text:
\section{Introduction}

The blackbody radiation temperature of the cosmic microwave background is $T_{0} = 2.725 \pm 0.002$ K at the present time \cite{Mather_1999}.
This temperature is regarded as the evidence for a hot Big Bang.
That hypothesis is supported by measurements of the radiation temperature--redshift $T$--$z$  relation \cite{Songaila_1994,Lu_1996,Ge_1997,Srianand_2000,Molaro_2002,Cui_2005,Srianand_2008,Luzzi_2009,Noterdaeme_2010,Noterdaeme_2011,Martino_2015,Luzzi_2015}.
The observations are consistent with a linear law, $T = T_{0} (1+z)$, as can be derived from standard cosmology, such as $\Lambda$CDM (lambda cold dark matter) models.
However, the $\Lambda$CDM model has several theoretical difficulties \cite{Weinberg1989,Weinberg1,Roy1,Bamba1_Miao1}, although it can elegantly explain the accelerated expansion of the late universe \cite{PERL1998ab,Riess1998_2004,Riess2007SN1,Suzuki_2012,Planck2013,Planck2015}.
To explain the acceleration, various models have been suggested,
such as $\Lambda (t)$CDM which assumes a time-varying cosmological term \cite{Freese-Lima,Sola_2009,Sola_2011a,Sola_2013b,Sola_2015c},
bulk viscous models which assume a bulk viscosity for the cosmological fluid \cite{Weinberg0,Murphy1,Barrow1986-Lima1988,Zimdahl1996-Nojiri2011},
and CCDM which assumes the creation of cold dark matter \cite{Lima_2000,Lima_1992b,Lima_1992e,Lima_2014b,Harko_2014,Harko_2013,Zimdahl_1993,Lima-Others1996-2008,Lima2010,Lima2010b,Lima2011,Lima2012,Jesus2014,Ramos_2014-2014b}.

The bulk viscous and CCDM models presume the existence of irreversible entropy in a homogeneous isotropic universe, unlike the $\Lambda$CDM and $\Lambda (t)$CDM models.
For example, in the CCDM model, irreversible entropy is generated from gravitationally induced particle creation in nonequilibrium thermodynamic states \cite{Prigogine_1988b,Prigogine1989}.
A possible equivalence of the bulk viscosity and matter creation dissipative mechanisms has been discussed in Ref.\ \cite{Lima_1992e}.
In addition, the connections between warm inflation \cite{Berera_1995-2014}, $\Lambda (t)$CDM models, and CCDM models have been debated in Ref.\ \cite{Connection}.
To examine the dissipative processes, a general radiative temperature law for adiabatic particle creation has been proposed by Lima \textit {et al.} \cite{Lima_2000,Lima_1992b,Lima_1992e,Lima_2014b}.
That law has been examined from various viewpoints \cite{Lima_2014b,Harko_2014,Harko_2013}.
In particular, a simple $T$--$z$ relation of the form $T = T_{0} (1+z)^{1 - \beta}$ is frequently compared with observations, where $\beta$ is a constant parameter (cf. Refs.\ \cite{Lima_2000,Noterdaeme_2011}).
The simple $T$--$z$ relation is obtained from a general radiative temperature law, if the effective equation-of-state parameter $w_{e}$ is constant \cite{Lima_2000,Lima_2014b}.
However, in the CCDM model, $w_{e}$ varies during the evolution of the universe.
Thus, a time-varying $w_{e}$ needs to be considered when the radiation temperature is discussed in the CCDM model.
However, the radiation temperature in the CCDM model has not yet been quantitatively examined from this viewpoint.
It is important to do so in order to acquire a deeper understanding of the CCDM model.

In the CCDM model, a \textit{negative sound speed} \cite{Lima2011} and the existence of \textit{clustered matter} \cite{Ramos_2014-2014b} are necessary to properly describe the growth rate for clustering related to structure formations.
Alternatively, a phenomenological dissipative model \cite{Koma7} has been proposed, in which the entropic force \cite{Easson12,Cai1-Costa1,Basilakos1,Sola_2014a,Koma4,Koma5,Koma6} is modified.
The model assumes constant terms that are equivalent to a nonzero cosmological constant $\Lambda$ and a dissipative process.
In previous work \cite{Koma7}, the dissipation rate was varied from $\tilde{\mu} =0$, corresponding to a nondissipative $\Lambda$CDM model, to $\tilde{\mu} =1$, corresponding to a fully dissipative CCDM model.
Low dissipation was found to correctly describe observations of structure formation.
The dissipation rate $\tilde{\mu}$ is expected to affect the $T$--$z$ relation, not only because the effective equation-of-state parameter $w_{e}$ depends on $\tilde{\mu}$, but also because $w_{e}$ varies during the evolution of the universe.

This model makes it possible to examine a dissipative universe systematically, ranging from a nondissipative $\Lambda$CDM model to a fully dissipative CCDM model.
To clarify the properties of the radiation temperature in a dissipative universe, the $T$--$z$ relation can be examined.
A general radiative temperature law is applied to the dissipative model, to formulate the $T$--$z$ relation.
Based on this formulation, the radiation temperature in the late universe is calculated numerically as a function of the rate of dissipation.
The present study provides new insights and a unique approach for examining a dissipative universe.

The remainder of the article is organized as follows.
In Sec.\ \ref{Thermodynamics}, the general radiative temperature law for adiabatic particle creation is briefly reviewed.
In Sec.\ \ref{Dissipation model}, a phenomenological modified dissipative model is proposed.
In Sec.\ \ref{Temperature law for the dissipation model}, the temperature law is applied to the modified model, and the $T$--$z$ relation is formulated.
In Sec.\ \ref{Results}, the radiation temperature in a dissipative universe is examined.
Finally, in Sec.\ \ref{Conclusions}, the conclusions are presented.
(The modified dissipative model is different from dissipative particle dark matter models examined in Ref.\ \cite{Lisa_Foot}.)

\section{General radiative temperature law for adiabatic particle creation}
\label{Thermodynamics}
In this section, the general radiative temperature law for adiabatic particle creation is reviewed, following the work of Lima \textit{et al.} \cite{Lima_2000,Lima_1992e,Lima_1992b,Lima_2014b}.
A homogeneous, isotropic, and spatially flat universe is initially considered.
The line element given by the Friedmann--Robertson--Walker (FRW) metric \cite{Lima_1992e,Lima_2014b} is
\begin{equation}
  d s^{2} = c^{2} dt^{2} - a^{2}(t) ( dr^{2} + r^{2} d \theta^{2} + r^{2} \sin^{2} \theta d \phi^{2} )
\label{eq:FRWmetric}
\end{equation}
where $c$ is the speed of light and $a(t)$ is the scale factor at time $t$.
The Hubble parameter $H$ is
\begin{equation}
   H \equiv   \frac{ \dot{a}(t) } {a(t)}  .
\label{eq:Hubble}
\end{equation}

In what follows, nonequilibrium thermodynamic states of the cosmological fluid in a FRW background are considered.
To this end, adiabatic particle creation is assumed \cite{Lima_1992e,Lima_1992b,Lima_2014b}.
The balance equations for the number of particles, entropy, and energy \cite{Lima_2014b} can then be written as
\begin{equation}
       \dot{n} + 3 H n   = n \Gamma   ,
\label{eq:NonEquil_1}
\end{equation}
\begin{equation}
       \dot{s} + 3 H s   = s \Gamma   ,
\label{eq:NonEquil_2}
\end{equation}
and
\begin{equation}
       \dot{\varepsilon} + 3 H ( \varepsilon + p + p_{c} )   = 0   ,
\label{eq:NonEquil_3}
\end{equation}
where $n$, $s$, $\varepsilon$, and $p$ are the particle number density, entropy density, energy density, and pressure, respectively.
Here, $\Gamma$ and $p_{c}$ are the particle production rate and the dynamic creation pressure, respectively.
When $\Gamma =0$ and $p_{c}=0$, the three balance equations reduce to the conservation law for equilibrium states in a standard cosmology.
Equation \ (\ref{eq:NonEquil_1}) can be rewritten as $\dot{N} / N = \Gamma$, using the total number $ N \propto n a^3 $ of particles in the comoving volume \cite{Lima_2014b}.
Keep in mind that the entropy per particle $\sigma = S/N$ is assumed to be constant \cite{Lima_1992b,Lima_2014b}, i.e., $\dot{\sigma} =0$, where $ S \propto s a^3 $ is the entropy in the comoving volume.
The constant value of $\sigma$ indicates $\dot{S} / S = \dot{N} / N = \Gamma$, which has been used for calculating the right-hand side of Eq.\ (\ref{eq:NonEquil_2}).
(Units are chosen so that $c = k_{B} =1$, unless otherwise stated. Here $k_{B}$ is the Boltzmann constant.)

The local Gibbs relation should be valid even for the nonequilibrium process considered here \cite{Lima_2014b}.
Accordingly, the thermodynamic quantities are related to the temperature $T$ by
\begin{equation}
         n k_{B} T d \left ( \frac{s}{n}    \right ) \equiv  n k_{B} T d\sigma  = d \varepsilon - \frac{\varepsilon + p}{n} dn      .
\label{eq:localGibbs}
\end{equation}
Substituting $\dot{\sigma} =0$ into Eq.\ (\ref{eq:localGibbs}), using both this result and Eq.\ (\ref{eq:NonEquil_1}), and rearranging Eq.\ (\ref{eq:NonEquil_3}), one obtains the dynamic creation pressure
\begin{equation}
        p_{c}  = - (\varepsilon + p) \frac{\Gamma}{3 H}  .
\label{eq:pc}
\end{equation}
The thermal evolution of matter creation can then be written as
\begin{equation}
       \frac{\dot{T}}{T}  =  \left ( \frac{\partial p}{\partial \varepsilon} \right )_{n}  \frac{\dot{n}}{n} =  \left ( \frac{\partial p}{\partial \varepsilon} \right )_{n} (\Gamma -3H) .
\label{eq:dotT_T}
\end{equation}
This equation has been derived by Lima \textit{et al.} \cite{Lima_2000,Lima_1992e,Lima_1992b,Lima_2014b}.
(More general formulations have been examined in Ref.\ \cite{Lima_2000}.
Harko has recently developed an equivalent formulation for a modified gravity theory with geometry-matter coupling \cite{Harko_2014}.)

