text:
\section{Introduction}

Up to now, lattice QCD remained the only practical method to investigate
QCD at finite temperature and baryon density \cite{1,2,3,4}. Recently it underwent a rapid
progress (\cite{4,5,6,7,8,9} and references therein). However, lattice QCD,
being a very specific regularization scheme, is primary aimed
at obtaining well-defined corresponding expressions in order
to get realistic numbers for physical quantities. One may therefore get numbers and curves
without understanding what the physics is behind them. Such an understanding can
only come from the dynamical theory, which is continuous QCD. For
example, any description of the quark-gluon plasma (QGP) has to be formulated within the
framework of a dynamical theory. The need for an analytical equation of state (EoS)
remains, but, of course it should be essentially non-perturbative
(NP), approaching the so-called Stefan-Boltzmann (SB) limit at
very hight temperatures. Thus the approaches of analytic NP QCD and lattice QCD
to finite-temperature QCD do not exclude each other; on the contrary, they should be complementary.
This is especially true at low temperatures where the thermal QCD lattice calculations suffer from
big uncertainties \cite{3,4,5,6,7,8,9}. On the other hand, any analytic NP approach has to correctly
reproduce thermal QCD lattice results at high temperatures (see papers cited above). There already
exist interesting phenomenological models based on the quasi-particle picture \cite{10,11,12,13,14,15,16,17,18,19}
(and references therein) to analyze results of $SU(3)$ lattice QCD calculations for the QGP EoS.

The main purpose of this paper is to complete the derivation of the NP analytical EoS for the gluon/glue plasma (GP), i.e., a system consisting purely of $SU(3)$ Yang-Mills (YM) fields without quark degrees of freedom. We have started it in our previous publication \cite{20} (part I). This gluon medium was called as gluon matter (GM) in part I.
However, here we prefer to call it GP, since we are going to include the free gluon contribution as well as the SB term in the present investigation, where the further analytical and numerical evaluation of the perturbative (PT) part of the gluon pressure will be also performed. The general formalism we use to generalize it to non-zero temperatures is the effective potential approach for composite operators \cite{21}. In the absence of external sources
it is nothing but the vacuum energy density (VED). This approach is NP from the very beginning, since
it deals with the expansion of the corresponding skeleton vacuum loop diagrams, and thus allows one
to calculate the VED from first principles \cite{20,21,22}.
The key element in this programme was the extension of our paper \cite{22} to non-zero temperatures. This makes it possible to introduce the temperature-dependent bag constant (pressure) as a function of the mass gap. It is this which is responsible for the large-scale structure of the QCD ground state. The confining dynamics in the GP will therefore be nontrivially taken into account directly through the mass gap and via the temperature-dependent bag constant itself, but other NP effects will also be present.

The paper is organized as follows. In sections 2 and 3 we summarize our results obtained in \cite{20} for the gluon pressure at zero (i.e., the above-mentioned VED) and non-zero temperatures, respectively. In section 3 the PT part of the gluon pressure is analytically derived as well. In section 4 we formulate a procedure how to include the free gluon contribution in the self-consistent way into the full GP pressure. A method of the simulating functions is proposed and analytical formulae for numerical simulations are introduced. This makes it possible to perform analytical and numerical simulations in order to determine the full GP pressure. In section 5 we display our numerical results for all the thermodynamic quantities, such as the entropy and energy densities, the heat capacity, etc., calculated with the help of the obtained GP pressure. We present also a brief discussion. In section 6 a description of the dynamical structure of the GP is given. In section 7 we summarize our conclusions. In appendix A the general expressions are given for the main thermodynamic quantities as functions of the pressure. In appendix B some analytical formulae for the GP thermodynamic quantities are derived, while in appendix C the latent heat is analytically and numerically evaluated. In appendix D the corresponding $\beta$-function for the confining effective charge is explicitly derived and briefly discussed.

\section{The gluon pressure at zero temperature}

In order to derive the gluon pressure at zero temperature $P_g$ in the first part of our investigation,
we have used the effective potential approach for composite operators \cite{21} to leading order,
the so-called log-loop level. Analytically the gluon pressure looks like \cite{20}

\begin{equation}
P_g = P_{NP} + P_{PT}= B_{YM} + P_{YM} + P_{PT},
\end{equation}
so that $P_{NP} = B_{YM} + P_{YM}$, while

\begin{equation}
B_{YM} = 16 \int^{q^2_{eff}} {\dd^4q \over (2\pi)^4} \left[ \ln [1
+ 3 d^{INP}(q^2)] - {3 \over 4} d^{INP}(q^2) \right],
\end{equation}

\begin{equation}
P_{YM} = - 16 \int {\dd^4q \over (2\pi)^4} \left[ \ln [1 + {3
\over 4} \alpha_s^{INP}(q^2)] - {3 \over 4} \alpha_s^{INP}(q^2)
\right],
\end{equation}
and

\begin{equation}
P_{PT}= - 16  \int_{\Lambda^2_{YM}} {\dd^4q \over (2\pi)^4} \left[ \ln \left(1 + {3
\alpha^{AF}(q^2) \over 4 + 3 \alpha_s^{INP}(q^2)} \right) - {3 \over 4}
\alpha^{AF}(q^2) \right].
\end{equation}
In the expression for the bag constant (2) symbolically shown $q^2_{eff}$ is the effective scale
squared, separating the soft momenta from the hard ones in the
integration over $q^2$, that's $0 \leq q^2 \leq q^2_{eff}$.

The intrinsically NP (INP) effective charge is

\begin{equation}
\alpha_s^{INP}(q^2) = {\Delta^2 \over q^2},
\end{equation}
where $\Delta^2 \equiv \Delta^2_{JW}$ is the Jaffe-Witten (JW) mass gap \cite{23}, mentioned above, which is responsible for the large-scale structure of the QCD vacuum, and thus for its INP dynamics. Let us note that how the regularized mass gap appears in QCD is shown in \cite{24}. It is worth emphasizing that in our recent works \cite{25,26} it has been proven that the INP effective charge (5) exactly reproduces the non-linear iteration solution for the full gluon propagator after the renormalization of the regularized mass gap is performed, {\it so it is not an ansatz}. The expression for the bag constant given in Eq.~(2) is free of all types of the PT contributions ("contaminations"). Details of its derivation can be found in \cite{22}. In the YM part of the NP gluon pressure
$P_{YM}$, given in Eq.~(3), the integration over variable $q^2$ goes from zero to infinity.

The PT part $P_{PT}$ of the gluon pressure shown in Eq.~(4) contains the contribution which is mainly determined by the asymptotic freedom (AF) part of the PT effective charge, though the dependence on the INP effective charge is also present. The AF running effective charge $\alpha^{AF}(q^2)$ is given
by the renormalization group equation solution, the so-called sum of the main PT logarithms \cite{26,27,28}, namely

\begin{equation}
\alpha^{AF}(q^2) ={ \alpha_s \over 1 + \alpha_s b \ln ( q^2 /
\Lambda^2_{YM})},
\end{equation}
and thus like the confining effective charge (5) {\it this is not an ansatz} either.
Here $\Lambda^2_{YM} = 0.09 \ \GeV^2$ \cite{29} is the asymptotic scale parameter for $SU(3)$ YM fields,
and $b=(11 / 4 \pi)$ for these fields, while the strong fine
structure constant is $\alpha_s = 0.1187$ \cite{30}. Let us also note that in Eq.~(6)
$q^2$ can not go below $\Lambda^2_{YM}$, which has been already symbolically shown in Eq.~(4).
In the $q^2 \rightarrow \infty$ limit from Eq.~(6) one recovers the well known AF expression
$\alpha_s^{AF}(q^2) = 1 / b \ln ( q^2 / \Lambda^2_{YM})$ \cite{27,28}.
The gluon pressure $P_g$ in Eq.~(1) is normalized to
the free PT vacuum to be zero, i.e., when the interaction is switched formally
off by putting $\alpha^{AF}(q^2)= \alpha_s^{INP}(q^2)=0$, then $P_{NP}=P_{PT}=0$, so that
$P_g=0$ as well. It comes from the initial normalization condition
of the free PT vacuum to be zero in the effective potential approach up to leading order \cite{20,21,22}.

\section{The gluon pressure at non-zero temperatures}

In the imaginary-time formalism \cite{31,32}, all the
four-dimensional integrals can be easily generalized to non-zero
temperatures $T$ according to the prescription (let us remind that in part I \cite{20} and in
the present investigation the signature is Euclidean from the very beginning)

\begin{equation}
\int {\dd q_0 \over (2\pi)} \rightarrow T \sum_{n=- \infty}^{+
\infty}, \quad \ q^2 = {\bf q}^2 + q^2_0 = {\bf q}^2 +
\omega^2_n = \omega^2 + \omega^2_n, \ \omega_n = 2n \pi T.
\end{equation}
In other words, each integral over $q_0$ of a loop momentum is to be
replaced by the sum over the Matsubara frequencies labeled by
$n$, which obviously assumes the replacement $q_0 \rightarrow
\omega_n= 2n \pi T$ for bosons (gluons). In frequency-momentum
space the effective charges (5) and (6) become

\begin{equation}
\alpha_s^{INP}(q^2) = \alpha^{INP}_s({\bf q}^2, \omega_n^2) =
\alpha^{INP}_s( \omega^2, \omega_n^2) = { \Delta^2 \over \omega^2
+ \omega_n^2},
\end{equation}
and

\begin{equation}
\alpha^{AF}(q^2) = \alpha^{AF}({\bf q}^2, \omega_n^2) =
\alpha^{AF} (\omega^2, \omega_n^2) = { \alpha_s \over 1 + \alpha_s b \ln ( \omega^2
+ \omega_n^2 / \Lambda^2_{YM})},
\end{equation}
respectively. It is also convenient to introduce the following notations:

\begin{equation}
T^{-1} = \beta, \quad \omega = \sqrt{{\bf q}^2},
\end{equation}
where, evidently, in all the expressions here and below ${\bf q}^2$ is the square of
the three-dimensional loop momentum, in complete agreement with the relations (7).

Introducing the temperature dependence into the both sides of
the relation (1), we finally obtain

\begin{equation}
P_g(T) = P_{NP}(T) + P_{PT}(T) = B_{YM}(T) + P_{YM}(T) + P_{PT}(T),
\end{equation}
where $P_{NP}(T)$ and $P_{PT}(T)$ are the NP and PT parts of the gluon pressure $P_g(T)$, respectively.
In what follows we will call them either the NP and PT gluon pressures or, for simplicity, the NP and PT
pressures.

