text:
\section{ Introduction}
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\vskip 0.5 cm
In the treatment of many-body systems there has been achieved much
success when assuming the bodies interact mainly in pairs. There are
many examples of this in atomic and nuclear physics with the use
of two-body Coulomb and nucleon-nucleon potentials.
 However, at a more basic level in the hadronic world, it is of interest to
 study the many-body system composed of  quarks interacting
via gluons carrying colour.
In this system, especially in view of the special features of the
colour interaction -- such as confinement -- , the continued use of simply
two-body potentials acting between constituent quarks can be questioned.
Unfortunately, even though
 QCD is now well accepted as the fundamental field theory underlying
hadron interactions, the dynamics of multi-quark systems is still too
difficult a problem to be treated at this fundamental level. As a
step towards tackling this problem,
in a series of recent papers\cite{MGP}-\cite{GMP2}, an attempt
has been made to understand the energies of systems of four infinitely heavy
quarks in terms of models  based on interquark potentials i.e. the quarks
are treated as fixed colour sources, and their kinetic energy neglected -- the
static approximation.
This programme requires knowing the following:
\begin{itemize}
\begin{enumerate}
\item Reliable estimates of the {\em exact} energies $(E_i)$ of four-quark
systems. Here the suffix will be taking on the two values 0 and 1 --
corresponding to the ground states and the first excited states respectively.
\item Reliable interquark two-body potentials $V_{qq}$.
\item A physically motivated and convenient model or prescription for
calculating the energies $(E'_i)$ of a four-quark
system in terms of the $V_{qq}$ -- the hope being that the $E'_i$ are a good
approximation to the exact $E_i$.
\end{enumerate}
\end{itemize}
The calculation of the energies $(E_i)$ of a four-quark system is not easy --
even though the basic equation describing the interaction between quarks
and gluons is governed by the well known QCD lagrangian.
 Traditionally, it is thought that the best way
to estimate the $E_i$ is to use a lattice formulation of this basic
lagrangian. To make this reasonably tractable
with present mainframe computers requires introducing the so-called quenched
approximation, in which the creation and annihilation of $q\bar{q}$
pairs is neglected -- details being given in
refs.\cite{GMP2}-\cite{Hunt}.
Over the years, a few such attempts \cite{Markum,Ohta}
have been made to extract the $E_i$ for four-quark systems. However, the
results have always had sufficiently large error bars that it has not proved
possible to carry out the type of analysis suggested in
refs.\cite{MGP}-\cite{GMP2} i.e. they were unable to distinguish between
theoretical models that differed very considerably.
 By taking advantage of advances in computer
technology and also recent theoretical developments in Monte Carlo lattice
simulations, the situation has now improved greatly. In addition,
considering the gauge group SU(2)
 -- and not SU(3) as in refs.\cite{Markum} and \cite{Ohta} -- further
improves the efficiency of the calculation. This use
of SU(2) means the distinction between quarks and antiquarks, in
general, disappears and so will be ignored in most of this paper -- except,
however, in section 5.
Overall, compared with
refs.\cite{Markum} and \cite{Ohta}, the total efficiency can be increased
 by several orders of magnitude, so that accurate and reliable estimates
of the $E_i$ can now be obtained in the static quenched approximation.
Without this latter approximation, present day computers would be unable to
give such estimates.  It should be added that the use of SU(2)
compared with the more realistic world of SU(3) seems to be
reasonable initially, since it has been demonstrated that
the ratios of physical quantities are very similar in SU(2) and SU(3)
e.g. the ratio $m_{GB}/\sqrt{b_S}$, where $m_{GB}$ is the glueball mass
and $b_S$ is the string tension, is close to 3.5 in both
cases \cite{chrism} -- see also ref.\cite{fing} for similar comparisons
involving $T_c$, the temperature for the deconfinement phase transition.

The second requirement mentioned above is the interquark potential
$V_{qq}(r)$, where $r$ is the distance between the two infinitely heavy
quarks in a colour singlet state.
This potential is quite well known and can be
accurately parametrized by essentially the form
\begin{equation}
\label{Vqqr}
V_{qq}(r)=-\frac{e}{r}+b_Sr.
\end{equation}
The parameters $e$ and $b_S$ can be extracted by fitting the observed masses
of heavy mesons or from Monte Carlo lattice simulations. Since the
four-quark energies ($E_i$) are calculated on a lattice, for consistency
 the $V_{qq}$ used are those calculated as a by-product of the {\em same}
lattice calculation. This improves the accuracy of the predictions, since the
main quantities of interest -- the $E_i$ -- are the {\em differences}  between
the total four-quark energies and that of two separated two-quark systems.
In other words, the $E_i$ can be thought of as the binding energy between
these two two-quark systems.

The final requirement is a model for generating four-quark energies $(E'_i)$
 in terms of the $V_{qq}$. Since the purpose of this work is to demonstrate
the relationship between $E_i$ and $E'_i$, it is appropriate to consider
first only very simple four-quark geometries.
If, in these extreme cases, a model does
not work (i.e. $E'_i\not =E_i$), then there is no reason to believe it will
succeed in more complicated cases. On the contrary, even if the model does work
in
these simple cases, then there is no guarantee of success elsewhere as the
situations become more complicated.
With this philosophy in mind, the first problems
to be treated will involve four infinitely-heavy quarks fixed into some
simple geometry.  This restricted model is consistent
with the corresponding Monte Carlo lattice simulation generating the $E_i$,
since the latter exploits the so-called static quenched approximation.
The restriction to ``simple four-quark
geometries'' is purely for convenience. In fact, in principle, any four-quark
geometry can be treated by the present methods. However, part of the idea
is to start with simple geometries -- such as rectangles or colinear -- in
order to extract systematics. The latter systematics are then parametrized
 in some simple way, with the hope that these parametrizations are
universal -- a feature that can then be checked by going to more complicated
geometries.

In ref.\cite{GMP2} the above programme has already been carried out for four
quarks at the corners of a rectangle in a lattice plane. The present paper, in
which
the accuracies of the two- and four-quark energies ($V_{qq}$ and $E_i$) are
checked and also  another simple four-quark geometry is considered,
 is -- in many ways -- a follow-up to ref.\cite{GMP2}. In section 2, a two-body
potential is extracted that is more consistent than the one used in
ref.\cite{GMP2}. In section 3, the dependencies of the $E_i$ on the size of
the lattice and also on the lattice spacing are checked. The outcome is that
both the earlier $V_{qq}$ and $E_i$ of ref.\cite{GMP2} are, in general,
sufficiently accurate for the present purposes. In section 4, a short review
is given of the model proposed in refs.\cite{MGP}-\cite{GMP2}.
In section 5, the programme of ref.\cite{GMP2} is repeated for the colinear
case of four quarks on a straight line. Section 6 contains some conclusions
and future prospects.

\section{ Two-body potentials}
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\vskip 0.5 cm

The form of the heavy quark potential displayed in eq.(\ref{Vqqr}) is a
theoretical
one. It incorporates both the perturbative limit at short distances and the
linear confinement limit. However, modifications to this picture are expected
from the creation of quark-antiquark pairs. Indeed, heavy quark spectroscopy
of the $c\bar{c}$ and $b\bar{b}$ systems, which is probing mainly the
range between 0.1 and 1.0 fm, seems to favour potentials of slightly different
shape \cite{buch}. The lattice calculations to be described in this article
are done in the quenched approximation, which ignores closed quark loops.
Therefore, one expects the form of eq.(\ref{Vqqr}) to be a reasonable
approximation, if one works throughout with the
lattice data instead of data from experiment.

The lattices to be measured are generated by the standard Wilson action
in Euclidean time
for SU(2) with $\beta$=2.4 and $\beta$=2.5 using $16^3 \times 32$ and
$24^3 \times 32$ lattice points respectively to keep the physical volume
approximately constant -- for a more detailed discussion see section 3.
 A heat bath method is employed for the
equilibration of the lattice.
Further updates are combinations of three over-relaxation sweeps and one
heat bath sweep. The measurement of the appropriate correlation functions
takes place always after such a cycle of four sweeps.
In order to make error estimates, successive
measurements are collected into a series of ``blocks''. For example, later
in this section each block will consist of 240 sweeps i.e. 60 measurements
per block, and in all 12 blocks will be created.

The potential between static fundamental colour sources is obtained
from the ratio
\begin{equation}
V_{qq}(r)=-\frac{1}{a} \lim_{t \rightarrow \infty} \frac{W(r,t)}{W(r,t-a)}
\end{equation}
where $W(r,t)$ is the  Wilson loop defined as the trace of a closed loop of
link variables $U$ (encircling a rectangle of size $r \times t$ ) and
$a$ is the lattice spacing. To get accurate results
with reasonably small time separations the spatial links are replaced
iteratively by smeared or fuzzed links \cite{CM1,CM2}
\begin{eqnarray}
U^0_\mu (n) & \rightarrow & U^1_\mu (n)=A^1_n \left[ cU^0_\mu (n)+
\sum_{\stackrel{\pm \nu \neq \mu}{\nu \neq 4}} U^0_\nu (n) U^0_\mu
(n+\hat{\nu})
U^{0 \dagger}_\nu (n+\hat{\mu}) \right], \nonumber \\
U^1_\mu (n) & \rightarrow & U^2_\mu (n)=A^2_n \left[ cU^1_\mu (n)+
\sum_{\stackrel{\pm \nu \neq \mu}{\nu \neq 4}} U^1_\nu (n) U^1_\mu
(n+\hat{\nu})
U^{1 \dagger}_\nu (n+\hat{\mu}) \right], \ldots
\end{eqnarray}
i.e. each spatial link gets replaced by a multiple of itself plus a sum
of the neighbouring spatial staples. Here the $A^i_n$ are normalization factors
chosen to project the $U^i_\mu (n)$ into SU(2) and $c$ is a free parameter
adjusted together with the level of fuzzing to optimize the path overlaps
with the ground state. In the current calculations $c=4$ has been used with
the fuzzing levels 12, 16 and 20.

In the calculation of V$_{qq}$(r) the different paths ($P_i$) between quarks
 obtained by different levels of fuzzing form a variational
basis for extracting the ground state and the first excited state
\cite{CM1,CM2}. The correlation between paths $P_i$ and $P_j$ parallel
transported by $N$ steps in the time direction is
\begin{equation}
W^N_{ij} = <P_i| \hat{T}^N | P_j>,
\end{equation}
where $\hat{T} = \exp {(-a\hat{H})}$ is the transfer matrix for one time
step with the Hamiltonian $\hat{H}$, and the paths $P_{i,j}$ are
constructed as products of fuzzed links. A trial wave function
$\psi = \sum_{i} a_i |P_i>$ leads to the eigenvalue equation
\begin{equation}
W^N_{ij} a^N_j = \lambda^{(N)} W^{N-1}_{ij} a^N_j,
\end{equation}
where $\lambda^{(N)} \rightarrow \exp{(-aV_{qq})}$ as $N \rightarrow \infty $
and which, for a single path, reduces to the form given in eq. (2.1).
Besides giving the possibility to extract also higher eigenvalues
in addition to the ground state, the method yields accurate information
about V already for small values of $N$ ($N < 5$) \cite{Hunt}.
It should be added that the higher eigenstates all have the same symmetry
as the ground state. In refs.\cite{CM1,CM2} they are refered to
as $A'_{1g}$ and are not those having $E_u$ or $A_{1u}$ symmetry.
Further discussion of the different technical details can be found in
refs. \cite{CM1,CM2,Hunt}.