For the nonequilibrium thermodynamic states considered here, a radiation temperature relation can be obtained from Eq.\ (\ref{eq:dotT_T}).
Substituting $p = \varepsilon / 3$ into Eq.\ (\ref{eq:dotT_T}), one finds
\begin{equation}
       \frac{\dot{T}}{T}  =    - \frac{\dot{a}}{a} +  \frac{\Gamma }{3}   \quad \textrm{(for radiation)} .
\label{eq:dotT_T_radiation}
\end{equation}
That can be rearranged as
\begin{equation}
       \frac{1}{aT} \frac{d (aT)}{dt}  =  \frac{\Gamma }{3}  \quad  \textrm{or} \quad      \frac{d (aT)}{aT}  =  \frac{\Gamma }{3} dt .
\label{eq:dotT_T_radiation2}
\end{equation}
Integrating Eq.\ (\ref{eq:dotT_T_radiation2}) from arbitrary time $t$ to the present time $t_{0}$ gives
\begin{equation}
     \int_{aT}^{a_{0}T_{0}}  \frac{d (a^{\prime} T^{\prime} )}{a^{\prime} T^{\prime} }  =  \int_{t}^{t_{0}}   \frac{\Gamma (t^{\prime}) }{3} dt^{\prime}  ,
\label{eq:dotT_T_radiation_int1}
\end{equation}
and therefore
\begin{equation}
     \ln  \left ( \frac{ a_{0}T_{0} }{ aT } \right )   =  \frac{1}{3} \int_{t}^{t_{0}}   \Gamma (t^{\prime})  dt^{\prime}
\label{eq:dotT_T_radiation_int2}
\end{equation}
where $a_{0}$ and $T_{0}$ are the present values of the scale factor and of the radiation temperature, respectively.
This equation can be rearranged as
\begin{equation}
     T = T_{0}  \left ( \frac{ a_{0} }{ a } \right ) \exp  \left [   - \frac{1}{3} \int_{t}^{t_{0}}   \Gamma (t^{\prime})  dt^{\prime} \right ]  ,
\label{eq:dotT_T_radiation_T-t}
\end{equation}
so that
\begin{equation}
     T = T_{0}  \left ( \frac{ a_{0} }{ a } \right ) \exp  \left [   - \frac{1}{3} \int_{\tilde{a}}^{1}   \Gamma (\tilde{a}^{\prime})  \frac{dt^{\prime}}{d\tilde{a}^{\prime}} d\tilde{a}^{\prime} \right ]  ,
\label{eq:dotT_T_radiation_T-a}
\end{equation}
or equivalently
\begin{equation}
     T = T_{0}   ( 1 + z ) \exp  \left [    \frac{1}{3} \int_{0}^{z}   \Gamma (z^{\prime})  \frac{dt^{\prime}}{dz^{\prime}} dz^{\prime} \right ]   ,
\label{eq:dotT_T_radiation_T-z}
\end{equation}
where the normalized scale factor $\tilde{a}$ and the redshift $z$ are
\begin{equation}
   \tilde{a} = \frac{a} { a_{0}}      \quad \textrm{and} \quad   z = \frac{ a_0 }{ a } -1   .
\label{eq:a_a0}
\end{equation}
Equations (\ref{eq:dotT_T_radiation_T-t}), (\ref{eq:dotT_T_radiation_T-a}), and (\ref{eq:dotT_T_radiation_T-z}) are the general radiative temperature law for adiabatic particle creation \cite{Lima_2014b}.
In this paper, that law is used to calculate the radiation temperature $T$ in a dissipative universe.
When $\Gamma =0$, Eq.\ (\ref{eq:dotT_T_radiation_T-z}) reduces to the linear law $T = T_{0}(1+z)$.
However, $\Gamma$ is not zero for the nonequilibrium states considered here.
Therefore, it is necessary to compute
$\int_{\tilde{a}}^{1}   \Gamma (\tilde{a}^{\prime})  ( dt^{\prime} / d\tilde{a}^{\prime} ) d\tilde{a}^{\prime} $ in Eq.\ (\ref{eq:dotT_T_radiation_T-a})
and $\int_{0}^{z}   \Gamma (z^{\prime})  (dt^{\prime} / dz^{\prime} ) dz^{\prime} $ in Eq.\ (\ref{eq:dotT_T_radiation_T-z}) to calculate $T$.
These integrals depend on the limits, the particle production rate, and the background evolution of the universe.
In Sec.\ \ref{Results}, the calculated $T$ is compared with observations in the late universe.
Accordingly, in the present study, the limits of integration in Eqs. (\ref{eq:dotT_T_radiation_T-t})--(\ref{eq:dotT_T_radiation_T-z}) correspond to the late universe.

A simple temperature relation is obtained from Eq.\ (\ref{eq:dotT_T_radiation}) for specific cases \cite{Lima_2000,Lima_2014b}.
For example, if $\Gamma = 3 \beta H = 3 \beta (\dot{a}/a)$, integrating Eq.\ (\ref{eq:dotT_T_radiation}) gives
\begin{equation}
       T = T_{0} \left ( \frac{a_{0}}{a} \right )^{1-\beta}   = T_{0}(1+z)^{1-\beta}
\label{eq:dotT_T_radiation_T-z_beta}
\end{equation}
where $\beta$ is taken to be constant.
This simple temperature relation has been studied in detail \cite{Lima_2000}.
In contrast, here a general radiative temperature law is considered for a dissipative universe.

\section{Modified dissipative model with constant terms}
\label{Dissipation model}

Entropic cosmology has been proposed to explain the accelerated expansion of the universe \cite{Easson12}.
In the entropic-force model, the horizon of the universe is assumed to have an associated entropy and an approximate temperature due to information holographically stored there \cite{Easson12}.
Recently, various entropic-force models have been examined in detail \cite{Easson12,Cai1-Costa1,Basilakos1,Sola_2014a,Koma4,Koma5,Koma6,Koma7}.
For example, in place of the Bekenstein entropy \cite{Bekenstein1}, the Tsallis-Cirto entropy \cite{Tsallis2012} based on nonextensive statistics \cite{Tsa0} has been applied to the horizon of the universe \cite{Koma5,Koma6}.
Basilakos \textit{et al.} \cite{Basilakos1,Sola_2014a} have shown that simple combinations of pure Hubble terms, such as $H^{2}$, $\dot{H}$, and $H$, are insufficient for a complete description of the cosmological data \cite{Koma7}.
Therefore, the constant term plays an important role.
Accordingly, a phenomenological entropic-force model that includes constant terms, an irreversible entropy $S_{\textrm{irr}}$, and a kind of reversible entropy $S_{\textrm{rev}}$ has been proposed \cite{Koma7}.
Using this model, a dissipative universe can be analyzed systematically, over the entire range from a nondissipative $\Lambda$CDM model to a fully dissipative CCDM model.
In the present work, it is called the modified dissipative model.
In what follows, it is briefly reviewed according to Ref.\ \cite{Koma7}.
Keep in mind that the entropic-force considered here is different from the idea that gravity itself is an entropic force \cite{Padma1,Verlinde1}.

The Friedmann, acceleration, and continuity equations for the modified dissipative model become
\begin{equation}
   H^{2}  = \frac{ 8\pi G }{ 3 } \rho     +  \alpha  H_{0}^{2}   ,
\label{eq:FRW01(g)}
\end{equation}
\begin{align}
  \frac{ \ddot{a}  }{ a }    &=  -  \frac{ 4\pi G }{ 3 } \left ( \rho  + \frac{3p}{c^2}  \right )                  +  \alpha  H_{0}^{2} +  \gamma_{\textrm{irr}}  H_{0}^{2}      \notag \\
                                    &=  -  \frac{ 4\pi G }{ 3 } \left ( \rho  + \frac{3           p_{e}  }{c^2}  \right )  + \alpha  H_{0}^{2}                        ,
\label{eq:FRW02(g)}
\end{align}
and
\begin{equation}
       \dot{\rho} + 3  \frac{\dot{a}}{a} \left (  \rho + \frac{p_{e}}{c^2}  \right )   = 0   ,
\label{eq:drho_General_01_pe}
\end{equation}
with two dimensionless constants
\begin{equation}
   \alpha     \geq 0   \quad  \textrm{and}  \quad     \gamma_{\textrm{irr}}    \geq 0   .
\end{equation}
Here $G$, $H_{0}$, and $\rho$ are the gravitational constant, the Hubble parameter at the present time, and the \textit{mass} density of the cosmological fluid, respectively \cite{Koma7}.
The mass density is $\rho \equiv \varepsilon /c^{2}$.
The effective pressure $p_{e}$ in Eqs. (\ref{eq:FRW02(g)}) and (\ref{eq:drho_General_01_pe}) is
\begin{equation}
    p_{e}   =  p + p_{c}
\end{equation}
where $p_{c}$ is a pressure derived from irreversible entropy related to dissipative processes.
In this study, $p_{c}$ is taken to be equivalent to the dynamic creation pressure in the CCDM model.
(For $\Lambda (t)$CDM models, nonzero terms related to reversible entropy appear on the right-hand side of the continuity equation \cite{Koma7}.
That model involves the transfer of energy between two fluids \cite{Barrow22,Amendola1Zimdahl01,Wang0102_YWang2014,Pavon_2005}.)

The modified dissipative model is used to calculate a radiation temperature $T$ from the general radiative temperature law in Eqs. (\ref{eq:dotT_T_radiation_T-t})--(\ref{eq:dotT_T_radiation_T-z}).
In Sec.\ \ref{Results}, the $T$--$z$ relation in the late universe is examined, to compare the calculated value of $T$ with observations.
The pressure of the cosmological fluid in the present model is negligible, $p = 0$.
A matter-dominated universe is assumed.
Consequently, the effective pressure $p_{e}$ is
\begin{equation}
   p_{e} =  p + p_{c} = p_{c} = - \frac{  c^{2} H_{0}^{2}} {  4\pi G }  \gamma_{\textrm{irr}}  .
\label{eq:pe_Q_1_alpha4}
\end{equation}
On the other hand, substituting $p_{c} = p_{e}$, $p = 0$, and $\varepsilon = \rho c^{2}$ into Eq.\ (\ref{eq:pc}), one obtains
\begin{equation}
  p_{e}   =   - \rho c^{2}   \frac{\Gamma}{3H}       .
\label{eq:pe_Gamma}
\end{equation}
Accordingly, from Eqs.\ (\ref{eq:pe_Q_1_alpha4}) and (\ref{eq:pe_Gamma}),
\begin{equation}
    \Gamma = \frac{3H}{4 \pi G}  \frac{  \gamma_{\textrm{irr}}  H_{0}^{2} }{ \rho }        .
\label{eq:Gamma_0_alpha4}
\end{equation}

Assume that the $\alpha  H_{0}^{2}$ terms in Eqs.\ (\ref{eq:FRW01(g)}) and (\ref{eq:FRW02(g)}) are equivalent to the cosmological constant $\Lambda /3$ in the standard $\Lambda$CDM model.
That is, $dS_{\textrm{rev}}=0$ for homogeneous systems in CCDM models \cite{Prigogine_1988b,Harko_2014}.
In contrast, $p_{e}$ is related to the irreversible entropy in dissipative processes because $p_{e} =p_{c}$.
In other words, the $\gamma_{\textrm{irr}}  H_{0}^{2}$ term in Eq.\ (\ref{eq:FRW02(g)}) is related to the irreversible entropy.
Accordingly, the preceding cosmological equations are equivalent to those for an extended $\Lambda$CDM model in a dissipative universe.
Therefore, the $\alpha  H_{0}^{2}$ term is interpreted as a modification of the Einstein tensor.
In contrast, $p_{e}$ is a modification of the energy--momentum tensor of the Einstein equation  \cite{Koma7}.
The properties of this model are equivalent to those of the modified entropic-force model examined in Ref.\ \cite{Koma7}.