\subsection{NP contribution}

One of the attractive features of the confining
effective charge (8) is that it allows an exact summation over the Matsubara
frequencies in the NP pressure $P_{NP}(T)$. Collecting all our analytical results obtained in our previous work \cite{20}, we can write

\begin{eqnarray}
P_{NP}(T) &=& B_{YM}(T) + P_{YM}(T) \nonumber\\
&=& {6 \over \pi^2} \Delta^2 P_1 (T) + {16 \over \pi^2} T
[P_2(T) + P_3(T) - P_4(T)],
\end{eqnarray}
and

\begin{equation}
P_1(T) = \int_{\omega_{eff}}^{\infty} \dd \omega {\omega \over
e^{\beta\omega} -1} = {\pi^2 \over 6} T^2 - \int^{\omega_{eff}}_0 \dd \omega {\omega \over
e^{\beta\omega} -1},
\end{equation}
while

\begin{eqnarray}
P_2(T) &=& \int_{\omega_{eff}}^{\infty} \dd \omega \ \omega^2
\ln \left( 1- e^{-\beta\omega} \right), \nonumber\\
P_3(T)&=& \int_0^{\omega_{eff}} \dd \omega \ \omega^2 \ln
\left( 1 - e^{- \beta\omega'} \right), \nonumber\\
P_4(T) &=& \int_0^{\infty} \dd \omega \ \omega^2 \ln \left( 1 -
e^{- \beta \bar \omega} \right),
\end{eqnarray}
where $\omega_{eff}=1 \ \GeV$ and the mass gap $\Delta^2= 0.4564 \ \GeV^2$ are
fixed \cite{20,22}. Then $\omega'$ and $\bar \omega$ are given by the relations

\begin{equation}
\omega' = \sqrt{{\bf q}^2 + 3 \Delta^2} = \sqrt{\omega^2 +
m'^2_{eff}}, \quad m'_{eff}= \sqrt{3} \Delta = 1.17 \ \GeV,
\end{equation}
and

\begin{equation}
\bar \omega = \sqrt{{\bf q}^2 + {3 \over 4} \Delta^2} =
\sqrt{\omega^2 + \bar m^2_{eff}}, \quad \bar m_{eff}= {\sqrt{3}
\over 2} \Delta = 0.585 \ \GeV,
\end{equation}
respectively. The NP part of the gluon pressure $P_{NP}(T)$ is shown in Fig. 1.

\subsection{ PT contribution}

In frequency-momentum space the PT part (4) of the gluon pressure (11) is

\begin{equation}
P_{PT}(T) = - { 8 \over \pi^2}  \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \ T
\sum_{n= - \infty}^{+ \infty} \left[ \ln \left( 1 + {3
\alpha^{AF}(\omega^2, \omega^2_n) \over 4 + 3
\alpha_s^{TNP}(\omega^2, \omega^2_n)} \right) - {3 \over 4}
\alpha^{AF}(\omega^2, \omega^2_n) \right].
\end{equation}
However, it is instructive to begin the evaluation of the PT contribution to the gluon pressure
(11) not from this expression, but from the very beginning, i.e.,  from Eq.~(4). There is an interesting observation
concerning the PT part (4). Let us consider the argument of its logarithm $\ln [ 1 + x(q^2)]$, where

\begin{equation}
x(q^2) =  {3 \alpha^{AF} (q^2) \over 4 + 3 \alpha_s^{INP}(q^2)} = {3 \alpha_s q^2 \over
(4q^2 + 3 \Delta^2) [ 1 +  \alpha_s b \ln ( q^2 / \Lambda^2_{YM})]},
\end{equation}
with the help of the expressions (5) and (6). In the integral (4) $q^2 \geq \Lambda^2_{YM}$, thus
at its lower limit $q^2 = \Lambda^2_{YM}$ the argument (18) numerically becomes

\begin{equation}
x(\Lambda^2_{YM}) = {3 \alpha_s \Lambda^2_{YM} \over (4 \Lambda^2_{YM} + 3 \Delta^2)} = 0.0185,
\end{equation}
and the numerical values of $\Delta^2$, $\alpha_s$ and $\Lambda^2_{YM}$
are as given above. The argument of the logarithm is really small and it will become smaller and smaller with $q^2$ going to infinity. This means that the logarithm $\ln [ 1 + x(q^2)]$ in the integral (4) is legitimated to expand in powers of $\alpha^{AF}(q^2)$ or, equivalently, $\alpha_s$. To the leading order one finally obtains

\begin{equation}
P_{PT}=  \alpha_s \Delta^2 9  \int_{\Lambda^2_{YM}} {\dd^4q \over (2\pi)^4} {1
\over \left( q^2 + {3 \over 4} \Delta^2 \right) \left[ 1 +  \alpha_s b \ln ( q^2 / \Lambda^2_{YM})
\right]} + O(\alpha^2_s).
\end{equation}
To say that this PT term is "contaminated" by the NP contributions or to
say that this NP term is "contaminated" by the PT contributions will be equally correct.
However, in any case it is suppressed by the order of $\alpha_s$ in comparison with the NP part
of the gluon pressure. In frequency-momentum space this term looks like

\begin{equation}
P_{PT}(T) =  \alpha_s \Delta^2 { 9 \over 2 \pi^2}  \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \ T
\sum_{n= - \infty}^{+ \infty} \left[ { (\bar \omega^2 + \omega_n^2)^{-1} \over [1 + \alpha_s b \ln ( \omega^2
+ \omega_n^2 / \Lambda^2_{YM})]} \right],
\end{equation}
where $\bar \omega$ is given in the relation (16). For convenience, we omit the term $O(\alpha^2_s)$.
Equivalently this integral can be presented as a sum of the two integrals, namely

\begin{equation}
P_{PT}(T) = P^{(1)}_{PT}(T) + P^{(2)}_{PT}(T),
\end{equation}
where

\begin{equation}
P^{(1)}_{PT}(T) = \alpha_s \Delta^2 { 9 \over 2 \pi^2}  \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \ T
\sum_{n= - \infty}^{+ \infty}  { 1 \over (\bar \omega^2 + \omega_n^2)},
\end{equation}
and

\begin{equation}
P^{(2)}_{PT}(T) = - \alpha_s^2 \Delta^2 { 9 b \over 2 \pi^2}  \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \ T
\sum_{n= - \infty}^{+ \infty} \left[ { (\bar \omega^2 + \omega_n^2)^{-1} \ln ( \omega^2
+ \omega_n^2 / \Lambda^2_{YM}) \over [1 + \alpha_s b \ln ( \omega^2
+ \omega_n^2 / \Lambda^2_{YM})]} \right].
\end{equation}

In the integral (23) the summation over the Matsubara frequencies can be performed analytically (i.e., exactly)
with the help of formula from part I of our investigation \cite{20}. Omitting all the derivation and dropping the $\beta$-independent terms \cite{31}, one obtains

\begin{equation}
P^{(1)}_{PT}(T) = \alpha_s \Delta^2 { 9 \over 2 \pi^2} \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \
{ 1 \over \bar \omega} { 1 \over e^{\beta \bar \omega}  - 1},
\end{equation}
where $\bar \omega$ is given in the relation (17).

Thus the so-called PT contribution (22) to the gluon pressure is

\begin{equation}
P_{PT}(T) = \alpha_s \Delta^2 { 9 \over 2 \pi^2} \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \
{ 1 \over \bar \omega} { 1 \over e^{\beta \bar \omega}  - 1} + O(\alpha^2_s),
\end{equation}
and it is numerically small in the whole temperature range, indeed (see Fig. 1).

\subsection{The gluon pressure $P_g(T)$}

Summing up the expressions (12) and (26) up to leading order, for the gluon pressure

\begin{equation}
P_g(T) = P_{NP}(T) + P_{PT}(T),
\end{equation}
defined in Eq.~(11), one finally obtains

\begin{equation}
P_g(T) = {6 \over \pi^2} \Delta^2 \left[ P_1 (T) + {3 \over 4} \alpha_s P'_1(T) \right]
+ {16 \over \pi^2} T[P_2(T) + P_3(T) - P_4(T)],
\end{equation}
where the terms of the $\alpha^2_s$ order are not shown, the integral $P'_1(T)$ is

\begin{equation}
P'_1(T) =  \int_{\Lambda_{YM}}^{\infty} \dd\omega \ \omega^2 \
{ 1 \over \bar \omega} { 1 \over e^{\beta \bar \omega}  - 1},
\end{equation}
and all other integrals are given in Eqs.~(13) and (14).

Let us recall that the gluon pressure (28) is normalized to be zero when the interaction is switched formally off, i.e., putting $\alpha_s= \Delta^2=0$. Hence each of its terms in Eq.~(27) becomes also zero in this case.
This is due to the initial normalization of the free PT vacuum to zero. The confining dynamics in Eq.~(28) is implemented via the bag pressure $B_{YM}(T)$ and the mass gap itself. However, other NP effects are also present
via $P_{YM}(T)$ and $P_{PT}(T)$, though in the PT part they are suppressed in the powers of $\alpha_s$ in Eq.~(26),
and, consequently, in Eq.~(28). The gluon pressure (28) is shown in Fig. 1.

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Pic1a.eps}
\caption{The gluon pressure (28), the NP pressure (12) and the PT pressure (26) all
properly scaled as functions of $T/T_c$. Effectively, all curves have a maximum at
$T_c = 266.5 \ \MeV$ (vertical solid line). As functions of $T$ they have slightly
different maxima (see text below).}
\label{fig:1}
\end{center}
\end{figure}

\section{The full GP EoS}

From our numerical results it follows that the
properly scaled (i.e., divided by $T^4/3$) gluon pressure (28) as a function of $T$ has a maximum at $T= 266.7 \ \MeV$, while the PT pressure (26) has a maximum at $T = 269.4 \ \MeV$. The NP pressure (12) has a maximum at some finite temperature, $T=T_c = 266.5 \ \MeV$. In what follows we will call this value "characteristic", since the NP pressure has been calculated exactly, whereas the PT pressure, and hence the gluon pressure, has been calculated up to the leading order in powers of $\alpha_s$. Moreover, all the pressures properly scaled as functions of $T/T_c$ effectively have maxima just at $T_c$, see Fig. 1. From this figure it clearly follows that the gluon pressure (28) will never achieve the thermodynamic SB limit (A.7) at high temperatures. This is not surprising, since it has been normalized to the free PT pressure to be zero from the very beginning \cite{20,21,22}.
First of all, this means that the SB term is non-explicitly present in the gluon pressure $P_g(T)$ (27) through the NP pressure $P_{NP}(T)$ (12) in the whole temperature range. From the integral $P_2(T)$ in Eq.~(14) it can be explicitly extracted as follows: $P_2(T) = \int_{\omega_{eff}}^{\infty} \dd \omega \ \omega^2 \ln \left( 1- e^{-\beta\omega} \right) = \int_0^{\infty} \dd \omega \ \omega^2 \ln \left( 1- e^{-\beta\omega} \right)
- \int^{\omega_{eff}}_0 \dd \omega \ \omega^2 \ln \left( 1- e^{-\beta\omega} \right)
= - ( \pi^2 / 8T) P_{SB}(T) - P'_2(T)$, where the integral
$P'_2(T) = \int^{\omega_{eff}}_0 \dd \omega \ \omega^2 \ln \left( 1- e^{-\beta\omega} \right)$. Multiplied further by
$(16/ \pi^2)T$, one finally obtains the contribution $- 2 P_{SB}(T)$ (along with terms describing another types of free gluons) to the NP pressure and hence to the gluon pressure. From the integrals $P_3(T)$ and $P_4(T)$ in Eq.~(14) it cannot be extracted explicitly in a such simple form. However, in the $T \rightarrow \infty$ limit we can neglect the difference between massless $\omega$ and massive $\omega', \ \bar \omega$ gluon excitations, and all the composition $[P_2(T) + P_3(T) - P_4(T)]$ in Eq.~(12) goes to zero in this limit. This means that the SB pressure (A.7) has been already subtracted from it, but in a very specific way. It is not surprising that the NP effects are important below $T_c$. From Fig. 1, however,
it follows that they are still important above $T_c$, at least, up to $3.75T_c = 1 \ \GeV = \omega_{eff}$.
The two other remarkable features of the gluon pressure (28) are to be also underlined. Firstly, below $T_c$ it goes exponentially down and up with $T$ going down and up, respectively. Secondly, above $T_c$ with increasing $T$ it goes down as $1 /T^2$ and not as $ 1 / T^4$ due to the explicit presence of the mass gap term $\Delta^2 T^2$ (see Eq.~(28) and the integral (13)), while the composition $[P_2(T) + P_3(T) - P_4(T)]$ goes to zero at high temperatures
as pointed out above. In general, this means that the gluon pressure (28), and hence the NP pressure (12), is
at high temperatures a polynomial in integer powers of $T$ up to $T^2$. All these features have been discussed in more detail in \cite{20}. The term $\sim T^2$ in the phenomenological EoS, which leads to the above-mentioned fall off at high temperatures, has been first discussed in \cite{33} (see also \cite{34,35,36}).

