdexact} as well. In any case, the difference between
both versions is small.
\subsection{Exact calculation of kaonic deuterium: results}
\label{Kd_accurate_sect}
The calculation performed in~\cite{our_Kdexact} was considered as
a first test of the method for the description of three-body hadronic atoms.
Due to this, the second three-body particle channel $\pi \Sigma N$ was not
directly included and no energy dependent potentials (exact optical
or chirally motivated one) were used. The $\bar{K}N$ and $NN$ interactions were
described using one-term separable complex potentials
with Yamaguchi form factors. Four versions of the $\bar{K}N$ potential
$V_I, V_{II}, V_{III}$ and $V_{IV}$, used in the calculations, give the $1s$ level
shift of the kaonic hydrogen within or close to the SIDDHARTA data and a reasonable
fit to the elastic $K^- p \to K^- p$ and charge exchange $K^- p \to \bar{K}^0 n$ cross-sections.
Parameters of the potentials can be found in Table I of~\cite{our_Kdexact}.
The nucleon-nucleon potential reproduces the $NN$ scattering lengths, low-energy phase shifts
and the deuteron binding energy in the $np$ state.
\begin{table*}
\caption{Exact $1s$ level shifts $\Delta E$ (eV) and widths $\Gamma$ (eV)
of the kaonic deuterium for the four complex $\bar{K}N$ potentials $V_I, V_{II}, V_{III}$,
and $V_{IV}$. The approximate results obtained using the corrected Deser formula and
the complex $K^- - d$ potential are also shown.}
\label{Kd_shiftExact.tab}
\begin{center}
\begin{tabular}{ccccccc}
\hline \noalign{\smallskip}
& \multicolumn{2}{c}{Corrected Deser} & \multicolumn{2}{c}{Complex $V_{K^- - d}$} &
\multicolumn{2}{c}{Exact Faddeev} \\
\noalign{\smallskip} \hline \noalign{\smallskip}
& $\Delta_{1s}^{K^- d}$ & $\Gamma_{1s}^{K^- d}$ &
$\Delta_{1s}^{K^- d}$ & $\Gamma_{1s}^{K^- d}$ &
$\Delta_{1s}^{K^- d}$ & $\Gamma_{1s}^{K^- d}$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
$V_I$ & $-675$ & $702$ & $-650$ & $868$ & $-641$ & $856$\\
$V_{II}$ & $-694$ & $740$ & $-658$ & $920$ & $-646$ & $888$\\
$V_{III}$ & $-795$ & $780$ & $-747$ & $1034$ & $-732$ & $980$\\
$V_{IV}$ & $-750$ & $620$ & $-740$ & $844$ & $-736$ & $826$\\
\noalign{\smallskip} \hline
\end{tabular}
\end{center}
\end{table*}
The results of the dynamically exact calculations of the kaonic deuterium are presented
in Table~\ref{Kd_shiftExact.tab}. The absolute values of the $1s$ level shift were
found in the region $641 - 736$ eV, while the width variates between $826 - 980$ eV.
Both observables are smaller than the accurate results from~\cite{ourKNN_I}, shown in
Table~\ref{Kd_aReff_EGam.tab}. However, it is necessary to remember that both
calculations differ not only by the three-body methods, but also by the two-body input.
To make the comparison reasonable, the two-body approximate calculation,
described in Section~\ref{Kdshift_approx_sect}, was repeated with the simple
complex $\bar{K}N$ potentials $V_I, V_{II}, V_{III}$, and $V_{IV}$. The corrected Deser
formula was also checked for these potentials. The approximate results are shown
in Table~\ref{Kd_shiftExact.tab}.
It is seen that the two-body approximate calculation, described in
Section~\ref{Kdshift_approx_sect}, makes $\le 2 \%$ error for the shift and $\le 5 \%$
for the width, so it is quite accurate. It is an expected result keeping in mind the relative
values of deuteron and Bohr radius of kaonic deuterium. The corrected Deser formula
Eq.(\ref{corDes}) leads to $2 - 8 \%$ error in the shift, and strongly, up to $25 \%$,
underestimates the width.
It is also possible to compare the approximate results obtained in \cite{our_Kdexact}
with the four complex $\bar{K}N$ potentials $V_I, V_{II}, V_{III}$, and $V_{IV}$
and in \cite{ourKNN_I} with the coupled-channel models of the antikaon-nucleon interaction
(the phenomenological $V^{1,SIDD}_{\bar{K}N-\pi \Sigma}$ and $V^{2,SIDD}_{\bar{K}N-\pi \Sigma}$
with one- and two-pole structure of the $\Lambda(1405)$ resonance respectively, and
the chirally motivated $V^{Chiral}_{\bar{K}N - \pi \Sigma-\pi \Lambda}$).
It is seen that the `'complex one-channel'' absolute values of the $1s$ level shift
and width shown in Table~\ref{Kd_shiftExact.tab} are smaller than the `'coupled-channel''
ones presented in Table \ref{Kd_aReff_EGam.tab}. The similar situation was observed
with the exactly evaluated characteristics of the strong pole in the $K^- pp$ system,
while a one-channel simple complex antikaon - nucleon potential led to more
narrow and less bound quasi-bound state than the coupled-channel version
(see Eqs.(\ref{2chOLDKpp},\ref{1chOLDKpp})). But the differences for the kaonic deuterium
are smaller than those for the $K^- pp$ quasi-bound state.
The very recent calculations~\cite{revai_exactKd} of the kaonic deuterium
$1s$ level shift were performed using the same Faddeev-type equations with Coulomb
interaction as in \cite{our_Kdexact}, but with energy-dependent $\bar{K}N$ potentials.
Namely, the exact optical versions of the one- and two-pole phenomenological
$\bar{K}N - \pi \Sigma$ potentials and of the chirally motivated $\bar{K}N-\pi \Sigma - \pi \Lambda$
interaction model were used. The predicted $1s$ level shifts ($800 \pm 30$ eV) and widths
($960 \pm 40$ eV) are larger by absolute value than the exact ones from Table~\ref{Kd_shiftExact.tab}
evaluated with the simple complex antikaon nucleon potentials.
Keeping in mind good accuracy of the results obtained with the exact optical
$\bar{K}N$ potentials for all three-body $\bar{K}NN$ observables, demonstrated in the present
paper, the predictions of \cite{revai_exactKd} for the kaonic deuterium must be the most
accurate ones up to date. The two-body approximation used in \cite{my_Kd_sdvig,ourKNN_I},
being compared to the more accurate approach of \cite{revai_exactKd}, gives very accurate
value of the $1s$ level shift (the error is $\le 2 \%$), while the error for the width is
larger ($\le 9 \%$).
\section{Summary}
\label{Summary_sect}
The three-body antikaon nucleon systems could provide an important information about
the antikaon nucleon interaction. It is quite useful since the two-body $\bar{K}N$ potentials
of different type can reproduce all low-energy experimental data with the same level
of accuracy. This fact was demonstrated on the example of the phenomenological $\bar{K}N-\pi \Sigma$
potentials with one and two-pole structure of the $\Lambda(1405)$ resonance together with
the chirally motivated $\bar{K}N - \pi \Sigma - \pi \Lambda$ potential. Being used
in the three-body calculations, the three $\bar{K}N$ potentials allowed to investigate
the influence of the $\bar{K}N$ model on the results.
It was found that while the quasi-bound state position in the $K^- pp$ and $K^- K^- p$ systems
strongly depends on the model of the $\bar{K}N$ interaction, the near-threshold observables
($K^-d$ scattering length, elastic near-threshold $K^- d$ amplitudes, $1s$ level shift and
width of kaonic deuterium) are almost insensitive to it. Therefore, some conclusions on
the number of poles of the $\Lambda(1405)$ resonance could be done only if a hight accuracy
measurement of $K^- pp$ binding energy and width will be done. Probably, one of the existing
experiments: by HADES~\cite{HADES} and LEPS~\cite{LEPS} collaborations, and in J-PARC
\cite{J-PARC_E15,J-PARC_E27} hopefully will clarify the situation with the $K^- pp$ quasi-bound
state, - will do it. While dependence of the three-body results on the $\bar{K}N$ potentials
is different for the different systems and processes, dependence on $NN$ and $\Sigma N$ interactions
is weak in all cases.
Comparison of the exact results with some approximate ones revealed the most accurate
approximations. In particular, the one-channel Faddeev calculations give results, which are
very close to the coupled-channel calculations if the exact optical $\bar{K}N$ potential
is used. This fact gives a hope for four-body calculations, which are already very complicated
without additional coupled-channel structure. It is necessary to note here that the ''exact optical''
potential is defined as an energy dependent potential, which exactly reproduces the elastic amplitudes
of the corresponding potential with coupled channels.
As for the kaonic deuterium, influenced mainly by Coulomb interaction, the shift of its $1s$ level
caused by the strong interactions is described quite accurately in the two-body approximation.
The $K^- - d$ complex potential should herewith reproduce the exact elastic three-body $K^- d$
amplitudes, and the Lippmann-Schwinger equation must be solved exactly with Coulomb plus the strong
potentials. Of cause, the exact calculation is more precise and, therefore, is preferable.
The predicted by the exact calculations $1s$ level energy could be checked by SIDDHARTA-2
collaboration \cite{SIDDHARTA2}.
The suggested $1/|{\rm Det(z)}|^2$ method of theoretical evaluation of an underthreshold
resonance is quite accurate for rather narrow and well pronounced resonances. It could supplement
the direct search of the pole providing the first estimation and working as a control. The method
is free from the uncertainties connected with the calculations on the complex plane, but it has
the logarithmic singularities in the kernels of the integral equations.
The next step in the field of the few-body systems consisting of antikaons and nucleons
should be done toward the four body systems. They could give more possibilities, but theoretical
investigations of them are much more complicated.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,630
|
November 22, 2017 11:00am Comment Doug Lynch
Google Introduces the Google Play Referrer API for Developers
Virtually anyone can publish an application or game onto the Play Store these days. Some people think the work ends there and that their product will either succeed or fail based on luck. However, there's a lot of work that goes into managing and marketing applications after they have been approved in the Play Store if you want them to succeed. Google has just introduced the Google Play Referrer API as a tool that can help developers track and measure application installs easily and securely.
A lot of the traffic monitoring and analysis that web developers have been doing for decades can help application developers as well. One key tool that analytic services tend to offer is a way to find out how people found out about your product. Knowing this type of information allows the developer to cater the content directly to that type of user which in turn can help to improve the user experience. Google's new Google Play Referrer API aims to offer this analytics solution for those who want to learn more about the people installing their applications.
Some application developers rely on app measurement companies and ad networks who offer ad attribution solutions based on referral data. So it's vital to know accurate install referral data for correctly attributing app installs, as well as discounting fraudulent attempts for install credit. This is where the Google Play Referrer API comes into the mix as it offers a reliable way to securely retrieve install referral content.
Application developers who end up using this new API will have access to precise information straight from the Play Store including the referrer URL of the installed package, the timestamp (in seconds) of when the referrer click happened, and the timestamp (again in seconds) of when the installation began. Developers who feel they would benefit from this new API can starting using it right now. Just be aware that this new API requires the device to have the Play Store app from version 8.3.73 and later.
Source: Android Developers Blog
Tags GoogleGoogle Play
XDA » Mini XDA » Google Introduces the Google Play Referrer API for Developers
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,682
|
@interface QLMPurchasedTwoFlowLayout : UICollectionViewFlowLayout
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,884
|
Események
Ibráhím bin Iljász örökli apjától Herát emírségét
Születések
Halálozások
február 4. – Hrabanus Maurus bencés apát, tudós, mainzi érsek (* 780 k.)
Iljász bin Aszad, Herát Számánida emírje
9. század
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,619
|
Intense action in Semifinals leads to Finals between France and Croatia | Kenyan_Report
Home World Intense action in Semifinals leads to Finals between France and Croatia
Intense action in Semifinals leads to Finals between France and Croatia
The Kenyan Report
By Sean Toomer—
Thrilling, exciting, disappointing, amazing, and heart-breaking. All of these terms can be used to describe the semifinal round of the FIFA World Cup. Hundreds of millions of people watched as France defeated Belgium in a close, defensive game, 1-0. Croatia came back against England, down 0-1, to best England 2-1. Both matches had very different game flows. All four teams have rosters loaded with talent but with very different styles of play.
The first match, France vs Belgium, remained scoreless in the first half with France aggressively attacking the Belgium goal but not scoring. Belgium seemed to be somewhat reckless with their physical style as they committed 16 fouls versus France's 6 fouls. France attacking nature lead to 18 total shots (5 shots on goal) and they were finally rewarded when defender Samuel Umiti scored during the 51st minute of play. His close-range header allowed France to pull ahead and put massive pressure on Belgium to score. Belgium possessed the ball for nearly 65% of the game but had only 8 shots (3 shots on goal) as France played excellent no-foul defence. Several yellow cards in the second half led to several scuffles between the teams. France never relinquished the lead after Umiti's goal and has advanced to the finals round.
The second match of the day was even more thrilling with England facing off versus Croatia. Within the first five minutes of play England defender, Kieran Trippier scored a goal from a Croatia misplay. This quick start seemed to shift the style of play for England as they did not aggressively attack Croatia's goal for the remainder of the match. On the opposite side, Croatia was not deterred and finally netted a goal during the 68th minute from midfielder Ivan Perišić. This tied the game at 1-1. And effectively reset the match. The remainder of the play would determine the winner and allow that team to advance to the championship round versus France. Both teams battled fiercely and were very physical with 37 total fouls (24, the game remained tied and went to extra minutes. Croatia relentlessly continued their attack on England (24 shots, 7 shots on goal) and scored a goal at the 109th minute by forwarding Mario Mandžukić. The comeback for Croatia was in full effect and transferred all the pressure to England was unable to get into favourable scoring positioning as they only had 10 total shots and a mere 2 shots on goal. This combined with a lack of aggression ultimately lead to a 2-1 defeat.
On this Sunday, July 15th at 11 AM EST, The finals game will be played between France and Croatia to determine the new world champion and World Cup, winner. This should be a great match featuring the two toughest, most resilient teams facing off against each other. Both teams have dynamic scores and aggressive attacking playstyles. Billions of eyes will be watching to see which country will hold the title of best football team for the next four years. Who do you think will win? Who are your favourite remaining players? Who are you rooting for? Leave your opinion in the comments below. Go LES BLUES!
Previous articleDP Ruto in Shambles as Uhuru Rewards Raila's Allies With State Jobs
Next articleEntire Management of Kenya Power Faces Jail Term as DPP Haji Cracks Whip
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,963
|
var http = require('http');
var url = require('url');
var net = require('net');
var repl = require("repl");
// servlets
var registerServlet = require('./lib/registerservlet');
var statusServlet = require('./lib/statusservlet');
var requestHandler = require('./lib/requesthandler');
var hubStatusServlet = require('./lib/hubstatusservlet');
var unregisterServlet = require('./lib/unregisterservlet');
var welcomeServlet = require('./lib/welcomeservlet');
registry = require('./lib/registry');
var models = require('./lib/models');
var parser = require('./lib/parser');
var log = require('./lib/log');
store = require('./lib/store');
var domain = require('domain');
var servletRoutes = {
"/selenium-server/driver" : rcRequestHandler,
"/grid/api/proxy" : statusServlet,
"/grid/register" : registerServlet,
"/grid/unregister" : unregisterServlet,
"/grid/api/hub" : hubStatusServlet,
"/wd/hub/session" : webdriverRequestHandler
};
var parseIncoming = function(req, res, cb) {
req.setEncoding('utf8');
var srvUrl = url.parse(req.url.toString(), true);
var servlet, route, common;
if (servletRoutes[srvUrl.pathname]) {
servlet = servletRoutes[srvUrl.pathname];
return servlet.handleRequest(req, cb, res);
} else if (common = servletRoutes[srvUrl.pathname.substring(0, 15)]) {
// webdriver and grid/register have the same 15 char base path
// this should be faster than the stuff below
return common.handleRequest(req, cb, res);
} else {
// slower lookup of routes
for (route in servletRoutes) {
if (route === srvUrl.pathname.substring(0, route.length)) {
servlet = servletRoutes[route];
return servlet.handleRequest(req, cb, res);
}
}
}
if (srvUrl.pathname === '/') {
return welcomeServlet.handleRequest(req, cb, res);
} else if (srvUrl.pathname === '/robots.txt') {
return welcomeServlet.robots(req, cb, res);
} else if (srvUrl.pathname === '/wd/hub/status') {
return welcomeServlet.status(req, cb, res);
}
return cb("Invalid endpoint: " + req.url.toString(), new models.Response(400, "ERROR Unable to handle request - Invalid endpoint or request. (" + req.url.toString() + ")", {'Content-Type': 'text/plain'}));
};
function main(args) {
store.setConfig(args);
var serverDomain = domain.create();
var server;
serverDomain.on('error', function(e) {
log.warn(e);
log.warn(e.stack);
});
serverDomain.run(function() {
server = http.createServer(function(req, res) {
var url = req.url.toString();
req.on("close", function(err) {
log.warn("!error: on close");
});
res.on("close", function() {
log.warn("!error: response socket closed before we could send");
});
var reqd = domain.create();
reqd.add(req);
reqd.add(res);
res.socket.setTimeout(6 * 60 * 1000);
res.socket.removeAllListeners('timeout');
req.on('error', function(e) {
log.warn(e);
});
reqd.on('error', function(er) {
log.warn(er);
log.warn(er.stack);
log.warn(req.url);
try {
res.writeHead(500);
res.end('Error - Something went wrong: ' + er.message);
} catch (er) {
log.warn('Error sending 500');
log.warn(er);
}
});
res.on('error', function(e) { log.warn(e); });
res.socket.once('timeout', function() {
try {
res.writeHead(500, {'Content-Type': 'text/plain'});
res.end('Error - Socket timed out after 6 minutes');
} catch (e) {
}
try {
res.socket.destroy();
} catch (e) {
}
});
parseIncoming(req, res, function(response) {
res.writeHead(response.statusCode, response.headers);
res.end(response.body);
});
}).listen(4444, '0.0.0.0');
});
server.httpAllowHalfOpen = true;
var manager = net.createServer(function(socket) {
repl.start({
prompt: "node via TCP socket> ",
input: socket,
output: socket,
useGlobal: true
}).on('exit', function() {
socket.end();
});
}).listen(4446, '127.0.0.1');
server.on('clientError', function(exception, socket) {
try {
if (socket.parser.incoming.url === "/grid/register") {
return;
}
} catch (e) {}
if (exception.message.indexOf('ECONNRESET') > -1) {
log.debug(exception);
return;
}
log.warn('!error: client error');
log.warn(exception);
log.warn(exception.stack);
log.warn(socket);
});
process.on('SIGTERM', function() {
if (registry.pendingRequests.length > 0) {
log.warn("Can't stop hub just yet, pending requests!");
// try now
registry.processPendingRequest();
return;
}
log.info("Stopping hub");
server.close();
});
process.on('uncaughtException', function(err) {
log.warn("! Uncaught Exception occurred");
log.warn(err);
log.warn(err.stack);
});
server.on('close', function () {
store.quit();
process.exit();
});
log.info("Server booting up... Listening on " + (parseInt(process.argv[2], 10) || 4444));
}
if (require.main === module) {
var args = parser.parseArgs();
main(args);
}
module.exports.run = main;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,315
|
{"url":"https:\/\/codeforces.com\/blog\/Akash_Roy","text":"### Akash_Roy's blog\n\nBy\u00a0Akash_Roy, history, 3 months ago,\n\nWe select members for first round dance. Then members of second round dance get automatically selected . Hence the factor nC(n\/2). Then we arrange both the round dance. This leads to nC(n\/2)*(n-1)!*(n-1)! {Circular permutation}. After this why are we dividing the answer only by 2? I feel we should divide final answer by 4 since there are two round dances. Since clockwise and anticlockwise arrangement does not matter, the final answer should be [nC(n\/2)*(n-1)!*(n-1)!]\/4. Why is it [nC(n\/2)*(n-1)!*(n-1)!]\/2 ?\n\nSorry for my poor number formatting.\n\n\u2022 +3\n\nBy\u00a0Akash_Roy, history, 4 months ago,\n\nAn O(n^2) solution gave me TLE on test 10. I came across one of the solutions below.\n\nmain() {\n\nint n,i,f,s;\ncin>>n;\nmap<int,int> mp;\npair<int , int> a[n];\nfor(i=0;i<n;i++)\n{\ncin>>f>>s;\nmp[f]++;\na[i]={f,s};\n}\n\nint m=n-1;\nfor(i=0;i<n;i++)\n{\na[i].first=m+mp[a[i].second];\na[i].second=m-mp[a[i].second];\n\ncout<<a[i].first<<\" \"<<a[i].second<<endl;\n}\n\n}\n\nI don't understand how the following two lines work.\n\na[i].first=m+mp[a[i].second];\n\na[i].second=m-mp[a[i].second];\n\nEach team plays n-1 home and n-1 away games . But what is purpose of mp[a[i].second]? Please help.","date":"2021-01-21 17:48:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.26214492321014404, \"perplexity\": 4325.514260515176}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703527224.75\/warc\/CC-MAIN-20210121163356-20210121193356-00371.warc.gz\"}"}
| null | null |
class GithubKeygen < Formula
desc "Bootstrap GitHub SSH configuration"
homepage "https://github.com/dolmen/github-keygen"
url "https://github.com/dolmen/github-keygen/archive/v1.302.tar.gz"
sha256 "cc703512ab67837a7025e7d018731e97ddc6a943f87ccd8620986d5757991a48"
head "https://github.com/dolmen/github-keygen.git"
bottle :unneeded
def install
bin.install "github-keygen"
end
test do
system "#{bin}/github-keygen"
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,142
|
## Suzanne McMinn
## SECRETS RISING
In tribute to my great-aunt Ruby
and her farmhouse—my haven.
## Contents
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Epilogue
## Chapter 1
There was a skull in her rose bed.
Keely Schiffer swiped the hair out of her eyes, felt the damp, cool smear of fresh dirt she'd applied to her cheek in the process, along with the sensation of her insides starting to crawl. The shovel dropped out of her other hand.
That wasn't really a skull, was it? It was probably just a rock. A big one. With gaping eye sockets—
Somebody screamed.
She realized that somebody had been her and that somehow she'd ended up about three feet back without being aware of her feet moving. Yes, that was a skull, a human skull. She should call the police. She hated calling the police. She'd called the police too many times lately. And they'd called her too many times to count. That's what happened when your husband had a lot of secrets you didn't know about and then got himself killed doing something stupid—
Oh, God. Was this another one of Ray's secrets?
It was Ray who'd had the bright idea of digging up this bed. Plant some roses, he'd said. Then he'd dug the crap out of it, torn out the old boxwood hedges, and left it in a big mess last fall right before—
Keely staggered back a few more steps then forced herself to move forward again to the hole she'd dug. She stood on trembling legs, her heart beating fast. Do something.
It took her ten seconds to decide. As if there was a decision to make.
Thunder rumbled in the distance. A sudden whip of wind tore through the West Virginia mountain hollow, buffeting leftover dead leaves from winter across the short grass. Another spring storm was on the way. She wasn't going to get the gorgeous hybrid tea roses planted in time. She wasn't going to get them planted today at all, not until the police were finished.... And she'd been so looking forward to this one day off to play outside in the dirt and sunshine. She looked back at the neatly lined-up roses, ready for planting. They had a short growing season here in the mountain region. She needed to get her bushes planted. That thought had seemed really important just about five minutes ago.
Her head reeled. She ran for the back door of the house, burst inside and reached for the phone. She punched 911 before the reality of the silent air hit her.
The phone was out.
The old house settled still and heavy around her. The farm was five miles outside Haven on a road so narrow, to pass another car one vehicle or the other had to pull over on the weedy shoulder. Sugar Run Farm had been in her family for four generations. It wasn't unusual for the phone to go out, storm or no storm. Inconveniences were par for the course in the boonies. There was no cell signal, no cable. They were lucky to have satellite TV. Dickie the mechanic provided personal service.... Including picking up his customers' vehicles on site then dropping them back off.
The windows on her ten-year-old Ford pickup weren't operating properly. Dickie had picked it up first thing this morning, promising it back by tonight. No phone. No vehicle. Not too big of a problem normally, especially on her day off.
Except for the skull in her rose bed and some really scary thoughts about how it might have gotten there.
But it wasn't an emergency, was it? It had been there since last fall.... Probably. And she'd been living right here all this time, sleeping soundly in spite of it. No reason for alarm now....
And yet, she was alarmed. Creeped out. She'd never minded being alone all the way out here.
Till now.
Row after row after row of timeworn family photographs stared down at her from the parlor walls as she cradled the portable phone back on its base. Through the large front window, she could see the day darkening swiftly. Wind crackled through the leaves on the two maple trees out front. Rain poured down, then the house gave a sudden shake.
Something crashed in the kitchen.
She ran the short distance, pulse thudding. A large cookie jar shaped like a windmill lay shattered on the floor, jostled off the shelf over the cabinets. Staring down at the broken cream-and-blue ceramic jar, she realized something small and shiny sat in the middle of the shards.
She paused for a long beat, glancing up at the ceiling. Had a tree struck the house? Nothing else made sense.... Dammit, she'd have to go check, make sure there wasn't water coming into the attic just in case it had been a tree.
Glancing down again, she reached for the foil-wrapped box. She had a second's instinctive temptation to play the childlike game of shaking it lightly, trying to guess what was inside. Her chest tightened. It was silver Christmas wrap with a tone-on-tone pattern of bells and ribbons, but the peel-and-stick holiday label with the bright caricature of Rudolph the Rednosed Reindeer on one end said Happy Birthday, Baby.
In Ray's writing.
So like Ray to not buy any real birthday wrapping, just use what he found in the house then stick something of possible value where probably even he would have forgotten it.
So not like Ray to buy her a gift that he had to have wrapped six months in advance since he'd died last fall and her birthday was tomorrow....
A pounding had the small wrapped box falling out of her hand and rolling onto the shards. It took her a minute to realize the noise was coming from the door, not the roof. God, she was on edge.
Finding a skull in your rose garden did that to a person. She didn't know whether she wanted to laugh or cry at that thought. As for the gift from the grave from Ray...
That was just weird. And sad. She'd figured out a long time ago that her marriage had been a result of youthful stupidity. But she had loved him nonetheless, in the way you can love a troublesome child, and she was determined to forget and forgive and move on. She hadn't been a perfect wife, either—
Whoever was at the door banged on it again. Impatient. She ran across the slightly slanted parlor floor—the foundation of the one-hundred-year-old farmhouse had shifted off-kilter more years ago than she knew about—and grabbed the handle. The carved wooden door swung inward, revealing a broad-shouldered figure, his profile shadowed on the porch overhang in the storm-darkened afternoon. Rain splattered down behind the man, puddles already forming in the yard. A very late model, very expensive-looking, very not-often-seen-around-these-parts sports car was pulled over and parked under the old oak by the cracked and crumbling concrete walk leading up to the house.
She found herself looking into the deepest green eyes she'd ever seen, fringed with incredible lashes. Near-black hair, on the long side, was plastered to his head, fanning the collar of his T-shirt. He hadn't escaped the burst of rain before he'd made it onto her porch.
"Keely Schiffer?"
He looked—and sounded—a little tense, even angry. Stubble shadowed his strong, well-defined jaw. He was dressed casually in faded jeans with a rip in one knee and a black tee under a leather bomber jacket, but there was nothing laid-back about his hard-edged demeanor.
He looked dangerous. And not in a good mood.
A shiver rippled up her spine and she couldn't decide if it was trepidation or, shockingly, attraction.
"Yes?" And you are—
"Jake Malloy," he said without her having to voice that question. "I was up at the Foodway and they said you hadn't left the keys to the Evans house. I was under the impression they would be there for me to pick up. Today."
Now she knew why he was mad. She'd set up the rental last week when he'd called. It was her fault for forgetting to leave the keys as arranged. And very out of character.
"Spring fever must have taken over my brain. I'm so sorry—"
He cut off her attempt at apology. "And I'm wet, miles out of my way, and I have other things to do. Do you have the keys?"
Well, now they had established that he was an ass. Good thing she hadn't noticed that he was also heartthrob material, especially since she was all done with men anyway. Not that she didn't like men. She liked plenty of males, mostly the ones who were related to her and were under the age of twelve. And yet she found herself remembering that she probably had dirt smeared on her face and she definitely had dirt on her jeans and the bright-yellow Haven Honeybees high school booster club T-shirt she was wearing.
He was probably six foot two, which had the effect of making her feel unusually feminine and petite at her five foot eight. That's all it was. And there was that bad-boy heartthrob thing, of course, that made her think of mindless sex.
Mindless sex with a stranger. Hot and raw and wild. One fantasy before she died.
Her pulse raced a little. Stop it, she warned herself. Really, she didn't even like him so far and she was thinking about having sex with him? Was she losing her mind? She had enough problems at the moment without making any new ones up.
Like, that skull in her rose bed....
"I'm sorry," she repeated. He could be rude; she couldn't. She had exactly two ways to earn money since Ray died—she'd taken over the small Foodway store in town he'd made the monumental mistake and command decision to mortgage her inheritance against, and she also handled local leasing properties as a sideline. Well, she let one of the neighbors run cattle on the farm and put up hay twice a year. But that didn't cut it, with the farm slipping through her fingers because of the store's sliding profit margin since the big warehouse-style grocery outlet had opened in the next town over last summer. "I have the keys. I forgot to leave them up at the store. I'll get them. It'll just take a sec."
She left him standing there and ran back to the kitchen. "So what kind of work do you do?" she called back to him through the screen. She'd left the main door open. Maybe he couldn't hear her over the rain lashing down, but he didn't answer. She remembered asking him why he was coming to Haven when he'd arranged the lease. He'd changed the subject then. And now—
The keys were on a hook on the wall with a little paper tag that read Evans. The rental house was, in fact, straight across the road from the Foodway, so no doubt he was extra annoyed that he'd had to drive all the way out in the country to find her. Or he was just an impatient ass and if he wasn't pissed off about one thing, he'd be pissed off about another.
She headed back to the door. "What brings you to Haven?"
"Business," he said briefly.
"What kind of business?" Did she really want to know or was she just being passive-aggressive at this point? She wasn't really sure. He didn't want to tell her anything, she was sure of that.
"I need to get going if you've got those keys."
She felt as if he'd smacked her hand. And maybe she just didn't really want him to go, even if he was an ass. She'd be alone again, just her and that skull and Ray's gift from the grave.
"Here you go." She leaned out between the doorjamb and the screen door just enough to pass him the keys. Their fingers brushed oh-so-briefly and she told herself to ignore the crackle of waking female libido that had no place in her life.
He was good-looking—so what? Good-looking and a little secretive. Even if she was interested in dating, which she was not, she'd had enough of the mysterious type. And he was surly to boot. And really, maybe he was a criminal. Drug activity had leaked out of the city, into rural communities. Maybe he was a thug. He certainly looked like one.
"My phone's out," she went on. "So if you tried to call from the store—" Not that she would have been able to drive the keys up there anyway since she didn't have her truck.
"Thanks for the keys." He didn't sound like he meant that and he was already turning away.
"Wait!" Keely bit her lip. How was she supposed to say this? "My phone is out, like I said, and my truck is in the shop—"
He stood there, still looking annoyed and impatient. Mr. Tall, Dark and Pissed Off didn't look helpful, and she really didn't want to tell this stranger why she needed to get the authorities out to her farm. I think my husband might have buried a body out back.... No. Not saying that.
"Never mind," she finished. "I hope you enjoy the house. You know what they say about Haven—it's just one letter short of Heaven." She gave him a polite smile that he didn't return. Jerk. She still didn't like him leaving, though.
She let the screen door shut. Alone again, just her and the dead body out back. The man turned to step off the porch.
Did it really matter if she contacted the police today? Her phone would be back, eventually. Or her truck would.
The skull and whatever else that went along with it in the garden wasn't going anywhere.... Not that this was a particularly comforting thought. She was just going to have to be a grown-up about the situation. She didn't need a man to take care of her, or so she'd decided. That meant handling anything that came her way.
Another huge gust swept down the mountain hollow and a crack tore the air, followed by a loud smack. And she realized Mr. Tall, Dark and Pissed Off wasn't going anywhere, either.
Not unless he was Superman and could pick an entire oak tree off his very expensive and probably very totalled car.
## Chapter 2
He couldn't believe his eyes. The car was going to be a complete loss.
Sort of like his day so far. And most of the past several months.
Jake Malloy tore his stunned gaze from the mangled vehicle and glanced back at the woman banging out the doorway of the farmhouse in that eye-popping yellow T-shirt of hers. Shoulder-length gold hair framed her suddenly pale face, making her milk-chocolate eyes stand out all the more.
She was sexy as hell and she'd been annoying him since the first time he'd talked to her on the phone last week about the rental. She asked too many questions—then and now. And he wasn't interested in providing any answers no matter how sexy she might be.
"Oh, my God," she gasped. "Your beautiful car! I'm so sorry!"
"Stop apologizing. You're not in charge of the wind, you know."
He sounded cold and rude, he knew. He was too filled with anger, too much negative emotion, for social niceties, that was all. Not too long ago, he'd had a successful career in the Charleston P.D. and he'd been a pretty decent guy. Then one fateful case had blown his life to hell and he'd spiraled into a black hole he was just beginning to dig his way out of. Supposedly, a little R&R was going to help.
"It's not your fault." He kept his voice ruthlessly hard as he went on. All he wanted now was to get the hell out of there and back to town. He moved, causing her to drop her hand from his arm, and turned to step off the porch. "I'll see if I can get my cell phone out of there and—"
"Cells don't work here. No signal."
He swore under his breath and wheeled back. She was staring at him, her pretty face and clear eyes looking fresh and innocent, and a little wary. If Haven was one letter short of Heaven, she was an angel.
But she was no angel, no matter how sweet she looked. And Haven was turning into sheer hell and he'd only been in town an hour.
"And my truck's in the shop," she reminded him.
"Where's the closest neighbor?"
"A mile that way." She nodded in one direction. "A mile and a half the other way." She indicated the other direction.
Rain poured in sheets. Wind blasted down the damn hollow, rattling leaves and jangling chimes hanging from one end of the porch. The warmer temperatures from earlier in the day were dipping quickly.
"And when it rains like this, the low water bridge flash floods," she added. "It's very dangerous, even if you wanted to get yourself soaked hiking off to find someone who would give you a lift. It wouldn't be smart to try it. The water rises fast, faster than people expect sometimes."
She looked fragile and worried suddenly. A little bit haunted. Generally, he was good at sizing people up. Decoding body language—every movement, every look and expression—was his business, which was also why he knew that the clues could be unreliable as hell. People smiled for all sorts of reasons and happiness was only one of them, and pathological liars could lie with flawless eye contact. The more information that could be gathered, the more likely the decoding would be accurate.
His instincts had him wondering what had brought that pained expression to Keely Schiffer's face, but he reminded himself that he didn't need to know and pushed the question aside.
Looking away from her, he stared out at the wild weather for a heavy beat. He didn't really give a rat's ass about the car other than its function as transportation. It used to be important to him, his pet, his baby, and he'd invested a ridiculous amount of his modest income in it. It didn't seem important now. But he did care about being stuck out here in the middle of nowhere, and for God knew how long. Another one of life's fun twists....
Jake breathed deeply, summoning the strength and willpower to push back his own pain and control the razor-sharp edge of his temper. She didn't deserve to bear the brunt of emotions that weren't her responsibility, though he had to wonder why the hell anyone lived out here in the sticks. He was a city boy, born and bred. This little trek to Haven hadn't been his idea, but he'd do anything, anything, to get his life back.
He looked back to find her still staring at him.
"If it doesn't keep up like this too long, the creek'll go down in a few hours," she said. "Dickie—he's the mechanic—will be back with my truck then, or the phones'll come back on. Or we can find you a ride."
She was talking as if this was her problem, too. For some reason, that stabbed him with a slice of hot hurt.
Wind blew a piece of her sunshine hair in her face. She brushed it out of the way, tucked it behind her ear. He could almost feel the small, soft curve of the shell of her ear beneath his fingertips.... And those eyes of hers. They were compelling, private yet vulnerable.
He forcibly reminded himself that he wasn't interested. Period.
But he wasn't getting away that easily. Not yet.
She waved him into the house. "Come on. Come inside. It's getting cold out here, and you're going to get wet. Wetter," she corrected.
He was already plenty wet, but she was right about one thing. Rain was blowing sideways onto the porch. And was he imagining it or was there something beseeching about her expression? As if she wanted him there. Almost as if she was relieved that he was stuck there for some reason.
She was hard to read, even for him, and that was bugging him.
"I'm a complete stranger. You don't know me from Adam." He had the stupid urge to tell her not to trust people. At all. Ever. He could go inside her house, and do anything he wanted to her after that. Not that he would. But a woman like her, alone out in this godforsaken countryside, shouldn't be asking strange men into her home. She seemed...nice. Genuinely nice, even if slightly annoying and nosy.
He felt an unexpected and uncomfortable sense of protectiveness toward her that he fought to shake off. It wasn't his concern if she was hopelessly naive about human nature.
"You don't look like a serial killer," she said flippantly, even as her sweet chocolate eyes studied him. "You're not in the big city now. You're in Haven. We're friendly here." She shrugged. "The people at the store sent you over here. You're not going to hurt me unless you're stupid. You're not stupid, are you? Anything happens to me today, my friends'll be looking for you, not to mention my family. Especially with your car sitting right out front."
His car that wasn't going anywhere. She had a point, but it was unrelated to why he really didn't want to go inside her house.
"I'll fix you something to drink," she said cheerily. "I owe you anyway for all the trouble of driving out here to get the keys, and if you hadn't had to do that, you wouldn't be stuck out here now with a tree trunk on top of your car."
She was already walking into the house, leaving the screen door to bang behind her and the front door open. Her slim, sexy figure disappeared through the shadowed parlor even as she kept talking, seeming to simply expect him to follow. He opened the screen door and stepped inside in spite of himself.
"You want water or tea?" she called back to him. "Or I've got some Coke."
He walked through the front room, a parlor with a slanted, scuffed hardwood floor. Rows of antique-looking photographs filled the room, solemn-faced eyes following him from the walls. The house smelled good, like cinnamon and sugar. Homey. Not that it was anything like the home he'd grown up in on the seedier streets of Charleston. Homey like...you saw on The Andy Griffith Show. He half expected to find Aunt Bea in the kitchen, pulling fresh-baked coffee cake out of the oven.
He arrived in the open doorframe between the kitchen and the parlor. She'd gotten down an amber-colored glass from a cabinet and was pulling every beverage known to man out of the fridge. Coke, iced tea, lemonade, milk...She'd probably offer him a cookie next.
And she was still talking.
"I've got sweet tea made, but if you don't like sweet, I can make some unsweetened. I don't mind."
"I'll just take some water." He didn't really care, truth be told. The whole scene suddenly felt terribly domestic. When was the last time he'd been in a kitchen with an attractive woman?
He didn't want to remember, but of course he could. Sheila had lived with him for two years in their nice, newly-constructed, cookie-cutter condo in South Charleston. She'd wanted to get married. He'd been in no hurry. Maybe he'd known all along it wasn't going to work out.
Sheila hadn't wasted any time when things had gone bad. Sooner was better than later, he figured. He and Sheila would have never made it anyway. She'd just been...convenient, for a while. He'd scarcely looked at a woman since. He liked being alone, detached.
And yet he found himself watching Keely Schiffer with a sort of odd and uneasy longing. Ghost pain, he thought wryly, like a patient who felt sensation in an amputated limb. He didn't think he missed Sheila, or her constant pressure.
He hadn't realized till now that he'd been missing anything at all other than work.
"Please sit down," she said when she finally gave him the glass. "Well, I hate to say it, but this rain is a good thing because we've had an awfully dry spring. I'm just so sorry about your car. Some welcome to Haven for you, huh?"
She pulled out a chair when he didn't. He scooted it around a pile of broken pottery he noticed on the floor as he sat. He placed the glass on the table.
"I was just about to clean that up." She disappeared for a minute into the next room then came back with a broom and dustpan. She bent down, picked something up, and he saw what he'd missed at first—some sort of small package. It was wrapped in silver foil and he read the label.
"Somebody's birthday?" There, his contribution to chitchat.
"Mine."
She glanced up from sweeping the shattered bits of cream and blue pottery. Her eyes looked huge in her slender face, and as he watched, she chewed on her full, unpainted lip. He looked away from her, to the box. Happy Birthday, Baby. She had a gift from somebody who called her Baby.
He carefully returned his gaze to Keely. "It's your birthday today?" he asked, and told himself he was not going to look or even think about her nibble-on-me lips. Maybe she was married. He didn't know why he'd assumed she lived way out here in the sticks alone. It didn't matter to him anyway.
"Tomorrow. The present was inside the cookie jar. It fell down off the shelf." She waved her hand vaguely toward the ledge over the cabinets. It was full of decorative glass items and various pieces of pottery. "I guess he was hiding it there. My husband, I mean. A branch must have hit the roof. I guess the jar was too close to the edge of the shelf. The house really shook and—" She stood, the pottery bits tidily swept into the dustpan in one hand. "I forgot. I need to get up in the attic and check it out. If rain's coming in, I'm in real trouble."
So she was married.
"You'll be in trouble when your husband finds out you stumbled onto his surprise." He was feeling suddenly much lighter, more in control.
She propped the broom in the corner of the kitchen and dumped the shards of pottery in the trash before replying. "He's not going to find out. He's dead. And he left me plenty of surprises. Most of them weren't good."
The look she gave him was flat and emotionless, then a shadow slid across her expression. She looked away quickly, as if afraid he had some kind of laser vision that would see something she didn't want him to see. Jake felt more uneasy than ever, and he wasn't certain if it was because she wasn't married after all or because he wanted to know what her deceased husband had done to hurt her, and he shouldn't want to know anything about her at all.
The muted patter of raindrops on the roof filled the kitchen. The storm was slowing down. Or at least, the rain was slowing down. Wind gusted against the house, strong as ever. The clapboard farmhouse creaked a bit in the storm.
"I'm sorry," he said.
She shook her head. "No, I am. I shouldn't have said that. You shouldn't speak ill of the dead." She grabbed the wrapped box off the table and turned away, pulled open a wide kitchen drawer, shoved it inside and slammed the drawer shut.
He heard a noise like thunder and suddenly the house shook so hard, he felt the floor move under his feet. The drawers in the kitchen banged open and Keely stumbled on her feet. Automatically, he shot up, grabbing hold of her upper arms. Glass hit the floor around them from the shelves over the cabinets. He heard pictures fall in the parlor.
"Oh, God, I knew I should have had that maple tree taken down." She sounded panicked. "It's too close to the house."
"I don't think that was a tree." He hadn't heard anything strike the roof.
There was no sound for a long beat, as if even the wind held its breath, and then came a roar. The house seemed to roll under them in waves. Jake fell against the table, still holding Keely, and together they crashed onto the floor. The sting of glass cut into his back. He could feel her breasts against his chest, her quivering belly and thighs, her breaths coming in shocky pants near his cheek. He stroked his hand down her spine, only meaning to soothe. She was soft—
The floor rocked violently beneath them. "We have to get out of the house," he grunted, pulling her up with him, both of them staggering as if they'd been transported to the deck of a storm-tossed ship. At the same time he realized the roof was coming down over them, the floorboards beneath them ripped apart and all he knew were eerie flashes of blinding red light, then plunging darkness.
## Chapter 3
Darkness closed in on her with terrifying completeness. Keely heard the boom of her heartbeat, the harsh sound of her breaths, in the sudden, awful quiet. Oh, God, oh, God. She waited for the rest of the kitchen, the rest of her house, to fall down on top of her.
Something shifted overhead, and crashed a foot away. She nearly jumped out of her skin.
Arms she hadn't realized were holding her tightened, as if ready to shield her from anything. She couldn't see a thing, not even the man she was clinging to. Fingers reached up, touched her face. She was on top of him, she realized. They'd hit hard, him protecting her with his body.
"Are you all right?" He was little more than a deep, disembodied voice in the terrible blackness.
"I think so." Her voice wobbled. Was the world still shaking? She bit her lip to keep from hyperventilating.
Jake Malloy grunted in pain, and she scrambled off him, pulling him up with her till they sat on debris. She could feel nothing but debris surrounding them.
"Where are we?" he asked her. "We fell into some kind of basement. Is this a cellar?"
She nodded, swallowed thickly, realizing then he couldn't see her.
"Yes. It's the cellar." Her head reeled. The kitchen ceiling had started coming down and the floor had opened up. The cold dampness of the cellar seeped through her then and she shivered. Shock. Maybe she was in shock. The cellar was low-ceilinged. They'd only dropped maybe seven feet.
And the ceiling boards from the kitchen must have covered the gap in the floor above. She felt as if her heart might pound out of her chest.
"There's a door, over here." She pushed to her feet, stumbling slightly on the uneven piles of wreckage she couldn't see. "The ground slopes down this way and the cellar's reached by a door below the rear of the house." In the thick darkness, she felt him reach for her hand. His hold felt strong and warm on hers. Oddly safe. Together, they took baby steps across the debris, guided only by her sense of direction, which was, at the moment, rocky.
She reached out with her hand, feeling her way. Her fingers brushed against the rough, peeling paint of the wooden door to the cellar. She pulled her other hand from his, heart thumping as she grabbed the handle. Debris in the cellar made opening the door nearly impossible.
"Wait."
She could feel the brush of Jake beside her, hear the sound of broken boards being tossed out of the way.
"Try now," he said.
The door creaked as she pulled it inward, still scraping across smaller bits of rubble. She pushed around it, reaching forward into the pitch-black. And stopping short at the sensation of rough, jagged material blocking the way.
No, no, no. There had been no light around the door, it hit her suddenly. No light not because it was dark outside but because part of the house must have fallen this way!
And who knew how much weight there was in wreckage blocking them from climbing back up into the kitchen. "We're trapped," she whispered in horror.
"We'll be all right. We'll get out of here."
He sounded so sure of himself, she almost believed him for a minute.
She swallowed hard. "How?"
"Rescue workers will be coming—"
"Do you know how long it will take them to get here, this far out of town?" If they even could. They'd have to wait to even try if the low water bridge was flooded. What was happening in the town? What about her store? What about her house? It was gone, clearly gone. And yet she still found that impossible to grasp. She loved her house in all its faded glory, from its American gothic farmhouse architecture to its walls teeming with family history. "Gemini" tea roses were her grandmother's favorite, that's why she was planting more of that specific variety. She was supposed to be planting roses right now. A normal day, planting roses, waiting for her truck to be done at the shop. She'd have fixed herself a sandwich for dinner, maybe a bowl of soup, and watched the news, followed by the latest season of her favorite amateur singing competition, and the new medical drama. She'd have gone to bed in her antique spool feather bed covered by a hand-sewn block quilt and read a magazine till she went to sleep.
Her life was boring, maybe, but she liked it. It was quiet and sensible.
Nothing made sense right now, especially how much she didn't want this stranger to let go of her. She clutched blindly at his shirt as she felt him turn.
"Are you all right?" he asked. "Did you cut yourself on anything?"
She felt his hands moving down her shoulders, her arms, as if checking. "No. I mean, yes. I'm all right. What happened? What could do this to my house?" She could barely stretch her mind around the horrifying reality of it. "Oh, God. That was—"
"An earthquake."
We don't have earthquakes in West Virginia. She hadn't realized she'd said that out loud till he answered her.
"Not very often. But if we weren't just at the epicenter of that one, I don't know what it was."
Her mind stumbled from the realization. That first time she'd felt the house shake and the cookie jar had fallen off the shelf had just been a precursor of what was coming.
There had been no tree hitting the roof at all.
"My house—"
One hundred years old, and it was in pieces over her head. Everything that had been in her family for generations—Her parents, Howard and Roxie Bennett, preferred their spacious home with all the modern conveniences and close to town. Her older sister and two brothers already had their own homes, too, by the time Granny Opal had died. Keely and Ray had needed a place, and so the farmhouse had gone to Keely, who had gladly accepted it. But now...What was she going to do without a house?
Another thought struck her. "What was that red light? Did you see it? Oh, my God. Was that fire?" No, it couldn't have been fire. They'd know if the house was on fire above them. So what—
"I don't know. Probably electricity snapping, who knows. Forget your house." A beat stretched, taut. "I'm sorry," he added, gentler. The unexpected kindness in his touch and voice sent her into a panic. She'd been hesitant to so much as ask him for a favor not too many minutes ago. Now she wanted to climb up his powerful, hard body and beg him not to leave her for a second in this pitch-black nightmare. He was the only other human being in her world, her touchstone to reality.
"You're okay, and that's all that matters," he continued. "Come on. You can worry about your house later."
As if he sensed she was an inch away from royally flipping out, he stroked his hands up and down her arms again. His touch was warm and strong and she didn't want him to stop. Probably, he didn't want to deal with a hysterical woman. He hadn't seemed this kind and patient earlier.
She was on fear overload, and she hated that. She was used to taking care of herself by now. She didn't need anyone, especially not a man. Get it together.
"I'm okay," she repeated back to him. The aching of her bones hit her, surfacing through the adrenaline. She was lucky she hadn't broken anything in the fall, even as short as it had been. Lucky he hadn't broken anything, either.
"You're okay, too, right?" she asked, to be sure.
"I'm fine. Tell me what's in here. Do you keep a flashlight somewhere?" He sounded steady, composed, organized.
"No such luck. But matches—maybe." She worked to catalog the cellar in her mind, recall what was where. She had to be strong now. Not fall apart.
One wall of the cellar had been lined with glass canning jars. Some empty now, others still packed with the fruits of last summer's gardening. Some old wooden stools. Boxes and antique trunks filled with forgotten items that had worked their way out of the farmhouse at one time or another. Tools that hadn't been used in ages. A couple of old tables. Basically, junk. The cellar was full of junk. The wall on the other side was storage. There were some candles somewhere, stashed on one of those over-packed shelves....
Vanilla-scented. They'd been a gift from a Christmas exchange party at church last year. She hated vanilla candles and she'd stuck them in the cellar, not able to bring herself to just throw them out. If they were still here...
She'd absolutely love, adore and worship the scent of vanilla right now.
Hadn't she left an old box of matches here somewhere? And a can of gasoline. She'd used the matches and gas to burn the brush pile last summer.
"This way." She moved along the wall to the right of the door. Away from the center of the cellar, there was less debris from above, but there was shattered glass everywhere. She walked carefully, but stumbled anyway when something creaked overhead.
Jake caught her as she made a strangled cry and she found herself dragged up against that hard, powerful body of his.
"Whoa." He held her for a long beat.
She couldn't see him, not even his eyes and they had to be only inches away. He smelled good. She hadn't noticed before, but she did now. He smelled really good. Woodsy and male. She was ready to cling on to him like he was some kind of life preserver. He definitely exuded some kind of raw masculine energy that was messing with her mind, which was hardly stable as it was. Her head was all over the place, reeling.
"Careful," he murmured.
"I'm trying." She'd better try harder.
The air in the cellar was suddenly thick with an odd tension. They were practically buried alive down here. He could be the last human being on earth she'd ever know. She was scared, terrified really, of dying before they were rescued. She'd lost her house, maybe her business for all she knew. But in the face of losing her life, suddenly it didn't amount to much.
Her family and friends—She had no idea what had happened to them. All she could do was desperately hope and pray. Images of her parents, her friends, flashed like photos in a slideshow in her mind.
A sob choked her throat and she swallowed it down. She couldn't do anything for anyone now but herself and this stranger beside her. She didn't realize she was crying till she felt something wet and cold trickle down her cheek.
"Hey. Come on. Let's find those matches."
She nodded, then tried to find her voice.
"Okay." She squeezed her eyes shut, struggling to stop the tears.
A hand touched her face. Jake Malloy's hand.
"Aww, now," he said, softer now, brushing at the tear.
Instinctively, she curled her arms around him, her hands sliding under his jacket, letting him pull her tight and comfort her there in that black void under her ruined house. She didn't understand how she could trust him like this, but it didn't matter. His arms wiped out that horrific fear.
Her head rested against his chest and she could hear his heartbeat, steady and sure as hers was not. He kept right on being rock solid even with the world falling apart around them.
And in the back of her mind, a crazy thought entered. She wondered what it would be like to kiss him. He moved against her, just slightly, as if checking his balance, and she shook herself and started to pull back.
Something cold and hard stuck out of the waistband of his jeans, beneath the cover of his jacket, as she drew her hands down and away. Something cold and hard and metal...
Her heart stopped and total fear slammed back in on her. She felt him freeze, and she moved quickly before he could, grabbed it in one hand and stumbled back, knowing what she held. Not needing any light.
She was holding a gun.
## Chapter 4
He felt the cold metal slide away from the waistband of his jeans and before he could move, she was gone, nothing but a ragged gasp in the darkness. He lunged forward then stopped cold at the low, fearful, shaking sound of her voice.
"Don't come near me. I'll shoot."
"No, you won't." He hoped she wouldn't. Hell, he didn't know. He didn't move.
He could feel his heart thumping hard against his ribs. And he could have sworn he could hear hers, thumping, too. She was scared and he didn't blame her, but he needed to get back in control of the situation. Scaring her more wasn't the way.
"I know how to use a gun," she said, that soft, low voice of hers still uneven. "What I don't know is what you're doing with one."
"It's licensed," he told her, keeping his voice steady, reassuring. "I have a right to carry it. There's no need to be afraid."
"I'm trapped here with you and a gun. I think I can decide for myself if I should be afraid or not."
"You already pointed out that everyone in town knows I'm here. My car is still outside. Rescuers will get here eventually. Why would I want them to find me here, in a cellar with a dead body and my gun? I'm not going to shoot you. I'm not stupid, remember?" Reason, he had to use reason on her. She was already frightened, for good cause, by the quake and the destruction of her house and their desperate situation.
She was silent for a beat and he could hear the house creak over them again. She could hear it, too, and he heard her feet shift on the rubble, knew she was unnerved even more, wondered if she was trying to decide whether she needed to hold his gun, or hold him, to feel most safe.
"Do you think I want to spend God knows how many hours alone down here, waiting for help, with a dead body?" he asked quietly. "I don't want to shoot you, Keely. I—"
His throat closed up a bit and the next words were hard to admit, but he had to make a choice, too. Risk a little of himself, or risk his life if he let a very frightened woman continue to point a gun at him. Situations got out of hand sometimes. He knew that too well.
"I like you," he finished finally. "Why would I want to shoot you?"
She didn't say anything for long seconds. He felt the electric pull of her even through the dark. She was thinking, he knew. Thinking about whether she could trust him or not.
It appeared she wasn't so naive, after all.
"You walked up to my door with a gun," she said. "And I want to know why." Her voice strengthened.
She was pulling herself together. That hadn't taken long, and it occurred to him that she had a tough spine inside that very sweet, hot, bombshell-quality body of hers.
He did like her, he realized with a shock, even though she had annoyed him quite a bit, from the first time he'd spoken with her on the phone about the rental, with questions he didn't want to answer. He liked her in spite of himself because she was nice. Even when he was rude to her, she was nice. In fact, she was too nice. Too nice for him. His instinct to get away from her as quickly as possible had been a good one.
Now he couldn't get away from her and she was going to take her opportunity to ask questions again, and he was going to have to give her answers whether he liked it or not. And he didn't like it at all.
"I'm a cop."
The house lay so still around them, he could hear the very low intake of her breath, sense the tension emanating from her body as his words sunk in. His gut tightened, waiting for her to respond.
"A cop?" She didn't sound like she really believed him.
He figured she'd thought he operated on the opposite side of the law, based on his appearance. He'd worked undercover most of the past few years and his wardrobe had suffered in keeping with his cases. Not that he cared or that it mattered. He was supposed to be resting and relaxing, not dressing for success.
In truth, he was just biding his time. He didn't need rest and relaxation. He needed to get back to work. The damn thing was, the chief wouldn't let him. The department shrink had said he wasn't dealing with his grief. Go to the country, the chief had ordered. Get some perspective. Unwind. One month. Then he'd let him come back to work. He'd suggested Haven. The chief had grown up here.
Jake had thought he was dealing with his grief just fine. How the hell was someone supposed to take it when they were responsible for their partner getting blown up right in front of them? And people had called him a hero. He'd just wanted to get back to work. He still wanted to get back to work. He wanted to bury himself in work. No thinking. No feeling. And certainly no consorting with the locals. He didn't want any entanglements.
But here he was in Haven, trapped in a cellar with a beautiful woman. How had that happened?
"Charleston Police Department," he told her.
"And I'm supposed to know that's the truth how...?" she asked.
"Because I'm telling you it's the truth...." he said. "I'm one of the good guys, Keely. I promise." He waited a beat. "If you don't mind, I don't really like it when people point guns at me," he said. "It makes me worry about whether I'm going to get to keep breathing. You stop pointing the gun at me and we find a candle, then I'll show you my badge and ID. Deal?"
He heard the soft click of the chamber pushing open.
"I'm going to take the bullets out. You don't mind, do you?" she asked.
"Not at all."
She hadn't lied about knowing how to operate a weapon. And she might believe him—or might not—but she wasn't going to leave the gun loaded. Again, not so naive, after all.
It wouldn't do her a whole lot of good if he wanted to wrestle the empty gun away from her and find the bullets, but it would buy her time. Better, he supposed she was figuring, than letting him wrestle the gun away from her loaded.
She'd probably just put the bullets in her back pocket. There weren't a whole lot of other options available.
He heard the gun drop on the debris behind her.
"I know you can pick it up," she said then. "I know you can get the bullets away from me. But," she added dryly, "I suppose you're right. You're pretty stuck if help comes and I'm laying here in a pool of blood. Wouldn't be too smart on your part. I just don't like loaded guns, so let's not keep it that way. Okay?" She still wasn't completely trusting him.
"Okay."
Tentative truce. Fragile, very fragile, he'd guess.
He'd take it.
"They'd know you did it," she added.
"Yes."
"You'd go to prison."
"Definitely."
"For the rest of your life."
"Probably."
"Or get the death sentence."
"There's no death sentence in West Virginia."
She was silent for a long beat. Disappointed, probably.
"You know what they do to guys like you in prison," she said finally.
In spite of himself, he felt a slow lift to his mouth. He actually almost laughed.
"Are you saying I'm cute?" What the hell was he doing now? Flirting with her?
He heard her blow out an irritated breath. Yeah, she thought he was cute. She probably hadn't meant to give that away.
"I'm not saying you're cute," she said tensely. "I'm not saying you're anything but on your way to the slammer if you try to hurt me."
He reminded himself that it wasn't important what she thought of him as long as she stopped holding a gun on him.
Sobering, he said, "I'm not going to shoot you, Keely. I don't want to hurt you in any way."
She was silent for another long stretch.
"I'd probably never find the candles and matches without you," he tacked on. "Plus, I'd be lonely down here waiting for help."
"Oh, yeah."
He heard her move, slowly, carefully, toward one wall of the cellar. Good. Back to business.
"There's a trunk over here, somewhere," she said.
He followed the sound of her voice and her footsteps. She'd knelt, was clearing debris from something. He went to work with her, removing boards and bits of plaster and who knew what else.
"This is it," she said, and her voice rose, confident, hopeful. The trunk lid creaked open and she fumbled around inside. "Here they are."
The box opened with a soft sound then she struck a long match, held it up.
She wasn't just a voice in the dark anymore. Her eyes glowed in the light from the flame, wary and still scared. He knelt there, close to her, close enough to fill his nostrils with her heady scent, feel overpowered for a second by the vulnerable look on her face.
Apple. She smelled like apple. Deliciously sweet.
He reached for his wallet, flipped out his ID and badge for her to see.
"You don't have to be scared of me," he said quietly, tucking the truth of his identity back in his pocket. "I really don't want to hurt you, Keely. I'm not going to. I promise."
She stared at him, and time locked, forever, it seemed, then she blinked and turned her gaze down, away from him.
"I don't believe in promises," she said so softly, it was nearly a whisper. "People lie all the time. So cut it out with the promises. I'm not interested."
The heart he wasn't supposed to feel tightened a little at the break in her voice. She'd been hurt, badly, he had no doubt now. Probably by that dead husband of hers.
But he wasn't responsible. It bothered him, anyway. That look in her eyes, that pain in her thready voice, bothered him. This was more than police instinct to read and study people. This was about her. And that wasn't good.
"I'm sorry anyone ever lied to you," he said, and it was too late to bite the words back even if he really wanted to.
She glanced back up and he saw emotion shining in her eyes. She cleared her throat, blinked back tears. "There are candles over there somewhere."
The long match was half-burned when she stood, moved to the other wall. Broken canning jars lay everywhere and she crouched again, searching. He went after her.
"Here they are." The joy in her voice was catching. "We have light!" She stuck one of the thick candles inside one of the intact jars and lit it with the match, then stood. The scent of warm vanilla rose around her, mixing with the ripe apple scent. She smelled good enough to eat and his libido was taking his brain in directions he didn't want to go.
He stood in front of her when she turned, the candle in the jar in her hand. He wanted to kiss her. Her mouth was right there, inches away. It was crazy, ridiculous. Her hair fell around her face in shimmery strands, like spun gold, wildly sexy and just begging for a man to tangle his fingers into it, pull her face close and—The strength of his very vivid fantasy shocked him and left him with a weird, edgy feeling as he reminded himself that he wasn't interested in any kind of relationship, with Keely Schiffer or anyone else.
"We're going to be okay, right?" she said then.
"Help will come. Your friends and your family will make sure of that." And he was sure she had friends and family that cared about her. He could just tell. She was all apple pie goodness through and through. A nice, wholesome country girl.
She couldn't have been more foreign to his experience if she'd hailed from another continent. Maybe that was the trouble. He was used to women who wielded their sexuality like a weapon. She was innocently sensual, naively seductive. She was killing him.
"If they're even okay." She bit her lip and he could hear the fear in her voice. "I don't know if they're okay."
"Faith," he offered. "You have to go on faith for now." He didn't know where that came from. He hadn't had much faith lately. He wanted her to have it, though. "We were lucky, you know? We may be trapped here, but we're all right. They were lucky, too. Just believe that for now. There's nothing else you can do."
He smoothed the hair back from her face even when he knew he shouldn't touch her more than necessary. He couldn't seem to stop touching her, but he forced himself to. He dropped his hand back to his side.
What the hell was wrong with him?
"You're right, I know," she whispered, her eyes holding him. "Stop being so nice," she said suddenly. "It's freaking me out."
He laughed, surprised by her remark, and loved it when she smiled through the shine of tears in her gaze. It was an unexpectedly satisfying reward.
"Sorry. I can go back to being an asshole if you want."
She laughed now. "No, I guess I don't want you to do that. I'm stuck with you here, after all." She cocked her head, studied him. "We're stuck with each other."
He nodded. "Looks that way."
"For who knows how long," she added. Her gaze moved, swept the cellar. In the flickering candlelight, the wreckage was stunning. "What now?"
## Chapter 5
Keely knew what she wanted to do. Run away. Jake Malloy, Mr. Tall, Dark and Pissed-Off, was more complicated than she'd expected. He made her feel safe at the same time he made her want to run away screaming.
There was no place to run, though, so she was going to have to trust him. He was a cop. Not a thug, a cop. A hot, delectable, adorable cop and here she was, trapped with him.
"We could keep standing here," he said, too close, way too close to her, looking dangerously sexy. "But that might get old."
Thank God he wasn't touching her anymore. She liked it too much when he touched her. She'd already made the big mistake of letting him hold her. Bad, Keely.
She'd be better off if she didn't remember how good that had felt.
"If we clean some of this up, there are some old tables here. We could sit on them, or stretch out on them to sleep. We could be here for a long time." She shivered, thinking of how long it could be. "We could be here for days."
"We won't be. Come on."
He set to work, and she was more than ready to occupy herself with something, anything, that would take her mind off the situation—the way they were trapped, and the heat she thought she saw simmering in Jake Malloy's eyes. Heat for her. It could be her too-vivid imagination, but she didn't think so, and as they worked, carefully moving the heavier debris, she worked, too, on stomping out any thoughts about it.
The past was always there, reminding her that a man in her life wasn't what she needed. She'd given herself way too easily to Ray. The sex had been okay, and she missed that, but it was the way he'd taken over her life that had really hurt.