The results of the 720 measurements  for $\beta=2.4$ (in 12 blocks of
240 sweeps each) are displayed in Table 1
and are denoted by $av_{MC}$ with {\bf r}=($x,y,0$). The off-axis
results ($y \neq 0$) correspond to L-shaped separations of the static
sources. When a fit to these data with a potential of the form given
in eq.(\ref{Vqqr}) is attempted, it must be remembered that the lattice has
a cubic rotational symmetry and so full rotational invariance is not
guaranteed. It is known that at larger $r$-values this symmetry is
restored to a good approximation at the present values of $\beta$.
At smaller values of $r$, the lattice effects can largely be taken into
account by replacing the standard Coulomb form by the lattice Coulomb
potential \cite{LR}
\begin{eqnarray}
\label{coul}
\left(\frac{e}{r}\right)_L = \frac{e \pi}{L'^3}
\sum_{\bf q} \frac{\cos{({\bf q} \cdot {\bf r})}}{\sum_{i=1,3}
\sin^2{(aq_i/2)}} \nonumber \\
aq_i=0,\frac{2\pi}{L'},\ldots,\frac{2\pi(L'-1)}{L'} \;\; {\rm and} \; \;
{\bf q} \neq 0,
\end{eqnarray}
where $L'=2L$ and $L$ is the number of spacial sites.
 Eq. (2.6) gives the potential only upto an additive constant
and, therefore, it is normalized by fixing it to the usual Coulomb
potential $e/r$ at some large value of $r$ such that the lattice artefacts
are expected to be negligible. Here the point {\bf r} = (6a,0,0) is chosen.
The potential to be used in the fits has the form
\begin{equation}
\label{coul1}
v_L(r_{ij})=-\left( \frac{e}{r_{ij}} \right)_L + b_S r_{ij} +v_0,
\end{equation}
where $r_{ij}$ is the distance between the lattice sites $i$ and $j$.
For the first $\chi^2$ fit, the ground state energies at $x/a = 2, \ldots ,6$
and $y=0$ are used as input. For the minimization, the CERN library
programme MINUIT \cite{JR} is employed. The resulting values for the
potential parameters for $\beta=2.4$ are
\begin{displaymath}
e=0.255; \; \; b_S a^2_{2.4}=0.0691; \; \; v_0 a_{2.4}=0.560
\end{displaymath}
with $\chi^2$=0.33 per degree of freedom. The corresponding lattice
potential ($v_L$) values
for $\beta=2.4$ are given in Table 1. There it is seen that $v_L$ values
are typically within 0.3 \% from the measured values $v_{MC}$ except for
$x/a=1$, where the lattice artefacts play an important role. The second
fit combines data from the $\beta=2.4$ and $\beta=2.5$ measurements. For
$\beta=2.5$ a much smaller sample of 10 blocks of 12 sweeps was collected.
The fit involves 5 parameters, because the $e$-value is common, and only
on-axis points with $x/a=2, \ldots ,6$ are taken as input. The result
of the fit is ($\chi^2$=0.86 per degree of freedom)
\begin{displaymath}
e=0.249; \; \; b_S a^2_{2.4}=0.0699; \; \; v_0 a_{2.4}=0.555;
\; \; b_S a^2_{2.5}=0.0338; \; \; v_0 a_{2.5}=0.549
\end{displaymath}
and Table 1 lists the corresponding potential values (for $\beta$=2.4).
In this case the
typical precision is about 0.5 \% for both sets of data. The $b_S$ values
can be used to extract the ratio of the lattice spacings with the result
$\rho=a_{2.4}/a_{2.5}=1.437(17)$, where the error is an underestimate, because
correlations between different $r$-values are ignored. Furthermore, if the
reasoning given in the Introduction allows for the identification of $b_S$ with
the experimental value 0.194 GeV$^2$ \cite{CM1} for the string tension,
the lattice spacing takes on the value $a_{2.4}\approx 0.12$ fm. The inclusion
of the off-axis points ($y=0, \ldots,3$) makes the $\chi^2$-value worse
(1.29 per degree of freedom), but the parameter values remain the same within
errors, e.g.
$a_{2.4}/a_{2.5}=1.432(9)$. The fact that an acceptable fit is found with a
common value of $e$ implies that the deduced continuum  potential $v(r)$ will
be the same at both $\beta$-values. An extension
to SU(3) of this type of an analysis can be found in ref. \cite{CM2}.

Later in sections 4 and 5, the interquark potential $V_{qq}$
is needed as input for a model to describe the energies of four-quark
configurations. The optimal situation would be to use only $v_{MC}$.
However, as seen from table 1, these are not calculated for
the largest values of $x,y$. Therefore, as a compromise, $v_{MC}$
is used for $x/a\leq 3$ with $y/a\leq 3$, and for larger values of $x$ and
$y$ the  fit $v_L(\beta=2.4)$  is taken. Other alternatives, such as
the use of $v_L$ for all values of $x$ and $y$, or the combined fit
to the $\beta=2.4$ and 2.5 Monte Carlo results, would lead to results that
do not differ in any significant way from those given by the above compromise.

\section{ The energies of four quarks in rectangular geometries}
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\vskip 0.5 cm

In ref.\cite{GMP2} the energies ($E_i$) of four quarks in the coplanar
rectangular
configurations shown in fig.1 were calculated with a $16^3\times 32$
lattice and $\beta=2.4$. These values of $E_i$ are {\em exact} energies
for the four-quark system on this particular lattice. However,  the results of
physical  interest are those ($E^c_i$) for the continuum system, which should
correspond to the case of an infinitely large lattice with an infinitely
small lattice spacing $a$. Any difference between the calculated $E_i$
and the desired $E^c_i$ is a so-called lattice artefact.
To check on such possible lattice artefacts in
the results of ref.\cite{GMP2}, two points should be studied more carefully.
\begin{itemize}
\begin{enumerate}
\item To what extent
are the results dependent on the size of the lattice

being $16^3\times 32$?
\item  To what extent
are the results dependent on the lattice spacing $a$, which can be varied
by changing the coupling through changing $\beta$?
\end{enumerate}
\end{itemize}
As mentioned in the introduction, the $E_i$ are the differences  between
the total four-quark energies $[V_i(4q)]$ and that of two separated
two-quark systems $[V_{qq}]$. In the present case of rectangles with sides $r$
and
$d$ -- with $r\geq d$ -- , this reduces to
\begin{equation}
\label{v4q}
E_i=V_i(4q)-2V_{qq}(d).
\end{equation}

\subsection{ Check on finite size of lattice}
\vskip 0.5cm

At first sight it may seem that finite size effects should appear
when the distance between the quarks $(r)$ is comparable to (or
greater than) $L/2$, since in this case there is also the possibility of
 joining the two quarks with a flux path of length $L-r$, which
{\em encircles} the spatial periodic boundary. However,
 for the 2-body potential the finite size effects are known to be
relatively small -- indeed it is feasible to measure $V_{qq}(r)$
for values of $r$ greater than $ L/2$. This arises, since there is
no mixing between the direct path
with flux of length $r$ and the above one of length $L-r$. This lack of
mixing comes from the
Z(N) invariance of the lattice Wilson action in the quenched
approximation -- see appendix A and ref.\cite{Hunt} for more details.

 Now for the 4-body case in SU(2), this Z(2) invariance does not forbid direct
mixing, because two flux paths then encircle the spatial boundary.
 Thus, a priori, there may be finite size effects at
$r \approx L/2$. To estimate the size of such effects is less
straightforward but the following calculations and arguments indicate that
they are relatively small.

In order to check the dependence of the earlier results on the size of the
lattice, the latter is increased from $16^3\times 32$ to $24^3\times 32$.
{}From a numerical point of view this is a considerable change, because it
involves the storage of over $28\times 10^6$
bytes of data compared with only $8\times 10^6$
earlier i.e. approximately an increase by  the expected  factor of
$(24/16)^3$.
 Since the computer programme is constructed in such a way
that this data is read-from and written-to files many times, the numerical
efficiency drops considerably with one measurement taking about 1700 seconds
compared with about 500 seconds earlier.
In view of this, only 20 blocks of 32 sweeps  are carried out for the square
configurations i.e. $r=d$. This results in 160 measurements per
configuration since the lattice is updated by three
over-relaxation sweeps plus a heat bath sweep, which is then followed by a
measurement of the appropriate correlation functions i.e. the ratio of
up-date sweeps to measurements is 4:1.
 This number of measurements turns out to be sufficient, since the
main aim is to see whether or not the {\em average} of each energy
changes -- less attention being paid to minimizing the errors associated with
each energy.  The finite size effects are checked only for {\em squares} with
sides upto $r=7a$, since the neighbouring rectangles have binding energies
that are at least an order of magnitude smaller and -- relatively -- have
much larger error bars.
The results are given in table 2 and fig.2.

In table 2 the columns $E^{16}_i(1600)$ show the ground $(i=0)$ and
excited $(i=1)$ state energies for the 1600 measurements on the
$16^4\times 32$ lattice of ref.\cite{GMP2}, whereas those headed
$E^{24}_i(160)$ are for the 160 measurements on the present
$24^3\times 32$ lattice.
The errors $(\epsilon )$ in the two sets of
measurements are seen to be, in general, consistent with each other, since
they should be approximately related by the expression
\begin{equation}
\epsilon ^{24}(N_{24})\approx\epsilon ^{16}(N_{16})
\sqrt{\frac{N_{16}16^3}{N_{24}24^3}}\approx2\epsilon ^{16}(N_{16}) .
\end{equation}
The square root  factor is simply the ratio of the total
number of points used in each case i.e. the product of the number of
measurements times the number of points per measurement. This expression
indicates that the errors in both cases are comparable -- with the tendency
for $\epsilon^{24}$ to be somewhat larger than $\epsilon^{16}$.
This feature is qualitatively observed in table 2.

It is now seen that both the ground state energies $(E_0)$ and that of the
first excited state $(E_1)$ are in agreement with the earlier results -- the
only possible exception being $E_0(r=d=7a)$. However, there are two possible
reasons for the latter apparent difference.  Firstly, there is a purely
numerical reason. When the actual contributions from each lattice are
studied, it is found that -- for the largest two squares -- the error
distributions  are non-gaussian, so that it is not
meaningful to ascribe any significance to the discrepancies between the
results for the $16^3$ and $24^3$ lattices.
 Secondly, a difference may not be unexpected,
since there are effectively two basic energies involved with a square of
size $r=d$.
\begin{itemize}
\begin{enumerate}
\item The unperturbed energy of state A or B i.e. $E_A=2V_{qq}(r)$ -- see
fig.1.
\item The energy of two flux tubes that go to the edge of the lattice
and then, because of the cyclic boundary conditions imposed, reappear on the
opposite edge. This state (F) has an unperturbed energy of

 $E_{F}=2V_{qq}(L-r)$.
\end{enumerate}
\end{itemize}
These possibilities are shown in fig.3.