The gluon pressure (28) may change continuously its regime only in the near neighborhood of $T_c$ in order for its full counterpart to reach the corresponding SB limit at high temperatures. However, the SB term cannot be simply added to Eq.~(28) above $T_c$, multiplied by the corresponding $\Theta$-function. The problem is that the different types of free gluons described by the NP pressure (12) are present in the GP in the whole temperature range, as it was discussed above. Thus in the right-hand-side of Eq.~(28) the free gluon contribution has to appear in a more sophisticated way. The most general form how this can be done is to add to Eq.~(28) the term $[\Theta((T / T_c) - 1)Y(T) + \Theta((T_c/T - 1))X(T)]$, valid in the whole temperature range, and the unknown yet functions $Y(T)$ and $X(T)$ are to be expressed in terms of $P_{SB}(T)$ and $P_{NP}(T)$ (see subsection below).

The previous Eq.~(28) then becomes

\begin{equation}
P_{GP}(T) = P_g(T) +  \Theta \left( {T_c \over T} - 1 \right) X(T)
+ \Theta \left( {T \over T_c} - 1 \right)Y(T),
\end{equation}
and its left-hand-side here and below is denoted as $P_{GP}(T)$ (the above-mentioned full counterpart).
The GP pressure (30) is continuous at $T_c$ if and only if

\begin{equation}
X(T_c) = Y(T_c),
\end{equation}
which can be easily checked. Due to the
condition (31), the dependence on the corresponding $\Theta$-functions disappears at $T_c$, and the GP pressure (30) remains continuous  at any point of its domain. The role of the function $X(T)$ is to change the
behavior of $P_{GP}(T)$ from $P_g(T)$ below $T_c$ especially in its near neighborhood. The function $Y(T)$ is aimed to change the behavior of $P_{GP}(T)$ from $P_g(T)$, as well as to introduce the SB term itself and its modification due to AF above and near $T_c$, respectively. These changes are inevitable indeed, since $P_{GP}(T)$ is different from $P_g(T)$ in the whole temperature range. The reason is that in the gluon pressure  $P_g(T)$ the SB term is missing.
However, as we saw in the GP pressure it cannot be restored in a trivial way. That is why the appearance
of the corresponding $\Theta$-functions in the GP pressure (30) is also inevitable, playing, nevertheless, only an auxiliary role.

\subsection{Analytical simulations}

Actual analytical and numerical simulations are the main subject of this paper in order to
derive the NP analytical EoS for the GP valid in the whole temperature range. Completing this program we will be able to compare our results and achieve the agreement with those of the thermal QCD lattice calculations above $T_c$ \cite{36,37}. The space of test functions, in terms of which the functions $X(T)$ and
$Y(T)$ should be found, has already been established. On a general ground therefore we can put

\begin{eqnarray}
X(T) &=& f_1(T)P_{SB}(T) + \phi_1(T) P_{NP}(T), \nonumber\\
Y(T) &=& f_2(T)P_{SB}(T) + \phi_2(T) P_{NP}(T),
\end{eqnarray}
where all the dimensionless functions $f_n(T), \ \phi_n(T), \ n=1,2$ will be called the simulating functions. The test functions $P_{NP}(T)$ and $P_{SB}(T)$ are independent from each other, and they are exactly known.
This also makes it possible to use in what follows the exact relations,
derived in our previous work \cite{20}, namely

\begin{equation}
\Bigl[ P_{SB}(T) - 2P_{NP}(T) \Bigr]_{T=T_c} = 0, \
\Bigl \{ {\partial \over \partial T} \Bigl[ P_{SB}(T) - 2P_{NP}(T) \Bigr] \Bigr\}_{T=T_c} = 0.
\end{equation}

In order to suppress the deep penetration of the SB term $P_{SB}(T)$ and the additional NP pressure $P_{NP}(T)$ below $T_c$, the simulating functions $f_1(T)$ and $\phi_1(T)$ are then convenient to choose as follows:

\begin{equation}
f_1(T) = A e^{- \alpha (T_c/T)}, \quad \alpha > 0, \quad \phi_1(T) = A_1 e^{- \alpha_1 (T_c/T)}, \quad
\alpha_1 \geq 0,
\end{equation}
and $A, \ A_1$ are the arbitrary constants at this stage. Evidently, these simulating functions mimic the asymptotic of the so-called gluon mean number $N_g = (e^{\beta\omega} -1)^{-1}$ \cite{31} in the $\beta \rightarrow \infty \ (T \rightarrow 0)$ limit. Replacing $\omega$ by $\alpha T_c$ and $\alpha_1 T_c$, the different numerical factors $A, \ A_1$ should appear in the corresponding simulating functions shown in the relation (34). From the GP pressure (30) below $T_c$ then it follows

\begin{eqnarray}
P_{GP}(T) &=& P_g(T) + f_1(T)P_{SB}(T) + \phi_1(T)P_{NP}(T)  \nonumber\\
&=& P_{PT}(T)  + [1 +  A_1 e^{- \alpha_1(T_c/T)}]P_{NP}(T)+ A e^{- \alpha(T_c/T)}P_{SB}(T),
\end{eqnarray}
where the relation (27) has already been used.
In the $T \rightarrow 0$ limit the additional contributions are exponentially
suppressed, and the asymptotic of the GP pressure $P_{GP}(T)$ is determined by the gluon pressure $P_g(T)$, as it should be. At the same time, these terms allow to change the value of the GP pressure from the gluon pressure near $T_c$, as it is expected.

For the simulating function $f_2(T)$ our choice is the standard, empirical one \cite{10,38}, recalculated only at $T_c=266.5 \ \MeV$, so it is

\begin{equation}
f_2(T) = 1 - \alpha_s(T) = 1 - { 0.36 \over 1 + 0.55 \ln (T/T_c)}, \quad T \geq T_c.
\end{equation}
Evidently, the second term in this expression mimics the AF formula for the effective charge (6) as a function of temperature, while the first one leads to the correct SB limit for the GP pressure (30).
The simulating function $\phi_2(T)$ has to be a regular function of $T$ as it goes to infinity; otherwise it remains arbitrary at this stage. The asymptotic of the GP pressure (30) at high temperatures thus becomes

\begin{eqnarray}
P_{GP}(T) &=& P_g(T)+ f_2(T)P_{SB}(T) + \phi_2(T) P_{NP}(T) \nonumber\\
&\sim& \left( 1 - { 0.36 \over 1 + 0.55 \ln (T/T_c)} \right) P_{SB}(T) \nonumber\\
&+& [1 + \phi_2(T)] P_{NP}(T) + P_{PT}(T) \rightarrow P_{SB}(T), \quad T \rightarrow \infty,
\end{eqnarray}
and we again use here the relation (27). Since the $P_{PT}(T)$ is very small, $[1 + \phi_2(T)]$ has to be negative above $T_c$, so the GP pressure approaches the SB limit from below, as it should be. The growth of the NP pressure $P_{NP}(T)$ at high temperatures is suppressed in comparison with the growth of the SB term $P_{SB}(T)$ itself. The first one increases as $\Delta^2T^2$ because of the explicit presence of the mass gap in the EoS (see Eq.~(12) and integral (13)). The SB pressure increases as $T^4$ in this limit. This means that the sum $[1 + \phi_2(T)]P_{NP}(T)$ describes the leading and next-to-leading order term as $T \rightarrow \infty$. We can approximate the function $\phi_2(T)$ by the corresponding expression to leading and next-to-leading order terms in powers of $(T_c/T)$.

The explicit expressions for the $X(T)$ and $Y(T)$ functions (32), via the chosen simulating functions
(34) and (36), are

\begin{equation}
X(T) = A e^{- \alpha(T_c/ T)} P_{SB}(T)  + A_1 e^{- \alpha_1 (T_c/ T)} P_{NP}(T),
\end{equation}
and

\begin{equation}
Y(T) =  \left( 1 -  { 0.36 \over 1 + 0.55 \ln (T/T_c)} \right)P_{SB}(T)
+  \phi_2(T) P_{NP}(T).
\end{equation}
From the relations (31) and (38)-(39), and using the relations (33), it follows that

\begin{equation}
2 A e^{- \alpha} + A_1 e^{- \alpha_1} = 2(1 - 0.36) + \phi_2(T_c) = 1.28 + \phi_2(T_c).
\end{equation}

The GP pressure (30), on account of the relations (38) and (39), then looks like

\begin{eqnarray}
P_{GP}(T) &=& P_g(T)  +  \Theta \left( {T_c \over T} - 1 \right)
\left[ A e^{- \alpha (T_c/ T)}P_{SB}(T) + A_1 e^{- \alpha_1 (T_c/ T)}P_{NP}(T) \right]
\nonumber\\
&+& \Theta \left( {T \over T_c} - 1 \right) \left[ \left( 1 -  { 0.36 \over 1 + 0.55 \ln (T/T_c)} \right)P_{SB}(T)
+ \phi_2(T) P_{NP}(T) \right]. \nonumber\\
\end{eqnarray}
Due the relation (40) from the previous expression at $T=T_c$, one obtains

\begin{eqnarray}
P_{GP}(T_c) &=& P_g(T_c) + [1.28 + \phi_2(T_c)] P_{NP}(T_c) \nonumber\\
&=& P_g(T_c) + [2 A e^{- \alpha} +   A_1 e^{- \alpha_1}] P_{NP}(T_c),
\end{eqnarray}
so that it depends on the number $\phi_2(T_c)$ only, since the values of $P_{NP}(T_c)$ and $P_g(T_c)$ are known from our calculations (see, for example Fig. 1).