Combining Eqs.\ (\ref{eq:FRW01(g)}) and (\ref{eq:FRW02(g)}), and using Eq.\ (\ref{eq:pe_Q_1_alpha4}), one finds
\begin{equation}
 \dot{H}    = -  C_{m} H^2    +   C_{ag}  H_{0}^{2}
\label{eq:dHC1C3hC4h(Cst)}
\end{equation}
where $C_{m}$ and $C_{ag}$ are dimensionless constants \cite{Koma7}
\begin{equation}
     C_{m} = 1.5   \quad   \textrm{and}  \quad  {C}_{ag}  =  \frac{ 3 \alpha + 2 \gamma_{\textrm{irr}}  }{   2   }    .
\label{eq:Cm=1.5_C4}
\end{equation}
Here $C_{m} =1.5$ corresponds to a matter-dominated universe in standard cosmology \cite{Weinberg1,Roy1}.
Solving Eq.\ (\ref{eq:dHC1C3hC4h(Cst)}), one obtains
\begin{align}
 \left ( \frac{H} {H_{0}} \right )^{2}   &=  \left ( 1-  \tilde{\Omega}_{\Lambda} \right )   \tilde{a}^{ - 3}   +  \tilde{\Omega}_{\Lambda}   \notag \\
                                                    &=                                      \tilde{\Omega}_{m}    \tilde{a}^{ - 3}   +  \tilde{\Omega}_{\Lambda}
\label{eq:H/H0(C1C4)_02}
\end{align}
where $\tilde{a}$ = $ a / a_{0}$ in Eq.\ (\ref{eq:a_a0}).
The two constant parameters are defined as
\begin{equation}
 \tilde{\Omega}_{\Lambda} \equiv  \frac{ C_{ag} }{ C_{m}  }   \quad   \textrm{and}   \quad  \tilde{\Omega}_{m}  \equiv  1 - \tilde{\Omega}_{\Lambda} .
\label{eq:tOmega_L}
\end{equation}
This solution is the same as that in the standard $\Lambda$CDM model \cite{Koma6,Koma7}.
Accordingly, the constant term $\tilde{\Omega}_{\Lambda}$ behaves as if it were $\Omega_{\Lambda}$ in the standard $\Lambda$CDM model.
Similarly, $\tilde{\Omega}_{m}$ behaves as if it were $\Omega_{m}$.
Here $\Omega_{m}$ and $\Omega_{\Lambda}$ are the density parameters for matter and for $\Lambda$, respectively.
The density parameter $\Omega_{r}$ for radiation is neglected in the late universe.

To study a dissipative universe quantitatively, define a dissipation rate \cite{Koma7}
\begin{equation}
      \tilde{\mu}  \equiv  \frac{ \gamma_{\textrm{irr}} }{ C_{ag}  }     =  \frac{ \tilde{\Omega}_{D} }{ \tilde{\Omega}_{\Lambda} }
\label{eq:dissipation_0}
\end{equation}
where $\tilde{\Omega}_{D}$ is a constant parameter related to dissipative processes,
\begin{equation}
      \tilde{\Omega}_{D}  \equiv  \frac{ \gamma_{\textrm{irr}} }{ C_{m} }      .
\label{eq:Omega_D_0}
\end{equation}
As discussed in Ref.\ \cite{Koma7}, when $\gamma_{\textrm{irr}} = 0 $, one obtains $\tilde{\mu} =0$ from Eq.\ (\ref{eq:dissipation_0}).
In this case, the present model is equivalent to the standard nondissipative $\Lambda$CDM model.
In contrast, when $\alpha =0$, one obtains $\tilde{\mu} =1$ from Eq.\ (\ref{eq:dissipation_0}) because $C_{ag} = \frac{ 3 \alpha + 2 \gamma_{\textrm{irr}}  }{2} = \gamma_{\textrm{irr}} $.
In that case, the present model is equivalent to the fully dissipative CCDM model proposed by Lima \textit{et al.} \cite{Lima2010}.
That is, $\tilde{\mu} =0$ corresponds to the nondissipative $\Lambda$CDM model, whereas $\tilde{\mu} =1$ corresponds to the fully dissipative CCDM model.
In this way, the extent of the dissipative universe is determined by the dissipation rate $\tilde{\mu}$.

Next, consider the effective equation-of-state parameter in the modified dissipative model.
From Eq.\ (\ref{eq:pe_Gamma}), $w_{e}$ becomes
\begin{equation}
  w_{e} \equiv \frac{p_{e}}{\rho c^{2}}      = - \frac{\Gamma}{3H}  .
\label{eq:we_p_alpha4}
\end{equation}
After some algebra, one finds
\begin{equation}
w_{e} = - \frac{  \tilde{\Omega}_{D}  \tilde{a}^{3}    }{    1-  \tilde{\Omega}_{\Lambda}     +  \tilde{\Omega}_{D}  \tilde{a}^{3}     }
        = - \frac{  \tilde{\Omega}_{D}  \tilde{a}^{3}    }{                   \tilde{\Omega}_{m}     +  \tilde{\Omega}_{D}  \tilde{a}^{3}     }
\label{eq:we-a_alpha4}
\end{equation}
where $\tilde{a}$ is the normalized scale factor $a/a_{0}$.
For details, see Ref.\ \cite{Koma7}.
Note that $w_{e}$ is not equal to the equation-of-state parameter $w$ for a generic component of matter, because $w$ is zero in a matter-dominated universe, for which $p=0$.
(A more general inhomogeneous equation of state has been examined in Ref.\ \cite{Odintsov_2006_b}.)

\section{Radiation temperature in the modified dissipative model}
\label{Temperature law for the dissipation model}

The general radiative temperature law in Eq.\ (\ref{eq:dotT_T_radiation_T-a}) is rearranged as
\begin{align}
     T  &= T_{0}  \left ( \frac{ a_{0} }{ a } \right ) \exp  \left [   - \frac{1}{3} \int_{\tilde{a}}^{1}   \Gamma (\tilde{a}^{\prime})  \frac{dt^{\prime}}{d\tilde{a}^{\prime}} d\tilde{a}^{\prime} \right ]   \notag \\
        &= T_{0}  ( 1 + z )                                   \exp  \left [   - \frac{1}{3} \int_{\tilde{a}}^{1}   \Gamma (\tilde{a}^{\prime})  \frac{dt^{\prime}}{d\tilde{a}^{\prime}} d\tilde{a}^{\prime} \right ]                  .
\label{eq:dotT_T_radiation_T-a_z}
\end{align}

First consider the calculation of $dt/d\tilde{a}$ in Eq.\ (\ref{eq:dotT_T_radiation_T-a_z}).
(For simplicity, the prime is omitted.)
As discussed in Sec.\ \ref{Dissipation model}, the background evolution in the modified dissipative model is the same as that in the standard $\Lambda$CDM model.
Accordingly, solutions of the standard $\Lambda$CDM model are used for a spatially flat universe.
The solution \cite{Sola_2009,Matsubara1} is
\begin{equation}
 \tilde{a} (t) =  \left (   \frac{ \tilde{\Omega}_{m}  }{  \tilde{\Omega}_{\Lambda}  } \right )^{1/3}   \sinh^{2/3} \left (    \frac{  3 H_{0} \sqrt{ \tilde{\Omega}_{\Lambda} } t }{2}    \right )
\label{eq:a-t_1}
\end{equation}
and equivalently
\begin{equation}
  t (\tilde{a}) =  \frac{2}{  3 H_{0} \sqrt{ \tilde{\Omega}_{\Lambda} }  }    \sinh^{-1} \left (    \sqrt { \frac{ \tilde{\Omega}_{\Lambda}  }{  \tilde{\Omega}_{m}  } }  \tilde{a}^{3/2} \right )   .
\label{eq:t-a_1}
\end{equation}
The density parameter for radiation is negligible in this late universe.
Differentiating Eq.\ (\ref{eq:a-t_1}) with respect to $t$, substituting Eq.\ (\ref{eq:t-a_1}) into the result, and rearranging, one obtains
\begin{equation}
   \frac{dt}{ d \tilde{a} } = \frac{  \sinh^{1/3} g(\tilde{a})  \cosh^{-1} g(\tilde{a})  }{  H_{0} \tilde{\Omega}_{\Lambda}^{1/6}  \tilde{\Omega}_{m}^{1/3}    }
\label{eq:dtda_1}
\end{equation}
where
\begin{equation}
    g(\tilde{a}) = \sinh^{-1}  \left (  \sqrt{   \frac{\tilde{\Omega}_{\Lambda}}{\tilde{\Omega}_{m}}  }   \tilde{a}^{3/2}    \right )    .
\label{eq:g(a)_1}
\end{equation}
Next, rearrange $\Gamma$ in Eq.\ (\ref{eq:dotT_T_radiation_T-a_z}).
Using Eq.\ (\ref{eq:we_p_alpha4}), it can be written as
\begin{equation}
\Gamma = - 3H w_{e} = - 3 \left ( \frac{H}{H_{0}} \right ) H_{0} w_{e}    .
\label{eq:Gamma3Hwe_1}
\end{equation}
Substituting Eqs.\ (\ref{eq:H/H0(C1C4)_02}) and (\ref{eq:we-a_alpha4}) into Eq.\ (\ref{eq:Gamma3Hwe_1}) leads to
\begin{equation}
 \Gamma =  \frac{   3  H_{0} \tilde{\Omega}_{D}  \sqrt{ \tilde{\Omega}_{m}  \tilde{a}^{ - 3}   +  \tilde{\Omega}_{\Lambda} }   }{  \tilde{\Omega}_{m}  \tilde{a}^{-3} +  \tilde{\Omega}_{D}   }   .
\label{eq:Gamma_2}
\end{equation}

Substituting Eqs.\ (\ref{eq:dtda_1}) and (\ref{eq:Gamma_2}) into Eq.\ (\ref{eq:dotT_T_radiation_T-a_z}), one finds
\begin{align}
T  = & T_{0}  (  1 + z )           \exp     \left [   - \frac{1}{3} \int_{\tilde{a}}^{1}   \Gamma (\tilde{a}^{\prime})  \frac{dt^{\prime}}{d\tilde{a}^{\prime}} d\tilde{a}^{\prime} \right ]   \notag \\
    = & T_{0}  (  1 + z )
                                          \exp     \Biggl[  \frac{  - \tilde{\Omega}_{D}   }{  \tilde{\Omega}_{\Lambda}^{1/6}   \tilde{\Omega}_{m}^{1/3}   }    \int_{\tilde{a}}^{1}  K(\tilde{a}^{\prime})   d\tilde{a}^{\prime}  \Biggr]
\label{eq:dotT_T_radiation_T-a_z_2}
\end{align}
where
\begin{equation}
  K(\tilde{a})  =  \frac{   \sqrt{ \tilde{\Omega}_{m}  \tilde{a}^{ - 3}   +  \tilde{\Omega}_{\Lambda} }   }{  \tilde{\Omega}_{m}  \tilde{a}^{-3} +  \tilde{\Omega}_{D}   }     \sinh^{1/3} g(\tilde{a})  \cosh^{-1} g(\tilde{a})
\label{eq:K}
\end{equation}
and $g(\tilde{a})$ is given by Eq.\ (\ref{eq:g(a)_1}).
Equation\ (\ref{eq:dotT_T_radiation_T-a_z_2}) is the radiation temperature--redshift relation for the modified dissipative model in the late universe.
The influence of dissipation is included in $\tilde{\Omega}_{D}$.
Using Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}), the radiation temperature can be determined as a function of the dissipation rate  $\tilde{\mu}$
where $\tilde{\mu}$ = $\tilde{\Omega}_{D} / \tilde{\Omega}_{\Lambda}$  from Eq.\ (\ref{eq:dissipation_0}).