For large $r$ these energies are dominated by the linear term
$b_S$ in eq.(\ref{Vqqr}). Therefore, the $16^3\times 32$ lattice (i.e. $L=16a$)
gives for $r=7a$ the energies

\noindent $E_A=2V_{qq}(7a) \ \ {\rm and} \ \ E_F=2V_{qq}(9a)$
with the difference $E_F-E_A\approx 4b_Sa \approx 0.3/a $.
 On the other hand, for the $24^3\times 32$ lattice
$ E_F=2V_{qq}(17a)$ with the difference
$E_F-E_A\approx 20b_Sa \approx 1.5/a$.
Having made this observation, it is still not easy to calculate accurately
the effect the F-states have had on the energies $E_0$ and $E_1$.

In appendix B, this discussion is pursued further in the framework of the
model to be introduced in section 4. There arguments are given that suggest
 the finite size correction is, indeed, unimportant for
$E_0(r=d=7a)$ on the $16^3 \times 32$ lattice.

\vspace{0.3cm}

\subsection{ Check on scaling}
\vskip 0.2cm
So far in this paper, all the four-quark measurements have been carried out
with
$\beta=2.4$. At this $\beta$ value the string tension
$b_Sa_{2.4}^2=0.0699$  in
eq.(\ref{Vqqr}), which corresponds to a lattice spacing of $a\approx 0.12$ fm.
 For the present results to be meaningful, since any self-energies cancel,
the values of the four-quark energies ($E_i$) should be
independent of $a$. In other words, for two different lattice spacings
$a_1$ and $a_2$ with $\rho=a_1/a_2$, the energies should satisfy
\begin{equation}
\label{scal}
E_i(a_1,r)= E_i(a_2,r).
\end{equation}
In order to isolate this scaling effect from a finite size effect,
it is convenient to use lattices which have approximately the  same
{\em physical} size. Since for $\beta=2.4$ the lattice spacing is 0.12 fm,
the $16^3$ lattice corresponds to $\approx 7 $ fm$^3$. Therefore, the $24^3$
lattice requires $a\approx 0.082$ fm, which corresponds to $\beta\approx 2.5$.
In practice, $\beta=2.5$ is used to give $\rho=1.437(17)$ -- see section 2 and
ref.\cite{CM1}.
As in the case of the finite size check, the use of a $24^3\times 32$ lattice
again prohibits numerically an exhaustive study. Therefore, only
11 blocks of 240 sweeps i.e. 660 measurements are made. The results are shown
in table 3 and fig.4.

Here it is seen that the hoped-for agreement expressed by
eq.(\ref{scal}) is only satisfied for the largest squares to within about
$20\%$. However, later in section 4, it will be seen that  these differences
between the results  for the  largest squares with the two values of $\beta$
 are very much smaller than the differences between these values
of $E_i$ and the predictions ($E'_i$) of a conventional model based on
pair-wise
interquark potentials i.e. $|E'_i-E_i|\gg |E_i(\beta =2.5)-E_i(\beta =2.4)|$.
For example, later in table 4 at $r=5a$ it will be shown that $aE'_0=-0.104$
and $aE'_1=0.312$, making $|E'_0-E_0|\approx 0.07/a$
and $|E'_1-E_1|\approx 0.21/a$  compared with
$|E_0(2.5)-E_0(2.4)|\approx 0.015/a(2.4)$
 and $|E_1(2.5)-E_1(2.4)|\approx 0.030/a(2.4)$.

Therefore, the conclusion to be drawn from this subsection is that, for the
present purposes,  scaling has been achieved with the value of
$\beta=2.4$. However, as discussed in ref.\cite{mich}, asymptotic
scaling -- which relates the results from differing $\beta$'s by
perturbative arguments -- should not be expected at these values of $\beta$.

\subsection{ Autocorrelation}
\vskip 0.5cm

Since the above type of simulation involves the generation of  successive
updated lattice configurations, one source of uncertainty in the error
estimation is the degree to
which these configurations are statistically
independent of each other. In the present work, the results are collected
into $N_B$ blocks each containing the average $E_i (i=1,..,N_B)$
 of $N_M$ measurements. From these numbers, the
quantities of interest are the  overall energy average, $\bar{E}$, and its
error $\sigma (\bar{E})$, which can be expressed as
\begin{equation} \bar{E}=\frac{1}{N_B}\sum_{i=1}^{N_B} E_i \ \ {\rm and } \ \
\sigma (\bar{E})=\sigma (E)\sqrt{\frac{1}{N_B}f(Auto)},
\end{equation}
where
\begin{equation}
\sigma (E)=\sqrt{\frac{1}{N_B}\sum_{i=1}^{N_B}(E_i-\bar{E})^2}
\end{equation}
is the standard deviation of the $E_i$'s and $f(Auto)$ is calculated as follows
 -- see ref.\cite{Priest}.
The normalised autocorrelation function between $E_i$ and $E_{i+k}$ is
defined as
\begin{equation}
\rho(k)=\frac{C(k)}{C(0)}=
\frac{\frac{1}{N_B-k}\left[ \sum_{i=1}^{N_B-k}(E_i-\bar{E})
(E_{i+k}-\bar{E})\right]}
{\frac{1}{N_B}\left[\sum_{i=1}^{N_B} (E_i-\bar{E})^2\right]},
\end{equation}
giving
\begin{equation}
\label{f(A)}
f(Auto)=1+2\sum_{k=1}^M(1-\frac{k}{N_B}) \rho(k).
\end{equation}
As discussed in ref.\cite{Mad}, $f(Auto)$ has the significance
of a decay time $\tau=\frac{1}{2}f(Auto)$.
The value of $M$ is then chosen to retain most of the ``signal'' but discard
most of the ``noise''. This is carried out by imposing the condition that $M$
need only be large enough to ensure it is several ($n\approx$ 3 or 4)
 times greater
than $\tau$ i.e. $M$ should be the smallest integer to
satisfy the equation $M\geq n\tau=n\frac{1}{2}f(Auto,M)$.
It is found that $M\approx 2$ in this particular problem.
The ratio $N_B/f(Auto)$ can then be interpreted as the equivalent number of
independent blocks.
In the present study, it is expected that only the correlation  between nearest
neighbours could possibly be of importance i.e. $E_i=cE_{i-1} + \epsilon_i$,
where $\epsilon_i$ is not correlated with other $E_i$'s
 and $|c|$ is assumed to be less than unity.
 In this idealised case, for large $M$ and $N_B$, this gives
$\rho(k)=c^k$ and $f(Auto)\approx 1+2(c+c^2+c^3+....)=(1+c)/(1-c)
=(1+\rho(1))/(1-\rho(1))$.

As typical illustrations of these estimates, there is shown in fig.5
the values of $\rho(k)$ for different values of $\beta$ and
different lattices. In all cases $\rho(1)$ is seen to be small
i.e. $|\rho(1)|\leq 0.2$. Also, since the limitation of $M$  in the sum
in eq.(\ref{f(A)}) is 1 or 2 to eliminate the ``noise'', it follows that
$\sqrt{f(Auto)}\approx 1.0\pm 0.2$.

The conclusion to be drawn from this subsection is that the error,
$\sigma(\bar{E})$, on the average value of the $E_i$ is adequately given by
simply
$\sigma (E)\sqrt{\frac{1}{N_B}}$ i.e. with the autocorrelation effect
expressed by $f(Auto)$ being negligible.

\section{  Review of model}
\setcounter{equation}{0}

\vskip 0.2 cm
So far two of the three original requirements listed in the introduction
have been calculated -- namely -- accurate and reliable values of
four-quark binding energies ($E_i$) and also the corresponding interquark
potential $V_{qq}$. It now remains to set up a
physically motivated and convenient model or prescription
for calculating the energies $(E'_i)$ of a four-quark
system in terms of the $V_{qq}$ -- the hope being that the $E'_i$ are a good
approximation to the exact $E_i$.

As discussed in refs.\cite{MGP}-\cite{GMP2}, the first model that comes to
mind is one in which configurations A and B in fig.1 are used as a basis for a
two-by-two matrix constructed directly from the potentials $V_{qq}$. In the
static approximation, for which the kinetic energy is zero, eigenvalues
$\lambda_i$ can then be obtained by diagonalizing
\begin{equation}
\label{Ham}
\left[{\bf V}-\lambda_i {\bf N}\right]\Psi_i=0,
\end{equation}
with
\begin{equation}
\label{NV}
{\bf N}=\left(\begin{array}{ll}
1&1/2\\
1/2&1\end{array}\right)\ \ {\rm and}\ \ {\bf V}=\left(\begin{array}{cc}
v_{13}+v_{24} & V_{AB}\\
V_{BA}&v_{14}+v_{23}\end{array}\right),
\end{equation}
where $v_{ij}=V_{qq}(r_{ij})$ and
\begin{equation}
\label{AVB}
<A|V|B>=V_{AB}=V_{BA}=
\frac{1}{2}\left(v_{13} +v_{24} +v_{14}+v_{23} - v_{12}-v_{34} \right).
\end{equation}
Here the colour factors are first determined using the form of $v_{ij}$
that is pure one-gluon-exchange. Only after this are the radial forms of
$v_{ij}$ taken from table 1 or eq.(\ref{coul1}) -- see ref.\cite{GMP2}.
The matrix {\bf N} arises since the configurations A and B, when expressed
purely
in terms of quark degrees of freedom, are not orthogonal but have the
overlap $<A|B>=1/2$. In general for SU(N), this overlap is 1/N.
{}From the eigenvalues $\lambda_i$ are now subtracted the internal energy
of the meson-like state with the lowest energy [i.e. $v_{13}(d)+v_{24}(d)$ for
$d\leq r$]. This gives the quantities
\begin{equation}
E'_i(d,r)=\lambda_i-[v_{13}(d)+v_{24}(d)]
\end{equation}
 that can be interpreted
 as the binding energy of two mesons of length $d$ a distance $r$ apart.
Since a very similar subtraction is also made in the lattice calculation
of section 3,
it is meaningful to compare $E'_i(d,r)$ with the $E_i(d,r)$ in tables 2 and
3. This is shown in table 4, where
it is seen that the $|E'_i|$ are always greater than the $|E_i|$.
The difference increases as $r$ or $d$ increase until, for the largest
squares, considered $|E'_i| \approx 10|E_i|$.