She was a sucker, but no more. Ray had cheated on her, stolen from her, lied to her. No more trusting men, not further than she could throw them. She wanted to throw Jake to the floor and throw herself after him. Oh, God, where had that thought come from?
"So why are you in Haven anyway?" she asked as they worked together. "I thought you said you were here on business. Are you here on a case? Undercover?" He looked like a man who lived for danger.
"I'm here on some R & R."
"Why Haven?"
He shrugged. "My chief recommended it. He's from around here. Jerry Overton."
"Oh, I know the Overtons," Keely said.
"Guess everyone knows just about everyone around here."
"Yeah. Pretty much. Small town. It's nice, though. Well, most of the time. People care. Are you from Charleston?"
She was just making conversation, passing time. She wasn't trying to get to know him better. No, not her.
"Grew up there. Born and bred. City boy."
"You'll love Haven. It's—"
"Friendly. One letter short of heaven. Uh-huh."
He didn't sound like he planned on loving Haven. "How long are you staying?" He'd put down a month's rent, that's all she knew.
"Just the month. Then I'm headed back to work. Back to the city."
Back to city lights and cell phone service, she guessed. He wouldn't be around long. Good thing she wasn't going to get attached to him.
They had a couple of tables cleared and pushed together, and with the jarred light on a nearby shelf, Keely dragged a couple of quilts out of the trunk and climbed up on one. She wrapped one of the quilts around herself, pulling her knees to her chest. It was getting colder. The cellar was always cold, but night had to be coming on and temperatures dipped hard in Appalachia even at this time of year once the sun went down.
Jake took the other blanket and did the same. His gaze caught hers with unflinching gravity. He'd said they'd be rescued soon, as if he had no doubt, but she didn't believe him. No telling how long they could be stuck down here.
"Are you hungry?" she asked. "There's food." They weren't going to starve, at least. Not all the canned goods had broken. There were green beans, jams, apples, pears, relishes.... Not exactly a square meal, but this wasn't a situation that called for being a picky eater.
"Not now. Did you can this stuff?"
She nodded. "I always grow a big garden."
"Gardening, canning, handling guns...You're a real all-around woman."
He was looking at her, and she felt herself blush. The candle was behind her, leaving her face in shadow, so she hoped he couldn't tell.
"I'm a country girl," she said.
"I noticed."
She wondered what he meant by that. Maybe he thought she was a hillbilly hick. Like she cared.
"We could try to get some sleep, I guess," she suggested.
"I don't sleep much lately," he said.
"You lie awake, stressing?" She did that, too.
"Sometimes."
He sounded guarded, as if he didn't want to admit to any vulnerability. She felt another twinge of something, a disquieting bonding with him that she never would have expected. She wondered why he was on R & R.
"I've had trouble getting used to sleeping alone," she admitted. "I don't miss much about Ray, but I miss that. Having someone there to hold me." She pulled the quilt tighter, felt the guilt hit her. "I shouldn't say that."
"Why not?" Jake asked quietly. "If it's how you feel, say it. It's not like Ray is going to hear it and be offended."
That almost made her laugh, but it was too sad. Even if she didn't miss much about their marriage, he'd been too young to die. He'd loved her in his way, hadn't he? He'd bought her a birthday present six months in advance. That had to say something. She'd probably never find that little box in the rubble that was left of her house now, but it was the thought that counted.
Of course, he may have left a dead body in her rose bed, too.
"I'm never getting married again," she said. He watched her, steady, and she had the weird sensation that they were the only two people left on Earth. Down here, trapped in the cellar, they might as well be. The flicker of the candle played light and shadow on the hard planes of his face, softening them. "You ever been married?"
Did he miss having someone to hold him at night?
He shook his head. "Never. Got close once, but I escaped."
"How?"
"She dumped me."
He was so matter-of-fact, she couldn't tell if he was upset about it or not.
"Do you miss her?"
A beat passed before he answered. "No. Now that sounds bad, doesn't it?"
"Dating anyone?"
"No." His smile was heart-stoppingly slow. "Why? Want to set me up?"
She laughed. "No!" Selfishly, no. But that wasn't something she needed to be thinking about, wanting him for herself. Even if she was more aware of him right now than she'd ever been aware of a man before in her entire life.
This conversation wasn't going anywhere good. She was starting to feel...hot. Maybe she was displacing her fears about the situation. But that was no excuse for being stupid. She didn't know Jake, not at all. Her first impression of him had been that he was angry and impatient. Then he'd seemed possibly dangerous. Now she didn't know what to think.
Other than that he was single and available, and she didn't need to be thinking about that.
"Wow," he finally said. "That was pretty certain." His lips curved further, proving he wasn't offended.
"It's none of my business. I shouldn't have asked." She sounded prim enough now, didn't she? "And I don't set people up anyway, so it's not personal."
"It's okay," he said. "I'm not interested in dating."
"Planning to be a monk?"
Jeez, what had made her ask that? She felt her face heat again, dammit. She was embarrassing herself.
"No, not exactly," he said, amusement tinging his voice. Then he seemed to sober. "Men and women, they make things too complicated, you know? Defining and analyzing what should be natural and easy."
"What do you mean?"
"Relationships. Sex. Marriage just gets courts and judges and lawyers involved."
If you got divorced it got courts and judges and lawyers involved. She got the idea Jake expected marriage to end in divorce.... Well, maybe if Ray had lived, that's where they would have ended up. If she'd finally gotten the courage to take the steps necessary.
"You don't believe in love?" she asked softly.
"I don't know about love," he answered. "I've seen what happens to people who think they're in it, though. I'm not interested."
She couldn't argue with him, even when she felt a disquieting twinge at his flat words. She felt the same, didn't she?
Then she heard it. And this time she knew what it was. It wasn't thunder. It was an aftershock. The house shook over their heads. Jake wasted no time even as the breath was so stalled in Keely's throat she hadn't thought past it.
Powerful arms hurled over her, rolling her down off the table, onto the floor where his body, again, protected her and then pulled her underneath. The quilt she'd been covered in, and his arms and legs, tangled around her. Debris crashed down from above and glass jars clattered from the shelves.
Darkness fell with almost violent suddenness.
The candle was gone, and as more and more material rained down, she knew so was everything else. Everything but Jake, here, holding her so tightly. Oh, God, oh, God.
Silence, then another creak, and another piece of her hundred-year-old house tumbled down onto the debris already filling the cellar. She could hear her heart beating hard, feel Jake's as he held her.
"We're going to die," she whispered.
"No, we're not. We're not going to die." His voice was raspy, close. She couldn't see him, but he was here, his breath warm against her ear.
"You don't know that."
"You're right," he admitted. "I don't know that."
She shuddered and he gathered her closer. He shifted, moving the quilt so that it was all the way around both of them, his hands inside it, stroking her, comforting her.
"We're safe as we can be, here. We'll stay under this table."
The table had shielded them from what had rained down. If not for his quick action—
"I'm not going anywhere," she whispered into his throat. How many aftershocks until the house crushed the table, crushed them, trapping them under its weight? She noticed he wasn't telling her they were going to be okay anymore.
"Neither am I. I'm right here."
And he was, he was right there, his arms around her. What if he hadn't come out to the farmhouse today? She would have been here by herself.
"I'm glad you're here because I don't want to be alone," she breathed.
"You're not alone."
She turned her head slowly. She couldn't see him but that didn't matter. She could feel him.
Then she could feel it again, another aftershock on the heels of the last one. And the terror was so all-consuming, it took over and she sucked in a breath to let out with a scream—but she couldn't, he wouldn't let her, and even as the world fell down on them again, he covered her mouth with his. And she wanted it, wanted him. She wanted to think of nothing but this moment and this man because everything else was so horrific.
He was safe, he was here, he was now, and now was all she had left.
Jake told himself he had no choice, he had to kiss her. She was losing it, totally, and that wasn't going to do her, or him, any good. They had hours left, hours or days, who knew, before they were rescued. If they were rescued.
If they didn't die any minute now.
He slid his hands down the length of her, pulling her near and she all but climbed up him, quivering and desperate, kissing him back. Then she tore free, and he could feel her gaze, inches over him, her breaths coming in shocky pants.
The shaking had stopped. The blackness around them was silent again, silent but for the pounding of her heart. Silent like a tomb.
She wasn't the only one losing it, he realized. He was, too, and what the hell difference did it make?
He leaned up, just a hair, and she leaned down and they were kissing again, hard and starving and oh, so sweet-hot. He wanted her.
Dear God, he wanted her. And given how she clung to him, kissing him back with just as much annihilating fever, she felt the same way.
She was soft, so damnably soft, and he let the taste of her take over. He pushed back the quilt, slid his hands down the denim covering her sweet little rear and up, inside her shirt, skimming the satiny heat of her skin. She let out a sound of need, ripped her mouth away from him.
"I'm scared," she whispered. "I don't want to die scared."
"I'm here. I'm right here. I promise." She needed him, and he needed a purpose. Or maybe he needed an excuse for needing her.
"Don't go away."
"I can't. And I don't want to." He moved one arm to reach up, touch her face, read her fear and need as if he were a blind man reading Braille. His other arm skimmed up along her spine.
She shifted, in the dark, moving beside him, where he'd pushed the quilt off of them. She lay there, breathing softly, as if waiting.... He'd moved with her, one arm wrapped over her middle where her shirt rose up, baring her flat belly.
He felt the strangest sensation. Awe. She was offering herself to him as if there were no tomorrow—because she didn't know if there was one. And if there wasn't...
Then she wanted to die in his arms.
"Jake?"
Her whispery voice wobbled just a bit.
"I'm here. I'm not going anywhere. I promised."
Keely wanted to tell him again not to make any promises, but then what did it matter now? She pulled at her shirt. She made it disappear then she was there again, beside him on the quilt.
"Keely—"
"Don't say anything," she begged softly. "Just kiss me till I can't think."
He did, he kissed her, tenderly this time, not hard, not fast, and it felt so natural and easy. His fingers swept up her torso as his tongue drove into her mouth and he swallowed the gasp of pleasure as his thumb and forefinger circled her nipple then squeezed it gently.
Her kiss turned hungry, fueling his, and his fingers slid down now, down to her pants. She ripped at his jacket, clawing at his shirt as he slid his hand inside her jeans and felt, knew, how much she wanted him, wanted this.
It was a bad idea, she thought suddenly. She was still thinking and she knew it was a bad idea. She didn't know him. She didn't do one-night stands.
She opened her mouth to say it was a bad idea, but she forgot why as she moved her hips against his hand, pressing that damp, pulsing part of her into his fingers. The thud of their heartbeats charged the air, interwoven with fear and hope, survival and desperation. Then her hands traveled their way back to his shoulders, pushing back his jacket, tearing at his shirt. He sat up, head down to keep from hitting the underside of the table.
Jacket—gone. Shirt—gone. Jeans...
Gone, too.
She heard him go through his pocket, the thud of what she guessed was a wallet hitting the floor. Protection. He'd remembered to think of protection while she could barely remember her name. Then he was back, slipping his hand down her bare belly...bare thighs.
Yes. This was what she wanted. And she knew it, couldn't pretend she didn't.
His fingers slid inside her and she gripped his shoulders hard, and a moan escaped her mouth just before he covered it with his. She wrapped her arms all the way around him, tugging him down over top of her. She felt the whole length of him against her, fitting perfectly, felt blood surging low in his body. She wanted him to bury himself inside her, right now. But first he kissed her eyelids, her nose, her chin. She wanted him to kiss her everywhere.
Trembling, she cupped the back of his head, pulled him down toward her breasts as she arched up to him now. He took one nipple, then another into his mouth and she sighed and writhed under him. His response was real, heat and hunger.
He tongued down her stomach, lower, to the V of her thighs where he placed his mouth there, at the swollen center of her, and sucked so hard, she cried out in a sound of pain and pleasure that begged him not to stop, begged him to make her forget everything but this intense thing happening right here between them.
Mindless sex.
One harmless fantasy before she died. And since things were pretty much headed in that direction, she was allowed.
She moved, lifting her legs, draping them over his shoulders, clearly begging for more, and he gave it, circling her then sucking again, harder, not stopping even as she clawed her fingers into his hair and made short, gasping sobs. Convulsions ripped through her as he replaced his tongue with his fingers and she felt so tight inside, so hot, she thought she'd die right there if he couldn't be there, right there, with her, all of him.
"Keely—"
"Don't leave me."
"I'm not leaving. I'm here."
"Closer."
He thrust inside her. Taut contractions gripped her as he sank in over and over, and she shattered again, and whatever was left of reality, of fear, was left far behind, lost to the world they made for themselves. The friction of his heat, her own pent-up and denied need, exploded and he shuddered into her as she wrapped her arms and legs tightly around him, going up into flames right along with him—again.
In what was left of her brain, a tiny piece of her mind broke loose from the sheer bliss of physical feeling to spin the thought through her that this was something different. Sex with Ray hadn't been like this.
But like so much else in her day, that thought made no sense.
She slept curled into his shoulder, and woke by surreal degrees to a terrible roar.
## Chapter 6
The roar was followed by dead silence.
She heard her name, called down through a crack in the debris above. A crack of light. That had been lifting equipment she'd heard. They were being rescued!
"I'm here! We're here!" she called back.
"Don't move," came the order back.
Jake's arm tightened around her. Jake. The stranger. The stranger she'd...
Had a mindless fantasy with before she died. Only she wasn't going to die. And she was so happy about that, but—
Complicated emotions ricocheted through her.
There were more sounds, debris being hand-removed as they closed in on the pocket of safety she and her oh-so-familiar-now stranger had found in the cellar under her demolished farmhouse. She felt a flash of fear as one piece of debris broke free and fell, crashing onto the pile of rubble she could now clearly see beyond the shelter of the table they lay beneath.
Then she realized she was naked. Jake was naked. Oh, God. She was going to be rescued naked.
Unlike her, he seemed totally unembarrassed by his nakedness, simply went about fixing it in the tightness of their quarters. Cool, composed, always.
He yanked on his jeans, handing over her shirt, separating out his and hers. Without a word. What was he thinking? What had she been thinking? It might have been natural and easy last night, but it didn't feel natural and easy now.
Had there ever been a worse morning after in the history of one-night-stand morning-afters?
In the awkwardness of the confined space, she managed to wiggle into her clothes. She utilized her nearly nonexistent acting skills to behave as if this was normal for her.
One-night stand. She hated that term. But what else could she call it? Comfort, need, shock, fantasy. Whatever the reason, she'd had a one-night stand with someone she did not intend to have a relationship with. A stranger, no less.
The rubble of the cellar surrounded them. For one moment, it was utterly still, utterly silent from above. Then something moved.
Something moved in the cellar.
At first, it was only the sound of movement that she knew. A soft thud, thud, thud, like something very small, something very light.
She could feel Jake behind her, half-sitting as she was.
Then she saw it, the box. The box from Ray. End over end, tumbling toward her. Moving on its own. But it couldn't move on its own. Another aftershock, another aftershock was coming—
A scream rose in her throat.
The box rolled over a pile of broken boards and stopped, a hand's reach away.
Stopped, just stopped. Her mind reeled, panic and confusion. No aftershock. Nothing. Just the box had moved, nothing but the box. And now it had stopped.
A steel ladder dropped, settled roughly in the debris. Boots came down after.
Jake moved, nudged her back.
"Go."
She got to her knees, dressed, thank God, and ready to crawl out from under the table, toward rescue. She grabbed the box. The rescue equipment must have dislodged it. Somehow. She looked back at Jake. It was morning, had to be. Time seemed elastic, stretching back and forth in her mind from the shock of the quake to the shock of what they'd done after, and hazy from sleep and the surreal comprehension that they'd survived, after all.
In a moment of high stress, they'd shared a bond. It was over now. A mindless little fantasy.
But dammit, she wasn't a one-night-stand kind of girl and was she crazy or had that been the most explosive sex ever? Maybe it was just the emotional thing, the fear and the drama of their circumstances that had heightened her awareness. Yes, that made sense. She needed something to make sense.
"Are we just going to...pretend this didn't happen?" she whispered. Oh, God, what did she even want him to say?
His unreadable gaze was tight on hers. "Is that what you want?"
She didn't know what she wanted. What did he want? "It was just a one-night stand. Right?" She needed him to clarify it for her. She needed him to say it then she'd be fine with it. "I mean, I've had them before. It's no big deal."
Oh, jeez, she'd just made herself sound like a slut. And it wasn't even true.
"Keely Schiffer?" The rescuer called her again.
"Go. Go find your family." Jake pushed her gently toward the light. "Everything's going to be okay."
She nodded, couldn't speak, wasn't sure what she'd say if she did. That she was so grateful that he'd been there for her in the frightening darkness? That she was embarrassed and could he possibly just forget it ever happened because that uninhibited woman who'd all but begged him to make love to her was as much of a stranger to her as he was? That she wanted to know if she'd ever see him again?
That last question was more frightening than the others combined. That grip she needed to get was troublingly elusive.
"Keely Schiffer?"
She turned to the rescue worker, his booted feet now planted firmly on the precarious debris.
"Anyone hurt?"
She shook her head.
"How many do you have down here?" the man asked, his flashlight beaming into the dark corners of the cellar even as with his other arm he took hold of her.
"Just one other person was in the house," she answered. "Jake Malloy." Already, Jake was crawling out from beneath the table, carefully rising to a stand on the debris beside her.
Another rescuer came down the ladder, grabbed for Keely's hand, and the first rescuer handed her over. The ladder shifted as she climbed, sending ripples of remembered shock through her, fear of another quake speeding her feet.
Everything happened quickly after that. The smashed remains of her house blurred across her vision as she rose from the huge pile that was left of her house. She tried to focus on the devastation, but her head reeled at the sight of it even as powerful arms reached out, helping her cross the debris to firm ground.
The world was all wrong.
There were the two maples that stood in front of her house and yet now they stood alone, towering over nothing but debris. She thought she should cry but no tears came. The early morning air was chilly and dew sparkled across the meadow behind the farmhouse.
Correction, she thought numbly, the meadow that used to be behind the farmhouse. Now the meadow was just...There. The woods beyond, hills rising above, remained. It all seemed so strange. She saw birds flit in the trees along the creek. The light over the hills glowed pink and gold. It was a pretty sunrise....
People swarmed toward her. Neighbors, relieved faces, arms reaching for her, embracing her, asking a thousand questions that rolled past her.
"We were so worried—"
"We drove by and saw your house and called—"
"My family..." she kept asking.
Nobody'd talked to her family—phones were out all over the place, but they told her the town was okay, shaken, no deaths reported, no buildings down. She couldn't believe it. She had to see for herself. The need to get there, right now, right this minute, burned through her.
A paramedic broke through, descended on her. She made light of her cuts and scrapes. There was an angry-looking scratch on her arm that she hadn't even realized was there.
"I'm fine," she said for the fourth time.
Her eyes caught Jake's briefly from a distance. A paramedic was checking him over, too. He looked different in the new day. Still dangerously sexy, maybe more so, even with grime from the contact with debris covering his clothes. Her heart gave a peculiar wrench and she struggled to keep her perspective. Get a grip, she reminded herself.
Then her mind swerved. The skull in the rose garden...
The thought of what she'd found, just before Jake Malloy and everything else that seemed so unreal about yesterday had hit, tumbled over her. The box still in her hand seemed oddly hot.
"Keely!"
Oh, God, her mom.
Keely wheeled in time to see her mother all but flying across the dew-laden grass, her father beyond her still getting out of their car where they'd pulled over on the side of the road between emergency vehicles.
In seconds, her mother's arms were around her. Her parents were okay. Her sister was okay, she found out quickly, as were her brothers and their families. Everyone was okay.
"But are you okay?" her mother repeated. "Are you really okay? To be trapped there, under the house, all that time and we didn't know—We couldn't call and the road was closed. There was a rock fall blocking the highway, no one could get through last night and after they cleared it they weren't letting anyone but emergency vehicles come through till just a little while ago."
Her parents had been in their own hell worrying about her just as she had about them. Her mother held her face in her hands now, looking up to her because she was smaller than Keely. Roxie Bennett's petite body, still slim despite her sixty-two years, had always been the one Keely had wished she'd inherited instead of her father's taller bone structure. Howard Bennett stood behind her mother now, watching her with the identical anxious look as in her mother's eyes.
"I'm okay, I promise." She hugged both of them again. "I wasn't alone, though. I had help." She turned, searched the faces of people still milling around the devastated property. "But I need to talk to the police. I need to—"
She couldn't see Jake Malloy anywhere.
An emergency triage had been set up at the community center in town. They'd taken a look at the cuts on his back, cleaned them out and applied some salve. Jake was in and out in under an hour. There was a serious amount of media attention, reporters roaming all over the parking lot and throughout the small center.
He hadn't said goodbye to Keely and that bothered him even when he knew it was for the best. They'd helped each other through a bad night. He'd done what he could to keep her safe, and even if he hadn't kept her safe from him, he could remedy things now, do the right thing, stay away from her.
She was with her family now. She didn't need him. He'd taken the trip into town with the paramedics just to save any awkwardness. She'd have probably offered him a ride with her family. No sense dragging things out that way, though he realized now he'd left so quickly he hadn't stopped to retrieve any of his luggage out of his smashed car. He'd have to find a way back later, see if he could pry a door open and get into the trunk.
The image of her in that bright yellow T-shirt when she'd opened the door of her farmhouse to him the evening before wormed its way into his mind. He knew what was under that T-shirt and those worn jeans now and her perfect body was forever branded on his mind. How she looked, how she smelled, how she felt beneath his hands...
She'd trusted him. Despite everything, she'd trusted him.
Don't go there, don't go there, don't go there.
Ruthlessly, he cut off all thoughts of Keely Schiffer. He didn't have time for a relationship. Or the desire for one. He was hanging out in this one-horse town for a month, if he could stand it that long, and that was it. He didn't want to leave a broken heart behind, and no matter what Keely said about not planning to get married again, she wasn't the type to fool around without risking her heart. She was soft and sweet under that fragile shell of hers.
He was hard and bitter and she didn't need him in her life.
The community center was maybe a half mile from the house he'd rented across from the Foodway. He skirted a reporter with a cameraman interviewing a woman clutching a small boy in her arms. She was wearing jeans and a ripped shirt with no shoes.
"He was outside playing in the creek when the quake hit," he heard the woman saying. "I went out looking for him and I couldn't find him. I couldn't find him all night." She started crying. "We found him wandering up the road this morning. He told me he was in a cloud of light, a red cloud of light."
"A number of residents called in reporting fire in the hollows. Do you think he was near a fire?"
"I don't know," the mother said. "I'm just glad my baby's safe. I didn't see any fire."
The journalist turned to face the camera.
"Despite numerous reports of fires and reddish lights or haze across the county throughout the evening hours, so far emergency personnel have yet to locate any fires. In this tiny rural county of fifteen thousand that was the epicenter of the four-point-three quake, the news is good with damage consisting chiefly of fallen chimneys, broken windows and rattled dishes. Ninety-eight people have reported for treatment at the temporary triage here at the Haven Community Center and over a hundred more have been treated at the local hospital. No deaths have been—"
The heavy door of the community center slammed shut behind him, cutting off the reporter's final words. He headed for the rental house, about a quarter of a mile up the road.
He hadn't paid much attention to the town of Haven the day before driving in. He'd been in a hurry. Why, he had no idea. He had nothing to do but twiddle his damn thumbs here. He was in the middle of nowhere and the city seemed far, far away. Haven was surrounded by thick woods full of oak and hickory and walnut, broken by the sloped pastures and quaint farmhouses of the Appalachian mountains. The town itself wasn't much more than a restored town square with a beautiful courthouse. Antique-style lampposts stood like sentinels along the cobbled sidewalks lined with businesses—a dress shop, a clock repair shop, lawyer offices, a craft and consignment retailer and a diner called Almost Heaven. A few side streets held a mix of Victorian-era homes. Another set of side streets held more modern brick businesses. A sign indicated a school up another road.
The reporter seemed correct in his statements about the damage in Haven. Main Street on the square led him back out to the quiet two-lane highway. After the demolishment of Keely's farmhouse he'd witnessed firsthand, he'd expected more devastation in town, but he saw little evidence of it.
What he saw as he reached the steps of the rental house was Keely, her slender, sexy, stop-traffic body unfolding from the passenger seat of a small sedan in front of the little grocery store across the narrow highway. He jammed his fingers into his front pocket, pulled out the house key he'd gotten from Keely what seemed like an eternity ago before all hell broke loose at the farmhouse.
He strode up the scuffed wooden steps of the house and onto the narrow front porch, refusing to look Keely's way. He had his own problems; she had hers. They'd had one night together, born of desperation and survival, and it was over.
And just because he could still taste her, feel her, smell her in his memory, didn't mean he'd get to ever do it again in reality.
Move on. Detach and focus.
It was three hours later when he managed to track down a car rental place within walking distance. He was able to rent a car there and he drove back to Keely's farm. The Jag was right where he'd left it—smashed under a tree. He'd saved and saved to buy the damn thing. Now it was toast.
The passenger side door was usable, so he pulled his bag of toiletries from the back then got the trunk open to get his suitcase and laptop. He stashed his things in the rental car.
The farm was quiet, deserted now after all the activity that morning. Even the equipment the rescuers had used had been moved off, no doubt needed elsewhere, as were the emergency workers. Despite what good shape the town appeared to be in, contrasting with the utter destruction of Keely's home, he'd been told by the car rental clerk that a number of roads were closed due to fallen rocks and trees.
There was a wooden sign swinging from a chain on a post near the road. Sugar Run Farm, Est. 1882.
Keely wasn't going to get over the loss of her house as quick as he'd gotten over his car, not considering the family history here. An old well house, looking like something out of a photograph, stood to one side of the ruined farmhouse, covered with ivy. The ground to one side was plowed, ready for a big vegetable garden. An empty chicken coop proclaimed this had been a real working farm in days gone by.
It was almost impossible for him to imagine the family history in this place. His father had taken off when he was nine, and after that, his mother had scraped by the best she could. He'd started working when he was fourteen. By the time he could really help her, she'd died of cancer. Extended family on either side was virtually nonexistent. Despite the rural nature of the state, he'd spent little time outside the city limits unless it was to go white-water rafting on one of West Virginia's wild rivers. He'd always headed straight back for the city afterward. He was comfortable there, with the traffic, the noise, the people who surrounded him but left him alone.
He'd just about always gone solo, so things weren't much different now. Keely was living in a whole different world. He didn't belong in it, but he felt an odd, edgy tightness as he thought of her with her family after they'd been rescued. The country was quiet, and his mind didn't know what to do with it. If he was supposed to find peace here, it wasn't working so far.
Restless, he walked toward the house, circling the pile of rubble. There was a barn beyond the house, and a meadow bottom leading down to a creek with woods beyond. Picturesque, despite the devastation, with spring wildflowers waving in the light air. Cows dotted a hillside in another distant direction. The breeze kicked up and carried the sound of a moo.
There was fresh-turned dirt in a small garden that lay just to the rear of the house, too, the soil still moist despite the bright sunshine. There was, in fact, a hole...a deep one.
Something trickled down his spine, an awareness of...Being watched.
He turned slowly, saw nothing but road and woods and hills beyond, heard nothing but the sigh of wind on the Appalachian air. His instincts told him otherwise, though. Somebody was here. He reached for his gun, held it down at his side. He'd replaced the bullets back at the rental house, on R&R but still operating on automatic.
Somebody was here and something was going on, he just didn't know what. He'd lay odds someone had been digging out behind Keely's house this afternoon.... Why?
He heard a noise and knew it came from the barn. He was maybe fifteen feet from it. No cell phone, no backup. No authority, frankly. An engine roared.
A thundering crash shocked him and he barely threw himself out of the way as a truck rammed straight through the barn doors, screaming toward the road. Jake hit the ground hard, rolled, scrambled to his feet in time to see nothing but the white tailgate of a late-model pickup disappearing around the sharp bend.
## Chapter 7
"Did you hear about Jud Peterson?"
"No. Could you grab the other end of this shelf?" Keely focused on the task at hand, only half listening to her friend and part-time assistant, who was usually in the office, but today she needed all available hands on restocking duty. The store was a wreck inside with things toppled off shelves in every aisle, but it was open. She had an obligation, being the only full-service grocer in town, but beyond that it was keeping her mind off what had happened to the farmhouse. And what had happened with Jake Malloy.
Lise Tanner was not deterred even as she helped Keely get a lower shelf back on its tracks. "I heard he called a paranormal detective."
Now that got Keely's attention. "What?"
"He said he got surrounded by some kind of weird red light out on Boscastle Road last night. Said it came down on the ground right over top of him and he ran and it followed him everywhere he went, like he was running in a ball of red light."
"He was drunk. He's always drunk." Jud got picked up about every other weekend for public intoxication. Everybody knew that. He was nice enough in between times and scraped out a living doing odd jobs around town. Keely hired him occasionally herself out of charity—he'd gone to school with Ray.
"Probably." Lise propped a box of detergent back up on a lower shelf. "He's an ass." Lise had dated Jud one summer in high school. Back then, he'd been a cute bad boy and Lise had sown her wild oats for a few months. "But you know he's not the only one saying weird stuff happened last night when the quake struck. Like, UFOs or something."
Keely felt a rush of something cold go down her spine. She'd seen a weird reddish light, too, and there'd been that strange way Ray's box had tumbled nearly into her hand right before they'd gotten rescued.
But unidentified flying objects or something paranormal...
That was just nonsense. Had to be. She'd been in shock, scared, and probably something electrical had snapped at the farmhouse when the quake struck, which would explain the flash of light, just like Jake had said.
And as for the box, the only thing weird about it was that Ray had bothered to plan that far in advance to give her a birthday gift. But then, didn't everyone do something out of character sometimes? Like, herself, for example. Last night.
Don't go there, she reminded herself even as her pulse jumped all by its rebellious self. A tingly feeling way down low reminded her that she'd had a good time, too.
Now she had to pay the price. Regret.
"Ray left me a present," she said abruptly.
The front door of the store dinged as another customer came in.
"What do you mean, a present?" Lise finished putting up the last box of detergent and stood, brushing off her jeans.
How she managed to look perfect no matter what she was doing was a mystery. Keely'd showered and changed into fresh clothes at her mom's before coming over to the store—against her mother's objections that she should rest after her "ordeal"—and yet she was already a mess from crawling around on the floor picking up grocery items.
Lise, on the other hand, looked like a fashion plate, per usual. She was three years older than Keely and her family had gone to church with hers, so Keely had often gotten her hand-me-downs. She'd never looked as good in them as Lise had.
Especially today when she was tired and felt hungover from stress and grief over the farmhouse. Everyone she loved was alive, though. That was what mattered.
Keely pushed up from her knees.
"A birthday present. Can you believe it? He bought me something, wrapped it up, left the box in the cookie jar. I found it yesterday."
Lise's perfectly manicured brows lifted. "Really? That's interesting. Hmm."
"Yeah. Hmm." Keely tried real hard not to say anything negative but gave up. "Maybe he thought I was going to find out he'd been sleeping around with Alexa Donner."
"Or Judy Applegate or Cherry Whitehead or—"
"Okay, okay." Keely held her hand up.
"I don't know why you get uptight whenever you start to say anything bad about him. He was a jerk, and personally I'm not sorry he's dead."
"Oh, my God, balance your chakra." Keely waved her hand in a zigzag motion. Mary O'Hurley used to be her best friend in kindergarten, but now she read palms and tarot cards out of her house and at the occasional school carnival. Even if Keely didn't believe in any of that stuff—truth was, Mary didn't, either—she'd picked up some of the New Age jargon just for fun.
"Ray needed something balanced and it wasn't his chakra," Lise said. "Maybe he even killed somebody! What are you going to say then?"
Keely'd tried to make a police report about the skull she'd found in the rose bed but every available state trooper was otherwise occupied on emergency detail. They told her they'd call her back later today and get out to the scene as soon as they had the manpower.
A skull in her garden wasn't top priority under the extraordinary circumstances.
"We don't know that," she pointed out. Not that she could think of any other explanation. She'd even thought about getting out the phone book and checking up on all the women, at least the ones she knew about, who Ray'd slept with.
But that was a little too embarrassing. Um, hello, this is Ray's widow, and I found a skull in my garden. Just wondering if you're okay....
Right.
Besides, she'd have heard if any of the women he'd slept with had gone missing. They were mostly local girls. The only one she knew about from Charleston was Cherry Whitehead, and surely she'd have heard about her if she was missing, too. Their local network news came out of the city.
Lise rolled her eyes. She was always quick to condemn Ray, which was one of the reasons Keely hated to say anything bad about him. Lise said enough.
Or maybe it just reminded her of how stupid she'd been to marry him. She had judgment trouble when it came to men. If anyone's energy centers were out of whack and needed balancing, it was hers.
"Whatever." Lise waved off the topic of Ray, to Keely's relief. "Now, about your birthday—"
"I don't want to do anything."
"Your mom said she was going to fix dinner at her house for everybody. Come on. That means I don't have to cook." Lise's husband, Tom, worked for the town and seemed pretty decent as far as men went. They'd been high-school sweethearts. Aside from that minor episode with Jud, Lise had better judgment in men than Keely. "Besides, you're staying there now anyway, aren't you?"
"Do you hate me? No." The thought of living with her parents made Keely want to choke. She loved them, but live with them? No, double no, triple no. They would drive her crazy, fussing over her. They were also seriously committed to getting her remarried and eventually they'd get back to their mission when the shock of the quake passed. "I'm going to use the apartment over the store. I haven't been able to rent it out, and now that's a good thing. But yes, okay, I'll go back over there for dinner." She wouldn't want to disappoint her mom about that, at least, though she knew it would mean a huge gathering of family and friends, and she wasn't really feeling up to it.
"So did you open the present?"
Back to that topic.
Keely shook her head. "Maybe later. I don't know. I still have it." She'd stuck it in her office desk drawer when she'd come over to the store earlier from her parents' house. Lise had brought clothes and a few necessities—Keely could get anything else she needed straight from the store shelves. She felt funny about opening the box. It was on the wrong side of strange, getting a gift from the grave. "I have a lot of other stuff on my mind."
"I'm sorry about the farmhouse." Lise touched her arm. "You were always so attached to it, and I know how much you wanted it. You can rebuild, you know. Insurance money. A new house will be so much easier to live in."
"I liked the old house."
"I know. I'm just glad you're okay, that's all." Lise pulled her into a quick hug then leaned back. "Everything's going to be okay, you know."
"I know," she lied.
"You're alive, that's all that matters. I'm glad you weren't alone. You haven't said much about this Jake Malloy guy who was with you."
And she wasn't going to. She'd given Lise the Reader's Digest version of her night trapped in the cellar, the same story she'd given her parents. Sometimes she told her friends everything and sometimes she didn't. This time it was the latter. She sensed that Lise didn't always tell her everything, either.
"There's nothing to tell. Look, it's nearly three o'clock. Weren't you supposed to get your mom at three? She wants to go help out at the community center, right? And you were supposed to give her a ride."
"Oh, yeah. Thanks. I wasn't watching the time." Lise narrowed her eyes. "But don't think I didn't catch that. I want to hear about this guy and you can't put me off forever."
"He's old and fat and has a wart," Keely said with a straight face. Lise could be as bad as her parents about trying to set her up. She was sure all Keely needed was a good man, like Tom. "He was nice, that's all. I was glad I wasn't alone all night. End of story."
"Hey, boss lady." Tammy Draper, the clerk, called her from the front of the store. "Somebody's looking for you."
Keely followed her friend up to the front of the store.
Tammy was ringing up another customer. There'd been a run on bottled water and Keely noticed it was again the most popular item at the cash register. A lot of the people out in the country had wells that ran on electricity and power was out in half the county. She counted herself lucky they still had it at the store and that the phones were working.
Bright afternoon light spilled in from outside and at first all she could see was a lean shadow against the glaring sun on the glass windows of the storefront. Then he turned and her eyes adjusted and the rest of her body went haywire, including her brain.
"I didn't mean for them to call you up here," Jake Malloy said. "I just wanted to know if you were in the store. I would have come to find you. I'd like to speak with you privately, if possible."
Privately? He wanted to speak to her privately?
Keely swallowed hard. Dammit. He was just as melt-your-knees hot as she'd remembered, and she'd rather not have even remembered much less had to face him. Unlike Lise, she looked like crap. Not that she cared what Jake Malloy thought of her.
Okay, she cared.
Her brain was definitely out of control. She had to fight a flashback of his hands all over her body. She'd lost her cool with him completely last night. Not again.
Deliberately, her gaze locked on him and held. She could handle this. He looked amazing in another plain T-shirt and jeans, but so what? Her insides quivered and she told herself she was probably just hungry. She hadn't stopped for lunch.
Then she remembered her friend, standing next to her. This was going to be trouble.
"Hi. I'm Lise Tanner, Keely's friend." Lise stuck out her manicured hand. "And you are..."
"Jake Malloy." He took her hand, shook it.
"Ah." Lise gave Keely a long look, then turned an amused gaze back on Jake. "I heard about you. I didn't recognize you without your wart."
Jake's brow furrowed. "My what?"
Lise grinned. "I'll let Keely explain." She gave an airy wave. "I've got to go grab my mom. You, babe, have got some 'splaining to do with me, too. See you at dinner." She turned, then swiveled back. "Hey. I know you're new in town and everything. And we're all so grateful about how you were there for Keely last night. It's her birthday and we're having a big dinner over at her folks' tonight. Family, and a bunch of her friends and—"
Oh, no. Keely knew where this was going and she didn't like it.
"Lise—"
"You're a friend now, that's for sure. Please join us. Dinner's at seven. Home cooking. Keely's mom's the best. I know she'd want you to come. We all would."
If she could have crawled into a hole, Keely would have so been there.
"Keely can get you directions." Lise sported a smug grin. "If you don't get your pickup back from Dickie by tonight, call me and I'll send Tom after you."
She took off into the bright sun, the bell over the door dinging behind her. The store suddenly felt really small, like airplane-bathroom small, just Keely and the man she'd had, what felt like fantasy sex with last night.
And people were watching. She could feel Tammy's eyes boring into her back. Curious.
"I have an office in the rear," Keely said. She couldn't imagine what Jake Malloy wanted to speak to her about, but best to get it over with quickly before her imagination went nuts with possibilities. And no way was she discussing Lise's wart remark.
As for the dinner invitation...
"I'm sorry about my friend. Don't feel like you have to be polite about that. I'm sure you'd be bored stiff at my parents'."
Her office was the size of a beanbag, but at least when she stuck her foot in her mouth, which she was bound to do at any moment, it wouldn't be reported on the gossip tree. Her desk was noticeably cluttered with snacks propped on top of every stack of paperwork. Half-eaten packages of donuts, chocolate pretzels, gummy bears, candy bars, you name it. It was a constant temptation, working all day in a grocery store full of goodies.
Keely sank into the chair behind the desk and motioned Jake into the metal folding chair against the wall by the door. He shut the door first.
She hadn't really wanted the door shut.
He filled up the tiny space, his expression dark and intense.
"That was nice of your friend," he said finally. "But you don't have to worry. I wouldn't want to intrude on your birthday dinner."
She felt like a heel now, a total heel.
"It's not that." Awkward, awkward, awkward. "I didn't mean that you would be intruding. It's just...I thought we decided it'd be better if we just went our separate ways." The conversation had been bad enough this morning. She didn't want to have it again. "What was it you wanted to speak with me about?" she prompted. Hurry. Her foot was already halfway down her throat.
He watched her like he could see right into her head and knew she was full of crap.
"Are you doing all right?" he asked.
"I'm fine. And that's really kind of you to ask, but—" But she had lots of things to do in the store and if he didn't stop staring at her, she was going to hyperventilate. She resisted the urge to reach for a donut. Sugar was not the answer to her problem.
A lobotomy might be.
"I'm glad," he said.
"Are you okay?" Polite. She could be polite. That's all he was being. It wasn't like he could possibly really care how she was doing beyond the social nicety of it. He certainly hadn't stuck around for chitchat after they'd been rescued. "Everything all right with the house? It wasn't damaged in the quake, was it?" The thought abruptly occurred to her. If the house was messed up, the only other rental she had to offer was the apartment over the store and she didn't really want to give that up.
"The house is fine. I noticed a few things out of place, some things that must have fallen out of shelves. No big deal."
"Oh, good." Relief washed her. "So..."
She waited, let her gaze drift away nonchalantly though the truth was she couldn't handle holding his eyes directly, wondering how he could carry on this casual conversation with her as if they hadn't stripped their clothes off last night and had wildly explosive sex. Or was she the only one who'd felt that way about it?
That was not a good thought. She noticed how his lean muscled shoulders pushed the limits of his T-shirt. She remembered how powerful those shoulders had felt under her touch. That was not a good thought, either.
Her mouth watered and this time she really did pick up a donut.
"Want one?" she offered. "I skipped lunch. I'm hungry."
He shook his head. "I went out to your place this afternoon. I rented a car."
His smashed car. She'd almost forgotten about it.
She swallowed a bite of donut. "Great. I'm still sorry about your car."
"Not your fault."
The office phone rang. Keely gladly reached for it. "Well, if that's all, it's really busy here." To show how terribly busy she was, she said, "Foodway, Keely Schiffer here," quite officiously into the phone, put her barely eaten donut down, wiped her fingers on a paper towel she deftly ripped off a roll on the filing cabinet behind her, and grabbed a pen and pad of paper as if ready to take down very important information. She was busy. Multitasking. Official.
"Hey, girl, how you doing?" Mary's perky voice bubbled across the phone line.
"Fine. But shouldn't you already know that?" Keely couldn't resist teasing her friend. She'd only once let Mary read her palm and that time had scared her so badly, she'd never let her do it again. Mary'd told her there was a black cloud surrounding her aura and that she saw a man and a woman as shadowy figures who were going to hurt her. It'd been a few years ago and it was right after that when Keely had discovered Ray was cheating on her.
She'd reminded Mary she'd only seen a man and one woman when it should have been at least four or five women.
"Brat," Mary said. "I'm just glad you're okay. I was having the weirdest feeling about you and I had to be sure."
"No more aura stuff," Keely interrupted. She avoided Jake's gaze though she could feel him watching her. He hadn't left the office.
He was waiting—for what?
"It's probably just your mom freaking me out when she told me about your house and everything," Mary said. "She told me you were at the store and I know I'll see you tonight, but I'm worried about you. I just had to hear your voice. I keep feeling like something's wrong with you."
"Nothing's wrong."
"You know, I saw those red lights last night that everybody's talking about," Mary said. "Did you? And today, Patsy Renniker came over and asked me to do a tarot reading on her. You know she has me do one once a week and I swear even an earthquake isn't going to stop her, and I got so scared, I told her to go home. She's got cancer. I know she's got cancer. I mean, I don't know that but it's like I could see it when I started doing her cards. I told her to go to the doctor, right away."
"You can't know that."
"It freaked me out, Keely. You know I don't ever tell anybody anything bad when I do a reading. I never thought anything bad before. I just tell everybody they're going to live to be a hundred. Good stuff. But now—It's like it's for real now. I'm not making it up anymore. And I keep getting this feeling about you."
"I'm fine." Actually, Mary was starting to freak her out. "What about you? What about your place?"
"Nothing messed up but my new gazebo. Fell over, completely, probably because somebody hadn't finished nailing the sides right. Eighty-five percent finished and he lets it sit there for the past six months. Now it's ruined."
"Eighty-five percent syndrome," Keely said. "Men only build things to eighty-five percent completion."
"No kidding. I get eighty-five percent on the way to an orgasm and Danny's finished," Mary said.
Keely started to laugh then accidentally caught Jake's drop-dead hot eyes when she leaned her head back and she swallowed so hard she nearly choked. No eighty-five percent completion rate for Jake.
He'd made her way too happy way too many times last night. Effortlessly, it seemed.
And still he sat here in her office, all steady and dangerously sexy-looking, as if he was waiting for something. He wasn't finished, whatever it was he'd come to speak to her about. And as long as he was here, she was going to have sex on the brain and Mary wasn't helping.
"I have to go. I'll call you later, okay?" She put the phone down after Mary'd said goodbye. "Um, is there something else you want with me?"
"I think something's going on, and I think you need to know."
"Didn't we already discuss this?" She felt a trickle of sweat between her breasts and the airconditioning was working, so that wasn't it. "It was just a one-night stand. I'm sure you've had them before. Who hasn't?" Liar.
"I wasn't talking about last night. Or about us."
Heat flushed her entire body. "Oh." Could she possibly be more stupid and one-track-minded?
She had a serious case of falling in lust with him, that's all. He was drop-dead, steal-your-breath handsome. But so what. Ray had been a looker, too. She mustered her self-control. Again. She forced herself to look for flaws, and found a few. His nose was slightly crooked. There was a scar along his jaw, and another one near his temple. He lived hard. He was a cop.
And there was, suddenly, a deadly serious cop look in his eyes.
A nervous prickle moved up her spine. "So what were you talking about?"
"Is there some reason," he said quietly, "that someone would be digging around in the back of your farmhouse?"
## Chapter 8
Keely went from a very pretty blush to white in the time it took Jake to blink. The bad feeling that had come calling out at her farm settled into permanent residence in his gut.
"What's wrong?"
"Nothing." She got up out of her chair, paced in the very tiny space behind her desk as if suddenly ready to crawl out of her skin. She stopped, her strangely desperate eyes locked on his. "Did you see anyone?"
"They tried to run me down, or at least they would have if I hadn't gotten out of the way. They were in a pickup truck in your barn."
"What?"
"The barn doors were shut and they just came smashing through the doors like a bat out of hell."
"Oh, my God."
"Know anybody with a white pickup?"
"I have a white pickup!"
"Where is it?"
"It's still at Dickie's. The mechanic," she explained.
"Are you sure?"
"I can find out. But wait. What's this about the ground being dug up?"
"There was freshly turned dirt in back of the house, where some pots of rosebushes were sitting like you were going to plant them. It was still damp and there was a hole. This isn't something that happened when the rescue guys were out there." And whoever had been digging, they hadn't wanted him to find them still on the place.
But why?
She turned even whiter, if that was possible. "Any police crime-scene tape, anything like that?"
Crime-scene tape?
"No. Why?"
She shook her head. "I was just wondering."
Just wondering if there was crime-scene tape around her house? Nope, he wasn't buying that blow-off.
"That wasn't the police barreling out of your barn, Keely."
She frowned. "Yes, I know." She looked confused, like she was struggling to put it all together. "I just was wondering if that was something separate from what you saw dug up. But that doesn't make sense, does it?"
None of it made sense to him.
He didn't want to be worried about her, but he was, and it was at least a welcome replacement to the unallowable beat of desire that had pulsed inside him when she'd jumped to the wrong conclusion about what he wanted to discuss.
She was wearing a weathered West Virginia University blue-and-gold T-shirt and a clean pair of jeans, and she looked more like she was about twenty-one than...Maybe thirty? Today was her birthday, he remembered.
She sat back down and picked up the phone, jabbed numbers in. "Hi, this is—" Her head turned up at Jake. "Do you mind?"
"Yes."
Exasperation tightened her features when he didn't leave, which she obviously wanted him to do. He wasn't leaving till he knew what was going on, why she'd reacted the way she had to what he'd told her, and if she was in any kind of danger.
He shouldn't give a damn. He'd slept with her, but so what? It was a one-night stand, just like she'd so pointedly said. Clearly, he was insane, but he wasn't leaving till he was sure she was all right. She was scared, and like last night, he was going to be there for her till he was sure she was safe.
As soon as he was sure of that, he could go back across the road and forget about her completely.
"—Keely Schiffer," she was saying into the phone. She'd swiveled her chair around as if by simply ignoring him, he'd disappear. "I called earlier about making a report. I was wondering—No, I know you said they were too busy to get out here to take my statement yet. But I was wondering if troopers had gone out to the farm? Jake Malloy told me that there had been some digging out there today. And someone tried to run him down coming out of my barn. Smashed the barn doors coming out."
A statement? She'd called earlier about making a statement to the cops. Jake watched her, the bad feeling gnawing harder at him. He remembered the way she'd seemed almost...desperate for him to come in her house the day before, how she'd seemed to not want to be alone. It had been a passing impression at the time, but now he wondered if it was connected to this statement she was waiting to make to the police. Something had happened at the farmhouse before he'd arrived, before the quake.
Something that had scared her.
Again, bad sign that he wasn't walking out the door. He would need to make his own statement to the police, though.
But the longer he hung around, the more he could be sucked in to whatever was going on in Keely's life. What if he did just walk away and then something happened to her? He couldn't live with that. He'd just have to be careful, watch his back while he was watching hers, and not get sucked in.
Keely's shoulders remained tight, tense. "Evidence could be disturbed," she was saying. "What if—What if it goes missing? I mean—" She pressed her hand into her forehead. "Okay. I know. You'll get out there as soon as you can. I know that." Another beat. "Jake Malloy." She gave the address of the rental house. "I'm sure he'd be happy to make a statement, as well. Thank you." She put the phone down.
"The cops been out there?"
"Not yet. She said someone was supposed to be out there this afternoon. They'll want a statement from you, too."
"Want to tell me what's going on?"
"No."
"You look scared, Keely."
"I'm okay." She didn't look okay, or sound okay, but she was damn sure going to try and fake it. She stood, walked around the desk and reached for the doorknob. She was throwing him out of her office.
He rose, fast, and put his hand over hers, stilling hers before she could turn the knob. She lifted bright, still scared but faking-it-for-all-she-was-worth eyes to his. He couldn't shake the bad feeling about what was happening to her.
"Look, thank you for telling me," she said. "Thank you for your concern. But this is my business. It's police business."
"The police don't seem to be helping."
"They're busy, that's all. We had an earthquake, remember?"
Oh, he remembered. And he couldn't help catching the defensive thread in her voice, or the way it slightly shook.
"Yes, I remember."
The flash of awkward intimacy in her eyes was his immediate and torturous reward. The apples of her cheeks pinkened again. He remembered how sweet-soft her skin had been....
He veered the subject away from that thought. "I'm a cop, too, Keely."
"You're not a cop here. And I have a store that needs my attention." Her voice came out slightly breathy.
She was upset about whatever was going on back at her farm, but she wasn't unaffected by his nearness, too. The buzz of awareness that had started humming between them last night hadn't diminished any—but she wasn't that same naive-seeming, friendly woman. She'd opened up to him last night in ways he'd never expected, and now she was all closed up again in whatever shell it was that she used to armor herself against the world. Or maybe just men.
And maybe this thing between them had a mind of its own because he could tell she was fighting it just like he was. And not doing so good. Like him on that, too.
"I just don't like it that you're scared," he admitted quietly. "I want to help if I can." Something about her was softening him, even if he didn't like it. He brought his hand up over her arm, barely realizing he was doing it.
"I'd thank you for that, but I'm not scared so much as I'm mad." Her voice rose.
"What are you mad about, Keely?"
He had a feeling she was holding a lot inside. She was sweet and good and didn't want to speak ill of the dead, but she'd been hurt in her life and she had to have some anger inside over it. He knew all about holding anger inside. Maybe he didn't want her to end up like him, bitter and isolated.
"I want to be your friend," he said. Friendship. That was easy. He could do friendship. "Tell me why you're mad. You deserve to be mad, about lots of things. You lost your house. You lost your husband. Maybe you've lost other things I don't even know about."
She blinked and he could see something moist in her eyes. "Yeah. I'm mad about my house falling down around me. You know how old that house was?"
"How old?"
"A hundred years. A hundred years old! And now it's gone. I'm the fourth generation in my family to live in that house, and the last one now."
"There were a lot of things in that house. Pictures, antiques. Probably a lot of sentimental value in those things, memories."
A tear swelled in her gaze. She nodded. "My great-grandmother's butter churns. My grandmother's nativity collection. My great-grandfather's Civil War rifle. The high chair my grandfather made for my father, carved it himself. It's all gone. And it just seems so senseless." Her shoulders sagged slightly. "A lot of things in life are senseless."
"I had a friend die recently," Jake said. "He was young, smart, had his whole life ahead of him." He sighed. "It wasn't just senseless, it was unnecessary."
"How'd he die?"
"We were undercover, cracking a drug ring. Things went bad. Brian got shot. The damn house was on fire, meth everywhere. There was a kid in the house and there was no way I could take out both of them at once. I saved the kid. Before I could go back in, the house blew." His chest banded. Guilt threatened to swallow him. They'd called him a hero for saving that little girl. He hadn't felt like much of a hero.
"I'm sorry." Her voice lowered, softened. "I'm so sorry."
He could have drowned in her eyes. Sympathy. He didn't want it, but the understanding in her gaze, understanding of pain and loss, felt good in some unexpected way at the same time that it made him uncomfortable.
"You lost your husband," he pointed out, changing the topic.
She said nothing.
"Talk to me, Keely. Tell me what happened to him."
"He made a bad decision, drove off in a downpour, flood conditions. His car got washed away at a low-water bridge. He drowned."
Now he knew why she'd been so adamant yesterday about the dangerous driving conditions in the storm.
"I'm mad about that," she said suddenly, sharply. "He knew better. He had bad judgment, about a lot of things."
"Did he hurt you, Keely?" He cared, more than he liked.
"He didn't hit me or anything, if that's what you mean."
"A man doesn't have to hit a woman to hurt her."
She was silent for a long breath. "He didn't work at all for years, went to school, but never finished. Then he hurt his back on the farm and decided he was going to be a writer. Supposedly. He never sold anything, or even tried to get anything published that I know about. He put us in debt when he bought the store. Then he didn't work it like he said he would. He was always gone. He'd come back with inventory for the store, antiques, small things for the jewelry case by the register, but I never knew where he got them. I was always scared they were stolen because I couldn't see where he had the money to buy anything. He said he got stuff at estate auctions, but I didn't believe him. And he cheated on me." Her chin lifted. "More than once. And I didn't do anything about it. I didn't leave him. I'm mad about that, too. I'm mad at myself."
"He's gone now. You get to start over." It blew his mind to think someone would cheat on this beautiful woman. No wonder she didn't like promises. She'd had plenty of them broken...important ones.
"Maybe. I'm mad that there's something buried in my garden back at the farm."
She bit her lip and he knew she hadn't meant to tell him that much. His mind stumbled over what she'd said, locked on.
No wonder she'd been nervous the day before.
"What do you think was buried there?"
He kept his hand on her arm. She stared back at him for a long beat then looked aside.
"I found a skull, a human skull, yesterday when I was getting ready to plant some roses." Her voice was low, slightly trembly now. "Ray's the one who dug that garden up last fall, tore some old bushes out. Maybe he put that skull there. Maybe my husband was a murderer. And maybe it's gone now because the state troopers didn't get out there soon enough. I'm mad about that, too." She gave a little laugh that didn't sound like she thought anything was funny. "I shouldn't get started on all I'm mad about."
"Maybe you should get it out more often." He waited until she looked back at him. "Stop keeping the anger inside." He was one to talk, he realized. "You don't have anything to be ashamed of, you know. You didn't do anything wrong. It was your husband who did."
If her husband wasn't already on his way to hell, he'd have been happy to give him a send-off. If it wasn't bad enough what Ray had done when he was alive, maybe he'd also left her a terrible secret to deal with after his death. A surge of protectiveness toward her rose inside him.
"I was a coward. I didn't want to admit I'd made a mistake, so I stayed with him."
Ah, hell. He wanted to pull her into his arms. He wanted to comfort her, make her feel better somehow.
"You're awfully hard on yourself."
"Somebody's got to be."
"Then maybe you should take a break, give somebody else a turn. And if nobody else steps up to the plate, maybe you don't deserve it." He'd seen her parents with her. Her friends. He didn't think anyone else was going to be hard on Keely the way she was on herself. People cared about her. She just didn't care about herself so much.
"You barely know me."
He realized he was still stroking his fingers up and down her arm and that she'd let go of the doorknob. She wasn't trying to escape him. He didn't mean to, but he pulled her closer and she let him.
Her nose and lips and seductive apple scent were that close, a breath away. He could kiss her and she wouldn't stop him, he sensed.
I want to be your friend. That was what he'd told her.
"I think we level-jumped on our relationship a little last night," he said. He forcibly made himself let go of her arm, take a half step back in the cramped space of her office. Strangers to lovers in a matter of hours. It was clouding the issue at hand, which was making sure Keely was safe. "I get to be concerned about you now, and I am concerned. Someone was out there at your farm today. Someone was looking for something. Who knew about you finding it?"
She straightened her shoulders, seemed to be shoring up her defenses, too, as if she was just as aware that things had almost gotten out of hand again.
"The police. My family and friends." She shrugged. "That's all. It's not something I really want to spread around."
"Your family and friends might have told someone. Other people might have overheard. Did you tell your parents back at the farm this morning?" They'd been surrounded by rescue workers then. Anyone could have overheard.
"No, at least not the whole story. I didn't tell them till later, back at their house when I called the police. But they could have mentioned it to someone, I guess. I want to go out to the farm. I want to see if the skull is still there."
"That wouldn't be safe, Keely."
"I guess. But I want to see what I can salvage from the house, too."
"Not safe," he repeated firmly. "No guessing about it. Someone was out there. And the dirt was freshly turned. See what you can salvage after the police are finished—they won't want you touching anything till then anyway. Someone's interested in what you found. Someone besides the police. And you know who that would be."
Someone who knew about the murder wanted it covered up.
"You're right. I know that. I won't go out there."
She looked scared enough that he believed her, but that didn't mean she was safe.
"But then..." She frowned. Her eyes lit slightly. "If there's somebody looking for it, then that would mean it wasn't Ray."
She didn't want to believe her husband had been a murderer. He could understand that. She was ashamed of Ray's actions, even though she wasn't responsible for them.
"Or it means Ray didn't act alone," he pointed out. That seemed more likely.
She swallowed hard and her mouth set grimly. "Yeah, that's a possibility, too."
"You should be careful," he said.
"I don't know anything. Whatever happened, whether it was Ray or someone else who put that skull there, I don't know anything about it. I just know Ray dug that garden up last fall. The ground out there wasn't touched till yesterday when I got ready to plant the roses. I don't know why he wanted to pull the bushes out last fall. He knew I wasn't going to plant anything new till spring, but he insisted and said he'd plant the roses for me, too. It wasn't really like him to do something in advance like that, before it had to be done, or to work in the garden at all. He had a bad back."
He'd bought her a birthday present in advance, too. Jake wondered if she'd opened that little box. But a gift from her dead husband seemed private. He hesitated to ask about it, but what if it was somehow connected to the murder?
"Maybe you know something and don't realize it, or maybe someone might just think you know something. You should be careful. Have you opened that present he left for you?"
She froze. "No."
"What if it has something to do with the murder?"
She gasped. "I don't think so. He wouldn't leave me a present that had anything to do with a murder."
"He left you a skull."
"We don't know that!"
Not yet. But Jake wasn't taking anything for granted.
"I'll open it later," she said quietly. "I'll give it to the police if it looks suspicious."
He nodded. "Thank you." He didn't like the anxious light in her eyes. "Where are you staying?"
"There's an apartment over the store. I'll be living there for now. The building is secure," she added. "The apartment can only be reached through the store, and the store has locks and I can lock the apartment, too."
Someone didn't have to get inside to get to her. They could catch her outside at night, or worse, set the building on fire. Anything was possible.
"Good. Don't forget." He reached for the door now. He was in too deep already and if he didn't get out of here, he might get in deeper. He might touch her again, and he might not be able to resist kissing her if he did.
"Umm...Jake?"
He looked back. She looked incredible, her spun-gold hair messy, bundled up on top of her head and falling down in rebellious tendrils that caressed her cheeks. She looked tired, too. Stressed out.
"Thank you."
"For what?"
"For being my friend." She chewed her lower lip and conflicted shadow passed over her gaze. She sighed, lowered her lashes to avoid his eyes while she continued on, blurting the next words. "I lied. I've never had a one-night stand before."
His hand slipped out, tipped her chin, while he valiantly resisted the urge to pull her to him and kiss her.
"I already knew that," he said.
He saw the way she fought for control. Her eyes brimmed, emotion threatening to spill out, but she blinked it back. "Okay. I just wanted to say thank you for being my friend because, well, because it's not like I share that kind of thing with just anyone. I'd really like it if we could be friends."
Right. Friends. No problem. It had been his own idea.
"Of course," he lied, to himself, to her. But he'd make it true. "I promise. We're friends."
She stared at him. "I don't like promises."
"I don't break mine."
A long, achy silence weighted the room.
"I almost believe you." She stepped back now. He dropped his hand. "If you want to, it'd be nice if you came to dinner tonight. My parents wanted to meet you, and Lise is right, you're new in town. Haven is a friendly place. You haven't had a very good welcome to it. And you are one of my friends now."
He should have said no. What good could come of getting more involved in Keely's life? If he hadn't come to Haven, if he hadn't rented a house from her, if he hadn't had to go out to her farm to get the key, if his car hadn't been crushed by that tree, if he hadn't happened to be there in her house when the quake had struck—
If he'd believed in things like fate, it'd be easy to think he was in Keely's life for a reason, and that that reason was bigger than whatever fears he had that were making him want to say no. He didn't believe in fate. But his insides twisted with a sense of urgency, and all reasoning vanished when she was near.
"It won't be a huge crowd or anything," she added. "Just my family and a few friends. But no present, okay? I know it's my birthday, but I don't want you to get me a present."
His well-famed self-discipline that had gotten him out alive from numerous dangerous situations was failing him badly. And he didn't mean the danger swirling around Keely. Keely herself was a danger—to his peace of mind.
She was making him feel strange and vulnerable in some unfamiliar way. His pulse was racing for no good reason.
"I promise, no present," he said. "I'd like to come to dinner with you. Thank you."
## Chapter 9
Primping. She was actually primping. And she was running late after a state trooper had stopped in at the store, finally, to take her statement. A forensics team had been called out of the city to take a look down at the farm and she'd be notified when the scene was cleared for her to attempt to salvage anything out of the pile that was left of her home. For now, it was a crime scene.
Her home, which, by the way, was demolished, was a crime scene. She could definitely use some distraction from that, and maybe it explained the primping.
Or maybe she was just kidding herself, big-time. There was something peculiar happening inside her whenever she looked into Jake's eyes. She got the feeling she was seeing a soul that matched her own—a little wounded, a little protected, a little scared, pretending to the world and themselves that they didn't want or need anyone else. That thought drew her up short. She was in serious danger of getting stupid about him.
Keely put her brush down and stared at herself in the mirror. Her lipstick looked perfect now, but it'd be all gone with the first bite of food. Somehow Lise could eat an entire meal without so much as smudging her lips, but Keely hadn't been born with that gene.
She couldn't remember the last time she'd worn anything but jeans and a T-shirt, even to the small community church she'd attended all her life. Haven was on perpetual casual Friday dress code. It was just that kind of town. Country folk.
But she was wearing Lise's clothes tonight, a bundle Lise had brought over for her this morning since all her own clothes were buried under a pile of rubble at the farmhouse. Lise tended to be a bit dressier than most people in Haven. She had an image to maintain, the town manager's wife with political aims beyond the local scene.
Keely didn't worry too much about her image, but it was her birthday, right? She didn't turn thirty every day.
You get to start over.
Maybe Jake had been right. Maybe letting her anger out today had been good for her. Venting out loud to someone.
But what did she really know about him? She'd made bad judgments about men before. She'd told him a lot more about herself than he'd told her about him.
Her instincts, if she could trust them, told her he was a good person. Anyway, he was just a friend, so what did it matter? She had told her mother so quite firmly when she'd called earlier to let her know he was coming with her.
So why did this tiny voice inside her keep reminding her that the way she felt about him had nothing to do with friendship? She worked to block it out.
She wasn't really dressing up that much, just a soft, black halter top instead of a T-shirt, and she'd still put on jeans—jeans with embroidery across the rear. Lise was a few pounds lighter than her so they fit snug and she felt sexy. It was good to be thirty and feel sexy.
It was bad to be ridiculously excited about seeing Jake again.
She left the tiny bathroom and walked back into the studio-style apartment on the second floor of the store. The rest of the upstairs was devoted to a sort of antique-store-type room, huge and open, where local crafters sold goods on consignment along with whatever antiques Ray had supposedly collected from estate auctions. The inventory still made her uncomfortable, but she didn't know what to do about it.