As a specific case, a simple temperature relation can be obtained from Eq.\ (\ref{eq:dotT_T_radiation}).
For example, if $w_{e} = - \Gamma / (3H)$ from Eq.\ (\ref{eq:we_p_alpha4}) is assumed to be constant, then
\begin{equation}
       T = T_{0}(1+z)^{1 + w_{e}}           .
\label{eq:dotT_T_radiation_T-z_we}
\end{equation}
This equation is equivalent to Eq.\ (\ref{eq:dotT_T_radiation_T-z_beta}), replacing $w_{e}$ by $ - \beta$.
In general, $w_{e}$ is not constant in a dissipative universe, as examined in the next section.
Accordingly, Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}) plays an important role in studying the dissipative universe.

\section{Evolution of the radiation temperature in a dissipative universe}
\label{Results}

\begin{figure} [b]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.31}{\includegraphics{fig1.eps}}
\end{center}
\end{minipage}
\caption{ (Color online). Dependence of the luminosity distance $d_L$ on the redshift $z$.
The three continuous curves represent the modified dissipative model for $\tilde{\Omega}_{\Lambda} = 0$, $0.685$, and $1$.
They are respectively equivalent to the standard $\Lambda$CDM model for $(\Omega_{m}, \Omega_{\Lambda}) = (1, 0), (0.315, 0.685)$, and $(0, 1)$ in a spatially flat universe.
The closed diamonds with error bars are supernova data \cite{Riess2007SN1}, for which $H_{0}$ is $67.3$ km/s/Mpc based on Planck 2013 results \cite{Planck2013}.
A similar $d_L$--$z$ relation has been discussed in entropic cosmology \cite{Koma4,Koma5,Koma6}.   }
\label{Fig-dL-z}
\end{figure}

\begin{figure} [t]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.3}{\includegraphics{fig2.eps}}
\end{center}
\end{minipage}
\caption{ (Color online). Dependence of the effective equation-of-state parameter $w_{e}$ on the redshift $z$ for the indicated dissipation rates $\tilde{\mu}$.
Zero dissipation corresponds to a nondissipative $\Lambda$CDM model, whereas $\tilde{\mu} =1$ corresponds to a fully dissipative CCDM model.
The background evolution of the universe in each case is equivalent to that in the fine-tuned standard $\Lambda$CDM model because $\tilde{\Omega}_{\Lambda}  = \Omega_{\Lambda} = 0.685$.
The dependence of $w_e$ on $\tilde{a}$ has been discussed in Ref.\ \cite{Koma7}.     }
\label{Fig-we-z}
\end{figure}

\begin{figure} [t]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.3}{\includegraphics{fig3.eps}}
\end{center}
\end{minipage}
\caption{ (Color online). The radiation temperature--redshift relation.
The vertical axis is $T/ (1+z)$.
The continuous curves represent the modified dissipative model for $\tilde{\mu} =0$, $0.05$, $0.1$, $0.5$, and $1$.
The background evolution of the universe for each value of $\tilde{\mu}$ is the same because $\tilde{\Omega}_{\Lambda}  = 0.685$.
The symbols with error bars are observed data points \ \cite{Mather_1999,Ge_1997,Srianand_2000,Molaro_2002,Cui_2005,Srianand_2008,Luzzi_2009,Noterdaeme_2010,Noterdaeme_2011,Martino_2015}.
The original data represented by the green squares, blue triangles, and red circles are from Refs.\ \cite{Ge_1997,Srianand_2000,Molaro_2002,Cui_2005}, \cite{Luzzi_2009}, and \cite{Srianand_2008,Noterdaeme_2010,Noterdaeme_2011}, respectively.
The recent data represented by the pink diamonds at low redshifts are from Ref.\ \cite{Martino_2015}.
The data point for $z=0$ is from Ref. \cite{Mather_1999}.
For the observed data points, see Table \ref{tab-T-z_obs} and Ref.\ \cite{Noterdaeme_2011}.
Note that several error bars shown here are asymmetric because the corresponding errors shown in Table \ref{tab-T-z_obs} have been asymmetric.  }
\label{Fig-T-z}
\end{figure}

\begin{table}[t]
\caption{ The observed temperature--redshift relation.
The measured values are based on the analysis of a fine structure of atomic carbon \cite{Ge_1997,Srianand_2000,Molaro_2002,Cui_2005}, the Sunyaev-Zel'dovich effect \cite{Luzzi_2009}, the rotational excitation of CO molecules \cite{Srianand_2008,Noterdaeme_2010,Noterdaeme_2011},
and a sample of X-ray selected clusters \cite{Martino_2015}.
The temperature at $z = 0$ is taken from Ref.\ \cite{Mather_1999}.     }
\label{tab-T-z_obs}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{1.5pc}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}lllll}
\hline
  $z$        &  $T \quad (\textrm{K})$                  &  $\textrm{Refs.}$      \\
\hline
\hline
$0.000$    & $2.725 \pm 0.002$          &  \cite{Mather_1999} \\
\hline
$0.023$    & $2.72   \pm 0.10$            &  \cite{Luzzi_2009} \\
$0.152$    & $2.90   \pm 0.17$            &  \cite{Luzzi_2009} \\
$0.183$    & $2.95   \pm 0.27$            &  \cite{Luzzi_2009} \\
$0.200$    & $2.74   \pm 0.28$            &  \cite{Luzzi_2009} \\
$0.202$    & $3.36   \pm 0.20$            &  \cite{Luzzi_2009} \\
$0.216$    & $3.85   \pm 0.64$            &  \cite{Luzzi_2009} \\
$0.232$    & $3.51   \pm 0.25$            &  \cite{Luzzi_2009} \\
$0.252$    & $3.39   \pm 0.26$            &  \cite{Luzzi_2009} \\
$0.282$    & $3.22   \pm 0.26$            &  \cite{Luzzi_2009} \\
$0.291$    & $4.05   \pm 0.66$            &  \cite{Luzzi_2009} \\
$0.451$    & $3.97   \pm 0.19$            &  \cite{Luzzi_2009} \\
$0.546$    & $3.69   \pm 0.37$            &  \cite{Luzzi_2009} \\
$0.550$    & $4.59   \pm 0.36$            &  \cite{Luzzi_2009} \\
\hline
$0.042$     & $2.857   \pm 0.018$        &  \cite{Martino_2015} \\
$0.077$     & $2.953   \pm 0.021$        &  \cite{Martino_2015} \\
$0.123$     & $3.072   \pm 0.026$        &  \cite{Martino_2015} \\
$0.169$     & $3.162   \pm 0.020$        &  \cite{Martino_2015} \\
$0.222$     & $3.326   \pm 0.015$        &  \cite{Martino_2015} \\
$0.274$     & $3.495   \pm 0.016$        &  \cite{Martino_2015} \\
\hline
$1.729$    & $7.5^{+ 1.6}_{-1.2}$           &  \cite{Noterdaeme_2011} \\
$1.774$    & $7.8^{+0.7}_{-0.6}$            &  \cite{Noterdaeme_2011} \\
$2.038$    & $8.6^{+1.1}_{-1.0}$            &  \cite{Noterdaeme_2011} \\
$2.418$    & $9.15   \pm   0.7$            &  \cite{Srianand_2008,Noterdaeme_2011} \\
$2.690$    & $10.5^{+0.8}_{-0.6}$          &  \cite{Noterdaeme_2010,Noterdaeme_2011} \\
\hline
$1.777$      & $7.2   \pm  0.8$            &   \cite{Cui_2005}         \\
$1.973$      & $7.9   \pm  1.0$            &   \cite{Ge_1997}          \\
$2.337$      & $10    \pm  4$              &   \cite{Srianand_2000} \\
$3.025$      & $12.1^{+1.7}_{-3.2}$       &   \cite{Molaro_2002}    \\
\hline
\hline
\end{tabular}\\
 \end{table}

The dissipation rate $\tilde{\mu}$ from Eq.\ (\ref{eq:dissipation_0}) is
\begin{equation}
      \tilde{\mu}  \equiv  \frac{ \gamma_{\textrm{irr}} }{ C_{ag}  }     =  \frac{ \tilde{\Omega}_{D} }{ \tilde{\Omega}_{\Lambda} }      .
\label{eq:dissipation_1}
\end{equation}
Zero dissipation corresponds to a nondissipative $\Lambda$CDM model, whereas $\tilde{\mu} =1$ corresponds to a fully dissipative CCDM model.
As discussed in Ref.\ \cite{Koma7}, $\tilde{\Omega}_{\Lambda}$ can be determined from the background evolution of the universe.
Accordingly, $\tilde{\Omega}_{\Lambda}$ = $\Omega_{\Lambda}$ from a fine-tuned standard $\Lambda$CDM model.
Consider a spatially flat universe in which $(\Omega_{m}, \Omega_{\Lambda}) = (0.315, 0.685)$ based on the Planck 2013 results \cite{Planck2013}.
That is, let $\tilde{\Omega}_{\Lambda} =\Omega_{\Lambda}  = 0.685$ \cite{Koma7}.
To confirm the background evolution of the universe in the present dissipative model, consider the luminosity distance $d_{L}$ \cite{Sato1} given by
\begin{equation}
  \left ( \frac{ H_{0} }{ c } \right )   d_{L}      =   (1+z)  \int_{1}^{1+z}  \frac{dy} { F(y) }    .
\label{eq:dL_00}
\end{equation}
The integrating variable $y$ and the function $F(y)$ are given by
\begin{equation}
  y = \frac{a_0} {a} = \tilde{a}^{-1}  \quad \textrm{and} \quad    F(y)  = \frac{ H }{ H_{0} }
\end{equation}
where $H/H_{0}$ is from Eq.\ (\ref{eq:H/H0(C1C4)_02}).
As shown in Fig.\ \ref{Fig-dL-z}, it is found that $d_{L}$ for $\tilde{\Omega}_{\Lambda} = 0.685$ agrees with the supernova data.
($\tilde{\Omega}_{\Lambda}  = 0.685$ used here is approximately equivalent to $\Omega_{\Lambda}  = 0.691$ for the recent Planck 2015 results \cite{Planck2015}.
The influence of $\tilde{\Omega}_{\Lambda}$ is investigated later.)

To examine the influence of the dissipation rate, $\tilde{\mu}$ is set to several typical values, $0$, $0.05$, $0.1$, $0.5$, and $1.0$ in turn.
The background evolution of the universe in each case is equivalent to that in the fine-tuned standard $\Lambda$CDM model.
As shown in Fig.\ \ref{Fig-dL-z}, the background evolution agrees with the observed supernova data because $\tilde{\Omega}_{\Lambda} =\Omega_{\Lambda}  = 0.685$.