To overcome the above differences between $E_i$ and $E'_i$, in
refs.\cite{MGP}-\cite{GMP2}
a modified version of the above model was proposed, in which the four-quark
energies $E_i$ could be understood in terms of the interquark potential
$V_{qq}$ and a quantity $f$ -- a phenomenological function of the spacial
coordinates of all four quarks. This function can be interpreted as an overlap
factor of the gluon fields mediating the interaction between the quarks.
Using again the configurations A and B in fig.1 as a basis, the problem now
reduces to diagonalising
\begin{equation}
\label{Hamf}
\left[{\bf V}(f)-\lambda_i(f) {\bf N}(f)\right]\Psi_i=0,
\end{equation}
with
\begin{equation}
\label{NVf}
{\bf N}(f)=\left(\begin{array}{ll}
1&f/2\\
f/2&1\end{array}\right)\ \ {\rm and}\ \ {\bf V}(f)=\left(\begin{array}{cc}
v_{13}+v_{24} & fV_{AB}\\
fV_{BA}&v_{14}+v_{23}\end{array}\right).
\end{equation}
{}From the eigenvalues $\lambda_i(f)$ are again subtracted the internal energy
of the meson-like state with the lowest energy [i.e. $v_{13}(d)+v_{24}(d)$ for
$d\leq r$]. This gives the quantities
\begin{equation}
E'_i(f;d,r)=\lambda_i(f)-[v_{13}(d)+v_{24}(d)].
\end{equation}
To ensure the equality $E'_i(f)=E_i$, the function $f$ is varied. In
practice, it is found that -- for each configuration $(d,r)$ -- a single
value of $f$, to be denoted by $\bar{f}$,
 is able to give an optimal fit to both $E'_0(\bar{f})=E_0$
and  $E'_1(\bar{f})=E_1$, indicating that the parametrization suggested
in eqs.(\ref{Hamf},\ref{NVf}) contains the most important features of
the more precise lattice calculation. The results are shown in table 4.

One important consequence of the above fit of $E'_i(\bar{f})$ to $E_i$ is that,
for
the larger squares, $\bar{f}$ is certainly {\em not} consistent with unity.
This means that
conventional ways \cite{cluster} of treating the interaction between quark
clusters purely in terms of the pair-wise interquark potential of
eq.(\ref{Vqqr}) are dubious.  However, as the size of the squares becomes
smaller, the value of $\bar{f}$ tends towards unity. For the smallest square
considered in this work i.e. $r=d=0.082$ fm -- the (1,1) case with $\beta=2.5$
in table
3 --
$E'_0(f=1)=-0.0626/a(2.5)$, $E'_1(f=1)=0.1878/a(2.5)$ and $\bar{f}=0.92$. This
is in
line with the expectation that the weak coupling limit
(i.e. one-gluon-exchange corresponding to $f=1$) emerges as $r$ and $d$ tend
to zero.

In table 4, for non-squares, the values of $\bar{f}$ are dictated mainly
by the energies of the excited states ($E_1$). The reason for this is clear
from the following argument. Use the notation of fig.1 and also

$v_1=v_{13}(d)=v_{24}(d), \ v_2=v_{14}(r)=v_{23}(r), \ v_d=v_{12}=v_{34}
, \  \Delta=2(v_2-v_1) \\ {\rm and} \ \bar{\Delta}=2(v_2-v_d).$
It is now seen that $\bar{\Delta}/\Delta\rightarrow 0$ as the rectangles become
more and more elongated. In this case, by expanding in powers of
$\bar{\Delta}/\Delta$,
\begin{equation}
\label{elong}
E'_0\approx -\frac{\bar{f}^2\bar{\Delta}^2}
{4\Delta}  \ \ {\rm and} \ \
E'_1\approx\frac{\Delta-\frac{\bar{f}^2\bar{\Delta}}{2}}
{1-\frac{\bar{f}^2}{4}}.
\end{equation}
In other words, to lowest order, when the coupling ($\bar{\Delta}$) vanishes,
$E'_0\approx 0$ and $E'_1\approx \Delta/ (1-\frac{\bar{f}^2}{4})$
i.e. $E'_1(\bar{f})=E_1$ gives a direct
measurement of $\bar{f}$. For example, even with the $d=a, \  r=2a$ case, from
table 1, it is seen that
$a\Delta=0.3766 $ and $a\bar{\Delta}=-0.0806$. The approximate expressions
in eqs.(\ref{elong}) then give -- for the optimal $\bar{f}=0.869$ from
table 4 -- the values
$aE'_0\approx-0.0030$ and $aE'_1\approx 0.502$ in good agreement with the Monte
Carlo results. This suggests that for non-squares it may be better to
concentrate on extracting the energy of the first excited state, since
this gives a much more reliable value of $f$ than $E_0$.
However, this is not without its problems. Firstly, in the present
lattice calculation the highest eigenvalue extracted $(E_1)$ is
influenced more than $E_0$ by the presence of even higher eigenstates.
Secondly, in the model proposed in eqs.(\ref{Hamf},\ref{NVf}) only two
configurations A and B are explicitly introduced to explain $E_0$ and $E_1$.
However, these two configurations and the factor $f$ may not be simulating
the effect of other configurations ( e.g. glue-ball exchange) in the same way
as these effects appear in the Monte Carlo energies
$E_0$ and $E_1$. Again, the higher eigenvalue
$E_1$ would be expected to be the less reliable.

The existence of a function $f$ that depends on the positions of all four
quarks is of interest but not particularly useful, unless it can be
parametrized in some convenient manner. In refs.\cite{MGP}-\cite{GMP2}
two such parametrizations were tested.
{}From refs.\cite{Morimatsu,Morimatsu2}
-- motivated by strong coupling arguments -- the phenomenological
form was taken to be
\begin{equation}
\label{f1}
f_1=\exp[-\alpha b_S S],
\end{equation}
where $b_Sa_{2.4}^2=0.0699$ is the string tension dictated by the interquark
potential of eq.(\ref{Vqqr}) and $S$ is the minimal area of the surface
bounded by the straight lines connecting the quarks and antiquarks.
The other form -- the one proposed in \cite{Masud} -- was
\begin{equation}
\label{f2}
f_2=\exp[-\frac{kb_S}{6}\sum\limits_{i<j}r^2_{ij}]
\end{equation}
i.e. the cut-down is governed by the average of the six links
present in a $q^2\bar{q}^2$ system.
In eqs.(\ref{f1},\ref{f2}) the $\alpha$ and $k$ are at present free
parameters to be determined by fitting $f_1$ and $f_2$ to the $\bar{f}$ in
table 4. It is then of interest to study whether either $\alpha$ or $k$ is
constant as the values of $r$ and $d$ vary.

Both of these parametrizations of $f$ accommodate the two
extreme models of:

1) Weak coupling, which has $f=1$ when all $r_{ij}=0$ .

2) Strong coupling, which has $f=0$ when any $r_{ij}\rightarrow \infty$.

\noindent For squares, these two parametrizations are equivalent with
$k=3\alpha/4$ -- so that only the few cases with $r\not =d$
( not included in the finite size and scaling tests of tables 2 and 3) could
possibly distinguish between $f_1$ and $f_2$. From ref.\cite{GMP2}, the
conclusion was that $\bar{k}=3\bar{\alpha}/4=0.5\pm 0.1$ achieved the best
overall fit to $\bar{f}$ -- with the error bars on the $E_i$ for the non-square
configurations being too large to separate $f_1$ from $f_2$.
However, with  the use of the more consistent set of $V_{qq}$ from table 1 and
also the realisation that fitting $E'_1(\bar{f})$ to $E_1$ is possibly more
informative for extracting a value of $\bar{f}$, the new results for $\bar{k}$
and
$\bar{\alpha}$
are shown in table 4. Here it is now seen that -- for squares --  $\bar{k}$ (
and
therefore $\bar{\alpha}$) shows a distinct tendency to decrease as the
squares become larger. Furthermore, for rectangles the values of
$\bar{\alpha}$ appear to decrease less for a fixed $d$ with increasing $r$.
This
suggests that the $f_1$ parametrization -- although not optimal -- is
superior to that of $f_2$.

Clearly, to get a better understanding of $f$, more configurations very
different from squares are necessary. Within the framework of
rectangles, geometries with $r\gg d$ would be suitable. However, as shown
 in table 4,

$E_0(d,r=d)\approx 10\times E_0(d,r=d+a)  \approx 100\times E_0(d,r=d+2a)$,

\noindent which rules out extracting $E_0$ for more elongated
geometries, since already
the results for $E_0(d,r=d+2a)$ are comparable to the error bars e.g.

$aE_0(1,1)=-0.0695(5),\ \ aE(1,2)=-0.0026(2), \ \ aE(1,3)=-0.0003(3)$.

\noindent However, as shown in eq.(\ref{elong}), in these cases it is $E_1$
that is of more importance.
Therefore, one extension to the above analysis could be to concentrate on $E_1$
 for configurations with $r\gg d$. In such cases, $f$ would be given
directly by eq.(\ref{elong}) as
$\bar{f}\approx 2\sqrt{1-\Delta/E_1}$. The restriction on $r$ would then
only be limited by the finite lattice size problem discussed earlier.
This possibility will not be considered here. Instead, in the next section
the even more extreme geometry of four colinear quarks will be studied.

\vspace{0.3cm}

\section{  The case of four colinear quarks}
\setcounter{equation}{0}

\vskip 0.2 cm
Another series of four-quark configurations -- that are very different from
rectangles and yet can be treated easily by a Monte Carlo simulation -- is
the geometry, shown in fig.6 of four quarks on a straight line.

In common with the configurations A and B in fig.1, all of the basic states
are constructed from paths that essentially involve a {\em single} spatial
dimension. For example, since they represent mesons in their ground state,
$M_{13}$ and $M_{24}$ in A can be constructed from links all in the
$y$-direction. Likewise, $M_{14}$ and $M_{23}$ need only links in the
$x$-direction. If the rectangular state
$C=M_{12}M_{34}=[q_1q_2]^0[q_3q_4]^0$ had been included, then L-shaped paths
involving both $x$- and $y$-links would have been necessary. In fact, in
table 1  such paths were considered in order to extract the
two-quark potential $V_{qq}(x,y)$ for $y\leq x$ -- quantities needed in
eq.(\ref{AVB}) when setting up the model of eqs.(\ref{Hamf},\ref{NVf}).
In ref.\cite{GMP2}, it was considered sufficient to  treat only the two
configurations
A and B in fig.1, since state C has a higher energy. However, in the linear
case the distinction between states A,B and C is less clear, since -- in the
domain where the two-quark potential of eq.(\ref{Vqqr}) is dominated
by the $b_Sr$ term -- states B and C have the {\em same} energy which
is always greater than that of state A.
 At this point,
it should be remembered that the four quarks are, in fact, fermions and
a more correct notation for states A,B and C is in terms of quarks and
{\em antiquarks}.
The
use throughout of SU(2) means that
 quarks and antiquarks belong to the same two dimensional
 representation of the colour group. However, in any discussion
 involving all three states A,B and C, it is important -- even with
SU(2) -- to keep track of phases.  It should be noted that,  in the limit when
all link matrices U are unit matrices,  the three off-diagonal overlaps of the
states A, B, and C are all $\pm 1/2$ where the signs are convention-dependent;
however,  independent of convention,  the product of all three signs is
negative.  This minus sign must be put in by hand in the $3\times 3$
calculation.

In ref.\cite{GMP2} the use of A and B as a basis in the Monte Carlo
simulation was considered from a variational point of view. In
principle, using only state A or B would lead to the same ground state
energy $E_0$, but this would require extrapolating further in Euclidean
time T. In addition, the extended variational basis of A and B enabled an
estimate to be made of the excited state energy $E_1$. In the light
of this discussion, it might be
thought that the even larger basis  of A, B and C would lead to a more rapid
convergence in T and also give the three lowest energies $E_{0,1,2}$.
However, because of the way -- in this work -- the states A, B and C are
constructed
from basic one-dimensional lattice links, for the case of colinear
configurations
these states are not linearly independent of each other. Therefore, the
inclusion of all three leads to a singular matrix in the numerical evaluation.
 This singularity would not arise in general.