\subsection{Analytical and numerical simulation of $\phi_2(T)$ }

Our aim here is to find the simulating function $\phi_2(T)$ by fitting lattice data at high temperatures above $T_c$ \cite{36}. For this, let us derive from the GP EoS (41) its values at $a = T/T_c, \ a > 1$ as follows:

\begin{equation}
P_{GP}(a) = B_a  P_{SB}(a) + P_{PT}(a) + [ 1 + \phi_2(a)] P_{NP}(a),
\end{equation}
where the relation (27) has been already used, and

\begin{equation}
B_a = 1 - \alpha_s(a) = \left( 1 -  { 0.36 \over 1 + 0.55 \ln a} \right).
\end{equation}
Adjusting our parametrization of the GP pressure (43) to that used in recent lattice simulations for the
YM $SU(3)$ case at $T=aT_c$ in \cite{36}, one obtains

\begin{equation}
{3 P_{GP}(T) \over T^4 } = {P_p (T) \over T^4 } \times 5.26,
\end{equation}
where $5.26$ is the general SB number shown in Eq.~(A.7) up to two digits only after the point. Let us note that, for simplicity, we will keep this for all our numbers through out this paper. So that if the calculated number, for example
is $1.177...$, we will approximate it by $1.18$, if it is $1.171...$, we will approximate it by $1.17$, and finally if it is $1.175...$, we will keep this value. At the same time, let us stress that we have carried out all our calculations keeping six digits after the point (thus we can calculate all numbers to any requested accuracy). $P_p(T)/T^4$ describes lattice data taken from the above-mentioned Panero's paper \cite{36}. They are normalized to the SB limit for the pressure (see Eq.~(A.7)), for example $3 P_{GP}(2T_c) / (2T_c)^4 = (P_p (2T_c) / (2T_c)^4 ) \times 5.26 = 0.61 \times 5.26$ and so on. Then from the relation (45) and Eq.~(43), one gets an equation for determining the numbers $\phi_2(a)$, namely

\begin{equation}
\{p\}_a = 5.26 B_a + [pt]_a +(np)_a [1 + \phi_2(a)],
\end{equation}
where the numbers

\begin{equation}
\{p\}_a = {P_p (aT_c) \over (aT_c)^4 } \times 5.26, \quad
(np)_a = {3P_{NP}(aT_c) \over (aT_c)^4}, \quad  [pt]_a = {3P_{PT}(aT_c) \over (aT_c)^4}
\end{equation}
are exactly known from our calculations (see, for example Fig. 1 for the approximate values of the numbers $(np)_a$ and $[pt]_a$) and $3P_{SB}(T) / T^4 = 5.26$ at any high temperature (see Eqs.~(A.7)). The numbers $\{p\}_a$ are to be taken as described above. From Eq.~(46) then it follows that

\begin{equation}
\phi_2(a) =  - 1 + { \{p\}_a - 5.26 B_a - [pt]_a \over (np)_a}.
\end{equation}

As we already know, the best way to choose the appropriate expression for the simulating function $\phi_2(a)$ is to mimic again the asymptotic of the gluon mean number $N_g = (e^{\beta\omega} -1)^{-1}$ \cite{31} but in the
$\beta \rightarrow 0 \ (T \rightarrow \infty)$ limit, which is equivalent to the $a \rightarrow \infty$ limit. Indeed, replacing $\omega$ by $\mu T_c$, one can write

\begin{equation}
\phi_2(a) = { B_2 a^{-1} \over e^{(\mu/a)} -1} = { B_2 a^{-1} \over \sum_{m=1}^{\infty} (1 / n!) (\mu/a)^n }
= \sum_{n=0}^{\infty} c_n a^{-n},
\end{equation}
so that this simulating function above $T_c$ becomes the series in inverse powers of $a = T /T_c$.

First of all, we are interested in

\begin{equation}
\phi_2(1) = \sum_{n=0}^{\infty} c_n,
\end{equation}
as it follows from the previous Eq.~(49), since $\phi_2(1)= \phi_2(T_c)$, which appears in Eq.~(42).
At values $a \approx 1$, we can put

\begin{equation}
\phi_2(1) \approx  - 1 + { \{p\}_a - 5.26 B_a - [pt]_a \over (np)_a}
\end{equation}
in Eq.~(48), and the values for $a$ in this equation are to be chosen from the low
temperatures region, $a=1.010530, \ ..., \ 1.244705$, \cite{36}.
The mean value of the numbers which appear in the right-hand-side of Eq.~(51) is $- 2.12$, and thus

\begin{equation}
\phi_2(1) = \sum_{n=0}^{\infty} c_n = - 2.12.
\end{equation}

On the other hand, we can write

\begin{equation}
\phi_2(a) = \sum_{n=0}^{\infty} c_n a^{-n} = \sum_{n=0}^m c_n a^{-n} + \
\sum_{n=m+1}^{\infty}
c_n a^{-n} = \sum_{n=0}^m c_n a^{-n} + O(a^{-m-1}),
\end{equation}
so that when $m$ and $a$ are big enough, we can approximate Eq.~(53) up to leading order

\begin{equation}
\phi_2(a) = \sum_{n=0}^m c_n a^{-n},
\end{equation}
and the values for $a$ in this equation are to be chosen from the high
temperature region, $a=3.006657, \ ..., \ 3.436657$, \cite{36}.
The mean value of the numbers which appear in the right-hand-side of Eq.~(48) is $- 1.53$, and thus

\begin{equation}
- 1.53 = \sum_{n=0}^m c_n a^{-n}.
\end{equation}
Then Eq.~(53) can be re-written as follows:

\begin{eqnarray}
\phi_2(a) &=& \sum_{n=0}^{\infty} c_n a^{-n} = \sum_{n=0}^m c_n a^{-n} + \sum_{n=m+1}^{\infty}
c_n a^{-n} \nonumber\\
&=& - 1.53 + c_{m+1} a^{-m-1} + \sum_{n=m+2}^{\infty} c_n a^{-n},
\end{eqnarray}
which makes it possible to estimate the contribution of the remaining terms at $a \approx 1$. Indeed, in this case
Eq.~(56), due to the value (52), yields

\begin{equation}
c_{m+1} + \sum_{n=m+2}^{\infty} c_n = - 0.59.
\end{equation}
Substituting it back to the previous equation, one obtains

\begin{equation}
\phi_2(a) = - 1.53 - 0.59 a^{-m-1} - a^{-m-1} \sum_{n=m+2}^{\infty} c_n  + \sum_{n=m+2}^{\infty} c_n a^{-n}.
\end{equation}
This means that the whole expansion (49) or, equivalently, (58) can be effectively replaced by the polynomial as follows:

\begin{equation}
\phi_2(a) = - 1.53 - 0.59 a^{-m-1},
\end{equation}
which obviously has correct values at $a = 1$ and when $a$ goes to infinity ($T \rightarrow \infty$).

The best fit to lattice data \cite{36} above $T_c$ is achieved at $m=3$ in the relation (59).
When $m \leq 2$ the GM pressure
curve lies slightly below the lattice curve in the high temperature limit. At $m \geq 4$ the GM pressure curve lies substantially above the lattice curve in the moderate temperatures region $T_c \leq T \leq 2 T_c$, see Fig. 2.

The final expression for the simulating function $\phi_2(T)$ is the two-term polynomial

\begin{equation}
\phi_2(T) = - 1.53 - 0.59 \left( { T_c \over T} \right)^4, \quad \phi_2(T_c) = - 2.12,
\end{equation}
determining this function up to leading and next-to-leading orders in the $T \rightarrow \infty$ limit.

Due to the value (60), from Eq.~(42) one obtains

\begin{equation}
P_{GP}(T_c) = P_g(T_c) - 0.84 P_{NP}(T_c) = 0.656
\end{equation}
in dimensionless units (45) and (47). At the same time, the relation (40) becomes

\begin{equation}
A = - [0.42 + 0.5 A_1e^{-\alpha_1}]e^{\alpha}.
\end{equation}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig5b.eps}
\caption{The GP pressure (65) as a function of $T/T_c$ is shown for different values of $m=2,3,4$ in the relation (59). The lattice  curve \cite{36} for $SU(3)$ GP pressure is also shown (lower thin line).}
\label{fig:1}
\end{center}
\end{figure}

In the connection with the choice of the simulating functions a few general things should be made perfectly clear. In principle, all the simulating functions can be chosen in many different ways. However, the request that the pressure is a growing continuous ({\bf class $C^0$}) function of temperature in the whole range is a rather strong restriction (this means that it is differentiable at any point of its domain, while at $T_c$ derivative itself may not be continuous, i.e., it may have a discontinuity at this point). The other strong condition which should be imposed on the simulating functions is that they should not undermine analytical and asymptotical properties of the corresponding test functions. The simulating functions may produce only small numerical corrections to the values determined by the
corresponding test functions. The justified choice of the simulating functions can only simplify the actual numerical calculations. i.e., they play only an auxiliary role. Let us remind that our choice of the simulating functions $f_1(T)$ and $\phi_1(T)$ has been made in accordance with the asymptotic of the gluon mean number $N_g = (e^{\beta\omega} -1)^{-1}$ in the $\beta \rightarrow \infty \ (T \rightarrow 0$) limit.
The simulating function $\phi_2(T)$ has been chosen in accordance with the asymptotic of the gluon mean number
in the $\beta \rightarrow 0 \ (T \rightarrow \infty$) limit up to leading and next-to-leading orders. The gluon mean number describes the distribution of gluons in the GP and is closely related to its pressure and thus to all other thermodynamical variables. The simulating function $f_2(T)$ has been chosen to introduce the SB term itself and to reproduce AF corrections to it above $T_c$ as well. The final result (the curve for the GP pressure (41)) will depend rather slightly on the numerical values of the parameters of such chosen simulating functions and details of their calculations. Thus this curve, and hence all the numbers behind it, will be uniquely determined up to the leading order (up to very small numerical uncertainties of the next-to-leading order). This is a general feature of the numerical simulations based on some analytical expressions.

\subsection{Numerical simulation of the GP pressure}

After the substitution of the relation (62), the GP pressure (41) becomes

\begin{eqnarray}
P_{GP}(T) &=& P_g(T) \nonumber\\
&-&  \Theta \left( {T_c \over T} - 1 \right)
\left[ [0.42 + 0.5 A_1 e^{-\alpha_1} ] e^{-\alpha((T_c/T)-1)} P_{SB}(T) - A_1 e^{-\alpha_1(T_c/T)} P_{NP}(T) \right] \nonumber\\
&+& \Theta \left( {T \over T_c} - 1 \right) \left[ \left( 1 -  { 0.36 \over 1 + 0.55 \ln (T/T_c)} \right)P_{SB}(T)
+ \phi_2(T) P_{NP}(T) \right], \nonumber\\
\end{eqnarray}
where $\phi_2(T)$ is completely determined and given in Eq.~(60).
This expression is convenient to simulate in order to continuously (and as much smoothly as possible) adjust both terms at $T_c$, which are  associated with the corresponding $\Theta$-functions. It depends on the three independent (free) parameters, $A_1$, $\alpha_1$ and $\alpha$. The fourth parameter $A$ is to be calculated through the
relation (62), and thus it plays no independent role.
The simulation procedure consists of the numerical variation of these parameters in order to achieve the adjustment in the requested way. In principle, there are no other constraints on these constants
apart from that all the derivatives of the GP pressure (63) should not gain negative values at low temperatures below $T_c$, i.e., they should exponentially approach zero from above.