Before studying the radiation temperature, the evolution of the effective equation-of-state parameter $w_{e}$ is examined.
To this end, $w_e$ for various values of $\tilde{\mu}$ is plotted in Fig.\ \ref{Fig-we-z}, where $w_e$ is calculated from Eqs.\ (\ref{eq:we-a_alpha4}) and (\ref{eq:a_a0}).
In this figure, $\tilde{\mu} =0$ corresponds to a nondissipative $\Lambda$CDM model, whereas $\tilde{\mu} =1$ corresponds to a fully dissipative CCDM model.
As shown in Fig.\ \ref{Fig-we-z}, $w_{e}$ for $\tilde{\mu} =0$ is always equal to $0$ because $p_{e} = 0$.
However, $w_{e}$ decreases with increasing $\tilde{\mu}$.
It is found that the dissipation rate $\tilde{\mu}$ affects $w_{e}$ even if the background evolution of the universe is not altered.
In addition, $w_{e}$ for $\tilde{\mu} >0$ gradually decreases with decreasing $z$ and eventually approaches $-1$.
That is, $w_{e}$ is not constant when $\tilde{\mu} >0$.
The varying value of $w_{e}$ indicates that the simple $T$--$z$ relation from Eq.\ (\ref{eq:dotT_T_radiation_T-z_we}) is not suited to describe the radiation temperature in the present model.
Accordingly, Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}) plays an important role.

Next, the radiation temperature in the modified dissipative model is examined for various values of $\tilde{\mu}$.
The radiation temperature $T$ in the late universe is numerically calculated from Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}).
For this calculation, the redshift is varied from $z= -0.98$ to $4$, i.e., the normalized scale factor is varied between $\tilde{a} =50$ and $\tilde{a} =0.2$.
To compare with observations, the data points are taken from Refs.\ \cite{Mather_1999,Ge_1997,Srianand_2000,Molaro_2002,Cui_2005,Srianand_2008,Luzzi_2009,Noterdaeme_2010,Noterdaeme_2011,Martino_2015}.
(The values are summarized in Table \ref{tab-T-z_obs}.
For details, see, e.g., Ref.\ \cite{Noterdaeme_2011}.)
In Fig.\ \ref{Fig-T-z}, the vertical axis is $T/ (1+z)$ so that a horizontal line can correspond to a linear law $T= T_{0} (1+z)$ in equilibrium.
From this figure, it is found that the radiation temperature $T$ for $\tilde{\mu} =0$ obeys the linear law and is consistent with the observed data.
However, with increasing dissipation $\tilde{\mu}$, $T$ gradually deviates from both the linear law and the observations.
In particular, $T$ for $\tilde{\mu} =1$ is far off the observed data points, due to the nonequilibrium dissipative processes.
Accordingly, a fully dissipative universe is constrained even if the background evolution of the universe is the same.
On the other hand, $T$ for $\tilde{\mu} =0.05$ agrees with the observed data points.
This agreement implies that a weakly dissipative universe fits the observed $T$-$z$ relation.
The weakly dissipative universe is discussed later.

In the preceding discussion, $\tilde{\Omega}_{\Lambda} = \Omega_{\Lambda} = 0.685$.
Accordingly, the background evolution of the universe is the same.
Consequently, a weakly dissipative model is consistent with observations.
Finally, to examine the influence of both $\tilde{\mu}$ and $\tilde{\Omega}_{\Lambda}$, a likelihood analysis for the radiation temperature is performed.
(A similar analysis was performed in Ref.\ \cite{Koma7}, in which a growth rate for clustering was examined.)
For this purpose, $\tilde{\mu}$ and $\tilde{\Omega}_{\Lambda}$ are treated as free parameters.
The chi-squared function for the radiation temperature then becomes
\begin{equation}
\chi^{2}_{\textrm{RT}} (\tilde{\Omega}_{\Lambda}, \tilde{\mu})
= \sum\limits_{i=1}^{29} { \left[ \frac{
                                                         T_{\textrm{obs}} (z_{i} ) - T_{\textrm{cal}} (z_{i}, \tilde{\Omega}_{\Lambda}, \tilde{\mu} )  }{ \sigma_{i}^{\textrm{RT}} }
                                     \right]^{2}   }
\label{eq:chi_mu}
\end{equation}
where $ T_{\textrm{obs}} (z_{i} )$ and $ T_{\textrm{cal}} (z_{i}, \tilde{\Omega}_{\Lambda}, \tilde{\mu} ) $ are the observed and calculated radiation temperatures, respectively, and $\sigma_{i}^{\textrm{RT}}$ is the uncertainty in the observed temperature.
The observed data points (numbered $i=1$ to $29$) are summarized in Table\ \ref{tab-T-z_obs}.
For the likelihood analysis, $\tilde{\Omega}_{\Lambda}$ and $\tilde{\mu}$ are sampled in the range 0 to 1 in steps of $0.005$.
Therefore, negative dissipation rates are not considered.
Using $\chi^{2}_{\textrm{RT}}$ from Eq.\ (\ref{eq:chi_mu}), the likelihood function $L_{\textrm{RT}}$ is \cite{Lima2010}
\begin{equation}
 L_{\textrm{RT}} \propto \exp ({- \chi^{2}_{\textrm{RT}} / 2}) .
\label{eq:L-chi}
\end{equation}
For simplicity, $L_{\textrm{RT}}$ is normalized.
Note that $\tilde{\Omega}_{\Lambda} =0$ and $\tilde{\Omega}_{\Lambda} = 1$ have not been sampled, in order to avoid a division by zero when $T$ is calculated from Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}).
Here $\tilde{\Omega}_{\Lambda} =1$ corresponds to $\tilde{\Omega}_{m} =0$ because $\tilde{\Omega}_{m} = 1 - \tilde{\Omega}_{\Lambda}$.

\begin{figure} [t]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.33}{\includegraphics{fig4.eps}}
\end{center}
\end{minipage}
\caption{ (Color online). Contours of the normalized likelihood $L_{\textrm{RT}}$ in the $(\tilde{\Omega}_{\Lambda}, \tilde{\mu})$ plane for the radiation temperature.
The contours of $L_{\textrm{RT}}$ for $0.2$, $0.4$, $0.6$, and $0.8$ are plotted.
The likelihood function is normalized using the maximum value, obtained for $(\tilde{\Omega}_{\Lambda}, \tilde{\mu}) = (\tilde{\Omega}_{\Lambda}, 0)$.
Note that the maximum value is the same for all values of $\tilde\Omega_\Lambda$ when $\tilde\mu =0$.  }
\label{Fig--L_mu_Lambda-T}
\end{figure}

\begin{figure} [t]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.33}{\includegraphics{fig5.eps}}
\end{center}
\end{minipage}
\caption{ (Color online).
Top: The normalized likelihood $L_{\textrm{SN}}$ for the supernova as a function of  $\tilde{\Omega}_{\Lambda}$.
Bottom: Contours of the normalized likelihood $L$ for both the radiation temperature and the growth rate.
To calculate $L_{\textrm{SN}}$, the \textit{Union 2.1} set of $580$ type Ia supernovae is used \cite{Suzuki_2012}.
(For details, see the text.)
The likelihood $L_{\textrm{GR}}$ for the growth rate is from Ref.\ \cite{Koma7}, whereas $L_{\textrm{RT}}$ for the radiation temperature is replotted from Fig.\ \ref{Fig--L_mu_Lambda-T}.
For clarity, the contours of $L =0.2$ and $0.8$ are plotted in the bottom panel.  }
\label{Fig--L_mu_Lambda-T_fc}
\end{figure}

Figure\ \ref{Fig--L_mu_Lambda-T} plots the contours of the normalized likelihood $L_{\textrm{RT}}$ in the $(\tilde{\Omega}_{\Lambda}, \tilde{\mu})$ plane.
In this figure, $(\tilde{\Omega}_{\Lambda}, \tilde{\mu}) = (\Omega_{\Lambda}, 0)$ corresponds to a pure $\Lambda$CDM model for $\Omega_{\Lambda}$.
Consider the contours of $L_{\textrm{RT}}$ for $\tilde{\Omega}_{\Lambda} \approx 0.7$.
It might be seen that low dissipation ($0 \leq \tilde{\mu} \lessapprox 0.01$) corresponds to high $L_{\textrm{RT}}$ regions when $\tilde{\Omega}_{\Lambda} \approx 0.7$.
That implies that a weakly dissipative Universe is compatible with the current data.
A similar result has been discussed previously \cite{Koma7}, in which the growth rate for clustering (related to structure formations) was examined.
Accordingly, a weakly dissipative universe is likely proposed.
This expectation may be confirmed by the bottom panel of Fig.\ \ref{Fig--L_mu_Lambda-T_fc} which shows the contours of the normalized likelihood $L$ not only for the radiation temperature but also for the growth rate.
To confirm the expectation in more detail, a joint likelihood analysis is performed later.

In the bottom panel of Fig.\ \ref{Fig--L_mu_Lambda-T_fc}, the normalized likelihood $L_{\textrm{GR}}$ for the growth rate is taken from Ref.\ \cite{Koma7}.
In that reference, a likelihood analysis for the growth rate in a modified entropic-force model was performed using the same method.
The modified entropic-force model is equivalent to the modified dissipative model considered here.
Therefore, $L_{\textrm{GR}}$ for the growth rate in the present model is the same as that in Ref.\ \cite{Koma7}.

In addition, using the same method, the normalized likelihood for the distance modulus (related to supernova data points) is discussed.
In the present study, we call this the normalized likelihood $L_{\textrm{SN}}$ for the supernova.
For the supernova data points, the \textit{Union 2.1} set of $580$ type Ia supernovae (up to redshift $z= 1.414$) is used \cite{Suzuki_2012}.
The chi-squared function is
\begin{equation}
\chi_{\textrm{SN}}^{2} (\tilde{\Omega}_{\Lambda}, H_{0})
= \sum\limits_{i=1}^{580} { \left[ \frac{
                                                         \mu_{\textrm{obs}} (z_{i} ) - \mu_{\textrm{cal}} (z_{i}, \tilde{\Omega}_{\Lambda}, H_{0} )  }{ \sigma_{i}^{\textrm{SN}} }
                                     \right]^{2}   }
\label{eq:chi_mu_SN}
\end{equation}
and the distance modulus $\mu$ is defined as
\begin{equation}
 \mu = 5 \log d_{L} + 25
\end{equation}
where the luminosity distance $d_{L}$ is from Eq.\ (\ref{eq:dL_00}).
Keep in mind that the distance modulus $\mu$ is not the dissipation rate $\tilde{\mu}$.
In this analysis, $\tilde{\Omega}_{\Lambda}$ and $H_{0}$ are treated as free parameters and sampled in steps of $0.001$ and $0.02$, respectively.
That is, the normalized likelihood $L_{\textrm{SN}}$ does not depend on the dissipation rate $\tilde{\mu}$.
Consequently, $L_{\textrm{SN}}$ in the $(\tilde{\Omega}_{\Lambda}, H_{0})$ plane is obtained.
Using $L_{\textrm{SN}}(\tilde{\Omega}_{\Lambda}, H_{0})$, the maximum value at each $\tilde{\Omega}_{\Lambda}$ can be determined.
The value is plotted as a function of $\tilde{\Omega}_{\Lambda}$ in the top panel of Fig.\ \ref{Fig--L_mu_Lambda-T_fc}.
In this panel, the maximum value is obtained at $\tilde{\Omega}_{\Lambda} = 0.722^{+0.030}_{-0.031}$ (with $H_{0} = 70.04^{+0.40}_{-0.38}$).
This result is consistent with that examined in Ref.\ \cite{Suzuki_2012}.