\subsection{ Results of the Monte Carlo simulation and their interpretation
in terms of the model in section 4}

\vskip 0.2cm

In the notation of fig.6, the colinear geometries considered have
the symmetric form $r_{13}=r_{24}=d$ and $r_{12}=r$ with $d/a=1,2,3$ and
$r=d+a,d+2a,d+3a$. As discussed in section 3-A,
this means that for $d=3a$ and $r=5a, \ 6a$ finite lattice artifacts could be
playing a role, since in these cases flux paths of length $8a$ and $9a$ enter.

The results for 20 blocks of 320 sweeps each, i.e. 1600 measurements, on
a $16^3\times 32$ lattice with $\beta=2.4$ are shown in tables 5 and 6.

Several points can be made about these tables.

1) $E_0(r,d)$ is  significantly non-zero only for $r=d+a$ i.e. for those
configurations in which the two mesons are within one lattice spacing
at their point of closest approach.

2) $E'_i(f=1)\not= E_i$ implying that $f\not=1$. Therefore, again -- as
in the rectangle case -- the conventional two-quark potential model
corresponding to $f=1$ is not successful.

3) Because $f\not=1$ is required, this implies the parametrization
$f_1$ in eq.(\ref{f1}) cannot be correct, since for linear configurations,
the area $S$ in the definition of $f_1$ is always zero -- to give
$f_1=1$ for all linear cases.

4) Since the accuracy of $E_1$ is considerably better than that for $E_0$, the
compromise value of $\bar{f}$ ensures $E'_1(\bar{f})=E_1$ to three significant
figures. On the other hand, the $E'_0(\bar{f})$ are less precisely
determined but still remain within the error bars of the $E_0$. The reason
for this was given in eq.(\ref{elong}). However, as discussed after that
equation, the use of $E_1$ does have problems not present with $E_0$.

5) As seen from fig.7, the values of $\bar{f}$ needed to fit $E_i(\bar{f})=E_i$
are
remarkably {\em constant} with the value $\bar{f}=0.86(2)$ covering the
whole range of colinear geometries considered here. This indicates that
it will be difficult to extract from $\bar{f}$ any precise geometry
dependence of the form suggested in eq.(\ref{f2}). Indeed,
the values of $\bar{k}$ needed to satisfy
$\bar{f}_2=\exp\left(-\bar{k}b_s<r_{ij}^2>\right)$ show a clear tendency to
decrease as the four-quark system gets larger -- see table 6 and fig.7.
The use of $(r,d)$ as the abscissa is not the most illuminating, since
then the value of $<r_{ij}^2>$ -- a more appropriate variable -- is not evenly
spaced. In fact, for (4,1) the value of $6<r_{ij}^2/a^2>$  is 68, whereas for
(3,2) it is only 52. However, this does not alter the conclusion that
$\bar{k}$ is {\em not } constant.

6) Another alternative parametrization of $f$ is
$\bar{f}_3=\exp\left(-\bar{p}\sqrt{b_s}L\right)$, where $L$ is the length
of the perimeter enclosing the area $S$ of $f_1$ in eq.(\ref{f1}).
 For the present colinear
case, $L=2(d+r)$ and the corresponding values of $\bar{p}$ are also given
in table 6 and figure 7. However, as expected from the constancy of
$\bar{f}$, this does not fare any better than the previous parametrization
$\bar{f}_2$ with $\bar{k}$. Again $\bar{p}$ has a clear tendency to decrease as
the
size of the system increases. Here a more appropriate abscissa variable
 would be the perimeter $L$ when plotting figures.  In this case the plot
of $\bar{p}$ would be somewhat distorted
with, for example, (4,1) and (3,2) corresponding to the same abscissa. But
again, this does not alter the conclusion that $\bar{p}$ is {\em not} constant.

The overall conclusion to be drawn from the above is that  a
reduction factor $\bar{f}$ is needed -- but that this factor is essentially
constant
at $\bar{f}=0.86(2)$ for all of the colinear geometries considered.

So far the Monte Carlo results and their subsequent analysis in terms of
the model reviewed in section 4 have been purely in terms of the states
A+B in fig. 6. However, similar analyses can be carried out with
A+C or B+C. As expected, in the Monte Carlo simulation these lead to exactly
the same values of $E_{0,1}$ in table 5. Also the three model energies
$E'_{0,1}$$(f=1$,A+B) \ , \ $E'_{0,1}$$(f=1$,A+C) and $E'_{0,1}$$(f=1$,B+C) are
all equal. But this symmetry does not persist, when the corresponding values
of $f$ are extracted to ensure $E'_i(f_{AC})=E_i$ or $E'_i(f_{BC})=E_i.$
This is shown in table 7, where the values of $f_{AB},f_{AC}$ and $f_{BC}$
 are given to ensure $E'_0(f)=E_0$ and in the brackets [...] to
ensure  $E'_1(f)=E_1$. However, in this case the use of the B,C combination
is clearly not very sensible, since it misses the lightest state A and then
attempts to reproduce its presence.

The following points can be made about table 7.

1) Since the $E_0$'s have larger error bars than the $E_1$'s, the resultant
errors on $f_{AB}(E_0)$ and $f_{AC}(E_0)$ are also larger than those on the
$f_{AB}(E_1)$ and $f_{AC}(E_1)$. In general, within these error bars
$f_{AB}(E_0)\approx f_{AC}(E_0)$.

2)  $f_{BC}(E_0)=1.0$ to within $1\%$ -- a result readily understood,
since from eqs.(\ref{Hamf},\ref{NVf}) it follows that
$f_{BC}(E_0)\approx 1-\frac{E_0}{4(r-d)}$, when the interquark potential
$v(r)$ is simply proportional to  $r$.

3) Those values of $f$ adjusted to fit $E'_1(f)=E_1$ show much more variation
 than those fitting $E'_0(f)=E_0$. Also, in some cases, no solution exists
for any value of the overlap $f$.

4) For most geometries, in cases A+B and A+C, a compromise value of $f$ can
be found that gives reasonable fits to both
$E_0$ and $E_1$. However, for B+C, the $f$'s that ensure
$E'_1(f(E_1))=E_1$ yield values of $E'_0(f(E_1))$ that are very far from
$E_0$ e.g. in geometry (3,1), $aE'_0(f=0.58)=0.352$ compared with the Monte
Carlo
number $aE_0=-0.0002(3)$. In other words, there is no compromise value of
$f$ for case B+C.

The above asymmetry between the use of A+B, A+C or B+C, when fitting
$E'_i$ to $E_i$ is not entirely satisfactory and has to be resolved, if
something like the 2$\times$ 2 model of section 4
is to be used as the basis of a dynamical
calculation as in ref.\cite{Masud}.

\subsection{ A test case of QCD in two dimensions (1+1)}

\vspace{0.2cm}

The above analysis unavoidably involves numerical results extracted
from Monte Carlo simulations on a 4-dimensional lattice. However,
 it is possible to study colinear colour sources in a simple
approximation for which exact theoretical results are known.
This is QCD in two dimensions (QCD2) -- the one spatial direction
 allowing colinear (and only colinear) configurations. For quenched QCD2,
the spectrum is known {\em exactly} even on a lattice. This
can be summarised as the requirement that each
link is in a representation of the colour group (i.e. SU(2) here).
Therefore, the links can be in the singlet ground state (J=0), or they can be
excited
to a fundamental representation (J=1/2), an adjoint representation
(J=1),.. etc. The energy per unit length for such links is given by
$b_J=\frac{4}{3}KJ(J+1)$.
For 4 colinear quarks, the lowest state (A in fig.6)
 has energy $2Kd$, while the first excited state has, in the middle, an adjoint
link of length $(r-d)$ resulting in an energy $2Kd+8K(r-d)/3$.  This exact
result
applies at any $\beta$-value, i.e. strong coupling or weak coupling.

A comparison with the above $f$-mixing model of
eqs.(\ref{Hamf},\ref{NVf}) shows that
there is  agreement
with the exact QCD2 results  provided $f=1$ -- independent of which
combination of states is used for the analysis A+B, A+C or B+C.
 This is also true in the more general case when $d_{13}\not= d_{24}$.
If the interpretation of $f$ being a gluon-field overlap factor
is correct, then it is easy
to understand that $f$ must be unity, since with only one spatial direction
the colour flux must overlap fully.
Earlier, the main motivation for the $f$-mixing model had been from
weak coupling arguments. So this agreement
between the mixing model and QCD2 does suggest that the model
is sensible even at large $\beta$. This adds support to
the claim that it may be a useful
phenomenological tool when the extension to $f\not = 1$ is made.

\section{  Conclusion}

In this paper, by considering a $24^3\times 32$ lattice and $\beta=2.5$,
 it has been shown that the use of a $16^3\times 32$
lattice with $\beta=2.4$ does not suffer from significant lattice
size or scaling effects. Therefore, the results for the smaller lattice and
$\beta$, quoted in tables 4 and 5, are expected to be a true reflection
of the four-quark energies $E_i$ in the {\em continuum} limit.

Compared with refs.\cite{GMP,GMP2} the case of four colinear quarks is in
addition considered.
Analysing the results using states A+B in fig.6, it is found that the
gluon field overlap factor $f$
is essentially constant at 0.86(2) for all of the colinear geometries
treated.

At this stage, it is not clear which parametrization of $f$ is optimal.
However, if $f$ is only needed to $\approx 15\%$ accuracy, then the
area dependent form of $f$ in eq.(\ref{f1}) appears to be superior.
Work is now in progress on other more elaborate four-quark
geometries and this, hopefully, will shed more light on a more
optimal form for $f$.

So far the Monte Carlo simulation results ($E_i$) have been interpreted by
 the $2\times 2$ model of section 4. However, as discussed at the end of
section 5, there are features of this model that can be considered as
unsatisfactory. In particular, the present formulation of the model does not
retain the symmetry between the states A,B and C that existed between the
corresponding states in the Monte Carlo simulation. In view of this, it is
 possibly too ambitious to present the $2\times 2$ analysis of section 4
as a fully justified model. It is probably more correct to think of this
analysis as simply a convenient and physically motivated
{\em prescription} for reconstructing the Monte Carlo results.

The authors wish to thank C.Montonen for useful discussions, C. Alexandrou and
M. Locher for useful comments on the manuscript and, in addition, one of the
authors (MES) thanks the Magnus Ehrnrooth Foundation for a grant.
The authors also wish to acknowledge that these  calculations were performed on
the CRAY X-MP machines at both Helsinki and RAL (UK) - computer time on
the latter being financed by a grant from SERC.