Our best values for them are:

\begin{equation}
A_1 = -1.1, \ \alpha_1 =0.006, \ \alpha=4.
\end{equation}
The value $\alpha=4$ means that the deep penetration of the SB term below $T_c$ is indeed suppressed,
as expected. The value $\alpha_1=0.006$ means that the NP pressure is not suppressed, but only the slope of its curve is decreasing below $T_c$, which is also expected. The number $A_1$ is, in principle, a dimensionless combination
of all our dimensional parameters: $\Delta^2, \ \omega_{eff}, \ T_c, \ \Lambda^2_{YM}$. Let us remind that only
$\Delta^2$ or, equivalently, $\omega_{eff}$ and $\Lambda^2_{YM}$ are independent, while the latter one is suppressed by $\alpha_s$, see Eq.~(26). However, how to assign to it a physical meaning (if any) is not clear; anyway this is not important.

Thus the GP pressure (63) finally becomes

\begin{eqnarray}
P_{GP}(T) &=& P_g(T) \nonumber\\
&+&  \Theta \left( {T_c \over T} - 1 \right)
\left[ 0.146 e^{-4((T_c/T)-1)} P_{SB}(T) - 1.1 e^{-0.006(T_c/T)} P_{NP}(T) \right] \nonumber\\
&+& \Theta \left( {T \over T_c} - 1 \right) \left[ \left( 1 -  { 0.36 \over 1 + 0.55 \ln (T/T_c)} \right)P_{SB}(T)
+ \phi_2(T) P_{NP}(T) \right], \nonumber\\
\end{eqnarray}
and it is completely determined now, since $\phi_2(T)$ is also determined and given by the relation (60).
All other thermodynamic quantities will be calculated just via this expression.

\section{Numerical results and discussion}

In Fig. 3 we present the GP pressure (65) as  well as the lattice pressure \cite{36}.
The main difference between them lies in the region of low temperatures below $T_c$. Within the NP analytic approach
the confining effective charge (8) is mainly responsible for the structure of the GP in this region.
It gives rise to the gluonic excitations of the dynamical origin which cannot be taken into account by the lattice simulations. The corresponding $\beta$-function (see appendix D) is always negative, as it is required by confinement \cite{26,27}. At the same time, the lattice $\beta$-function is always positive and almost constant \cite{36,37}. The discrepancy between NP analytic GP and lattice pressures seen in Fig. 3 just above $T_c$ up to $2T_c$ is also due to other NP effects which are still important in this region as pointed out above (see discussion
in connection with Fig. 1).

The GP entropy and energy densities are shown in Fig. 4. The size of the discontinuity in the energy density, the so-called latent heat (LH) is

\begin{equation}
\epsilon_{LH} = 1.54
\end{equation}
in dimensionless units (see appendices B and C for its definition and analytical/numerical derivation, respectively). This means that the first-order phase transition in the GP is
analytically confirmed for the first time, in complete agreement with thermal $SU(3)$ YM lattice simulations.
The physical reason of such sharp changes at $T_c$ in the derivatives of the GP pressure is that its
exponential rise below $T_c$ is changing to the polynomial rise above $T_c$ in order to reach finally the thermodynamic SB limit. Our value (66) for the LH is consistent with various thermal $SU(3)$ YM lattice calculations: in \cite{37} $\epsilon_{LH} \approx 2$, in \cite{36} $\epsilon_{LH} = 1.4$, in \cite{39} $\epsilon_{LH} = 1.413$ and in \cite{40} $\epsilon_{LH} = 1.54$.
Note, that our value is slightly bigger than the recent lattice values in \cite{36,39}, and coincides with the value in \cite{40} (for simplicity, we do not specify
the lattice parameters such as $N_t$, $N_s$, etc.). This agreement is rather remarkable, taking into account
the completely different natures of the NP analytical and lattice approaches to thermal QCD.

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig5.eps}
\caption{The GP pressure (65) as a function of $T/T_c$ (upper thick line). The lattice data \cite{36}
for $SU(3)$ GP pressure is also shown (lower thin line). Both pressures are scaled in the same way
in accordance with Eq.~(45).}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig5a.eps}
\caption{The GP pressure, the entropy and energy densities all
properly scaled are shown as functions of $T/T_c$. The finite jumps in densities are clearly seen, and
the LH is $\epsilon_{LH} = 1.54$. Their common SB limit (A.7) is approaching rather slowly.}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig6.eps}
\caption{The trace anomaly
properly scaled is shown as a function of $T/T_c$ (solid line). The gluon condensate at zero temperature scaled in the same way is also shown (dot-dashed line).}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig7.eps}
\caption{The properly scaled gluon condensate at non-zero temperatures
is shown as a function of $T/T_c$. It approaches zero from below at high temperatures.}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Picf1.eps}
\caption{The NP gluon condensate scaled by $T^4$ as a function of $T/T_c$.
The zero temperature gluon condensate scaled by $T^4$ is also shown.
At high temperatures the NP gluon condensate scales as $1/T^2$. }
\label{fig:10}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig8a.eps}
\caption{$P(\epsilon)$ EoS (solid line) and approach to $conformality = (1/3)$ (thin line).}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig9.eps}
\caption{Conformity $C(T)=P(T)/ \epsilon(T)$ as a function of $T/T_c$.
It is zero at $T=0$, and has a jump at $T_c$ due to a jump in the energy density at this point.
It shows the non-trivial dependence on $T$ below $T_c$, and approaches the SB limit $1/3$ at high
temperatures rather quickly.}
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig10.eps}
\caption{The heat capacity as a function of $T/T_c$. It has the $\delta$-type singularity (an essential discontinuity) at $T_c$. It approaches its SB limit (A.7) (horizontal dashed line) at high temperatures very slowly.  }
\label{fig:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{Fig11.eps}
\caption{The velocity of sound squared as a function of $T/T_c$.
It shows the non-trivial dependence on $T$ below $T_c$, while at $T=0$ and at $T=T_c$ it is zero.
It approaches the SB limit $1/3$ at high temperatures rather quickly. }
\label{fig:1}
\end{center}
\end{figure}

The trace anomaly relation, which is especially sensitive to the NP effects including the conformal
symmetry breaking, is shown in Fig. 5.
Its two most interesting features are: the rapid rise of the peak due to the LH in the energy density (see Fig. 4)
is exactly placed at $T_c$, while in lattice calculations it peaks at about $1.1T_c$ \cite{35,36,37,40}.
We interpret this lattice artefact as a clear evidence that by lattice simulations one cannot get correct
numbers for all the thermodynamic quantities near $T_c$ and at $T_c$ itself. As it follows from Fig. 3, the lattice
pressure below $T_c$ is almost zero and at $T_c$ it is about $0.1$, while our pressure is $0.656$ (in dimensionless
units). That was the reason why we simulate the function $\phi_2(T)$ in subsection (4.2) not using the lattice value at $T_c$. It is worth underlining once more that lattice simulations cannot take into account the NP dynamical effects, especially confinement, neither at zero nor at non-zero temperatures. They dominate the structure of the QCD vacuum at
zero temperature and will remain important at non-zero temperatures as well (up to $3.75T_c$ within our approach \cite{20}). In other words, they cannot be so negligible small below $T_c$ as it is shown by all lattice results for all the thermodynamic quantities.
However, above $T_c$ at sufficiently high temperatures $T \geq 2 T_c$ our results coincide because of the adjusted procedure necessary performed in subsections 4.2 and 4.3. The second interesting feature is that like in all lattice simulations (see for example \cite{35,36,37,40}) the trace anomaly falls off as $1/T^2$ and not as $1/T^4$ at high temperatures. As emphasized above in section 4 of the present investigation and in \cite{20}, this is due to the explicit presence of the
mass gap term $\Delta^2 T^2$ in the EoS (65). The terms proportional to $T^4$ in the trace anomaly cancel each other at high temperatures due to Eq.~(A.9), and hence the mass gap term comes out into play in this limit.
It is the mass gap which violates the conformal symmetry from the very beginning within our approach.

In close connection with the trace anomaly is the gluon condensate defined by Eq.~(A.6) and shown in
Fig. 6. It approaches zero from below, so it gains negative values at high temperatures (fixed also by lattice simulations in \cite{37}) due to the behavior of
the trace anomaly in this region. In this connection, a few remarks are in order. The gluon condensate at zero temperature has been defined and calculated by the subtraction of all types of the PT contributions ("contaminations") in \cite{22}. Here the temperature-dependent gluon condensate has been calculated with the help of the GP pressure (65), which contains explicitly the PT SB terms in different combinations. If the temperature-dependent gluon condensate is to be defined in the same way as its zero-temperature counterpart, then only the NP pressure (12) should be used for its calculation. This has been already done in \cite{20} and shown in Fig. 7.
It is always positive and at high temperatures it scales as $1/T^2$.

The GP pressure versus the GP energy density, the so-called EoS  $P(\epsilon)$, is present in Fig. 8.
The size of the LH and the rather quick approach to conformality are clearly seen. Let us note in advance that we distinguish between the conformality here and conformity defined in Eq.~(A.4), though numerically in the limit of high temperatures they are the same. The conformity itself is shown in Fig. 9. It has a finite jump at $T_c$ because of a jump in the energy density at this point, and approaches its SB limit (A.8) at high temperatures rather quickly. However, its most interesting feature is its non-trivial dependence on $T$ below $T_c$, which has been fixed explicitly in $SU(3)$ GP for the first time. It is due to the fact that the conformity is the ratio of the independent thermodynamic quantities, and therefore is not obliged to be always a growing function of $T$ below $T_c$,
though above $T_c$ it is always growing.

The last independent thermodynamic quantity, the heat capacity defined in Eq.~(A.2), is shown in Fig. 10.
It is always a growing function of $T$ below and above $T_c$, while at $T_c$ it has
a $\delta$-type singularity (an essential discontinuity) due to the expression (B.6). It approaches its SB limit
(A.7) at high temperatures very slowly. The velocity of sound squared is shown in Fig. 11. Like conformity it is the ratio of the independent thermodynamic variables, see Eq.~(A.3). It behaves very similar to conformity (Fig. 9), since the latter one mimics its properties. The principal difference from conformity is that at $T_c$ it is zero because of the heat capacity having the above-mentioned $\delta$-type singularity at this point. It approaches its SB limit (A.7) rather quickly.