A weakly dissipative universe likely describes the observed data, as mentioned previously.
To confirm this expectation, a joint likelihood analysis is performed, using the three normalized likelihood functions.
For the joint likelihood analysis, a combined likelihood function $L_{\textrm{total}}$ is defined by
\begin{equation}
L_{\textrm{total}} = L_{\textrm{RT}} \times L_{\textrm{GR}} \times L_{\textrm{SN}}  .
\label{eq:L-joint}
\end{equation}
In this analysis, $\tilde{\Omega}_{\Lambda}$ and $\tilde{\mu}$ are sampled in steps of $0.005$.
The obtained $L_{\textrm{total}}$ is normalized.
The contours of $L_{\textrm{total}}$, corresponding to $1 \sigma$, $2 \sigma$, and $3 \sigma$ confidence levels, are plotted in Fig.\ \ref{Fig--Lt_mu_Lambda-T}.
The region surrounded by the likelihood contours is close to zero dissipation, i.e., $\tilde{\mu} =0$, when $\tilde{\Omega}_{\Lambda} \approx 0.7$.
In fact, the maximum value of $L_{\textrm{total}}$ is obtained for $(\tilde{\Omega}_{\Lambda}, \tilde{\mu}) = (0.725, 0)$,
where an upper limit of the dissipation rate is $\tilde{\mu} \approx 0.01$ at the $1 \sigma$ confidence level.
In this sense, a weakly dissipative model proposed here is very similar to the standard $\Lambda$CDM model.

\begin{figure} [t]
\begin{minipage}{0.495\textwidth}
\begin{center}
\scalebox{0.33}{\includegraphics{fig6.eps}}
\end{center}
\end{minipage}
\caption{ (Color online). Contours of the normalized likelihood $L_\textrm{total}$ for the joint likelihood analysis in the $(\tilde{\Omega}_{\Lambda}, \tilde{\mu})$ plane.
The likelihood function is normalized using the maximum value, obtained for $(\tilde{\Omega}_{\Lambda}, \tilde{\mu}) = (0.725, 0)$.
The contours of $1 \sigma$, $2 \sigma$, and $3 \sigma$ confidence levels (corresponding to $- 2 \ln L_\textrm{total}$ equal to $2.30$, $6.16$, and $11.81$, respectively \cite{Sola_2015c}) are plotted.
That is, the contours correspond to $ L_\textrm{total} = 3.17 \times 10^{-1}$, $4.60 \times 10^{-2}$, and $2.73 \times 10^{-3}$, respectively.  }
\label{Fig--Lt_mu_Lambda-T}
\end{figure}

As has been noted in Refs.\ \cite{SZ_1970etc,Chluba_2014}, adiabatic photon production does not affect a blackbody spectrum if radiation fields are not thermalized.
Also, Chluba has recently examined the influence of a CMB (cosmic microwave background) spectral distortion and adiabatic photon production processes on the $T$--$z$ relation in detail \cite{Chluba_2014}.
Consequently, it is found that the photon production process does not affect a blackbody spectrum except when the process has a very special energy dependence.
In contrast, the CMB spectral distortion affects the CMB temperature.
The CMB distortion can be constrained using COBE/FIRAS limits.
(For details, see Ref.\ \cite{Chluba_2014}.)
Accordingly, if adiabatic photon production for CCDM models \cite{Lima_2000} is assumed,
a weakly dissipative model discussed here should be further constrained because the CMB distortion is neglected in this study.
That is, the dissipation in the universe is expected to be smaller than the above mentioned one especially when the radiation temperature is discussed.
Similarly, the recent data \cite{Martino_2015,Luzzi_2015,Planck2015} imply a more weakly dissipative model.
For example, the simple $T$--$z$ relation of the form $T = T_{0}(1+z)^{1-\beta} $ has been investigated in the Planck 2015 results \cite{Planck2015}.
In that reference, one finds $\beta = (0.2 \pm 1.4) \times 10^{-3}$, where a recombination redshift of $z=1100$ is adapted.
Substituting $z =1100$, $\beta = (0.2 \pm 1.4) \times 10^{-3}$, and $T_{0} = 2.725 \pm 0.002$ K (from Table\ \ref{tab-T-z_obs}) into the simple $T$--$z$ relation, we have $T=2965 \sim 3028$ K.
Therefore, when $z =1100$, upper limits of the dissipation rate can be estimated from this result, using Eq.\ (\ref{eq:dotT_T_radiation_T-a_z_2}).
The upper limit is approximately $\tilde{\mu} = 0.018$ \cite{C100}.
This constraint is consistent with the confidence levels shown in Fig.\ \ref{Fig--Lt_mu_Lambda-T}.

\section{Conclusions}
\label{Conclusions}

The radiation temperature--redshift relation has been examined in a dissipative universe.
A phenomenological modified dissipative model has been developed that includes two constant terms, assuming a homogeneous, isotropic, and spatially flat universe.
The model is equivalent to an extended $\Lambda$CDM model in a dissipative universe.
Therefore, it behaves as if a nonzero cosmological constant $\Lambda$ and a dissipative process are operative.
A general radiative temperature law \cite{Lima_2000,Lima_1992e,Lima_1992b,Lima_2014b} has been used in the model, to deduce a $T$--$z$ relation for a dissipative universe.
That relation has been computed in the late universe as a function of a dissipation rate ranging from $\tilde{\mu} =0$, corresponding to a nondissipative $\Lambda$CDM model, to $\tilde{\mu} =1$, corresponding to a fully dissipative CCDM model.

The results confirm that the $T$--$z$ relation for $\tilde{\mu} =0$ obeys a linear law in equilibrium.
However, the calculated radiation temperature $T$ gradually deviates from the linear law with increasing $\tilde{\mu}$, even if the background evolution of the universe is not altered.
In particular,  $T$ for $\tilde{\mu} =1$ is nonlinear because the effective equation-of-state parameter $w_{e}$ varies with time in a dissipative universe.
In contrast, $T$ for low $\tilde{\mu}$ agrees with observations when $\tilde{\Omega}_{\Lambda}  = 0.685$ (i.e., the background evolution is equivalent to that of a fine-tuned pure $\Lambda$CDM model).
This agreement indicates that low dissipation describes the radiation temperature--redshift relation.
That is, the dissipation rate is constrained by the observed $T$--$z$ relation, even if density perturbations are not treated.
The present study thus provides new insights into a dissipative universe.

To examine the influence of $\tilde{\Omega}_{\Lambda}$, a likelihood analysis has been performed.
A low-$\tilde{\mu}$ high-$\tilde{\Omega}_{\Lambda}$ universe and a high-$\tilde{\mu}$ low-$\tilde{\Omega}_{\Lambda}$ universe have high likelihoods, consistent with previous work \cite{Koma7} in which a growth rate for clustering (related to structure formations) was examined.
However, higher and lower $\tilde{\Omega}_{\Lambda}$ values are inconsistent with supernova data.
Accordingly, a weakly dissipative universe ($\tilde{\mu} \lessapprox 0.01$) for $\tilde{\Omega}_{\Lambda}  \approx 0.7$ is a viable scenario.
To examine this scenario in more detail, a joint likelihood analysis has been performed.
Consequently, a weakly dissipative model proposed here is found to be very similar to the standard $\Lambda$CDM model because the expected dissipation rate is small.
Interestingly, recent works of the radiation temperature \cite{Planck2015,Martino_2015,Luzzi_2015,Chluba_2014} imply a more weakly dissipative model.
However, the properties of a weakly dissipative model differ from those of a nondissipative $\Lambda$CDM model.
Therefore, further observations are necessary to determine whether a low-dissipation model is valid.
It should be noted that a fully dissipative CCDM model (for which $\tilde{\mu} =1$) agrees with observations of the growth rate if a negative sound speed \cite{Lima2011} and the existence of clustered matter \cite{Ramos_2014-2014b} are assumed.

\begin{acknowledgements}
The authors wish to thank Professors J. Chluba, G. Luzzi, and I. de Martino for very valuable comments.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


Preheat broiler. Put chiles on a baking sheet and broil 4 in. from heat until blackened all over, turning as needed, about 15 minutes. Let chiles cool, then remove and discard stems, seeds, and skins. Coarsely chop chiles and set aside. Preheat oven to 400°.
Stir together flour, cornmeal, sugar, baking powder, and salt in a large bowl. In another bowl, beat eggs, buttermilk, and butter to blend. Pour egg mixture into flour mixture; add chiles, corn, and onions; and stir just until blended. Spread in a buttered 9- by 13-in. baking dish.
Bake until a toothpick inserted in center comes out clean, 25 to 30 minutes. Let cool at least 20 minutes. Serve warm or cool, cut into pieces.
*Buy in a grocery store or natural-food store.
Make ahead: Up to 1 day. Wrap in foil and reheat in a 350° oven about 10 minutes (set bread on top of something if oven is full).
Note: Nutritional analysis is per piece.


The Community Impact Model (or CIM) is intended to promote better land use decision-making through a better understanding of the fiscal impacts of land development. The model makes it possible to quickly and easily estimate the fiscal impacts of a proposed development on local governments.
Fiscal impact analysis is a tool for making better land use and urban planning decisions. The fiscal impact is the financial impact of a project, policy, or decision on one or more units of government. For local land use decisions, the impact on local governments (and local taxpayers) is typically the primary interest.
The CIM performs a fiscal impact analysis of a proposed land development's impacts and can be used to inform public policy decisions related to proposed land use changes, comprehensive plan amendments, and land conservation opportunities.
Any type, size, or combination of residential, commercial, industrial, and mixed-use development can be modeled. Once basic information about the development, local taxes, and fees is obtained, the information can be entered and the modeling completed very quickly. The model automatically generates a complete output report with summary tables and charts that can be easily printed.
The model is intended to provide an initial estimate of the fiscal impacts of a land development proposal and is not intended for calculation of local development impact fees, or for any other legal purpose, which requires greater precision.
Thanks to a grant, the CIM was initially made available for public use, along with instructions and detailed model description. However, due to continued costs to maintain and update the model, it is only available through Fodor & Associates. Please contact us if you are interested in this fast and affordable approach to fiscal impact analsyis.


We brew a range of Belgian inspired beers.
Our location is idyllic, with beautiful mountain views from our stoep overlooking a secure child friendly play area.
Our facilities are wheelchair and pet friendly.
Honingklip Farm also has a 4x4 Route and Mountain Bike Trails.