\vskip 1.0 cm

redpajama_set_name: RedPajamaArXiv


text:
« Thoroughly Modern Muny
Dream Come True »
The Road to Muny Magic
July 13, 2012 by Michelle Kenyon ("Snoop")
Music by Alan Menken, Lyrics by Howard Ashman, Tim Rice and Chad Beguelin, Book by Chad Beguelin
Directed by Gary Griffin
Choreographed by Alex Sanchez
The Muny, St. Louis
As a longtime fan of animated films, I have to admit that Aladdin was not one of my favorite of Disney's films, even though I did enjoy it. I wasn't sure what to expect when I heard that the Muny would be presenting a new stage production based on the movie, but I figured it would probably be a crowd-pleaser, and it certainly has turned out to be just that. In many ways, this production is the ideal Muny show, and it's another excellent entry in this season of the newer, revitalized Muny.
Book writer Chad Beguelin has tweaked the story of the film to make it work better onstage and to make it longer. The story is now reminiscent of the old Bob Hope/Bing Crosby "Road" movies, with a trio of narrators: Omar (Jason Graae), Babkak (Eddie Korbich) and Kassim (Francis Jue), who make a grand entrance on real camels (gotta love the Muny) to introduce the setting and then reappear throughout the story to serve as narrators and commentators on the action. They also are traveling musicians and bandmates with Aladdin (Robin DeJesus), the young "diamond in the rough" who finds adventure and meets a princess (Samantha Massell as Jasmine), falls afoul of the evil Jafar (Thom Sesma) and gets help from a campy, wisecracking, larger than life Genie (John Tartaglia) along the way. This is basically the plot of the movie without the animal sidekicks and a few extra characters and plot twists. Even Jafar's crony Iago (Curtis Holbrook) who isn't really given much to do, doesn't seem to be a bird in the show as he was in the film. He's just a colorfully dressed, parrot-like human henchman.
This production is a great example of the power of spectacle and great performances, because even though the story is simplified in some ways from the film and there aren't any over-the-top special effects, this production is anything but dull. Its big, colorful sets (most notably the Cave of Wonders) and bright costumes help set the mood, but the performances are really what drive the show. Tartaglia as the Genie (even though the role seems smaller than it was in the film) owns the stage from his first entrance from the audience on a motorcycle (more Muny magic at work). His over-the-top, flamboyant characterization sets the tone for the show, with many pop-culture references and asides to the audience, including a great deal of Muny in-jokes. He's like a human Disney World ride with all his energy, and he makes the most of every moment he's onstage. His introductory number "Friend Like Me" is a true showstopper, and even though Tartaglia is backed by the excellent Muny ensemble, he almost doesn't need them since all eyes in the house are on him. It's a great comic performance, and the rest of the cast almost match him in their energy and enthusiasm. DeJesus as Aladdin, Massell as Jasmine and the always excellent Ken Page as the Sultan all give convincing performances, as do the trio of narrators and and the rest of the cast, but Tartaglia really is the centerpiece.
The musical numbers are a combination of songs from the film, songs that were written for the film but cut during production (such as the moving "Proud of Your Boy", movingly performed by DeJesus), and songs that were written specifically for the stage show. Of the last category, the most notable is the rousing, Vaudeville-styled "Somebody's Got Your Back", which is performed with gusto by Aladdin and his trio of bandmates. The large Muny ensemble is put to good use, with some big, bright, energetic dance numbers. The only slight disappointment is in what is perhaps the most famous song from the film, "A Whole New World" which, while beautifully sung by DeJesus and Massell, was distinctly underwhelming visually as Aladdin and Jasmine are simply spotlighted on the dark stage, sitting on the magic carpet with a background of just a few stars behind them. Still, even with that small let-down, it was an extremely entertaining production, and the cast, crew and creative team obviously pulled out all the stops to deliver such an elaborate and fun show.
I'm somewhat of two minds reviewing this show, because as a performance I really enjoyed it, but structure-wise I think it needs a little bit of revision before it can play on Broadway, which is apparently the ultimate aim of the show's producers. There are a few issues that I think should be addressed with the story–the role of the three narrators can get confusing as they pop in and out of the action, some characters are given very little to do, and I think there's a little too much breaking the fourth wall, to the point where it can take the audience too far out of the story– but for the most part it's an engaging presentation of the story from the film with a few entertaining additions. I think the show does need some work in the writing , but the performers give their all, the production looks good, and it's a whole lot of fun to watch. In many ways, this is the ultimate Muny show, and it makes for a great evening of music, laughter and spectacle.
Posted in St Louis Theatre | Tagged Aladdin, alan menken, Chad Beguelin, disney, howard ashman, John Tartaglia, muny, Robin DeJesus, st louis, Tim Rice | Leave a Comment

redpajama_set_name: RedPajamaCommonCrawl


I had a divorced mother call me a while back and wanted to know how to control her son who was addicted to his electronic games/phone. He was withdrawn and spent a great deal of his time avoiding people. He used his games on his computer and cell phone to pass the day away and his mother was concerned. This is a common problem for most parents today. Try sitting in any restaurant and see what's happening at the table beside you. The parent is usually looking at their cell phone and the kid is playing games on his electronic devices. There is no communication whatsoever.
I can remember growing up when kids parents would keep tabs on what they were doing. Mom and dad knew what was going on in the inside the house and outside as well. I don't see that so much now. With the electronics that are available to kids these days is amazing. You can find the kids in their bedrooms laying on their bed for hours playing all kinds of games. Parents don't have to go very far to find them in the house. Unfortunately most parents don't know what kinds of games their kids are playing. I have heard from parents themselves that some of these games are so violent that they are worried about the effects they are having on their children.
Parents need to take control of all electronics their kids own or use. This does not only apply to electronic games, but their computers (Internet), cell phones, television shows and videos. Too many parents are trusting their kids with the things they watch or play. A random courtesy check is appropriate and necessary to make sure your kids are not getting themselves into trouble. Some parents restrict the time their kids are on the computer or watching television. I think that is a great idea. I know we are living in the electronic age, I get that! But there needs to be time for any family to sit down and interact with each other instead of being glued to the computer screen or the television screen.
You can't develop family relationships when the your kids best friend is a fictional character on some video game. Kids will often withdrawal from family dinners, or are so preoccupied with their electronics on family vacations that they have no family connection or attend any planned events. It's hard to believe some parents are perfectly OK with this kind of behavior. If you're a parent who is sick and tired of your kids being addicted to their electronic games take heed. You may need to set boundaries with your children when it comes to electronics. Here are my tips to help you regain control of outta control kids on electronics.
limit the time you will allow your kids to play with their computers or smart phones (set boundaries for your children).
Have a curfew when lights go out and electronics are shut down.
Time outs can apply with all their electronics.
No smart phone or games while eating meals with family.
Home work first, chores second, then some time playing games or being on the smart phone.
If you've been disrespected or your set rules have been broken..take privileges away.
Poor grades, or poor attendance..good reason to take video games/phone away.
These are pretty simple rules to follow and they will help set your expectations for your kids. Stick to the game plan and your kids should follow your ground rules. Be consistent and remind the children there are consequences for bad behavior. Electronics can be very useful in helping kids learn school projects and doing homework…as long as it's not abused.
Are you children unable to break away from their electronic devices and you need help in making that happen. Does you child avoid eating meals with you because of the games they are playing? If you answered yes to any of these questions..call Dr. Mike he can help you.


WHAT TO DO IF YOUR VISA IS REFUSED?
If your visa is refused, it is important to get visa advice ASAP. We highly recommend you call our office and book in with one of our Migration Agents in Perth ASAP!
Most visa applications which are made in Australia and are refused are able to be appealed. When you are notified that your visa application is refused by the Department of Immigration and Border Protection, you will be advised of your Review rights.
It is important to note that there are strict time limits for making an appeal. Time limits depend on the type of decision and whether you are in immigration detention. The time limit cannot be extended.
We cannot reaffirm how important it is to seek advice before applying for a review. Depending on the type of application refused, current waiting times sit for nearly 1.5 years. It may be easier and cheaper to simply re-lodge the application with additional evidence to address the reason for refusal. With that said, this sometimes isn't possible, as an 'Onshore' refusal, for example, may lead to a Section 48 Bar – stopping a person making another onshore application and requiring them to leave the country to lodge a subsequent visa application. Please call us today on 08 6364 3782.
WHAT TO DO IF YOUR VISA IS CANCELLED?
You are assessed as having a substantial risk of engaging in future, unacceptable conduct or criminal behaviour.
WHAT HAPPENS IF MY VISA IS CANCELLED?
If your visa is cancelled you cannot remain in Australia unless you get another type of visa and you may not be able to apply for any other visa except a protection (refugee) visa or a bridging visa. If sentenced to a term in prison, once you have finished your sentence, you will be removed from Australia and returned to the country of which you are a citizen. After you have been removed from Australia you may never be able to return. As such, you do have the ability to Apply to the Administrative Appeals Tribunal to review the decision. Similar to refusals, you have an extremely short time limit to appeal the decision. You have nine days to make the application from the date you have been deemed to receive the application. If the decision was posted you have 7 days from the date listed in the letter.


How rich is Steven Blum?
Steven Blum Net Worth: Steven Blum is an American voice actor that has a net worth of $4.8 Million. He's spent the greater portion of his career giving to Anime, animations, and game titles, though he is no stranger to movies. Some of his controlling character voices include Spike Spiegel, Eikichi Onizuka, Wolverine, and Captain Foley. He holds a Guinness World Record for his gaming voice acting, amounting to 261 appearances as of May 10, 2012. Lovers refer to him as the "king of Anime," for he's brought to English versions of Naruto, Digimon: Digital Monsters and MegaMan NT Warrior. Blum also goes by the names David Lucas and Andrew Watton. His present agent is Arlene Thornton and Associates and his most famous films include Cowboy Bebop: The Movie and Hulk Vs. He has two children by the names of Jeremy and Brandon.
Is Steven Blum's Net Worth Deserved?