\section{The dynamical structure of $SU(3)$ GP. A brief description}

A rich dynamical structure of the GP emerges in the framework of our approach. The NP part (12)
of the GP pressure (65)
tells us that in the whole temperature range we have two different massive gluonic excitations
(or, equivalently, effective gluonic degrees of freedom \cite{10,41})
$\omega'$ and $\bar \omega$ with the effective masses $m'_{eff}=
1.17 \ \GeV$ and $\bar m_{eff} = 0. 585 \ \GeV$, respectively. Both effective
masses are due to the mass gap $\Delta^2$, which is responsible
for the large-scale structure of the QCD ground state. It is dynamically generated by the
nonlinear interaction of massless gluon modes \cite{24,25,26}. The first
massive excitations can be interpreted as glueballs, since $m'_{eff}$ is compared to the masses
of scalar glueballs \cite{42}. The second one
$\bar m_{eff}$, might be identified with an effective gluon mass of about $(500-800) \ \MeV$, which arises
in different quasi-particle models (see again \cite{42} and references therein).

We also have two different massless gluonic excitations $\omega$: the NP massless gluons, propagating in
accordance with the integral $P_1(T)$ in Eq. (13), and almost SB massless gluons, propagating in accordance
with the integral $P_2(T)$
in Eqs. (14). Let us remind that the integral (13) should be multiplied by $(6 /\pi^2) \Delta^2$, and all other integrals (14) are to be multiplied by $(16 / \pi^2) T$, when one speaks about different NP contributions to the pressure (12). However, this is not the whole story yet. The NP pressure (12) enters the GP pressure (65) in three different places, one through the gluon pressure (27). This means that all the above-mentioned
massive and massless gluonic excitations propagate in the GP in two different ways: below and above $T_c$ they propagate in accordance with the terms in Eqs.~(35) and (37) with parameters
and the function $\phi_2(T)$ fixed by the relations (64) and (60), respectively.

The PT part (26) of the gluon pressure (27) assumes the presence of the NP massive gluonic excitations $\bar \omega$
in the GP in the whole temperature range. Having the same effective masses, nevertheless, they propagate in a completely different way, which is determined by the mass gap $\Delta^2$ and $\Lambda_{YM}$ (compare the integrals (26) and (14)). However, their propagation is suppressed by the order of $\alpha_s$. The terms of the order of $\alpha_s^2$ and higher have been left undetermined. In any case, numerically they are very small. Let us emphasize once more that all these NP massive and massless gluonic excitations have not been introduce by hand; contrary, they are of the dynamical origin, and only they (i.e., of the dynamical origin) can be described and accounted for within our approach to thermal QCD.

In the GP in the whole temperature range the free gluons context within our approach is to be described as follows:
we definitely have the SB free massless gluons far away from $T_c$, while near $T_c$ their propagation is
corrected by AF. Below $T_c$ the deep penetration of the SB term itself is suppressed. So we have three different types of the PT free gluons, apart from those discussed above and at the beginning of section 4.

The exponential rise of all the independent thermodynamic quantities in the transition region
$(0.6-1)T_c$ \cite{20} clearly seen in Figs. 4 and 10 indicates that near $T_c$ a dramatic increase
in the number of effective gluonic degrees of freedom will appear (for example, the glueballs will begin
to dissolve rapidly). This will lead to drastic changes in the structure of the GP. A change in this number is enough to generate pressure gradients, but not enough to affect the
pressure itself. It varies slowly and therefore remains continuous
in this region. At the same time, the pressure gradients such as
the energy and entropy densities, etc., undergo sharp changes in their behavior, having different types of
discontinuities at $T_c$. Thus $SU(3)$ GP has a first-order phase transition within our approach, in complete
agreement with the thermal QCD lattice calculations (see \cite{36,37} and references therein).
Of course, not all the glueballs will be dissolved in the transition region. Some of them will remain
above $T_c$, along with other massive and massless gluonic excitations, forming thus a possible
mixed and extended mixed phases around $T_c$ from its both sides \cite{20,43}.
One can conclude that the NP physics of the mixed phases (the temperature interval approximately
$(06-3.75)T_c$) in $SU(3)$ GP is well understood within our approach.
At very high temperatures starting with $3.75T_c$ the NP effects become very small,
and the structure of the GP will be mainly determined by the SB relations (A.7)-(A.9) between
all the thermodynamic quantities.

\section{Conclusions}

The effective potential approach for composite operators \cite{21}
has been generalized to non-zero temperatures in order to derive the NP
analytical EoS for pure $SU(3)$ YM fields from first principles.
In its NP part (12) there is no dependence on the coupling constant, only
a dependence on the mass gap, which is responsible for the
large-scale structure of the QCD ground state. A key element of
this part is the generalization of the expression for the bag constant at zero
temperature \cite{20,22} to non-zero temperatures. Its PT part does analytically depend on the fine-structure
constant of strong interactions as well. We also propose and formulate a method how to include the free gluon contribution into the GP pressure (65) in a self-consistent way.

Our main quantitative and qualitative results in this investigation are:

{\bf (i)}. In the GP EoS (65) the confining dynamics at non-zero temperatures (8) is taken into account through
the $T$-dependent bag constant and the mass gap $\Delta^2$ itself. The GP pressure (65) compared to
lattice pressure \cite{36} is shown in Fig. 3.

{\bf (ii)}. Other NP effects are also taken into account via the YM and PT parts of the GP
pressure, though the latter one is suppressed by the order of $\alpha_s$.

{\bf (iii)}. The mass gap $\Delta^2$ or, equivalently, $\omega_{eff}$
and the asymptotic scale parameter for YM fields $\Lambda^2_{YM}$ are only
two independent input scale parameters in our approach.

{\bf (iv)}. The characteristic temperature $T_c = 266.5 \ \MeV$
is a temperature at which the maximum of the NP part of the GP pressure is achieved (Fig. 1).

{\bf (v)}. The presence of the mass gap $\Delta^2 T^2$ term in the GP pressure (65).

{\bf (vi)}. Due to this the trace anomaly and the NP
gluon condensate will go down as $1/T^2$ at high temperatures.
The trace anomaly has a peak just at $T_c$ (Figs. 5,6 and 7).

{\bf (vii)}. The presence of the two different types of massive gluonic excitations of the NP origin
with an effective masses $0.585 \ \GeV$ and $1.17 \ \GeV$. They propagate in different ways
below and above $T_c$. There is no place for some other types of massive gluonic excitations in our picture.

{\bf (viii)}. The presence of the different types of massless gluonic excitations
including the free PT or, equivalently, SB gluons far away from $T_c$.

{\bf (ix)}. Below $T_c$ all the independent thermodynamic quantities exponentially go down and up with $T$
 going down and up, respectively (Figs. 4 and 10). Above $T_c$ this regime is changed to a polynomial growing one.

{\bf (x)}. The thermodynamic quantities being the ratios of the corresponding independent ones such as
conformity and the velocity of sound squared demonstrate the non-trivial similar dependence on $T$ below $T_c$
(Figs. 9 and 11). At high temperatures their approach to the SB limit (A.8) is accelerated by a factor of 3.

{\bf (xi)}. At $T_c$ the GP pressure is continuous, while the energy and entropy densities have finite jumps.
The size of the discontinuity in the energy density is: $\epsilon_{LH} = 1.54$ (in dimensionless units), see Fig. 4. The heat capacity has an essential discontinuity of a $\delta$-type singularity at $T_c$, see Fig. 10.

{\bf (xii)}. The first-order phase transition at $T_c$ in $SU(3)$ GP is thus analytically proven.

{\bf (xiii)}. Since the NP effects are still important up to $3.75T_c$, all the independent thermodynamic quantities approach their perspective SB limits rather slowly.

{\bf (xiv)}. Because of this the behavior of $SU(3)$ GP in this region is substantially different from
the behavior of a gas of free massless gluons.

The NP analytical approach to thermal QCD provides a detailed numerical and dynamical description of
$SU(3)$ GP in the whole temperature range, and especially in the region of low temperatures below $T_c$. Something that's not accessible for and missing by lattice approach to thermal QCD. Our next and final step is to include the quark degrees of freedom into the formalism of the effective potential approach for composite operators at non-zero temperatures. This will make it possible to derive the NP analytical EoS for the QGP.

\ack{This paper is dedicated to the memory of the late Prof. J Zimanyi, who has initiated this investigation.
Support by HAS-JINR Scientific Agreement (P Levai) is to be acknowledged. The authors are especially grateful
to M Panero for providing the lattice $SU(3)$ YM numerical values of various thermodynamic quantities.
We thank R Pisarski for bringing our attention to reference \cite{33}.
We would like also to thank T Bir\'{o}, T Csorg\"{o}, P V\'an, G Barnaf\"{o}ldi,
A Luk\'acs and J Nyiri for useful discussions, comments, remarks and help.
One of the authors (V G) is grateful to V K and A V Kouzushins for constant support and interest.}