Associations between published quality ratings of skilled nursing facilities and outcomes of medicare beneficiaries with heart failure.
Assuring the adequacy of staffing of long-term care, strengthening the caregiving workforce, and making long-term care a career destination of choice: from mission impossible to mission critical?
Bundle Up for Value-Based Heart Failure Care.
Contemporary trends and predictors of postacute service use and routine discharge home after stroke.
Development and implementation of the TrAC (Tracking After-hours Calls) database: a tool to collect longitudinal data on after-hours telephone calls in long-term care.
Discharge to a skilled nursing facility and subsequent clinical outcomes among older patients hospitalized for heart failure.
Factors associated with discharge to home versus discharge to institutional care after inpatient stroke rehabilitation.
Home health and skilled nursing facility use: 1982-90.
Long-Term Outcomes Among Elderly Survivors of Out-of-Hospital Cardiac Arrest.
Medicare's 3-Day Rule: Time for a Rethink.
Patient expectation is the most important predictor of discharge destination after primary total joint arthroplasty.
Percent body fat more associated with perioperative risks after total joint arthroplasty than body mass index.
Residence in Skilled Nursing Facilities Is Associated with Tigecycline Nonsusceptibility in Carbapenem-Resistant Klebsiella pneumoniae.
The American College of Surgeons National Surgical Quality Improvement Program Surgical Risk Calculator Has a Role in Predicting Discharge to Post-Acute Care in Total Joint Arthroplasty.
The association of discharge destination with 30-day rehospitalization rates among older adults receiving lumbar spinal fusion surgery.
Transitional care in skilled nursing facilities: a multiple case study.
Transitional care of older adults in skilled nursing facilities: A systematic review.
Use of Medicare services before and after introduction of the prospective payment system.
Use of a standardized assessment to predict rehabilitation care after acute stroke.
Where patients with mild to moderate heart failure die: results from the Sudden Cardiac Death in Heart Failure Trial (SCD-HeFT).


May has arrived. For many college students, it signals the end of a grueling semester and a welcome summer vacation. For seniors though, it means the end of an era. With mere days until graduation, those of us awaiting our diplomas are officially growing up and entering the real world. But if you're like me, you're not quite ready to let go of your youth. Take a listen to this playlist of childhood throwbacks for the grads that don't want to grow up.
Drug-deals and HIV didn't quite register when we sang this tune at recess. Honestly, I was just impressed that they spent the whole music video dancing on water.
Now that we're older, it turns out that Christina wasn't singing about Aladdin after all. Bummer.
You still can't understand what he's saying? Me neither. But the fun of this song has always been attempting to mumble as fast as you can until you reach "Chinese chicken," right?
The quintessential boy band anthem, this tune had all the girls swooning and all the boys wishing they had the same effect.
Now I get it. Semi-charmed life, like graduating with honors, but without any job offers.
When you found out that your crush didn't "like-like" you, the boys of N*SYNC understood your pain so well.
Beyonce has been a diva from the very beginning, teaching us all the definition of sass with this single.
You know how this one goes. Middle school dance, arms wrapped around each other, swaying back and forth while "leaving room for the Holy Spirit" – young love at its finest.
Years before Kevin Federline, baby drama and a shaved head, the sweet, teenage Britney raised the bar for Catholic school girls everywhere with her first and most-enduring single.
It doesn't matter that their hair was longer and their voices were higher than mine, the Hanson brothers stole my heart at seven-years-old. Mmmbop? Mmmbaby!


Here at Appliance Retailer we are big fans of complete kitchen renovations; the kind of knock 'em down, rebuild 'ems from the ground up type jobs that require a whole new matching set of appliances, often costing northwards of $50,000.
In the recent past, we've profiled the amazing work done by Lee Hardcastle and Eileen Middleton — both recent winners at the Kitchen and Bathroom Design Institute (KBDi) Awards — while also highlighting the great support Smeg continues to give designers and architects as patrons of several awards and event programs.
Of course, not every one is well-minted enough to fit out a brand new kitchen with handmade Swiss appliances from V-Zug or the latest touchscreen controlled masterpieces from the Miele Generation 6000 range. That's why we were interested to read these six tips to a great new kitchen without bankrupting yourself in the process.
Most kitchens have limited wall space so it's usually affordable to add a decorative wall mural or feature panel using glass mosaic tiles. Splashbacks introduce colour, texture and a tactile finish, creating a fresh new look for any kitchen.
White kitchen cabinets give you the freedom to choose almost any of the finishes available and even allow you to mix and match and choose a style that works within the theme of your kitchen. Chrome with brushed nickel or oil rubbed bronze with brass go well, as long as the finishes complement each other.
New light fixtures can instantaneously transform your kitchen. A tradesman can remove the old lighting and replace it with contemporary and subtle recessed lighting. Or install pendants over islands, tables and sinks for a bolder, modern feel.
Upgrading to a solid, low maintenance benchtop will add a contemporary new look to your home. Natural stones like granite add a touch of class and cleanliness to your kitchen.
If you're working within a limited budget, think about investing in one great statement piece for your kitchen. Be it a fridge or an oven, modern appliances will be sure to impress and draw attention away from less flattering kitchen features.
This really is a great way to change the look of your kitchen without the hassle of ripping apart the entire space. Simply replace old cupboard doors with new ones to create a stylish look at a fraction of the cost of a new kitchen.
Thanks to Jessie Harvey for these tips. Share them with your retail customers and, remember, the more money customers save on the kitchen fixtures, the more they can spend on appliances!


Categories > The Open Forum > Anyone perform stage shows?
It seems like there are a lot of close up people here.
My wife and I perform separate and together. My wife juggles, stilt walks, performs magic, face paints, balloon animals, and sideshow stunts.
I perform magic, mind reading, bizarre magick and seances.
Together we call ourselves "Psychic Sideshow." It's a family friendly show filled with Magic, Mind-Reading, Carnival Cons, Sideshow Stunts and More.
The show is like a walk down the carnival midway. I like this because it gives us a lot of leeway as far as adding material into our show. BUT, it has to fit within the theme of the show so not everything makes it.
We are well-known for our "Breadstick of Death" routine. This is where Colleen the Sideshow Queen whips a breadstick out of my mouth piece by piece until a fraction of an inch remains or it gets flush with my lips. And yes, I've been whacked a few times but she has a 99.5% success rate, which is amazing. She's the only one I trust aiming a whip at the space in front of my face.
This pic got us a spot on The Tonight Show with Jay Leno.
We sent them the pic. They wanted video. Then they wanted us.
Then the writers strike happened.
But.... Once the strike was over, they called and flew is out there.
We even made it to the Magic Castle while we were out there.
Didn't perform at the Castle, just went there.
Awesome. My wife and I are actually starting a two person mental act. Mostly light-hearted fun targeted at youth/teens. Our first show is booked for this summer, and we're trying to get our act together (pun intended) now so we can be ready to start hustling to book more shows.
I must ask, do you guys do any Renaissance festivals? I ask because it seems like the show could be adapted to that environment pretty easily.
We did one Ren Faire a while back. Haven't done any since because we're booked for other events.
A friend of ours is trying to get something booked this year at a local venue. Ren Faire is one of the events they hold on one of their weekends.
We do a lot of local festivals, private parties and we've been doing a monthly show at DiDonna's South Shore Restaurant on Saratoga Lake.
This is my first post here since being invited by Tommy Ellison of Ring 35 in Birmingham, Alabama.
My wife Lucy and I do perform stage magic. We have been performing together for 15 years.
We also host Magic Valley Magic each October at our home Magic Valley Ranch. It is a family reunion of entertainers: recording artists, magician, and clown. The one for 2016 will be our seventh year of holding the event.
We try to make it to 12-14 magic conventions a year. Hopefully we can meet all on the road.
This pic got us a spot on The Tonight Show with Jay Leno. We sent them the pic. They wanted video. Then they wanted us. Then the writers strike happened. But.... Once the strike was over, they called and flew is out there. We even made it to the Magic Castle while we were out there. Didn't perform at the Castle, just went there.
Haha! That is a great photo, Dan!
As I said in another thread...I'm very happy that you've joined us, Bob!
Thanks for inviting him here, Tommy!


SUFFERN - Good Samaritan Hospital has found an "ap-pealing" way to celebrate each milestone in the fight against cancer.
After patients receive a final radiation treatment, they can tug the rope to ring the "New Beginning Bell" in the hospital's Radiation Oncology Unit. When the bell peals, it touches off a celebration signifying the completion of a patient's treatments.
The bell came to Good Sam thanks to Isabella Spar as part of the Ardsley teen's bat mitzvah project, "Project Bell."
Spar, 15, who belongs to Temple Beth Abraham in Tarrytown, was inspired by her mother's successful radiation therapy for a benign brain tumor at a Boston hospital.
Spar sells homemade bracelets on her website, ProjectBell.org, for $18 each. The proceeds go toward providing other cancer treatment centers with similar bells.
On Friday, the hospital's "Ringing of the Bell" ceremony honored Clifford Nyman, who recently finished five weeks of radiation treatment after being diagnosed with prostate cancer last summer.
"When you go through this and you ring the bell, it's like, 'Thank you Lord, we did it …we did it, " said Nyman, a retired dentist from Eastchester.


text:
Q: Could not write JSON: For input string: "PARENT"; nested exception is com.fasterxml.jackson.databind.JsonMappingException: Any hint or help would be greatly appreciated it.
I am using Java 17 and jpa with spring boot.
I get the following error:
nested exception is
org.springframework.http.converter.HttpMessageNotWritableException: Could not write JSON: For input string: "PARENT"; nested exception is com.fasterxml.jackson.databind.JsonMappingException: For input string: "PARENT" (through reference chain: com.bank.account.model.ChoreEntity["createUserId"]->com.bank.account.model.UserEntity["role"]->com.bank.account.model.RoleEntity$HibernateProxy$LSkoyC50["roleName"])
ChoreEntity.java:
                @JsonIgnoreProperties({"hibernateLazyInitializer", "handler"})
                @ManyToOne(fetch = FetchType.EAGER, optional = false)
                @JoinColumn(name = "create_user_id", nullable = false)
                @OnDelete(action= OnDeleteAction.CASCADE)
                private UserEntity createUserId;

                @JsonIgnoreProperties({"hibernateLazyInitializer", "handler"})
                @ManyToOne(fetch = FetchType.EAGER, optional = false)
                @JoinColumn(name = "assign_user_id", nullable = false)
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                private UserEntity assignUserId;

                Hibernate:
                insert
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                values
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In this posting:
No serializer found for class org.hibernate.proxy.pojo.bytebuddy.ByteBuddyInterceptor
In the ChoreEntity.java I added based on the above stackoverflow:
@JsonIgnoreProperties({"hibernateLazyInitializer", "handler"})
In the application.properties file I also added:
spring.jackson.serialization.fail-on-empty-beans=false
This error occur after line 36:

redpajama_set_name: RedPajamaStackExchange


One of the leading manufacturer and exporter of Antique Reproduction furniture is located in Jepara, Central Java. Present many product ranges such as Antique Reproduction, Colonial, Contemporary and Old Country Style, also available in any types.
Jepara international furniture is manufacturer and exporter of antique furniture reproduction. Selected carefully, controlled by expert, we provide our customer with wooden antique reproduction furniture with the best raw material.
Have a small garden in your house or maybe a tiny backyard shouldn't limit how much you can relax in your backyard. So, lets go to furniture online and find your ides to make it all fit together. Make sure you choose the fit wood material for those.