text:
The appeal of narcissistic leaders is also their downfall
October 21, 2015 5.30am EDT
Ross Roberts, Chin Wei Ong, Tim Woodman, Bangor University
Ross Roberts
Lecturer, School of Sport, Health and Exercise Sciences and Co-Director, Institute for the Psychology of Elite Performance, Bangor University
Chin Wei Ong
PhD Candidate, Institute for the Psychology of Elite Performance, Bangor University
Tim Woodman
Professor and Head of the School of Sport, Health and Exercise Sciences, Bangor University
The authors do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.
Bangor University provides funding as a member of The Conversation UK.
Republican presidential hopeful Donald Trump has been called a 'textbook narcissist'. Chris Keane/Reuters
From the sports field to the battlefield, from business to politics, ineffective leaders often shoulder the blame when things go wrong. Perhaps we should be more careful about who we put in charge. Our research has found that your personality – and how narcissistic you are – is linked to how effective you are as a leader. We found that narcissists may appear to be good leaders early on, but they soon fall out of favour.
As we choose the leaders around us, we often think we are making informed choices about who is most effective. But our research suggests that this is not always the case. In fact, we are more likely to select as leaders those people who display narcissistic traits.
Those who score highly in narcissism tests believe they are special people who are superior. They also report high levels of confidence, are focused on themselves at the expense of others, and are vain. These overly positive views of themselves help narcissists to perform very well in situations that offer them an opportunity for personal glory, such as performing under pressure, performing tasks that are difficult, and doing things in the presence of others.
But when they perceive that there is no such opportunity, narcissists withdraw their effort and perform poorly. Because narcissists are so focused on personal glory they can be difficult team members; yet they might make good leaders. Positions of leadership provide an opportunity to gain glory from others and so are likely to be attractive to the narcissist.
The leader ship is sinking
Others have researched and written about the idea of narcissists as leaders, but until now there has been no evidence of whether or not narcissists actually do make effective leaders in the long term.
Not as good as he looks. Looking in mirror via Minerva Studio/www.shutterstock.com
In two studies, we assessed people's narcissism using the Narcissistic Personality Inventory – a standard narcissism questionnaire used in psychology research. Example items include: "If I ruled the world it would be a much better place" and "I am an extraordinary person". People were asked to score themselves against these items on a scale of 0 to 40, with higher scores indicating higher levels of narcissism. Our mean scores were just under 14 for both the studies which is consistent with most research using similar participants.
We then asked people to work in small groups, completing weekly tasks for 12 weeks. Examples of tasks included naming all the Team GB medallists at the 2012 London Olympics and identifying the states of the USA on a blank map. In the first study (using 112 first-year students, 71 men and 41 women, working in 24 groups in their first semester at university) we deliberately allocated people to groups so that they would be unlikely to know each other. In the second study, we used individuals who knew each other reasonably well (152 final year students, 96 men and 56 women, working as part of 29 groups) and let them choose their own groups.
Both during and at the end of the 12 weeks, the participants rated each other on their leadership effectiveness. The results were striking. Initially, the people who had scored highest on the narcissism test were rated as highly effective, but as time went on these positive perceptions waned until eventually narcissists were seen as very ineffective leaders. Although we expected narcissists not to last long as leaders, we were amazed by how rapidly they lost favour with their group, and how negatively they were viewed by the end. Over time, the narcissistic leaders' ships sank.
Are narcissistic leaders doomed to fail?
Our results showed that the group was initially attracted to the narcissist's charisma and vision, which allowed the narcissists to rise as the "natural" leaders. But over a very short time, narcissistic leaders failed to provide their followers with appropriate levels of challenge or support. This ultimately led to their downfall.
Although our data painted a rather negative picture for narcissists in the long run, it is not all doom and gloom for the narcissistic leader. The challenge for them is to be able to harness their charisma and combine it with other factors such as humility or empathy, which should enable them to be seen as effective leaders over time. An extreme narcissist may not care what others think of them and may be doomed to fail in leadership roles. But there are other narcissism traits that may be more effective and even necessary, in some forms of diplomacy for example – such as narcissistic charm.
Being able to choose between leaders who we "like" in the short term and those who we believe will get the job done and be effective over time is not necessarily an easy task. Dealing with this paradox is vital to be able to ensure effective leadership in the long term.
Research has found some children become narcissists because their parents teach them they are more special than other kids. Shutterstock
Why some children think they're more special than everyone else
Tag me, tag me. OK I'll do it myself. Ambuj Saxena
Narcissism on social media tells us a lot about ourselves
Mentally ill, or just a bit in love with himself?
Stop press: research discovers 'narcissists' are only human
'I'm just normal?!' Sweetshop by Shutterstock
Just because you think your children are extraordinary, doesn't mean they are


text:
PMC Policy Papers
HomeClinical InformationPerformance MeasuresPrevention and Wellness
Group/Practice
Individual Clinician
Tobacco Use and Help with Quitting Among Adolescents
The percentage of adolescents 12 to 20 years of age with a primary care visit during the measurement year for whom tobacco use status was documented and received help with quitting if identified as a tobacco user.
Date Reviewed: November 19, 2017
Integrated Delivery System
Percentage of women 21-64 years of age who were screened for cervical cancer using either of the following criteria:
Women age 21-64 who had cervical cytology performed every 3 years.
Women age 30-64 who had cervical cytology/human papillomavirus (HPV) co-testing performed every 5 years.
Do Not Support: Uncertain Validity
Alcohol Screening and Follow-up for People with Serious Mental Illness
The percentage of patients 18 years and older with a serious mental illness, who were screened for unhealthy alcohol use and received brief counseling or other follow-up care if identified as an unhealthy alcohol user.
Note: The proposed health plan measure is adapted from an existing provider-level measure for the general population (NQF #2152: Preventive Care & Screening: Unhealthy Alcohol Use: Screening & Brief Counseling). It was originally endorsed in 2014 and is currently stewarded by the American Medical Association (AMA-PCPI).
Date Reviewed: July 21, 2018
Non-Recommended Cervical Cancer Screening in Adolescent Females
The percentage of adolescent females 16-20 years of age who were screened unnecessarily for cervical cancer.
Medical Assistance with Smoking and Tobacco Cessation
he three components of this measure assess different facets of providing medical assistance with smoking and tobacco use cessation:
Advising Smokers and Tobacco Users to Quit: A rolling average represents the percentage of patients 18 years of age and older who are current smokers or tobacco users and who received advice to quit during the measurement year.
Discussing Cessation Medications: A rolling average represents the percentage of patients 18 years of age and older who are current smokers or tobacco users and who discussed or were recommended cessation medications during the measurement year.
Discussing Cessation Strategies: A rolling average represents the percentage of patients 18 years of age and older who are current smokers or tobacco users and who discussed or were provided cessation methods or strategies during the measurement year.
The percentage of patients 50-75 years of age who had appropriate screening for colorectal cancer.
Initiation and Engagement of Alcohol and Other Drug Dependence Treatment
The percentage of adolescent and adult patients with a new episode of alcohol or other drug (AOD) dependence who received the following.
- Initiation of AOD Treatment. The percentage of patients who initiate treatment through an inpatient AOD admission, outpatient visit, intensive outpatient encounter or partial hospitalization within 14 days of the diagnosis.
- Engagement of AOD Treatment. The percentage of patients who initiated treatment and who had two or more additional services with a diagnosis of AOD within 30 days of the initiation visit.
Age Appropriate Screening Colonoscopy
The percentage of patients greater than 85 years of age who received a screening colonoscopy from January 1 to December 31.
Date Reviewed: November 4, 2018
SUB-3a Alcohol & Other Drug Use Disorder Treatment Provided or Offered at Discharge and SUB-3a Alcohol & Other Drug Use Disorder
The measure is reported as an overall rate which includes all hospitalized patients 18 years of age and older to whom alcohol or drug use disorder treatment was provided, or offered and refused, at the time of hospital discharge, and a second rate, a subset of the first, which includes only those patients who received alcohol or drug use disorder treatment at discharge. The Provided or Offered rate (SUB-3) describes patients who are identified with alcohol or drug use disorder who receive or refuse at discharge a prescription for FDA-approved medications for alcohol or drug use disorder, OR who receive or refuse a referral for addictions treatment. The Alcohol and Other Drug Disorder Treatment at Discharge (SUB-3a) rate describes only those who receive a prescription for FDA-approved medications for alcohol or drug use disorder OR a referral for addictions treatment. Those who refused are not included.
These measures are intended to be used as part of a set of 4 linked measures addressing Substance Use (SUB-1 Alcohol Use Screening ; SUB-2 Alcohol Use Brief Intervention Provided or Offered; SUB-3 Alcohol and Other Drug Use Disorder Treatment Provided or Offered at Discharge; SUB-4 Alcohol and Drug Use: Assessing Status after Discharge [temporarily suspended]).
Appropriate Follow-Up Interval for Normal Colonoscopy in Average Risk Patients
Percentage of patients aged 50 years to 75 years receiving a screening colonoscopy without biopsy or polypectomy who had a recommended follow-up interval of at least 10 years for repeat colonoscopy documented in their colonoscopy report.
As part of an ongoing effort to improve user experience, we have made some updates to the Performance Measures section of ACP. Tell us what you think!


text:
20,000 soldiers set to deploy for DEFENDER-Europe 20
by Elizabeth Aslakson
A 3rd Infantry Division Soldier performs a convoy brief prior to the movement of military vehicles on Feb. 5, 2020 in Savannah, GA. This movement of equipment is in preparation for the DEFENDER-Europe 20 exercise which takes place in multiple European countries. DEFENDER-Europe 20 will build strategic-level readiness and demonstrate the U.S. military's ability to rapidly deploy a large combat-credible force in support of the U.S. National Defense Strategy. Photo by Pfc. Carlos Cuebas Fantauzzi.
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The largest U.S.-based Army exercise of forces to Europe since the Cold War is currently underway.
DEFENDER-Europe 20, a joint, multi-national, U.S. Army Europe-commanded exercise, takes place this spring, with the deployment of 20,000 soldiers from across all components.
Brig. Gen. Sean Bernabe, Deputy Chief of Staff for Operations for U.S. Army Europe, explained the intent of the largest exercise in 25 years.
"Headquarters Department of the Army directed U.S. Army Europe led exercise designed to build strategic readiness for U.S. Army," Bernabe said. "Twenty-thousand U.S. Army Soldiers, with their assigned equipment, will deploy from the U.S. to Europe and go to training areas around Europe to participate in other smaller exercises."
Other smaller exercises include:
Exercise Swift Response, an annual multinational parachute operation across Europe;
Exercise Saber Strike, a biannual exercise featuring ground maneuver training based in Poland and the Baltic States;
Exercise Allied Spirit, which is focused on interoperability at brigade level and below between allies and partners;
Exercise Dynamic Front, an annual exercise designed to improve interoperability between allied and partner artillery units.
Bernabe added, "Once these smaller exercises are complete, Defender–Europe 20 will redeploy the 20,000 U.S. forces back to the U.S. for their next mission."
Soldiers with 3rd Infantry Division begin convoy operations to Savannah, Georgia, in February as part of preparation for the DEFENDER-Europe 20 exercise which takes place in multiple European countries. Photo by Pfc. Carlos Cuebas Fantauzzi.
Maj. Traun Moore, USAREUR spokesperson, further detailed the exercise's intent.
"The purpose of DEFENDER-Europe 20 is to demonstrate the U.S. military's ability to rapidly deploy a large combat-credible force from the U.S. to Europe and, alongside our allies and partners, quickly respond to a potential crisis. Key objectives include units drawing Army prepositioned stock; receiving, staging, moving and integrating units; commanding and controlling a division-size force, rehearsing response options; demonstrating U.S. ability to leverage alliances and partnerships; and testing new equipment," Moore said.
The exercise will take place from April to May 2020, with personnel and equipment movements occurring from now through July. About 37,000 U.S., allied, and partner nation troops will participate. Of the 20,000 troops deploying from the continental U.S., 7,000 will be Army National Guard soldiers from 12 states and 750 U.S. Army Reserve soldiers. Additionally, 9,000 U.S. troops based in Europe, from the Army, Navy, Air Force and Marines will participate.
The main objectives of DEFENDER-Europe 20 are the military's refocus on strategic readiness and continued improvement of interoperability.
According to Moore, the military has concentrated on tactical readiness over the last several years. However, emphasis has shifted to building strategic readiness.
"Strategic readiness includes the ability of the U.S. military to dynamically project force and set the theater by mobilizing and deploying forces, sustaining them in a crisis, and redeploying them when their mission is complete," Moore explained, also clarifying that this exercise is not in response to a specific threat or adversary.
Moore continued, "Interoperability is a state of readiness that we collectively strive to improve … Although this will be the largest movement of U.S. forces to Europe in over 25 years, we are working closely with our allies and partners to ensure a common understanding and enhance our combined readiness and interoperability capabilities. Living and training alongside our allies and partners in Europe helps the U.S. maintain the relationships and trust that are essential for ensuring global security. Interoperability is not only about technology; it is about people and processes — about all participating nations' militaries developing basic battlefield-critical capabilities like communicating with and understanding allies and partners."
U.S. Army Europe hosted a Rehearsal of Concept drill in February, as a final conditions check for the execution of exercise DEFENDER-Europe 20. Photo by Robert Sekula.
As with any major military operation, units involved — including NATO and host nations — will be organizing and practicing ahead of time. Also, planning conferences will take place prior and throughout each stage of the exercise.
"All participants are in constant communication, to ensure that there is a shared understanding of the key objectives and the steps that are being taken to successfully execute pre-deployment, exercise execution and redeployment operations," Moore more.
For example, the U.S. Army Europe team is already working with host nation teams "to plan and execute pre-exercise senior leader engagements in participating countries in an effort to further share information about DEFENDER-Europe 20, alongside their host nation counterparts."
Impact on families
In terms of how this exercise may impact military families, Moore says families should be prepared for their service members to engage in additional training prior to the exercise, as well as resetting when redeploying from Europe. Though most activity will occur in April and May, pre-deployment training timelines are dependent on the individual unit's objective. Furthermore, troops and families stationed in Europe and not involved in the exercise, should see minimum impact on their daily lives and experience no interruption of services.
Follow updates from U.S. Army Europe at https://www.eur.army.mil/DefenderEurope.
Tags: Army National GuardDEFENDER-Europe 20InteroperabilityUS ArmyUS Army EuropeUS Army Reserve
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Elizabeth Aslakson
Elizabeth Aslakson is a freelance writer whose work revolves around military issues as well as her own Army wife adventures and love of history. She is married to a cyber soldier and they have three sons. Her idea of fun is hiking with their German Shepherd, Diana Gabaldon novels, and Acorn TV. She holds a BS in Psychology through the University of Maryland and MFA in English and Creative Nonfiction Writing through Southern New Hampshire University.
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text:
using System;
using Lunt.Descriptors;
using Lunt.Diagnostics;
using Lunt.IO;
using Lunt.Runtime;