There was no proof the items were hot. She'd stickered them at bargain basement prices after Ray's death and she donated the profit beyond what covered her living expenses to the church. She'd be glad when it was all gone.
The apartment was dusty and she hadn't had a chance to clean it with her time focused on straightening up the store, but she'd brought fresh sheets to put on the pullout sofa bed tonight. There was a mini-fridge in the pint-size kitchen that was really just a wall of countertop with a hot plate against one side of the apartment. The bachelor who'd owned the store before Ray bought it had lived in the apartment for years, so it was comfortable if spartan, all the necessities in place. A long-dead philodendron sat in the window that overlooked the road.
She missed the farmhouse.
Keely sat on the worn but serviceable sofa and stared out the window, across the road. There were lights on inside the rental house. It was a tiny, quaint, white clapboard home, perched near the road with a cliff dropping down behind it. She saw the front door of the house open, and Jake emerge.
She rose, looking back at the small marble-topped coffee table in front of the sofa. The box from Ray sat there.
What if it was connected to a murder? She couldn't believe that, or maybe didn't want to any more than she wanted to believe Ray had been involved in a murder at all. She didn't want to believe any of the events surrounding that skull she'd found.
Almost not wanting to touch it, she took the box from the table. The air in the room seemed to rush around her. Her vision swirled and her pulse rocked.
Her heart hammered painfully. She heard a thunk and realized she'd dropped the box. Her vision cleared and she felt as if some kind of energy sucked back from her, shaking her on her feet.
She blinked several times. She should have eaten lunch. She was hungry and tired and—She bent down, reached for the box. A shiver ran through her.
Picking up her purse, she quickly tucked the box inside. She didn't plan on opening it in front of her parents, but she didn't like leaving it just sitting here, either.
What was she going to do, though? Carry it around—for how long?
She was scared to open it, she confessed to herself. It was silly. If it proved to be some sort of evidence connected to the skull, she'd give it to the police. If it was just a guilt gift from a philandering husband to his wife, then she'd deal with that, too, and all the emotions it brought up. She knew she had to open it alone. She didn't want to listen to her family and friends putting down Ray while she was receiving a gift from him.
She'd end up defending him out of some sense of obligation, dammit, and she didn't want to defend him. Letting out that anger today had been good for her.
Downstairs, the store was still busy. Customers moved around the aisles, but up front the line of people at the checkout stood still, heads all craned to the TV over the counter, which was tuned to cable news. Tammy, ample and fortyish, had her fingers frozen over the register. Video footage of Haven's main street stopped Keely short.
"The four-point-three shock wasn't all that took rural residents of the small West Virginia county by surprise last night," the female announcer stated. "Panicked homeowners reported bursts of horizontal light and a reddish haze in the air. Volunteer fire trucks responded to a variety of locations, but found no flames to douse. One resident called a paranormal detective after a four-year-old boy was found, scratched and confused, along a roadside this morning. The boy claimed to have been trapped inside a red ball of light.
"What's behind all these strange reports? Is it panic? Shara Shannon from PAI, the Paranormal Activity Institute, is on the ground in Haven, the epicenter of the quake, with town mayor Johnny Southern."
Oh. My. God.
Keely couldn't believe her ears. She stood, rooted, staring at the TV along with the other customers. She knew Johnny Southern wanted to ramp up Haven's profile—turn the sleepy farmtown into some kind of artsy-craftsy tourist mecca, taking advantage of its close proximity to the city. He'd been pumping his ambitious plan for years.
She wasn't against anything that would boost her own bottom line at the grocery store, but along with most residents, she didn't want to see Haven lose its small town charm.
Making the town sound like a loony bin wasn't charming. She hoped his appearance didn't mean he was willing to go that far.
"What's going on here in Haven, Mayor Southern?" the cable anchor asked. "Panic or paranormal activity? Some people have even suggested the possibility of UFOs in the area."
"An event like we had last night here in Haven takes everyone by surprise," the mayor said, his gravelly voice steady with a slight edge of nervousness. "Haven has never been known to have an earthquake. Emergency officials inside and outside the town are to be commended for their quick response and efficient clearing of roads and bridges. Volunteers have been out in droves today providing water and other necessities, particularly to our older residents. Most buildings received minimal damage and we plan on having the town back to normal as soon as possible."
"But things don't sound normal in Haven, Mayor Southern," the anchor pointed out. "What do you think is behind these bursts of horizontal light?"
Johnny Southern's expression remained noncommittal. He doesn't know which way the wind is going to blow on this, Keely thought. He's playing politics—happy to gain attention for himself and the town, reluctant to take a real stand till he knows how local residents are going to react.
"We did have a lightning storm last night, hitting right before the quake—" the mayor started.
"I believe I can answer that question," Shara Shannon, the PAI spokesperson, cut in. Her hair was a deep auburn, sleek, brushing the shoulders of her stylishly cut suit. She didn't look like a crazy person, but—
The bell over the door dinged.
"Let me start by saying that UFOs aren't behind what witnesses saw in the Haven area. Conditions last night, the low pressure and dense moisture, combined with an earthquake of that particular magnitude, form the 'perfect storm,' if you will, of atmospheric conditions to release positive ions into the air," Shara Shannon explained. "Positive ions trigger supernatural wavelengths, and those bursts of reddish light reported by residents are in line with what PAI has long tracked in other parts of the world as foundational movement."
"Foundational movement?"
"Yes. Foundational movement for oncoming paranormal activity."
"What type of paranormal activity can residents of Haven and the surrounding areas expect?"
Keely couldn't tell if the anchor was taking this interview seriously or not.
"Anything can happen in Haven now," Shara Shannon replied. Her eerily bright green eyes gazed directly into the camera. "And probably will."
Keely had the urge to roll her eyes but a sudden pounding headache stopped her. Or maybe it was the chill fingering down her spine. Shara Shannon was a nut. Nobody in their right mind would take her seriously.
"On that note, we're out of time, but please come back again, Ms. Shannon," the cable anchor said. "Our thanks as well to Haven mayor Johnny Southern."
A commercial flashed on, replacing Shara Shannon's perfectly groomed, seriously spooky face on the screen.
"I'm locking my doors tonight," one of the customers, an older lady with her hair up in a tight bun, said.
"People have been saying all day that they saw those lights," Tammy said.
"I didn't see nothing," a man in overalls put in.
A little girl, ice-cream cone dripping down her hand from the short-order counter at the back, leaned around her mother.
"What are positive tryons?"
"Ions," her mother said. "Positive ions. I don't know." She looked uneasy. "But I don't like it. I wonder if I should let you go to school tomorrow."
"Tammy." Keely nodded pointedly at the cash register. People were standing in line and the line wasn't moving. And she was afraid if they stood here much longer dwelling on that news interview, there was going to be a panic.
Earthquakes were rarely felt in West Virginia. The quake had turned everyone's world upside down by surprise and people were vulnerable. Even she had gotten a little shaky thinking about that flash of light right before the house had caved in on her. And the way that box had tumbled into her hand. But it was all explainable in the realm of reality.
"Okay, okay." Tammy got back to work.
Keely turned and all but bumped right into Jake Malloy's hard chest. She hadn't even realized he'd been in the store.
The chill inside her turned hot. Was that her heart thundering in her ears?
"Hi," she managed.
He was wearing the leather jacket again, this time over a white button-down shirt. He'd dressed for dinner, too, and looked more devastatingly handsome than ever. Even the most hardened of women would have to notice him. Small comfort.
"Ready to go?"
"My truck's not back. It is still at Dickie's, by the way." That had been a relief. It wasn't her truck that had busted out of the barn earlier. "Can we go in your car?"
"Sure."
She glanced back at Tammy, who unfortunately was not the most hardened of women and most definitely was noticing him along with the entire checkout line. Trouble with a small town was the gossip. She'd just become the center of some of that, she figured.
"I'm going to my folks' for dinner. Lock up, okay?" The store closed at nine and the way her family liked to talk, she didn't think she'd be back in time.
Outside, Jake turned to her.
"Did you hear from anyone about what you found out at the farm?"
She nodded. "A trooper came out and took my statement. You?"
"Yes. I was just wondering if you heard anything further."
"A forensics team will be handling the site. They're supposed to notify me when it's cleared so I can see if there's anything I can salvage from the rubble. I don't know anything else, like if the skull was still there, or more bones. I guess they'll let me know."
"Okay. Good."
She stopped on the edge of the road to take a look both ways, then tried not to notice the way Jake's palm lightly brushed her back as if watching out for her as they crossed the two-lane highway. It was an automatic gesture, chivalrous but meaningless. She liked it, though. Liked the feeling of his arm around her.
"I guess you saw all that about the paranormal stuff." She was making conversation, hoping the ball of nerves in her throat would go away. What was she, sixteen years old and afraid of being alone with a boy?
Not that Jake was anybody's idea of a boy. He was a man, tough as nails, but quietly kind despite first impressions. Her pulse raced with stupid nerves. This was no date and yet she felt like it was.
"Do you know what positive ions are?"
He shrugged. "Some kind of electrically charged particles."
"What did you think about all that?"
They'd reached his car parked in front of the rental house across the road. He had his hand on the door, set to open it for her.
"I think they need something to talk about when they carry news twenty-four hours a day." He shrugged. "Of course, anything's possible."
He'd surprised her. "You believe what that woman was saying from the Paranormal Activity Institute?"
"I didn't say that." He opened the door for her. Keely slid into the passenger seat of the economy-size lease car. "I just stopped thinking I knew it all a while back. Whenever you start thinking you know everything, that's when life hits you on the ass."
He shut the door and walked around the car to the driver's side and got in.
"Which way?" he asked when he'd keyed the engine.
She gave him directions as he pulled onto the highway. She watched his rock-hard profile, wondering about his comment.
"You get hit on the ass by life much?" she couldn't resist asking. He'd told her about his partner. That had to hurt, and maybe that was why he'd been sent away on R&R.
"Doesn't everybody?" His voice was even, giving nothing away.
"Not really. My friend Lise. She's married to her high-school sweetheart. Tom. Her life's pretty perfect, I'd say. She doesn't get hit on the ass much."
"It's good that you aren't jealous."
"I am not!" Dammit. "Well, okay, maybe a little bit."
Jake's mouth quirked and it was hard to be defensive. He wasn't judging her.
She turned her gaze from his profile to the narrow country road she'd directed him to turn onto off the highway. "I like my life. Mostly. Hers just seems a little simpler."
"Nobody's life is simple to them."
"I know." She looked back at Jake. "You like your life, Jake?"
"When I get one, I'll let you know," he said.
She felt a flush of embarrassment at how much she'd opened up to him. Obviously, it wasn't a two-way street.
"I thought we were friends." Jeez, she sounded pissy now and she hated that.
He shot her a look. "We are friends, Keely."
"Then tell me who you are. I don't mean your name, but who you are. Who is Jake Malloy?" She realized they were almost at their destination. "Turn there." She pointed to a fork in the road. "Then the first drive is my parents' house."
Jake made the turn then pulled down the drive and stopped behind a black SUV.
"Who is Jake Malloy? What do you want to know, Keely?" He gazed at her and she knew he was trying to decide how much to tell her. "I grew up in Charleston. My dad died in a crash ten years ago out in Pocahontas County, right after I graduated from WVU. My mom had cancer and passed a while back, too. They were divorced. She was married and divorced five times by the time I was out of high school."
"I'm sorry."
"That stuff, like most things you apologize for, isn't your fault."
"I know that. I'm just sorry you had to go through it."
"It's all right. I'm tough."
He gave her a cocky smile, as if he didn't have a care in the world. He was covering up, though. She could see the pain underneath. She knew all about covering up.
"What else do you want to know? I watch the History Channel and I like math. I count things."
"Count things?" Now he'd thrown her.
"To get stuff off my mind. I count things. Steps, passing cars, people, anything. I count at night. I used to count backward, but now I count forward because you run out of numbers the other way."
She remembered their conversation about trouble sleeping.
"Why are you in Haven on R & R?"
Silence stretched taut. There was no more cocky smile on his face. "They say I'm not dealing with my partner's death very well. I think I'm doing fine."
He didn't seem that fine to her. "You have trouble sleeping."
"So do lots of people."
That was true. She did, for example. But she didn't think it was quite the same. Or maybe it was.... "Do you feel guilty about his death?" She remembered that he'd said he'd saved the kid. He'd been too late to save his partner. "Do you think you did something wrong?"
"Nothing ever goes down perfect," he said. "I should have known that. The backup didn't get there in time. Brian's cover got blown. I wasn't there in time. I couldn't save him." His voice was hoarse suddenly.
"Maybe if you'd been there in time, you would have died, too," she pointed out gently. "Then you wouldn't have been able to save the child. You saved a child. That's a good thing. Brian would have wanted you to save the child first."
"He told me to get the girl. He told me to get her first."
"Then forgive yourself. I'm sure Brian does."
He was silent, his chiseled face in stark profile now. She could almost feel his pain, stretching across the short space between them in the shadows of the car. Seeing the way he blamed himself made her take another look at her own thought patterns.
"I felt responsible for Ray's death," she said quietly. "Still do sometimes. We had a fight right before he drove off that day. I felt like it was my fault that he did something that stupid, but it wasn't my fault. He did a lot of stupid things and none of them were my fault."
Jake looked back at her. "You're one of the nicest people I've ever met. Nothing you could have ever done justified how he treated you."
"I'm not all that nice."
He reached out, touched her face, skimmed his fingers along her jaw. "Yeah, you are. You're damn nice."
She saw all the pain and need she was feeling reflected back at her in his eyes. She understood him, too.
Quiet stretched between them.
"Why did you come here with me tonight?" she asked.
"Nothing else to do in this one-horse town, is there?" He added more seriously, "I'm worried about you. I'm worried that you're in danger."
He was worried about her. That was unbearably sweet. And dammit, she was wanting him again.
Her so-called friend.
Her stomach clenched. What was wrong with her? Why couldn't she keep this to a friendship? She didn't want a man in her life, especially one who clearly had no lasting interest in the community she adored. And she had plenty of problems, including a possible murder.
And yet...
Yeah, she was in danger, but not the kind Jake meant.
## Chapter 10
The front door of the house opened.
"I think we've been made," he said. "Ready?"
She looked oddly reluctant to go inside, and he wasn't all that eager himself.
"I think I should tell you something," Keely said. She put her hand on his arm, stopped him before he could get out of the car. "The people who love me are hell-bent on setting me up. That's what the wart thing was about."
"Wart thing?" He vaguely recalled her friend making mention of something along those lines.
"Lise was bugging me. Wanted to know all about you. I told her you were old and fat and had a wart."
He had to laugh. "Thanks."
"Sorry." She blushed.
"It's too late for me to grow one," he said, working to keep things light. "Or get fat and old."
"That's okay." She laughed now, too, and he could see the tension break in her eyes. "I'm just letting you know so that you'll be prepared. They won't be content with your being my friend. We'll just have to be firm about it, and you might have to be a little patient with their pushiness. They're good people."
He hoped she was right. A large, rugged-looking man was heading down the steps from the front of the house. Jake and Keely met him halfway and Jake reached out to shake the hand offered to him. The older man's grip was firm, his gaze level. He wore crisp slacks and a golf shirt, casual yet clean-cut.
"Dad, this is Jake Malloy."
"Howard Bennett," he introduced himself. "I want to thank you for being there with our Keely last night when the quake hit. She told us you were a big help to her. Good to meet you."
"It's good to meet you, too, but I didn't really do anything," Jake responded.
"Just being there was important to her," Howard argued. "And important to us."
"We're so glad to have you here. Roxie, Keely's mother." The woman who came up behind Howard Bennett was petite, much slighter than Keely's tall build. "I'm just thrilled that you're having dinner with us."
Both of her parents were regarding him with obvious interest. They both seemed young and physically fit for their age. Roxie Bennett was attractive, her slender face carefully made up, looked younger than her years.
"Thank you, Mrs. Bennett."
"No, now you just call me Roxie."
Howard slapped an affable arm around Jake's back and guided him into the house. It smelled like fried chicken and fresh baked bread inside and his mouth watered just a little. It'd been a long time since he'd had home cooking. A long time since he'd been part of any kind of family gathering.
He looked back for Keely. Her mother seemed to be fussing over her, but he couldn't hear what they were saying as they entered the house to the sound of a baby's wail.
A woman walked into the airy foyer of the home, a chubby baby dressed in a sailor suit draped over one shoulder.
"Hi there," she said over the child's crying. She jiggled the boy on her shoulder and he quieted to a sobbing hiccup. "Paul, come meet Keely's friend," she called over her shoulder. "I'm Sherry, Keely's brother's wife."
She positively beamed at Jake.
A man appeared behind her, another sailor-suited baby in his arms. "I'm Paul." He shook Jake's hand while cradling the baby in his other arm. He was tall and bore a striking resemblance to his father.
More people than he could imagine fitting into the house started filling the foyer, all with curious looks at him while calling birthday greetings to Keely. She'd been just about swamped in hugs. He learned that Keely had two brothers, as well as a sister. There were several cousins and friends, with children and spouses.
"Danny, hey, thanks for coming," Keely was saying as she stepped around him, brushing against him in the press of family and friends. She gave the man who was as big as a linebacker a warm hug, then said, "Where's Mary?"
"Out back with the kids," Danny said.
"Mary's a good friend of mine," Keely explained. "This is her husband, Danny. He teaches at the high school and coaches football."
"Now let's not crowd in here when we've got a whole house," Roxie Bennett said, shooing everyone back into the living room. The furnishings were comfortable, informal cottage-style with lots of florals and checks. Jake went with the flow as they spilled out onto a spacious back deck where he saw at least one person he recognized—Lise Tanner.
"You came! I'm so glad," Lise welcomed him. "Meet my husband, Tom."
Premature sparks of silver at his temples, wire-rimmed eyeglasses and a slender frame gave Tom Tanner an academic air. He reached his hand out to grip Jake's, then turned to Keely. "Happy birthday. Some birthday, eh?"
Keely's friend Mary, he discovered, was the woman in the flowing purple-patterned summer dress with beads draped around her neck and a couple of kids hanging onto her ankles.
He thought about his own family and old friends, split up about a hundred different ways. He had some stepsiblings from his mother's later marriages. He wasn't close to any of them and hadn't been in touch with old school friends in years. Haven was the kind of place where people grew up and stayed together. It was all foreign to him.
"Sit down, sit down. Now. Tell us about yourself," Roxie invited as she sat down on a padded redwood chair across from Jake, who found himself shooed to a seat next to Keely. There was a stack of beautifully wrapped birthday gifts on the umbrella-topped outdoor dining table. This crowd wasn't going to fit around it but he could see that there were plenty of seats, some nice redwood and other extra plastic chairs that had probably been brought out for the occasion. Flowers overflowed from pots and wind chimes hung from poles on the ends of the deck.
Keely had one of the bouncing baby boys on her lap. One of her cousin's daughters was running laps around a bird feeder in the smoothly cut sloping lawn that stretched to a line of trees. The sun had dipped below the trees and shadows crossed the lawn. A chorus of cicadas beat the crisp, clear air.
"What brings you to Haven?" Roxie prompted when Jake didn't reply right away.
Keely's gaze slid from the giggling baby to Jake. She looked gorgeous with a baby, he thought unexpectedly. She also looked extremely curious, waiting for his response.
"Country living," he came up with. "Peace and quiet." It was true enough, though not his idea.
That brought a laugh from Keely's father. "And we greet you with an earthquake and a media frenzy. What kind of work do you do?"
"I'm not working right now."
"I could help you find some work if you like," Tom put in. "I work for the town."
Well, Jake had managed to give the impression that he was out of work and slightly better than a bum. He figured that would help Keely out with the matchmaking problem.
"Tom helps everybody," Lise put in. "You should see him at Christmas. We hardly even see him because he's out buying food and presents for the families of everybody who comes through the local shelter."
"Girls, would you help me in the kitchen?" Roxie asked, and Keely's sister, sisters-in-law and friends exited, one of them with her arm swung happily around Keely's shoulders.
The people who love me...Yep, she had a lot of people who loved her, that was clear. He wondered if they were all really as nice as they seemed. Or maybe that was just his suspicious, bitter-cop side coming out.
He found himself hoping they were all as nice as they seemed. The scene before him was like some fantasy family gathering. He didn't fit in, didn't belong.
Keely's mother was no doubt dragging Keely off to point out how unsuitable he was as husband material, bringing along the rest of the women for added support. And he was unsuitable, just not for the reasons Keely's family knew about. Her friend Mary was the last to go, grabbing up the lap-running three-year-old from the lawn on her way. Another little girl, maybe six, ran behind her. Sherry had left the twin babies with her husband, who looked somewhat pained as he attempted to juggle the two of them.
"You all live around here?" Jake asked Tom and Danny, who were both seated near him.
Danny took another beer out of a big cooler on the deck and offered one to Jake. "We've got a place out on Black Hollow Road," Danny said. "It'd take a forklift for me to get Mary out of Haven. When we have kids, she wants to raise them here, where she grew up."
"We're out on Lick Fork," Tom said. "Just built a new house last year."
The town manager was doing pretty good, Jake guessed as he opened his beer. Keely's parents were doing all right, too, judging by the nice home he'd passed through on his way to the deck. It was well-furnished, comfortable, very modern and full of light. Nothing like Keely's farmhouse, he thought as the men's conversation turned to the disaster.
"It's too bad about what happened to the old family farmhouse," Jake said.
"Hopefully we can salvage something out of it," Howard said. "It's a huge loss, in terms of family treasures, but—"
"Maybe Keely can move on now," Tom said. "We tried to talk her out of living out there when Ray died. I never thought she was safe there by herself. And now, with what she says she found—"
"I went out there today to get something out of my car," Jake said casually. "My car got hit by a tree in the storm, right before the shock hit. Someone'd been tampering with the site before the police got there. The dirt was freshly turned out behind the house and someone rammed a white pickup truck out of the barn, blew the doors down, nearly ran me over."
He'd taken careful note of the vehicles in the driveway and parked in the yard when he'd driven up. None of them had been white pickups. Of course, there he went, being paranoid and suspicious again.
"Now that worries me," Howard said. "I don't want Keely out there at all."
"I agree," another of her brothers—David? Jake couldn't remember his name—added.
"I'm just wondering who knew about what she found," Jake said.
"Unfortunately, probably a lot of people," Tom said. "I heard about it this afternoon from my secretary, who'd had lunch with one of the dispatchers. Of course, she knew it was my sister-in-law in this case, but people talk. They shouldn't, but they do, and finding a skull in your rose bed is certainly good gossip around here. We don't get a lot of murders in Haven. But for all we know, might not be human bones. Could just be animal bones."
"Keely said she saw a human skull."
"You know she probably took off screaming the second her shovel hit it," Danny said jokingly. "You know how women are. Could've just been another animal digging around since she'd uncovered bones."
"I still don't like the sound of this whole thing," Howard said. "I'd rather Keely was staying here now."
"I don't think we need to overreact until we have more information," Tom said. "Let the police do their job."
Lise popped her head out of the kitchen. "Tom! Phone." She spoke over her shoulder to her husband as he headed past her to the phone inside. "Do not give him any money! It's Jud Peterson again. I'm tired of his freeloading and now he's tracking you over here, for Pete's sake. Used to, he'd at least do a half-ass drunken odd job for a few dollars but now he just wants handouts." Tom was already gone with the phone. She turned to Howard. "Tom is such an easy touch. We've got a new house and bills to pay and he's financing Jud Peterson's binges. Anyway, dinner's ready."
Done venting, she disappeared in a huff.
"Come on, Jake, don't be shy," Roxie called from the dining table. "There's plenty, so eat up," she went on as Jake lined up where directed and picked up a plate. The crowd filled the dining room and one by one they carried full plates outside. Everyone seemed to talk at once.
"We've got spice cake with whipped cream frosting, Keely's favorite, for dessert," Roxie said, "and then we'll open presents."
"Have you opened that present from Ray?" Lise asked Keely as she sat down next to Jake in one of the plastic chairs.
"No."
Tom came back out to the deck with his plate, having finished his phone conversation. Jake noticed Lise gave him the cold shoulder when he sat down beside her.
Roxie did a double take. "What present from Ray?"
"It's nothing," Keely said. "I found a little box, all wrapped up, with happy birthday written on it, from Ray. I'll open it when I'm ready. I don't want to talk about it."
"Okay, honey." Roxie Bennett's fork froze over her mashed potatoes for a beat before she went on, turning to Jake. "Keely says you two are just friends."
Keely didn't look any happier with this topic. "Mom—"
"Good plan," Mary quipped to Jake. "She's shy. You don't want to scare her off."
Keely seemed focused on her plate, determined to ignore her family.
"Why don't you join us at church on Sunday?" Roxie invited Jake. "It's the Haven Community Church on the main highway west of town."
"I bet Tom can get you a line on some work," Lise said. "With the damage in the county, there's going to be a lot of reconstruction. You look like you could handle a hammer."
What was he now, a charity project?
"People really don't need to wait a year after they're widowed to start dating," Roxie said. "I keep telling Keely that went out of style a long time ago. It's fine."
"Especially since Ray was such a cheating, lying bastard," Lise added.
Keely's jaw tightened. "We're not dating."
Mary's head swiveled to her friend. "Then why did you sleep with him?"
Keely went beet-red.
The whole deck went silent. A bird chirped in the woods.
Mary clapped her hand over her mouth. "Oh my God," she breathed into the charged air. "I really am psychic."
"That went well."
Jake backed the car up the dark driveway. Keely didn't respond for a moment. She felt shy and embarrassed, and the even-keeled way he'd dealt with everything from the earthquake to dinner made her both more attracted and more scared.
"I'm sorry. That was awkward for you. I don't know what made Mary say that. I never said anything to make her think that. Thank God everyone seemed to take my word for it that she was wrong. She's not psychic, you know. She does this palm-reading and tarot card thing, mostly at local carnivals, sometimes out of her house, private readings. But even she doesn't take it seriously. It's her home business, she calls it."
"It's okay. Don't worry about it."
"Thanks. No wonder my family likes you." How could they help liking him? "They don't even care that you made it sound as if you were out of work." Hot, that's what Mary had called him.
"I tried to help you out," Jake said, sliding her a grin she could barely make out in the lights from the dash. The night was dark and still. Electricity was back on tonight in most of the county and as they drove, lights twinkled from the occasional farmhouse.
They reached the highway and headed into Haven proper. Businesses were closed except for the small gas station. The community building was dark, whatever emergencies remained being shuttled up for care at the hospital. The peak of the crisis had passed. Haven could return to normal now. The Foodway was dark inside, the lighted sign illuminating the front.
"Thank you," Keely said as Jake pulled up in front of the store. She gathered up the bag of presents she'd brought back—some new and now desperately needed clothes, some of the homemade candles her sister made, some pottery she'd admired at the fair in Cedar Lakes last year when she'd gone with her mother.
"I could walk up with you, make sure everything's all right." Jake regarded her seriously. "I'm worried."
"You don't need to be. I'll be fine." If she couldn't even walk into her apartment without someone holding her hand, she was going to have a hard time getting through the rest of her life. "Maybe Tom is right and all I saw were animal bones."
The intensity in his look didn't flinch. "I'll wait out here until I know you're inside and everything's okay."
"Whatever you want to do." She frowned. "That didn't come out right. I appreciate your concern. I just don't want to be a bother."
"You're not a bother. I'm your friend, remember?"
She avoided his gaze by staring at his shoulders in the darkened car. He had very broad shoulders. His arms had felt so good wrapped around her last night when she'd been scared. She couldn't look at Jake Malloy without him arousing tender, hungry feelings that had no place in her life. He made her think of the type of man she'd wished she'd married. Someone not at all like Ray.
"Then I'm just bothering myself," she said and pushed the car door open. "Maybe I don't know how to be your friend."
Oh, damn. Now she'd gone and said something more stupid than ever.
"Hey—"
"Good night." She wrestled with her purse, thank God finding her keys immediately, and was in the door and had it shut behind her before Jake could think twice about coming after her. If he was thinking about coming after her. He was probably starting to think she belonged in an asylum the way she ran hot and cold on him.
If she wasn't confusing him, she was definitely confusing herself.
Night-lights lent enough illumination for her to find the stairs that led to the apartment and antique/consignment shop. She unlocked the apartment and reached for the switch beside the door.
She never knew what hit her.
## Chapter 11
Jake leaned against the car, the engine turned off, waiting to see the light flick on inside the upstairs apartment. Movement from inside the darkened store caught the corner of his eye. A shadowy blur raced through the store, knocking down a display near the register, heading for the back.
A blur that was way too big to be Keely.
His heart kicked into overdrive. He leaped the few steps to the front of the store. Locked! He rattled the door then banged on it, swore roundly and raced for the back, toward where he'd seen the shadow heading. He reached for his gun as he tore through the pitch-black around the side of the building.
Keely. Dammit, if anything had happened to her—The rush of emotion was so fierce, it nearly stopped him. He charged on, pushing away the mindblowing fear he felt for her.
He heard a crash from the back of the building as he reached it and a shadow dashed between two Dumpsters, heading for a high chain-link fence. Breath seared Jake's lungs as he leaped, grappling at—
Air.
A thud from the other side of the fence told him he'd been too late. The shadow was gone, in a split second, into the trees behind the store. And he could either chase after him, and most likely lose him in the twist of residential streets and woods that whoever the hell that had been probably knew way better than Jake did, or he could find Keely—
There was no second thought to that one.
He whipped around, raced for the open rear door of the store. He found himself in the kitchen, stumbling against what he realized was a stove. The short-order kitchen.
The store was dark, dimly lit by low night lights near the front counter.
"Keely?" he shouted, hoarse from running, as he tore up the stairs to where he knew the apartment would be. It was even darker here. No light at all. He found a doorway, open, and reached for what he hoped would be a light switch—
Bright light from an overhead fixture temporarily blinded him. Then he saw her.
Soft, tangled hair. She lay on the floor, still. Blood pounded in his veins. He didn't even feel his feet move the steps it took to reach her.
"Keely." He bent down, pushed the hair from her face. There was no blood that he could see, but she was so still—"Keely!"
Her lashes fluttered. His heart nearly exploded in his chest. She blinked, her eyes dark in her pale, shell-shocked face. She stared at him for an agonizing beat, as if trying to figure out what had just happened.
"Someone was in my apartment," she whispered raggedly. "They hit me. Oh, God." She pushed up on her elbows, slowly, painfully, edged backward, struck the edge of a coffee table.
"He's gone. I saw him run through the store. I went after him, saw him run out of the back of the store."
Dammit, he wished he hadn't lost him. Whoever had done this to Keely, he could kill him with his bare hands right now. He wouldn't need a gun.
"Are you all right?"
She didn't look all right. She looked terrified, and she'd been hit. He could see a bruise blooming on her temple.
"He slammed me to the floor. I think. How long was I out? I don't understand what happened." She sounded so confused, it hurt to hear her speak.
"Just a few minutes."
"How could he have gotten in? Everything was locked."
"The back door was open, through the kitchen. Either he had a key, or he broke in, or someone left it unlocked. I'm calling the police."
He saw a phone on an end table and grabbed for it, punched 911 as he went back to Keely. He wasn't leaving her side. He made the report quickly then put the phone down. Her quiet devastation broke what was left of his restraint. He placed his hand on her head, stroked the tangled hair there.
Thank God. Thank God she was alive.
"They'll be here soon," he said. "And I'm not leaving."
Sobs shook her suddenly, and there was no question in his mind. He wrapped his arms around her, just held her. She'd wanted him to go away a few minutes before, when she'd gotten out of his car, but she wasn't asking him to go away now.
She burrowed her face into his shirt and the urge to pull her even closer blindsided him. She was shaking and he realized he was, too.
"Thank you," she choked out through tears. "Thank you for watching out for me."
His chest wound tighter. "I told you it wasn't a bother. I'm not going to let anyone hurt you." He meant it. He wasn't going to let any arm come to her, not if he could help it.
She lifted her face to him, her mouth a mere whisper away, and he moved, just so, and touched his lips to hers. She made a sound in her throat, needy, aching, and he forgot all about self-discipline. He kissed her, needing to taste her, hot and sweet and alive. And she was responding. She wanted him just as he wanted her.
He shifted his weight, nestling her closer against him, kissing her deeper. She pushed herself snugly against him and he knew this was a mistake, a bad, bad mistake. She wasn't a one-night-stand woman, he knew that, and two nights wouldn't make it any more right. He'd leave Haven before long and she'd be left with a broken heart.
Pulling back, he stared down at her, desperate need fighting with his better sense.
"I don't want to hurt you," he said, low, fierce. "I don't want to take advantage of you. I'm sorry."
Keely held Jake's gaze, wishing—Oh, God, she was wishing for more. Wishing for last night all over again. And he was afraid he was taking advantage of her. He didn't really want her, not beyond a physical intersection between their two lives, what they'd had last night in the farmhouse. It was abundantly clear.
"You're not taking advantage of me." She tipped her chin, pain shooting through her temples at the movement. "It's no big deal. You don't have to explain." He'd just been comforting her, that was all.
"Just...don't go anywhere yet." She sounded pathetic and she hated that.
"I'm not going anywhere."
She blinked, swallowed hard over the lump of uncomfortable, confused emotion in her throat. What she really wanted was for him to hold her again and never let go. What happened to never wanting a man in her life again? Yeah, that had always been a lie. She was just the cowardly lion, like always. Afraid just because Ray had been a bastard she'd never do any better, so she'd stuck with him, and once he was gone, she'd figured she'd rather be alone than risk getting hurt again.
It wasn't what she really wanted. Not if she had any guts.
"Maybe I should see if anything was taken." Yeah, that would keep her occupied, she thought desperately. She started going over the apartment.
"If they were here to steal, why would they be up here when there is a store full of anything they could want right down the stairs?" Jake asked.
She looked back him. His gaze was unreadable, dark. If he was as moved as she by that kiss, he wasn't showing it.
"They wanted something from you, Keely."
She shivered, froze.
"What?"
"The present from Ray?" he suggested. "It's just a thought. Where is it?"
Her purse. It was there, by the door, where she must have dropped it.
"It's in my purse. Maybe they didn't even see it. I was carrying that bag of stuff with me." The presents were scattered and she saw the pottery her mother had given her—in pieces on the floor. "Aw, hell."
Emotion welled up again. She didn't care that much about the pottery even though she'd admired it at the fair. It was just—
Everything. She was losing everything lately.
Even things she'd never had. Like Jake.
"They were probably in a rush to get out of here when you came in," Jake theorized. "But what else could they have been after?"
"I don't know." She blinked harder, willing back the tears that kept threatening.
The phone rang. She hurried toward it, pressed a hand to her temples as pain seared her head again. She grabbed the phone.
"Hello."
"Hey. I just wanted to apologize again."
"It's okay, Mary."
"It's not okay. I can't believe I said that in front of your whole family! I don't know where it came from. It just burst out of me. I feel terrible. I just want you to know that."
"It's okay. Really. It's okay."
Her friend was silent for a beat. "Are you okay? I keep having this awful feeling. I don't like this. I don't like these feelings I'm getting about you. I really don't want to know anything! I swear to God, I liked being a fake psychic. You are sleeping with that guy, aren't you?"
"Mary..." She didn't want to have this conversation. Usually, she told Mary almost everything, but she was too tired right now. "Someone broke in. There was an intruder in my apartment when I got back to the store. I have to go. The police are on the way."
That distracted Mary.
"Oh, no! What happened?"
"I walked in and someone knocked me out. That's all I know. Jake was here. He'd just dropped me off. He tried to chase after him but he lost him."
"Is he still there? You shouldn't be alone."
"Yes, he's here." She turned, saw Jake still standing there, looking so dangerous and protective all at once. "I'll be fine."
"Are you sure? Are you all right? Maybe you have a concussion. Maybe—"
"I'm okay, I promise."
"You shouldn't stay there by yourself tonight. I can come get you."
"No, don't do that. I'll figure something out. If I need someone I'll call you back. Okay?" There was a sound from the front of the store, downstairs. "I think the police are here. I have to go."
"I'll let them in, if you'll get me the key to the front," Jake said.
She took her purse and got out the key. Her knees felt wobbly and she was grateful for the help. In a minute, he was back with a uniformed trooper, the same trooper who'd come out to the store earlier to take her statement about the skull in her garden.
Briefly, Jake explained about the shadow figure he'd chased, and the trooper asked Keely if anything was missing. She hadn't had time to look around much, but she did now while the officer watched. She hadn't brought much personal to the apartment so far. If anything was missing that had been here before, she had no way of knowing.
"I had a bag of gifts with me," she said, pointing to the mess on the floor. "And my purse. But he didn't take any of that."
"Looks like you took him by surprise and he just wanted to get away," the trooper surmised. "I'll need a list of everyone who has a key to the building. I'll take some prints here, then downstairs at the back. We'll see what we can find out behind the building where you saw someone." He looked at Jake then back to Keely. "He's long gone by now, I'm sure."
"What about my farm?" Keely asked. "Have you heard anything about that, about the skull I found?"
The officer nodded. "I was about to get to that. The forensics team didn't find anything, but I called in a few favors so they're going to keep working on it. They haven't cleared the site yet, so they're asking you not to go out and work on salvaging anything from the house yet. But so far, nothing."
"The skull was right there. I hit it with my shovel. It was uncovered." She felt, and sounded, panicked. She had seen a skull, she knew she had. "Couldn't they see that the scene had been disturbed?"
"They didn't see it before, ma'am, so they really can't tell," the trooper pointed out. "There was a lot of activity out there this morning, too. It might not have been disturbed by anyone on purpose. We just don't know right now."
Jake's gaze tightened. "I was out there this morning and saw the earth freshly turned, as if someone had been digging this afternoon. That was before the forensics people were out there and after the rescue team had left. And you know someone was out there. They tried to run me down."
"We just don't know," the trooper repeated. "It could have been something that happened this morning—during the rescue operation. There were a lot of people out there. And we don't know who was out there or why they were out there. Might have had nothing to do with what Ms. Schiffer saw. Someone could have been there trying to take something from the rubble. If we had a plate number—"
"It happened too fast," Jake said. "But the soil was damp. The sun was hot yesterday—the soil would have been dried if it had been something that had been disturbed in the morning. I think whoever broke in here could have been looking for the box Keely's husband left her."
"What box?"
"A birthday present," Keely explained. "I guess we were thinking maybe it was connected to the murder." A tingle of fear rippled up her spine.
The trooper shook his head. "We don't have a murder yet."
"Someone took the skull," Jake said.
The trooper stared at him. "As I said, they haven't cleared the site yet, but they haven't found anything." He looked back at Keely. "Even if they do, we'd be a long way from connecting your husband to it, though we'd certainly start there. I don't know what we could do with your box at this point."
"Investigate?" Jake suggested.
She could see the frustration in the rigid set of his jaw. She was frustrated, too. She felt helpless and she was tired of feeling like her life was out of control.
"Here." She dug the box out of her purse and handed it to the trooper. Nerves jangled up her spine and the box felt cold in her hand. She was glad to drop it into his. "Why don't you open it and see what it is."
The trooper took the box. He tore the wrapping and pulled off the lid.
"It's a necklace," he said. "Just a necklace."
He tilted the box so she could see. A silver-and-garnet-encrusted heart lay inside, a long chain tangled beneath it.
"I don't see what we can make of this," the officer said. "I don't have a murder. I don't have a case to file evidence on. It looks like a gift to me."
It looked like a gift to her, too. She felt silly suddenly. And yet...Someone had broken in here tonight for something. If it wasn't the box from Ray, then what was it?
"If you'll make me that list while I get prints," he said, "I'll get out of your way."
She looked around for some paper and a pen. Her head hurt and she resisted letting out her frustration on the trooper. He was doing his job as he saw it. She just didn't like the feeling that no one was taking her seriously about what she'd found in the garden. No one but Jake.
He was right about one thing, at least—if someone was out there to get that skull, then Ray hadn't acted alone in whatever was going on. And maybe that person had been right here in her apartment tonight.
Or maybe she'd imagined that skull had been human and the whole thing was nonsense. She didn't think so, but she was so confused, she didn't know what to think anymore.
The officer went around the apartment taking prints, then downstairs. When he came back, he took the list of employees with keys.
"I'll give you a call if anything turns up with the prints," he said. "And I'll let you know when the site is cleared out on your farm."
Jake followed the officer downstairs. She could hear the sound of the bolt sliding back, the key turning in the lock.
His tall, broad-shouldered frame filled the doorway of the apartment again. She met his gaze across the room.
"You're not staying here alone tonight," he said. "In fact, you shouldn't stay here at all. They might come back."
That wasn't a comforting thought. She was pretty much over the independent attitude with which she'd turned down Jake's offer to walk her upstairs earlier.
"I could go to my parents' or Lise's or Mary's...."
But then she might be bringing danger to their door, she thought with a sick, hollow dread in her stomach. Fear rippled through her. She wanted Jake to hold her again, but she was scared of that, too. He was a stranger. How could he be the one she wanted to run to for security? She felt like her life had become some kind of creepy roller-coaster ride in the dark and she didn't know when, or if the next frightening drop would occur.
She just knew that if it did, she wanted him to be the one seated next to her when she screamed.
His hard, comforting gaze seared right through her, as if he knew what she was thinking.
"I think you should stay with me," he said.
## Chapter 12
"It's gonna rain again." The air was heavy, damp. The night stirred with a soft breeze and Keely shivered as she turned after locking up the store. "I love rain." Her expression was painfully wistful.
Jake guessed she was thinking about her normal life, the one she didn't have anymore.
"Yeah," he agreed. "I think it's gonna rain. Come on." He left his rental car parked in front of the store and they walked across the road even as the first splashes of the oncoming storm hit them.
Maybe someone was watching them. He wasn't sure that they were that much safer across the road than they were at the store except that there was a good chance whoever had gotten in the store had gotten in with a key. At least at his rental house, they'd have to break in, make some racket.
He'd be ready if they did.
Inside the house, he flipped the switch by the door that turned on a lamp in the front room. Keely was quiet, her expression exhausted and troubled. He didn't know what to do to make her feel better. She sat on the brown-and-red-checked couch, seeming to try to make herself as small as possible. The house was simply decorated, comfortable, with cozy cream walls and serviceable furniture. An antique German clock ticked softly from across the room.
"Nobody believes me about the skull I saw," she said quietly.
"I believe you."
He saw the flare of fear in her eyes and he wished he could do something to take that fear away. He hated to scare her more, but she needed to see the reality of her situation.
"Someone put that skull in your garden. Last fall, we can guess, if that's when the old shrubs were torn out. And most likely Ray was involved. What was going on last fall, before he died?"
She turned her hollowed gaze to him. "He'd had a string of affairs. The women—I know their names, well, the ones I know about, I guess. There could be others for all I know." She looked away, at her hands fisted in her lap. "Otherwise, I don't know. He'd bought the store last year and we both worked there. Nobody's gone missing around here lately that I know of."
"We don't know when the murder took place."
Her gaze rose to him again. "But it couldn't have been any longer ago than last fall."
"Unless the body was being moved. But you only saw a skull, right?"
She nodded. It was possible that more of the skeleton could have been buried than she'd uncovered, or someone had only moved the skull. Maybe by tomorrow the forensics team would dig deeper and find further evidence out at the farm.
He watched as Keely opened her purse and pulled out the small box. She set it on the coffee table in front of her as if she didn't like touching it.
"I guess they weren't after this," she said. "It's just a necklace."
She lifted the lid again. The silver heart inlaid with garnets gleamed in the low lamplight from its nest inside the box. It was a necklace, just a necklace.
Keely didn't move to pick it up.
The air in the room remained taut somehow.
"It's just a necklace," she repeated finally, low. "It looks kind of old. There's a little tarnish on it. Maybe he got it on one of his antiquing forays."
"Maybe." There were other possibilities, including a link to murder. There was no proof of it, but he wasn't excluding the idea yet.
"I feel silly," Keely said. "Really, I think this whole situation has made me a little nuts. I think the whole town is going a little nuts. Positive ions. Flashing lights. Earlier, when I picked up the box, I felt..." She looked up at Jake. "I felt strange, like this weird rush of air." She gave a forced laugh. "The power of suggestion, you know? I mean, I think I was just hungry, feeling a little faint. Look at Mary. She thinks she's really psychic now."
None of that explained the skull she'd seen in her rose bed, though. That was before the quake. He didn't think Keely was the type of person to be swept up in a paranormal hysteria, either.
"None of this makes sense, that's all," she said. "And I don't like it." She shivered again, visibly. "I'm cold," she said. "I'm really, really cold. I just want to go to sleep."
"You can use my bed." There was only one bedroom in the little rental house. It wasn't more than eight hundred square feet total, just the bedroom, kitchen, front parlor and bathroom.
"No. I'll take the couch. You're paying me to rent this house. I'm not taking your bed."
He wanted to argue with her, but he could see her obstinate exhaustion. Her shining eyes stabbed him.
"I'll get you a blanket." He went to the bedroom, pulled the blanket from the bed, along with a pillow, and brought them out to her. "I'd rather you took the bed," he said. "But you can do what you want."
"I'd feel better about it this way." She took the blanket and wrapped it around herself. She was pale and he could still see her shaking.
The temperature in the house couldn't be below seventy-five. There was no air-conditioning in the house, and the spring night was cool, but not that cool.
He didn't want to leave her alone, and he had no intention of sleeping. She needed sleep, though. She looked almost sick. He wanted to do something, but he felt helpless.
"I'll be in the kitchen, if you need me," he told her. "I'll turn on the heat. I can fix you some coffee, if you like." He'd picked up a few things before he'd left the store earlier.
She shook her head. "I just need to get some sleep." She put her head down on the couch, stretching her legs.
Outside, he could hear rain, steady now. He felt the pull of her, the desire inside him to find a way to comfort her and make her feel safe.
The best thing he could do for her, and himself, was to let her sleep.
He adjusted the thermostat, turned off the lamp and headed for the kitchen. He sat down at the laptop he'd brought with him and opened it. Connecting to the Internet, he did a search for the local newspaper. Six months ago, Ray had dug up a rose bed and now there was a skull in it. Something happened six months ago—either the murder itself, or the location change of the body. That was the theory he was going on.
He hunched over the screen, tapping through articles. Crime was low in Haven. No reports of missing persons. No murders. A few domestic violence reports, a suspicious fire, a case of animal neglect, minor burglaries, drug possession charges...He searched other news stories. Controversy over an iron bridge that needed repair or replacement, local elections, youth sports reports, a cancer run, a baby beauty contest, 4-H meetings...He would have thought life in a small town would have put him to sleep, but an unfamiliar tug gnawed at him as he scanned the stories.
The simple life. It was oddly appealing.
The stray thought took him by surprise. So far, Haven had been quite a few letters short of heaven. He should hate Haven at this point and he sure as hell had no business even thinking about feelings of attachment to either Haven or Keely.
Detach. Focus.
The edgy feeling he'd had before came back. He didn't feel right in his own skin anymore.
Pushing aside the uncomfortable train of thought, he clicked on the next story. A two-hundred-acre abandoned farm the town had tried to buy. Tom Tanner had fought hard to convince the town to fork over the funds needed to purchase the land for use as a nature preserve. A developer had sneaked in under the wire and bought it out from under the town's nose with a higher bid. Flipping forward in time, he found that construction on a small subdivision of midpriced homes was underway.
His nape tingled. Reason to move a body?
It was a possibility, but there was certainly no evidence to suggest he was doing anything other than wildly speculating.
The site had a search function. He searched on missing persons, and came up with an Alzheimer's patient who'd wandered off, a few small children, a runaway—all found alive. A sixty-year-old local businessman had disappeared and been found dead, the killer apprehended and sent to prison. Nothing he could connect up to what Keely had uncovered back at her farm.
He pushed back from the table and started to walk toward the front room to check on Keely. Then he started running.
She was screaming.
Raging wind. Cold. Lights. Lights in the dark, roaring straight for her. She couldn't run fast enough. White light cut a swath right through her. Run! Terrified, she stumbled, hit hard—
"Keely!"
"No!"
"Keely! I'm not going to hurt you."
Gentle arms cradled her, holding her down. She'd hit him. She'd hit—Jake.
Her head reeled. She blinked, awareness coming at her in sickening increments. She still half felt as if she were somewhere else.
Someone else.
Where had that thought come from?
"I won't hurt you, I promise. Shh." He whispered to her, rocking her against his solid, warm body.
She was cold, so cold.
"I was—" She struggled to get the words out, her teeth chattering. "I had a bad dream?" She didn't mean it to be a question but she heard the confusion in her own voice. She struggled to focus on her surroundings, on reality. The dream had seemed so real....
"It's okay now. I'm here and I'm not going anywhere, I promise. You're okay. You fell off the couch."
She felt her heart pounding violently. Jake shifted and she gripped his shoulders, afraid he'd leave her. His gaze on her was deep, unwavering, shockingly patient, and she told herself to stop panicking. He wasn't going to leave her, he'd promised.
Promised. She hated promises, but she believed his. That was scarier than her dream.
He held her tighter. She was scared. He was comforting her. But that had nothing to do with the dangerous need causing her to practically climb up his body.
"Tell me about your dream," he said softly, still rocking her. "Tell me."
"I don't know." Already, the dream was disintegrating in her mind, disjointed images fading in and out. "It was like I was on a road. I think it was a road. I fell and it was hard. Lights were coming at me, like a car."
"You were afraid you were going to be killed."
"I guess so. I don't know. I guess all of this, it's getting to me. I'm sorry."
"There's nothing to be sorry about."
Yes, there was. There was plenty to be sorry about. She wanted Jake, not as a friend, and maybe he even felt the same, at least physically. He desired her. She'd seen that last night, and even now, she could swear that wasn't just her heart she heard pounding. It was his, too.
But he didn't want to take advantage of her. And he'd said that because what had happened between them wasn't anything more than sex to him.
Emotion choked her throat. She reached up, felt something hard against her chest....
Her fingers closed around it. She pulled it away from her, staring down at it in the slash of light that streamed in from the kitchen.
The hair rose on the back of her neck and it took everything inside her to keep from screaming.
"What is this doing on me? I didn't put it on. I didn't put it on!" She held the silver-and-garnet heart in her fingers.
Lights beaming. Run!
Surreal images flashed through her. She was dreaming, only she was wide awake. The chain felt hot in her cold hands. Burning hot. But at the same time, she was so cold.
She didn't feel herself pushing to her feet. She only knew she was there, standing, staggering backward, clawing at the chain. She heard Jake's voice, as if from a distance.
"Keely—"
"I have to take it off!" She struggled, ripping the necklace over her head.
The heart hit the ground, thudding softly.
Her veins nearly exploded. "Put it back in the box. I don't even want to see it."
Jake reached for it, plucked it up, stuck it in the box on the table. "You don't remember putting it on?"
"No, I don't remember putting it on! I don't want to put it on." She didn't want anything from Ray, especially not this necklace.
She really didn't want this necklace.
And she really didn't want to believe there was anything weird about it. She couldn't even put the words together to say what she was thinking. But the words were forming all on their own and she was terrified.
Anything can happen in Haven.... And probably will.
Anything like...A necklace that was possessed?
Jake probably already thought she was nuts, the way she was behaving.
"Are you sure you're all right?" His gaze turned on her, worried, intense.
"I'm scared," she whispered. "Jake, what's happening?"
## Chapter 13
Keely looked close to a nervous breakdown. And cold. Freezing cold. Her lips were nearly blue and she was shaking.
"I don't know." He hated the helpless feeling inside him. He took his leather jacket, wrapped it around her and pulled her into his arms, held her, and still she shook.
The necklace.
He watched it over Keely's shoulder. "Make it go away," she whispered, her teeth chattering through the words. She turned in his arms, stared back at the piece of jewelry, then back to him. "Put it back in the box."
Stepping around her, he closed the distance to the box, scooping the necklace back inside it in one quick move. He went to the kitchen, grabbed tape out of a drawer, came back and taped the hell out of the box. Keely stood there, looking like she was ready to freak out. He remembered how the box had tumbled toward her out of the rubble. The red lights they'd seen as the quake struck. The voice of Shara Shannon.
Conditions last night, low pressure and dense moisture, combined with an earthquake of that particular magnitude, form the perfect storm.
He'd told Keely he believed anything was possible, but he hadn't taken it that seriously. Now...
"Those red lights—" Keely breathed. She was thinking it, too. She was thinking the same thing he was. "Foundational movement." He saw her throat move. "Foundational movement for paranormal activity, that's what she said. No, that's insane."
Jake didn't know what to tell her. He wrapped his arms around her, held her tightly.
"It's possessed, Jake." She lifted her head, her dark, terror-filled eyes searching his, searching for sanity. "I didn't put it on! It came to me, somehow, in my sleep. Anything can happen in Haven now. Anything. That's what she said."
The mere idea that the necklace held some kind of life of its own was crazy. It shook the very core of everything Jake knew to be true about the natural world.
But this wasn't natural. If what she said was true, this was supernatural.
Keely lifted her face to him.
"Tell me you believe me," she whispered. "Tell me I'm not imagining this. Tell me I'm not losing my mind!"
"No." He couldn't believe what was happening, but he knew she needed him to believe her. Could she have put the necklace on in her sleep? All he knew was what she believed and that she needed him to believe her, too. She said she felt strange when she held the necklace, and she was clearly and quite oddly freezing cold. Could she be getting sick? He could find other explanations if he tried.... But he didn't believe them any more than Keely did and that shocked him. Something about that dream of hers seemed too strange when he put it together with the skull she'd found in her garden. "You're not crazy. Or at least if you are, you aren't going crazy alone."
Her world had been taken over by the unbelievable. From the moment Jake had arrived on her doorstep and the quake had struck, nothing had been the same. Ray had been involved in a murder. And he'd left her this necklace.
And maybe the two of them were connected.
"Sit down," Jake said. "Before you fall down. Come on. You're freezing. We've got to get you warmed up."
He'd become her rock in her world gone mad and she wanted to cling to him when he moved away, adjusted the thermostat again. She sat on the couch. He came back, pulled the blanket up around her. Even with the jacket and the blanket she was shivering.
"That dream," Jake said. "You woke and found the necklace on."
She nodded, swallowed hard. "Why me? Why is the necklace after me?" It lay in the box now, taped up, still, as if content that they weren't trying to get rid of it, or leave the house. Not till it got what it wanted.
What did it want?
"Ray gave the necklace to you," Jake said slowly. "It's yours now."
"I don't want it!" she cried.
Silence beat heavy. "I don't think you have a choice."
Events beyond her control were taking over her life. Again. She felt anger twine with the relentless fear drumming through her blood.
"What if the necklace is connected to the skull you found?" Jake suggested again. "What if the necklace had been on the body? It belonged to the victim. Somebody killed her—Assuming it was a woman."
The necklace suggested he was correct about the sex of the victim. If there was a victim. If there was even a murder.
"Why would Ray give it to me?"
She wondered if Jake was patronizing her, trying to calm her down by pretending to take her seriously. But the grim light in his eyes belied that thought. He trusted her, and almost as unbelievable as what was happening now, she trusted him.
"Guilt?" he theorized. "He was cheap? It was a convenient gift? Who knows. He got rid of it, wrapped it up like a birthday present. Maybe he didn't even plan on giving it to you, maybe he was just stashing it away and the best way to make sure you didn't poke around in the box if you happened on it was to make it look like a birthday present. You certainly weren't going to open it right away, you'd wait for your birthday and maybe by then he was going to do something else with it. Who knows. He didn't plan on dying. All we know is that Ray got the necklace, and let's assume he got it off the body. We know there was a body and we know he was digging around in your garden last fall. Tell me about your dream again."
Keely's heart thumped. Was she dreaming. No, not dreaming, seeing murder? "I was on a road. It was dark. Lights were coming at me. I knew I was going to be run down. I knew I was going to die."
She knew it like she was there. Like she was seeing it with her own eyes. And she was cold like she was...Dead.
Horror burned through her again. She was seeing a murder and she'd woken with that silver-and-garnet heart around her neck.
"Somebody killed her," she gasped. "Somebody ran over her. And they buried her in my garden?" Somebody. Ray. Her heart thumped harder. "The police. We should call the police." The uselessness of that idea hit her immediately. Call the police and tell them what? That her dead husband had left her a demonic necklace? Then they could put her in the crazy bin.
The grim light in Jake's eyes told her he was thinking the same thing. She stared hopelessly at him. "Then what are we going to do?"
"Maybe tomorrow they'll find something more out at your farm," he said. "Maybe those prints they took at your apartment will match up to someone."
"We could be completely offtrack," she said, her mind racing. "Maybe it was just animal bones, like they said. Maybe the break-in at my apartment was random. Maybe—"
"Maybe that necklace didn't jump onto your throat?" Jake held her eyes. "You didn't put it on."
That he believed her was just about the only thing keeping her sanity holding on by a bare thread.
"So what do we do?" she repeated.
He put his hand on hers. "We get through the night." He squeezed her hand, his warmth seeping into her. "We keep you warm."
She swallowed hard. She needed to pull herself together. He put his arms around her, pulling her closer, touching her shoulders, her hair, her face. He was trembling, she realized, as much as she. She wanted him to tell her he could make all this go away.
But he held her and said nothing. He was there for her, that was all he could do for now.
She gazed up at him desperately. "I'm scared of going to sleep again."
He shook his head, his grim eyes shining through the low-lit room. "I won't leave you alone."
Their faces were only inches apart.
This was madness, on some level she knew that. She shouldn't make love to him again. But was it any more mad tonight than it had been last night? She needed him tonight, just as she had before. Only now there was no fooling herself that her heart wouldn't be on the table and that she could get hurt. His gaze was hungry, like hers, and she knew he was thinking the same thing. She was, she realized, his rock in this strange new world, too. With him, she felt warm and alive, not cold and dead.
Then he kissed her and it didn't matter how she would feel later. It only mattered how she felt now.
His fingers ran through her hair, his hands streamed down her back, her body pressed against him. When at last his lips left hers, his eyes gazed into hers, haunted and crackling with a desire that she saw rocked him the way it rocked her. She felt that energy humming between them.
"Make love to me," she whispered, kissing him again. His response was clear from the intensity of his answering kiss. Her shivers fell away, replaced by a deep, hot quivering that shook her just as much.
He pulled back just slightly and the heat of his gaze seared straight through the haunting chill in her bones.
"You'll regret this," he said. "And maybe I will, too, but you will and I can't live with that, Keely."
He was telling her this was just another one-night stand. Okay, two-night stand. He wasn't looking for a permanent relationship, at least not in Haven, maybe not anywhere, anytime.
"I'm not asking for promises," she told him simply. "I'm just asking for tonight."
Rain beat down outside. Wind rattled the small frame of the house. His heartbeat thudded against her own and she knew this was destiny, somehow, some way.
This was wrong, so wrong, but for the life of him Jake couldn't recall why. He'd given up hope since the day he'd watched Brian die. It wasn't just his partner that had blown up in his face. His life had blown up, too. He'd become a machine, a living, breathing machine that couldn't feel, wouldn't let himself feel. Maybe he'd been headed that way for a long time beforehand, too.
And as crazy as the situation was in which he'd found Keely, she was breathing life into him again. Hope and passion and hunger. And he wanted it like he couldn't remember wanting anything before.
She was hurting and scared, but it wasn't just him comforting her—she comforted him, too. He stood there, barely able to breathe, his heart banging so hard against the wall of his chest. He scooped her into his arms, barged through the open door that connected to his bedroom, leaving everything behind—his fears, hers, that damned necklace.
It hit him that the necklace was what had drawn them together, tonight, and maybe last night, too. Was it an accident they'd been trapped in that cellar or part of some supernatural plan he couldn't yet see? Then the thought was lost as he tumbled onto the bed with her and when she reached up to him, there were no more questions. Her hands moved down his back, pulling him closer and she moaned low in her throat as he reached between them to touch his hand to her breast through her shirt.
She pulled back just enough to tear the jacket and shirt off her, baring her taut nipples to his touch. There was no light in the room except for the flashes of lightning coming through the thinly shrouded windows. Greedily, he claimed one tight peak, then another, and she writhed beneath him, those sexy moans driving him wild. Then she was tearing at his clothes, fumbling at his zipper, pushing at his shirt. He helped her rip them off, then she sat up, tugged at her own jeans, leaving nothing but scant panties he could barely see. He dipped his fingers along the sides and pulled them down and off and then she was there, on her back, her skin so smooth and naked and waiting for him.
He possessed her with his hands, his mouth, and she responded with her body arching into him. The smell, the touch, the taste of her wiped everything else away. He claimed her mouth with all the fevered need raging in his soul. Reaching between them, he felt the hot, wet need of her.
Hot, not cold now, as if his heat transferred to her, not just emotionally but physically, warming her to the very core. He felt her release in the way she cried out against his mouth and her body surged against his hand, then she reached for him, desperate, shaking, guiding him inside her.
Take it slow, he told himself. God, he wanted this to be slow. He wanted this to last forever.
The shock of that thought jammed hard into his chest, then she rocked into him, sweeping him away in the gut-wrenching sweetness of her cries. He swallowed them with his mouth, plunging deeper inside her, as deep as he could go. They rocked together and he came hard and fast and they lay panting, clinging together as if for survival "I can't get enough of you, Keely," he breathed raspily against her ear. She pulled over top of him, straddling him, clinging to him still, clinging to his heat, and he couldn't believe how quickly he was ready for her again. She sheathed him inside her, ready, too, and he thrust upward, softer, gentler, longer, the rhythmic pattern taking them higher, every breath a whimper, a low moan of perfect pleasure, until they climbed the stars together.
She collapsed, damp and sweet and warm, in the crook of his shoulder. He wrapped his arms around her and held her close. Nothing was going to happen to her tonight.
Light laced through the thin curtains barely veiling the morning fingers of dawn. She was...In Jake's house. In Jake's bed.
In Jake's arms.
Turning her head, she saw him beside her, sleeping soundly. He looked wonderful naked, all powerful shoulders and bare, sculpted chest tapering to a lean waist, and lower.... Remembered desire drummed through her. He stole her senses and probably he'd steal her peace of mind for a good, long time to come. And yet she felt some odd peace anyway. He'd shown her that she could live again. She didn't have to hold herself alone forever just because Ray hadn't been the man she'd needed.
And if she needed Jake and he didn't choose to be that man...She would survive. She felt the first pang of heartbreak. She'd thought one night was enough, but now she knew two was not enough, either.
Their clothes were scattered on the floor. His handgun lay on the nightstand beside him, ready. He was ready to protect her, ready for anything.
She shut her eyes suddenly against the flash of renewed fear, the puzzlelike pieces of the night before ramming into her then scattering in a kaleidoscope of colors, unsolved. The skull, the strange white truck bursting out of her barn, the break-in at her apartment, the necklace...
None of it made sense. Ray had left her this nightmare. But at least Jake was by her side to solve it.
They had to solve it. The possessed necklace was possessing her. They could hardly take a nightmare and a crazy tale of a supernatural piece of jewelry to the police. The necklace belonged to her now, had been given to her. The necklace wanted her to find the truth. Justice, maybe?
The chill from last night crept into her again as she sat up in the bed. The heaviness around her neck seeped into her consciousness. Something hard and cold, a cold that she felt even now seeping down deep inside her, hung between her breasts. She reached up her hand, grasped the silver-and-garnet heart and gasped. It was hanging around her neck. Hanging. The chain had repaired itself. And the necklace had come to her again.
Jake opened his eyes as she swung her gaze to his, and held it for a horror-stretched beat.
"It's back," she whispered.
## Chapter 14
The stark, haunted light of her eyes tore Jake's heart out. She held herself very still, as if afraid to move. But she was moving, involuntarily. She was starting to shiver. Last night, he'd warmed her up. But this morning, the necklace was back and it was freezing her again.
"I'm going to take it off you," he said slowly, steadily, knowing she was so scared right now, any sudden move could frighten her more. He wanted to protect her, and he didn't know how because he didn't know what he was protecting her from.
He reached for the chain and she turned her head, pulling at the thick gold-spun hair to move it out of his way. The nape of her neck was ice to his touch. He unclasped the necklace, reaching around to let it fall into his palm.
Her skin was smooth and pale, naked. The bedsheet twisted at her waist, her breasts falling heavy and beautiful against her chest.
"You're cold," he said. "Take the blanket." He shrugged the sheet from him along with the blanket, tucking it all around her. She scrunched eagerly into it, still shivering.