redpajama_set_name: RedPajamaArXiv


text:
http://www.discovernikkei.org/ja/journal/2022/1/11/ruth-sato-reinhardt-2/
グレッグ・ロビンソン / 2022年1月11日
Read Part 1 >>
Ruth in Sweet and Low, 1930
After 1940, Ruth Sato's life was transformed, and she shifted her career from showgirl to more serious pursuits. It must be said that she had always had artistic and intellectual interests.
During the 1930s, she dated writer John O'Hara, with whom she discussed literature. DownBeat magazine reported in 1939 that Sato was a "hep chick" and jazz enthusiast, who had a major record collection and knew all the members of the major swing bands. In September 1939, Sato published an odd "open letter" in DownBeat. While she rather unjustly took a fellow letter writer for anti-black bias, she insightfully pointed to the social barriers that she experienced as a "half-Japanese and half-Irish" and the positive role of jazz in breaking them down. "As for prejudice, jazz is a common means for eliminating any so-called bugaboo."
During 1941, Sato wrote a gossip column, "Could Be," for the jazz magazine Swing. At some time in this same period, she began dating William "Bill" Reinhardt, a jazz clarinetist 4 years her junior.
Sato recalled that she went through a period of unemployment with the coming of the war—she noted drily that demand for Japanese exotic dancers went way down after Pearl Harbor. She also asserted that she was followed by the FBI, because of her late father's importance to the Japanese community, and dyed her hair blonde in order to pass. Even her name was a liability.
In January 1942, popular columnist Walter Winchell, who had been a leading prewar publicist for Sato, stated facetiously that "the Eurasian entertainer (half Jap, half Yankee Doodle) will change her name to Pearl Haba." She demanded a retraction, under threat of lawsuit. Winchell published her denial the next week, but with poor grace, and after that never again mentioned her in his column.
In response to the pressure, she left New York—columnist Danton Walker proclaimed that she had disappeared from Broadway—and moved to Virginia to be with Bill Reinhardt, who had been sent there. On June 6, 1942, the two were married in Norfolk. Ruth Reinhardt (as she was henceforth known) gave differing versions of her life during the three years that followed.
In a 1948 profile, she stated that she had spent the war years restoring a seven-room farmhouse in Williamsburg, Virginia, while Bill was stationed at Peary Naval Training Station. In a later interview, she revealed that she had taken over a small farm on Florida's Gulf Coast, where she passed her time killing snakes. "I would zap 'em with Lysol. Nobody ever taught me how to use a gun. I was from New York, where anything suspicious got it with Lysol right away."
During this period, Reinhardt contributed a series of columns to DownBeat magazine which displayed her talents as music critic and comedic writer. For example, Reinhardt's column "Saxes Should Be Played Not Tossed For a Goal," in the December 15, 1943 issue, features the author's outspoken criticism of the "pseudo showmanship" of the saxophone players she watched who wrestled furiously with their instruments in playing. "One man looked like a huge mastiff tugging at a large, brass bone."
Curiously, the columns also hint at a New York connection. Not only did "Saxes" bear a New York dateline, but the September 15, 1944 issue featured Reinhardt's "Random Ramblings from Rhythm Row," a gossip column about the New York jazz scene whose items strongly suggested that the author had been spending some time in the city.
Whichever place(s) Ruth Reinhardt actually resided during the war years, once Bill Reinhardt returned to the United States and secured his discharge from the Navy, the two spouses faced the decision of where to settle and what to do. Ruth Reinhardt later stated that she considered taking over a sponge fishing business in Tarpon Springs.
In the end, the couple moved to Bill's hometown of Chicago, where they decided to try making a living out of music by opening a jazz club. Reinhardt recalled that at first, the couple were so naïve that they placed a newspaper advertisement offering a $1000 reward for information on a suitable establishment, but received no responses. Instead, Bill Reinhardt went out walking one day and found a basement space that seemed ideal. The property was located at 11 East Grand Ave. on the Near North Side—the main center of Japanese American resettlement in Chicago. The couple settled on Jazz, Ltd. as the name of their club (their lawyer vetoed the original name, "Jazz, Inc.").
Jazz, Ltd. opened in June 1947. It offered New Orleans and Chicago style Dixieland music. The music was entirely instrumental: to avoid an extra 20 percent night-club tax, the Reinhardts did not permit singing in the club. Bill led the house band, a five-piece ensemble with four regulars (notably Doc Evans) plus a special guest star.
Ruth acted as hostess, and greeted all customers personally. She also took up the role of manager and administrator, bringing to the task a hardheaded business skill and capacity for hard work: she was present in the club every night, 6 days a week, from 7 p.m. to 4 a.m. She also took up the unpleasant task of asking undesirables to leave.
As she remarked in a column in DownBeat, "I guess I have become quite a character. One drunk told me that that I should be a schoolmarm rather than a saloonkeeper. My shell is thick, and after all we are not selling me—we are selling jazz." Writing in the Chicago Tribune, columnist Savage described her as the club's guiding hand, "a combination of modified Texas Guinan [a famed speakeasy operator in 1920s New York] and implacable house dick."
From the beginning, the club offered a different experience to its patrons. To begin with, it was the first jazz club outside the African American district of Bronzeville without a color bar, and from the beginning its audiences were integrated. Conversely, the Reinhardts imposed strict rules to underline the club's status as a respectable institution: reservations were encouraged; patrons had to wait behind a velvet rope until seated; gentlemen were required to wear jackets; unaccompanied women were not admitted; and no women were served at the bar under any circumstances. The serving staff was all male.
In order to stay "clean" of racketeers, the Reinhardts refused to install jukeboxes or cigarette vending machines. Ruth Reinhardt also made use of her flair for publicity. Rather than run expensive newspaper or radio ads, the Reinhardts maintained an extensive mailing list and made use of columnists such as then-disc jockey Dave Garroway to spread the word. They shrewdly began the practice of making live recordings from the club and selling them on site, in the process gaining both money and publicity.
Jazz, Ltd, was immediately popular. It became even more in its second season, when the Reinhardts signed legendary New Orleans jazzman Sidney Bechet to an exclusive contract, and he began a residency at the club. With encouragement from Ruth Reinhardt, newspapermen began discussing Bechet in news stories and gossip items, providing extra publicity for the club.
At Jazz, Ltd. club November 1947. Left to Right: Bill Reinhardt, Danny Alvin, Sidney Bechet, Mel Grant, Munn Ware
Unfortunately, the Reinhardts' relationship with the musician soon proved contentious. In May 1949 Bechet left for Europe without alerting the club owners, leaving them to scramble to fill his place. When he returned in November of that year, he pledged to return to Jazz, Ltd. However, he played an outside gig in Chicago, violating his exclusive agreement with the club, whereupon Ruth Reinhardt successfully sued him for breach of contact.
Even after Bechet's departure, Jazz, Ltd. continued to prosper. It attracted notable patrons such as writer Nelson Algren. While the club moved to a larger location in the 1960s, the Reinhardts continued to manage it closely. By the time it closed its doors in February 1972, a casualty of declining public interest in jazz, it had been open for nearly 25 years, and claimed to be the oldest traditional-jazz club in the world. Following the demise of her club, Ruth Sato Reinhardt went into retirement. She died on December 9, 1992.
Ruth Sato Reinhardt once described herself as "tone-deaf" and stated that her main interest in her club was in making money. All the same, she made an important contribution to Chicago's music scene, and provided a space for interracial contact. Her newspaper writings, though limited in number, reveal both her real passion for music and her talent as a prose stylist.
Chicago club Jazz Ltd. post-war Ruth Reinhardt Ruth Sato
グレッグ・ロビンソン
(Greg Robinson)
ニューヨーク生まれのグレッグ・ロビンソン教授は、カナダ・モントリオールの主にフランス語を使用言語としているケベック大学モントリオール校の歴史学教授です。ロビンソン教授には、以下の著書があります。
『By Order of the President: FDR and the Internment of Japanese Americans』(ハーバード大学出版局 2001年)、『A Tragedy of Democracy; Japanese Confinement in North America』 ( コロンビア大学出版局 2009年)、『After Camp: Portraits in Postwar Japanese Life and Politics』 (カリフォルニア大学出版局 2012年)、『Pacific Citizens: Larry and Guyo Tajiri and Japanese American Journalism in the World War II Era』 (イリノイ大学出版局 2012年)、『The Great Unknown: Japanese American Sketches』(コロラド大学出版局、2016年)があり、詩選集『Miné Okubo: Following Her Own Road』(ワシントン大学出版局 2008年)の共編者でもあります。『John Okada - The Life & Rediscovered Work of the Author of No-No Boy』(2018年、ワシントン大学出版)の共同編集も手掛けた。 最新作には、『The Unsung Great: Portraits of Extraordinary Japanese Americans』(2020年、ワシントン大学出版)がある。連絡先:robinson.greg@uqam.ca.

redpajama_set_name: RedPajamaCommonCrawl


"Our objective is to meet the needs of every pupil in accordance with his particular abilities, aptitudes, interests and personality"
The Department of Education and its co-partners - the teachers, trustees and parents - have done much over the past ten years to up date and up grade the quality and content of education, in spite of insurmountable financial problems. For the first time, given an adequate budget, the Department of Education, with the assistance and cooperation of all sectors of the province will be able to develop a total plan to meet the educational needs of our time.
Read more about "Our objective is to meet the needs of every pupil in accordance with his particular abilities, aptitudes, interests and personality"
My son Oliver started grade 7 at Birchwood Intermediate School this fall and he decided that he wanted to buy his lunches in the school cafeteria. There's a daily special offered by the cafeteria each day, and the school helpfully publishes this on the school website so it's been easy for us to keep track of what he's having for lunch every day. But we didn't really know anything about the nutrition of the daily specials, and I didn't really know where to find this out.
Celebrate National Teacher/Staff Appreciation Week - Celebrating since 1988!
The 30th National Teacher/Staff Appreciation Week (T/SAW), celebrated February 13 - 17, 2017, by the Prince Edward Island Home and School Federation, and local Home and School Associations encourages all parents, students and the wider community to recognize the important contributions, both personal and professional, of teachers and school staff to the lives of children and their education.
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Need tips on how to help your child with elementary math homework?
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Check out these Raegan and Monty's wedding images!
Check out these images from Alyssa and Bobby's wedding!
Check out these two! Ann and Matt are looking amazing at the Dallas Arboretum.
Geoff and Meagan's wedding was so fantastic. I could literally do an entire blog post on just the beautiful detail shots they had at their ceremony and reception but I will leave that to Pintrest. ha! Their wedding was right around Christmas and with all the holiday hustle I forgot to blog about their wedding. So its Christmas in April apparently! Enjoy their beautiful story!