namespace Lunt
{
    internal sealed class AssetBuilder : IDisposable
    {
        private readonly IFileSystem _fileSystem;
        private readonly DescriptorRegistry _registry;
        private readonly IBuildLog _log;
        private readonly IHashComputer _hasher;
        private bool _disposed;

        public IHashComputer Hasher
        {
            get { return _hasher; }
        }

        public AssetBuilder(IFileSystem fileSystem, IPipelineScanner scanner, IHashComputer hasher, IBuildLog log)
        {
            _fileSystem = fileSystem;
            _registry = new DescriptorRegistry(scanner.Scan());
            _hasher = hasher;
            _log = log;
        }

        public void Dispose()
        {
            if (!_disposed)
            {
                _hasher.Dispose();
                _disposed = true;
            }
        }

        public AssetBuildResult Build(Asset asset, BuildConfiguration configuration)
        {
            object importedData;

            var result = BuildAsset(asset, configuration, out importedData);
            if (importedData != null)
            {
                var disposable = importedData as IDisposable;
                if (disposable != null)
                {
                    disposable.Dispose();
                }
            }

            return result;
        }

        private AssetBuildResult BuildAsset(Asset asset, BuildConfiguration configuration, out object obj)
        {
            obj = null;

            // Create the manifest for the asset.
            var context = new Context(_fileSystem, configuration, _hasher, _log, asset);
            var manifest = new BuildManifestItem(asset);

            // Makes ure the asset has a file extension.
            var extension = asset.Path.GetExtension();
            if (extension == null)
            {
                var message = new Message("The asset {0} has no file extension.", asset.Path);
                return AssetBuildResult.Failure(manifest, message);
            }

            // Get the source file and make sure it exist.
            var sourcePath = asset.GetSourceFilePath(configuration);
            var sourceFile = _fileSystem.GetFile(sourcePath);
            if (!sourceFile.Exists)
            {
                var message = new Message("Could not find the file '{0}'.", sourcePath.FullPath);
                return AssetBuildResult.Failure(manifest, message);
            }

            AssetBuildResult result;
            if (!BuildAsset(asset, configuration, manifest, context, sourceFile, ref obj, out result))
            {
                return result;
            }

            _log.Information("Built {0}", asset.Path);

            manifest.Checksum = _hasher.Compute(sourceFile);
            manifest.Length = sourceFile.Length;
            manifest.Dependencies = context.GetDependencies();

            return AssetBuildResult.Success(manifest);
        }

        private bool BuildAsset(Asset asset, BuildConfiguration configuration, BuildManifestItem manifest,
            Context context, IFile sourceFile, ref object obj, out AssetBuildResult result)
        {
            if (!ImportAsset(asset, manifest, context, sourceFile, ref obj, out result))
            {
                return false;
            }
            if (!ProcessAsset(asset, manifest, context, ref obj, out result))
            {
                return false;
            }
            if (!WriteAsset(asset, configuration, manifest, context, obj, out result))
            {
                return false;
            }
            return true;
        }

        private bool ImportAsset(Asset asset, BuildManifestItem manifest, Context context,
            IFile sourceFile, ref object obj, out AssetBuildResult error)
        {
            // Get the importer from the registry.
            var importerDescription = _registry.GetImporter(asset);
            if (importerDescription == null)
            {
                var message = new Message("Could not find an importer for {0}.", asset.Path);
                error = AssetBuildResult.Failure(manifest, message);
                return false;
            }

            _log.Verbose("{0}: Importing {1}", importerDescription.Name, asset.Path);
            obj = importerDescription.Importer.Import(context, sourceFile);
            if (obj == null)
            {
                var message = new Message("Import of {0} resulted in null.", asset.Path);
                error = AssetBuildResult.Failure(manifest, message);
                return false;
            }

            error = null;
            return true;
        }

        private bool ProcessAsset(Asset asset, BuildManifestItem manifest,
            Context context, ref object obj, out AssetBuildResult error)
        {
            // Get the processor (if any) from the registry.
            var processorDescription = _registry.GetProcessor(asset);
            if (processorDescription != null)
            {
                // Make sure the obj type and the processor source type match.
                // We do this since it's possible to directly implement
                // the ILuntImporter and return an object of a type that's
                // inconsistent with the type that the importer claims it handle.
                var importedType = obj.GetType();
                var processorSourceType = processorDescription.SourceType;
                if (importedType != processorSourceType)
                {
                    const string format = "Cannot process {0} since the data is of the wrong type ({1}). Processor expected {2}.";
                    var message = new Message(format, asset.Path, importedType.FullName, processorSourceType.FullName);
                    error = AssetBuildResult.Failure(manifest, message);
                    return false;
                }

                _log.Verbose("{0}: Processing {1}", processorDescription.Name, asset.Path);
                obj = processorDescription.Processor.Process(context, obj);
                if (obj == null)
                {
                    var message = new Message("Processing of {0} resulted in null.", asset.Path);
                    error = AssetBuildResult.Failure(manifest, message);
                    return false;
                }

                if (obj.GetType() != processorDescription.TargetType)
                {
                    const string format = "Value returned from processor for {0} does not match expected target type ({1}).";
                    var message = new Message(format, asset.Path, processorDescription.TargetType.FullName);
                    error = AssetBuildResult.Failure(manifest, message);
                    return false;
                }
            }

            if (!string.IsNullOrWhiteSpace(asset.ProcessorName) && processorDescription == null)
            {
                const string format = "Cannot process {0} since the processor '{1}' wasn't found.";
                var message = new Message(format, asset.Path, asset.ProcessorName);
                error = AssetBuildResult.Failure(manifest, message);
                return false;
            }

            error = null;
            return true;
        }

        private bool WriteAsset(Asset asset, BuildConfiguration configuration, BuildManifestItem manifest,
            Context context, object obj, out AssetBuildResult error)
        {
            // Get the writer from the registry.
            var writerDescription = _registry.GetWriter(obj.GetType());
            if (writerDescription == null)
            {
                var message = new Message("Could not find a writer for {0}.", asset.Path);
                error = AssetBuildResult.Failure(manifest, message);
                return false;
            }

            // Get the target file.
            var targetPath = asset.GetTargetFilePath(configuration);
            var targetFile = _fileSystem.GetFile(targetPath);

            // Does the target file directory path exist?
            var targetDirectory = _fileSystem.GetDirectory(targetPath.GetDirectory());
            if (!targetDirectory.Exists)
            {
                if (!targetDirectory.Create())
                {
                    const string format = "Could not create target directory '{0}'.";
                    var message = new Message(format, targetDirectory.Path);
                    error = AssetBuildResult.Failure(manifest, message);
                    return false;
                }
            }

            _log.Verbose("{0}: Writing {1}", writerDescription.Name, asset.Path);
            writerDescription.Writer.Write(context, targetFile, obj);

            error = null;
            return true;
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Q: Loop through multiple txt files and count frequencies of chosen word in Python I have a exercise problem where I am asked to write a function that loops through 50 text files and counts the frequency of a chosen word in each text file. My code at the moment looks like this:
def count(term):
    frequencies = 0

    work_dir = "C:/my_work_directory"
    for i in range(1, 51):
        name = "chapter-{i}.txt".format(i=i)
        path = os.path.join(work_dir, name)
        with io.open(path, "r") as fd:
            content = fd.read()

        chapter = io.StringIO(content)
        line = chapter.readline()
        print(chapter)
        while line:
            lower = line.lower()
            cleaned = re.sub('[^a-z ]','', lower)
            words = cleaned.strip().split(' ')
            for word in words:
                if word == term:
                    frequencies += 1
            line = chapter.readline()

        print(frequencies)

The output I want is that, if I enter count("Man"), I get 50 different frequencies of how often the word "Man" came up in each text file. However, all I am getting at the moment is 50 zeros. I am fairly sure this is because I have initialised the variable 'frequencies' at 0 and then haven't done anything to it. Can anyone help me fix this issue or tell me where I am going wrong? Any help would be greatly appreciated, thank you.

A: Well, your 'Man' has a capital letter, and all your words are lower case. So the first thing would be to call the lower() function on the term variable. The second thing that is wrong, and you would only notice it later, is that you're keeping a running count, instead of a per-file count. So move the initialization of the frequency variable into the for loop. So it should look something like this.
def count(term):
    term = term.lower()

    work_dir = "C:/my_work_directory"
    for i in range(1, 51):
        frequencies = 0

        name = "chapter-{i}.txt".format(i=i)
        path = os.path.join(work_dir, name)
        with io.open(path, "r") as fd:
            content = fd.read()

        chapter = io.StringIO(content)
        line = chapter.readline()
        print(chapter)
        while line:
            lower = line.lower()
            cleaned = re.sub('[^a-z ]','', lower)
            words = cleaned.strip().split(' ')
            for word in words:
                if word == term:
                    frequencies += 1
            line = chapter.readline()

        print(frequencies)

A: I runned it and it works fine after i change work_dir="" (So it looked at local). So i think you should check work directory path or assert if term parameter is correct

redpajama_set_name: RedPajamaStackExchange