He stared down at the necklace in his hand. He hadn't examined it closely the night before. The silver was slightly tarnished, as she'd mentioned, as if it had been cleaned, perhaps by Ray, but not with attention to detail. There were five little garnets encrusted into the silver. They were small, not worth much, he guessed, though he was no connoisseur of jewels. It looked harmless in the dawn sun, but it held secrets. What did the necklace want from them, from Keely?
Turning the heart over, he saw the initials I.L.K.
"What is it?" Keely asked.
He shifted the back of the heart toward her so that she could see it in the light. "Initials. I.L.K." He raised his gaze to her. "Know anyone with those initials?"
She shook her head.
He ran his fingers over the heart, looking for a clue, anything that would tell them more—
A thin ridge rode all along the sides, clear around the heart. His finger stopped on a tiny indentation.
"It's a locket," he said suddenly.
"What?"
"It's a locket." His pulse pounded. The heart was so slim, so flat, that hadn't occurred to him. He dipped the edge of his nail, pressed hard against the indentation, and the heart sprang open.
The picture inside was faded, but the faces were visible. The sides were black, making him think of those photo booths at carnivals where you paid five bucks and got a series of pictures in strips. The faces were smiling at the camera, heads together. Teenagers. A girl and a boy, maybe seventeen or eighteen. A sense of giddiness pervaded the photo. The girl was dark-haired, doe-eyed, her eyes turned toward the boy at her side. The boy gazed straight-on at the camera, through glasses, and despite the years that had to have passed between then and now, Jake recognized him. He'd met him last night.
"That's Tom," Keely breathed.
Tom Tanner, town manager of Haven, husband of one of Keely's closest friends...
"Tom and who?" Jake asked.
"I don't know." He could see her swallow hard, see her confusion. "Tom dated Lise all through high school, then all through college. Except for a few weeks when they broke up and Lise went out with Jud Peterson. That was their senior year, I think. I was a few years behind them. They ran around together and I wasn't so much a part of their crowd then. I knew Lise more from church. Her family was friends with my family. I used to get her hand-me-downs...." She put shaking fingers to her mouth. "What does this mean? Tom dated whoever this girl is, and this is the girl who was killed, the girl whose skull I found in my garden?"
"Maybe." Jake worked to process what they knew so far. "Assuming this is the girl who owned the necklace—and I think that's pretty safe—then we guess that's her body that was buried in your garden. If your visions are real, then she was killed, run down on the road. Maybe it was even an accident. Maybe Ray helped him take care of it, hide the body. Then they moved it last fall...."
Jake dropped the necklace on the bed, got up to grab fresh jeans and a shirt from the bag he hadn't even unpacked yet.
"I did some checking on the Internet last night, while you were sleeping," he went on, zipping his pants as he continued. He was speculating, maybe still wildly, but they were getting closer to something, maybe the truth. "Tom Tanner pushed the town hard to buy some land last fall. The town lost out to a developer."
"The new Maple Creek subdivision," Keely said. Her eyes followed him as he paced the room. She didn't touch the necklace, as if she still feared it and the secrets it held, secrets it seemed to want to convey.
"Maybe they'd buried the body out there. Maybe that's why they moved it. Maybe Ray took care of the job."
"Why would Ray do it if Tom was the one who—" Suggesting Tom was a killer seemed no less easy for Keely than that it could have been her late husband.
Tom was still alive and Tom was married to her friend. An old friend. A friend whose somewhat perfect life might be about to shatter.
"Ray must have known about it. Somehow. Maybe he was in the car with Tom that night, or maybe Tom went to him for help. Was he part of the crowd that Tom ran around with?"
"Yes. Tom and Ray, sometimes Jud, till Tom and Lise broke up and Lise went out with Jud for awhile. She and Tom got back together around the time I started seeing Ray. That's when I started getting closer with Lise."
Jake remembered how upset Lise had been last night with Tom for giving money to Jud. He used to do a drunken half-ass job for money, but now Jud wanted nothing but handouts.
"Jud found out about the murder. Maybe he was blackmailing Tom," he suggested. "Maybe Ray was, too. You said he had money to buy things at estate sales to stock at the store but you had no idea where the money came from. And Jud and Ray had been friends. Maybe he clued Jud in about the body. Maybe he even helped Ray move it."
Blood money. Maybe Ray had had blood money.
Keely paled. "But if Tom knew about the locket—"
"He knew Ray had given you a gift. Everyone at the party last night knew Ray had given you a gift. Then someone broke in to your apartment."
"Tom and Lise were still at the party when we left," Keely pointed out.
"Tom talked to Jud on the phone."
"That was before the gift from Ray was brought up."
"Lise brought it up. She could have mentioned it to Tom earlier."
"You just said maybe Jud was blackmailing him," Keely argued. "Now you think he asked Jud to break into my apartment?"
"He was already paying him. Maybe he wanted more for his money."
"But if Tom knew this girl, whoever she was, had this locket on her when she died, a locket with a picture of him with her, why would he have buried her with the locket in the first place?"
"Maybe he didn't realize she had it on her. It was nighttime, dark, in your dream, right?"
Keely nodded.
"Maybe he had no idea at the time. He was in a hurry—and in a panic. Maybe he called on Ray to help him cover it up, help him get rid of the body. That would be manslaughter and a boy like Tom with college plans ahead of him would see those dreams being ruined if he didn't cover it up. But he knew where the body was and he didn't want it going to a developer—and when it did, he arranged to have the body moved, called on his old friend again.
"And this time maybe," he went on, "it wasn't dark and maybe Ray found the locket and decided the body hadn't been safe once and if it ever wasn't safe again, he was going to make sure he had some evidence linking it back to Tom, evidence he could use for his own purposes, blackmail. He wrapped it up, hid it in the house like a birthday present, probably planned to do something else with it later but then he died...."
Keely drew a shaky breath. "You have a lot of maybe's there."
"Yeah," he said grimly. A lot of maybe's. "But the biggest maybe is that maybe it's all true."
Keely rubbed her palm over the steamy mirror in the apartment bathroom. She'd opened the store and called Tammy to come in early. Picking up her comb, she drew it through the tangles in her hair. Depression weighed her down and she was annoyed with herself.
She'd gone into that bedroom with Jake last night with her eyes wide open. He wasn't looking for a relationship, and truth was, neither was she. That she was falling in more than lust with Jake was a side issue she'd deal with on her own. He cared about her, obviously, but that wasn't enough. She'd had a relationship that was less than enough before. He'd made her want a relationship again. Shocking. But she wasn't settling. She couldn't settle. She couldn't put herself through that kind of hell again.
She wanted his love and he'd made it plain he didn't even believe in the concept.
He was waiting for her outside the bathroom when she finished dressing. He lifted those dark, smoldering eyes to her that made her feel hot inside even when she was still cold. She'd dressed warmly despite the spring day. A pervasive chill carried clear to her bones.
"You can't be my bodyguard," she said. "I don't want one and I don't need one."
The apartment phone, the same line that connected with the store office downstairs, rang. Keely picked it up.
"This is Trooper Nielson. Keely Schiffer?"
Keely's pulse thudded automatically. Trooper Nielson was the officer who'd handled the break-in call the night before. "Yes?"
"The prints came up quickly," he said. "We had them right here in our office, didn't even have to put them through the state's. They belonged to Jud Peterson."
Keely's stomach dipped. "Jud Peterson? He's the one who broke into my apartment last night?"
If it had been Jud last night, then that was one maybe of Jake's that was true and what did that mean about the rest of his maybe's?
"We went out to his home this morning," the trooper continued. "We found Jud Peterson dead. Gunshot wound to the head. He also had a white pickup truck, with front side dents. We're checking tire patterns to see if that was the vehicle that Jake Malloy reported trying to run him down out at your farm."
Keely gasped. "Oh, my God." She lifted a shaking hand to her lips, moved the phone to whisper to Jake, "Jud's dead. Someone shot him. He had a white truck!" And where would Jud get the money for a truck? He certainly didn't earn it....
"Let me talk to the officer."
Keely stood by while Jake asked a few more questions, experiencing a mix of comfort and anxiety as he morphed into grim cop mode.
"We have reason to believe the victim out at the farm could have had the initials I.L.K. and that Tom Tanner was involved," Jake was saying. He told them about the possible connection to the property sold to the developer last fall. "Bring in Tom Tanner for questioning."
They'd had nothing but an unbelievable paranormal experience and conjecture before. Now the police had a new murder. Now they'd pay attention.
"What'd they say?" she asked when he hung up.
"They found some bones out at your farm this morning," he told her.
"What do we do? Do they believe us? Are they going to talk to Tom?"
"Now that they have two bodies on their hands, a new one and an old one, they're taking what we have to say more seriously. They're going to pick up Tom and I'm going to take the necklace to the station. They want it to take in to evidence now. They're going to show it to Tom, see what they can shake out of him."
"He might not talk. Or we might even be wrong."
"Either way, whoever did it will find out you don't have the necklace anymore. You'll be safe, out of it. If Tom was responsible for the death of I.L.K., it's going to be difficult to prove after all this time. But if he was responsible for Jud Peterson's death, that's a different story and the necklace, coming from Ray and the bones being found in your garden, could tie the two crimes together."
"He took a big chance, if he killed Jud."
"If he's the one who killed I.L.K., then he's got a lot to lose," Jake pointed out. "Then and now. What's one more murder? In for a penny, in for a pound. He knows there's still a chance he'll get away with it. He has no choice but to take the chance if he knows this locket could connect him to those bones. Now we know the necklace was what they were after from you. I'll give it to the police, and you make sure everyone in this town knows you don't have it anymore. Ray didn't leave you anything else, did he?"
Keely laughed harshly. "Debts."
The silence in the room felt heavy.
The phone rang.
Keely picked it up.
"Hey. Do you need me to come into the store today?"
She put her hand over the phone. "It's Lise," she whispered. Hurt waved through her. If Tom had done everything they thought he had, Lise's life was going to be shattered terribly.
"Tell her to come in," Jake said. "In case Tom goes nuts, it'll get her out of harm's way."
Keely asked Lise to come in at ten then hung up. "I hope she's okay. Oh, God, what I hope is that it wasn't Tom, after all."
Jake nodded. "I hope so, too."
He left for the station with the necklace, still in the box, and Keely was unbelievably relieved to see it go. Whatever had happened with that necklace last night, she wanted it to be over. She wanted that necklace out of her life.
The store remained busy, though not quite as bad as the day before. She made a big point of telling people she'd turned a necklace Ray left her over to the police. She plowed her way through the morning, feeling warmer as the day progressed. Thank God. The necklace was gone and so was its strange power over her. The more hours passed, the less she could even believe what had happened, with the necklace and with Jake.
When they needed extra hands at the short-order counter, she went to the kitchen, happy to be busy.
She took a full bag of trash out back to the Dumpster. It was nearly noon and the birds were singing and the air felt good. She was still alive, and as long as she was alive, she had hope. Jake had at least shown her that she could live again, even love again.
Back inside, she found Mary waiting outside her office.
"Hey, brat," Mary said. "What's up?"
Keely had a hard time suddenly controlling the tears. The happiness fell away as reality hit her. It'd been hard keeping things to herself all day with Lise at the store, knowing how her life could be about to shatter. She needed to confide in someone.
"Let's go in my office," she said.
Mary came in, shutting the door behind her. "What's wrong?"
Keely refrained from teasing her friend that she should already know, seeing as how she was psychic. Then Mary surprised her.
"I know something's wrong," she said gravely. "There's something about that skull back at your farm."
"They found more bones today. It's real." Keely shivered.
"That necklace...It was a locket, wasn't it? I keep seeing a man and a woman, and there's some kind of danger, Keely. Danger to you!"
Keely swallowed hard. "I'm scared," she admitted. "The police have the necklace now." She told Mary about waking with the locket around her neck, about the vision. If there was anyone who would believe her in this town, it was Mary. She was afraid to say anything about Tom Tanner. There was always Danny.... He'd run around with that crowd, too. She didn't want to scare Mary.
Mary already looked scared. "I think there's more. What aren't you telling me?"
Maybe she really was psychic now. Anything was believable.
"I'm seeing something," Mary said suddenly. Her eyes widened. "There's something, papers, that's it. Papers. Papers from Ray. Do you have any papers from Ray?"
"Papers? No, just, well, his manuscripts. But that was fiction."
"Are you sure? Where are they?"
"He kept them in a bank deposit box." She'd always thought that was strange. Fear raced straight to her bones. What if it was more than strange? He'd never seemed to do anything with his writing. What if it hadn't all been fiction? What if he'd written down what happened in case anything ever happened to him? She'd completely forgotten about them.
Ray didn't leave you anything else, did he?
## Chapter 15
"Oh, my God," Keely breathed. "I have to go get those papers."
"I'm going with you," Mary said.
The bank was within walking distance. She told Tammy where they were going and once inside the bank, she explained that she didn't have the key. She had no idea what Ray had done with it, but she didn't have it and if it was in the house, she'd never find it now. Thank God for small towns and people who'd known her since birth and broke the rules.
The box contained one slim legal-size envelope, not the piles of manuscripts she might have expected.
Mary was waiting for her when she came out of the vault. "Let's go back and get your car," Keely said. "I want to take this to the police station. I'll open it there."
Mary nodded. "Okay. Good idea."
They got back to the store parking lot and found Lise by her car. She turned and Keely saw the tears in her eyes.
"Tom's at the police station," Lise choked out. "I have to get there." She started sobbing.
Oh, God she was in no shape to drive. And she knew. She knew about Tom. The news wasn't going to get any better, either. Keely put her arms around her friend, looking at Mary over Lise's shoulder. "Come on. Come with us."
Lise got in the back. It was only a mile up the road to the police station. Keely looked back at Lise after they pulled out of the parking lot. "Are you o—"
She wasn't crying anymore. "Just drive," Lise said, the small pistol she'd pulled out of somewhere suddenly jammed against the back of Mary's head. "And keep driving."
It took four hours for Tom Tanner to crack under police examination while Jake, by professional courtesy, was allowed to watch and listen behind a concealed observation window, and when he finally did crack Jake knew he'd made the biggest mistake of his life. And once again, someone was going to die.
Someone he felt more than protectiveness toward, and the fear he'd felt about that was completely gone in the face of danger. He was going to lose her.
"I really hate to do this to you guys."
She had a gun. Lise had a gun. Her friend Lise had a gun and she was pointing it at the back of Mary's head while she continued to drive. The car swung wide, nearly running into a guardrail as they wildly rounded a sharp curve.
Keely's heart pounded. "Then maybe you shouldn't do this," she said. What was Lise going to do?
"No, I'm going to do it. I just feel bad about it, I want you to know that."
Lise sounded shockingly calm. Like this was no big deal. And she didn't really sound like she felt bad about it.
"Why?" Her mouth felt thick. Fear. Fear was taking over her. She was almost afraid she'd pass out from it, but she had to keep her wits about her. She had to do something. What? They were careening down the highway out of Haven at fifty miles an hour down a winding, sharply curving road through mountain hollows. They whipped by a two-story farmhouse. A horse munched grass behind a wooden post fence in a field beside it. A trembling Mary just barely missed hitting the mailbox by the road. The serene country scenery contrasted sickly with the nightmare playing out inside the car.
"You know why, you idiot," Lise said. "You found that damn locket. If we hadn't been in such a drunken mess that night, we'd have realized she was wearing it."
"We? Who's we?" Who all was involved? Keely's head reeled.
"Me and Tom and Ray and Jud. We were drunk, driving around, too many of us piled in a car with too much alcohol. Stupid! And then Tom tells us that moron Ilene Klasko was pregnant—he was going to leave me even though we'd just gotten back together."
Ilene Klasko. I.L.K. Keely didn't know an Ilene Klasko, but she had to have been a couple classes ahead of her in school, like Danny and Tom and Ray and Jud. Mary had been in Keely's class, but Mary had started dating Danny long before Keely'd hooked up with Ray—
"I swear to God, it was an accident," Lise said, angry now, the hand holding the gun shaking. Shaking but still pointing at Mary. "I wanted to go see her. I told Tom to tell her to get an abortion. We went to her house and he told her, then we backed out, turned back around and Ilene was standing in the damn road. It was so dark, we didn't see her till the last minute. She was running and I was trying to brake. I was trying! I was drunk. And I was angry, dammit. I hit her and—"
She broke off.
Lise had hit her. Lise had been driving.
"It was all Tom's fault," Lise snarled. "He should have gotten that property for the town. Then it would have been a nature preserve and we wouldn't have had to do a thing. Everything got screwed up then. We buried her there, all of us. We were scared, okay? Scared!"
Her eyes burned. "We wrote letters to her mom, said she'd run away, signed them from Ilene. Tom had some notes from her, so we faked her handwriting. Her mom's a drug addict, and she was in jail the next year. Ilene's older sister ran away the year before. Nobody ever expected Ilene to do any different. Nobody cared! Ilene disappeared and nobody cared. Her mom didn't even make a police report. It was perfect."
Perfectly horrible. Keely's mind spun. All this time, her friends had been involved in this awful secret. Ray, too.
Then Ray had up and died in an accident, leaving that locket and his manuscript. She hadn't even looked at it yet, but she knew what it had to be. Mary knew, somehow.
"What's in those papers of Ray's?" Keely asked, wanting to hear Lise say it.
"He said if anything ever happened to him, if any of us ever did anything to hurt him—He said he'd left something behind that would be found after his death. He said he'd written it all down. We weren't that worried about it. We knew you'd think it was his damn fiction. Then you found the skull. Then everything changed."
They knew she wouldn't think it was fiction now. Her throat all but closed up in horror. It could have stayed secret forever if she hadn't found Ilene's skull. And if the locket hadn't revealed too much truth...
Jake was with the police. He was with Tom. Maybe Lise didn't think Tom would break, but maybe he would.
But even if he did, it could be too late for her.
"That developer bought that old farm," Lise was saying. "So Ray said he'd move the body. He wouldn't tell anybody what he'd done with it, but he started talking about the locket, getting money out of Tom. Then Jud wanted money, too. Tom was the only one who had any, at least that kind of money. He sent Jud to your apartment last night but he screwed up again and he was going to go to the cops when Tom threatened him. Tom killed him. He didn't tell me he did, but I know he did. It's the second time Jud screwed up—he was supposed to dig up the rest of the bones but he didn't get them all. He got scared when that damn Jake Malloy came out there and nearly ran him down trying to get away. Jud was a liability. I had to fix it. I have to fix everything."
Keely knew she was a liability now, too. Now that Lise had Ray's papers, Keely was expendable.
"Tom's at the police station," she said sharply. "You aren't going to get away with this, Lise. If you hurt us, you aren't going to get away with it. They'll show Tom the picture in the locket. They know he was involved with Ilene somehow. And they found those bones at my farm. They'll start putting it together, and there'll be some evidence, some kind of DNA, that'll connect up to Jud's murder. It's all falling apart. Don't make it worse!"
She wanted to beg, but she didn't think that would work. She had to try reason.
And she had to get out of this car. Wherever Lise was taking them, nothing good could happen there. They'd make her and Mary disappear, just like they'd made Ilene Klasko disappear. And Lise was crazy.
"They're going to find out, Lise. Tom's going to tell them everything!"
"They're not going to know. Tom isn't going to break and tell them. He knows I was going to figure out some way to get Ray's papers. Nobody's going to tell. I'm going to fix it."
They'd been covering up this crime for so long, Lise couldn't believe it would all come apart. And she was going to kill them. Hopelessness swamped Keely. A car came toward them from the other direction. She needed help, but she had no idea how to get it.
"I'm sorry, I really am, but I'll get over it," Lise said with a hard laugh. "I'll even feel bad about it for a little—Oh, my God!"
Keely jerked her head to the front windshield of the car, following Mary's suddenly horrified gaze. She couldn't see anything, just winding blacktop and the other car, but she felt a biting wave of cold rushing over her bones.
"Get out of the road," Lise screamed. "Oh, my God! Ilene!"
The car spun out of control. All Keely knew then was the rush of trees coming at her, too fast, too blinding.
She had no idea how much time passed before she woke. She felt heat, terrible heat. Opening her eyes painfully, she saw Mary, head thrown on the steering wheel, completely still. Lise, too, was still—thrown clear out of the front window.
Something hissed.
The engine was going to catch on fire. The engine. Keely's head reeled with pain and she felt herself passing out. She couldn't keep her eyes open. Arms reached for her. The door of the car was open, somehow. Arms reached for her, pulled her to safety, depositing her gently on the sloping pine-covered bank. They were over the river. She could hear the sound of water swishing through rocks over the hissing of the car. Only a line of trees had stopped them from careening into the water.
She opened her eyes. A girl she recognized smiled down at her. Keely blinked, hard, her vision wavering. She reached up, tried to touch the girl, but her hand went...Right through her.
Ilene Klasko. The name burst through her head.
"Ilene?" she whispered brokenly.
The girl smiled and the vision of her fell apart, disintegrating. She sensed Mary laying down next to her. And then all she could see was smoke and flames and all she could hear was the sound of sirens. The other car...Someone had seen the accident. Someone had called for help.
She struggled to sit up, still afraid. The car was going to explode. She crawled, desperate.
Then something grabbed at her ankle and she felt the business end of a gun at the back of her head.
Jake scrambled down the embankment, past trees, sliding, down to the bottom where smoke choked the air and seared his lungs. "Keely!"
He hadn't waited for the officers coming behind him when the call had come in reporting an accident. The car they'd described fit with Mary's. Mary had gone to get Keely and Ray's papers, they knew that from calling the store to look for Keely. Tom had spilled everything. Lise had run down Ilene Klasko years ago, and he had killed Jud after he'd failed to fix things at the farmhouse yesterday morning. They'd all participated in covering up the old crime. Tom had transformed from a coolly elegant, almost academic, politician to a quivering, weeping mess. Jake had no sympathy for him.
He could only pray he wasn't too late for Keely and Mary, the innocents in this complicated, dark drama. The sirens came closer and he stopped short feet beside the car, searching inside.
There was no one there.
He swung around, backing away. The car could explode and only the fear that Keely was inside had brought him that close.
"Jake!"
He pivoted, saw them through the trees, Lise shoving Keely ahead of her. He could see Mary now, on the ground near them, not moving. Lise grabbed Keely by the hair when she didn't get up fast enough, swung her arm around and he saw the gun pointed straight at him.
Keely rose up suddenly, shoved Lise forward. The other woman lost her footing, dropping the gun. Keely was on it in two seconds, tackling Lise, putting that same gun to the back of her friend's neck. Her shocked gaze jerked to Jake's.
"Jake!" Keely screamed.
It was too late when he realized why she was screaming. Keely was okay, but he wasn't.
The force of the vehicle's explosion knocked him to the ground even as he saw Keely throwing herself down atop Mary and covering her own head from the blast. He heard officers shouting. As he fought off blackness, he knew Keely was okay.
"Jake," he heard her crying. It was Keely, he knew it was Keely. He heard shouts, officers barking orders. They were taking Lise into custody and getting medics for Mary.
But Keely, she was here, by his side, despite the flaming heat still too near.
He couldn't get up. God, he wanted to get up. He had to work to lift his eyelids, focus hard to make even one bit of his body obey his directions.
Her sexy, drop-dead, drown-in-me eyes seared his gaze.
"Don't die on me now," she whispered. "Don't you dare die on me now. I need you," she went on sweetly. God, he loved her voice. "I need you." She wasn't in danger anymore, but she still needed him....
"I'm not going to die," he promised. "But I think I'm going to black out." He grabbed her arm, focusing desperately on her, keeping himself conscious by sheer force of his will. He felt himself floating. "I just have to know one thing."
"Anything."
"I'm thinking about falling in love with you." Maybe he was in love with her already. And it felt right. Real. Perspective. He was about to pass out and he'd finally found perspective. "Is that okay?"
He heard the sob Keely caught in her throat. "I thought you didn't believe in love."
A laugh rose inside him but it hurt too much. He hurt everywhere. He'd hit the ground hard. And he was losing it fast. He had to tell her. She had to believe, the way she'd made him believe.
"You're not going to start arguing now, are you?" he said. "I want you to believe me."
"I do," she choked out, "I believe you."
"Anything can happen in Haven," he whispered roughly. "Anything."
Her tears were falling fast. He felt first one then another hit his face as she leaned down to kiss him. "And probably will," she cried. "And probably will."
## Epilogue
Recovery came in little bits, a day at a time, for Keely, for Jake, for the whole town. She'd lost a lot...her house, Tom and Lise. Ray's writings had been a fictional account of the night Ilene Lauren Klasko had been run down, in a drunken accident by a car full of teenagers who'd panicked and covered it up.
The manuscript hadn't been finished. The ending was still working its way through the courts. But for all Keely had lost, most days she woke thinking of what she'd gained. A new sense of her self, a new courage and confidence. And Jake.
They'd level-jumped on their relationship, as he'd said once. They took things slow now. They dated. He quit his job on the force in Charleston and picked up with the Haven P.D. The small force needed his big-city experience and was glad to have him.
Sometimes they talked about marriage and the future, but mostly they talked about today. What they didn't talk about much was the necklace, though she'd told Jake about the girl, the ghost of Ilene, and how she'd saved her. She'd known Jake would believe her. Mary believed her, of course. Mary had seen Ilene, too, and was struggling now with her own demons as she recovered from the ordeal. She'd stopped doing psychic readings. It all had become too real for Mary. Keely wasn't so sure about the rest of the town, though she heard rumors of other people having strange experiences. Whispers. Gossip at the store. Some people believed what Shara Shannon had said about the quake. Some people didn't. Some people, like Keely, kept it to themselves. Looking back, it seemed impossible, too impossible to be real.
Her life had been changed by that necklace. It had brought her truth, and it had brought her Jake. Whatever had happened, if she'd just been a little sick with chills and if she'd put that necklace on in her sleep...If that vision had just been a coincidental dream...And if she and Mary had gotten out of that about-to-explode car on their own...
It didn't matter. Somehow, it had all brought her life together with Jake's. She knew what she believed and it didn't matter what anyone else thought about it.
Anything could happen in Haven. Even love.
ISBN: 978-1-4268-0346-8
SECRETS RISING
Copyright © 2007 by Suzanne McMinn
All rights reserved. Except for use in any review, the reproduction or utilization of this work in whole or in part in any form by any electronic, mechanical or other means, now known or hereafter invented, including xerography, photocopying and recording, or in any information storage or retrieval system, is forbidden without the written permission of the editorial office, Silhouette Books, 233 Broadway, New York, NY 10279 U.S.A.
This is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental.
This edition published by arrangement with Harlequin Books S.A.
® and TM are trademarks of Harlequin Books S.A., used under license. Trademarks indicated with ® are registered in the United States Patent and Trademark Office, the Canadian Trade Marks Office and in other countries.
Visit Silhouette Books at www.eHarlequin.com
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\section{Introduction}
Any communication system can be considered as a particular case of
general dynamical system formed by many interacting units. If the
system components are permitted to freely interact without
\emph{strict} external control, then such unreduced interaction
process leads inevitably to complex-dynamical, essentially
nonlinear and chaotic structure emergence, or generalised
(dynamically multivalued) self-organisation
\cite{kir:1,kir:2,kir:3}, extending the conventional, basically
regular self-organisation concept. The usual technology and
communication practice and paradigm rely, however, on very strong
human control and totally regular, predictable dynamics of
controlled systems and environment, where unpredictable events can
only take the form of undesirable failures or noise.
Growing volumes and complication of communication system links and
functions lead inevitably to increasing probability of undesirable
deviations from the pre-programmed regular behaviour, largely
compromising its supposed advantages. On the other hand, such
increasingly useful properties as intrinsic system creativity and
autonomous adaptability to changing environment and individual
user demands should certainly involve another, much less regular
and more diverse kind of behaviour. In this paper we analyse these
issues in a rigorous way by presenting the unreduced,
nonperturbative analysis of an arbitrary system of interacting
entities and show that such \emph{unreduced interaction process}
possesses the natural, dynamically derived properties of
chaoticity, creativity (autonomous structure formation ability),
adaptability, and exponentially high efficiency, which can be
consistently unified into the totally universal concept of
\emph{dynamic complexity} \cite{kir:1}. This concept and
particular notions it unifies represent essential extension with
respect to respective results of the usual theory always using one
or another version of perturbation theory that strongly reduces
real interaction processes and leads inevitably to regular kind of
dynamics (even in its versions of chaoticity). We shall specify
these differences in our analysis and demonstrate the key role of
unreduced, interaction-driven complexity, chaoticity and
self-organisation in the superior operation properties, as it has
already been demonstrated for a large scope of applications
\cite{kir:1,kir:2,kir:3,kir:4,kir:5,kir:6,kir:7,kir:8}.
We start, in Sect. 2, with a mathematical demonstration of the
fact that the unreduced interaction process within \emph{any} real
system leads to intrinsic, genuine, and omnipresent
\emph{randomness} in the system behaviour, which can be realised
in a few characteristic regimes and leads to the
\emph{universally} defined \emph{dynamic complexity}. We outline
the change in strategy and practice of communication system
construction and use, which follows from such unreduced analysis
of system interactions. The \emph{universality} of our analysis is
of special importance here, since the results can be applied at
various \emph{naturally entangled} levels of communication system
operation. In particular, we demonstrate the complex-dynamic
origin of the huge, \emph{exponentially high efficiency growth} of
the unreduced, causally random system dynamics, with respect to
the standard, basically regular system operation (Sect. 3).
Finally, the dynamically derived, \emph{universal symmetry, or
conservation, of complexity} is introduced as the new guiding
principle and tool of complex system dynamics that should replace
usual, regular programming. The \emph{paradigm of intelligent
communication systems} is thus specified, since we show also
\cite{kir:1,kir:5} that the property of \emph{intelligence} can be
consistently described as high enough levels of the unreduced
dynamic complexity. This ``intelligent communication" is the most
complete, inevitable realisation, and in fact a synonym, of the
truly \emph{autonomous} communication dynamics and its expected
properties.
\section{Complex Dynamics of Unreduced Interaction Process}
We begin with a general expression of multi-component system
dynamics (or many-body problem), called here \emph{existence
equation}, fixing the fact of interaction between the system
components, and generalising various model equations:
\begin{equation}\label{eq:1}
\left\{ {\sum\limits_{k = 0}^N {\left[ {h_k \left( {q_k } \right)
+ \sum\limits_{l > k}^N {V_{kl} \left( {q_k ,q_l } \right)} }
\right]} } \right\}\Psi \left( Q \right) = E\Psi \left( Q \right)\
,
\end{equation}
where $h_k \left( {q_k } \right)$ is the ``generalised
Hamiltonian" of the $k$-th system component in the absence of
interaction, $q_k$ is the degree(s) of freedom of the $k$-th
component (expressing its ``physical nature"), $V_{kl} \left( {q_k
,q_l } \right)$ is the (generally arbitrary) interaction potential
between the $k$-th and $l$-th components, $\Psi \left( Q \right)$
is the system state-function, $Q \equiv \left\{ {q_0 ,q_1 ,...,q_N
} \right\}$, $E$ is the eigenvalue of the generalised Hamiltonian,
and summations are performed over all ($N$) system components. The
generalised Hamiltonian, eigenvalues, and interaction potential
represent a suitable measure of dynamic complexity defined below
and encompassing practically all ``observable" quantities (action,
energy, momentum, current, etc.) at any level of dynamics.
Therefore (\ref{eq:1}) can express the unreduced interaction
configuration at any level of communication network of arbitrary
initial structure. It can also be presented in a particular form
of time-dependent equation by replacing the generalised
Hamiltonian eigenvalue $E$ with the partial time derivative
operator (for the case of explicit interaction potential
dependence on time).
One can separate one of the degrees of freedom, e.g. $q_0 \equiv
\xi $, corresponding to a naturally selected, usually
``system-wide" entity, such as ``embedding" configuration (system
of coordinates) or common ``transmitting agent":
\begin{equation}\label{eq:2}
\left\{ {h_0 \left( \xi \right) + \sum\limits_{k = 1}^N {\left[
{h_k \left( {q_k } \right) + V_{0k} \left( {\xi ,q_k } \right)}
\right] + } \sum\limits_{l > k}^N {V_{kl} \left( {q_k ,q_l }
\right)} } \right\}\Psi \left( {\xi ,Q} \right) = E\Psi \left(
{\xi ,Q} \right),
\end{equation}
where now $Q \equiv \left\{ {q_1 ,...,q_N } \right\}$ and $k,l \ge
1$.
We then express the problem in terms of known free-component
solutions for the ``functional", internal degrees of freedom of
system elements ($k \ge 1$):
\begin{equation}\label{eq:3}
h_k \left( {q_k } \right)\varphi _{kn_k } \left( {q_k } \right) =
\varepsilon _{n_k } \varphi _{kn_k } \left( {q_k } \right)\ ,
\end{equation}
\begin{equation}\label{eq:4}
\Psi \left( {\xi ,Q} \right) = \sum\limits_n {\psi _n \left( \xi
\right)} \varphi _{1n_1 } \left( {q_1 } \right)\varphi _{2n_2 }
\left( {q_2 } \right)...\varphi _{Nn_N } \left( {q_N } \right)
\equiv \sum\limits_n {\psi _n \left( \xi \right)} \Phi _n \left(
Q \right),
\end{equation}
where $\left\{ {\varepsilon _{n_k } } \right\}$ are the
eigenvalues and $\left\{ {\varphi _{kn_k } \left( {q_k } \right)}
\right\}$ eigenfunctions of the $k$-th component Hamiltonian $h_k
\left( {q_k } \right)$, forming the complete set of orthonormal
functions, $n \equiv \left\{ {n_1,...,n_N } \right\}$ runs through
all possible eigenstate combinations, and $\Phi _n \left( Q
\right) \equiv \varphi _{1n_1 } \left( {q_1 } \right)\varphi
_{2n_2 } \left( {q_2 } \right)...\varphi _{Nn_N } \left( {q_N }
\right)$ by definition. The system of equations for $\left\{ {\psi
_n \left( \xi \right)} \right\}$ is obtained then in a standard
way, using the eigen-solution orthonormality (e.g. by
multiplication by $\Phi _n^
* \left( Q \right)$ and integration over $Q$):
\begin{equation}\label{eq:5}
\begin{array}{rcl}
\left[ {h_0 \left( \xi \right) + V_{00} \left( \xi \right)}
\right]\psi _0 \left( \xi \right)&+&\sum\limits_n {V_{0n} \left(
\xi \right)} \psi _n \left( \xi \right) = \eta \psi _0 \left(
\xi \right)\\
\left[ {h_0 \left( \xi \right) + V_{nn} \left( \xi \right)}
\right]\psi _n \left( \xi \right)&+&\sum\limits_{n' \ne n}
{V_{nn'} \left( \xi \right)} \psi _{n'} \left( \xi \right) =
\eta _n \psi _n \left( \xi \right) - V_{n0} \left( \xi
\right)\psi _0 \left( \xi \right),
\end{array}
\end{equation}
where $n,n' \ne 0$ (also below), $\eta \equiv \eta _0 = E -
\varepsilon _0$, $\eta _n = E - \varepsilon _n$, $\varepsilon _n =
\sum\limits_k {\varepsilon _{n_k } }$,
\begin{equation}\label{eq:6}
V_{nn'} \left( \xi \right) = \sum\limits_k {\left[ {V_{k0}^{nn'}
\left( \xi \right) + \sum\limits_{l > k} {V_{kl}^{nn'} } }
\right]}\ ,
\end{equation}
\begin{equation}\label{eq:7}
V_{k0}^{nn'} \left( \xi \right) = \int\limits_{\Omega _Q }
{dQ\Phi _n^ * \left( Q \right)} V_{k0} \left( {q_k ,\xi }
\right)\Phi _{n'} \left( Q \right)\ ,
\end{equation}
\begin{equation}\label{eq:8}
V_{kl}^{nn'} \left( \xi \right) = \int\limits_{\Omega _Q } {dQ\Phi
_n^ * \left( Q \right)} V_{kl} \left( {q_k ,q_l } \right)\Phi
_{n'} \left( Q \right)\ ,
\end{equation}
and we have separated the equation for $\psi _0 \left( \xi
\right)$ describing the generalised ``ground state" of the system
elements, i. e. the state with minimum complexity. The obtained
system of equations expresses the same problem as the starting
equation (\ref{eq:2}), but now in terms of ``natural", dynamic
variables, and therefore it can be obtained for various starting
models, including time-dependent and formally ``nonlinear" ones
(see below for a rigorous definition of \emph{essential}
nonlinearity).
We try now to approach the solution of the ``nonintegrable" system
of equations (\ref{eq:5}) with the help of the generalised
effective, or optical, potential method \cite{ded}, where one
expresses $\psi _n \left( \xi \right)$ through $\psi _0 \left(
\xi \right)$ from the equations for $\psi _n \left( \xi \right)$
using the standard Green function technique and then inserts the
result into the equation for $\psi _0 \left( \xi \right)$,
obtaining thus the \emph{effective existence equation} that
contains \emph{explicitly} only ``integrable" degrees of freedom
($\xi$) \cite{kir:1,kir:2,kir:3,kir:4,kir:5,kir:6,kir:7,kir:8}:
\begin{equation}\label{eq:9}
h_0 \left( \xi \right)\psi _0 \left( \xi \right) +
V_{{\rm{eff}}} \left( {\xi ;\eta } \right)\psi _0 \left( \xi
\right) = \eta \psi _0 \left( \xi \right)\ ,
\end{equation}
where the operator of \emph{effective potential (EP)},
$V_{{\rm{eff}}} \left( {\xi ;\eta } \right)$, is given by
\begin{equation}\label{eq:10}
V_{{\rm{eff}}} \left( {\xi ;\eta } \right) = V_{00} \left( \xi
\right) + \hat V\left( {\xi ;\eta } \right),\ \ \hat V\left( {\xi
;\eta } \right)\psi _0 \left( \xi \right) = \int\limits_{\Omega
_\xi } {d\xi 'V\left( {\xi ,\xi ';\eta } \right)} \psi _0 \left(
{\xi '} \right),
\end{equation}
\begin{equation}\label{eq:11}
V\left( {\xi ,\xi ';\eta } \right) = \sum\limits_{n,i}
{\frac{{V_{0n} \left( \xi \right)\psi _{ni}^0 \left( \xi
\right)V_{n0} \left( {\xi '} \right)\psi _{ni}^{0*} \left( {\xi '}
\right)}}{{\eta - \eta _{ni}^0 - \varepsilon _{n0} }}}\ ,\ \ \
\varepsilon _{n0} \equiv \varepsilon _n - \varepsilon _0\ ,
\end{equation}
and $\left\{ {\psi _{ni}^0 \left( \xi \right)} \right\}$,
$\left\{ {\eta _{ni}^0 } \right\}$ are complete sets of
eigenfunctions and eigenvalues of a \emph{truncated} system of
equations:
\begin{equation}\label{eq:12}
\left[ {h_0 \left( \xi \right) + V_{nn} \left( \xi \right)}
\right]\psi _n \left( \xi \right) + \sum\limits_{n' \ne n}
{V_{nn'} \left( \xi \right)} \psi _{n'} \left( \xi \right) =
\eta _n \psi _n \left( \xi \right)\ .
\end{equation}
One should use now the eigenfunctions, $\left\{ {\psi _{0i} \left(
\xi \right)} \right\}$, and eigenvalues, $\left\{ {\eta _i }
\right\}$, of the formally ``integrable" equation (\ref{eq:9}) to
obtain other state-function components:
\begin{equation}\label{eq:13}
\psi _{ni} \left( \xi \right) = \hat g_{ni} \left( \xi
\right)\psi _{0i} \left( \xi \right) \equiv \int\limits_{\Omega
_\xi } {d\xi 'g_{ni} \left( {\xi ,\xi '} \right)\psi _{0i} \left(
{\xi '} \right)}\ ,
\end{equation}
\begin{equation}\label{eq:14}
g_{ni} \left( {\xi ,\xi '} \right) = V_{n0} \left( {\xi '}
\right)\sum\limits_{i'} {\frac{{\psi _{ni'}^0 \left( \xi
\right)\psi _{ni'}^{0*} \left( {\xi '} \right)}}{{\eta _i - \eta
_{ni'}^0 - \varepsilon _{n0} }}}\ ,
\end{equation}
and the total system state-function, $\Psi \left( {q_0 ,q_1
,...,q_N } \right) = \Psi \left( {\xi ,Q} \right)$ (see
(\ref{eq:4})):
\begin{equation}\label{eq:15}
\Psi \left( {\xi ,Q} \right) = \sum\limits_i {c_i } \left[ {\Phi
_0 \left( Q \right) + \sum\limits_n {\Phi _n } \left( Q
\right)\hat g_{ni} \left( \xi \right)} \right]\psi _{0i} \left(
\xi \right)\ ,
\end{equation}
where the coefficients $c_i$ should be found from the
state-function matching conditions at the boundary where
interaction effectively vanishes. The measured quantity,
generalised as structure density $\rho \left( {\xi ,Q} \right)$,
is obtained as the state-function squared modulus, $\rho \left(
{\xi ,Q} \right) = \left| {\Psi \left( {\xi ,Q} \right)} \right|^2
$ (for ``wave-like" complexity levels), or as the state-function
itself, $\rho \left( {\xi ,Q} \right) = \Psi \left( {\xi ,Q}
\right)$ (for ``particle-like" structures) \cite{kir:1}.
Since the EP expression in the effective problem formulation
(\ref{eq:9})-(\ref{eq:11}) depends essentially on the
eigen-solutions to be found, the problem remains ``nonintegrable"
and formally equivalent to the initial formulation (\ref{eq:1}),
(\ref{eq:2}), (\ref{eq:5}). However, it is the effective version
of a problem that leads to its unreduced solution and reveals the
nontrivial properties of the latter
\cite{kir:1,kir:2,kir:3,kir:4,kir:5,kir:6,kir:7,kir:8}. The most
important property of the unreduced interaction result
(\ref{eq:9})-(\ref{eq:15}) is its \emph{dynamic multivaluedness}
meaning that one has a \emph{redundant} number of different but
individually complete, and therefore \emph{mutually incompatible},
problem solutions, each of them describing an \emph{equally real}
system configuration. We call each such locally complete solution
(and real system configuration) \emph{realisation} of the system
and problem. Plurality of system realisations follows from the
unreduced EP expressions due to the nonlinear and self-consistent
dependence on the solutions to be found, reflecting the physically
real and evident plurality of possible combinations of interacting
eigen-modes
\cite{kir:1,kir:2,kir:3,kir:4,kir:5,kir:6,kir:7,kir:8}. It is
important that dynamic multivaluedness emerges only in the
unreduced problem formulation, whereas the standard theory,
including EP method applications (see e.g. \cite{ded}) and the
scholar ``science of complexity" (theory of chaos,
self-organisation, etc.), resorts invariably to one or another
version of perturbation theory, whose approximation, used to
obtain an ``exact", closed-form solution, totally ``kills"
redundant solutions by eliminating just those nonlinear dynamical
links and retains \emph{only one}, ``averaged" solution, usually
expressing only \emph{small} deviations from initial,
pre-interaction configuration. This \emph{dynamically
single-valued}, or \emph{unitary}, problem reduction forms the
basis of the whole canonical science paradigm.
Since we have many \emph{incompatible} system realisations that
tend to appear from the same, driving interaction, we obtain the
key property of \emph{causal, or dynamic, randomness} in the form
of permanently \emph{changing} realisations that replace each
other in the \emph{truly random} order. Therefore dynamic
multivaluedness, rigorously derived simply by unreduced, correct
solution of a real many-body (interaction) problem, provides the
\emph{universal dynamic origin} and \emph{meaning} of the
\emph{omnipresent, unceasing} randomness in the system behaviour,
also called \emph{(dynamical) chaos} (it is essentially different
from any its unitary version, reduced to an ``involved regularity"
or \emph{postulated} external ``noise"). This means that the
genuine, truly complete \emph{general solution} of an arbitrary
problem (describing a \emph{real} system behaviour) has the form
of \emph{dynamically probabilistic} sum of measured quantities for
particular system realisations:
\begin{equation}\label{eq:16}
\rho \left( {\xi ,Q} \right) = \sum\limits_{r = 1}^{N_\Re } {^{^
\oplus} \rho _r \left( {\xi ,Q} \right)}\ ,
\end{equation}
where summation is performed over all system realisations, $N_\Re$
is their number (its maximum value is equal to the number of
system components, $N_\Re = N$), and the sign $\oplus$ designates
the special, dynamically probabilistic meaning of the sum
described above. It implies that any measured quantity
(\ref{eq:16}) is \emph{intrinsically unstable} and its current
value \emph{will} unpredictably change to another one,
corresponding to another, \emph{randomly} chosen realisation. Such
kind of behaviour is readily observed in nature and actually
explains the living organism behaviour \cite{kir:1,kir:4,kir:5},
but is thoroughly avoided in the unitary theory and technological
systems (including communication networks), where it is correctly
associated with linear ``noncomputability" and technical failure
(we shall consider below this \emph{limiting} regime of real
system dynamics). Therefore the universal dynamic multivaluedness
thus revealed by the rigorous problem solution forms the
fundamental basis for the transition to ``bio-inspired" and
``intelligent" kind of operation in artificial, technological and
communication systems, where causal randomness can be transformed
from an obstacle to a qualitative advantage (Sect. 3).
The rigorously derived randomness of the generalised EP formalism
(\ref{eq:9})-(\ref{eq:16}) is accompanied by the \emph{dynamic
definition of probability}. Because the elementary realisations
are equivalent in their ``right to appear", the dynamically
obtained, \emph{a priori probability}, $\alpha _r$, of an
elementary realisation emergence is given by
\begin{equation}\label{eq:17}
\alpha _r = \frac{1}{{N_\Re }}\ ,\ \ \ \sum\limits_r {\alpha _r
} = 1\ .
\end{equation}
However, a real observation may fix uneven groups of elementary
realisations because of their multivalued self-organisation (see
below). Therefore the dynamic probability of observation of such
general, compound realisation is determined by the number, $N_r$,
of elementary realisations it contains:
\begin{equation}\label{eq:18}
\alpha _r \left( {N_r } \right) = \frac{{N_r }}{{N_\Re }}\ \ \
\left( {N_r = 1,...,N_\Re ;\ \sum\limits_r {N_r } = N_\Re }
\right),\ \ \ \sum\limits_r {\alpha _r } = 1\ .
\end{equation}
An expression for \emph{expectation value}, $\rho _{\exp } \left(
{\xi ,Q} \right)$, can easily be constructed from
(\ref{eq:16})-(\ref{eq:18}) for statistically long observation
periods:
\begin{equation}\label{eq:19}
\rho _{\exp } \left( {\xi ,Q} \right) = \sum\limits_r {\alpha _r
\rho _r \left( {\xi ,Q} \right)}\ .
\end{equation}
It is important, however, that our dynamically derived randomness
and probability need not rely on such ``statistical", empirically
based result, so that the basic expressions
(\ref{eq:16})-(\ref{eq:18}) remain valid even for a \emph{single}
event of realisation emergence and \emph{before} any event happens
at all.
The realisation probability distribution can be obtained in
another way, involving \emph{generalised wavefunction} and
\emph{Born's probability rule}
\cite{kir:1,kir:3,kir:5,kir:8,kir:9}. The wavefunction describes
the system state during its transition between ``regular",
``concentrated" realisations and constitutes a particular,
``intermediate" realisation with spatially extended and ``loose"
(chaotically changing) structure, where the system components
transiently disentangle before forming the next ``regular"
realisation. The intermediate, or ``main", realisation is
explicitly obtained in the unreduced EP formalism
\cite{kir:1,kir:3,kir:5,kir:8,kir:9} and provides, in particular,
the causal, totally realistic version of the quantum-mechanical
wavefunction at the lowest, ``quantum" levels of complexity. The
``Born probability rule", now also causally derived and extended
to any level of world dynamics, states that the realisation
probability distribution is determined by the wavefunction values
(their squared modulus for the ``wave-like" complexity levels) for
the respective system configurations. The generalised wavefunction
(or distribution function) satisfies the universal Schr\"odinger
equation (Sect. 3), rigorously derived from the dynamic
quantization of complex dynamics
\cite{kir:1,kir:3,kir:5,kir:8,kir:9}, while Born's probability
rule follows from the \emph{dynamic} ``boundary conditions"
mentioned in connection to the state-function expression
(\ref{eq:15}) and actually satisfied just during each system
transition between a ``regular" realisation and the extended
wavefunction state. Note also that it is this ``averaged",
weak-interaction state of the wavefunction, or ``main"
realisation, that actually remains in the dynamically
single-valued, one-realisation ``model" and ``exact-solution"
paradigm of the unitary theory, which explains both its partial
success and fundamental limitations.
Closely related to the dynamic multivaluedness is the property of
\emph{dynamic entanglement} between the interacting components,
described in (\ref{eq:15}) by the dynamically weighted products of
state-function components depending on various degrees of freedom
($\xi, Q$). It provides a rigorous expression of the tangible
\emph{quality} of the emerging system structure and is absent in
unitary models. The obtained \emph{dynamically multivalued
entanglement} describes a ``living" structure, permanently
changing and probabilistically \emph{adapting} its configuration,
which provides a well-specified basis for ``bio-inspired"
technological solutions. The properties of dynamically multivalued
entanglement and adaptability are further amplified due to the
extended \emph{probabilistic fractality} of the unreduced general
solution \cite{kir:1,kir:4,kir:5}, obtained by application of the
same EP method to solution of the truncated system of equations
(\ref{eq:12}) used in the first-level EP expression (\ref{eq:11}).
We can now consistently and universally define the unreduced
\emph{dynamic complexity}, $C$, of any real system (or interaction
process) as arbitrary growing function of the total number of
\emph{explicitly obtained} system realisations, $C = C\left(
{N_\Re } \right),\ \ {{dC} \mathord{\left/ {\vphantom {{dC}
{dN_\Re > 0}}} \right. \kern-\nulldelimiterspace} {dN_\Re >
0}}$, or the rate of their change, equal to zero for the
unrealistic case of only one system realisation, $C\left( {\rm{1}}
\right){\rm{ = 0}}$. Suitable examples are provided by $C\left(
{N_\Re } \right) = C_0 \ln N_\Re$, generalised energy/mass
(proportional to the temporal rate of realisation change), and
momentum (proportional to the spatial rate of realisation
emergence) \cite{kir:1,kir:5,kir:8,kir:9}. It becomes clear now
that the whole \emph{dynamically single-valued} paradigm and
results of the canonical theory (including its versions of
``complexity" and \emph{imitations} of ``multi-stability" in
\emph{abstract}, mathematical ``spaces") correspond to exactly
\emph{zero} value of the unreduced dynamic complexity, which is
equivalent to the effectively zero-dimensional, point-like
projection of reality in the ``exact-solution" perspective.
Correspondingly, \emph{any} dynamically single-valued ``model" is
strictly regular and \emph{cannot} possess any true, intrinsic
randomness (chaoticity), which should instead be introduced
artificially (and inconsistently), e.g. as a \emph{regular}
``amplification" of a ``random" (by convention) \emph{external}
``noise" or ``measurement error". By contrast, our unreduced
dynamic complexity is practically synonymous to the equally
universally defined and genuine \emph{chaoticity} (see above),
since multiple system realisations, appearing and disappearing
only in the \emph{real} space (and \emph{forming} thus its
tangible, changing structure \cite{kir:1,kir:3,kir:5,kir:8}), are
redundant (mutually incompatible), which is the origin of
\emph{both} complexity and chaoticity. The genuine dynamical chaos
thus obtained has its complicated internal structure (contrary to
the ill-defined unitary ``stochasticity") and always contains
\emph{partial regularity}, which is dynamically, inseparably
entangled with truly random elements.
The universal dynamic complexity, chaoticity, and related
properties involve the \emph{essential, or dynamic, nonlinearity}
of the unreduced problem solution and corresponding system
behaviour. It is provided by the naturally formed dynamical links
of the developing interaction process, as they are expressed in
the (eventually fractal) EP dependence on the problem solutions to
be found (see (\ref{eq:9})-(\ref{eq:11})). It is the
\emph{dynamically emerging} nonlinearity, since it appears even
for a formally ``linear" initial problem expression
(\ref{eq:1})-(\ref{eq:2}), (\ref{eq:5}), whereas the usual,
mechanistic ``nonlinearity" is but a perturbative approximation to
the essential nonlinearity of the unreduced EP expressions. The
essential nonlinearity leads to the irreducible \emph{dynamic
instability} of any system state (realisation), since both are
determined by the same dynamic feedback mechanism.
Universality of our description leads, in particular, to the
unified understanding of the whole diversity of existing dynamical
regimes and types of system behaviour \cite{kir:1,kir:2,kir:5}.
One standard, limiting case of complex (multivalued) dynamics,
called \emph{uniform, or global, chaos}, is characterised by
sufficiently different realisations with a homogeneous
distribution of probabilities (i.e. $N_r \approx 1$) and $\alpha
_r \approx {1 \mathord{\left/ {\vphantom {1 {N_\Re }}} \right.
\kern-\nulldelimiterspace} {N_\Re }}$ for all $r$ in
(\ref{eq:18})) and is obtained when the major parameters of
interacting entities (suitably represented by frequencies) are
similar to each other (which leads to a ``strong conflict of
interests" and resulting ``deep disorder"). The complementary
limiting regime of \emph{multivalued self-organisation, or
self-organised criticality (SOC)} emerges for sufficiently
different parameters of interacting components, so that a small
number of relatively rigid, low-frequency components ``enslave" a
hierarchy of high-frequency and rapidly changing, but
configurationally similar, realisations (i.e. $N_r \sim N_\Re$ and
realisation probability distribution is highly inhomogeneous). The
difference of this extended, multivalued self-organisation (and
SOC) from the usual, unitary version is essential: despite the
rigid \emph{external} shape of the system configuration in this
regime, it contains the intense ``internal life" and \emph{chaos}
of permanently changing ``enslaved" realisations (which are
\emph{not} superposable unitary ``modes"). Another important
advance with respect to the unitary ``science of complexity" is
that the unreduced, multivalued self-organisation unifies the
extended versions of a whole series of separated unitary
``models", including SOC, various versions of ``synchronisation",
``control of chaos", ``attractors", and ``mode locking". All the
intermediate dynamic regimes between those two limiting cases of
uniform chaos and multivalued SOC (as well as their multi-level,
fractal combinations) are obtained for intermediate values of
interaction parameters. The point of transition to the strong
chaos is expressed by the \emph{universal criterion of global
chaos onset}:
\begin{equation}\label{eq:20}
\kappa \equiv \frac{{\Delta \eta _i }}{{\Delta \eta _n }} =
\frac{{\omega _\xi }}{{\omega _q }} \cong 1\ ,
\end{equation}
where $\kappa$ is the introduced \emph{chaoticity} parameter,
$\Delta \eta _i$, $\omega _\xi$ and $\Delta \eta _n \sim \Delta
\varepsilon$, $\omega _q$ are energy-level separations and
frequencies for the inter-component and intra-component motions,
respectively. At $\kappa \ll 1$ one has the externally regular
multivalued SOC regime, which degenerates into global chaos as
$\kappa$ grows from 0 to 1, and the maximum irregularity at
$\kappa \approx 1$ is again transformed into a multivalued SOC
kind of structure at $\kappa \gg 1$ (but with a ``reversed" system
configuration).
One can compare this transparent and universal picture with the
existing diversity of separated and incomplete unitary criteria of
chaos and regularity. Only the former provide a real possibility
of understanding and control of communication tools of arbitrary
complexity, where more regular regimes can serve for desirable
direction of communication dynamics, while less regular ones will
play the role of efficient search and adaptation means. This
combination forms the basis of any ``biological" and
``intelligent" kind of behaviour \cite{kir:1,kir:4,kir:5} and
therefore can constitute the essence of the \emph{intelligent
communication paradigm} supposed to extend the now realised
(quasi-) regular kind of communication, which corresponds to the
uttermost limit of SOC ($\kappa \to 0$). While the latter
\emph{inevitably} becomes inefficient with growing network
sophistication (where the chaos-bringing resonances of
(\ref{eq:20}) \emph{cannot} be avoided any more), it definitely
lacks the ``intelligent power" of unreduced complex dynamics to
generate meaning and adaptable structure development.
\section{Huge efficiency of complex communication dynamics
and the guiding role of the symmetry of complexity}
The \emph{dynamically probabilistic fractality} of the system
structure emerges naturally by the unreduced interaction
development itself \cite{kir:1,kir:4,kir:5}. It is obtained
mathematically by application of the same EP method
(\ref{eq:9})-(\ref{eq:14}) to solution of the truncated system of
equations (\ref{eq:12}), then to solution of the next truncated
system, etc., which gives the irregular and
\emph{probabilistically moving} hierarchy of realisations,
containing the intermittent mixture of global chaos and
multivalued SOC, which constitute together a sort of
\emph{confined chaos}. The total realisation number $N_\Re$, and
thus the power, of this autonomously branching interaction process
with a \emph{dynamically parallel} structure grows
\emph{exponentially} within any time period. It can be estimated
in the following way \cite{kir:5}.
If our system of inter-connected elements contains
$N_{{\rm{unit}}}$ ``processing units", or ``junctions", and if
each of them has $n_{{\rm{conn}}}$ real or ``virtual" (possible)
links, then the total number of interaction links is $N =
n_{{\rm{conn}}} N_{{\rm{unit}}}$. In most important cases $N$ is a
huge number: for both human brain and genome interactions $N$ is
greater than $10^{12}$, and being much more variable for
communication systems, it will typically scale in similar
``astronomical" ranges. The key property of \emph{unreduced,
complex} interaction dynamics, distinguishing it from any unitary
version, is that the maximum number $N_\Re$ of realisations
actually taken by the system (also per time unit) and determining
its real ``power" $P_{{\rm{real}}}$ (of search, memory, cognition,
etc.) is given by the number of \emph{all possible combinations of
links}, i.e.
\begin{equation}\label{eq:21}
P_{{\rm{real}}} \propto N_\Re = N! \to \sqrt {2{\rm{\pi }}N}
\left( {\frac{N}{e}} \right)^N \sim N^N \gg \gg N\ .
\end{equation}
Any unitary, sequential model of the same system (including its
\emph{mechanistically} ``parallel" and ``complex" modes) would
give $P_{{\rm{reg}}} \sim N^\beta$, with $\beta \sim 1$, so that
\begin{equation}\label{eq:22}
P_{{\rm{real}}} \sim \left( {P_{{\rm{reg}}} } \right)^N \gg \gg
P_{{\rm{reg}}} \sim N^\beta\ .
\end{equation}
Thus, for $N \sim 10^{12}$ we have $P_{{\rm{real}}} \gg
10^{10^{13} } \gg 10^{10^{12} } \sim 10^N \to \infty $, which is
indeed a ``practical infinity", also with respect to the unitary
power of $N^\beta \sim 10^{12}$.
These estimates demonstrate the true power of complex
(multivalued) communication dynamics that remains suppressed
within the unitary, quasi-regular operation mode dominating now in
man-made technologies. The huge power values for complex-dynamical
interaction correlate with the new \emph{quality} emergence, such
as \emph{intelligence} and \emph{consciousness} (at higher levels
of complexity) \cite{kir:5}, which has a direct relation to our
\emph{intelligent} communication paradigm, meaning that such
properties as \emph{sensible}, context-related information
processing, personalised \emph{understanding} and autonomous
\emph{creativity} (useful self-development), desired for the new
generation networks, are inevitable \emph{qualitative}
manifestations of the above ``infinite" power.
Everything comes at a price, however, and a price to pay for the
above qualitative advantages is rigorously specified now as
irreducible \emph{dynamic randomness}, and thus unpredictability
of operation details in complex information-processing systems. We
only rigorously confirm here an evident conclusion that
\emph{autonomous} adaptability and genuine \emph{creativity}
exclude any detailed, regular, predictable pre-programming in
principle. But what then can serve as a guiding principle and
practical strategy of construction of those qualitatively new
types of communications networks and their ``intelligent"
elements? We show in our further analysis of complex-dynamic
interaction process that those guiding rules and strategy are
determined by a general law of complex (multivalued) dynamics, in
the form of \emph{universal symmetry, or conservation, of
complexity} \cite{kir:1,kir:3,kir:5}. This universal ``order of
nature" and evolution law unifies the extended versions of all
(correct) conservation laws, symmetries, and postulated principles
(which are causally derived and realistically interpreted now).
Contrary to any unitary symmetry, the universal symmetry of
complexity is \emph{irregular} in its structure, but always
\emph{exact} (never ``broken"). Its ``horizontal" manifestation
(at a given level of complexity) implies the actual, dynamic
symmetry between realisations, which are really taken by the
system, constituting the system dynamics (and evolution) and
replacing the abstract ``symmetry operators". Therefore the
conservation, or symmetry, of system complexity totally determines
its dynamics and explains the deep ``equivalence" between the
emerging, often quite dissimilar and chaotically changing system
configurations \cite{kir:3}.
Another, ``vertical" manifestation of the universal symmetry of
complexity is somewhat more involved and determines emergence and
development of different levels of complexity within a real
interaction process. System ``potentialities", or (real) power to
create new structure at the very beginning of interaction process
(before any actual structure emergence) can be universally
characterised by a form of complexity called \emph{dynamic
information} and generalising the usual ``potential energy"
\cite{kir:1,kir:3,kir:5}. During the interaction process
development, or structure creation, this potential, latent form of
complexity is progressively transformed into its explicit,
``unfolded" form called \emph{dynamic entropy} (it generalises
kinetic, or heat, energy). The universal \emph{conservation of
complexity} means that this important transformation, determining
every system dynamics and evolution, happens so that the sum of
dynamic information and dynamic entropy, or \emph{total
complexity}, remains unchanged (for a given system or process).
This is the absolutely universal formulation of the symmetry of
complexity, that includes the above ``horizontal" manifestation
and, for example, extended and unified versions of the first and
second laws of thermodynamics (i.e. conservation of energy and its
permanent degradation). It also helps to eliminate the persisting
(and inevitable) series of confusions around the notions of
information, entropy, complexity, and their relation to real
system dynamics in the unitary theory (thus, really expressed and
processed ``information" corresponds rather to a particular case
of our generalised dynamic entropy, see \cite{kir:1,kir:5} for
further details).
It is not difficult to show \cite{kir:1,kir:3,kir:5,kir:8} that
the natural, universal measure of dynamic information is provided
by the (generalised) action $\cal A$ known from classical
mechanics, but now acquiring a much wider, essentially nonlinear
and causally complete meaning applicable at any level of
complexity. One obtains then the universal differential expression
of complexity conservation law in the form of generalised
Hamilton-Jacobi equation for action ${\cal A} = {\cal A} (x,t)$:
\begin{equation}\label{eq:23}
\frac{{\Delta \cal A}}{{\Delta t}}\left| {_{x = {\rm const}} }
\right. + H\left( {x,\frac{{\Delta \cal A}}{{\Delta x}}\left| {_{t
= {\rm const}} ,t} \right.} \right) = 0\ ,
\end{equation}
where the \emph{Hamiltonian}, $H = H(x,p,t)$, considered as a
function of emerging space coordinate $x$, momentum $p = \left(
{{{\Delta \cal A} \mathord{\left/ {\vphantom {{\Delta A} {\Delta
x}}} \right. \kern-\nulldelimiterspace} {\Delta x}}} \right)\left|
{_{t = {\rm const}} } \right.$, and time $t$, expresses the
unfolded, entropy-like form of differential complexity, $H =
\left( {{{\Delta S} \mathord{\left/ {\vphantom {{\Delta S} {\Delta
t}}} \right. \kern-\nulldelimiterspace} {\Delta t}}} \right)\left|
{_{x = {\rm const}} } \right.$ (note that the discrete, rather
than usual continuous, versions of derivatives and variable
increments here reflect the naturally quantized character of
unreduced complex dynamics \cite{kir:1,kir:3,kir:5,kir:8}). Taking
into account the dual character of multivalued dynamics, where
every structural element contains permanent transformation from
the localised, ``regular" realisation to the extended
configuration of the intermediate realisation of generalised
wavefunction and back (Sect. 2), we obtain the universal
Schr\"odinger equation for the wavefunction (or distribution
function) ${\mit \Psi} (x,t)$ by applying the causal, dynamically
derived quantization procedure
\cite{kir:1,kir:3,kir:5,kir:8,kir:9} to the generalised
Hamilton-Jacobi equation (\ref{eq:23}):
\begin{equation}\label{eq:24}
\frac{{\partial {\mit \Psi} }}{{\partial t}} = \hat H\left(
{x,\frac{\partial }{{\partial x}},t} \right){\mit \Psi}\ ,
\end{equation}
where ${\cal A}_0$ is a characteristic action value (equal to
Planck's constant at quantum levels of complexity) and the
Hamiltonian operator, $\hat H$, is obtained from the Hamiltonian
function $H = H(x,p,t)$ of equation (\ref{eq:23}) with the help of
causal quantization (we also put here continuous derivatives for
simplicity).