text:
The "At Ease With the Media" Blog
Valuable insights for media relations professionals and their spokespeople
#mediatraining Tweets
"You're Just Blowing Smoke"
This is the second instalment in a series on media relations measurement.
To help shed some light on what the state-of-the-art in media relations measurement should be, I thought I'd turn to Wilma Mathews, ABC, a long-time colleague and friend, and author of Media Relations: A Practical Guide for Communicators. Wilma has been practicing media relations for … well, let's just say quite a few years.
When it comes to media relations evaluation and measurement, Wilma says our industry is certainly better off than it was even five or ten years ago. For many years, media relations practitioners relied on the simplistic output measures of counting clips and adding up circulation.
From there, the process evolved into impressions which, from her perspective, means pretty much the same thing as circulation and viewing audience. Next, the advertising value equivalency (AVE) was born, which she points out is a term that's not even listed in theDictionary of Public Relations Measurement and Research.
"But over the years, as PR people, agencies and companies have gotten a little savvier, they've said that what we're asking you as media people to do is sell a product, get people to come to an event, change their minds or vote for someone," Wilma explains. "In short, we're asking you to change behaviour of a certain audience. And that's a little harder to do than counting clips."
She believes the AVE was adopted as a matter of convenience (and I suspect she would say something similar about Media Relations Rating Points). It was a simple way to state some perceived value of media relations to management groups. But to her the AVE is a completely abstract number that has no correlation to any activity because advertising and media relations simply cannot be compared.
"You control everything about advertising," she explains. "You control nothing about the editorial side of the media. But (the AVE) was a way to say to clients 'if you had purchased advertising, it would have cost you X amount of dollars, and we prevented you from having to do that.' And it sounded good at the outset."
She makes a clear distinction between evaluation and measurement in media relations. "You can evaluate your media relations work and still not measure whether or not it worked," she explains. "In other words, if a media relations practitioner wanted a positive story on the front page of the business section with a quote from their CEO — and they wanted it to appear before the product launch — if they got all of that it says their process worked. It says nothing about whether that helped sales."
To her, measurement is the end outcome — from an attitudinal or behavioural perspective. Did people buy the product? Did they vote the way you wanted? Did they form an opinion or change their minds?
"If that didn't happen and all you've got to show for it is advertising value equivalents or impressions," she points out, "you're just blowing smoke."
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Perhaps you are getting children and want to remodel the present bedrooms in your house. You may want to develop a Jack as well as Jill lavatory so the two children you propose on getting can talk about the bathroom, but feel like they've their own lavatory.
You can sell your new items with Craigslist, eBay or even with the local resale shop. You really need to clean and sterilize anything you discover and make this look brand new. If you find factors of value this won't be not easy to sell these phones someone to make easy money. I know have offered things with Craigslist generating fast cash.
On renting a waste bin on your trash removal needs, you may set yourself away from the waste disposal troubles. Dumpster will accomplish your job spot at your given regular time periods and carry away this dump with no demanding your intervention at the same time. you can have the ability of placing the dumpster some time in the underground room which is from customer's view. You will practical experience no setbacks in the program provided, this dumpster will be obtained by the Large Dumpster For Rent in Littleton, WV corporation Guam at the time specified by you. The assistance of garbage bin hire company Guam can break your misconception with regards to trash bin rental.
Having a Large Dumpster For Rent in Littleton, WV to help dispose of dangerous waste materials is a simple and yet effective means of eliminating unwanted chemicals from the standard living area. The need to be more cash conscious in the methods we get rid of waste materials are so widely sought after by the average person that even the United States Federal government is offering significant benefits to tax payers who reuse. People are finding the beauty of performing their part as it has to do with waste management. What if there were no easy ways to get rid of waste materials in a very neighborhood, neighborhood, city, and so forth.
Cleaning out your garage can seem like an frustrating task, however chances are that when investing in started, you will be on a move and get yourself organized quickly. It might help break down the job into a handful of easy steps to help you guide the method.
Have you viewed your garage or underground room lately? Almost all of us use these spots as everlasting home hard drive for factors we think we might eventually work with. But you can always find more factors we could perhaps need in the foreseeable future, and these spaces fill up swiftly, often until we couldn't find one of the things we need among the clutter.
There are lots of companies that present rolloff Large Dumpster For Rent in Littleton, WV. A rolloff dumpster is a that gets brought to your property, driveway or perhaps alley, and then gets obtained again if you are done. You will work out a nominated rental period while using company (ordinarily a week or two). An individual fill this dumpster, they figure it out. Your project helps to keep rolling with schedule and you don't have to be worried about trash mounting up. It's a win-win situation.
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If you're in product design and style, software executive, construction or any type of other market, there is certainly generally even more than a single way to get issues done. In terms of projectmanagement, both most high profile methods pertaining to getting tasks done will be the classic waterfall style of job operations as well as the cutting edge kid in the wedge, vif job supervision. To decide which can be best just for the needs you have, you need to produce a good comprehension of the advantages and limitations of each and every type of project management strategy. Below we all check to see precisely what is different about the two schools of thought, and assess side by side the huge benefits and limitations of each.
Design project supervision imitates the standard work process in any manufacturing or structure project since it is a sequential process. Each of the stages occurs in remote location, and once finished, the team begin the subsequent level in the range. Waterfall task management is dependent upon careful record making. This means there exists a sharp paper trek to adhere to, permitting the process to be white and increased in the future. From the beginning, the consumer will have a clear idea of what can be going to happen during job delivery. They may know roughly what the cost, timescales and size of the job will probably be, and will have a great idea of what to expect finally. Once a level in the method has been completed, at this time there is no way to return and change tasks without scrapping the whole job and beginning again. The full method relies upon sturdy first requirements; in cases where these kinds of are mistaken then the job is bound to failing from the outset. The product is created found in phases and only analyzed fully by the end, meaning bugs may be so historical eventually merchandise that they can be extremely hard to eliminate. Finally, this kind of job operations will not enable adjustments to the brief, consequently if the client understands they should improve the simple half way through, surrenders will need to always be made in conditions of finances and timescales.
Kbvkj was when recognized seeing that the perfect solution to many of this problems in waterfall task management. Rather than following a collection of measures in remote location, this procedure depends upon a great phased strategy to the project delivery. Project teams start off with an incredibly simple notion of where they are going, and then work with highly discreet web theme in short 'sprints'. At the end of each and every sprint the modules are tested to seek out any drive or blemishes and customer opinions is gathered prior to the next sprint occurs. Plenty of changes can be manufactured after preliminary plans happen to be developed, the truth is rewrites and major improvements are almost predicted. This will make it less difficult to add features and maintain abreast of changes in the industry, actually whilst the task will be shipped. Regular testing assures weak spots will be founded early on, which means that the item can be looking forward to release quicker and is very likely to be a quality item. This technique needs a good project manager to preserve stuff on track and balance out creativeness with task delivery requirements. Because of the haziness of the classic project program, the last product may often turn out to be wildly different to what was at first expected.
Each methods of job management both have their as well as place, plus the one which in turn is best for the needs will be based entirely in your specific simple. When you know very well what a final item should end up being and therefore are confident your client won't need to modification the opportunity half method through, waterfall project management is the friend. On the other hand in the event acceleration of production is more essential than the quality of the completed merchandise and the customer likes to be capable to transform scope throughout, agile provides for more flexibility and is definitely one of the things every single job supervisor ought to know about. To read more browse in this article restaurantalarz.com .


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Tag: stalwart
Buhari didn't order Nigerians to be shot summarily, says Tinubu
All Progressives Congress (APC) stalwart Asiwaju Bola Ahmed Tinubu yesterday cautioned against misinterpreting President Muhammadu Buhari's remarks on ballot box snatchers. Asiwaju Tinubu said the President never ordered security agencies...
Gunmen abduct APC stalwart in Bayelsa
Gunmen have kidnapped a stalwart of All Progressives Congress (APC) in Bayelsa State, Mr. Robert Desmond. The victim, a supporter of ex-Governor Timipre Sylva in Brass Local Government, is said...
How to make power devolution work, by APC stalwart
A Chieftain of the All Progressives Congress (APC)in Lagos State, Chief Remi Williams, has lent his support to the agitation for devotion of powers to the states. He said though devolution...
PDP stalwart, supporters, defect to APC in Enugu
A stalwart of the Peoples Democratic Party (PDP), Amechi Ozoemena and hundreds of his supporters, have defected to the All Progressives Congress in Enugu State. Ozoemena, who is an Abuja...
Bayelsa rerun: We'll checkmate PDP, says APC stalwart
A stalwart of the All Progressives Congress (APC) in Bayelsa State, Chief Bodi Arerebo, has said that his party was adequately prepared to resist alleged plot by the Peoples Democratic...
Who killed APC stalwart in Lagos?
Rights activists have called for an independent investigation to unravel the killers of a member of the All Progressives Congress (APC), Mr Eyitayo Peters, on Orile Road, Agege, a Lagos...


I have been a client at Body Sense since 2007. I initially came to the salon for a facial as I had read about Dermalogica products and wanted to try them. I have been coming back ever since, every month and my skin has never looked better. I suffered from acne through my teens and twenties but my skin is now more balanced and I rarely have breakouts anymore.
Kerry has been my regular therapist but I have seen the other girls when she was off on maternity leave and they are all excellent and knowledgeable. I have recently tried the Crystal Clear COMCIT facial which has delivered amazing results and I have also just had my sparse eyebrows filled in with micro blading. Bodysense employs fantastic therapists who are all well trained and knowledgeable about their products and services. I thoroughly recommend a visit!


Set of 5 vector bride illustrations with floral ornaments for wedding invitation cards, stationery designs, announcements and other wedding designs. Format: EPS stock vector clip art and illustrations. Free for download.
Set name: "Bride illustrations wedding" for Adobe Illustrator.
Theme tags: wedding invitations, bride and groom, wedding cards.
Posted on 06/09/2012, in Free Vector Graphics | Tags: cards, floral, invitation, templates, wedding.


For the old timers reading this, you remember TeleCon. Once a year in Anaheim, 20,000 or so of your industry friends would gather to learn about video conferencing, audio conferencing and collaboration.
The conference was electric (and Disneyland and Universal were soooo close)….lots of ideas, lots of stuff, lots of great people….wow!
Then….at some point earlier in this decade (a mental block is keeping me from remembering the exact year….it must be the enormous amounts of beer we consumed the day we were laid off), Pat Portway sold "TeleCon" to someone who used us for a year then trashed almost all of us. That's ok, TeleCon died shortly thereafter….what goes around, comes around.
Anyway….since then, a "unified" conference related to visual communications was hard to find (The Wainhouse Summits were great, but, on a smaller scale). And some of the big conferences (which were now dwindling in size) had video conferencing stuff as part of it, but, they were lacking the excitement for our cool technology.
This year (coming up October 10-12, 2011 in The Big Easy) is the Second Annual VCI-Group Conference that, to me, promises to bring back the old electricity, and focus on OUR industry, that TeleCon had way back in the day.
Keynote speakers will be there from: Polycom, Cisco, HP (?), and LifeSize. There will be a conference track and an exhibit hall and meeting people like you face to face in New Orleans…what could be better!???
1. Over the past few years (well maybe more than a few) conferences have been declining in attendance. What made you start up a conference at this time? I understand this is #2?
There are really no conferences out there that focus on the End User and the challenges they are having. It was an opportune time to begin an organization that was not associated with any manufacturer as we had found that with the emergence of telepresence (immersive today) most organizations were no longer a one (1) manufacturer shop. To this date 50% of our User base is new to conferencing – this is the only organization which enables these "Newbies" to learn, network and meet with many of the vendors in a one-to-one non-sales format.
2. Is this conference geared for "beginners" or all levels of knowledge…ie would I be totally bored or would I find cool useful stuff?
This conference focuses on all levels from the first timer to the experienced over 20 years in the industry. Since this industry is constantly evolving with mergers, equipment and technology many of our End Users come year after year.
3. Is it mostly for educators?
Absolutely not, our End User base is a mix of Enterprise (corporate), Healthcare, Government and Education. There are sessions that are geared to each of the verticals; as well as, Special Interest Groups (SIGs).
4. Are you going to offer virtual attendance at some of the keynotes, presentations, or the exhibit hall? Brainstorm: Imagine having the Giraffe or Vgo robots for "rent" and I could walk around the exhibit hall….THAT would be cool!
As much as we would be open to that, we have not been able to secure this for this event as yet. We are however doing a pre and post virtual conference through vcopius which will be the group's first look into virtual conference and we are very excited.
The Analyst panel on Wednesday afternoon will discuss the industry and the fact that it is currently a landmine and who will be the front runners in the industry.
6. I will look more, but, I see Polycom and Lifesize….what about Vidyo? Radvison? Cisco/Tandberg? VeaMea?
Cisco/Tandberg and HP will also be there. We feel that we have all the major manufacturers at this conference and we are very excited to have them there under one "umbrella" for our End Users to see their equipment, demonstrations; but, most importantly speak to them.
If we dust some more cobwebs from our old timer memories, we will recall that TeleCon was held in San Jose a couple of years before moving back to Annaheim.


Skype's Internet phone and video chat application for Apple's iPad was available late Monday, but was quickly pulled from Apple's online stores. Skype said in a Twitter message that it had gone live prematurely.
"We know you've been eagerly awaiting Skype for iPad and apologize for the inconvenience", the Luxembourg-based company said in a second message on Twitter.
It is not clear when Skype on iPad, which was reported to go live over a month ago, will now be available. A video of the application running on the iPad was posted online in June.
In a June interview with Apple blog TUAW, Skype's vice president of consumer and product design, Rick Osterloh said the iPad version will have chat, audio and video calling. A few features in the PC/Mac app such as file sharing did not make it into the iPad version because of the way it handles files, he added.
Microsoft said in May that it had agreed to acquire Skype in an $8.5 billion deal. It said it would continue to support Skype's product on rival platforms. Versions of the Skype application are already available on a variety of platforms including the Mac and iPhone.