Equations (\ref{eq:23})-(\ref{eq:24}) represent the universal
differential expression of the symmetry of complexity showing how
it directly determines dynamics and evolution of any system or
interaction process (they justify also our use of the Hamiltonian
form for the starting existence equation, Sect. 2). This
universally applicable Hamilton-Schr\"odinger formalism can be
useful for rigorous description of any complex network and its
separate devices, provided we find the \emph{truly complete}
(dynamically multivalued) general solution to particular versions
of equations (\ref{eq:23})-(\ref{eq:24}) with the help of
unreduced EP method (Sect. 2).
We have demonstrated in that way the fundamental, analytical basis
of description and understanding of complex (multivalued) dynamics
of real communication networks and related systems, which can be
further developed in particular applications in combination with
other approaches. The main \emph{practical proposition} of the
emerging intelligent communication paradigm is to open the way for
the \emph{free, self-developing structure creation} in
communication networks and tools with strong interaction
(including self-developing internet structure, intelligent search
engines, and distributed knowledge bases). The liberated,
autonomous system dynamics and structure creation, ``loosely"
governed by the hierarchy of system interactions as described in
this report, should essentially exceed the possibilities of usual,
deterministic programming and control.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,642
|
Safe drivers, confident in today's traffic situations, often find the amount of practice driving before the road test had made a positive difference when they took the road test. Before you take the test, it is very important that you have at least 50 hours of driving practice, with at least 15 hours after sunset.
Before you can make a road test appointment, you must first complete an approved pre-licensing safe driving course. This requirement is automatically fulfilled as part of every high school or college driver education course. All other drivers must complete this requirement by taking a special five-hour course available at most professional driving schools licensed by State of New York.
Student must have an MV -278 certificate at the time of road test appointment otherwise he/she will not be allowed to take the road test. Students under 18 years of age prior to taking the Road Test must present the form MV -262 in addition to the MV -278 certificate. Students must have an MV-262 certificate at the time of road test appointment otherwise he/she will not be allowed to take the road test.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 518
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namespace Microsoft.TeamFoundation.Migration.Tfs2010WitAdapter.Linking
{
public class Tfs2008LinkProvider : TfsLinkingProviderBase
{
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,522
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Prospalta cyclicicoides är en fjärilsart som beskrevs av Max Wilhelm Karl Draudt 1950. Prospalta cyclicicoides ingår i släktet Prospalta och familjen nattflyn. Inga underarter finns listade i Catalogue of Life.
Källor
Nattflyn
cyclicicoides
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,104
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#ifndef THIRD_PARTY_BLINK_RENDERER_CORE_STYLE_STYLE_FETCHED_IMAGE_SET_H_
#define THIRD_PARTY_BLINK_RENDERER_CORE_STYLE_STYLE_FETCHED_IMAGE_SET_H_
#include "third_party/blink/renderer/core/loader/resource/image_resource_observer.h"
#include "third_party/blink/renderer/core/style/style_image.h"
#include "third_party/blink/renderer/platform/geometry/layout_size.h"
#include "third_party/blink/renderer/platform/weborigin/kurl.h"
#include "third_party/blink/renderer/platform/wtf/casting.h"
namespace blink {
class CSSImageSetValue;
class ImageResourceObserver;
// This class represents an <image> that loads one image resource out of a set
// of alternatives (the -webkit-image-set(...) function.)
//
// This class keeps one cached image from the set, and has access to a set of
// alternatives via the referenced CSSImageSetValue.
class StyleFetchedImageSet final : public StyleImage,
public ImageResourceObserver {
USING_PRE_FINALIZER(StyleFetchedImageSet, Dispose);
public:
StyleFetchedImageSet(ImageResourceContent*,
float image_scale_factor,
CSSImageSetValue*,
const KURL&);
~StyleFetchedImageSet() override;
CSSValue* CssValue() const override;
CSSValue* ComputedCSSValue(const ComputedStyle&,
bool allow_visited_style) const override;
// FIXME: This is used by StyleImage for equals comparison, but this
// implementation only looks at the image from the set that we have loaded.
// I'm not sure if that is meaningful enough or not.
WrappedImagePtr Data() const override;
bool CanRender() const override;
bool IsLoaded() const override;
bool ErrorOccurred() const override;
FloatSize ImageSize(const Document&,
float multiplier,
const LayoutSize& default_object_size,
RespectImageOrientationEnum) const override;
bool HasIntrinsicSize() const override;
void AddClient(ImageResourceObserver*) override;
void RemoveClient(ImageResourceObserver*) override;
scoped_refptr<Image> GetImage(const ImageResourceObserver&,
const Document&,
const ComputedStyle&,
const FloatSize& target_size) const override;
float ImageScaleFactor() const override { return image_scale_factor_; }
bool KnownToBeOpaque(const Document&, const ComputedStyle&) const override;
ImageResourceContent* CachedImage() const override;
void Trace(Visitor*) override;
private:
bool IsEqual(const StyleImage& other) const override;
void Dispose();
// ImageResourceObserver overrides
String DebugName() const override { return "StyleFetchedImageSet"; }
Member<ImageResourceContent> best_fit_image_;
float image_scale_factor_;
Member<CSSImageSetValue> image_set_value_; // Not retained; it owns us.
const KURL url_;
};
template <>
struct DowncastTraits<StyleFetchedImageSet> {
static bool AllowFrom(const StyleImage& styleImage) {
return styleImage.IsImageResourceSet();
}
};
} // namespace blink
#endif // THIRD_PARTY_BLINK_RENDERER_CORE_STYLE_STYLE_FETCHED_IMAGE_SET_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,947
|
In 2014 both the interior and exterior of the home were renovated and 3/4ths of the house is brick. The remainder of the home has new siding with insulation. The home has all new windows that are energy efficient. There are three carports, two of the carports are taller so larger vehicles can be accommodated.
The home is over 1500 square feet with three bedrooms, kitchen, 1 full bath room, 1 half bath, living room, family room, and dining area.
The master bedroom has a queen size bed with a chest, dresser, and large closet.
The second bedroom has a queen size bed with a dresser.
The third bedroom has two twin beds with luxury memory foam mattresses and two tables. This room has a modem with an internet connection that uses satellites. During a storm there may not be an internet connection. Guests can connect to the internet with their laptops. There is a router with WiFi that is connected to the modem. Guest who wish to use WiFi can turn on the router. Guests who are electrically sensitive can turn the WiFi off or only turn it on when they are using it. The router with WiFi only works when there is an internet connection.
There is a self inflatable queen size mattress for two additional people. Children count as people.
The full bathroom has a large sink with a elevated faucet, an exhaust fan with a digital timer, lights on the wall, light in the ceiling with a separate switch, bath tub with shower with two heads (one head can be held in your hand), and shelving with towels, etc.
The living room has a fireplace (do not use – the flu has been closed) and a sofa and chairs with a dining area. There is a built in desk with an IMac that is connected to the modem.
Since this is our home part of the year, you will find many amenities such as a crock pot, rice cooker, etc. You will have the comforts of home.
The kitchen has new counter tops, a new dishwasher that is raised so it is easier to use, two built in ovens, a microwave, cabinets, dishes and a roll-about kitchen cart. One oven was installed in 2014.
The family room has an old wood cook stove, which is not to be used. (The flue is closed.) The old wood cook stove was used on the farm for many years. Also, in the family room is a television on an old wash table from a girl's school in Baltimore, Maryland. The family room has a love seat, a reclining chair with a step stool, a rocking chair, etc.
The half bath has a large sink with an elevated faucet.
The dead trimmed sassafras tree in the front yard is left for the enjoyment of wood peckers.
The home has a whole house surge protector, which provides added protection in the event of an electrical surge.
The home has a geothermal heating and air conditioning system (installed in 2014) that also heats the hot water. In place of a regular thermostat there is a computer that provides much information in addition to the temperature. For example, one of the screens shows the temperature of the water coming into and going out of the house. The geothermal system is a variable speed system. The computer on the wall shows the current speed, which can vary from o\0% to 100%. Often the system operates between 0% and 10%. A separate electric meter shows the amount of electricity that the geothermal system is using.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,649
|
\section{Introduction}
\label{sec1}
The properties of Quantum Chromodynamics (QCD) and its phase transition from a quark gluon plasma (QGP) to hadronic matter is one of the main fields of research in heavy ion physics.
It is known from lattice QCD that for small baryon chemical potential ($\mu_{B}$), the thermodynamics properties of QCD becomes a smooth crossover \cite{PRD71034504,PRD85054503}.
As for QCD matter at large baryon chemical potentials ($\mu_{B}>400~\text{MeV}$), created in nuclear collisions at lower incident energies, various theoretical studies have suggested that the QCD phase transition may be of first order \cite{npa504668,plb442247,prc62054901,PRD78074507,ppnp72992013}.
Several experimental programs at the Brookhaven National Laboratory \cite{bes2}, European Organisation for Nuclear Research (CERN) and future Facility for Antiproton and Ion Research (FAIR) \cite{epja5360} and Nuclotron-based Ion Collider fAcility (NICA) \cite{NICA whitepaper} have been devoted to search for signals of the QCD critical point and phase transition and to study the properties of the QGP. One of the main goals of the Beam Energy Scan (BES) program at the Relativistic Heavy Ion Collider (RHIC) is to study the various aspects of the QCD phase diagram.
The BES Phase-\Rmnum{1} has been carried out at collision energies from $\sqrt{s_{NN}}= 200$ to 7.7 GeV, corresponding to baryon chemical potentials from 20 to 420 MeV.
BES-\Rmnum{1} has produced a large number of exciting results (see, e.g., \cite{PhysRevLett.120.062301,PhysRevC.93.014907,PhysRevLett.114.162301,STAR86054908,jpg38.124023,jpg38.124145}), and has restricted the region of interest to the collision energies below $\sqrt{s_{NN}}=20~\text{GeV}$.
A particularly interesting result of this experimental program was the high precision measurement of the elliptic flow of identified hadrons and their antiparticles, produced in the midrapidity region of the center-of-mass collision of the two heavy ions \cite{PhysRevC.88.014902}.
It was found that the elliptic flow of particles and their antiparticles shows a distinctive splitting which increases with decreasing beam energy or increasing net baryon density.
The anisotropic flow is an observable commonly used to study the properties of matter created in heavy ion collisions (HICs), sensitive to the equation of state (EoS) in the hot and dense early stage of the HICs. Various explanations have been proposed by several theoretical groups \cite{prc74064908,prc81044906,prc85041901,prc86044903,prc88064908,prl112012301,prd92114010,prc91024903x,prc91024914,prc91064901,EPJA52247} to account for the observed $v_{2}$ splitting of particles and corresponding antiparticles.
For instance, based on an extended multiphase transport model \cite{prc61067901,prc72064901} which includes mean field potentials in both the partonic and hadronic phase, the experimental data could be reproduced reasonably well at $\sqrt{s_{NN}}=7.7~\text{GeV}$ \cite{prc94054909}, if repulsive interactions for quarks were considered. In Ref.\cite{prc86044903}, within the framework of the ultrarelativistic quantum molecular dynamics (UrQMD) hybrid model, which combines the fluid dynamical evolution of the fireball with a transport treatment for the initial state and the final hadronic phase, the difference of elliptic flow is a result of the equilibrium hydrodynamical phase but is washed out by the subsequent transport phase through antiparticle annihilation. By tracing the number of initial quarks in the protons \cite{prc86044901}, it was proposed that the difference of $v_{2}$ between produced quarks and transported quarks may also contribute to the splitting \cite{CPC43054106}.
The purpose of this paper is to study the splitting of elliptic flow between protons and antiprotons at $\sqrt{s_{NN}}=5-12~\text{GeV}$ within the purely hadronic framework of the UrQMD model. Our goal is to show the effect of the well understood nuclear potentials of lower incident beam energies on the $v_{2}$ splitting at higher beam energies. Thus, providing an alternatively explanation which solely relies on lower energy (high density) physics, that only slowly subsides at higher beam energies. In the following we will show that also a purely hadronic description is able to reproduce the observed effect, if realistic hadronic interactions are taken into account. Predictions at even lower beam energies are made to check that indeed hadronic interactions are mainly responsible for the observed splitting.
\section{model}
\label{sec2}
In the following study the UrQMD model in its cascade (UrQMD/C, version 2.3) and a modified mean field mode (UrQMD/M) are employed. In order to accumulate a sufficient statistical accuracy, more than 10 million events are simulated for each energy and each mode. In the mean field mode of the UrQMD model \cite{ppnp41255,JPG251859}, hadrons are represented by Gaussian wave packets in phase space which read as:
\begin{equation}\label{Gau}
\phi_{i}(\textbf{r},t)=\frac{1}{(2\pi L)^{3/4}} \text{exp}\left(-\frac{(\textbf{r}-\textbf{r}_{i})^{2}}{4L}\right)\text{exp}\left(\frac{i\textbf{p}_{i}\cdot\textbf{r} }{\hbar}\right),
\end{equation}
here $L=2~\text{fm}^{2}$ is the width parameter of the wave packet. The Wigner distribution function of the $i\text{th}$ hadron is represented as:
\begin{equation}
f_{i}(\textbf{r},\textbf{p})=\frac{1}{(\pi\hbar)^{3}}\text{exp}\left(-\frac{(\textbf{r}-\textbf{r}_{i})^{2}}{2L}\right)\text{exp}\left(-\frac{(\textbf{p}-\textbf{p}_{i})^{2}\cdot2L}{\hbar^{2}}\right).
\end{equation}
The coordinate $\textbf{r}_i$ and momentum $\textbf{p}_i$ of hadron $i$ are propagated according to Hamilton's equation of motion:
\begin{eqnarray}
\dot{\textbf{r}}_{i}=\frac{\partial \langle H \rangle }{\partial\textbf{ p}_{i}},~~
\dot{\textbf{p}}_{i}=-\frac{\partial \langle H \rangle}{\partial \textbf{r}_{i}}.
\end{eqnarray}
The Hamiltonian $H$ consists of the kinetic energy $T$ and the effective interaction potential energy $U$, $H=T+U$.
The density distribution function of a single particle can be obtained from the Eq. (\ref{Gau}):
\begin{equation}
\rho_{i}(\textbf{r},t)=\frac{1}{(4\pi L)^{3/2}}\text{exp}\left(-\frac{(\textbf{r}-\textbf{r}_{i})^2}{4L}\right).
\end{equation}
In order to study HICs at intermediate energies, a density- and momentum-dependent potential was implemented \cite{AICHELIN1991233,JPG32151}. In our study, both formed hadron and pre-formed hadron (string fragments that will be projected onto hadron states later) potentials are taken into account within the UrQMD model \cite{plb659525,plb623395,JPG36015111,mpla27,scpma59632001}.
The concept of pre-formed hadron can be found in the process of hadron production in deep inelastic scattering on nuclei \cite{NPA735277,NPA782224}.
In previous studies it was found that the description of observables such as nuclear stopping, elliptic flow and the Hanbury-Brown-Twiss interferometry (HBT) parameters of pions can be significantly improved by the inclusion of pre-formed hadron potential. Please see, e.g., \cite{plb659525,plb623395,JPG36015111,mpla27,scpma59632001} for more details.
For formed hadrons, density-, momentum-dependent and Coulomb potentials are included. The density-dependent potential can be written as:
\begin{equation}
U=\alpha\left(\frac{\rho_{b}}{\rho_{0}}\right)+\beta\left(\frac{\rho_{b}}{\rho_{0}}\right)^{\gamma}.
\end{equation}
Where $\alpha=-268~\text{MeV}$, $\beta=345~\text{MeV}$, and $\gamma=1.167$ \cite{prc72064908}, corresponding to the nuclear incompressibility $K=314~\text{MeV}$.
The normal nuclear matter saturation density is given as $\rho_{0}=0.16$ fm$^{-3}$, and $\rho_{b}$ is the density of formed baryon and antibaryon.
The momentum-dependent term reads as:
\begin{equation}
U_{md}=\sum_{k=1,2}\frac{t_{md}^{k}}{\rho_{0}}\int d\textbf{p}_{j}\frac{f(\textbf{r}_{i},\textbf{p}_{j})}{1+[(\textbf{p}_{i}-\textbf{p}_{j})/a_{md}^{k}]^{2}},
\end{equation}
where $t_{md}$ and $a_{md}$ are parameters, a detailed description
of the implementation can be found in \cite{prc72064908}.
For pre-formed hadrons, only a similar density dependent term as that for the formed baryons is used and the momentum-dependent term is neglected, which read as
\begin{equation}
U=\alpha\left(\frac{\rho_{h}}{\rho_{0}}\right)+\beta\left(\frac{\rho_{h}}{\rho_{0}}\right)^{\gamma}.
\end{equation}
The parameters $\alpha$, $\beta$ and $\gamma$ are taken the same values as for formed baryons. $\rho_{h}=\sum_{i\neq j}c_{i}c_{j}\rho_{ij}$ is the hadronic density. For both formed and pre-formed baryons $c_{i,j}=1$, while $c_{i,j}=2/3$ for pre-formed mesons due to the difference of quark number, and 0 for formed mesons. Beside that, there are no interaction between pre-formed baryons and formed baryons. For formed mesons, no nuclear potential is considered. Since pre-formed hadrons are usually created in a dense environment, the interaction between these pre-formed hadrons will be mainly repulsive.
In the high energy region, the Lorentz-covariant treatment of the mean-field potentials is considered since the Lorentz contraction effect is significant. As done in \cite{PTP96263,prc72064908,prc97064913,EPJA5418}, the covariant prescription of the mean-field from the RQMD/S is implemented. RQMD/S uses much simpler and more practical time fixation constraints than the full RQMD \cite{RQMD192}, and can give almost the same results for the transverse flow as the original RQMD \cite{PTP96263}. The Hamiltonian, which reads \cite{prc72064908}
\begin{equation}
H =\sum_{i=1}^{N}\sqrt{\mathbf{p}^{2}_{i} + m_i^2 + 2m_iV_i}.
\end{equation}
where $V_{i}$ is the effective potentials felt by the $i$th particle. And the equations of motion then become
\begin{eqnarray}
\frac{d\mathbf{r}_{i}}{dt} &\approx& \frac{\partial H}{\partial \mathbf{p}_{i}}=\frac{\mathbf{p}_{i}}{p_i^0}+\sum_{j=1}^{N}\frac{m_{j}}{p_j^0}\frac{\partial V_{j}}{\partial\mathbf{p}_{i}}, \\
\frac{d \mathbf{p}_{i}}{dt} &\approx& -\frac{\partial H}{\partial \mathbf{r}_{i}}=-\sum_{j=1}^{N}\frac{m_{j}}{p_j^0}\frac{\partial V_{j}}{\partial\mathbf{r}_{i}}.
\end{eqnarray}
The relative distance $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ and $\mathbf{p}_{ij}=\mathbf{p}_{i}-\mathbf{p}_{j}$ in the two-body center-of-mass frame should be replaced by the squared four-vector distance with a Lorentz scalar,
\begin{eqnarray}
\tilde{\mathbf{r}}_{ij}^{2} &=& \mathbf{r}_{ij}^{2}+\gamma_{ij}^{2}(\mathbf{r}_{ij}\cdot \mathbf{\beta}_{ij})^{2}, \\
\tilde{\mathbf{p}}_{ij}^{2} &=& \mathbf{p}_{ij}^{2}-(p_i^0-p_j^0)^{2}+\gamma_{ij}^{2}\left(\frac{m_{i}^{2}-m_{j}^{2}}{p_i^0+p_j^0} \right)^{2},
\end{eqnarray}
where the velocity $\beta_{ij}$ and the $\gamma_{ij}$-factor between the $i$th and $j$th particles are defined as
\begin{equation}
\beta_{ij}=\frac{\mathbf{p}_{i}+\mathbf{p}_{j}}{p_i^0+p_j^0},~~~~~\gamma_{ij}=\frac{1}{\sqrt{1-\mathbf{\beta}_{ij}^{2}}}.
\end{equation}
It is known that new hadronic degrees of freedom (hyperons, mesons, and quarks) are expected to appear in addition to nucleons in high densities, e.g., in HICs and in the neutron star interior. It has been argued that the different potentials for different particles can play an important role in the understanding of the EoS of neutron star matter \cite{14125722,APJ831,EPJA53121,PRSA474,epja5227}, as well as HICs matter. On another hand, for heavy-ion collisions in the energy range $\sqrt{s_{NN}}=5$ to 12 GeV studied in this work, the early dynamic processes is not so much determined by the Lambda-nucleon potential as nucleon-nucleon interaction, since the yield of Lambda is still small relative to that of the nucleon \cite{prl93022302,prc73044910,prc78034918,prc83014901,arxiv190800451}. Thus, it is expected that the inclusion of the Lambda potential would have a subleading contributions to the $v_2$ splitting between protons and anti-protons. Where the main contribution to the $\Lambda$ (and other hypeons) would also come from a reduced absorption, due to the lower density.
In the cascade mode of the model, no potential interactions are present and the hadrons interact only through binary scattering according to a geometrical interpretation of elastic and inelastic cross sections.
\section{results}
\label{sec3}
The anisotropic flow coefficients are defined by the Fourier decomposition of the azimuthal distribution of particles with respect to the reaction plane, which can be written as \cite{prc581671,PhysRevC.88.014902}:
\begin{equation}
E\frac{d^{3}N}{d^{3}p}=\frac{1}{2\pi}\frac{d^{2}N}{p_{t}dp_{t}dy}\left[1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\phi-\Psi_{RP})]\right],
\end{equation}
where $\phi$ is the azimuthal angle of the particles, $\Psi_{RP}$ is the azimuthal angle of the reaction plane, which is defined by the line joining the centers of colliding nuclei and the beam axis.
$\Psi_{RP}$ is fixed at zero in this work by definition. $v_{n}$ is the Fourier coefficient of harmonic $n$. The first harmonic coefficient of the Fourier expansion $v_1$ is called directed flow, the second coefficient $v_2$ is called elliptic flow and the third $v_3$ is called triangular flow. In the following, we will discuss the elliptic flow ($v_2$), which defined as:
\begin{equation}
v_{2}\equiv\langle \cos[2(\phi-\Psi_{RP})]\rangle=\left\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{x}^{2}+p_{y}^{2}}\right\rangle,
\end{equation}
here $p_{x}$ and $p_{y}$ are the two components of the transverse momentum $p_{t}=\sqrt{p_{x}^{2}+p_{y}^{2}}$. The angular brackets denote an average over all considered particles from all events.
For beam energies of ($1-11A$ GeV), the elliptic flow results from a competition between the early squeeze-out and the late-stage in-plane emission. The magnitude and the sign of the elliptic flow depend primarily on two factors \cite{E895831295,prl812438,science298}, the pressure of the compressed region and the passage time of the spectator matter.
At beam energies below $4A $ GeV, negative values for $v_{2}$ are observed \cite{E895831295,PLB612173,epja3031,NPA8761}, reflecting a preferential out-of-plane emission, as spectator matter is present and blocks the path of participant matter which try to escape from the fireball zone.
At higher energies the expansion occurs after the spectator matter has passed the compressed zone, and therefore the elliptic flow is mainly caused by the initial asymmetries in the geometry of the system produced in a non-central collision. An additional, very important, feature of increasing beam energy is the decreased amount of baryon stopping in the central rapidity region of the collision. At the highest beam energies available, at the Large Hadron Collider (LHC), one essentially observes a full particle antiparticle symmetry in all observables as no incoming baryons are stopped. Only as the beam energy decreases, more and more baryons are stopped in the experiments acceptance and therefore can have an effect on the difference between particles and antiparticles. Thus, it is natural to assume that any effect sensitive to the net baryon density will disappear slowly with increasing beam energy.
\begin{figure}[t]\centering
\includegraphics[width=0.5\textwidth]{fig1.eps}
\caption{(Color online) Collision energy dependence of the elliptic flow $v_{2}$ of protons and antiprotons in $^{197}{\rm Au}+^{197}{\rm Au}$ collisions in midcentral ($10-40\%$) collisions with $|{\rm y}|<0.1$ and $p_{t}<2~\text{GeV/c}$. The results are compared to data from different experiments for midcentral collisions. For E895 \cite{E895831295} and NA49 \cite{NA4968034903} there is the elliptic flow for protons. For E877 \cite{E877}, CERES \cite{CERES698253,0109017} and STAR \cite{STAR81024911} there is the $v_{2}$ for charged particles. Simulations with a pure cascade mode UrQMD/C (dashed lines with circles) are compared to results from UrQMD/M (solid lines with squares).}
\label{fig1}
\end{figure}
In our previous work, the elliptic flow of charged particles and protons for Pb+Pb collisions at Super Proton Synchrotron (SPS) energies was studied within the UrQMD model \cite{prc74064908}.
It was found that, the NA49 experimental data in the energy region below $E_{\text{beam}}=10A~\text{GeV}$ can be reasonably described by the UrQMD model with the inclusion of nuclear potentials. However, the model, based on the cascade mode, underpredicts the elliptic flow above $40A~\text{GeV}$. As an attempt to better describe the experimental data, the pre-formed particle potential was further considered \cite{plb659525,plb623395,JPG36015111,mpla27,scpma59632001}.
A strong repulsion is generated at an early stage with the inclusion of mean field potentials.
The repulsive interactions for the baryons make them expand faster, decreasing the local density and thus antiprotons have a smaller probability of being annihilated, which, furthermore enhanced the yield of antiprotons. Thus measurable yields of antiprotons can be quantitatively explained fairly well \cite{plb659525,mpla27}.
Also, the experimental data of protons and antiprotons elliptic flow in $^{197}{\rm Au}+^{197}{\rm Au}$ collisions at $\sqrt{s_{NN}}=7.7~\text{GeV}$ can be reasonably described \cite{scpma59632001}. In this work, we extend the model to describe the difference in $v_{2}$ between protons and antiprotons at $\sqrt{s_{NN}}=5-12~\text{GeV}$.
\begin{figure}[t]\centering
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{(Color online) Elliptic flow of protons (line with solid symbols) and antiprotonss (line with open symbols) as a function of collision energy for $0-80\%$ central $^{197}{\rm Au}+^{197}{\rm Au}$ collisions with $p_{t}<2$ GeV/c and $|\eta|<1$.}
\label{fig2}
\end{figure}
Fig.\ref{fig1} depicts the UrQMD results for the energy dependence of the elliptic flow of protons and antiprotons for $10-40\%$ central $^{197}{\rm Au}+^{197}{\rm Au}$ collisions with $|{\rm y}|<0.1$ and $p_{t}<2~\text{GeV/c}$.
The calculated results from standard UrQMD cascade (UrQMD/C) mode and the UrQMD with mean field potential (UrQMD/M) mode are shown together with the experimental data from E895 \cite{E895831295}, NA49 \cite{NA4968034903}, E877 \cite{E877}, CERES \cite{CERES698253,0109017} and STAR Collaborations \cite{STAR81024911} for comparison.
As many experimental data as possible in this energy region are collected, although they are for various species of particles with different centrality and rapidity cuts.
The NA49 data \cite{NA4968034903} of protons from mid-central ($5.3-9.1~\text{fm}$) ${\rm Pb}+{\rm Pb}$ collisions is analysed by the cumulant method.
The STAR data \cite{STAR81024911} was taken with acceptance $p_{t}<2~\text{GeV/c}$, and averaged over pseudorapidity region $|\eta|<1$ from $0-60\%$ central Au+Au collisions. It is important to note that the STAR data are actually for charged particles and obtained using the event plane rather than the reaction plane which could lead to different results. Therefore this comparison should be taken with some caution.
For the elliptic flow of protons (line with solid symbols), the UrQMD/M mode which includes the pre-formed particle potential, is in line with experimental data, while the UrQMD/C mode overestimates the data at lower energies.
As for the $v_{2}$ of antiprotons (line with open symbols), the result from the UrQMD/M mode steadily increases as the energy increases.
Including nuclear potential, the difference in $v_{2}$ between protons and antiprotons, at midrapidity ($|{\rm y}|<0.1$) from $10-40\%$ central simulations, decreases with increasing beam energy, which is consistent with the trend of the SATR data \cite{PhysRevC.93.014907}. In the UrQMD/C pure cascade simulations, the $v_{2}$ values of antiprotons and that of protons are identical within errors. As discussed earlier, in the UrQMD/M mode, both the $v_2$ of protons and anti-protons will be enhanced at the very early stage ($\sim$4 fm/$c$ for $\sqrt{s_{NN}}=7.7~\text{GeV}$, before most hadrons are formed), since a stronger early pressure is supplied by the potential of pre-formed particles.
With increasing time, the $v_{2}$ of protons is further increased by the repulsive potentials and a large number of two-body collisions.
Nevertheless, the $v_{2}$ of anti-protons keeps unchanged at the same period in time. Because most of anti-protons have been pushed out of the fireball and survived without a further annihilation process due to the decreased net-density, as well as potential modifications. And more detailed description of the early time dynamics, see our previous work \cite{scpma59632001}.
\begin{figure}[t]\centering
\includegraphics[width=0.5\textwidth]{fig3.eps}
\caption{(Color online) Pseudorapidity dependence of $v_{2}$ of protons and antiprotons as well as the absolute $v_{2}$ difference between protons and antiprotons for $\sqrt{s_{NN}}=6.0\sim11.5~\text{GeV}$ $10-40\%$ central Au+Au collisions. Only simulated with UrQMD/M model.}
\label{fig5}
\end{figure}
The influence of the acceptance windows on the energy dependence of the difference in $v_{2}$ between particles and corresponding antiparticles cannot be ignored.
In Ref. \cite{PhysRevC.93.014907}, the experimental data on the centrality dependence of the $v_{2}$ difference were presented.
The difference is larger for midcentral ($10-40\%$) collisions than for central ($0-10\%$) and peripheral ($40-80\%$) collisions.
In Fig.\ref{fig2} a larger centrality ($0-80\%$) and rapidity (here the so-called pseudorapidity $|\eta|<1$) bin is adopted. Again one can clearly observe that the elliptic flow of protons and antiprotons in the cascade simulation (UrQMD/C) are almost identical, which is similar to the result shown in Fig.\ref{fig1}.
However, the difference in $v_{2}$ between protons and antiprotons still exists in the simulations within the UrQMD/M mode, but the energy dependence is weaker, due to the larger centrality and rapidity windows. Taking a larger rapidity window decreases the effect for stopping, as even at higher beam energies regions with larger baryon density then fall into the acceptance. This finding supports the idea that the observed effect is mainly sensitive to the net baryon density of the system.
The influence of the rapidity cut on the difference in $v_{2}$ between protons and antiprotons is investigated further in Fig. \ref{fig5}.
The pseudorapidity ($\eta$) dependence of the elliptic flow $v_{2}$ of protons and antiprotons is respectively shown in panels (a) and (b) of Fig. \ref{fig5}.
The $v_{2}$ difference between protons and antiprotons is shown in the panel (c).
The elliptic flow of protons as a function of $\eta$ is qualitatively consistent with previous measurements \cite{NA4968034903,PRL94122303}.
At higher energy ($\sqrt{s_{NN}}>7.0~\text{GeV}$), the $v_{2}$ of antiprotons increases quicker than that of protons at midpseudorapidity.
And the difference in $v_{2}$ at midpseudorapidity decreases with increasing collision energy, mainly due to the $v_{2}$ of antiprotons at midpseudorapidity increases with increasing energy. However, it is opposite at the projectile and target pseudorapidity regions.
Thus in a wide pseudorapidity range $|\eta|<1$, the difference between $v_{2}$ of protons and antiprotons is only weakly dependent on the beam energy. As the net baryon density decreases in midrapidity the observed splitting effect slowly disappears.
The relative $p_{t}$-integrated elliptic flow difference between protons and antiprotons, defined by $[v_{2}(p)-v_{2}(\bar{p})]/v_{2}(p)$, is shown in Fig.\ref{fig3}, in which panel (a) is for $10-40\%$ central and panel (b) is for $0-80\%$ central Au+Au collisions, the experimental data taken from Ref. \cite{PhysRevC.93.014907} and Ref. \cite{jpg38.124023}, respectively.
In panel (a), the simulated results calculated from UrQMD/M mode are shown for three cases.
The UrQMD/M-\Rmnum{1} (solid line with solid squares) is the relative elliptic flow difference at midrapidity $|{\rm y}|<0.1$, while the UrQMD/M-\Rmnum{2} (solid line with open squares) is the relative $v_{2}$ difference at $|\eta|<1$.
The UrQMD/M-\Rmnum{3} (solid line) shows the relative $v_{2}$ difference at $|\eta|<1$ normalized by $v_{2}^{\text{norm}}$, the proton elliptic flow at $p_{t}=1.5~\text{GeV/c}$ as done in Ref. \cite{PhysRevC.93.014907}.
By comparing the results obtained with UrQMD/M-\Rmnum{1} and UrQMD/M-\Rmnum{2}, which differ only in the pseudorapidity window, we see that the relative $v_{2}$ difference decreases quickly with increasing energy at midrapidity. However, over a wider pseudorapidity range ($|\eta|<1$) this energy dependence becomes weaker, as discussed above.
When the same normalization as Ref. \cite{PhysRevC.93.014907} is employed in UrQMD/M-\Rmnum{3}, a weak energy dependence is observed, which is in qualitative agreement with the result of UrQMD/M-\Rmnum{2}.
In the simulations, the $v_{2}^{\text{norm}}$ is larger than the $p_{t}$-averaged $v_{2}$ of protons which used in UrQMD/M-\Rmnum{2}, thus the solid line is lower than the solid line with open squares.
\begin{figure}[t]\centering
\includegraphics[width=0.5\textwidth]{fig4.eps
\caption{(Color online) The relative $p_{t}$-integrated elliptic flow difference between particles and antiparticles versus the collision energy. The top panel is for $10-40\%$ central collisions which is simulated by UrQMD/M model. The bottom panel is for $0-80\%$ central collisions, calculated with UrQMD/C and UrQMD/M model. And the experimental data from STAR Collaboration is from Ref. \cite{PhysRevC.93.014907,jpg38.124023}.}
\label{fig3}
\end{figure}
In the panel (b) of Fig. \ref{fig3}, the relative $v_{2}$ difference at $|\eta|<1$ from $0-80\%$ central collisions within UrQMD/M and UrQMD/C model are shown.
In the UrQMD/C mode the relative $v_{2}$ difference is essentially zero.
The model containing potentials can quantitatively describe the STAR data \cite{jpg38.124023}.
At energies below $\sqrt{s_{NN}}= 7.7~\text{GeV}$ the difference seems to saturate and does not obviously depend on the beam energy.
By comparing the solid line with open squares in the panel (a) and the solid line with solid squares in the panel (b), both being the relative $v_{2}$ difference at $|\eta|<1$ but for different centralities, the energy dependence of the difference is gradually weakened with a larger range of the impact parameter.
We therefore suggest to measure the difference of $v_{2}$ between protons and antiprotons at various centralities and rapidity bins at lower beam energies as an indicator to explore the nuclear potential in this beam energy range. We note here that besides the $v_2$ splitting between protons and anti-protons, the $v_2$ splitting between other particles and anti-particles (e.g., $\Lambda$, $\Xi$, $\pi$, $K$) also have been measured by the STAR Collaboration \cite{STAR86054908,jpg38.124023}. The splitting for the $\Lambda$ seems similar to that of protons, indicating that the effect stems mainly from the bulk density and not so much from different potentials for different baryons.
\section{Summary and outlook}
\label{sec4}
To summarize, we have studied the elliptic flow of protons and antiprotons in heavy ion collisions at $\sqrt{s_{NN}}=5-12~\text{GeV}$, within the UrQMD model.
Two different modes of the model where employed: a pure cascade and the mean field mode.
The energy dependence of the elliptic flow of protons and the relative difference in elliptic flow $v_{2}$ between protons and antiprotons can be well reproduced within this purely hadronic description.
A stronger repulsion generated by the potential in the early stage leads to an earlier freeze out of the antiprotons.
The energy dependence of the $v_{2}$ difference between protons and antiprotons is gradually weakened with increasing the range of the impact parameter.
In addition, the difference in $v_{2}$ between protons and antiprotons in a narrow rapidity window is more sensitive to the beam energy variation than in a wide rapidity range. This indicates that the observed effect is strongly dependent on the net baryon number density.
Thus, the effect disappears and the nuclei become more transparent at higher beam energies. An interesting measurement would be the verification of this effect at large rapidities of higher beam energies.
In our simulations the elliptic flow splitting for $0-80\%$ central Au+Au collisions with $|\eta|<1$ still exists below the BES Phase-\Rmnum{1} energy region, and the splitting does not strongly depend on the collision energy. Meanwhile, we propose that the $v_2$ splitting of particles and anti-particles should be measured at existing and planned heavy ion experiments to highlight the importance of nuclear interactions even at the BES Phase-\Rmnum{2} ($\sqrt{s_{NN}}=7.7-19.6~\text{GeV}$) at RHIC, CBM ($\sqrt{s_{NN}}=2.7-4.9~\text{GeV}$) at FAIR and NICA ($\sqrt{s_{NN}}=4-11~\text{GeV}$).
Although our results explain the experimental data reasonably well, the difference in elliptic flow between particles and anti-particles might receive contributions from other effects, e.g., the different potentials for different particles \cite{prc85041901,prc94054909}, the chiral magnetic effect \cite{PRL107052303}, which are not included in the present work. However, these effects seem to be dominated by the nucleon potentials for all hadronic species. It is of particular interest to improve the present model by including these effects, to understand more deeply the energy-dependent difference in elliptic flow between particles and anti-particles for $\Lambda$, $\Xi$, $K$ and $\pi$. Only in such a study it can be truly understood whether the effect is due to the net-baryons density or also sensitive to different hadronic potentials.
Recently, we have noticed that in \cite{1907.03860v1}, J. Aichelin $et~al$ presents the novel microscopic n-body dynamical transport approach PHQMD (Parton-Hadron-Quantum-Molecular-Dynamics). They modified single-particle Wigner density $\tilde f$ of the the nucleon $i$ by accounts for the Lorentz contraction effect. It is more practical and time-saving to simulate the reaction in realistic heavy-ion calculation, and also provides reference and inspiration to our next investigations.
\section*{Acknowledgments}
This work has been supported by the National Natural Science Foundation of China under Grants No. 11875125, No. 11847315, No. 11675066, the Zhejiang Provincial Natural Science Foundation of China under Grants No. LY19A050001, No. LY18A050002, and the ``Ten Thousand Talent Program" of Zhejiang province. The authors thank the computer facility of Huzhou University (C3S2). JS thanks the Samson AG and the Walter Greiner Gesellschaft zur F\"{o}rderung der physikalischen Grundlagenforschung e.V. for their support.
|
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{"url":"https:\/\/discourse.julialang.org\/t\/how-to-express-an-objective-with-l1-norm-in-jump-jl\/91387","text":"# How to express an objective with L1-norm in JuMP.jl?\n\nI learned about the NormOneCone, which can be used to add L1 constraints to a model. How to minimize the L1 norm of a vector of residuals? Tried @objective(model, Min, sum(abs.(x))) but it is not recognized by JuMP.jl.\n\n1 Like\n\nI\u2019m literally adding this to the documentation right now: [docs] add more tips and tricks for linear programs by odow \u00b7 Pull Request #3144 \u00b7 jump-dev\/JuMP.jl \u00b7 GitHub\n\n@variable(model, t)\n@constraint(model, [t; x] in MOI.NormOneCone(1 + length(x)))\n@objective(model, Min, t)\n\n\nI recall jump understanding this automatically a long time ago, is there a reason it doesn\u2019t doesn\u2019t nowadays? If feels like a mechanical transformation that a mathematical modeling language could do for the user?\n\n1 Like\n\nJuMP used to recognize norm{x} as a second-order cone. But we don\u2019t do that anymore. I don\u2019t think we every supported abs.\n\nRecognizing and reformulating arbitrary nonlinear expressions is a tricky business. Currently, our main problem is that we don\u2019t have a data structure to store sum(abs.(x)). But WIP: [Nonlinear] begin experiments with NonlinearExpr by odow \u00b7 Pull Request #3106 \u00b7 jump-dev\/JuMP.jl \u00b7 GitHub is a work-in-progress.\n\n1 Like\n\n@odow can you please confirm the syntax for minimizing f(x) + \\lambda ||x||_1? The snippet of code you shared above does not consider the L1-norm as a penalty term with penalty \\lambda > 0.\n\nAny lessons to learn from disciplined convex programming and Convex.jl in particular? It would be super nice if JuMP.jl had more sugar syntax for all kinds of terms (when that is possible).\n\nFrom Tips and tricks \u00b7 JuMP, t is the L1-norm, so you\u2019d do something like:\n\n@objective(model, Min, f(x) + lambda * t)\n\n1 Like\n\nAny lessons to learn from disciplined convex programming and Convex.jl in particular? It would be super nice if JuMP.jl had more sugar syntax for all kinds of terms (when that is possible).\n\nConvex builds an expression tree of the entire problem, with each node in the tree annotated with metadata like bounds, convexity, and monotonicity, and then it uses that information to deduce convexity of the entire problem. Because of the metadata, you\u2019re only allowed to use a fixed number of \u201catoms\u201d that Convex.jl has defined the metadata for.\n\nIt\u2019s more of a tricky engineering challenge than a tricky conceptual challenge. At the moment, the design of Convex.jl is orthogonal to that of JuMP.\n\n3 Likes","date":"2023-01-30 10:50:56","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\": 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\": 0.33953559398651123, \"perplexity\": 2347.23259820911}, \"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\/1674764499816.79\/warc\/CC-MAIN-20230130101912-20230130131912-00720.warc.gz\"}"}
| null | null |
# **BRAZEN
CREATURE**
## AKRON SERIES IN POETRY
**AKRON SERIES IN POETRY**
_Mary Biddinger, Editor_
Anne Barngrover, _Brazen Creature_
Matthew Guenette, _Vasectomania_
Sandra Simonds, _Further Problems with Pleasure_
Leslie Harrison, _The Book of Endings_
Emilia Phillips, _Groundspeed_
Philip Metres, _Pictures at an Exhibition: A Petersburg Album_
Jennifer Moore, _The Veronica Maneuver_
Brittany Cavallaro, _Girl-King_
Oliver de la Paz, _Post Subject: A Fable_
John Repp, _Fat Jersey Blues_
Emilia Phillips, _Signaletics_
Seth Abramson, _Thievery_
Steve Kistulentz, _Little Black Daydream_
Jason Bredle, _Carnival_
Emily Rosko, _Prop Rockery_
Alison Pelegrin, _Hurricane Party_
Matthew Guenette, _American Busboy_
Joshua Harmon, _Le Spleen de Poughkeepsie_
Titles published since 2010.
For a complete listing of titles published in the series,
go to www.uakron.edu/uapress/poetry.
# **BRAZEN
CREATURE**
## ANNE
BARNGROVER
Copyright © 2018 by The University of Akron Press
All rights reserved • First Edition 2018 • Manufactured in the United States of America.
All inquiries and permission requests should be addressed to the publisher,
The University of Akron Press, Akron, Ohio 44325-1703.
ISBN: 978-1-629220-80-2 (paper)
ISBN: 978-1-629220-81-9 (ePDF)
ISBN: 978-1-629220-82-6 (ePub)
Library of Congress Cataloging-in-Publication Data
Names: Barngrover, Anne, author.
Title: Brazen creature / Anne Barngrover.
Description: Akron, Ohio : The University of Akron Press, [2017] | Series: Akron series in poetry |
Identifiers: LCCN 2017041052 (print) | LCCN 2017051332 (ebook) | ISBN 9781629220819 (ePDF) | ISBN 9781629220826 (ePub) | ISBN 9781629220802 (softcover : acid-free paper)
Classification: LCC PS3602.A8344 (ebook) | LCC PS3602.A8344 A6 2018 (print) | DDC 811/.6—dc23
LC record available at <https://lccn.loc.gov/2017041052>
∞The paper used in this publication meets the minimum requirements of ANSI/NISO z39.48–1992 (Permanence of Paper).
Cover image: "Erinys" by Alyson Thiel. Cover design by Amy Freels.
_Brazen Creature_ was designed and typeset in Vendetta by Amy Freels and printed on sixtypound natural and bound by Bookmasters of Ashland, Ohio.
Contents
Hallucinate the House, Hallucinate the Woods
Egg and Ash
He Vows to Follow Me Even After He Leaves Me
Prudesville, USA
Science of Uncertainty
The Waiting Girl Sobers Up in the Same Room as Before
Finding Out the Lie One Year Later
If That Isn't a Sign Then God Knows What Is
Questions for When We Meet Again
Every Time an Ex-Boyfriend Calls You _Irrational_ , a Woodchuck Gains Its Wings
Erasure | Phenomenon
Self-Portrait with Brown Recluse
Bastille Day
Hideaway
Fourteen
Thief Hallow Branch, Arkansas
He Takes Her
If I Start Talking About It Now I Won't Stop Hollering
The Waiting Girl Sweats Out a Fever
The one drag show in town is closing
Show-Me State
Home Video, 1987
Dock and Withers
Rather than run into you at a bar, I drove out for an hour on gravel roads
A Beautiful Night for a Biscuit Truck
The month after the cruelest month
Incident in Northwestern Missouri
Hypochondria
Survival Tactics
White and Rain
After Yet Another Man Tells Me _I Don't Want to Hurt You_
Disassociation
Sapwood, Heartwood, Pith
You apologized to me in the passive voice
He Hates What I Do
Little Birds
The Waiting Girl Feels a Pulsing Where There Is No Child
Your Name in My Boot
Still Haunted
In Defense of Not Getting Over It, at Least Not for Now
Salt Creek, Missouri
Midwestern Litany
Site Fidelity
Play the Fool
The Encounter
Vow
_Acknowledgments_
## Hallucinate the House, Hallucinate the Woods
I should have known that this was coming: my home
was not my home long before the crickets found
their way inside with the rain, the smell of a gas leak,
the man I did not know who circled in the poison ivy
and pressed his hands full-palmed against my windowpanes
without apology, until one, still locked, gave. I don't care
about making amends any longer: to forgive is to betray.
A woman's home is her house and her house is her body,
and so on nights when the sky is lit like a violet lantern,
when there are no stars, small animals from the woods
claw a dark labyrinth in the walls and I wake to a bomb
going off inside my own head and the ghosts of flashlights
glow against the windowpanes of my brain—a parasomnia
so rare doctors won't bother to record. I feel like a wasp
nest nailed to a door, all the stingers dried to rose thorns. Oh,
I am the fool mouse and the fool bird. And if you are brave
enough to love me, then close your eyes and count the horses
made black by nightfall in the pastures eight houses beyond
my own. At night they are muscle and fog. Leave your party
early with a plastic bag full of strawberries because I only
want you near if your palms are stained and your tongue
saccharine with promises too dull to keep, with words
meaningless as the sound of my name being called out loud
by no one as I wake from dreams. What else could I do
for love but answer? Bombs go off all the time now. All night
long, I hear the animals finding their way back to the wild.
## Egg and Ash
Sometimes a ghost is not a ghost
but a sink of water filled to the brim.
My friend paints over the nicotine
stains that her husband left behind
as her dog whines in rusty circles.
I spend too much on too little
sympathy until there's none. Spring,
tiny losses fizz the pond's electric
scrim. Birds feel it first—eggshells
litter the sidewalk in snapped chalk
and blue, those marbles full of rain.
If a ghost cannot follow by its nature
then what voices keep in these walls?
Their sound is sugar grains dissolving.
Their sound is a mouth of broken teeth
that still remembers meat and bone.
## He Vows to Follow Me Even After He Leaves Me
Winter, I stirred sugar and cream. I cut your hair
and counted spoons, tamed the river birds
into umbrellas and taught the dog to sit
for a piece of apple. I wanted to unmake a fox earth
into a home. You thanked me a little less
every day, never saw my love scatter
off like shingles. As a child, I played a game
where I hid as the smallest creature.
I trained my eye to spot places where
I'd disappear and flee. Too late for you
to track me down. Where would you find
me now? In booths, in burrows, an attic,
a woodpile, your pocket, the milk jug, under bridges,
under porches, under a moss-eaten log?
If trust is to hem your promises
into my jacket lining like folded dollars
in an ice storm, then I have trusted all my life.
I could vanish in a white rain,
as I try to sluice your words from my clothes,
sodden with ink. I've been meaning to tell you:
I cannot comprehend how you change—
from balm to shovel, from padlock to light snow.
The moment I look back, I see again
your lips shaping the words,
_I will follow you_ , each syllable tender as teeth.
## Prudesville, USA
Even my angst isn't punk rock anymore.
Truth be told, I am too restless for seasons.
I can't abide the long-awaited pleasure
of spring, to endure the months of noodle casseroles and gray
snow encrusted with trash and motor oil
for a yard of mud and mousy daffodils.
Don't tell me I need to be more patient.
I just want it hot all the time, the humidity prickling like lust
under my skin. I used to live in a place like this.
Once, when I thought I would return
to a man who was hurting me, I bought
myself a bouquet of flowers from the grocery store. I filled
a vase with water, cut the stems, and arranged
the bouquet on my desk, next to where
I'd stuck a Post-it note for myself,
_I love you, no matter what_. The flowers were Gerbera daisies dyed
a garish neon green. The next morning
I drove back home down a street lined
with blooming crepe myrtle, all
those tiny pink hands waving at me under a bright banner
sky, my one-woman parade of foolery.
What was I thinking? I must have been crazy
all those nights I bent over
his gas stove, transfixed by the blue-orange rings
I could turn on and off while he raged
like a house fire in the next room, like a kind
of lightning that wakes you suddenly
like the dawn, and then it is dawn, and all is quiet, and I am
quiet though years have passed in a flash,
and still I am nobody's bride in the swamps
or in the fields. You'd think I would
have learned: even now I can't help myself from leaning in closer,
no matter how clearly I am not wanted,
no matter the smell of my own singed hair.
## Science of Uncertainty
As the season browns
like an apple I watch
the spider lace her web
paler than smoke
in my porch corner.
Against the inside-
outside world she waits
for what catches
in light. They come
to her and stay—
thistle, fly, mosquito,
cottonwood seed,
black thread, hangnail,
littlest measuring
spoon, a knot
of golden hair.
Against the wool-heat
of the season I wait
for the man who kept
on the lantern.
When he asked where
I learned to kiss
that way, how could I say
it was to keep
another man from leaving?
Each flick
was a cry of desperation,
each cry a measure
clung to rancid air.
How could I say
that he left anyway,
found another girl
and married her?
I learned nothing
is sweet enough to keep
from casting aside.
## [The Waiting Girl Sobers Up
in the Same Room as Before](05_Contents.xhtml#r6)
Another one with your spell in my blood.
Another one with you leaving. After the flood.
After the song. Another one with dry wine
at my table. Your dark smell lingers, heavy
as walls. A pear rots in the carpet. Moths fuzz
the chandelier. After the red tide.
After the chorus line. A doorknob chunks
inside my skull. My nails stiffen with clay.
After the conjuring. After the over-
share. Another one with everyone in khakis
squinting through the windows, murmuring,
_Why are you so alone_? Another one
where I like that. Sunset flushes seeded glass,
pink as citrus. After the hangover. After
the nicotine stains. Another one where
I show too much thigh. What else could I do
but clear my throat of smoke, tap my wrists
with oil, and arrange my face
like a plate of odd salads? All night, the long shadows.
Another one with sudden rain in my hair.
Another one where the ceiling caves in.
## Finding Out the Lie One Year Later
After my friends and I set off bottle rockets into the trees,
my hair smells like gunpowder, and I don't want to wash it away.
I am no stranger to impulsivity. How good it felt to seize
a glass of wine and throw it, to use my mouth on him and betray
what little dignity I had left, to make him stay another night
even when he told me that he was bad and I just made him worse.
I can smell his memory on me as I walk past aching streetlights
that hang their necks in shame. Cowards. In my dreams, cursed,
he appears and asks if I still wanted what broke
all those years ago. How can I say I want shrieking lines of fire?
He wants a chimney cowl, a curved stone. I want smoke
thick as color in blue and white and gold. And I'd be a liar
tonight if I didn't wonder: before fireworks were shaped like flowers,
if one woman ever thought to make them weapons, and how.
## If That Isn't a Sign Then God Knows What Is
How my sister moves in with the man she loves
and a guy on the street croons to me, _Imma buy you dinner_ —
but where the hell's my toasted ravioli?
How the frog that wants sex
calls out like the door that wants WD-40.
How if you're an asshole drunk you're
an asshole. How I smell cigarette smoke and my lips ready
for a kiss. How the cool girls in this town
bobble around like creepy starved birds.
How I don't like this one girl,
and I don't like her face either. How I run
into the street without looking and I step
onto road kill. How I put my foot through
a rotted pumpkin with a mushed yellow skull.
How I put my fist though salt water, through sand,
scoop out cone shells, and inside
there are tiny crabs. How I put two
in my palm, roll them close, and make them
fight. How close are you now to the mountains?
How close are you to the city? Remember we spent
that night in a nice hotel.
Remember we ate fried chicken
in one bed and laid down in the other.
Remember you slept in one bed and I curled up
on the bathroom floor. Someone was throwing
a party through the walls. Someone was breaking
bottles till it sounded like the ocean.
## Questions for When We Meet Again
Remember that time I asked you to be nicer to me
and you said no? Who does that? Says no? How would you react
if you got uninvited from a weekly potluck
that was your idea in the first place? All those smug
photos of egg noodles and mini pot pies getting around
every Sunday night? You know how I got kicked out
of a Bible study when I was eighteen? How the Campus
Crusade leader pursed her lips in a very straight line
so that the actual lips disappeared and her mouth
became just a creepy line of pale skin
(which must have taken great concentration on her part)
then she squeaked, _I think you're being very brave_
_to say what's on your mind, but there's no place for that kind_
_of talk among Godly women_? Are you practicing
selective memory? Can you teach me how
to do that? Remember when we slept in that room
with all the bow ties? I mean, like, a hundred
goddamn bow ties? Isn't it weird
how all the houses in the town where we lived
interchanged? How your new girl
lives in the same house where I taught you how to open
a crawdad? Remember how much you loved
crawdad once you learned how to open them? Did you ever feel
the same about me? Are houses ever loyal to owners
who dust the hard-to-reach places or power-wash the siding?
Do you think we could ever form an unlikely
animal friendship? Like a bulldog and a stray kitten?
Or a monkey and a dove? Something to show a pal
to cheer her when she's having a rough day? Remember when
you told me to shut up in front of all my friends
at the oyster bar? Do you even remember the thing I'd said
to make you do that? How my friends' eyes all widened
over the tops of their beers, and someone muttered, _Dang_ ,
and I can only imagine how much I'd looked
like a fool, and how you stared me dead in the eye
and said, _I am not apologizing for that_ ,
then took another swig? And then how the night just went on?
Can your new girl stomach seafood? Even raw?
Is she one of those people who complains that fish tastes
fishy? Like that's a surprise? Does she believe
in salt lamps and Myers-Briggs online assessments?
Is she symmetric and proportional? Does her hair smell
like Christmas? Are her freckles endearing and her music
taste on point? Does she bother you
with incessant chattering when you're trying to fall asleep?
Will she be Mrs. Hamburger Helper?
Will you have some tomboy peapods with exchangeable
boy-girl names? When you cuddle in bed does she ever
run her fingertips down your arm, gaze up and smile,
then ask you to please be nicer to her? Just sometimes?
If it's not too much trouble? Do you still say no?
## [Every Time an Ex-Boyfriend Calls You _Irrational_ ,
a Woodchuck Gains Its Wings](05_Contents.xhtml#r10)
Rattled & thrown, I leave my bed with its sheets like cold cuts,
each room dimmed the gray of molted feathers. Oh, I'm so calm
until I'm red-tailed & bedeviled in the weeds. I take the rut
instead of the trail. The pond holds November sky serene as palm
against brow, & my God, you sure know how to hollow a heart
of seeds & sinew. So aloof, so cavalier: life must be so easy
for you. Wind loves you! Whistle-pigs love you! The heron parting
from solitary confinement (not so unlike you) loves you! Please
explain this to me: what is it about you? For silence, I stuff
my mouth with milkweed. For stillness, I become a deer
stalking red boughs. I go crazy with your brand of stupid tough.
When I miss you, you write, _I gathered that you want to sever_.
When I raise hell, you say, _Sorry it's been hard_. In the bare trees:
three bluebirds dart the cold. Wait a bit longer for me.
## Erasure | Phenomenon
For years and years all I did was say
the hard things
and say them again. But you don't
know how winter
is quiet and loud here.
Your muted
eyes don't see the way
the shadows of trees
render the white slopes
a zebra and what was
water for animals is now ice
and the harried sound
of geese. I wish I could
tell you _you look_
_like the scum of the earth today_.
Wild things
once circled a town no one
would let them in
and I had to take a class this year
I didn't have a choice
where they taught me to make
words into trees
clauses into forests where nests
knot branches I drew
birds I drew bluebirds and blackbirds
those were the only
colors I had and they took flight
from the page I made
stories with my hands when
I was a child until
I learned to write them down
and the last time
I walked away from a party
this was recently
where everyone was having fun
I cursed my poor
person sweater and I cursed
my boring tears
and have you ever seen a field
where winter moves
in tongues like it's singing out
it's violent and singing
out it's fragile because I have.
It's a delicate process
to roll snowflakes over the ice
the wind must be strong
enough to move them but not
to move them apart.
## Self-Portrait with Brown Recluse
Tonight, I have no one to speak to but you,
leggy scab that vanishes in the daylight hours. Dear toxic
listener, you of six eyes, you the color of a bourbon
slow-poured, where do you hold your sadness?
Is it in your venom, your belly, your eight legs? In the mark
of a fiddle on your back, pleading instrument that makes
no sound? I think of my fed-up high school
math teacher mimicking a tiny bow and strings with his hands
while taunting, _Here's the world's smallest violin, playing_ " _my heart_
_bleeds for you_." Nobody cared. Fiddleback, I'm glad
that you don't have a spine. One of us shouldn't, anyway.
And just as the black widow cradles vengeance in the red
hourglass of her abdomen, so I hold a deep quiver
inside my mouth that never goes away. My boots are heeled
and these nights I drink only river water as I sidestep the fallen
crabapples glistening dark as organs along the road.
I have slept with revenge for years now. Noxious companion,
I confess I wouldn't know if you sought my skin and bit down.
You're the fellow hiding in my shower, the prick
resting inside my shoe, you whose god nestles in the rotting
bark of a woodpile, you who feeds on firebrats and soft-bellied
insects, who is hunted by wolf spiders and centipedes,
who moves throughout the secret crevices and corners of my home
and trails no hoary spark I can find along baseboards, unused
beds, and piled clothes. Our hearts are nothing
but liars and lilac bruises. Old friend, we both want
each other dead tonight. I, too, am learning
to recognize sadness from poison in the dark.
## Bastille Day
There are many unplanned ways to say
goodbye: stealing away, making a show,
lying drunk and intertwined in lawns. Against
the washing machine in our friends' sunroom,
this may be the last time I will feel your hands
rub down my legs, even if just to use up
the last drops of bug spray. You touch me
as though I were piebald with bruises. Rebels,
we were too rancorous to celebrate the Fourth,
too jaded for the rural comforts of white bun
hot dogs and watered beer. No matter, I leave
you in a week's time. It has been a day
of crepes and heavy cream, the evening's marvel
smack against clear and green wine bottles.
The humid air is a dress of cigarette smoke.
We eat escargot and frog legs, each their own
rubber squeal. I do not understand love's
cruel assemblage: how from across the room
your face still brightens to me like the rainbowed
coin of sunlight sweetening up a blank wall.
Why on earth am I still pretending? What good
is free will when all the fiddlers inside my chest
are causing such an uproar? How desperate
am I to leave my mark on you even when this day
already feels as distant as a history lesson,
pressing and surreal as a briny mouthful of stars.
## Hideaway
A map tells me I am just shy
of a thousand miles from the sand
pine nodding outside your window.
Each state between us jangles
like a coin box. Each city: an insect-
hole. I am too broke to come visit
you, and even so I have become
as small to you as the leak inside
a cup of water. A clean getaway
is just a row of bridges walled in
by fog's cold mushroom. I imagine,
now, how you gaze at someone new:
her eyes, the bright keys, her auburn
hair, the soft pages of that book
you'll tell her you love. It is time
that I must let this little fish
slip from my hand, even as I am
blinded by all the salt of this world.
## Fourteen
and bandy-legged, I passed time
among hills and hay, swam in dirty
pools. The summer skies were green-
eyed, the color of witches and grassy
cow pies. I was greedy for blue.
In the fields, in the high school with no AC,
men with big teeth scolded girls' skin.
Shirts must cover shoulders.
Shorts must be longer than arms can swing.
Our bodies were symmetry or else
scandal. After the horse
crushed my toenail, I chewed the rim
off a Dixie cup of Kool-Aid
while a clutch of girls prayed
my foot would heal. I wished
the horse in the field would feel pain.
Later a boy I didn't know
grabbed my ponytail at a football game
and breathed in my hair in front
of my father, who described it
to everyone around: _he was intoxicated_
_by her, that boy_ , and I was mortified.
I had powers yet to be harnessed,
ones I couldn't name. I tell you,
I willed it, and that horse: it went lame.
## Thief Hallow Branch, Arkansas
Sometimes a ghost is not a ghost
but a column of light. A man's voice
mists in the cottonwoods, speaks
her name over and over, smooth
as egg and butter pie. A girl can't wear
a skirt above her knees in this town.
Watch the red hens, how they squat
and murmur in their scratching
when they fear roughed hands, how
she can't walk the night bridges alone.
Watch the first forsythia creeping up
the road, petals yellow as a thumb-
pressed bruise. By evening, a faint
smell of smoked ribs. Here, spring
keeps the women cold for so long.
Rain curls its fingers into hooks.
## He Takes Her
The story is an old one—
a man wants a girl and so
he takes her. Girls are locked inside houses. Girls crawl
naked from dog leashes,
Duct tape slapped across wet
mouths. _I hacked my hair with garden shears, the limp_
_pigtails bright as snakes in the grass_.
A man sits in jellied luxury
and watches, his shoulders rounded as beef rump.
There's only so much shame
a body can bleed
out in the yard where even cheap lights can't shine.
Girls do not speak.
## [If I Start Talking About It Now
I Won't Stop Hollering](05_Contents.xhtml#r18)
A man once told me sweetness
was the highest quality a woman
could own then put my name
down as _Trouble_ in his phone.
And here I've gone
and said too much already,
for this is the country of fat
threading through muscle,
the land that bleeds corn syrup
and brown rivers that flow
in directions I can never
recall. _Stop speaking. Now smile_.
This is how you keep
a story from being told—
mothers teach daughters
what knowledge they write
in bloodlines and what they must
trace into silt or wet snow,
each letter erasing itself as soon
as it's exposed. I have learned
that sweetness is love
thrown back in my face.
I have spread a map across
my knees and dreamt of all
the places I could flee—
the better states (not many
after all) with better laws
(my body still not my body
wherever the corn oil sun
rises and falls) and then I saw
a picture of myself as if
from afar listening again
to your endless white boy
search for God—oh, there's
the God in your weekly
poker game, God in your
grandfather's barn, God
in your goddamn manifesto
on human consciousness,
God in your marriage
to a much younger girl
who believes you've hung
the moon, and there's God
in your painted bedroom,
God in the bleakest heart
of your coldest Midwestern
woods—where any woman
would be a fool to go alone—
the God who has been there
for you all along, who was
made to look like you,
who never wanted me.
They tell me, _Trouble_ ,
_let down your hair. Use it to hide_
_your eyes_. Tell me how,
then, do I still see shame
as it courses through
the aching silence of lineage,
this rusted river that stains
its color on my hands?
## The Waiting Girl Sweats Out a Fever
After the rain you return to the house
whose drawers fill with mud and fish
scales. When I talk to women, we walk
through ragged weeds and pinch
wild strawberries from stem. They taste
like air. We boil dumplings, sweaty
as hands. When I talk to men, we drink
whiskey. We drink thundered glass. I want
to throw a bottle against the wall.
I want to punch somebody in the head.
I cannot survive the drought this way—
sucking rocks for water, crushing
pine needles into spice, curling in puddled
shade. Without rain the night is polished
as bone. You are the mosquito stinging
through my poisoned dress. You are cheap
cloth puckered into dimples, and everyone
sees. I sleep wet in the scalp and legs.
I dream that a man dies of thirst in a cave
and his body becomes a cat's. His eyes
are dark as a horse's and round as stones.
I press them down, yet they will not close.
## The one drag show in town is closing
and tonight the snow is a performance,
the way it changes light and sound.
Lit windows poke out, fish eyes
under the alien sky.
Our hostess sashays across the floor,
leans over our chairs and hooks
our purse straps with one finger.
_This is the cheap bitches table over here!_
she hoots to the crowd. We laugh
at ourselves. _This shit looks like_
_you got it at the airport!_ Praise be
her silicon cheekbones, perfect
scoops of cream. Praise be
her russet wig and feral cat
eyes, her dress made out
of neon straws. We dance.
Champagne bleeds from plastic
flutes, pools around the table.
From here we can forget
our circumference of soy and corn,
blind as we are to the smoke
stacks at the edge of town,
their smog sometimes white,
sometimes gray, sometimes gold.
This is mid-winter in mid-country:
luxuriating in banality,
the iron trees, clouds the color
of gasoline, austere fields leaking
pleasure. How I long to be undressed
by unshaking hands.
I look into your eyes,
and I am denied. My voice
becomes grass seeds caught
in the wind. But I am a good dog.
Halved pills, stress rash, stolen
television: I don't cry. I don't cry,
though the coyote crouches
on my chest and howls. This club
is bankrupt. No one here pays
for sequins flashing at the throat,
and this is the Midwestern way:
a man who has everything
believes he is bad, sits at home
alone and feels good about it.
The rest of us will dance
tonight with our eyes closed.
The rest of us are in pain.
## Show-Me State
A bicycle strung up between trees
in the woods beyond the soybean field
held by wire its wheels unblinking eyes
they can see back before each wildflower
was tagged with a wooden stake green
plastic tied in ribbons back before a snake
long dead was nailed to a telephone pole
back when people were like the river
when they turned back to the woods
for what hurt them knew jewelweed a balm
for stinging nettles and bur oaks for bad
hearts that basswood made ropes and shoes
that creeper vine's blue fruit brought down
a fever and the yellow lady slipper called
whippoorwill's shoe eased troubled minds
during restless nights I know the hour
when the night birds sing with morning
someone told me old souls are born in winter
closest to the new year someone told me
very young that red berries could kill us
they are meant for the birds they are meant
for snakes alone in the field the yellow
flowers are brown-eyed the fence birds
are blue I asked the last boy when would you
have said you loved me he said I was waiting
for you I said what would you have done
if I did he said I would not have said it back
I am looking for a scene cut with colors
blue storm over wheat field the black trees
beyond I am waiting for a man who pulls
back the woods where the owls cry out
in day for the way he pulls my hair my mouth
away from his oh doesn't he know I want
everything at once I want to be spinning
in trees I want the wire to break I want
the moment before the moment that I fall—
## Home Video, 1987
The apple tree in our backyard truly served no role. We did not gather
the fruit, so they dropped, a carmine heft in dank grass, willed
to summon sweat bees and wasps. I was nearly two, my hair
a white-gold flare, my body not yet trained for how to stand still.
Like an actress in the wings, I waited, hopping on tiptoe,
for my father's voice to ask, _Pick me an apple, Annie_. He was proud
of the fresh knowledge in my mind. I was eager to show
him I could choose from the tree that sprawled over our fence, boughs
laden with maroon bulbs. I plucked one and rushed to him, my face bright
with glee. He murmured, _Doesn't look too good, does it_? and palmed
it like a baseball. Member of the rose family, skin rough and tight,
their color less claret and more brown, the fruit in our yard was flawed.
Still I brought him all the bruised ones, the ones torn with rot and bugs.
No matter. He would've let me take down the whole tree with his love.
## Dock and Withers
Nothing's about what it's about anymore—
Trailers populate
the horse pasture. Sheep clump the hillside
and cows wade into low pools.
A hoof is living, an intricate
lantern: coffin bone, cannon bone,
sensitive and insensitive
frogs. A boy slaps a frog in a water jar. I talk to you
in my sleep about Jackson and I remember
the gray horse in Tennessee
that watched me swing until I became too grown
for the swing,
the years blurring a sealed window. There are so many
things you are silent about. Watch me unzip
weeds along the fence line.
What's wild can wait in glass.
## [Rather than run into you at a bar,
I drove out for an hour on gravel roads](05_Contents.xhtml#r24)
to a nonprofit wolf preserve, or so
it had been advertised, but what I found
was the owner's husband—his face purpled,
his eyes dead-black, hands in flight
like hostile crows—outside the visitors'
center while she went back in to run
the gift shop of Sunday school needlepoints
and Icee Pops. The man was a Bible-
thumper, a Vietnam vet who preached at us,
an odd gaggle of tourists and locals,
for an hour in the mid-afternoon July sun,
first about the Revolutionary War,
then about Second Amendment rights
(as everyone but me nodded along)
and then finally about the true moral compass
of the wolf pack, as _the ideal family, the closest_
_thing to God_ , because didn't we know
that _wolves don't cheat on their spouses_ ,
_wolves don't molest, don't steal or join gangs_ ,
_wolves don't do drugs_ , and _Bill Clinton_
_could learn a lot from these wolves, and that fellow_
_John Edwards could, too_. There was not much
I could do but pay a twenty to run
my fingers through a tamed Gray's coat
as it passed by, its fur coarse and greasy
and unlike any dog's, even though he said
it would be _just like your dog's at home_.
But when the man told us to howl, shapes
I had not seen before began to shift
among the trees without meaning to—a trick
of the light, a flicker of rain—save
for what rose in their blood, after all this time
asleep, for what couldn't be undone.
## A Beautiful Night for a Biscuit Truck
Who does well in this town where night flaunts
her ragged satin, her snaggletoothed mouth
of stars and moon? Not the processed steak
encased in dirty ice, nor the crocuses
unfurled amid street clutter. Not the rabbits,
those gray disciples haloed in headlights,
nor the terrified buck that hurled itself
through parking lots, its legacy a bloody
hoof print stamped into a truck's hood.
I never hear trains anymore. In the square,
the dancer's statue dulls from lack
of eyes. Another night drags on, the biscuit
truck's beam a satellite among the bars
that spit out drunk college kids who scatter
like minnows. In this town where hands
are too polite to roam, I remember the place
where night slashed herself on palm fronds
and swayed in a heat that knows the open lips
of a loaded gun, where the guilt I devoured
dripped with honey, and I clung on to whatever
man I loved, as if at any moment my life
could be wrenched from my hands and had.
## The month after the cruelest month
is silk and velvet, redbuds and forsythia,
lace-white pear trees backlit
in a streetlamp's planetary glow.
A grinning dog chases cars in tall grass's
gold tassels, and some fool
burns wet green wood in the near
distance, the rising smoke in the trees
with a bad smell that creates no heat,
no clear purpose. How I no longer feel
_out-of-love_ but simply _not-loving_.
I established this pattern years ago.
For one month I believe
I'm someone's dream girl. I fall
for someone's charm like a migrating bird—
the bright flicker of feathers, the rare
trill threading the dogwoods—then gone.
I'm down on my luck again, pissing off
every man around. I'm no one's
dream; therefore I am everyone's
foe. Call me _jaded_ —it fits
me like a dress that's so tight
I can't properly sit down. Every woman
must come to a crossroads. Oh, charmer,
I have learned your bright alphabet
of night-blooming flowers.
There will always be dirt in your nails
and smoke on your breath.
There will always be smoke in the trees.
## Incident in Northwestern Missouri
> _—For Daisy Coleman, who, at age 14, was drugged, raped_ ,
>
> _and left for dead in freezing temperatures in Maryville, Missouri_ ,
>
> _the night of January 8, 2012, and survived_.
Who's to say I haven't thrown back
what storm was left in me and tried not to visit God,
but the night they burned down
your house, Daisy, I wasn't thinking
about prayer. You could be anywhere
( _very red and very bruised_ ) almost any girl
( _no socks no coat no shoes_ ) whose family
dogs were trained to bark for a child
of a rabbit, a child of a deer, and you—
left for dead, bloodletting the snow in your front lawn
( _frostbite on hands and feet_ ,
_icy chunks in hair_ ). Daisy, more and more
I wake to something sharp
as gunshots, to the garbled plastic sounds
of what happens after words choke
and dissolve and I'm the only one who hears.
The night you tried to kill yourself
for the second time you saw ghosts but no one
believed you, did they?
A girl scarcely lives before she's a lie.
In my new house, I scatter cinnamon
like a séance and rub down the cabinets with vinegar.
I claim it's to rid the ants but ritual,
Daisy, is what keeps us going and I don't need
to tell you what it looks like underneath a scar.
It's a lake, a forest, or worse—because we could
be anywhere but we're here
in Missouri where fences don't appear electric
yet warn against hands, houses smolder
from matches lit by politicians' sons and football kings
and where a godforsaken heart
is a self-fulfilled prophecy for those
who've forgotten how to pray, and Daisy,
could we blame them? You know how forgiveness
is crumbled starlight in the trees
and salvation ( _she is nothing!_ ) rises ugly as healing skin.
## Hypochondria
My body needs something to be wrong. You
filled me
like a cup and then you
left. I scrub mouse blood
from the baseboards. I scrape frozen bird shit
from the front door.
The smell of death lifts
up the walls. Still small animals want in.
An orange cat tosses a rag of a rabbit, whiskers it
with claws.
If you knew how the body can clench
and hold: a quickening
of the guts and lungs.
I once had succulents—the man
who broke
in smashed them from the windowsill, and now I
eat a little less, a little
less not because I want
to be a light bulb but because I need to be lantern. Forever
winter, the sky looks cold, pink as a clot in the mouth.
## Survival Tactics
Bless the June rains rolling down the Big Muddy,
the river-choked fields dried out in smear
and clod. Bluebirds baffle barbed wire. A puppy flops
in a vegetable plot. Bless the snake and worm
together, their bodies a crooked lecture. Before you met me,
I made my heart a basket of empty laundry. I drank
something pink and feral, watched as I drove away
and everyone stood behind in the gravel. How easy now
the trees perform an alien ballet, front-lit to neon.
Some families play music, see ghosts. I do not know
where my family comes from or why maps pucker up
for this place where the cottonwoods cast out their seeds
to strike water in such brazen display—oh, bless them.
## White and Rain
Sometimes a ghost is not a ghost
but a trail of white footprints
down a hall. From Ohio to Florida
valleys flood with mud and salt,
the magnolia flowers dirty as wet
newspapers, too heavy for arms.
Brides dip backwards, feet perfect
as snails. They cut damp cakes
into squares. West of Mississippi,
heat lightning sizzles the dry grass.
Drought makes us lighter, and I
can see through everything now:
sparrow bones, possum bones,
bones of the heart wound tight
as a clock, its key a rain-shaped
word I don't yet understand.
## [After Yet Another Man Tells Me
_I Don't Want to Hurt You_](05_Contents.xhtml#r31)
Because after we slaughtered the lamb, I stayed
and helped to skin the head and cut out the tongue.
Because my tongue weighs heavy as a key, and I, too, can be
just as brass. Because I sleep near a steak knife
where condoms should go. Because I'm only missing
a rib, which I like to think a bird of prey flew off
with in its talons—a curved arrow, a piece of moon, a fang.
Call me _salted away_ and not _withheld_. Call me a rabbit
and not a speeding car. And you're the gallant one,
aren't you? So noble, always doing the right thing. Why,
you're a regular squire! Never lily-livered. Never false-
hearted. Because it's not like every woman wasn't already
stunned from the moment a man rolled off her
and called her a mistake, from the moment she was forced
to lie there like an old farm tool because what else
could she do, really, and then later she walked
down the street to her job and another man barked
at her, _Smile!_ Because wouldn't she look so much prettier?
Because no wonder he left—and how kind of him
not to lead her on, how valiant to know his own heart
right from the get-go. Because you're a constellation
that light pollution has faded over time. Call me
a table built from barn wood. Call me a bucket of ashes
and rain set out by the front door. And call me tender-
hearted, if you must, because when I sliced off
the dead lamb's ear, I did so lovingly, which means
without flinching. Because it was one of the only parts
we didn't freeze for winter, didn't right then and there
boil or fry up or stew. Because some things are meant to be
discarded, and because you just can't help the way you feel,
like how I cut the skin from the bone and it yielded
to my knife, but I wouldn't call it soft.
## Disassociation
Oh, every season is long for you now
especially in Missouri, where you're bound
to be restless in the honeysuckle
light of June evenings. But things happen
every day without your knowing:
your oven ticks as it warms, a slug
drowns in your bathtub, the chopper
flies so low your windowpanes rattle
(whom was it looking for?), your lover
and your friend text each other,
_I hate you_. You've had a rough go.
It's hard to regain one's reputation
after throwing a wig into a dancing
crowd, but no one can tell you, _Be more_
_approachable_ , when you must dip
your nails into your beer to test
for roofies (there's a polish for that now)
and all the windows in your house
are cricketed with alarms. Look at you
here in your thin cotton dress, pulling
at the threads that shorten your hem.
Look at the scuffmarks on your new
white boots. You refuse to ever be
the girl next door. Still you blame
the cracks in the walls for the poison
you spray every morning then wait
with the door open for it to dry. It's true
what they say: the men who've known
you before see you differently now.
A man asks, _What's your name_? and you're
the one to respond, _It doesn't matter_.
They don't want to remember
and you'll admit it: I've occupied
the darkest times in their lives.
## Sapwood, Heartwood, Pith
I went on my own for a while. You couldn't
find me and you didn't dare try. On my way
to school one day I stopped and watched
two workers cut a large oak into six-foot logs.
A rope hoisted one man around his waist;
the other directed from below. I stood
close enough to smell their chewing tobacco
and the sawdust that flaked the heap of branches
with leaves green as bottle shards. I held my breath
as each log thudded to the ground, their hits
colossal and earth-strong. Each time, I clenched
my fists and jaw. That whole day I felt sad
for useless things and how you taught me to see
my own body—parceled, some parts
that you desired, some parts that you jeered,
and how I couldn't un-see myself that way,
even after years. But then the next day I walked
the same path and there it was, ancient disease
darkly worming the stump's rough-hewn maw.
I tell you, after you left me with something
like sickness, something like shame,
I waited for so long to feel whole again
while you went on not worrying, not thinking,
really, at all, and I'm not supposed
to write about what happened.
In my house where I live alone,
I touch my foot to the floorboard
where the wood has rotted away.
I say: here is where I will try
and here is where I will try again.
## You apologized to me in the passive voice
and when I pointed it out, you scrubbed me
from your story. I'm not perfect, but at least I try
to always make my voice direct: subject, verb,
object. One of the earliest lessons
in language, owning your shit. Even animals
know that. Sheep can't eat copper
but they will. Chickens can't eat apple cores,
potatoes, or tomatoes. A flash of blood winks
in the yolk, grows cold, and won't ever
become an embryo. Move on. Donkeys protect
the herd, will kill a coyote or a dog. Deer lie
down like dogs (have you ever noticed?)
and a donkey brays like a porch swing swung
too hard. When we were together, you did
things to me but never really saw me.
So you did things to yourself, your past
and your future, to another girl with brighter hair,
another girl who wasn't there. Sheep can read
faces better than you. They know when it's time
to feed or slaughter. A lamb bucked and ran
up the hillside, a pail's metal handle looped
around its neck. I tried to chase it back down
but there was no give, no take. And I've read
that green eyes are the rarest, but people who look
at me directly are pretty rare, too. If you scare
away the herd, the lone sheep or goat will follow.
If you truly want to become a better person,
you should at least drink a different beer
than the one you had before. I'm so sick of you,
I don't even like you, and we had to off
the rooster because it hurt three hens so bad,
their feathers torn out from their backsides,
pimpled skin glowing through. And now they have
to walk around opened up like that while the rest
of the flock stays whole. I remember how
you said to me, _I'm sorry you got hurt. I'm sorry_ ,
_but the damage is done_. The damage is always done.
I try my best to ignore you, but here you are
again, coming back to do me in. Old like nature.
Old like a country song. My heart's only a skin
that's bitten through, a young peach I mistook
for a nectarine. Sometimes animals sound
like they are speaking or rather mocking
the way people make animal sounds. Let's see
how you like it: we'll go on and have it your way.
You're active, and I'm passive. I tried
to call you, but my phone was filled with water.
I tried to talk to you, but I was choked.
## He Hates What I Do
I am insufferably cruel.
I stand in the yard, bruise-legged
with wet shoes.
The rough man at the bar—
he hates what I do.
Smell the sick human of his skin.
Taste the sour winter whiskering his breath.
He lectures me and I see tin, gray fur,
a splotch of beans on the floor. _I am better_
_than McCarthy and Faulkner. No one is greater_
_than Nabokov_. He curses. _Poetry is shit compared to prose_.
I laugh the laugh that hurts the next day.
I laugh the laugh of a cocktease, not a whore.
*
I hollowed out for a man when I was twenty-four.
*
The one I loved mocked my hipbones
jutting up like knives. He'd had
a married woman once ( _a very poor girl_ )
then his boss, two students ( _they were sort-of former_ ).
He'd won a steak dinner
( _a gentleman's agreement_ )
for which housemate would be first to fuck
their landlord's brown-skinned daughter,
he bragged to me as he threw a dart
against a door. How I hate the chiseled meat
with its tar gristle and lines of fat gleaming
like worms. I don't want to forget
the deer carcass I found decaying
in a creek bed, her ribcage tacky with blood.
*
I had to hollow out to echo.
I had to squeeze my heart to become strong.
*
_You're wasting your talents on poetry_ , the rough man
slurs to me at the bar. He's wanted me
for three years now. I feel his hands
root around in my dress. He's lost something
inside me. He doesn't get
what he doesn't wait for. How forlorn
it is to stare into night's bald eye and be alone.
He hates the selfish thing that I do.
*
Hens peck seeds from my outstretched hand.
My legs are painted black and yellow—
my shins took the brunt
when I went down. I'm the lucky one. Stairs
have no words. I was not pushed. I fell.
The hens eat with a light touch, soft pinches
that do not hurt me after all.
## Little Birds
What is it about the December woods
that makes me lonesome? The sky glows
a pale maple syrup, parceled in stained
glass between branches. My breath snags
on cold as I run through. Every house I pass
slumbers peaceful as a cat, every lamp
murmuring _fine_ , _fine_. Frozen creeks
lumber under bridges, the long back
of some forsaken creature who'd rather not
weigh in. It's hard to be both large and shy.
And my heart is a spilled
glass of wine—blot it with paper towels,
throw salt on it, sponge it with shampoo, dab it
with white vinegar. I cannot reason the silver
linings—aluminum foil scrunched
around saplings' roots to keep away
slugs and sunscald—the snap of a book closing, the aloof
space in bed, the words I dread that come
after a chair's resolved groan:
_I need to talk to you_. I turn at a fluorescent light
stamped into a wooden post. Truly, I cannot fathom
the dodged bullets that dotted deer hides
instead of mine. Finches charm dead
leaves, rucked as gloveless hands. Tonight, let me
gather everything dead or dying in my arms. The sky
is blank as a pie plate, yet it will not snow.
## [The Waiting Girl Feels a Pulsing
Where There Is No Child](05_Contents.xhtml#r37)
When they say I am nothing like my family,
everywhere I begin to see my son.
The days loll into clammy pages, stain
my hands indigo. This is what cleaning
affords me: a soup lid in a trash bag slices
my leg. Blood smears dirty plastic.
I smash the coffee pot on the counter.
If glass is bound to break, then why
am I surprised when shards glitter
with dust and prick my bare soles?
In the scorched bar yard a long-legged
rabbit darts between pizza crusts
and cigarette butts. The sky churns
pewter and olive, drips muddy into day-
drinks. Little bunny, this fairy tale
is an ugly one. There won't be bonnets,
no nursery painted cream and blue.
Those that I have loved, they've made
themselves strangers to me now. My child,
wait for me in darkness. These godless
men are not worthy of your name.
## Your Name in My Boot
I put your name in my boot. I wrote it
on a piece of paper and ground it with my heel.
Spider-bitten, knee-bruised,
the year began. I looked up and found
the Milky Way, a ghost pinned
to the ceiling, its eyes as sure as stars.
I spilled ink on your fortune
and then you appeared at my door,
a bodiless wind. In what's left of the wild,
there grows a weed called doll's eyes.
A man asked, _Have you been here this whole time_?
I was invisible until I drew on my eyes.
So is it you or me who cannot see clearly?
I picked the weed named Adam and Eve.
Blue-eyed grass, blue-eyed Mary, ghost plant,
woman's tobacco, touch-me-not, false
sunflower also called yellow star—you blooms
are all a tease. I was born for root-food,
corn-planting, hot sun, short harvest, red plum.
A man asked, _Did that all really happen to you_?
I thought you were a good man, but you have
less and less to say. I traced your name
in the river. I drew your name on the wall.
Northern fires stain the pink moon red,
and the year begins its ancient end—
leaf-falling, deer-mating, fur-pelts, wolves
revised to coyotes I only see beside the road.
Sometimes they're dead, but sometimes
they're a flash of copper and gold.
I froze your name into the snow.
My hips ached. My leather split, and my heel
caught a slick wooden stair. Blood
matted my hair, and I don't remember
leaving. Blood moon, hunter's moon,
oak's moon, cold moon, long night's moon—
it's been the longest time here.
A man asked, _Why are you always alone_?
I put your name in my boot as I walk away.
You'll never forget my own.
## Still Haunted
A stranger to this winter city flushed blue
with fallen snow, I touch metal to make light
to walk until I think I break a train. A rough
voice jeers at me, _What the hell is wrong with you_?
Mathematics informs me I no longer want
you. Still, I remain the one who slinks
down to a wet and wretched creature
when I see your face: still, my voice
catches in my throat hard as a coin. Still,
my hands shake pallid as a specter even
after two years without your body near.
On the north side of town, littlenecks
coddle in warm butter while girls in black
boots tromp through curbside pots of slush
and gasoline. Over wine sparkling in tavern
firelight, my most logical friend tells me
how she once met a ghost on the stairs
of the old house we shared years ago,
how she first heard the footsteps of nothing
at all, then felt a gust of cold air as it passed
her by. Her eyes fearful, she did not deserve
this witching hour affliction from a spirit
she has since learned was a girl scorned
by her family for running off with an artist,
her unseen phantom only women now
can sense, and in a house of eight women,
only my friend. Why wasn't it lovelorn me?
New city, new lovers, nothing matters. Spirit,
on a snow-laced street, you are the lamppost
caught wild in flame. This you still know:
if I sensed your footsteps racing towards me,
if I felt your rush and if I heard you call out
my name, Spirit, still, I would turn.
## [In Defense of Not Getting Over It,
at Least Not for Now](05_Contents.xhtml#r40)
An artist friend told me that her favorite subjects to draw
were those she could not depict all at once: the drunken geometry
of an old house, a majestic pine, our town's river in its lawful
currents—the sky reflected in water the dull-bright colors of money,
but pure. She came back every day to a new vantage point, a new angle,
and that became her blessing and her curse, the way she must return.
Summer, I washed dishes to pay a fee. There was one pot I'd wrangle
with for hours—the bane of my evenings, a Herculean feat earned
by its height that reached my knees; its girth, my embrace could not hold.
Someone always burned sauce at the bottom, a thick char that gleamed
despite a soak with dish soap, a dusting of baking soda, a controlled
boil on the stove. There's beauty in it, though: the slow work of dreams
unreached, revision's stench and steam. It's all meant to train one's heart.
I have learned how to build something first by how to take it apart.
## Salt Creek, Missouri
Sometimes a ghost is not a ghost
but a woman who hates me a thousand
miles eastward. There is nothing
I can do here but palm the weight
of hailstones, wear the green rain
as a veil. Two hens named for country
stars attack a hen named for a queen.
There is no one I can ever be
to her than the brat her husband loved
first. Trees wring bare hands: winter.
Dog-tooth violets flirt with soft earth:
nearly. I once had a plan to forget him.
How futile our backstory, how
precious her grudge. Today, I drive out
to the field with five white horses
just to make sure they haven't been sold.
## Midwestern Litany
Wouldn't give my heart to the railroad.
Wouldn't give my heart to an old dog.
Not for rusted lace or seed pearls.
Not for bone china or barely dented silver.
Not even this old house, the walls flaking off like hardened sugar.
Wouldn't give my heart to hog corn.
Not for the promise of dyed flowers.
Wouldn't give it to the bridge over the river.
And no, not for penicillin.
Not for coffee hot in Styrofoam or the smell of burning leaves.
Not for dark meat.
Not for pelicans in a soybean field stubbled by winter
that follow the river currents north, then south once more.
Not for baked apples in nutmeg and butter.
Not in this land exhausted of color.
Not for beer the color of dirt, for dirt the color of blood.
Wouldn't give my heart to the ghosts of deer
or for violence in Illinois—I saw that doe throw her body
in front of an eighteen-wheeler—she died in a pinwheel
explosion—she was alive and leaping
and then she wasn't—she didn't even look.
And it's not easy to hide in a flattened land
with hearts beating and stopping on every billboard.
Wouldn't give my heart for the threat of hell.
Not for dead leaves curled like fists on the vine.
Not for my body within my body.
Not for rain that turns the fields to ice, my car to ice, the trees—
the highway
that would carry me to the mountains,
to the swamps, the canyons, the sea—if I let it.
Wouldn't give my heart to the telephone wires.
Wouldn't even give my heart to the sun
melting the whitened fields and revealing what is blue,
so why would you think
why would you think
why would you think (I didn't even look)
that I would give my heart to you?
## Site Fidelity
I never heard them until I did.
Coyotes slunk out
from cornfields and creek beds
their haunches a gold-brown
rumor in tall grass. The pup
I found was an offering
for cluster flies, scarcely breathing
in catalpa's shade.
A starved creature
begins to resemble its prey: long
rabbit ears, crooked
fawn legs. These days
my eyes are painted in shadows.
I have stared into the black
river. I have slept
beneath the old moon and woken
to the ghost of a man
who turns from me,
fading back into earth's hard soil.
These days I walk
around with a secret
deeper than skin: a bitten cheek,
a burnt tongue.
By autumn's pale sun,
the coyotes return to the fields.
Through the day
through the night
I can still hear their mournful
howls. Believe me:
I tried to save a life.
I carried a pan of water.
I shooed away the flies.
## Play the Fool
The road was red again. I trespassed
and a horse with a rope halter
but no saddle followed me down
the driveway as would a dog.
I picked up my gait, started to run.
I didn't want it to get too close,
didn't want to lead it to the road.
Once somebody called me
a _shit-magnet_ , though lovingly,
and I couldn't disagree. I am prone
to self-injury, predisposed
to locate the most trifling man
in any room. Even when I try
not to host a party I host the party.
And three years later I think
I have finally figured out why
you didn't want me in the way
I wanted you. You asked me
if I needed to hear the reason
and I responded, _Don't tell me_ ,
because I knew one day I would
find it in a place I shouldn't be.
While foraging for blackberries
in the Tennessee woods, I came
across an abandoned shed rusted
olive, its walls stabbed through
with old tools, and on its deck
out front stood two goats
expectedly as if they were my
neighbors and I had stopped
by to borrow an axe or some eggs.
Animals have no reason to be kind
and really, neither did you. Ducks
will drown a chicken in its own pen.
A ewe will head-butt her own lamb
to feed. Don't mistake obedience
for a gentle heart. I understand
now what you wanted. The horse
reached the fence post by the road.
I hid behind the tulip poplar
and when it couldn't see me any
longer, it swung around as though
in two parts: first the gray neck,
then the dappled torso, and turned
back up the driveway to its home,
despondently, but with purpose.
## The Encounter
Late in the day, late into spring, the sky a lilac gauze
over the last flowering trees—their petals curled
as if singed, the new green pushing beyond—
I turned on the path and, for a moment, could not discern
what I'd just come upon: three surprised faces, a trio
of baby possums latched to their mother's back, her head
down, drinking from a rain puddle, then at once keen
and lifted, teeth bared, emitting a low snarl. Wide-eyed,
their ears like button mushrooms, they were nearly
too big to all fit, the third one trailing, barely hanging
on. Frogs sang from their pond. I was not fearful,
though she growled at me as I passed by, brazen
creature who refused to shrink back into the thrum
of new evening, into the wild from where she'd come.
## Vow
I will mistake the plop of dew on canvas
for raindrops for footfall. It will be dawn
before dawn. The sky will be no color. The birds'
wings will be pinned in trees, their warbles
unplanned, bright flares, not yet song. I will wake
with a start. I will walk to the river, its water
like watered milk, the mist thick as a layering
of hands, the fishermen in prayer. I will wait there.
It will be a bouquet of dried violets, ornaments
carved from green soap, the blueberry bush
in my childhood backyard, its wire gown glinting
in the sun and the deer gazing on, the berries
before they are berries, pink as rosebuds,
the shells buried beneath sand in hurricanes
that have turned over the sea for centuries,
for a thousand years, the grains of sand colors
beyond colors, their beauty hard-won.
It will be written in salt lines. It will be as old
as every omen, as the music of dying stars,
the soft wings unfurling from a common bird
in the known purpose of her song. I have worlds
of things to tell you. I will give you my word.
## Acknowledgments
Many thanks to the editors of the journals where these poems first appeared, some in slightly different versions:
_Academy of American Poets_ : "Midwestern Litany"
_The Adroit Journal_ : "Salt Creek, Missouri," "The month after the cruelest month"
_Blackbird_ : "The one drag show in town is closing"
_Catch Up_ : "If That Isn't a Sign Then God Knows What Is"
_The Columbia Review_ : "Little Birds"
_Connotation Press: An Online Artifact_ : "Science of Uncertainty," "The Waiting Girl Sobers Up in the Same Room as Before"
_Copper Nickel_ : "In Defense of Not Getting Over It, At Least Not for Now"
_Crazyhorse_ : "If I Start Talking About It Now I Won't Stop Hollering," "Prudesville, USA"
_DIAGRAM_ : "Thief Hallow Branch, Arkansas"
_Ecotone_ : "After Yet Another Man Tells Me _I Don't Want to Hurt You_ "
_Fugue_ : "Vow"
_Grist_ : "Egg and Ash," "White and Rain"
_Gulf Coast_ : "Bastille Day"
_Harpur Palate_ : "Disassociation"
_Memorious_ : "Hypochondria"
_Meridian_ : "Every Time an Ex-Boyfriend Calls You _Irrational_ , a Woodchuck Gains its Wings"
_Mid-American Review_ : "Dock and Withers," "Site Fidelity," "Survival Tactics"
_Midway Journal_ : "Hideaway," "The Waiting Girl Feels a Pulsing Where There Is No Child"
_Nashville Review_ : "A Beautiful Night for a Biscuit Truck"
_North American Review_ : "Hallucinate the House, Hallucinate the Woods"
_Paper Darts_ : "He Vows to Follow Me Even After He Leaves Me"
_Phoebe_ : "Fourteen"
_Sonora Review_ : "Self-Portrait with Brown Recluse"
_Third Coast_ : "Your Name in My Boot"
_Thrush Poetry Journal_ : "The Waiting Girl Sweats Out a Fever"
_Tinderbox Poetry Journal_ : "He Takes Her," "Still Haunted"
_Tupelo Quarterly_ : "The Encounter"
_Vox Magazine_ : "Show-Me State"
_Washington Square Review_ : "Play the Fool," "You apologized to me in the passive voice"
"Incident in Northwest Missouri" first appeared on _Thethe Poetry_ as part of the "Political Punch" poetry series and was then reprinted in the anthology _Political Punch: Contemporary Poems on the Politics of Identity_ , edited by Erin Elizabeth Smith and Fox Frazier-Foely (Sundress Publications, 2016).
*
I am grateful for the mentorship and wisdom of my doctoral committee, Aliki Barnstone, Cornelius Eady, Alex Socarides, and Mary Jo Neitz, and for the gifts of time and beauty from the residencies at the Vermont Studio Center and the Sundress Academy for the Arts at Firefly Farms, where several of these poems were written.
Thank you to my dear friends and readers Rachel Inez Lane, Brandi Nicole Martin, Greg Allendorf, Heather McGuire, J.D. Smith, Avni Vyas, Candice Wuehle, Amanda Bales, Melissa Range, Austin Segrest, and Vedran Husić, as well as Alyson Thiel, whose art always has eyes and wings. To my Reynolds kin for the brightest fire and friendship. And to my family, Nancy and Scott Barngrover, Kristy and James Clear, and to my grandparents for their unwavering love and support.
To Mary Biddinger, Amy Freels, Jon Miller, and the University of Akron Press: endless gratitude for your exquisite care, your vision, and your steadfast declaration that, no matter what, "Poetry Lives!"
Finally, this book is dedicated to the memories of Monica A. Hand and Naira Kuzmich. As far as being brazen goes, they taught me everything I need to know.
Photo: Heather McGuire
Anne Barngrover is the author of _Yell Hound Blues_ (Shipwreckt Books, 2013) and coauthor, with poet Avni Vyas, of the chapbook _Candy in Our Brains_ (CutBank, 2014). Her poems have appeared in _Ecotone, Crazyhorse, Copper Nickel, Indiana Review_ , and others. Anne earned her MFA from Florida State University and her PhD from University of Missouri. She is an assistant professor of creative writing at Saint Leo University and lives in Tampa, Florida.
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namespace StockSharp.Messages
{
using System;
/// <summary>
/// The interface describing an message with <see cref="IsSubscribe"/> property.
/// </summary>
public interface ISubscriptionMessage : ITransactionIdMessage, IOriginalTransactionIdMessage
{
/// <summary>
/// Message contains fields with non default values.
/// </summary>
bool FilterEnabled { get; }
/// <summary>
/// Start date, from which data needs to be retrieved.
/// </summary>
DateTimeOffset? From { get; set; }
/// <summary>
/// End date, until which data needs to be retrieved.
/// </summary>
DateTimeOffset? To { get; set; }
/// <summary>
/// The message is subscription.
/// </summary>
bool IsSubscribe { get; set; }
/// <summary>
/// Skip count.
/// </summary>
long? Skip { get; set; }
/// <summary>
/// Max count.
/// </summary>
long? Count { get; set; }
/// <summary>
/// Data type info.
/// </summary>
DataType DataType { get; }
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/blogs.mathworks.com\/cleve\/2014\/04\/01\/reverse-singular-value-decomposition\/","text":"# Reverse Singular Value Decomposition2\n\nPosted by Cleve Moler,\n\nEmploying a factorization based on the least significant singular values provides a matrix approximation with many surprisingly useful properties. This Reverse Singular Value Decomposition, RSVD, is also referred to as Subordinate Component Analysis, SCA, to distinguish it from Principal Component Analysis.\n\n### Contents\n\n#### RSVD\n\nThe Singular Value Decomposition of a matrix $A$ is computed by\n [U,S,V] = svd(A);\nThis generates two orthogonal matrices $U$ and $V$ and a diagonal matrix $S$ with diagonal elements $\\sigma_k$ that provide the expansion $$A = \\sigma_1 E_1 + \\sigma_2 E_2 + ... + \\sigma_n E_n$$ where $E_k$ is the rank one outer product $$E_k = U(:,k) V(:,k)'$$ Traditionally, the singular values are arranged in descending order. In contrast, the Reverse Singular Value Decomposition Approximation of rank $r$ is obtained by arranging the singular values in ascending order, $$0 \\le \\sigma_1 \\le \\sigma_2 \\le ...$$ and then using the first $r$ terms from the expansion. Here is a function that computes the RSVD for square or rectangular matrices with at least as many rows as columns.\n type rsvd\n\nfunction X = rsvd(A,r)\n% RSVD Approximation by the Reverse Singular Value Decomposition\n% rsvd(A,r) approximates A by a matrix of rank r obtained from\n% the r least significant singular values in ascending order.\n\n[m,n] = size(A);\n[U,S,V] = svd(A,'econ');\nk = n:-1:n-r+1;\nX = U(:,k)*S(k,k)*V(:,k)';\n\n\n#### Roundoff Error\n\nIn certain situations, the RSVD can reduce or even eliminate roundoff error. For example, according to its help entry the elmat function hilb attempts to compute\n hilb(N) is the N by N matrix with elements 1\/(i+j-1)\nBut the function can only succeed to within roundoff error. Its results are binary floating point numbers approximating the reciprocals of integers described in the help entry. Here is the printed output with n = 5 and the default format short.\n format short\nH = hilb(5)\n\nH =\n\n1.0000 0.5000 0.3333 0.2500 0.2000\n0.5000 0.3333 0.2500 0.2000 0.1667\n0.3333 0.2500 0.2000 0.1667 0.1429\n0.2500 0.2000 0.1667 0.1429 0.1250\n0.2000 0.1667 0.1429 0.1250 0.1111\n\n\nWe are seeing the effects of the output conversion as well as the underlying binary approximation. Perhaps surprisingly, the reverse singular value decomposition can eliminate both sources of error and produce the exact rational entries of the theoretical Hilbert matrix.\n format rat\nH = rsvd(H,5)\n\nH =\n\n1 1\/2 1\/3 1\/4 1\/5\n1\/2 1\/3 1\/4 1\/5 1\/6\n1\/3 1\/4 1\/5 1\/6 1\/7\n1\/4 1\/5 1\/6 1\/7 1\/8\n1\/5 1\/6 1\/7 1\/8 1\/9\n\n\n\n#### Text Processing\n\nThe RSVD is capable of uncovering spelling and typographical errors in text. My web site for Numerical Computing with MATLAB has a file with the text of Lincoln's Gettysburg Address. gettysburg.txt I've used this file for years whenever I want to experiment with text processing. You can download the file and then use the MATLAB data import wizard with column delimeters set to spaces, commas and periods to produce a cell array, gettysburg, of individual words. There are 278 words. The longest word has 11 characters. So\n A = cell2mat(gettysburg)\nconverts the cell array of strings to a 278-by-11 matrix of doubles. It turns out that an RSVD approximation of rank nine finds three spelling errors in the original text.\n B = rsvd(A,9);\nk = find(sum(A,2)~=sum(B,2))\ndisp([char(A(k,:)) char(B(k,:))])\n weather whether\nconsicrated consecrated\ngovenment government\nIn all the years that I have been using this data, I have never noticed these errors.\n\n#### Image Processing\n\nWe have also found that the RSVD is capable of softening the appearance of aggression in photographs. A matrix is obtained from the JPEG image by stacking the RGB components vertically. Roughly 90% of the small singular values then produce a pleasant result.\n RGB = imread('creature.jpg');\nA = [RGB(:,:,1); RGB(:,:,2); RGB(:,:,3)];\n[m,n] = size(A);\nB = rsvd(A,ceil(0.90*n));\nm = m\/3;\nC = cat(3,B(1:m,:),B(m+1:2*m,:),B(2*m+1:3*m,:))\nimage(C)\n\nGet the MATLAB code Published with MATLAB\u00ae R2014a\n\n43 views (last 30 days) \u00a0| |","date":"2019-11-17 10:08:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5522743463516235, \"perplexity\": 1089.1653098413585}, \"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\/1573496668910.63\/warc\/CC-MAIN-20191117091944-20191117115944-00095.warc.gz\"}"}
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\section{Introduction}
\label{intro}
The quantization of the gravitational force is an outstanding open problem in theoretical high-energy physics.
In this context, Asymptotic Safety, first proposed by Weinberg \cite{Weinberg:1980gg,Weinproc1} and recently reviewed in \cite{Niedermaier:2006wt,Codello:2008vh,Litim:2011cp,Percacci:2011fr,Reuter:2012id,Reuter:2012xf,Nagy:2012ef}, may provide an attractive mechanism for obtaining a consistent and predictive quantum theory for gravity within the framework of quantum field theory. Given that asymptotic safety is a rather general concept whose applicability is not limited to the gravitational interactions it was soon realized that this mechanism may also be operative once the gravitational degrees of freedom are supplemented by matter fields \cite{Percacci:2002ie,Percacci:2003jz}. In this way the asymptotic safety scenario may also provide a framework for unifying all fundamental forces and matter fields populating the universe within a single quantum field theory.
The key ingredient underlying the asymptotic safety mechanism is a renormalization group (RG) fixed point which controls the behavior of the theory at ultra-high energies. The fixed point then ensures that all dimensionless coupling constants remain finite, preventing the occurrence of unphysical UV divergences. Provided that it also comes with a finite number of eigendirections along which the flow is dragged into the fixed point for increasing energy this construction has the same predictive power as a perturbatively renormalizable quantum field theory. For the case where
the gravitational degrees of freedom are encoded in fluctuations of the (Euclidean) spacetime metric, defining the so-called metric approach to Asymptotic Safety, the
existence of a suitable non-Gaussian fixed point (NGFP) has been demonstrated in a vast number of works including the projection of the gravitational RG flow onto the Einstein-Hilbert action \cite{Reuter:1996cp,Lauscher:2001ya,Reuter:2001ag,Litim:2003vp,Fischer:2006fz,Donkin:2012ud,Nagy:2013hka},
$f(R)$-type actions build from finite polynomials constructed from the curvature scalar $R$ \cite{Lauscher:2002sq,Codello:2007bd,Machado:2007ea,Codello:2008vh,Falls:2013bv,Falls:2014tra,Demmel:2014hla,Falls:2016msz}, and including the square of the Weyl tensor \cite{Codello:2006in,Benedetti:2009rx,Benedetti:2009gn,Groh:2011vn}. Moreover, Ref.\ \cite{Gies:2016con} established that this NGFP also persists once the perturbative two-loop counterterm found by Goroff and Sagnotti \cite{Goroff:1985th} is included in the projection. A complementary class of approximations which also keeps track of the fluctuation fields, corroborates this picture \cite{Manrique:2009uh,Manrique:2010mq,Manrique:2010am,Christiansen:2012rx,Codello:2013fpa,Christiansen:2014raa,Becker:2014qya,Christiansen:2015rva}. Moreover, approximations including an infinite number of scale-dependent coupling constants are currently under development \cite{Codello:2010mj,Benedetti:2012dx,Demmel:2012ub,Dietz:2012ic,Demmel:2013myx,Dietz:2013sba,Benedetti:2013nya,Demmel:2014sga,Percacci:2015wwa,Borchardt:2015rxa,Demmel:2015oqa,Ohta:2015efa,Ohta:2015fcu,Henz:2016aoh,Labus:2016lkh,Dietz:2016gzg}. Starting from \cite{Percacci:2002ie,Percacci:2003jz} it has also been shown that the Asymptotic Safety mechanism may also play a key role in the high-energy completion of a large class of gravity-matter models \cite{Zanusso:2009bs,Vacca:2010mj,Harst:2011zx,Eichhorn:2011pc,Dona:2013qba,Dona:2014pla,Meibohm:2015twa,Oda:2015sma,Dona:2015tnf,Meibohm:2016mkp,Eichhorn:2016esv,Henz:2016aoh}.
An open question in the metric approach to Asymptotic Safety is the so-called ``problem of time'' (see \cite{Isham:1992ms} for review). While quantum mechanics and quantum field theory in a fixed Minkowski background possess a natural notion of time, the notion of time in a dynamical (and possibly fluctuating) spacetime becomes rather involved. A way to address this question in general relativity is the Arnowitt-Deser-Misner (ADM)-formalism. In this case the spacetime metric is decomposed into a Lapse function $N$, a shift vector $N_i$, and a metric $\sigma_{ij}$ which measures distances on the spatial slices $\Sigma_t$ defined as hypersurfaces where the time-variable $t$ is constant. The foliation structure then leads to natural time-direction.
A functional renormalization group equation (FRGE) for the effective average action \cite{Wetterich:1992yh,Morris:1993qb,Reuter:1993kw,Reuter:1996cp} tailored to the ADM-formalism has been constructed in \cite{Manrique:2011jc,Rechenberger:2012dt}. A first evaluation of the resulting RG flow within the Matsubara-formalism provided strong indications that the UV fixed point underlying the Asymptotic Safety program is robust under a change from Euclidean to Lorentzian signature \cite{Manrique:2011jc,Rechenberger:2012dt}. Moreover, the FRGE in ADM-variables provides a powerful tool for studying RG flows within Ho\v{r}ava-Lifshitz gravity \cite{Horava:2009uw} since it allows for anisotropic scaling relations between spatial and time-directions by including higher-derivative intrinsic curvature terms \cite{Contillo:2013fua,D'Odorico:2014iha,D'Odorico:2015yaa}.
The purpose of the present work is twofold. Firstly, it provides all technical details underlying the construction of RG flows from the ADM-formalism on a flat Friedmann-Robertson-Walker background studied in \cite{Biemans:2016rvp}. Here the key ingredient is the gauge-fixing scheme presented in Sect.\ \ref{sect.3a} which leads to regular propagators for \emph{all} component fields including the lapse function and the shift vector thereby avoiding the pathologies encountered in temporal gauge. Secondly, we initiate the study of matter effects in this setting, by computing the scale-dependence of Newton's constant $G_k$ and the cosmological constant $\Lambda_k$ for foliated gravity-matter systems containing an arbitrary number of minimally coupled scalars, $N_S$, vector fields $N_V$, and Dirac fermions $N_D$. The inclusion of the matter fields leads to a two-parameter deformation of the beta functions controlling the flow of $G_k$ and $\Lambda_k$ in the pure gravity case. Analyzing the beta functions of the gravity-matter systems utilizing these deformation parameters allows classifying their fixed point structure of the model independently of a specific choice of regulator in the matter sector. The fixed point structure found for a specific gravity-matter model can then determined by evaluating the map relating its field content to the deformation parameters. In particular, we find that the matter content of the standard model of particle physics as well as many of its phenomenologically motivated extensions are located in areas which give rise to a single UV fixed point with real critical exponents. These findings provide a first indication that the asymptotic safety mechanism encountered in the case of pure gravity may carry over to the case of gravity-matter models with a realistic matter field content, also in the case where spacetime is equipped with a foliation structure.
This work is organized as follows. Sect.\ \ref{sect.2} reviews the ADM-formalism and the construction of the corresponding FRGE \cite{Manrique:2011jc,Rechenberger:2012dt}. Our ansatz for the effective average action and the evaluation of the resulting RG flow on a (Euclidean) Friedmann-Robertson-Walker (FRW) background is presented in Sect.\ \ref{sect.3}. In particular, Sect.\ \ref{sect.3a} summarizes the construction of our novel gauge-fixing scheme leading to regular propagators for all component fields. Limiting the analysis to $D=3+1$ spacetime dimensions, the flow equation in the context of pure gravity are analyzed in Sect.\ \ref{sect.42} while the fixed point structure appearing in gravity-matter systems is discussed in Sect.\ \ref{sect.43}.
We provide a short summary and discussion of our findings in Sect.\ \ref{sect.7}. Technical details about the background geometry, the construction of the Hessians in a flat Friedmann-Robertson-Walker background, and the evaluation of the operator traces entering the FRGE are relegated to the App.\ \ref{App.A}, App.\ \ref{App.B}, and App.\ \ref{App.C}, respectively.
\section{Renormalization group flows on foliated spacetimes}
\label{sect.2}
The functional renormalization group equation on foliated spacetimes has been constructed in \cite{Manrique:2011jc,Rechenberger:2012dt} and we review the formalism in the following section. For a pedagogical introduction to the $3+1$-formalism the reader is referred to \cite{Gourgoulhon:2007ue}.
\subsection{Arnowitt-Deser-Misner decomposition of spacetime}
\label{sect.2a}
We start from a $D$-dimensional Euclidean manifold ${\cal M}$ with metric $\gamma_{\mu\nu}$, carrying coordinates $x^\alpha$. In order to be able to perform a Wick rotation to Lorentzian signature, we define a time function $\tau(x)$ which assigns a specific time $\tau$ to each spacetime point $x$. This can be used to decompose ${\cal M}$ into a stack of spatial slices $\Sigma_{\tau_i} \equiv \left\{ x: \tau(x) = \tau_i \right\}$ encompassing all points $x$ with the same value of the ``time-coordinate'' $\tau_i$. The gradient of the time function $\partial_\mu \tau$ can be used to define a vector $n^\mu$ normal to the spatial slices, $n_\mu \equiv N \partial_\mu \tau$ where the lapse function $N(\tau,y^i)$ is used to ensures the normalization $\gamma_{\mu\nu} \, n^\mu \, n^\nu = 1$. Furthermore, the gradient can be used to introduce a vector field $t^\mu$ satisfying $t^\mu \partial_\mu \tau = 1$. Denoting the coordinates on $\Sigma_\tau$ by $y^i$, $i = 1, \ldots, d$ the tangent space on a point in ${\cal M}$ can then be decomposed into the space tangent to $\Sigma_\tau$ and its complement. The corresponding basis vectors can be constructed from the Jacobians
\begin{equation}\label{proj1}
t^\mu = \left . \frac{\partial x^\mu}{\partial \tau} \right|_{y^i} \, , \qquad e_i{}^\mu = \left. \frac{\partial x^\mu}{\partial y^i} \right|_\tau \, .
\end{equation}
The normal vector then satisfies $\gamma_{\mu\nu} \, n^\mu \, e_i{}^\nu = 0$.
The spatial coordinate systems on neighboring spatial slices can be connected by constructing the integral curves $\gamma$ of $t^\mu$ and requiring that $y^i$ is constant along these curves. A priori $t^\mu$ is neither tangent nor orthogonal to the spatial slices. Using the Jacobians \eqref{proj1} it can be decomposed into its components normal and tangent to $\Sigma$
\begin{equation}\label{tdec}
t^\mu = N \, n^\mu + N^i \, e_i{}^\mu \, .
\end{equation}
where $N^i(\tau,y^i)$ is called the shift vector. Analogously, the coordinate one-forms transform according to
\begin{equation}
dx^\mu = t^\mu d\tau + e_i{}^\mu dy^i = N n^\mu d\tau + e_i{}^\mu \, (dy^i + N^i d\tau) \, .
\end{equation}
Defining the metric on the spatial slice $\sigma_{ij} = e_i{}^\mu \, e_j{}^\nu \, \gamma_{\mu\nu}$ the line-element $ds^2 = \gamma_{\mu\nu} \, dx^\mu dx^\nu$ written in terms of the ADM fields
takes the form
\begin{equation}\label{fol1}
ds^2 = \gamma_{\alpha\beta} \, dx^\alpha dx^\beta
= N^2 d\tau^2 + \sigma_{ij} \, (dy^i + N^i d \tau)
(dy^j + N^j d \tau) \, .
\end{equation}
Note that in this case the lapse function $N$, the shift vector $N^i$ and the induced metric on the spatial slices $\sigma_{ij}$ depend on the spacetime coordinates $(\tau, y^i)$.\footnote{This situation differs from projectable Ho\v{r}ava-Lifshitz gravity where $N$ is restricted to be a function of (Euclidean) time $\tau$ only.} In terms of metric components, the decomposition \eqref{fol1} implies
\begin{equation}\label{metcomp}
\gamma_{\alpha\beta} = \left(
\begin{array}{cc}
N^2 + N_i N^i \; \; & \; \; N_j \\
N_i & \sigma_{ij}
\end{array}
\right) \, , \qquad
\gamma^{\alpha\beta} = \left(
\begin{array}{cc}
\frac{1}{N^{2}} \; \; & \; \; - \frac{N^j}{ N^{2}} \\
- \frac{N^i}{ N^{2}} \; \; & \; \; \sigma^{ij} + \, \frac{ N^i \, N^j}{ N^{2}}
\end{array}
\right) \,
\end{equation}
where spatial indices $i,j$ are raised and lowered with the metric on the spatial slices.
An infinitesimal coordinate transformation $v^\alpha(\tau,y)$ acting on the metric can be expressed in terms of the Lie derivative ${\cal L}_v$
\begin{equation}\label{diffeo1}
\delta \gamma_{\alpha\beta} = {\cal L}_v \, \gamma_{\alpha\beta} \, .
\end{equation}
Decomposing
\begin{equation}\label{vdec}
v^\alpha = \left(f(\tau,y), \zeta^i(\tau,y)\right)
\end{equation}
into its temporal and spatial parts, the transformation \eqref{diffeo1} determines the transformation properties of the component fields under Diff(${\cal M}$)
\begin{equation}\label{eq:gaugeVariations}
\begin{split}
\delta N &= \partial_\tau (f N ) + \zeta^k \partial_k N - N N^i\partial_i f \, , \\
\delta N_i &= \partial_\tau( N_i f) + \zeta^k\partial_k N_i + N_k\partial_i\zeta^k
+ \sigma_{ki}\partial_\tau \zeta^k
+ N_k N^k\partial_i f + N^2\partial_i f \, , \\
\delta\sigma_{ij} &= f\partial_\tau \sigma_{ij} + \zeta^k\partial_k \sigma_{ij} + \sigma_{jk}\partial_i\zeta^k + \sigma_{ik}\partial_j\zeta^k + N_j\partial_i f + N_i\partial_j f \, .
\end{split}
\end{equation}
For completeness, we note
\begin{equation}\label{Nui}
\delta N^i = \partial_\tau(N^i f) + \zeta^j\partial_j N^i - N^j\partial_j \zeta^i + \partial_\tau \zeta^i - N^i N^j \partial_j f + N^2 \sigma^{ij}\partial_j f \, .
\end{equation}
Denoting expressions in Euclidean and Lorentzian signature by subscripts $E$ and $L$, the Wick rotation is implemented by
\begin{equation}\label{Wickrot}
\tau_E \rightarrow - i \tau_L \, , \qquad N^i_E \rightarrow i N^i_L \, .
\end{equation}
The (Euclidean) Einstein-Hilbert action written in ADM fields reads
\begin{equation}\label{EHaction}
S^{\rm EH} = \frac{1}{16 \pi G} \int d\tau d^dy \, N \sqrt{\sigma} \left[ K_{ij} \, {\cal G}^{ij,kl} \, K_{kl} - {}^{(d)}R + 2 \Lambda \right] \, .
\end{equation}
Here ${}^{(d)}R$ denotes the intrinsic curvature on the $d$-dimensional spatial slice,
\begin{equation}\label{Kext}
K_{ij} \equiv \frac{1}{2 N} \left( \partial_\tau \sigma_{ij} - D_i N_j - D_j N_i \right) \, , \quad K \equiv \sigma^{ij} K_{ij}
\end{equation}
are the extrinsic curvature and its trace, and $D_i$ denotes the covariant derivative constructed from $\sigma_{ij}$. The kinetic term is determined by the Wheeler-de Witt metric
\begin{equation}
{\cal G}^{ij,kl} \equiv \sigma^{ik} \, \sigma^{jl} - \lambda \, \sigma^{ij} \, \sigma^{kl} \, .
\end{equation}
The parameter $\lambda = 1$ is fixed by requiring invariance of the action with respect to Diff(${\cal M}$) and we adhere to this value for the rest of this work.
When studying the effects of matter fields in Sect.\ \ref{sect.43}, we supplement the gravitational action \eqref{EHaction} by $N_S$ scalar fields, $N_V$ abelian gauge fields and $N_D$ Dirac fields minimally coupled to gravity
\begin{equation}
S^{\rm matter} = S^{\rm scalar} + S^{\rm vector} + S^{\rm fermion} \, ,
\end{equation}
where
\begin{equation}\label{matter}
\begin{split}
S^{\rm scalar} = & \frac{1}{2} \sum_{i=1}^{N_S} \int d\tau d^dx N \sqrt{\sigma} \left[ \, \phi^i \, \Delta_0 \, \phi^i \right] \, , \\
S^{\rm vector} = & \frac{1}{4} \sum_{i=1}^{N_V} \int d\tau d^dx N \sqrt{\sigma} \left[ \, g^{\mu\nu} g^{\alpha\beta} F_{\mu\alpha}^i F_{\nu\beta}^i \right] \, + \frac{1}{2\xi} \sum_{i=1}^{N_V} \int d\tau d^dx \bar{N} \sqrt{\bar{\sigma}} \left[ \, \bar{g}^{\mu\nu} \bar{D}_\mu A_\nu^i \right]^2 \\
& + \sum_{i=1}^{N_V} \int d\tau d^dx \bar{N} \sqrt{\bar{\sigma}} \left[ \, \bar{C}^i \, \Delta_0 \, C^i \, \right] \, , \\
S^{\rm fermion} = & \, i \, \sum_{i=1}^{N_D} \int d\tau d^dx N \sqrt{\sigma} \left[ \bar{\psi}^i \, \slashed{\nabla} \, \psi^i \right] \, .
\end{split}
\end{equation}
The summation index $i$ runs over the matter species and we adopt Feynman gauge setting $\xi=1$. In the context of Asymptotic Safety, matter sectors of this type have been discussed in the context of the covariant approach in \cite{Percacci:2002ie,Percacci:2003jz} with extensions considered recently in \cite{Dona:2013qba,Labus:2015ska,Dona:2015tnf,Meibohm:2015twa}. In particular, our treatment of the Dirac fermions follows \cite{Dona:2012am,Dona:2013qba}.
All matter actions are readily converted to by using the projector \eqref{proj1}. In order to retain compact expressions, we refrain from giving this decomposition explicitly, though.
\subsection{Functional renormalization group equation}
\label{sect.2b}
The first step in deriving the FRGE for the effective average action $\Gamma_k$ \cite{Wetterich:1992yh,Morris:1993qb,Reuter:1993kw,Reuter:1996cp}
specifies the field content of the model. For foliated spacetimes, it is natural to encode the gravitational degrees of freedom in terms of the ADM-fields $\{N, N_i, \sigma_{ij}\}$. Additional matter degrees of freedom
are easily incorporated by including additional fields in the construction.
The construction of $\Gamma_k$ makes manifest use of the background field method. Following \cite{Rechenberger:2012dt} we use a linear split of the ADM fields into background fields (marked with an bar) and fluctuations (indicated by a hat)\footnote{Strictly speaking, the fields appearing in the effective average action are the vacuum expectation values of the classical fields introduced in the previous subsection. In order to keep our notation light, we use the same notation for both fields, expecting that the precise meaning is clear from the context.}
\begin{equation}\label{linearsplit}
N = \bar{N} + \hat{N} \, , \qquad N_i = \bar{N}_i + \hat{N}_i \, , \qquad \sigma_{ij} = \bar{\sigma}_{ij} + \hat{\sigma}_{ij} \, .
\end{equation}
Conveniently, we will denote the sets of physical fields, background fields and fluctuations by $\chi$, $\bar{\chi}$, and $\hat{\chi}$, respectively. I.e., $\chi = \{N, N_i, \sigma_{ij}, \ldots \}$ where the dots indicate ghost fields and potentially additional matter fields.
The effective average action is then obtained in the usual way. Starting from a generic diffeomorphism invariant action $S^{\rm grav}[N,N_i,\sigma_{ij}]$, one formally writes down the generating functional
\begin{equation}
Z_k[J; \bar{\chi}] \equiv \int {\cal D} \hat{N} {\cal D} \hat{N}_i {\cal D} \hat{\sigma} \, \exp\left[-S^{\rm grav} - S^{\rm gf} - S^{\rm ghost} - \Delta_kS - S^{\rm source}\right] \, ,
\end{equation}
where $S^{\rm grav}$ is supplemented by a suitable gauge-fixing term $S^{\rm gf}$, a corresponding ghost action $S^{\rm ghost}$ exponentiating the Faddeev-Popov determinant, and source terms $S^{\rm source}$ for the fluctuation fields. The crucial ingredient is the infrared regulator
\begin{equation}\label{regulator}
\Delta_k S \equiv \tfrac{1}{2} \int d\tau d^dy \sqrt{\bar{\sigma}} \bar{N} \, \left[ \, \hat{\chi} \, {\cal R}_k[\bar{\chi}] \, \hat{\chi} \, \right] \, ,
\end{equation}
where the matrix-valued kernel ${\cal R}_k[\bar{\chi}]$ is constructed from the background metric and provides a scale-dependent mass term for fluctuations with momenta $p^2 \lesssim k^2$. Based on the partition function, we define the generating functional for the connected Green functions
\begin{equation}
W_k[J; \bar{\chi}] \equiv \log\left[ Z_k \right] \, .
\end{equation}
The effective average action is then obtained as
\begin{equation}
\Gamma_k[\hat{\chi};\bar{\chi}] \equiv \widetilde \Gamma_k[\hat{\chi};\bar{\chi}] - \Delta_kS[\chi;\bar{\chi}]
\end{equation}
where $\widetilde \Gamma_k$ is the Legendre-transform of $W_k$. In general $\Gamma_k$ consists of a (generic) gravitational action $\Gamma_k^{\rm grav}$ supplemented by a suitable gauge-fixing $\Gamma_k^{\rm gf}$, ghost action $\Gamma_k^{\rm ghost}$ and, potentially, a matter action
\begin{equation}\label{Gform}
\Gamma_k = \Gamma_k^{\rm grav} + \Gamma_k^{\rm gf} + \Gamma_k^{\rm ghost} + \Gamma_k^{\rm matter} \, ,
\end{equation}
with the gauge-fixing constructed from the background field method.
The key property of $\Gamma_k$ is that its scale-dependence is governed by a formally exact FRGE
\begin{equation}\label{FRGE}
k \partial_k \Gamma_k = \frac{1}{2} \, {\rm STr} \left[ \left( \Gamma_k^{(2)} + {\cal R}_k \right)^{-1} \, k \partial_k {\cal R}_k \right] \, .
\end{equation}
Here $\Gamma_k^{(2)}$ denotes the second variation of $\Gamma_k$ with respect to the fluctuation fields $\hat{\chi}$, STr contains a graded sum over component fields and an integration over loop momenta, and the matrix-valued IR regulator ${\cal R}_k$ has been introduced in \eqref{regulator}. The interplay between the regularized propagator $\left( \Gamma_k^{(2)} + {\cal R}_k \right)^{-1}$ and $k \partial_k {\cal R}_k$ ensures that the right-hand-side of the FRGE is actually finite. Moreover, the FRGE realizes Wilson's idea of renormalization in the sense that the flow of $\Gamma_k$ is essentially driven by fluctuations located in a small momentum-interval situated at the RG scale $k$.
At this stage, the following remark is in order. Owed to the non-linearity of the ADM decomposition, the transformation of the ADM fields under the full diffeomorphism group is non-linear. In combination with the linear split \eqref{linearsplit} this entails that $S^{\rm gf}$ and $\Delta_kS$ preserve a subgroup of the full diffeomorphism group as a background symmetry only. Inspecting eqs.\ \eqref{eq:gaugeVariations} and \eqref{Nui} one sees, that restricting $f = f(\tau)$ eliminates the quadratic terms in the transformations laws. This indicates that the flow equation \eqref{FRGE} respects foliation preserving diffeomorphisms where, by definition, the transformation $f = f(\tau)$ is independent of the spatial coordinates. Also see \cite{Rechenberger:2012dt} for a detailed discussion.
\section{RG flows on a Friedmann-Robertson-Walker background}
\label{sect.3}
In this section we use the FRGE \eqref{FRGE} to determine the beta functions encoding the scale-dependence of Newton's constant and the cosmological constant in the context of pure gravity and gravity minimally coupled to non-interacting matter fields. The key ingredient in the construction is a novel gauge-fixing scheme introduced in Sect.\ \ref{sect.3a} where all ADM-fields acquire a relativistic dispersion relation. Our discussion primarily focuses on the gravitational sector of the flow, incorporating the contributions from the matter sector at the very end only.
\subsection{The Einstein-Hilbert ansatz}
\label{sect.3mod}
Finding exact solutions of the FRGE \eqref{FRGE} is rather difficult. A standard way of constructing approximate solutions, which does not rely on the expansion in a small coupling constant, is to restrict the interaction monomials in $\Gamma_k$ to a specific subset and subsequently project the RG flow onto the subspace spanned by the ansatz.
In the present work, we will project the full RG flow onto the Einstein-Hilbert action written in terms of the ADM-fields
\begin{equation}\label{GammaEH}
\Gamma_k^{\rm grav} \simeq \frac{1}{16 \pi G_k} \int d\tau d^dy \, N \sqrt{\sigma} \left[ K_{ij} K^{ij} - K^2 - {}^{(d)}R + 2 \Lambda_k \right] \, .
\end{equation}
This ansatz contains two scale-dependent couplings, Newton's constant $G_k$ and the cosmological constant $\Lambda_k$. Their scale-dependence can be read off from the coefficient multiplying the square of the extrinsic curvature and the spacetime volume, respectively.
In order to facilitate the computation, it then suffices to work out the flow on a background which allows to distinguish between these two interaction monomials. For the ansatz \eqref{GammaEH} it then suffices to evaluate the flow on a flat (Euclidean) Friedmann-Robertson-Walker (FRW) background
\begin{equation}\label{FRWback}
\bar{g}_{\mu\nu} = {\rm diag} \left[ \, 1 \, , \, a(\tau)^2 \, \delta_{ij}\right] \qquad \Longleftrightarrow \qquad
\bar{N} = 1 \, , \quad \bar{N}_i = 0 \, , \quad \bar{\sigma}_{ij} = a(\tau)^2 \, \delta_{ij} \, ,
\end{equation}
where $a(\tau)$ is a positive, time-dependent scale factor. Evaluating \eqref{GammaEH} on this background using \eqref{curvatures} yields
\begin{equation}\label{FRGElhs}
\left. \Gamma_k^{\rm grav} \right|_{\hat{\chi} = 0}
= \frac{1}{16 \pi G_k} \int d\tau d^dy \, \sqrt{\bar{\sigma}} \left[ - \tfrac{d-1}{d} \, \bar{K}^2 + 2 \Lambda_k \right] \, ,
\end{equation}
where $\hat{\chi}$ denotes the set of all fluctuation fields. Thus the choice \eqref{FRWback} is sufficiently general to distinguish the two interaction monomials encoding the flow of $G_k$ and $\Lambda_k$. Note that we have not assumed that the background is compact. In particular the ``time-coordinate'' $\tau$ may be taken as non-compact.
\subsection{Hessians, gauge-fixing, and ghost action}
\label{sect.3a}
Constructing the right-hand-side of the flow equation requires the Hessian $\Gamma_k^{(2)}$. Starting with the contribution originating from $\Gamma_k^{\rm grav}$ it is convenient to introduce the building
blocks
\begin{equation}\label{imon}
\renewcommand{\arraystretch}{1.2}
\begin{array}{ll}
I_1 \equiv \int d\tau d^dy \, N \sqrt{\sigma} \, K_{ij} K^{ij} \, , \qquad &
I_2 \equiv \int d\tau d^dy \, N \sqrt{\sigma} \, K^2 \, , \\
I_3 \equiv \int d\tau d^dy \, N \sqrt{\sigma} \; {}^{(d)}R \, , \qquad &
I_4 \equiv \int d\tau d^dy \, N \sqrt{\sigma} \, , \\
\end{array}
\end{equation}
such that
\begin{equation}
\Gamma_k^{\rm grav} = \frac{1}{16 \pi G_k} \left( I_1 - I_2 - I_3 + 2 \Lambda_k \, I_4 \right) \, .
\end{equation}
Expanding this expression around the background \eqref{FRWback},
the terms quadratic in the fluctuation fields then take the form
\begin{equation}\label{GammaEH2}
\delta^2 \Gamma_k^{\rm grav} = \frac{1}{16 \pi G_k} \left( \delta^2 I_1 - \delta^2 I_2 - \delta^2 I_3 + 2 \Lambda_k \, \delta^2 I_4 \right) \, ,
\end{equation}
with the explicit expressions for $\delta^2 I_i$ given in \eqref{varI1}.
The FRW-background then makes it convenient to express the fluctuation fields in terms of the component fields used in
cosmic perturbation theory (see, e.g., \cite{Baumann:2009ds} for a pedagogical introduction).
Defining $\Delta \equiv - \bar{\sigma}^{ij} \partial_i \partial_j$, the shift vector is decomposed into its transverse and longitudinal parts according to
\begin{equation}\label{TTshift}
\hat{N}_i = u_i + \partial_i \, \tfrac{1}{\sqrt{\Delta}} \, B \, , \qquad \partial^i \, u_i = 0 \, .
\end{equation}
The metric fluctuations are written as
\begin{equation}\label{TTmet}
\hat{\sigma}_{ij} = h_{ij} - \left( \bar{\sigma}_{ij} + \partial_i \partial_j \, \tfrac{1}{\Delta} \right) \psi + \partial_i \partial_j \, \tfrac{1}{\Delta} \, E + \partial_i \tfrac{1}{\sqrt{\Delta}} v_j + \partial_j \, \tfrac{1}{\sqrt{\Delta}} \, v_i \, , \quad \hat{\sigma} \equiv \bar{\sigma}^{ij} \hat{\sigma}_{ij} \, ,
\end{equation}
with the component fields subject to the differential constraints
\begin{equation}
\partial^i \, h_{ij} = 0 \, , \qquad \bar{\sigma}^{ij} h_{ij} = 0 \, , \qquad \partial^i v_i = 0 \, .
\end{equation}
The result obtained from substituting these decompositions into eq.\ \eqref{varI1} is given in eqs.\ \eqref{I1res}, \eqref{I2res}, and \eqref{I3res}. On this basis it is then rather straightforward to write down the explicit form of \eqref{GammaEH2} in terms of the component fields.
At this stage it is instructive to investigate the matrix elements of $\delta^2 \Gamma^{\rm grav}_k$ on flat Euclidean space, obtained by setting $\bar{K} = 0$.
The result is summarized in the second column of Table \ref{Tab.1}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c|l|l|} \hline \hline
Index & matrix element $32 \pi G_k \, \delta^2\Gamma_k^{{\rm grav}}$ & matrix element $32 \pi G_k \, \left(\delta^2\Gamma_k^{{\rm grav}} + \Gamma_k^{\rm gf} \right)$ \\ \hline
$h \, h$ & $\Box - 2 \Lambda_k$ & $\Box - 2 \Lambda_k$ \\ \hline
$v \, v$ & $ 2 \big[-\partial_\tau^2 - 2 \Lambda_k \big]$ & $\Box - 2 \Lambda_k$ \\ \hline
$E \, E$ & $- \Lambda_k$ & $\tfrac{1}{2}(\Box - 2 \Lambda_k)$ \\ \hline
$\psi \, \psi$ & $ - (d-1) (d-2) \big[ \Box - \tfrac{d-3}{d-2} \, \Lambda_k \big] $ & $ - \tfrac{(d-1) (d-3)}{2} \big[ \Box - 2 \, \Lambda_k \big] $ \\ \hline
$\psi \, E$ & $ - (d-1) \big[-\partial_\tau^2 - 2 \Lambda_k \big]$
& $ - (d-1) \big[\Box - 2 \Lambda_k \big]$ \\ \hline \hline
$ u \, u$ & $ 2 \, \Delta$ & $ 2 \, \Box $ \\ \hline
$ u \, v $ & $-2 \, \partial_\tau \sqrt{\Delta}$ & 0 \\ \hline
$ B \, \psi $ & $2 \, (d-1) \sqrt{\Delta} \, \partial_\tau$ & 0 \\ \hline
$ \hat{N} \, \psi $ & $2 \, (d-1) \big[ \Delta - \Lambda_k \big]$ & $ \, (d-1) \big[ \Box - 2 \Lambda_k \big]$\\ \hline
$ \hat{N} \, E $ & $ - 2 \, \Lambda_k$ & $ \Box - 2 \Lambda_k$
\\ \hline
$\hat{N} \hat{N}$ & $0$ & $2 \, \Box$ \\ \hline \hline
\end{tabular}
\end{center}
\caption{\label{Tab.1} Summary of the matrix elements appearing in $\delta^2\Gamma_k$ when expanded $\Gamma_k$ around flat Euclidean space. The column ``index'' identifies the corresponding matrix element in field space, $\Delta \equiv - \bar{\sigma}^{ij} \partial_i \partial_j$ is the Laplacian on the spatial slice, and $\Box \equiv -\partial_t^2 - \bar{\sigma}^{ij} \partial_i \partial_j$. For each ``off-diagonal'' entry there is a second contribution involving the adjoint of the differential operator and the order of the fields reversed.}
\end{table}
On this basis, one can make the crucial observation that the component fields do not possess a relativistic dispersion relation. One may then attempt to add a suitable gauge-fixing term $\Gamma_k^{\rm gf}$. A suggestive choice (also from the perspective of Ho\v{r}ava-Lifshitz gravity) is proper-time gauge \cite{Dasgupta:2001ue}. This gauge choice eliminates the fluctuations in the lapse and shift vector $\hat{N} = 0$, $\hat{N}_i = 0$ by choosing
\begin{equation}
\Gamma_k^{\rm gf, proper-time} = \lim_{\alpha \rightarrow 0} \, \frac{1}{2\alpha} \, \int d\tau d^dy \sqrt{\bar{\sigma}} \, \left[ \hat{N}^2 + \hat{N}_i \, \bar{\sigma}^{ij} \, \hat{N}_j \right] \, .
\end{equation}
At the level of the component fields \eqref{TTshift} this choice entails $\hat{N} = 0$, $u_i = 0$ and $B=0$. This eliminates the last six entries from Tab.\ \ref{Tab.1}, essentially restricting quantum fluctuations to the components of the spatial metric. Tab.\ \ref{Tab.1} then indicates that the sector containing the fluctuations of the spatial metric (first five entries) contains propagators which do not include a spatial momentum dependence. On this basis
proper-time gauge may not ideal for investigating the quantum properties of the theory in an off-shell formalism like the FRGE.
Motivated by the recent investigation \cite{Barvinsky:2015kil} it is then natural to investigate if there is a different gauge choice ameliorating this peculiar feature. Inspired by the decomposition \eqref{vdec} the gauge-fixing of the symmetries \eqref{eq:gaugeVariations} may be implemented via two functions $F$ and $F_i$
\begin{equation}\label{gf:ansatz}
\Gamma_k^{\rm gf} = \frac{1}{32 \pi G_k} \int d\tau d^dy \, \sqrt{\bar{\sigma}} \, \left[ F_i \, \bar{\sigma}^{ij} F_j + F^2 \right] \, ,
\end{equation}
where $F$ and $F_i$ are linear in the fluctuation fields. The integrand entering $\Gamma^{\rm gf}_k$ may also be written in terms of a $D$-dimensional vector $F_\mu \equiv (F, F_i)$ and the background metric \eqref{FRWback} exploiting that $F_\mu \, \bar{g}^{\mu\nu} \, F_\nu = F^2 + F_i \, \bar{\sigma}^{ij} \, F_j$. The most general form of $F$ and $F_i$ which is linear in the fluctuation fields $\hat{N}, \hat{N}_i, \hat{\sigma}_{ij}$ and involves at most one derivative with respect to the spatial or time coordinate is given by
\begin{equation}\label{gauge1}
\begin{split}
F = & \, c_1 \, \partial_\tau \, \hat{N} + c_2 \, \partial^i \, \hat{N}_i + c_3 \, \partial_\tau \, \hat{\sigma} + d \, c_8 \, \bar{K}^{ij} \, \hat{\sigma}_{ij} + c_9 \, \bar{K} \hat{N} \, , \\
F_i = & \, c_4 \, \partial_\tau \, \hat{N}_i + c_5 \, \partial_i \, \hat{N} + c_6 \, \partial_i \, \hat{\sigma} + c_7 \, \partial^j \, \hat{\sigma}_{ji} + d \, c_{10} \, \bar{K}_{ij} \hat{N}^j \, .
\end{split}
\end{equation}
The $c_i$ are real coefficients which may depend on $d$ and the factors $d$ are introduced for later convenience. Following the calculation in App.\ \ref{App.B3}, rewriting the gauge-fixing \eqref{gf:ansatz} in terms of the component fields yields \eqref{d2I5} and \eqref{d2I6}. Combining $\delta^2\Gamma_k^{\rm grav}$ with the gauge-fixing contribution one finally arrives at \eqref{eqS1B}. The coefficients $c_i$ are then fixed by requiring, firstly, that \emph{all component fields come with a relativistic dispersion relation} and, secondly, that the resulting gauge-fixed Hessian does not contain square-roots of the spatial Laplacian $\sqrt{\Delta}$. It turns out that these two conditions essentially fix the gauge uniquely, up to a physically irrelevant discrete symmetry:
\begin{equation}\label{gffinal}
\begin{array}{lllll}
c_1= \epsilon_1 \, , \qquad & c_2= \epsilon_1 \, , \qquad & c_3=- \tfrac{1}{2} \epsilon_1 \, , \qquad & c_8=0 \, , \qquad & c_9= \tfrac{2\,(d-1)}{d} \,\epsilon_1 \, , \\[1.1ex]
c_4= \epsilon_2 \, \qquad & c_5=- \epsilon_2 \, , \qquad & c_6= - \tfrac{1}{2} \epsilon_2 \, , \qquad & c_7= \epsilon_2 \, , \qquad & c_{10}= \tfrac{d-2}{d} \, \epsilon_2
\end{array}
\end{equation}
where $\epsilon_1=\pm 1$ and $\epsilon_2=\pm 1$. Since $\Gamma^{\rm gf}_k$ is quadratic in $F$ and $F_i$ it depends on $\epsilon_i^2$ only and the choice of sign does not change $\Gamma^{\rm gf}_k$.
Combining \eqref{GammaEH2} with the gauge choice \eqref{gf:ansatz} with \eqref{gffinal} finally results in the gauge-fixed Hessian
\begin{equation}\label{eqS2Frank}
\begin{split}
& \, 32 \pi G_k \Big( \tfrac{1}{2} \delta^2 \Gamma^{\rm grav}_k + \Gamma_k^{\rm gf} \Big) = \\ & \,
\int_x \Big\{
\tfrac{1}{2} \, h^{ij} \left[ \Delta_2 - 2 \Lambda_k - \tfrac{2(d-1)}{d} \dot{\bar{K}}
- \tfrac{d^2-d+2}{d^2} \bar{K}^2 \right] \, h_{ij} \\ & \qquad
%
+ u^i \left[\Delta_1 -\tfrac{d-1}{d} \dot{\bar{K}} - \tfrac{1}{d} \bar{K}^2 \right] u_i
%
+ v^i \left[ \Delta_1 - 2 \Lambda_k - \dot{\bar{K}} - \tfrac{5d-7}{d^2} \bar{K}^2 \right] v_i \\ & \qquad
%
+ B \, \left[ \Delta_0 - \tfrac{d-1}{d}\,\dot{\bar{K}}-\tfrac{d-1}{d^2}\bar{K}^2 \right] B
+ \hat{N} \left[ \Delta_0 - \tfrac{2(d-1)}{d} \dot{\bar{K}} {-\tfrac{4(d-1)}{d^2}}\, \bar{K}^2 \right] \hat{N} \\ & \qquad
%
%
+ \hat{N}\, \Big[
\Delta_0 - 2\Lambda_k - {
\tfrac{5d^2-12d+16}{4d^2}} \,\bar{K}^2
\Big] \big( (d-1) \psi + E \big) \\ & \qquad
%
%
- \tfrac{(d-1)(d-3)}{4} \, \psi \, \Big[ \Delta_0 - 2\Lambda_k - {\tfrac{2(d-1)}{d} } \,\dot{\bar{K}} - { \tfrac{d-1}{d}} \bar{K}^2 \Big] \psi
\\ & \qquad
%
+ \tfrac{1}{4} \, E \, \Big[ \Delta_0 -2 \Lambda_k - {\tfrac{2(d-1)}{d}} \,\dot{\bar{K}} - {\tfrac{d-1}{d}} \,\bar{K}^2
\Big] E \\ & \qquad
%
- \tfrac{1}{2} (d-1) \, \psi \Big[ \Delta_0 - 2 \Lambda_k - { \tfrac{2(d-1)}{d} \dot{\bar{K}} } - {\tfrac{d-1}{d}} \,\bar{K}^2
\Big] E
%
\Big\} \, .
\end{split}
\end{equation}
Here the operators $\Delta_i$ are defined in \eqref{LapDala} and the diagonal terms in field space have been simplified by partial integration. Setting $\bar{K} = 0$, the matrix elements resulting from this expression are shown in the third column of Tab.\ \ref{Tab.1}. On this basis, it is then straightforward to verify that all fluctuation fields acquire a relativistic dispersion relation.
This condition fixes the gauge-choice \emph{uniquely} \cite{Biemans:2016rvp}.
The ghost action exponentiating the Faddeev-Popov determinant is obtained from the variations \eqref{eq:gaugeVariations} by evaluating \eqref{ghs}. The ghost sector then comprises one scalar ghost $\bar{c}, c$ and one spatial vector ghost $\bar{b}^i, b_i$ arising from the transformation of $F$ and $F_i$, respectively. Restricting to terms quadratic in the fluctuation field and choosing $\epsilon_1 = \epsilon_2 = -1$, the result is given by
\begin{equation}\label{Gghost}
\begin{split}
\Gamma_k^{\rm ghost} = \int d\tau d^dy \sqrt{\bar{\sigma}} \, \Big\{ & {\bar{c} \left[\Delta_0 + \tfrac{2}{d} \bar{K} \partial_\tau + \dot{\bar{K}} \right] c } \\ & \,
+ \bar{b}^i \left[ \Delta_1 + \tfrac{2}{d} \bar{K} \partial_\tau + \tfrac{1}{d} \dot{\bar{K}} + \tfrac{d-4}{d^2} \bar{K}^2 \right] b_i \Big\} \, .
\end{split}
\end{equation}
Notably, the ghost action does not contain a scale-dependent coupling. The results \eqref{eqS2Frank} and \eqref{Gghost} then complete the construction of the Hessian $\Gamma_k^{(2)}$.
At this stage the following remark is in order. Projectable Ho\v{r}ava-Lifshitz gravity \cite{Horava:2009uw} restricts the lapse function $N(\tau, y) \rightarrow N(\tau)$ to a function of time only while the symmetry group is restricted to foliation preserving diffeomorphisms $f(\tau,y) \rightarrow f(\tau)$. This structure suggests a Landau-type gauge-fixing for the lapse-function, setting $F = \hat{N}$. Retaining the most general (local) form of $F_i$ given in \eqref{gauge1}, a quick inspection of eq.\ \eqref{eqS1B} with $\hat{N} = 0$ and $c_1 = c_2=c_3 = 0$ reveals that there is no set of parameters $c_i$ which would bring the dispersion relations of the remaining component fields into the relativistic form displayed in Table \ref{Tab.1}. Thus the extension of the present off-shell construction to Ho\v{r}ava-Lifshitz gravity is not straightforward.
\subsection{Evaluating the operator traces}
\label{sect.3b}
Notably, the Hessians arising from \eqref{eqS2Frank} and \eqref{Gghost} contain $D$-covariant Laplace-type operators only and can thus be evaluated using standard heat-kernel techniques (see the Appendix of \cite{Codello:2008vh} for details). Resorting to a Type I regulator \cite{Codello:2008vh}, implicitly defined by
\begin{equation}
\Delta_s \mapsto P_k = \Delta_s + R_k \, ,
\end{equation}
and choosing the profile function $R_k$ providing the $k$-dependent mass term for the fluctuation modes, to be of Litim-form $R_k=(k^2-\Delta_i)\,\theta(k^2-\Delta_i)$, the computation uses the
heat-kernel techniques detailed in App.\ \ref{App.A}. Combining the intermediate results obtained in App.\ \ref{App.C},
the flow of Newton's constant and the cosmological constant
is conveniently expressed in terms of
the dimensionless quantities
\begin{equation}\label{defdimless}
\eta \equiv (G_k)^{-1} \partial_t \, G_k \, , \qquad
\lambda_k \equiv \Lambda_k \, k^{-2} \, , \qquad
g_k \equiv G_k \, k^{d-1} \, .
\end{equation}
Here $\eta$ is the anomalous dimension of Newton's constant.
In order to write down the beta functions in a compact form,
it is moreover useful to define
\begin{equation}
B_{\rm det}(\lambda) \equiv (1-2\lambda)(d-1-d\lambda)\, .
\end{equation}
The final form of the beta functions is\footnote{The beta functions given here differ from the ones used in \cite{Biemans:2016rvp} by a different form of the regulator in the transverse-traceless and vector sectors of the decomposition \eqref{TTshift} and \eqref{TTmet}.}
\begin{equation}\label{betafunction}
\begin{split}
\beta_g = & \, (d-1+\eta) \, g \, , \\
%
\beta_\lambda = & \, (\eta - 2) \lambda + \tfrac{2g}{(4 \pi)^{(d-1)/2}} \, \tfrac{1}{\Gamma((d+3)/2)}
\Big[
%
\big( d +
\tfrac{d^2 + d -4}{2(1-2\lambda)}
+ \tfrac{3d-3 - (4 d-2) \lambda }{B_{\rm det}(\lambda)}
\big)
%
\big( 1 - \tfrac{\eta}{d+3} \big) \,
\\ & \, \qquad \qquad
%
-2(d+1) + N_S + (d-1) N_V - 2^{\left[(d+1)/2\right] } \, N_D \Big] \, ,
\end{split}
\end{equation}
with anomalous dimension of Newton's constant given by
\begin{equation}\label{etaflow}
\begin{split}
\eta = \frac{16 \pi g \, B_1(\lambda)}{(4\pi)^{(d+1)/2} + 16 \pi g \, B_2(\lambda)} \, :
\end{split}
\end{equation}
The functions $B_1(\lambda)$ and $B_2(\lambda)$ depend on $\lambda$ and $d$ and are given by
\begin{equation}
\begin{split}
B_1(\lambda) \equiv &
%
- \tfrac{d^5 + 17 d^4 + 41 d^3 + 85 d^2 + 174 d - 78}{24 \, d ( d - 1) \, \Gamma(( d+5)/2)}
+ \tfrac{d^4-5d^2+16d+48}{12 \, d (d-1)\, (1-2\lambda) \, \Gamma((d+1)/2)}
\\ & \,
- \tfrac{d^4-15d^2+28d-10}{2 d (d-1) \, (1-2\lambda)^2 \, \Gamma((d+3)/2)}
%
+ \tfrac{3d-3 - (4d-2) \lambda }{6 \, B_{\rm det}(\lambda) \, \Gamma(( d+1)/2)}
%
+ \tfrac{c_{1,0} + c_{1,1} \lambda + c_{1,2} \lambda^2}{4 \, d \, B_{\rm det}(\lambda)^2 \, \Gamma((d+3)/2)} \\ & \, + \tfrac{1}{6 \, \Gamma((d+1)/2)} \left[N_S + \tfrac{d^2 - 13}{d+1} N_V - \tfrac{1}{4} \, 2^{\left[ (d+1)/2\right]} N_D \, \right] \, .
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
B_2(\lambda) = & \,
\tfrac{d^4-10d^3+21d^2+6d+6}{24 \, d (d-1)\, \Gamma((d+5)/2)}
+ \tfrac{d^4-5d^2+16d+48}{24 \, d (d-1)\, (1-2\lambda) \, \Gamma((d+3)/2)}
%
- \tfrac{d^4-15d^2+28d-10}{4 \,d(d-1) \, (1-2\lambda)^2 \, \Gamma((d+5)/2)} \\ & \,
%
+ \tfrac{3d-3 - (4d-2)\lambda}{12 \, B_{\rm det}(\lambda) \, \Gamma((d+3)/2)}
%
+ \tfrac{c_{2,0} + c_{2,1} \lambda + c_{2,2} \lambda^2 }{8 \, d \, B_{\rm det}(\lambda)^2 \, \Gamma((d+5)/2)} \, .
%
\end{split}
\end{equation}
The coefficients $c_{i,j}$ are polynomials in $d$ and given by
\begin{equation}
\begin{array}{ll}
c_{1,0} = - 5 d^3 + 22 d^2 - 24 d + 16 \, , \qquad &
c_{1,1} = 4 \left(d^3 - 10 d^2 + 16 d -16 \right)
\, , \\
c_{1,2} = 4 \left( d^3 + 6 d^2 - 16 d + 16 \right) \, , &
c_{2,0} = - 5 d^3 + 22 d^2 -24 d + 16 \, ,
\end{array}
\end{equation}
together with $c_{1,1} = c_{2,1}$ and $c_{1,2} = c_{2,2}$.
Notably $B_2$ is independent of the matter content of the system, reflecting the fact that the matter sector \eqref{matter} is independent of Newton's constant. The result \eqref{betafunction} together with the explicit expression for the anomalous dimension of Newton's constant \eqref{etaflow} constitutes the main result of this section.
\section{Properties of the RG flow}
\label{sect.4}
In this section, we analyze the RG flow resulting from the beta functions
\eqref{betafunction} for a $D=3+1$-dimensional spacetime. The case of pure gravity, corresponding to setting $N_S = N_V = N_D = 0$, is discussed in Sect.\ \ref{sect.42} while the classification of the fixed point structure appearing in general gravity-matter systems is carried out in Sect.\ \ref{sect.43}. Our results complement the findings reported in \cite{Biemans:2016rvp}.
\subsection{Pure gravity}
\label{sect.42}
The beta functions \eqref{betafunction} constitute a system of coupled first-order differential equations. In general such systems do not admit analytical solutions and one has to resort to numerical methods. Nevertheless, the general theory of dynamical systems allows to determine possible long-term behaviors of the flow \eqref{betafunction} by determining its fixed points (FPs) $(g_*,\lambda_*)$ satisfying
\begin{equation}\label{fpcond}
\beta_g(g_*,\lambda_*)=0\; , \qquad \beta_\lambda(g_*,\lambda_*)=0 \, .
\end{equation}
Such fixed points may control the long-term behavior of the theory in the limit $k\to\infty$ (UV completion) or $k\to0$ (IR limit). By linearizing the system \eqref{betafunction} around its FPs, the stability matrix $\mathbf{B}_{ij} \equiv \left. \partial_{g_j} \beta_{g_i} \right|_{g = g_*}$ encodes the $k$-dependence of the couplings near the fixed point. In particular, the scaling of the couplings is characterized by the critical exponents $\theta_i$, defined by (minus) the eigenvalues of $\mathbf{B}_{ij}$:
eigendirections coming with ${\rm Re}(\theta_i) > 0$ are dragged into the fixed point for $k\to\infty$ while directions with ${\rm Re}(\theta_i) < 0$ are repelled in this limit. The former then constitute the relevant directions of the fixed point.
For $d=3$ spatial dimensions the system \eqref{betafunction} possesses a unique NGFP with positive Newton's constant,
\begin{equation}\label{NGFP1}
\mbox{NGFP:} \qquad \; g_* = 0.785 \, , \qquad \lambda_* = 0.315 \, , \qquad g_* \lambda_* = 0.248 \, ,
\end{equation}
coming with a complex pair of critical exponents,
\begin{equation}\label{critexp}
\theta_{1,2} = 0.503 \pm 5.377 i \, .
\end{equation}
The positive real part, Re($\theta_{1,2}) > 0$, indicates that the NGFP acts as a spiraling UV attractor for the RG trajectories in its vicinity. Notably, this is the same type of UV-attractive spiraling behavior encountered when evaluating the RG flow on foliated spacetimes using the Matsubara formalism \cite{Manrique:2011jc,Rechenberger:2012dt}, and a vast range of studies building on the metric formalism \cite{Souma:1999at,Lauscher:2001ya,Reuter:2001ag,Litim:2003vp,Fischer:2006fz,Codello:2013fpa,Nagy:2013hka,Christiansen:2015rva,Codello:2006in,Codello:2007bd,Benedetti:2009rx,Benedetti:2009gn,Demmel:2014hla,Machado:2007ea,Manrique:2010am,Christiansen:2012rx,Christiansen:2014raa,Falls:2014tra,Falls:2015cta,Donkin:2012ud,Lauscher:2002sq,Groh:2010ta,Manrique:2009uh,Manrique:2010mq,Becker:2014qya,Gies:2015tca,Eichhorn:2009ah,Rechenberger:2012pm,Eichhorn:2010tb,Nink:2012vd,Becker:2014pea,Becker:2014jua}.
Subsequently, it is instructive to determine the singular loci of the beta functions \eqref{betafunction} where either $\beta_g$ or $\beta_\lambda$ diverge. For finite values of $g$ and $\lambda$ these may either be linked to
one of the denominators appearing in $\beta_\lambda$ becoming zero or a divergences of the anomalous dimension of Newton's constant.
Inspecting $\beta_\lambda$, the first case gives rise to two singular lines in the $\lambda$-$g$--plane
\begin{equation}\label{singlin1}
\begin{split}
\lambda^{\rm sing}_1 = \tfrac{1}{2} \, , \qquad \lambda^{\rm sing}_2 = \tfrac{d-1}{d} \, .
\end{split}
\end{equation}
The singular lines $\eta^{\rm sing}(g,\lambda)$ associated with divergences of the anomalous dimension $\eta$ are complicated functions of $d$. For the specific cases $d=2$ and $d=3$ the resulting expressions simplify and are given by the parametric curves
\begin{equation}\label{etasing2}
\begin{split}
\begin{array}{lll}
d = 2: \; \qquad & \eta^{\rm sing}: \qquad & g = - \frac{45 \pi (1-2\lambda)^2}{2 \, (76 \lambda^2 - 296 \lambda + 147 ) } , \\[1.2ex]
%
d = 3: \; \qquad & \eta^{\rm sing}: \qquad & g = - \frac{144 \pi (6 \lambda^2 - 7 \lambda + 2)^2}{144 \lambda^4 -1884 \lambda^3 +3122 \lambda^2 -1688 \lambda + 279 } \, .
\end{array}
\end{split}
\end{equation}
The position of the singular lines \eqref{singlin1} and \eqref{etasing2} are illustrated in Fig.\ \ref{Fig.sing}. Focusing to the domain $g \ge 0$, it is interesting to note that the singularities bounding the flow of $\lambda_k$ for positive values are of different nature in $d=2$ and $d=3$: in $d=2$ the domain is bounded to the right by a fixed singularity of $\beta_\lambda$ and $\eta$ remains finite throughout this domain while in $d=3$ the singular line $\lambda_1^{\rm sing}$ is screened by a divergence of $\eta$. Notably, the position of the singular lines is independent of $N_S$, $N_V$, and $N_D$ and thus also carries over to the analysis of gravity-matter systems.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.48\textwidth]{betasingd2} \;
\includegraphics[width=0.48\textwidth]{betasingd3}
\caption{Singularity structure of the beta functions \eqref{betafunction} in the $\lambda$-$g$--plane in $d=2$ (left diagram) and $d=3$ (right diagram). The blue lines indicate fixed singularities of $\beta_\lambda$, eq.\ \eqref{singlin1}, while the red lines illustrate the curves \eqref{etasing2} and where $\eta$ develops a singularity. \label{Fig.sing} }
\end{center}
\end{figure}
Finally, we note that the point $(\lambda,g) = (1/2,0)$ is special in the sense that the beta functions \eqref{betafunction} are of the form $0/0$. In particular the value of the anomalous dimension $\eta$ depends on the direction along which this point is approached. We will denote this point as ``quasi-fixed point'' $C\equiv(\tfrac{1}{2},0)$ in the sequel.
Upon determining the fixed point and singularity structure relevant for the renormalization group flow with a positive Newton's constant, it is rather straightforward to construct the RG trajectories resulting from the beta functions \eqref{betafunction} numerically. An illustrative sample of RG trajectories characterizing the flow in $D=3+1$ spacetime dimensions is shown in Fig.\ \ref{flow2d3d}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\textwidth]{phasediagram3d}
\caption{Phase diagram of the RG flow originating from the beta functions \eqref{betafunction} in $D=3+1$ spacetime dimensions. The flow is dominated by the interplay of the NGFP (point ``A'') controlling the flow for ultra-high energies and the GFP (point ``O'') governing the low-energy behavior. The flow undergoes a crossover between these two fixed points. For some of the RG trajectory this crossover is intersected by the singular locus \eqref{etasing2} (red line). The arrows indicate the direction of the RG flow pointing from high to low energy.
\label{flow2d3d} }
\end{center}
\end{figure}
Notably, the high-energy behavior of the flow is controlled by
the NGFP \eqref{NGFP1}. Following the nomenclature introduced in \cite{Reuter:2001ag}, the low-energy behavior can be classified according to the sign of the cosmological constant:
\begin{equation}\label{Typeclass}
\begin{array}{lll}
\mbox{Type Ia:} \qquad \quad & \lim_{k \rightarrow 0} (\lambda_k, g_k) = (- \infty, 0) \, , \qquad & \Lambda_0 < 0 \, , \\[1.1ex]
\mbox{Type IIa:} & \lim_{k \rightarrow 0} (\lambda_k, g_k) = (0, 0) \, , & \Lambda_0 = 0 \, , \\[1.1ex]
\mbox{Type IIIa:} & \mbox{terminate at} \; \; \eta^{\rm sing} & \Lambda_{k_{\rm term}} > 0 \, . \\
\end{array}
\end{equation}
These three phases are realized by the RG trajectories flowing to the left (Type Ia), to the right (Type IIIa), and on top of the bold blue line emanating from the GFP ``O'' (Type IIa). Once the trajectories enter the vicinity of the GFP, characterized by $g_k \ll 1$, the dimensionful Newton's constant $G_k$ and cosmological constant $\Lambda_k$ are essentially $k$-independent, so that the trajectories enter into a ``classical regime''. For trajectories of Type Ia this regime extends to $k = 0$. Trajectories of Type IIIa terminate in the singularity $\eta^{\rm sing}$ (red line) at a finite value $k_{\rm term}$.
The high-energy and low-energy regimes are connected by a crossover of the RG flow. For some of the trajectories, this crossover cuts through the red line marking a divergence in the anomalous dimension of Newton's constant.
This peculiar feature can be traced back to the critical exponents \eqref{critexp} where the beta functions \eqref{betafunction} lead to a exceptionally low value for Re($\theta_{1,2}$). Compared to other incarnations of the flow, which come with significantly higher values for Re($\theta_{1,2}$), this makes the spiraling process around the NGFP less compact. As a consequence the flow actually touches $\eta^{\rm sing}$. Since this feature is absent in the flow diagrams obtained from the Matsubara computation \cite{Manrique:2011jc,Rechenberger:2012dt}, the foliated RG flows studied in \cite{Biemans:2016rvp}, and in the flows obtained in the covariant formalism \cite{Reuter:2001ag}, it is likely that this is rather a particularity of the flow based on \eqref{betafunction}, instead of a genuine physical feature.
\subsection{Gravity-matter systems}
\label{sect.43}
In order to classify the fixed point structures realized for a generic gravity-matter system, we first observe that the number of minimally coupled
scalar fields, $N_S$, vectors, $N_V$, and Dirac spinors $N_D$ enter the gravitational beta functions \eqref{betafunction} in terms of the combinations\footnote{The precise relation between the parameters $d_g$, $d_\lambda$ and the matter content may depend on the precise choice of regulator employed in matter traces, see App.\ \ref{App.C3}. Carrying out the classification of fixed point structures in terms of the deformation parameters shifts this regulator dependence into the map $d_g(N_S,N_V,N_D), d_\lambda(N_S,N_V,N_D)$ allowing to carry out the classification independently of a particular regularization scheme.}
\begin{equation}
d_g \equiv N_S + \frac{d^2-13}{d+1} N_V - \tfrac{1}{4} \, 2^{(d+1)/2} N_D \, , \quad
d_\lambda \equiv N_S + (d-1) N_V - 2^{(d+1)/2} N_D
\, .
\end{equation}
For $d=3$ these definitions reduce to
\begin{equation}\label{dldgdef}
d_g = N_S - N_V - N_D \, , \quad
d_\lambda = N_S + 2 N_V - 4 N_D
\, .
\end{equation}
The relation \eqref{dldgdef} allows to assign coordinates to any matter sector. For example, the standard model of particle physics comprises $N_S =4$ scalars, $N_D = 45/2$ Dirac fermions and $N_V = 12$ vector fields and is thus located at $(d_g, d_\lambda) = (- 61/2, - 62)$. For $N_S$ and $N_V$ being positive integers including zero and $N_D$ taking half-integer values in order to also accommodate chiral fermions, $d_g$ and $d_\lambda$ take half-integer values and cover the entire $d_g$-$d_\lambda$--plane.
The beta functions \eqref{betafunction} then give rise to a surprisingly rich set of NGFPs whose properties can partially be understood analytically. The condition $\beta_g|_{g = g_*} = 0$ entails that any NGFP has to come with an anomalous dimension $\eta_* = -2$. This relation can be solved analytically, determining the fixed point coordinate $g_*(\lambda_*; d_g)$ as a function of $\lambda_*$ and $d_g$. Substituting $\eta_* = -2$ together with the relation for $g_*$ into the second fixed point condition, $\beta_\lambda|_{g = g_*} = 0$, then leads to a fifth order polynomial in $\lambda$ whose coefficients depend on $d_g, d_\lambda$. The roots of this polynomial provide the coordinate $\lambda_*$ of a candidate NGFP. The fact that the polynomial is of fifth order then entails that the beta functions \eqref{betafunction} may support at most five NGFPs, independent of the matter content of the system.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.7\textwidth]{fpstructure1}
\caption{Number of NGFPs supported by the beta functions \eqref{betafunction} as a function of the parameters $d_g$ and $d_\lambda$. The colors black, blue, green, and red indicate the existence of zero, one, two, and three NGFPs situated at $g_* > 0$, $\lambda_* < 1/2$, respectively. \label{Fig.7a}}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.489\textwidth]{fpstructure2} \;
\includegraphics[width=0.487\textwidth]{fpstructure3}
\caption{\label{Fig.7} Classification of the NGFPs arising from the beta functions \eqref{betafunction} in the $d_g$-$d_\lambda$--plane, following the color-code provided in Table \ref{Tab.4}. The left diagram classifies the stability behavior of the one-fixed point sector. In particular, the black region does not support any NGFP while the regions giving rise to a single, UV-attractive NGFP with complex and real critical exponents are marked in blue and green, respectively. The field content of the standard model is situated in the lower-left quadrant, $(d_g, d_\lambda) = (-61/2, -62)$, and marked with a bold green dot. The gray area, supporting multiple NGFPs is magnified in the right diagram with empty and filled symbols indicating the existence of two and three NGFPs, respectively.}
\end{center}
\end{figure}
The precise fixed point structure realized for a particular set of values $(d_g, d_\lambda)$ can be determined numerically. The number of NGFPs located within the physically interesting region $g_* > 0$ and $\lambda_* < 1/2$ is displayed in Fig.\ \ref{Fig.7a}, where black, blue, green and red mark matter sectors giving rise to zero, one, two, and three NGFPs, respectively. On this basis, we learn that systems possessing zero or one NGFP are rather generic, while matter sectors giving rise to two or three NGFPs are confined to a small region in the center of the $d_g$-$d_\lambda$--plane.
\begin{table}[t!]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c||c||c|c|c||c|} \hline \hline
class & NGFPs & NGFP$_1$ & NGFP$_2$ & NGFP$_3$ & color code \\ \hline \hline
Class 0 & 0 & $-$ & $-$ & $-$ & black region \\ \hline \hline
%
Class Ia & 1 & UV, spiral & $-$ & $-$ & blue region \\ \hline
Class Ib & 1 & UV, real & $-$ & $-$ & green region \\ \hline
Class Ic & 1 & saddle & $-$ & $-$ & magenta region \\ \hline
Class Id & 1 & IR, spiral & $-$ & $-$ & red region \\ \hline
Class Ie & 1 & IR, real & $-$ & $-$ & orange region \\ \hline \hline
%
Class IIa & 2 & UV, real & IR, real & $-$ & open circle \\ \hline
Class IIb & 2 & UV, real & IR, spiral & $-$ & open square \\ \hline
Class IIc & 2 & UV, spiral & IR, spiral & $-$ & open triangle \\ \hline
Class IId & 2 & UV, spiral & UV, real & $-$ & open diamond \\ \hline
\hline
%
Class IIIa & 3 & UV, real & saddle & IR, real & filled circle \\ \hline
Class IIIb & 3 & UV, real & saddle & IR, spiral & filled square \\ \hline
Class IIIc & 3 & UV, spiral & saddle & IR, spiral & filled triangle \\ \hline
\hline
\end{tabular}
\end{center}
\caption{\label{Tab.4} Color-code for the fixed point classification provided in Fig.\ \ref{Fig.7}. The column NGFPs gives the number of NGFP solutions while the subsequent columns characterize their behavior in terms of 2 UV-attractive (UV), one UV-attractive and one UV-repulsive (saddle) and 2 IR-attractive (IR) eigendirections with real (real) and complex (spiral) critical exponents.}
\end{table}
The classification of the NGFPs identified in Fig.\ \ref{Fig.7a} according to their stability properties is provided in Fig.\ \ref{Fig.7} with the color-coding explained in Table \ref{Tab.4}. The left diagram provides the classification for the case of zero (black region) and one NGFP. Here
green and blue indicate the existence of a single UV-attractive NGFP with real (green) or complex (blue) critical exponents. Saddle points with one UV attractive and one UV repulsive eigendirection (magenta) and IR fixed points (red, orange) occur along a small wedge paralleling the $d_\lambda$-axis, only. The gray region supporting multiple NGFPs is magnified in the right diagram of Fig.\ \ref{Fig.7a}. All points in this region support at least one UV NGFP suitable for Asymptotic Safety while there is a wide range of possibilities for the stability properties of the second and third NGFP. Clearly, it would be interesting to study the RG flow resulting from the interplay of these fixed points. Since it will turn out (see Table \ref{Tab.3}), however, that there is no popular particle physics model situated in this regions, we postpone this investigation to a subsequent work.
The classification in Fig.\ \ref{Fig.7a} establishes that the existence of
a UV-attractive NGFP suitable for Asymptotic Safety is rather generic and puts only mild constraints on the admissible values $(d_g, d_\lambda)$. At this stage, it is interesting to relate this classification to phenomenologically interesting matter sectors including the standard model of particle physics (SM) and its most commonly studied extensions.\footnote{For a similar discussion within metric approach to Asymptotic Safety see \cite{Dona:2013qba}.} The result is summarized in Table \ref{Tab.3}. The map \eqref{dldgdef} allows to relate the number of scalars $N_S$, vector fields $N_V$ and Dirac fermions $N_D$ defining the field content of a specific matter sector to coordinates in the $d_g$-$d_\lambda$--plane. The resulting coordinates are given in the fifth and sixth column of Table \ref{Tab.3}.
\begin{table}[t!]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|p{3.94cm}||c|c|c||c|c||c|c||c|c|} \hline \hline
model & $N_S$ & $N_D$ & $N_V$ & $d_g$ & $d_\lambda$ & $g_*$ & $\lambda_*$ & $\theta_1$ & $\theta_2$ \\ \hline \hline
%
pure gravity & 0 & 0 & 0 & 0 & 0 & $0.78$ & $+\,0.32$ & \multicolumn{2}{c|}{$0.50 \pm 5.38 \, i$} \\ \hline
%
Standard Model (SM) & 4 & $\tfrac{45}{2}$ & 12 & $-\,\tfrac{61}{2}$ & $-\,62$ & $0.75$ & $-\,0.93$ & 3.871 & 2.057 \\ \hline
%
SM, dark matter (dm) & 5 & $\tfrac{45}{2}$ & 12 & $-\,\tfrac{59}{2}$ & $-\,61$ & $0.76$ & $-\,0.94$ & 3.869 & 2.058 \\ \hline
%
SM, $3\,\nu$ & 4 & 24 & 12 & $-\,32$ & $-\,68$ & $0.72$ & $-\,0.99$ & 3.884 & 2.057 \\ \hline
%
SM, $3\,\nu$, dm, axion & 6 & 24 & 12 & $-\,30$ & $-\,66$ & $0.75$ & $-\,1.00$ & 3.882 & 2.059 \\ \hline
%
MSSM & 49 & $\tfrac{61}{2}$ & 12 & $+\,\tfrac{13}{2}$ & $-\,49$ & $2.26$ & $-\,2.30$ & 3.911 & 2.154 \\ \hline
%
{SU(5) GUT} & {124} & {24} & {24} & $+\,76$ & $+\,76$ & $0.17$ & $+\,0.41$ & 25.26 & 6.008 \\ \hline
%
{SO(10) GUT} & {97} & {24} & {45} & $+\,28$ & $+\,91$ & $0.15$ & $+\,0.40$ & 19.20 & 6.010 \\ \hline \hline
\end{tabular}
\end{center}
\caption{\label{Tab.3} Fixed point structure arising from the field content of commonly studied matter models. All models apart from the minimally supersymmetric standard model (MSSM) and the grand unified theories (GUT), sit in the lower--left quadrant of Fig.\ \ref{Fig.7}. All matter configurations possess a single ultraviolet attractive NGFP with real critical exponents.}
\end{table}
Correlating these coordinates with the data provided by Fig.\ \ref{Fig.7} yields two important results: Firstly, all matter models studied in Table \ref{Tab.3} are located in regions of the $d_g$-$d_\lambda$--plane which host a single UV-attractive NGFP with real stability coefficients. Secondly, we note a qualitative difference between the standard model and its extensions (first five matter sectors) and grand unified theories (GUTs). The former all belong to the green region in the lower left part of the $d_g$-$d_\lambda$--plane while the second class of models sits in the upper-right quadrant. As a result, the corresponding NGFPs possess very distinct features. The NGFPs appearing in the first case have a characteristic product $g_* \lambda_* < 0$. Their critical exponents show a rather minor dependence on the precise matter content of the theory and have values in the range $\theta_1 \simeq 3.8 - 3.9$ and $\theta_2 \simeq 2.0$. In contrast, the NGFPs appearing in the context of GUT-type models come with a positive product $g_* \lambda_* > 0$. Their are significantly larger $\theta_1 > 19$ than in the former case and show a much stronger dependence on the matter field content. Thus while all matter sectors investigated in Table \ref{Tab.3} give rise to a NGFP suitable for realizing Asymptotic Safety the magnitude of the critical exponents hints that the SM-type theories may have more predictive power in terms of a lower number of relevant coupling constants in the gravitational sector.
At this stage it is also instructive to construct the phase diagram resulting from gravity coupled to the matter content of the standard model.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\textwidth]{phasediagramSM}
\caption{Phase diagram depicting the RG flow of gravity coupled to the matter content of the standard model in $D=3+1$ spacetime dimensions. Similarly to the pure gravity case, the phase diagram is dominated by the interplay of the NGFP (point ``A'') controlling the flow for ultra-high energies and the GFP (point ``O'') governing its low-energy behavior. The singular locus \eqref{etasing2} is depicted by the red line and arrows point towards lower values of $k$.
\label{flowSM} }
\end{center}
\end{figure}
Following the strategy of Sect.\ \ref{sect.42}, an illustrative sample of RG trajectories obtained from solving the beta functions \eqref{betafunction} for $(d_g, d_\lambda) = (- 61/2, - 62)$ is shown in Fig.\ \ref{flowSM}. Similarly to the case of pure gravity, the flow is dominated by the interplay of the NGFP situated at $(g_*,\lambda_*) = (0.75,-0.93)$ and the GFP in the origin. The NGFP controls the UV behavior of the trajectories while the GFP is responsible for the occurrence of a classical low-energy regime. The classification of possible low-energy behaviors is again given by the limits \eqref{Typeclass}. A notable difference to the pure gravity case is the absence of the inspiraling behavior of trajectories onto the NGFP. This reflects the property that the NGFPs of the gravity-matter models come with real critical exponents. Moreover, the shift of the NGFP to negative values $\lambda_*$ entails that the singularity \eqref{etasing2} (red line) no longer affects the crossover of the trajectories from the NGFP to the GFP. Notably, other matter sectors located in the lower-left green region of Fig.\ \ref{Fig.7} give rise to qualitatively similar phase diagrams so that the flow shown in Fig.\ \ref{flowSM} provides a prototypical showcase for this class of universal behaviors.
Owed to their relevance for cosmological model building, we close this section with a more detailed investigation of the fixed point structures appearing in gravity-scalar models with $N_V = N_D = 0$. For illustrative purposes we formally also include negative values $N_S$ in order to capture the typical behavior of matter theories located in the lower-left quadrant of Fig.\ \ref{Fig.7}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.45\textwidth]{scalargfix} \; \;
\includegraphics[width=0.45\textwidth]{scalarlambdafix} \\[2ex]
\includegraphics[width=0.45\textwidth]{scalarrethetafix} \; \;
\includegraphics[width=0.45\textwidth]{scalarimthetafix}
\caption{Position (top) and stability coefficients (bottom) of the UV NGFPs appearing in gravity-scalar systems as a function of $N_S$. The fixed point structure undergoes qualitative changes at $N_S \approx -6$ and $N_S \approx 46$ where the critical exponents change from real to complex values.\label{scalarfp}}
\end{center}
\end{figure}
Notably, all values $N_S$ give rise to a NGFP with two UV attractive eigendirections. The position $(\lambda_*, g_*)$ and stability coefficients of this family of fixed points is displayed in Fig.\ \ref{scalarfp}. The first noticeable feature is a sharp transition in the position of the NGFP occurring at $N_S \simeq -6$: for $N_S \le -6$ the NGFP is located at $\lambda_* < 0$ while for $N_S > -5$ one has $\lambda_* > 0$. For $N_S \rightarrow \infty$ the fixed point approaches $C\equiv(1/2,0)$, which can be shown to be a fixed point of the beta functions \eqref{betafunction} in the large $N_S$ limit. The value of the critical exponents shown in the lower line of Fig.\ \ref{scalarfp} indicates that there are two transitions: for $N_S \le -6$ there is a UV-attractive NGFP with two real critical exponents $\theta_1 \simeq 4$ and $\theta_2 \simeq 2$. These values are essentially independent on $N_S$. On the interval $-6 \le N_S \le 46$ the critical exponents turn into a complex pair. In particular for $N_S = 0$, one recovers the pure gravity fixed point NGFP \eqref{NGFP1}. For $N_S > 46$ one again has a UV-attractive NGFP with two real critical exponents with one of the critical exponents becoming large. Thus we clearly see a qualitatively different behavior of the NGFPs situated in the upper-right quadrant (relevant for GUT-type matter models) and the NGFPs in the lower-left quadrant (relevant for the standard model) of Fig.\ \ref{Fig.7}, reconfirming that the Asymptotic Safety mechanism realized within these classes of models is of a different nature.
\section{Summary and outlook}
\label{sect.7}
This work uses the functional renormalization group equation (FRGE) for the effective average action $\Gamma_k$ \cite{Wetterich:1992yh,Morris:1993qb,Reuter:1993kw,Reuter:1996cp} adapted to the Arnowitt-Deser-Misner (ADM) formalism \cite{Manrique:2011jc,Rechenberger:2012dt} to study the renormalization group flow of Newton's constant and the cosmological constant for minimally coupled gravity-matter models. As an important conceptual advantage the resulting construction equips spacetime with a natural foliation structure.
The resulting distinguished ``time''-direction may be used to implement a Wick rotation from Euclidean to Minkowski signature.
The ADM-formalism expresses the metric degree's of freedom in terms of a Lapse function, a shift vector and a metric measuring distances on spatial slices, see eq.\ \eqref{fol1}. The key difficulty in obtaining a well-defined off-shell flow equation for this case originates from the fact that the Lapse function and the shift vector appear as Lagrange multipliers. Implementing ``proper-time gauge'' \cite{Dasgupta:2001ue}, using the freedom of choosing a coordinate system to eliminating the fluctuations in the Lapse function and shift vector, leads to non-canonical propagators for the remaining fluctuation fields, see Table \ref{Tab.1}. Following \cite{Biemans:2016rvp}, our work bypasses this obstruction by implementing a new Feynman-type gauge fixing for the ADM-fields.
The main virtue of the construction is that all fields, including the Lagrange multipliers and ghosts, obtain regular, relativistic dispersion relations. Moreover, all component fields propagate with the same speed of light when the dispersion relations are evaluated in a Minkowski background. This condition fixes the gauge choice uniquely up to a physically irrelevant ${\mathbb Z}_2 \times {\mathbb Z}_2$ symmetry. The construction is reminiscent to Feynman gauge in quantum electrodynamics where the gauge-fixing provides a suitable kinetic term for the time-component of the gauge-potential.
In this work we apply the resulting flow equation to study the scale-dependence of Newton's constant and the cosmological constant for gravity minimally coupled to an arbitrary number of free scalar, vector, and Dirac fields. In this case, it suffices to evaluate the general FRGE on a flat Friedmann-Robertson-Walker background.
The beta functions encoding the projected flow are encoded in the volume factor and extrinsic curvature terms constructed from the background. In this way our construction bypasses on of the main limitations of the Matsubara-type computations \cite{Manrique:2011jc,Rechenberger:2012dt} where time-direction was taken compact.
Our central result for the case of pure gravity is the phase diagram shown in Fig.\ \ref{flow2d3d}. Structurally, the result matches the phase diagrams obtained from studying similar RG flows in the metric formulation \cite{Reuter:2001ag,Litim:2003vp,Donkin:2012ud,Nagy:2013hka} and
from the evaluation of Lorentzian RG flows based on the Matsubara-formalism \cite{Manrique:2011jc,Rechenberger:2012dt}. In particular,
we recover the key element of Asymptotic Safety, a UV-attractive non-Gaussian fixed point (NGFP).
Subsequently, we classify the fixed point structure for gravity minimally coupled to an arbitrary number of free matter fields. We observe that the contribution of the matter sector can be encoded in a two-parameter deformation of the beta functions resulting from the pure gravity case and we give an explicit map between the field content of the model and the deformation parameters. In terms of the deformation parameters, it is found that the occurrence of a NGFP suitable for Asymptotic Safety is rather generic (see Fig. \ref{Fig.7}). In particular the field content of the standard model (and also its most commonly studied extensions) gives rise to a UV fixed point with real critical exponents. Moreover, our classification reveals that certain models with a low number of massless matter fields also admit an additional infrared fixed point which could provide the completion of the RG flow at low energy.
Our findings complement earlier studies based on the metric formalism \cite{Dona:2013qba,Meibohm:2015twa} by clearly demonstrating that the NGFPs responsible for Asymptotic Safety appearing for gravity coupled to the matter content of the standard model and grand unified type theories are qualitatively different.
The setup developed in this work provides an important stepping stone for future developments of the gravitational Asymptotic Safety program. Conceptually, the results reported in this work remove one of the main obstructions for computing transition amplitudes between spatial geometries at different instances in time. Moreover, they provide a solid starting ground for computing real time correlation functions by combining the FRGE for the ADM formalism with the ideas advocated in \cite{Floerchinger:2011sc,Pawlowski:2015mia}. On the phenomenological side, our work evaluates the flow equation \eqref{FRGE} on a flat Friedmann-Robertson-Walker background and captures the gravitational fluctuations through the component fields typically used in cosmic perturbation theory. This setup is easily extended by including scalar fields and we showed explicitly that the Asymptotic Safety mechanism remains intact for this case. These features make the present framework predestined for studying the scale-dependence of cosmic perturbations within the Asymptotic Safety program also in the context of single-field inflationary models. We hope to come back to these points in the future. \\[2ex]
\paragraph*{Acknowledgements.}
We thank N.\ Alkofer, A.\ Bonanno, W.\ Houthoff, A.\ Kurov, R.\ Percacci, M.\ Reuter, and C.\ Wetterich for helpful discussions. The research of F.~S.
is supported by the Netherlands Organisation for Scientific
Research (NWO) within the Foundation for Fundamental Research on Matter (FOM) grants 13PR3137 and 13VP12.
\begin{appendix}
\section{The flat Friedmann-Robertson-Walker background}
\label{App.A}
Throughout this work, we evaluate the flow equation on a flat (Euclidean) Friedmann-Robertson-Walker background
\begin{equation}\label{FRWback2}
\bar{g}_{\mu\nu} = {\rm diag} \left[ \, 1 \, , \, a(\tau)^2 \, \delta_{ij}\right] \qquad \Longleftrightarrow \qquad
\bar{N} = 1 \, , \quad \bar{N}_i = 0 \, , \quad \bar{\sigma}_{ij} = a(\tau)^2 \, \delta_{ij} \, .
\end{equation}
In this background the projectors \eqref{proj1} take a particularly simple form
%
\begin{equation}\label{proj2}
t^\mu = \big( \, 1 \, , \, \vec{0} \, \big) \, , \qquad
e_i{}^\mu = \big( \vec{0} \, , \, \delta_i^j \big) \, ,
\end{equation}
%
implying that $t^\mu$ is always normal to the spatial hypersurface $\Sigma_\tau$.
The extrinsic and intrinsic curvature tensors of this background satisfy
\begin{equation}\label{curvatures}
\bar{K}_{ij} = \tfrac{1}{d} \, \bar{K} \, \bar{\sigma}_{ij} \, , \qquad \bar{R} = 0 \, , \qquad \bar{D}_i = \partial_i
\end{equation}
where $\bar{K} \equiv \bar{\sigma}^{ij} \, \bar{K}_{ij}$. Moreover, the Christoffel-connection on the spatial slices vanishes such that $\bar{D}_i = \partial_i$.
In order to evaluate the operator traces appearing in the flow equation it is useful to resort to heat-kernel techniques with respect to the background spacetime \eqref{FRWback2}. For this purpose, we observe that \eqref{proj2} entails that there is a canonical ``lifting'' of vectors tangent to the spatial slice to $D$-dimensional vectors
\begin{equation}\label{uplift}
v^i(\tau,y) \quad \mapsto \quad v^\mu(\tau,y) \equiv (0 \, , \, v^i(\tau,y))^{\rm T} \, .
\end{equation}
The $D$-dimensional Laplacian $\Delta_s \equiv - \bar{g}^{\mu\nu} \bar{D}_\mu \bar{D}_\nu$ ($s=0,1,2$) naturally acts on these $D$-vectors. In order to rewrite the variations in terms of $D$-covariant quantities, we exploit that $\Delta_s$ can be expressed in terms of the flat space Laplacian $\square \equiv - \partial_\tau^2 - \bar{\sigma}^{ij} \partial_i \partial_j$ and the extrinsic curvature.
For the Laplacian acting on $D$-dimensional fields with zero, one, and two indices one has
\begin{equation}\label{LapDala}
\begin{split}
\Delta_0 \phi = & \, \Big( \square - \bar{K} \partial_\tau \Big) \phi \, , \\
%
\Delta_1 \phi_\mu = & \, \Big( \square - \tfrac{d-2}{d} \, \bar{K} \partial_\tau + \tfrac{1}{d} (\partial_\tau \bar{K}) + \tfrac{1}{d} \bar{K}^2 \Big) \phi_\mu \, , \\
%
\Delta_2 \phi_{\mu\nu} = & \,\Big( \square - \tfrac{d-4}{d} \, \bar{K} \partial_\tau + \tfrac{2}{d} (\partial_\tau \bar{K}) + \tfrac{2(d-1)}{d^2} \bar{K}^2 \Big) \phi_{\mu\nu} \, . \\
\end{split}
\end{equation}
When evaluating the traces by covariant heat-kernel methods, we then use the embedding map \eqref{uplift} together with the completion \eqref{LapDala} to express the operator $\square$ in terms of $\Delta_i$.
The operator traces appearing in \eqref{FRGE} are conveniently evaluated using standard heat-kernel formulas for the $D$-dimensional Laplacians \eqref{LapDala}
\begin{equation}\label{heat1}
{\rm Tr}_i \, e^{-s \left( \Delta_i + E \right) } \simeq \frac{1}{(4\pi s)^{D/2}} \int d^Dx \sqrt{g} \, \Big[ {\rm tr}_i \, {\mathbb 1} + s \left( \tfrac{1}{6} \, {}^{(D)}R \, {\rm tr}_i \, {\mathbb 1} - {\rm tr}_i \, E \right) + \ldots \Big] \, .
\end{equation}
Here ${\rm tr}_i$ is a trace over the internal space and the dots indicate terms build from four and more covariant derivatives, which do not contribute to the present computation. For the FRW background, the spacetime curvature ${}^{(D)}R$ can readily be replaced by the extrinsic curvature, evoking
\begin{equation}\label{rep1}
\int d^Dx \sqrt{\bar{g}} \; {}^{(D)}R = \int d\tau d^dy \sqrt{\bar{\sigma}} \left[ \tfrac{d-1}{d} \bar{K}^2 \right] \, .
\end{equation}
Combining the diagonal form of the projectors \eqref{proj2} with the $D$-dimensional heat-kernel expansion \eqref{heat1} allows to write operator traces for the component fields. On the flat FRW background these have the structure
\begin{equation}\label{heat2}
{\rm Tr}_i \, e^{-s \Delta_i} = \frac{1}{(4\pi s)^{D/2}} \, \int d\tau d^dy \sqrt{\bar{\sigma}} \, \Big[ a_0 + a_2 \, s \, \bar{K}^2 + \ldots \Big] \, ,
\end{equation}
The coefficient $a_n$ depend on the index structure $i$ of the fluctuation field and are listed in Table \ref{Tab.heat}.
\begin{table}[t!]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \hline
& $S$ & $V$ & $T$ & $TV$ & $TTT$ \\ \hline
$a_0$ & $1$ & $d$ & $\tfrac{1}{2}d(d+1)$ & $d-1$ & $\tfrac{1}{2}(d+1)(d-2)$ \\
$a_2$ & $\frac{d-1}{6d}$ & $\frac{d-1}{6}$ & $\frac{(d-1)(d+1)}{12}$ &
$\frac{d^3-2d^2+d+6}{6d^2}$ & $\frac{d^4-2d^3-d^2+14d+36}{12d^2}$ \\
\hline \hline
\end{tabular}
\caption{\label{Tab.heat} Heat-kernel coefficients for the component fields appearing in the decompositions \eqref{TTshift} and \eqref{TTmet}. Here $S$, $V$, $T$, $TV$, and $TTT$ are scalars, vectors, symmetric two-tensors, transverse vectors, and transverse traceless symmetric matrices, respectively. }
\end{center}
\end{table}
This result \eqref{heat2} is the key ingredient for evaluating the operator traces of the flow equation on a flat FRW background.
\section{Hessians in a Friedmann-Robertson-Walker background}
\label{App.B}
The evaluation of the flow equation \eqref{FRGE} requires the Hessian $\Gamma_k^{(2)}$. The technical details of this calculation
are summarized in this appendix. In the sequel, indices are raised and lowered with the background metric $\bar{\sigma}_{ij}$. Moreover, we introduce the shorthand notations
\begin{equation}\label{shorthand}
\int_x \equiv \int \, d\tau \, d^dy \, \sqrt{\bar{\sigma}} \; , \qquad \mbox{and} \qquad \hat{\sigma} \equiv \bar{\sigma}^{ij} \, \hat{\sigma}_{ij}
\end{equation}
to lighten the notation and use $\Delta \equiv - \bar{\sigma}^{ij} \partial_i \partial_j$ to denote the Laplacian on the spatial slices.
\subsection{Hessians in the gravitational sector: decomposition of fluctuations}
\label{App.B1}
When constructing $\Gamma_k^{(2)}$, it is convenient to consider \eqref{GammaEH} as a linear combination of the interaction monomials \eqref{imon}. These monomials are then expanded in terms of the fluctuation fields according to
\begin{equation}
N = \bar{N} + \hat{N} \, , \qquad N_i = \bar{N}_i + \hat{N}_i \, , \qquad \sigma_{ij} = \bar{\sigma}_{ij} + \hat{\sigma}_{ij} \, .
\end{equation}
As an intermediate result, we note that the expansion of the extrinsic curvature \eqref{Kext} around the FRW background is given by
\begin{equation}\label{extvar}
\begin{split}
\delta K_{ij} = & \, - \hat{N} \, \bar{K}_{ij} + \tfrac{1}{2} ( \partial_{\tau} \hat{\sigma}_{ij} -\partial_i \hat{N}_j - \partial_j \hat{N}_i ) \, , \\
\delta^2 K_{ij} = & \,
{2}\, \hat{N}^2 \, \bar{K}_{ij}
- \hat{N} \left( \partial_{\tau} \hat{\sigma}_{ij} -\partial_i \hat{N}_j - \partial_j \hat{N}_i \right)
+ \hat{N}^k \left( \partial_i \hat{\sigma}_{jk} + \partial_j \hat{\sigma}_{ik} - \partial_k \hat{\sigma}_{ij} \right) \, ,
\end{split}
\end{equation}
were $\delta^n$ denotes the order of the expression in the fluctuation fields.
For later reference, it is also useful to have the explicit form of these expressions contracted withthe inverse background metric
\begin{equation}\label{traceextvar}
\begin{split}
\bar{\sigma}^{ij} \left( \delta K_{ij} \right) = & \, - \hat{N} \bar{K} + \tfrac{1}{2} \bar{\sigma}^{ij} (\partial_\tau \hat{\sigma}_{ij}) - \partial^i \hat{N}_i \, , \\
%
\bar{\sigma}^{ij} \left(\delta^2 K_{ij} \right) = & \, {2} \, \hat{N}^2 \bar{K} - \hat{N} \bar{\sigma}^{ij} \left( \partial_\tau \hat{\sigma}_{ij} \right) + 2 \hat{N} \partial^i \hat{N}_i + \hat{N}^k \left(2 \, \partial^i \, \hat{\sigma}_{ik} - \partial_k \, \hat{\sigma} \right) \, .
\end{split}
\end{equation}
Expanding the interaction monomials \eqref{imon}, the terms quadratic in
the fluctuation fields are
\begin{equation}\label{varI1}
\begin{split}
\delta^2 I_1 = & \int_x \Big[
2 (\delta K_{ij}) \bar{\sigma}^{ik} \bar{\sigma}^{jl} (\delta K_{kl})
+ \tfrac{2}{d} \, \bar{K} \, \bar{\sigma}^{ij} \left( \delta^2 K_{ij} + (\delta K_{ij}) (2 \hat{N} + \hat{\sigma} ) \right) \\ & \qquad
- \tfrac{8}{d} \, \bar{K} \, \hat{\sigma}^{ij} \, (\delta K_{ij})
+ \tfrac{1}{d} \, \bar{K}^2 \left( \tfrac{d-4}{d} \hat{N} \hat{\sigma} + \tfrac{d-8}{4d} \hat{\sigma}^2 - \tfrac{d-12}{2d} \hat{\sigma}_{ij} \hat{\sigma}^{ij} \right)
\Big] \, , \\
%
\delta^2 I_2 = & \int_x \Big[ 2 (\bar{\sigma}^{ij} \, \delta K_{ij})^2 + 2 \bar{K} \, \bar{\sigma}^{ij} \left( \delta^2 K_{ij} + (\delta K_{ij}) (2 \hat{N} + \hat{\sigma} ) \right) - 4 \bar{K} \hat{\sigma}^{ij} \, (\delta K_{ij}) \\ & \qquad
+ \bar{K}^2 \left( \tfrac{d-4}{d} \hat{N} \hat{\sigma} + \tfrac{d^2-8d+8}{4d^2} \hat{\sigma}^2 - \tfrac{d-8}{2d} \hat{\sigma}_{ij} \hat{\sigma}^{ij} \right)
{ - \tfrac{4}{d} \bar{K} \hat{\sigma} \bar{\sigma}^{ij} \delta K_{ij}} \Big] \, , \\
%
\delta^2 I_3 = & \, \int_x \left[ \big( 2 \hat{N} + \hat{\sigma} \big)\big(\partial_i \partial_j \hat{\sigma}^{ij} + \Delta \hat{\sigma} \big) - \tfrac{1}{2} \hat{\sigma}_{ij} \Delta \hat{\sigma}^{ij} - \tfrac{1}{2} \hat{\sigma} \Delta \hat{\sigma} + \big(\partial_i \hat{\sigma}^{ik}\big) \big(\partial_j \hat{\sigma}^j{}_k \big) \right] \, , \\
%
\delta^2 I_4 = & \, \int_x \left[ \hat{N} \hat{\sigma} + \tfrac{1}{4} \hat{\sigma}^2 - \tfrac{1}{2} \hat{\sigma}^{ij} \hat{\sigma}_{ij} \right] \, .
\end{split}
\end{equation}
In order to arrive at the final form of these expressions, we integrated by parts and made manifest use of the geometric properties of the background \eqref{curvatures}.
In order to develop a consistent gauge-fixing scheme and to simplify the structure of the flow equation it is useful to carry out a further transverse-traceless decomposition of the fluctuation fields entering into
\eqref{varI1}. A very convenient choice is provided by the standard decomposition of the fluctuation fields used in cosmic perturbation theory (see, e.g., \cite{Baumann:2009ds} for a detailed discussion) where the shift vector and the metric on the spatial slice are rewritten according to
\begin{equation}\label{TTdec}
\begin{split}
\hat{N}_i = & \, u_i + \partial_i \, \tfrac{1}{\sqrt{\Delta}} \, B \, , \\
\hat{\sigma}_{ij} = & \, h_{ij} - \left( \bar{\sigma}_{ij} + \partial_i \partial_j \, \tfrac{1}{\Delta} \right) \psi + \partial_i \partial_j \, \tfrac{1}{\Delta} \, E + \partial_i \tfrac{1}{\sqrt{\Delta}} v_j + \partial_j \, \tfrac{1}{\sqrt{\Delta}} \, v_i \, .
\end{split}
\end{equation}
The component fields are subject to the constraints
\begin{equation}
\partial^i \, u_i = 0 \, , \qquad \partial^i \, h_{ij} = 0 \, , \quad \bar{\sigma}^{ij} h_{ij} = 0 \, , \quad \partial^i v_i = 0 \, ,
\end{equation}
indicating that $u_i$ and $v_i$ are transverse vectors and $h_{ij}$ is a transverse-traceless tensor. Notably, the partial derivatives and $\Delta$ can be commuted freely, since the background metric is independent of the spatial coordinates. The normalization of the component fields has been chosen such that the change of integration variables does not give rise to non-trivial Jacobians. This can be seen from noting
\begin{equation}\label{aux1}
\begin{split}
\hat{N}_i \, \hat{N}^i = & \, u_i \, u^i + B^2 \, , \\
\hat{\sigma}_{ij} \, \hat{\sigma}^{ij} = & \, h_{ij} h^{ij} + (d-1) \, \psi^2 + E^2 + 2 \, v_i \, v^i \, , \\
\end{split}
\end{equation}
implying that a Gaussian integral over the ADM fluctuations leads to a Gaussian integral in the component fields which does not give rise to operator-valued determinants.
The final step expresses the variations \eqref{varI1} in terms of the component fields \eqref{TTdec}. The rather lengthy computation can be simplified by using the identities
\begin{equation}\label{aux2}
\begin{split}
\hat{\sigma} = - (d-1) \psi - E \, , \qquad
\partial^i \hat{\sigma}_{ij} = - \partial_j E - \sqrt{\Delta} \, v_j \, , \qquad
\partial^i \partial^j \hat{\sigma}_{ij} = \Delta E \, .
\end{split}
\end{equation}
together with the relations \eqref{aux1} and \eqref{aux2}.
Starting with the kinetic terms appearing in
$\delta^2 I_1$ and $\delta^2 I_2$,
\begin{equation}\label{kinterms}
{{\mathbb{K}}}_1 \equiv \, 2 \int_x \,
(\delta K_{ij}) \, \bar{\sigma}^{ik} \bar{\sigma}^{jl} \, (\delta K_{kl}) \, , \qquad
{{\mathbb{K}}}_2 \equiv \, 2 \int_x (\bar{\sigma}^{ij} \, \delta K_{ij})^2 \, ,
\end{equation}
the resulting expressions written in terms of component fields are
\begin{equation}
\begin{split}
{\mathbb{K}}_1 = & \, \int_x \Big[
- \tfrac{1}{2} \, h^{ij} \left( \partial_\tau^2 + \tfrac{d-4}{d} \bar{K} \partial_\tau \right) h_{ij}
- \tfrac{d-1}{2} \, \psi \left(\partial_\tau + \tfrac{d-2}{d} \bar{K} \right) \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \psi \\ & \qquad \;
- \tfrac{1}{2} E \left(\partial_\tau + \tfrac{d-2}{d} \bar{K} \right) \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) E
- v^i \left( \partial_\tau + \tfrac{d-3}{d} \bar{K} \right) \left( \partial_\tau + \tfrac{1}{d} \bar{K} \right) v_i \\ & \qquad \;
- 2 \, B \, \sqrt{\Delta} \, \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) E
- 2 \, u^i \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \sqrt{\Delta} \, v_i + 2 B \Delta B \\ & \qquad \;
+ u_i \Delta u^i - \tfrac{4}{d} \bar{K} \hat{N} \, \sqrt{\Delta} \, B
+ \tfrac{2}{d} \bar{K} \hat{N} \, \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \, \big((d-1) \psi + E \big) \\ & \qquad \;
+ \tfrac{2}{d} \, \bar{K}^2 \, \hat{N}^2
\Big] \, ,
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
{\mathbb{K}}_2 = & \, \int_x \Big[ - \tfrac{1}{2} \big((d-1)\psi + E\big) \left(\partial_\tau + \tfrac{d-2}{d} \bar{K} \right) \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \big((d-1)\psi + E\big) \\ & \qquad \;
-2 B \sqrt{\Delta} \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \big((d-1)\psi + E\big) + 2 B \Delta B \\
& \qquad \;
- 4 \bar{K} \hat{N} \sqrt{\Delta} \, B
+ 2 \bar{K} \hat{N} \left( \partial_\tau + \tfrac{2}{d} \bar{K} \right) \big((d-1)\psi + E\big)
+ 2 \bar{K}^2 \hat{N}^2
\Big] \, .
\end{split}
\end{equation}
On this basis one finds that
\begin{equation}\label{I1res}
\begin{split}
\delta^2 I_1 = {\mathbb{K}}_1 - \, \int_x \Big[ \; & + \tfrac{(d-4)(d-8)}{4d^2} \, \bar{K}^2 \hat{N} \, \big( (d-1)\psi + E \big) \\
&
+ \tfrac{4}{d} \, \bar{K} \, h^{ij} \big( \partial_\tau + \tfrac{d-12}{8d} \bar{K} \big) h_{ij} \\
&
- \tfrac{4}{d} \, \bar{K} \, \big( u^k \, \sqrt{\Delta} \, v_k + B \, \sqrt{\Delta} \, E - v^i \, ( \partial_\tau + \tfrac{d-8}{4d} \, \bar{K}) \, v_i \big) \\
&
- \tfrac{1}{d} \, \bar{K} \, \big((d-1) \psi + E \big) \big(\partial_\tau + \tfrac{1}{4} \bar{K} \big) \big((d-1) \psi + E \big) \\
&
+ \tfrac{4}{d} \, \bar{K} \, E \, \big(\partial_\tau + \tfrac{d+4}{8d} \, \bar{K} \big) \, E
+ \tfrac{4(d-1)}{d} \, \bar{K} \, \psi \,\big(\partial_\tau + \tfrac{d+4}{8d} \, \bar{K} \big) \, \psi
\Big]
\end{split}
\end{equation}
and
\begin{equation}\label{I2res}
\begin{split}
\delta^2 I_2 = {\mathbb{K}}_2 - \, \int_x \Big[ \; & - \tfrac{d-4}{d} \, \bar{K}^2 \hat{N} \big( (d-1) \psi + E \big) + 2 \, \bar{K} \, h^{ij} \, \big(\partial_\tau +\tfrac{d-8}{4d} \, \bar{K} \big) \, h_{ij} \\
&
- \bar{K} \, \big((d-1) \psi + E \big) \big(\tfrac{d-2}{d} \, \partial_\tau + \tfrac{d^2-8}{4d^2} \, \bar{K} \, \big) \big((d-1) \psi + E \big) \\
&
+ 2\, \bar{K} \, E \, \big(\partial_\tau +\tfrac{1}{4} \, \bar{K} \big) \, E
+ 2 (d-1) \bar{K} \, \psi \, \big(\partial_\tau +\tfrac{1}{4} \, \bar{K} \big) \, \psi \\
&
+2 \bar{K} v^i \, \big(\partial_\tau +\tfrac{d-6}{2d} \, \bar{K} \big) \, v_i {- \tfrac{4}{d} \bar{K} \left( (d-1) \psi + E \right) \sqrt{\Delta} B }
\Big] \, .
\end{split}
\end{equation}
Finally, $\delta^2 I_3$ and $\delta^2 I_4$ written in terms of the component fields are
\begin{equation}\label{I3res}
\begin{split}
\delta^2 I_3 = & \, \int_x \Big[
\tfrac{ (d-1) (d-2)}{2} \, \psi \Delta \psi - \tfrac{1}{2} \, h_{ij} \Delta h^{ij} - 2 \, (d-1) \, \hat{N} \Delta \psi
\Big] , \\
%
\delta^2 I_4 = & \int_x \Big[ \tfrac{(d-1)(d-3)}{4} \psi^2 + \tfrac{(d-1)}{2} \psi E - \tfrac{1}{4} E^2
- \hat{N} \big( (d-1) \psi + E \big) - \tfrac{1}{2} h_{ij} h^{ij} - v_i v^i \Big] .
\end{split}
\end{equation}
Combining these variations according to \eqref{GammaEH} one
arrives the matrix entries for $\delta^2\Gamma_k^{\text{grav}}$. In the flat space limit, where $\bar{K}=0$, these entries are listed in the second column of Table \ref{Tab.1}.
\subsection{Gauge-fixing terms}
\label{App.B3}
Following the strategy of the previous subsection it is useful to also decompose the gauge-fixing terms \eqref{gf:ansatz} into two interaction monomials
\begin{equation}\label{I56def}
\delta^2 I_5 \equiv \int_x \, F^2 \, , \qquad \delta^2 I_6 \equiv \int_x F_i \, \bar{\sigma}^{ij} \, F_j \, .
\end{equation}
Since the functionals $F$ and $F_i$ defined in eq.\ \eqref{gauge1} are linear in the fluctuation fields, the gauge-fixing terms are quadratic in the fluctuations by construction. This feature is highlighted by adding the $\delta^2$ to the definition of the monomials.
Substituting the explicit form of $F$ and $F_i$ and recasting the resulting expressions in terms of the component fields \eqref{TTdec}
one finds
\begin{equation}\label{d2I5}
\begin{split}
\delta^2 I_5 = & \, \int_x \Big[ c_2^2 \, B \Delta B
- \hat{N} \, \left(c_1 \partial_\tau + (c_1 - c_9) \bar{K} \right) \left( c_1 \partial_\tau + c_9 \bar{K} \right) \hat{N} \\ & \quad
- \big((d-1)\psi + E\big) \left(c_3 \partial_\tau + (c_3 - c_8) \bar{K} \right) \left( c_3 \partial_\tau + c_8 \bar{K} \right) \big((d-1)\psi + E\big) \\ & \quad
- 2 \, c_2 \, B \, \sqrt{\Delta} \, \left(c_1 \, \partial_\tau + c_9 \bar{K} \right) \, \hat{N}
\\ & \quad
+ 2 \hat{N} \, \left(c_1 \partial_\tau + (c_1-c_9) \bar{K} \right) \left(c_3 \partial_\tau + c_8 \bar{K} \right) \big((d-1)\psi + E\big)
\\ & \quad
+2 \, c_2 \, B \sqrt{\Delta} \, \left(c_3 \partial_\tau + c_8 \bar{K} \right) \, \big((d-1)\psi + E\big)
\Big]
\end{split}
\end{equation}
and
\begin{equation}\label{d2I6}
\begin{split}
\delta^2 I_6 = & \, \int_x \Big[ c_5^2 \, \hat{N} \Delta \hat{N}
- \, u^i \, \big(c_4 \, \partial_\tau + ( \tfrac{d-2}{d} c_4 - c_{10} ) \, \bar{K} \big) \big( c_4 \partial_\tau + c_{10} \bar{K} \big) \, u_i \\ & \; \; \qquad
- B \big(c_4 \, \partial_\tau + ( \tfrac{d-1}{d} c_4 - c_{10} ) \, \bar{K} \big) \big( c_4 \partial_\tau + (\tfrac{1}{d} \, c_4 + c_{10}) \, \bar{K} \big) \, B
\\ & \; \; \qquad
+2 \, c_5 \, \hat{N} \big( c_4 \partial_\tau + (\tfrac{2}{d} c_4 + c_{10}) \, \bar{K} \big) \sqrt{\Delta} B \\ & \; \; \qquad
%
-2 \, c_6 \, \big((d-1)\psi + E\big) \big(c_4 \partial_\tau + (\tfrac{2}{d} c_4 + c_{10}) \bar{K} \big) \sqrt{\Delta} B
\\ & \; \; \qquad
-2 \, c_7 \, E \big(c_4 \partial_\tau + (\tfrac{2}{d} c_4 +c_{10}) \bar{K} \big) \sqrt{\Delta} B \\ & \; \; \qquad
- 2 \, c_7 \, v^i \sqrt{\Delta} \, ( c_4 \partial_\tau + c_{10} \bar{K} ) \, u_i
-2 c_5 c_6 \, \hat{N} \Delta \, \big((d-1)\psi + E\big)
\\ & \; \; \qquad
-2 \, c_5 \, c_7 \, \hat{N} \Delta E
+ c_6^2 \, \big((d-1)\psi + E\big) \Delta \big((d-1)\psi + E\big)
\\ & \; \; \qquad
+ 2 c_6 c_7 \, \big((d-1)\psi + E\big) \Delta E
+ c_7^2 \, \big(E\Delta E + v^i \Delta v_i \big)
\Big] \, .
\end{split}
\end{equation}
Here again we made use of the geometric properties of the background and integrated by parts in order to obtain a similar structure as in the gravitational sector.
Combining the results \eqref{I1res}, \eqref{I2res}, \eqref{I3res}, \eqref{d2I5}, and \eqref{d2I6}, taking into account the relative signs between the terms and restoring the coupling constants according to \eqref{GammaEH2} gives the part of the gauge-fixed gravitational action quadratic in the fluctuation fields. The explicit result is rather lengthy and given by
\begin{equation}\label{eqS1B}
\begin{split}
& \, 32 \pi G_k \Big( \tfrac{1}{2} \delta^2 \Gamma^{\rm grav}_k + \Gamma_k^{\rm gf} \Big) = \\ & \,
\int_x \Big\{ - \hat{N} \left[ (c_1 \partial_\tau +(c_1 - c_9) \bar{K})(c_1 \partial_\tau + c_9 \bar{K}) - c_5^2 \, \Delta + {\tfrac{2(d-1)}{d} \, \bar{K}^2} \right] \hat{N} \\ & \qquad
%
- B \, \left[ (c_4 \partial_\tau + ( \tfrac{d-1}{d} c_4 - c_{10}) \bar{K})(c_4 \partial_\tau +( \tfrac{1}{d} c_4 + c_{10})\bar{K}) - c_2^2 \, \Delta \right] B \\ & \qquad
%
- 2 \, B \, \sqrt{\Delta} \left[ (c_1 c_2 + c_4 c_5) \partial_\tau
+ (c_2 c_9 + c_4 c_5 \, \tfrac{d-2}{d} - c_5 c_{10} -
\tfrac{2(d-1)}{d}) \bar{K} \right] \hat{N} \\ & \qquad
%
+ 2 \, \hat{N} \Big[
(c_1 \partial_\tau + (c_1 - c_9)\bar{K})(c_3 \partial_\tau + c_8 \bar{K}) - \tfrac{d-1}{d} \bar{K} \partial_\tau
- c_5 (c_6 + c_7) \Delta \\ & \qquad \qquad \quad
- {\tfrac{5d^2-12d+16}{8d^2}} \bar{K}^2 - \Lambda_k
\Big] E \\ & \qquad
%
+ 2 (d-1) \, \hat{N} \Big[
(c_1 \partial_\tau + (c_1 - c_9) \bar{K})(c_3 \partial_\tau +c_8 \bar{K}) - \tfrac{d-1}{d} \bar{K} \partial_\tau \\ & \qquad \qquad \quad
+ (1 - c_5 c_6) \Delta - {\tfrac{5d^2-12d+16}{8d^2}} \bar{K}^2 - \Lambda_k
\Big] \psi \\ & \qquad
%
+ 2 \, B \sqrt{\Delta} \left[ \big(c_2 c_3 + c_4(c_6+c_7) \big) \partial_\tau +
\big( c_2 c_8 + (c_6 + c_7)(\tfrac{d-2}{d} c_4 - c_{10}) \big) \bar{K} \right] E \\ & \qquad
%
%
+2 (d-1) B \sqrt{\Delta} \left[
\big(1 + c_2c_3+c_4c_6\big) \partial_\tau + \big( c_2 c_8 + \tfrac{d-2}{d} c_4 c_6 - c_6 c_{10} \big)
\bar{K}\right] \psi \\ & \qquad
%
- (d-1) \, \psi \, \Big[ (d-1)\left((c_3 \partial_\tau + (c_3 -c_8)\bar{K})(c_3 \partial_\tau + c_8 \bar{K}) - c_6^2 \Delta \right)
\\ & \qquad \qquad \quad
+ \tfrac{d-2}{2} ( - \partial_\tau^2 + \Delta - {\tfrac{2}{d} } \dot{\bar{K}} ) +
{\tfrac{d^2 - 10d +14}{2d} } \bar{K} \partial_\tau + {\tfrac{d^2-8d+11}{4d} } \, \bar{K}^2 - \tfrac{d-3}{2} \Lambda_k
\Big] \psi
\\ & \qquad
%
+ E \Big[ (c_6 +c_7)^2 \Delta -(c_3 \partial_\tau + (c_3 - c_8) \bar{K})(c_3 \partial_\tau + c_8 \bar{K})
\\ & \qquad \qquad \quad
- \tfrac{1}{2} \Lambda_k + { \tfrac{d-1}{d}} \bar{K} \partial_\tau + {\tfrac{d-1}{4d}} \, \bar{K}^2 \Big] E \\ & \qquad
%
+ (d-1) \, \psi \Big[ 2 c_6 (c_6 + c_7) \Delta -2 (c_3 \partial_\tau + (c_3 -c_8)\bar{K})(c_3 \partial_\tau + c_8 \bar{K})
\\ & \qquad \qquad \quad
+ \partial_\tau^2 + {\bar{K} \partial_\tau } + {\tfrac{d-1}{d} \dot{\bar{K}} } + {\tfrac{d-1}{2d}} \bar{K}^2 + {\Lambda_k}
\Big] E
\\ & \qquad
%
- u^i \left[ \big(c_4 \partial_\tau + (\tfrac{d-2}{d} c_4 -c_{10})\bar{K}\big) \big(c_4 \partial_\tau + c_{10} \bar{K} \big) - \Delta\right] u_i \\ & \qquad
+ v^i \left[ - \partial_\tau^2 + \tfrac{d-2}{d} \bar{K} \partial_\tau - \tfrac{1}{d} \dot{\bar{K}} + \tfrac{d^2-8d+11}{d^2} \bar{K}^2 + c_7^2 \, \Delta - 2 \Lambda_k \right] v_i \\ & \qquad
- 2 \, u^i \left[ \big(1-c_4 c_7\big) \partial_\tau
+ c_7 \big( c_{10} - \tfrac{d-2}{d} c_4 \big) \bar{K} \right] \, \sqrt{\Delta} \, v_i \\ & \qquad
%
+ \tfrac{1}{2} \, h^{ij} \left[ -\partial_\tau^2 + \tfrac{3d-4}{d} \, \bar{K} \partial_\tau + \tfrac{d^2 - 9d+12}{d^2} \, \bar{K}^2 + \Delta - 2 \Lambda_k \right] \, h_{ij}
\Big\} \, .
\end{split}
\end{equation}
Based on this general result, one may then search for a particular gauge fixing which, firstly, eliminates all terms containing $\sqrt{\Delta}$ and, secondly, ensures that all component fields obey a relativistic dispersion relation in the limit when $\bar{K} = 0$. A careful inspection of eq.\ \eqref{eqS1B} shows that there is an essentially unique gauge choice which satisfies both conditions. The resulting values for the coefficients $c_i$ are given in eq.\ \eqref{gffinal}. Specifying the general result to these values finally results in the gauge-fixed Hessian appearing in the gravitational sector \eqref{eqS2Frank}. Taking the limit $\bar{K} = 0$, the propagators resulting from this gauge-fixing are displayed in the third column of Table \ref{Tab.1}. In this way it is straightforward to verify that the gauge choice indeed satisfies the condition of a relativistic dispersion relation for all component fields.
The gauge-fixing is naturally accompanied by a ghost action exponentiating the resulting Faddeev-Popov determinant. For the gauge-fixing conditions $F$ and $F_i$ the ghost action comprises a scalar ghost $\bar{c}, c$ and a (spatial) vector ghost $\bar{b}^i, b_i$. Their action can be constructed in a standard way by evaluating
\begin{equation}\label{ghs}
\Gamma_k^\text{scalar ghost}=\int_x \bar{c} \, \, \frac{\delta F}{\delta \hat{\chi}^i} \, \, \delta_{c,b_i} \chi^i \; , \qquad
\Gamma_k^\text{vector ghost}=\int_x \bar{b}^{\, j} \, \, \frac{\delta F_j}{\delta \hat{\chi}^i} \, \, \delta_{c,b_i} \chi^i \, .
\end{equation}
Here $\frac{\delta F}{\delta \hat{\chi}^i}$ denotes the variation of the gauge-fixing condition with respect to the fluctuation fields $\hat{\chi} = \left[ \hat{N},\hat{N}_i,\hat{\sigma}_{ij} \right]$ at fixed background and
the expressions $\delta_{c,b_i} \chi^i$ are given by the variations \eqref{eq:gaugeVariations} with the parameters $f$ and $\zeta_i$ replaced by the scalar ghost $c$ and vector ghost $b_i$, respectively. Taking into account terms quadratic in the fluctuation fields only, the resulting ghost action is given in eq.\ \eqref{Gghost}. Together with the Hessian in the gravitational sector, eq.\ \eqref{eqS2Frank}, this result completes the construction of the Hessians entering the right-hand-side of the flow equation \eqref{FRGE}.
\section{Evaluation of the operator traces}
\label{App.C}
In Sect.\ \ref{sect.3} the operator traces have been written in terms of the standard $D = d+1$-dimensional Laplacian $\Delta_s \equiv - \bar{g}^{\mu\nu} D_\mu D_\nu$ where $s = 0,1,2$ indicates that the Laplacian is acting on fields with zero, one or two spatial indices. In this appendix, we use the heat-kernel techniques detailed, e.g., in \cite{Reuter:1996cp,Codello:2008vh} and \cite{Benedetti:2010nr} to construct the resulting contributions to the flow.
\subsection{Cutoff scheme and master traces}
The final step in the construction of the right-hand-side of the flow equation is the specification of the regulator ${\cal R}_k$. Throughout this work, we will resort to regulators of Type I, which are implicitly defined through the relation that the regulator dresses up each $D$-dimensional Laplacian by a scale-dependent mass term according to the rule
\begin{equation}\label{Rscheme}
\Delta_s \mapsto P_k \equiv \Delta_s + R_k \, .
\end{equation}
Here $R_k$ denotes a scalar profile function, providing the $k$-dependent mass term for the fluctuation modes. The prescription \eqref{Rscheme} then fixes the matrix-valued regulator ${\cal R}_k$ uniquely. In this course, we first notice that the matrix elements $\Gamma^{(2)}_k$ found in App.\ \ref{App.B} take the form
\begin{equation}
\left. \Gamma^{(2)}_k \right|_{\hat{\chi}_i \hat{\chi}_j} = \left(32 \pi G_k \right)^{-\alpha_s} \, c \, \left[ \Delta_s + w + v_1 \, \bar{K}^2 + v_2 \, \dot{\bar{K}} + v_3 \, \bar{K} \partial_\tau \right] \, ,
\end{equation}
where $\alpha_s = 1,0$ depending on whether the matrix element arises from the gravitational or the ghost sector and $w$ encodes a possible contribution from a cosmological constant. Moreover, $c$ and the $v_i$ are $d$-dependent numerical coefficients whose values can be read off from eqs.\ \eqref{eqS2Frank} and \eqref{Gghost} Applying the rule \eqref{Rscheme} then yields
\begin{equation}
\left. {\cal R}_k \right|_{\hat{\chi}_i \hat{\chi}_j} = \left(32 \pi G_k \right)^{-\alpha_s} \, c \, R_k \, .
\end{equation}
Subsequently, one has to construct the inverse of $(\Gamma_k^{(2)} + {\cal R}_k)$. Given the left-hand-side of the flow equation \eqref{FRGElhs} it thereby suffices to keep track of terms containing up to two time-derivatives of the background quantities, i.e., $\bar{K}^2$ and $\dot{\bar{K}}$. This motivates the split
\begin{equation}\label{PVdec}
\left(\Gamma_k^{(2)} + {\cal R}_k\right) \equiv {\cal P} + {\cal V} \, ,
\end{equation}
where the propagator-matrix ${\cal P}$ collects all terms containing $\Delta_s$ and $\Lambda_k$ and the potential-matrix ${\cal V}$ collects the terms with at least one power of the extrinsic background curvature $\bar{K}$. The inverse $ (\Gamma_k^{(2)} + {\cal R}_k)^{-1}$ can then be constructed as an expansion in ${\cal V}$. Retaining terms containing up to two powers of $\bar{K}$ only
\begin{equation}\label{exp}
\big({\cal P} + {\cal V}\big)^{-1} = {\cal P}^{-1} - {\cal P}^{-1} \, {\cal V} \, {\cal P}^{-1} + {\cal P}^{-1} \, {\cal V} \, {\cal P}^{-1} \, {\cal V} \, {\cal P}^{-1} + {\cal O}(\bar{K}^3) \, .
\end{equation}
Typically, ${\cal P} + {\cal V}$ has a block-diagonal form in field space. At this stage it is instructive to look at a single block for which we assume that it is spanned by a single field (e.g., $h_{ij}$). In a slight abuse of notation we denote the propagator and potential on this block by ${\cal P}$ and ${\cal V}$ as well. From the structure of the Hessians one finds that the propagator has the form
\begin{equation}\label{propagator}
{\cal P}^{-1} = (32 \pi G_k)^{\alpha_s} \, c^{-1} \, \left( \Delta_s + R_k + w \right)^{-1} \, ,
\end{equation}
while the potential ${\cal V}$ is constructed from three different types of insertions
\begin{equation}\label{pottypes}
{\cal V}_1 = (32 \pi G_k)^{-\alpha_s} \, c \, \bar{K}^2 \, , \quad
{\cal V}_2 = (32 \pi G_k)^{-\alpha_s} \, c \, \dot{\bar{K}} \, , \quad
{\cal V}_3 = (32 \pi G_k)^{-\alpha_s} \, c \, \bar{K} \partial_\tau \, .
\end{equation}
The structure \eqref{exp} can be used to write the right-hand-side of the flow equation in terms of master traces, which are independent of the particular choice of cutoff function. Defining the profile function $R^{(0)}(\Delta_s/k^2)$ through the relation $R_k = k^2 R^{(0)}(\Delta_s/k^2)$, it is convenient to introduce the dimensionless threshold functions \cite{Reuter:1996cp}
\begin{equation}
\begin{split}
\Phi^p_n(w) \equiv \frac{1}{\Gamma(n)} \int_0^\infty dz \, z^{n-1} \, \frac{R^{(0)}(z) - z R^{(0)\prime}(z)}{[z+ R^{(0)}(z) + w]^p} \, , \\
%
\widetilde{\Phi}^p_n(w) \equiv \frac{1}{\Gamma(n)} \int_0^\infty dz \, z^{n-1} \, \frac{R^{(0)}(z)}{[z+ R^{(0)}(z) + w]^p} \, .
\end{split}
\end{equation}
For a cutoff of Litim type, $R_k=(k^2-\Delta_s)\,\theta(k^2-\Delta_s)$, to which we resort in the main part of the paper the integrals in the threshold functions can be evaluated analytically, yielding
\begin{equation}\label{thresholdfcts}
\Phi^p_n(w) \equiv \frac{1}{\Gamma(n+1)} \, \frac{1}{(1+w)^p} \, , \qquad
\widetilde{\Phi}^p_n(w) \equiv \frac{1}{\Gamma(n+2)} \, \frac{1}{(1+w)^p} \, .
\end{equation}
The right-hand-side of the flow equation is then conveniently evaluated in terms of the following master traces. For zero potential insertions one has
\begin{equation}\label{master0}
\begin{split}
{\rm Tr}\left[{\cal P}^{-1} \partial_t {\cal R}_k \right] = \tfrac{k^{D}}{(4\pi)^{D/2}} \int_x & \Big[ a_0 \left( 2 \Phi^1_{D/2}(\tilde{w}) - \eta \, \alpha_s \, \widetilde{\Phi}^1_{D/2}(\tilde{w}) \right) \\ & + a_2 \, \left( 2 \Phi^1_{D/2-1}(\tilde{w}) - \eta \, \alpha_s \, \widetilde{\Phi}^1_{D/2-1}(\tilde{w}) \right) \tfrac{\bar{K}^2}{k^2} \Big] \, .
\end{split}
\end{equation}
The case with one potential insertion gives
\begin{equation}\label{master1}
\begin{split}
{\rm Tr}\left[\, {\cal P}^{-1} \, {\cal V}_1 \, {\cal P}^{-1} \, \partial_t {\cal R}_k \right] = & \,
\tfrac{k^{D}}{(4\pi)^{D/2}} \int_x \, a_0 \left( 2 \Phi^2_{D/2}(\tilde{w}) - \eta \, \alpha_s \, \widetilde{\Phi}^2_{D/2}(\tilde{w}) \right)\tfrac{\bar{K}^2}{k^2} \, , \\
{\rm Tr}\left[\, {\cal P}^{-1} \, {\cal V}_2 \, {\cal P}^{-1} \, \partial_t {\cal R}_k \right] = & \, - \,
\tfrac{k^{D}}{(4\pi)^{D/2}} \int_x \, a_0 \left( 2 \Phi^2_{D/2}(\tilde{w}) - \eta \, \alpha_s \, \widetilde{\Phi}^2_{D/2}(\tilde{w}) \right) \tfrac{\bar{K}^2}{k^2} \, , \\
{\rm Tr}\left[\, {\cal P}^{-1} \, {\cal V}_3 \, {\cal P}^{-1} \, \partial_t {\cal R}_k \right] = & \, 0 \, .
\end{split}
\end{equation}
At the level of two insertions only the trace containing $({\cal V}_3)^2$ contributes to the flow. In this case, the application of off-diagonal heat-kernel techniques yields
\begin{equation}\label{master2}
{\rm Tr}\left[({\cal V}_3)^2 \, {\cal P}^{-3} \, \partial_t {\cal R}_k \right] = - \tfrac{1}{2}
\tfrac{k^{D}}{(4\pi)^{D/2}} \int_x \, a_0 \left( 2 \Phi^2_{D/2+1}(\tilde{w}) - \eta \, \alpha_s \, \widetilde{\Phi}^2_{D/2+1}(\tilde{w}) \right) \tfrac{\bar{K}^2}{k^2} \, .
\end{equation}
Here $a_0$ and $a_2$ are the spin-dependent heat-kernel coefficients introduced in App.\ \ref{App.A} and $\tilde{w} \equiv w k^{-2}$. Note that once a trace contains two derivatives of the background curvature, all remaining derivatives may be computed freely, since commutators give rise to terms which do not contribute to the flow of $G_k$ and $\Lambda_k$.
\subsection{Trace contributions in the gravitational sector}
At this stage, we have all the ingredients for evaluating the operator
traces appearing on the right-hand-side of the FRGE, keeping all terms contributing to the truncation \eqref{defdimless}. In order to cast the resulting expressions into compact form, it is convenient to combine the threshold functions \eqref{thresholdfcts} according to
\begin{equation}\label{qfct}
q_n^p(w) \equiv 2 \, \Phi^p_n(w) - \eta \, \widetilde{\Phi}^p_n(w) \, ,
\end{equation}
and recall the definition of the dimensionless quantities \eqref{defdimless}. Moreover, all traces include the proper factors of $1/2$ and signs appearing on the right-hand-side of the FRGE.
We first evaluate the traces arising from the blocks of $\Gamma^{(2)} + {\cal R}_k$ which are one-dimensional in field space. In the gravitational sector, this comprises the contributions of the component fields $h_{ij}, u_i, v_i$, and $B$. Applying the master formulas \eqref{master0} and \eqref{master1} and adding the results, one has
\begin{equation}\label{tracegrav}
\begin{split}
{\rm Tr}|_{hh} = & \tfrac{k^D}{2 \, (4\pi)^{D/2}}
\int_x \Big[
\tfrac{(d+1)(d-2)}{2} \, q^1_{D/2}(-2 \lambda)
+ \tfrac{d^4-2d^3-d^2+14d+36}{12d^2} \, q^1_{D/2-1}(-2 \lambda) \, \tfrac{\bar{K}^2}{k^2} \\ & \qquad \qquad \quad
- {\tfrac{(d-2)^2(d+1)^2}{2 \, d^2} } \, q^2_{D/2}(-2\lambda) \tfrac{\bar{K}^2}{k^2} \
\Big] \, , \\
%
{\rm Tr}|_{vv} = & \tfrac{k^D}{2 \, (4\pi)^{D/2}}
\int_x \Big[ (d-1) \, q^1_{D/2}(-2 \lambda)
+ \tfrac{d^3-2d^2+d+6}{6d^2} \, q^1_{D/2-1}(-2 \lambda)\, \tfrac{\bar{K}^2}{k^2} \\ & \qquad \qquad \quad
- \tfrac{(d-1)(d^2-5d+7)}{d^2} \, q^2_{D/2}(-2\lambda) \, \tfrac{\bar{K}^2}{k^2} \
\Big] \\
%
{\rm Tr}|_{uu} = & \tfrac{k^D}{2 \, (4\pi)^{D/2}}
\int_x \Big[ (d-1) \, q^1_{D/2}(0)
+ \tfrac{d^3-2d^2+d+6}{6d^2} \, q^1_{D/2-1}(0)\, \tfrac{\bar{K}^2}{k^2} \\ & \qquad \qquad \quad
- \tfrac{(d-1)(d-2)}{d} \, q^2_{D/2}(0) \, \tfrac{\bar{K}^2}{k^2} \
\Big] \\
%
{\rm Tr}|_{BB} = & \tfrac{k^D}{2 \, (4\pi)^{D/2}}
\int_x \Big[ q^1_{D/2}(0)
+ \tfrac{d-1}{6d} \, q^1_{D/2-1}(0)\, \tfrac{\bar{K}^2}{k^2}
- \tfrac{(d-1)^2}{d^2} \, q^2_{D/2}(0) \, \tfrac{\bar{K}^2}{k^2} \
\Big]
\end{split}
\end{equation}
The evaluation of the traces in the ghost sector follows along the same lines. In this case one also has a contribution from the third master trace \eqref{master2}. The total contributions of the scalar ghosts is then given by
\begin{equation}\label{gh1}
- {\rm Tr}|_{\bar{c}c} = - \tfrac{k^D}{(4\pi)^{D/2}} \int_x
\Big\{ 2 \, \Phi^1_{D/2} + \tfrac{\bar{K}^2}{k^2} \left[ \tfrac{d-1}{3d} \Phi^1_{D/2-1} + 2 \Phi^1_{D/2} - \tfrac{4}{d^2} \Phi^1_{D/2+1} \right] \Big\} \, ,
\end{equation}
where all threshold functions are evaluated at zero argument.
Recalling that the vector ghost $b_i$ is not subject to a transverse constraint, the trace evaluates to
\begin{equation}\label{gh2}
- {\rm Tr}|_{\bar{b}b} = - \tfrac{k^D}{(4\pi)^{D/2}} \int_x
\Big\{ 2d \, \Phi^1_{D/2} + \tfrac{\bar{K}^2}{k^2} \left[ \tfrac{d-1}{3} \, \Phi^1_{D/2-1} + \tfrac{8}{d} \, \Phi^1_{D/2} - \tfrac{4}{d} \, \Phi^1_{D/2+1}\right] \Big\}.
\end{equation}
The last contribution of the flow is provided by the three scalar fields $\xi = (\hat{N}, E, \psi)$. Inspecting \eqref{eqS2Frank}, one finds that the block $\Gamma^{(2)} + {\cal R}_k$ appearing in this sector is given by a $3\times 3$-matrix in field space with non-zero off-diagonal entries. Applying
the decomposition \eqref{PVdec} the matrix ${\cal P}$ resulting from
\eqref{eqS2Frank} is
\begin{equation}
\renewcommand{\arraystretch}{1.2}
\begin{split}
{\cal P} = & (32 \pi G_k)^{-1}
\left[\begin{array}{ccc}
\Delta_0 & \tfrac{1}{2} \left(\Delta_0 - 2 \Lambda\right) & \tfrac{d-1}{2} \left(\Delta_0 - 2 \Lambda\right) \\
\tfrac{1}{2} \left(\Delta_0 - 2 \Lambda\right) & \tfrac{1}{4} \left(\Delta_0 - 2 \Lambda\right) & - \tfrac{d-1}{4} \left(\Delta_0 - 2 \Lambda\right) \\
\; \tfrac{d-1}{2} \left(\Delta_0 - 2 \Lambda\right) \; & \; - \tfrac{d-1}{4} \left(\Delta_0 - 2 \Lambda\right) \; & \; - \tfrac{(d-1)(d-3)}{4} \left(\Delta_0 - 2 \Lambda\right) \;
\end{array}\right] \, , \\
\end{split}
\end{equation}
while the matrix ${\cal V}$ is symmetric with entries
\begin{equation}
\begin{array}{ll}
{\cal V}_{11} = - \tfrac{2(d-1)}{d^2} \left(2 \bar{K}^2 + d \dot{\bar{K}} \right) \, , \qquad &
{\cal V}_{12} = - \tfrac{5d^2-12d+16}{8d^2} \bar{K}^2 \\
{\cal V}_{22} = - \tfrac{d-1}{4d} \left(\bar{K}^2 + 2 \dot{\bar{K}} \right) \, , \quad &
{\cal V}_{13} = - \tfrac{(d-1)(5d^2-12d+16)}{8d^2} \bar{K}^2
\\
{\cal V}_{33} = \tfrac{(d-3)(d-1)^2}{4d} \left(\bar{K}^2 + 2 \dot{\bar{K}} \right) \, , \quad &
{\cal V}_{23} = \tfrac{(d-1)^2}{4d} \, \left(\bar{K}^2 + 2 \dot{\bar{K}} \right) \, .
\end{array}
\end{equation}
Applying \eqref{Rscheme}, the cutoff ${\cal R}_{k}$ in this sector is given by
\begin{equation}
\renewcommand{\arraystretch}{1.2}
{\cal R}_k = (32 \pi G_k)^{-1} \, R_k \,
\left[\begin{array}{ccc}
1 & \tfrac{1}{2} & \tfrac{d-1}{2} \\
\tfrac{1}{2} & \tfrac{1}{4} & - \tfrac{d-1}{4} \\
\; \tfrac{d-1}{2} \; & \; - \tfrac{d-1}{4} \; & \; - \tfrac{(d-1)(d-3)}{4} \;
\end{array}\right] \, .
\end{equation}
The master traces \eqref{master0} and \eqref{master1} also hold in the case where ${\cal P}$ and ${\cal V}$ are matrix valued. Constructing the inverse of ${\cal P}$ on field space explicitly and evaluating the corresponding traces, the contribution of this block to the flow is found as
\begin{equation}\label{scalars0}
\begin{split}
{\rm Tr}|_{\xi\xi} = \tfrac{k^{D}}{2(4\pi)^{D/2}} & \int_x \Big[
2 \, q^1_{D/2}(-2\lambda) + q^1_{D/2}\left(-\tfrac{d}{d-1}\lambda\right) \\ &
+ \tfrac{d-1}{6d} \, \left( 2 \, q^1_{D/2-1}({-2\lambda}) + q^1_{D/2-1}\left(-\tfrac{d}{d-1}\lambda\right) \right) \tfrac{\bar{K}^2}{k^2}
%
\\
& - \left( \tfrac{2(d-1)}{d} \, q^2_{D/2}\left(-2\lambda\right) - \tfrac{3 d^3 + 6d^2-16d + 16}{4 d^2 (d-1)}
q^2_{D/2}\left(-\tfrac{d}{d-1}\lambda\right) \right) \tfrac{\bar{K}^2}{k^2}
\Big] \, .
\end{split}
\end{equation}
The traces \eqref{tracegrav}, \eqref{gh1}, \eqref{gh2}, and \eqref{scalars0} complete the evaluation of the flow equation on a flat FRW background. Substituting these expressions into the FRGE \eqref{FRGE} and retaining the terms present in \eqref{FRGElhs} then leads to the beta functions \eqref{betafunction} where the threshold functions are evaluated with a Litim type regulator \eqref{thresholdfcts}.
\subsection{Minimally coupled matter fields}
\label{App.C3}
At the level of the Einstein-Hilbert truncation \eqref{GammaEH}, including the contribution of the matter sector \eqref{matter} to the flow of Newton's constant and the cosmological constant is rather straightforward. When expanding the matter fields around a vanishing background value, the Hessian $\Gamma^{(2)}$ arising in the matter sector contains variations with respect to the matter fields only and all Laplacians reduce to the background Laplacians. The resulting contributions of the matter trace are then identical to the ones obtained in the metric formulation \cite{Percacci:2002ie,Percacci:2003jz,Dona:2012am}. The trace capturing the contributions of the the $N_S$ scalar fields $\phi$ yields
\begin{equation}\label{scalarmattertrace}
{\rm Tr}|_{\phi\phi} = N_S \, \tfrac{k^D}{(4\pi)^{D/2}} \int_x \Big\{ \Phi^1_{D/2}(0) + \tfrac{d-1}{6d} \, \Phi^1_{D/2-1}(0) \, \tfrac{\bar{K}^2}{k^2} \Big\} \, .
\end{equation}
The gauge sector, comprising $N_V$ gauge fields $A_\mu$ and the corresponding Faddeev-Popov ghosts $\bar{C}, C$ contributes
\begin{equation}\label{Atrace}
{\rm Tr}|_{AA} = N_V \, \tfrac{k^D}{(4\pi)^{D/2}} \int_x \Big\{ (d+1) \, \Phi^1_{D/2}(0) + \tfrac{(d-1)(d^2+2d-11)}{6d \, (d+1)} \, \Phi^1_{D/2-1}(0) \, \tfrac{\bar{K}^2}{k^2} \Big\} \, ,
\end{equation}
and
\begin{equation}\label{CCbtrace}
- {\rm Tr}|_{\bar{C}C} = N_V \, \tfrac{k^D}{(4\pi)^{D/2}} \int_x \Big\{ 2 \, \Phi^1_{D/2}(0) + \tfrac{d-1}{3d} \, \Phi^1_{D/2-1}(0) \, \tfrac{\bar{K}^2}{k^2} \Big\} \, .
\end{equation}
Adding eqs.\ \eqref{Atrace} and \eqref{CCbtrace} gives the total contribution of the gauge fields to the flow
\begin{equation}\label{gaugemattertrace}
{\rm Tr}|_{\rm GF} = N_V \, \tfrac{k^D}{(4\pi)^{D/2}} \int_x \Big\{ (d-1) \, \Phi^1_{D/2}(0) + \tfrac{(d-1)(d^2-13)}{6d \, (d+1)} \, \Phi^1_{D/2-1}(0) \, \tfrac{\bar{K}^2}{k^2} \Big\} \, .
\end{equation}
When evaluating the contribution of the fermionic degrees of freedom, we follow the discussion \cite{Dona:2012am}, resulting in
\begin{equation}\label{diracmattertrace}
{\rm Tr}|_{\psi\psi} = - \tfrac{N_D \, 2^{(d+1)/2} \, k^D}{(4\pi)^{D/2}} \int_x \Big\{\Phi^1_{D/2}(0) + \tfrac{d-1}{d} \left[ \left( \tfrac{1}{6} - \tfrac{r}{4} \right) \Phi^1_{D/2-1}(0) - \tfrac{1-r}{4} \Phi^2_{D/2}(0) \right] \tfrac{\bar{K}^2}{k^2} \Big\} \, .
\end{equation}
Here $r$ is a numerical coefficient which depends on the precise implementation of the regulating function: $r=0$ for a Type I regulator while the Type II construction of \cite{Dona:2012am} corresponds to $r=1$. In order to be consistent with the evaluation of the other traces in the gravitational and matter sectors, we will resort to the Type I regulator scheme, setting $r=1$.
Adding the results \eqref{scalarmattertrace}, \eqref{gaugemattertrace}, and \eqref{diracmattertrace} to the contribution from the gravitational sector gives rise to the $N_S$, $N_V$, and $N_D$-dependent terms in the beta functions \eqref{betafunction}.
\end{appendix}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Cylichna mongii is een slakkensoort uit de familie van de Cylichnidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1826 door Audouin.
Cylichnidae
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{
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/22402-normal-distributions-integrals.html","text":"# Math Help - Normal Distributions and Integrals\n\n1. ## Normal Distributions and Integrals\n\nThe question says that suppose f(x) is a density function of a normal distribution with mean u and standard dev o(sigma). Sow that u= integral from -infinity to infinity of x(fx)dx.\n\n= f(x)\n\nWhat I did was substitue 0 for u and 1 for sigma since its a normal distribution. THen took the integral using substitution and got the integral =0. I then said since u=o and the integral =0 then u=integral. Is that correct?\n\n2. Originally Posted by clipperdude21\nThe question says that suppose f(x) is a density function of a normal distribution with mean u and standard dev o(sigma). Sow that u= integral from -infinity to infinity of x(fx)dx.\n\n= f(x)\n\nWhat I did was substitue 0 for u and 1 for sigma since its a normal distribution. THen took the integral using substitution and got the integral =0. I then said since u=o and the integral =0 then u=integral. Is that correct?\nYou want to show that:\n\n$\nu = \\int_{-\\infty}^{\\infty} x \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-(x-u)^2\/(2\\sigma^2)}~dx\n$\n\nPut:\n\n$\nI(u) = \\int_{-\\infty}^{\\infty} x \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-(x-u)^2\/(2\\sigma^2)}~dx\n$\n\nNow consider:\n\n$\nI(u)-u=\\int_{-\\infty}^{\\infty} (x-u) \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-(x-u)^2\/(2\\sigma^2)}~dx\n$\n$\n=\\frac{1}{\\sigma \\sqrt{2 \\pi}} \\left[ -2 \\sigma^2 e^{-(x-u)^2\/(2\\sigma^2)} \\right]_{x=-\\infty}^{\\infty}=0\n$\n\n(the first step is using the fact that $u \\int_{-\\infty}^{\\infty} f(x)~dx = u$)\n\nHence: $I(u)=u$\n\nRonL\n\n3. Hi Thanks Captain Black... I have one question though. When you substracted the u from each side. How did you know you could put it inside the integral and make it integral of (x-u)f(x)dx?\n\nThanks!","date":"2015-11-27 20:50:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 6, \"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\": 0.9746045470237732, \"perplexity\": 1077.4985063128813}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398450559.94\/warc\/CC-MAIN-20151124205410-00089-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
Q: Combining Google Maps MarkerClusterer v3 and Viewport Marker Management I've successfully setup both MarkerClusterer v3 and Viewport Marker Management ( performing an ajax call to gather only markers visible with viewport and rendering those whenever the map is 'idle') separately.
However when I combine them, they only seem to work together when the page first loads and not afterward.
When zooming or panning the initial clusters remain and the markers are for the entire map are rendered unclustered, but leaves the previously clustered markers.
The original clustered markers still behave properly though when you zoom in/out, but the new markers provided when the viewport bounds are changed aren't added to them or clustered.
Code Below:
function drawMap(swLat, swLng, neLat, neLng){
//Load the map data
$.ajax({
type: "POST",
url: "readMapInfo.php",
cache: false,
data:{S:swLat, W:swLng, N:neLat, E:neLng},
dataType: "xml",
success: function(data) {
if(markerArray.length >0){
for (i in markerArray) {
markerArray[i].setMap(null);
}
drawMarker(data); //takes the info provided and performs "markerArray.push(marker);"
mc = new MarkerClusterer(map, markerArray, clusterOptions);
} else {
drawMarker(data); //takes the info provided and performs "markerArray.push(marker);"
mc = new MarkerClusterer(map, markerArray, clusterOptions);
}
});
}
google.maps.event.addListener(map, 'idle', function() {
bounds = map.getBounds();
sw = bounds.getSouthWest();
ne = bounds.getNorthEast();
swLat = sw.lat();
swLng = sw.lng();
neLat = ne.lat();
neLng = ne.lng();
drawMap(swLat, swLng, neLat, neLng);
});
A: Your description of your problem is detailed and thorough, but it would be easier if there were also a URL to a page that demonstrates the problem(s). Of course, that may not be possible in this specific scenario. That said, I will take a crack at helping out:
I believe you need some additional cleanup code at the beginning of your ajax success callback:
if( markerArray.length > 0 ) {
// For loop logic is unchanged
// It removes the markers from the Map, but leaves them in markerArray
for (i in markerArray) {
markerArray[i].setMap( null );
}
// New code to truncate the Array; the markers should be removed
markerArray.length = 0;
// New code to clear the clusterer; the markers should be removed
mc.clearMarkers();
// Original line of code unchanged
drawMarker(data); //takes the data and performs markerArray.push(marker)
// Commented out, because the cluster is cleared each time, not recreated
//mc = new MarkerClusterer(map, markerArray, clusterOptions);
// New code to refill the cluster, rather than recreate the cluster
// The original clusterOptions are retained
mc.addMarkers( markerArray );
} else {
// Original line of code unchanged
drawMarker(data); //takes the data and performs markerArray.push(marker)
// Original line of code unchanged
// Necessary here, because the clusterer does not yet exist
mc = new MarkerClusterer(map, markerArray, clusterOptions);
}
I believe this will help move you forward. Please let me know if this resolves the problem or at least helps.
After you have your immediate challenges resolved, I also suggest that you take a look at MarkerClustererPlus; it is described in the question: Is there any way to disable the Marker Clusterer for less than defined marker counts?.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,656
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Q: Quotient norm and actual norm I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed.
In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< \infty$ and the idea is to show that this series without the norm has a limit. Therefore it is said that we can assume that we have $||x_k||\le ||[x_k]||+2^{-k}$ for every $k \in \mathbb{N}$ and I do not see why.
Does anybody here know why this is true?
A: Since the quotient norm is defined as $\|[x]\| = \inf_{u \in U} \|x+u\|$, then for all $\epsilon >0$ , there is some $u \in U$ such that $\|x+u\| > \|[x]\| +\epsilon$. Since $u \in U$, we have $[x] = [x+u]$, and so $\|[x]\|= \|[x+u]\|$.
So given the sequence $x_k$ and $\sum_k \|[x_k]\|< \infty$, one can find $u_k \in U$ such that $\|x_k+u_k\| > \|[x_k]\| +2^{-k}$. If we let $x'_k = x_k+u_k$, then we have $x'_k$ with $\sum_k \|[x'_k]\|< \infty$.
The book is just saying that you might as well start with the $x'_k$ in the first place.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Replace multiple captured groups in regex VB2005: I've been looking at regex for some hours now and cant seem to get my head around the .Replace for my case. I'm looking for two fields and then I want to replace those fields with new values. So my string looks like so:
Dim myInputString as string ="RTEMP MIN<240 MAX<800"
My regex is
dim ptn as string = "RTEMP\s{17}MIN<(?<min>(\d|\s){1,3})\s{1,3}MAX<(?<max>(\d|\s){1,3})\s{1,12}"
Dim MyRegex As Regex = New Regex(ptn, RegexOptions.IgnoreCase)
and that works well and it captures my two fields.
Now I have new values
dim newMin as integer = 300
dim newMax as integer = 999
But cant seem to figure out how to replace the two values in one swoop
Dim result As String = MyRegex.Replace(myInputString, MyRegexReplace)
What do I put in MyRegexReplace? This is a simple two value replace but Im going to have possibly more so was thinking there has got to be a way to do this but need help.
Thanks
AGP
A: Since you have 2 distinct values to swap into those 2 fields, wouldn't you want to use 2 separate Regex operations?
But if you want to use one Regex operation, you could use a MatchEvaluator:
Dim result As string = MyRegex.Replace(myInputString, ReplaceField)
Private Function ReplaceField(match As Match) As String
' Use the Index property of the Match to determine what value to use as replacement
End Function
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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John Bower (November 8, 1940 – June 6, 2017) was an American nordic combined skier who competed in the 1960s and later went on to become a coach of the American nordic skiing team for the 1976 and 1980 Winter Olympic team. He also became the first non-European to ever win at the Holmenkollen Ski Festival in Norway with his 1968 victory in the Nordic combined event, winning the prestigious King's Cup.
A native of Auburn, Maine, Bower attended Middlebury College in Vermont, where he won the NCAA national championship in Nordic combined in 1961. After graduating from Middlebury in 1963, he joined the United States Army and serve during the mid-1960s. Bower also won the national Nordic combined event four times (1963, 1966–8). Competing in two Winter Olympics, Bower finished 15th in the Nordic combined event at Innsbruck in 1964 and 13th in the same event at Grenoble in 1968. After his retirement from Nordic combined competition, Bower went on to coach the Nordic skiing team for both the 1976 and 1980 Winter Olympics and later served as program director for the U.S. Nordic Combined Ski Team.
After retirement, Bower and his wife Bonnie moved to Park City, Utah, where Bonnie started the Park City Winter School to allow skiers to attend school in the summer while competing in the winter. John became the first director of the Utah Olympic Park when it first opened in 1989. Utah Olympic Park would later host several competitions during the 2002 Winter Olympics. Bower's son Ricky (born 1980) won the Snowboarding half-pipe World Championships in Germany in 1999.
In 1999, Sports Illustrated magazine ranked him 19th among Maine's 50 Greatest Athletes of the 20th century. He was the first of five Americans to win a Nordic combined event at the Holmenkollen Ski Festival, considered the premier event in nordic combined. Other American King's Cup winners include Kerry Lynch (1983), Todd Lodwick (1998), Bill Demong (2009) and Bryan Fletcher (2012).
References
"Christian Science Monitor" article on John Bower winning King's Cup
Christian Science Monitor article on Bowers' son Ricky
Middlebury College's skiing program listing Bower
Olympic nordic combined individual results: 1948-64
Olympic nordic combined individual results: 1968-84
Sports Illustrated 50 Greatest Maine Sports Figures - December 27, 1999
John Bower's obituary
1940 births
2017 deaths
American male cross-country skiers
American male Nordic combined skiers
Holmenkollen Ski Festival winners
Middlebury College alumni
Cross-country skiers at the 1964 Winter Olympics
Cross-country skiers at the 1968 Winter Olympics
Nordic combined skiers at the 1964 Winter Olympics
Nordic combined skiers at the 1968 Winter Olympics
Olympic cross-country skiers of the United States
Olympic Nordic combined skiers of the United States
Sportspeople from Auburn, Maine
People from Park City, Utah
United States Army soldiers
Sportspeople from Vermont
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Q: Word for "Programmer Analyst"? (Spanish: "Analista Programador") I really would like to know the correct translation of this IT role into English, because I believe that "Programmer Analyst" or "Analyst Programmer" do not make sense in English and it is a very specific role in the IT world.
In Computer Science hierarchy, from top to bottom:
*
*Analyst
*Analyst Programmer (?)
*Senior Developer
*Junior Developer
When you are Analyst/Programmer in Spain it means that you have some expertise and that you are given some analysis and software engineering tasks, while you still program or develop like a regular developer.
When you become an Analyst, you only coordinate the dev team and seldomly or never type a line of code again.
That should help to understand the concept I am trying to isolate. I have also heard terms like "Software Gardener" but they do not sound convincing either.
A: Software developer/ software engineer depending on whether you only code or perform other tasks from the SDLC
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,090
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#include <numeric>
#include <iomanip>
#include "logging.hpp"
#include "LIEF/MachO/hash.hpp"
#include "LIEF/MachO/RelocationObject.hpp"
#include "LIEF/MachO/EnumToString.hpp"
#include "LIEF/MachO/Section.hpp"
#include "MachO/Structures.hpp"
namespace LIEF {
namespace MachO {
RelocationObject::RelocationObject(const RelocationObject& other) = default;
RelocationObject::~RelocationObject() = default;
RelocationObject::RelocationObject() = default;
RelocationObject& RelocationObject::operator=(RelocationObject other) {
swap(other);
return *this;
}
RelocationObject::RelocationObject(const details::relocation_info& relocinfo) :
is_pcrel_{static_cast<bool>(relocinfo.r_pcrel)}
{
address_ = static_cast<uint32_t>(relocinfo.r_address);
size_ = static_cast<uint8_t>(relocinfo.r_length);
type_ = static_cast<uint8_t>(relocinfo.r_type);
}
RelocationObject::RelocationObject(const details::scattered_relocation_info& scattered_relocinfo) :
is_pcrel_{static_cast<bool>(scattered_relocinfo.r_pcrel)},
is_scattered_{true},
value_{scattered_relocinfo.r_value}
{
address_ = scattered_relocinfo.r_address;
size_ = static_cast<uint8_t>(scattered_relocinfo.r_length);
type_ = static_cast<uint8_t>(scattered_relocinfo.r_type);
}
RelocationObject* RelocationObject::clone() const {
return new RelocationObject(*this);
}
void RelocationObject::swap(RelocationObject& other) {
Relocation::swap(other);
std::swap(is_pcrel_, other.is_pcrel_);
std::swap(is_scattered_, other.is_scattered_);
std::swap(value_, other.value_);
}
bool RelocationObject::is_pc_relative() const {
return is_pcrel_;
}
size_t RelocationObject::size() const {
if (size_ < 2) {
return (size_ + 1) * 8;
}
return sizeof(uint32_t) * 8;
}
bool RelocationObject::is_scattered() const {
return is_scattered_;
}
uint64_t RelocationObject::address() const {
const Section* sec = section();
if (sec == nullptr) {
return Relocation::address();
}
return address_ + section()->offset();
}
int32_t RelocationObject::value() const {
if (!is_scattered()) {
LIEF_ERR("This relocation is not a 'scattered' one");
return -1;
}
return value_;
}
RELOCATION_ORIGINS RelocationObject::origin() const {
return RELOCATION_ORIGINS::ORIGIN_RELOC_TABLE;
}
void RelocationObject::pc_relative(bool val) {
is_pcrel_ = val;
}
void RelocationObject::size(size_t size) {
switch(size) {
case 8: size_ = 0; break;
case 16: size_ = 1; break;
case 32: size_ = 2; break;
default: LIEF_ERR("Size must not be bigger than 32 bits");
}
}
void RelocationObject::value(int32_t value) {
if (!is_scattered()) {
LIEF_ERR("This relocation is not a 'scattered' one");
return;
}
value_ = value;
}
void RelocationObject::accept(Visitor& visitor) const {
visitor.visit(*this);
}
bool RelocationObject::operator==(const RelocationObject& rhs) const {
if (this == &rhs) {
return true;
}
size_t hash_lhs = Hash::hash(*this);
size_t hash_rhs = Hash::hash(rhs);
return hash_lhs == hash_rhs;
}
bool RelocationObject::operator!=(const RelocationObject& rhs) const {
return !(*this == rhs);
}
bool RelocationObject::classof(const Relocation& r) {
return r.origin() == RELOCATION_ORIGINS::ORIGIN_RELOC_TABLE;
}
std::ostream& RelocationObject::print(std::ostream& os) const {
return Relocation::print(os);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,976
|
<?xml version="1.0" encoding="UTF-8"?>
<tableInfo>
<headerDetails headerStyle="drawtablerowboldnoline"/>
<row totWidth="100" bottomLineRequired="true">
<column label="FeeType" value="feeName" valueType="method" columnType="text">
<columnDetails rowStyle="drawtablerow" colWidth="50" />
</column>
<column label="Amount" value="amount" valueType="method" columnType="text">
<columnDetails rowStyle="drawtablerow" colWidth="50"/>
</column>
</row>
</tableInfo>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,127
|
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