text:
Home › Reviews › Boosey & Hawkes "Concert" Double Bass anno 1938 – Review
Boosey & Hawkes "Concert" Double Bass anno 1938 – Review
What will I like about this Hawkes Concert?
In a word I'd describe this Hawkes as an imposing instrument. With it's violin outline, full swell back and outer linings - the model is grand and impressive.
The instrument looks very pretty too.
Yes it certainly does. The spruce used for the table shows a nice even straight grain. The maple wood used for the back and ribs is very well flamed and one can even see the grain lines running downwards.
What about the sound – that's much more important to me?
Simply huge - on top of which there are lush-dark tonal qualities.
I am an orchestral player / I am a jazz player.
That's great. You'll absolutely love this instrument.
What is the overall condition of the instrument like?
We'd describe it as being in pretty exceptional condition. On the table there are only four minor cracks that run up from the lower edge of the table. Amongst the six ribs – there is only one small crack in the bottom treble side rib. That's it – apart from these now well-restored cracks – the instrument is mint.
Is the varnish original?
Yes it is and what's more the quality and texture of the spirit-based varnish is really glorious. The colour is chestnut brown over a yellow ground and there is an ever-so-slight craquelleur to the surface of the varnish, which is truly fabulous.
Have you upgraded the instrument at all?
Yes - internally the instrument sports a new bass bar of the correct length and depth while externally there is a set-up par-excellence.
Is the instrument labelled?
Yes. It bears it's origional label which states: The Concert Bass, (Registered), No1 1938 – Boosey & Hawkes Ltd, 295 Regent Street, London W1. A second label gives the number 634 in large figures.
Do you have any general info on Hawkes & Son?
Yes - the instruments manufactured by Hawkes & Son during the latter quarter of the nineteenth century and the first half of the twentieth century were often supplied to the many army and military bands that existed especially in England. There were three grades of instruments - namely Professor, Concert and Panormo - that were based on the unsurpassable form of Vincenzo Panormo's instruments.
All the models are characterized by their violin outline, deep ribs and generous proportions which are by no means too large to play easily. The Professor was a flat back model with an upper angle break and often outside linings - while the Concert and Panormo were produced with full swell backs.
To meet the demand for these robust and well-constructed instruments a great majority were imported from Germany and France - the Concert and Panormo models being superior in their choice of timber, workmanship and finish than those of the Professor. In 1924 the four string Professor instruments were being advertised at £22, the four string Concert at £26 and the four string Panormo model was £36. For the three string versions of these same models prices were £20, £24 and £32 respectively. The advert aptly proving that in England there was still some demand for the three stringed instruments even at this late date.
It is a well-known fact that the majority of Hawkes double basses had a very hard life - so to find this particular example in a near mint state - is quite remarkable indeed. With its massive sound and full-tone this really is an attractive option for an upwardly mobile, budget-conscious orchestral professional. Fitted with a set of Thomastic Spirocore meds or softs - it really does produce the kind of juicy-groove sounds that you jazz guys & chicks are all after. In terms of investment potential – this fashionable piece of bass making history is sure to provide excellent dividends.

redpajama_set_name: RedPajamaCommonCrawl


text:
The Gender Pay Gap Remains
On March 15, Equal Pay Day was observed in the United States, a day meant to raise awareness of the gap in pay between men and women in this country. Begun by the National Committee on Pay Equity in 1996, the changing date "symbolizes how far into the year women must work to earn what men earned in the previous year."
To commemorate this year's Equal Pay Day, President Biden issued a proclamation, stating, "Equal pay is a matter of justice, fairness, and dignity—it is about living up to our values and who we are as a Nation."
"This year, Equal Pay Day falls on March 15, the earliest we have ever marked the occasion. The earlier that Equal Pay Day arrives, the closer our Nation has come to achieving pay fairness," the proclamation continues. "But while we should celebrate the progress we have made, as I have said in the past, we should not be satisfied until Equal Pay Day is no longer necessary at all."
Though March 15 has come and gone, there remain other Equal Pay Days left to observe. According to the American Association of University Women (AAUW), Asian American, Native Hawaiian and Pacific Islander Equal Pay Day will come on May 3, with those groups making 75 cents for every dollar earned by white men. Black Women's Equal Pay Day is September 21, with Black women making 58 cents for every dollar. Equal Pay Day does not come until November 30 and December 8 for Native American women and Latina women, respectively, with Native American women earning 50 cents for every dollar and Latina women earning 49 cents.
According to the Pew Research Center, progress in eliminating the gender pay gap has slowed, with the gap remaining relatively unchanged for 15 years. Their most recent report states, "In 2020, women earned 84% of what men earned, according to a Pew Research Center analysis of median hourly earnings of both full- and part-time workers. Based on this estimate, it would take an extra 42 days of work for women to earn what men did in 2020."
Gains made in past decades have come in areas more easily measured, according to the report, as women have increased educational attainment, found increasingly diverse employment opportunities in previously male-dominated sectors, and had greater work experience. But even with these gains, true gender pay equity has remained elusive.
"Even though women have increased their presence in higher-paying jobs traditionally dominated by men, such as professional and managerial positions, women as a whole continued to be overrepresented in lower-paying occupations relative to their share of the workforce," the report states.
Women of color are especially overrepresented among those lower-paying occupations, while at the same time heavily underrepresented in higher-paying occupations, with discrimination and lack of adequate legal protections contributing to the gender pay gap and preventing women of color from successfully breaking into those occupations, according to the AAUW.
Meanwhile, motherhood presents problems to women in their careers, the Pew report states. Mothers with children under 18 were more likely to say they, "needed to reduce their work hours, felt like they couldn't give full effort at work, and turned down a promotion because they were balancing work and parenting responsibilities," compared with fathers. Further, around 20 percent of mothers said they had been passed over for assignments or promotions, while 27 percent said they have been treated as less committed to their work.
The professional and financial impact on mothers in the workforce has sometimes been dubbed the "motherhood penalty.," and is in part due to the lack of affordable child care. According to the Center for American Progress (CAP), "Child care challenges have become a barrier to work, especially for mothers, who disproportionately take on unpaid caregiving responsibilities when their family cannot find or afford child care…Too often, mothers must make job decisions based on child care considerations rather than in the interest of their financial situation or career goals."
Their survey found that if there was more access to affordable childcare, mothers said they "would increase their earnings progress in their careers by finding a higher-paying job, applying for a promotion, seeking more hours at work, or finding a job in the first place."
Along with providing affordable and high-quality childcare, CAP believes that raising the minimum wage and raising the tipped minimum wage would help women, especially women of color, overrepresented in lower-paying industries. Supporting fair scheduling practices would allow mothers flexibility in their schedules for family needs without fear of losing their jobs, and pay transparency would give women knowledge of pay discrimination between themselves and their male colleagues. Paid sick days legislation and paid family and medical leave insurance would support mothers and caregivers in the workforce from losing income due to children or family issues, with CAP noting that the United States remains the only developed country that does not guarantee women paid maternity leave.
Parts of the problem remain harder still to measure quantitatively. Another Pew Research Center analysis of the gender pay gap posed several questions about factors that aid career and pay advancement behind the scenes and that remain without a definitive answer.
"How much is due to gender stereotypes that contribute to lower aspirations before they even reach the job market?" the report reads. "Or is it due to weaker professional networks once they look for work? Do women lose out because they do not push as hard as men for raises and promotions? And what is the role of discrimination, which turns up in experiments where people are asked to rate identical resumes from mothers and fathers?"
There are signs of progress, but a large amount of work remains to be done. In March, a new report from the Pew Research Center based on Census data found that women working full-time under the age of 30 earned as much as or slightly more than men in 22 of 250 metropolitan areas, including New York, Washington D.C., and Los Angeles metropolitan areas. The pay gap is smaller for younger women, but as they advance into their careers those stumbling blocks to pay equity remain.
According to The Washington Post, "Men and women start more or less on an even footing but wage disparities widen over time, according to economists who study gender compensation. Small pay differentials are magnified as raises and promotions stack up over years and decades."
Over the course of a lifetime, this gap causes women to lose out on a significant amount of income, impacting them and their families for years to come. With women joining the workforce in increasingly large numbers since the middle of the last century, there are already too many wages and income lost for women and their families. In the decades since, the gap has measurably shortened—but women remain behind men in their earnings with women of color the most affected. There is still much more to be done.


Search for the Hidden Treasure is Volume 2 in the 'Chompy & the Munchy Bunch' Series, about a carrot with superpower eyesight! A secret treasure is hidden deep inside the mysterious Peppa Mansion. Chompy, Salty and Weiner set out to solve the mystery...but will Candybar shake up their plans? Action-packed fun for kids ages 4 - 9!


Since industrial lasers appeared in the 1970s, one of the most promising application possibilities has been removal of coatings. A laser's concentrated energy easily and cleanly strips paints, sealants, adhesives, corrosion, any coating really, from any kind of substrate.
The obstacle has been that any laser powerful enough to remove coatings quickly can also damage substrates, if it continues to "fire" after the coating has been removed. So, for applications where preserving a pristine substrate is important – say, the wing of an airplane, or the hull of a submarine – laser "ablation" has never been practical.
Their focus was on pulsed lasers, with fiber optic beam delivery onto any work surface, and customized scanning optics to optimize efficiency and safety. The company introduced its first system in 1998 and quickly followed with development of real-time color-selective beam controls. This control system interrogates the coated surface before each pulse and permits the next pulse to fire only if preset conditions are met. One example condition is that the coating to be removed is still present; there are others. This closed-loop control prevents damage to substrates, enabling the use of high-powered industrial lasers to deliver concentrated laser energy to quickly ablate coatings.
Since 2001, GLC has developed an expanding patent portfolio based on its control and beam delivery technologies. In 2009, after passing a thorough test program monitored by the Federal Aviation Administration (FAA), the GLC closed-loop process received FAA endorsement for use on Boeing 737 "classic" aircraft. In 2010 the FAA further advised that it has no objections to use on the aluminum airframes of any commercial aircraft. Our ablation system is the only laser coating removal process to have earned this critical FAA qualification..
To remove red paint from an aluminum fuselage, for example, the operator sets our system to search for red. Our color recognition system enables each laser pulse to fire only if red paint is found. As the laser ablates the red paint, exposing the fuselage underneath, the laser is suppressed until the moving workhead finds more red paint, at which time the laser is once again enabled. This detection and suppression is instantaneous, so that laser energy never impacts uncoated substrate. The energy pulse directed onto the coating being removed is completely absorbed in the ablation process, with virtually none passed through to the substrate.
This patented color recognition capability is unique to Lasertronics systems.
Our system can be combined with an air capture system to trap ablated material in a HEPA filter. This sequestration dramatically reduces the complexity and cost of disposal of the toxic substances often found in coatings.


Year 10 & 11 students! If you have a passion for STEM (Science, Technology, Engineering and Maths) subjects and are possibly thinking of a career in this field, then there is an amazing opportunity to get yourself involved in.
The below link details several courses you can become involved in over various half terms throughout the year.


Abstract: During the breeding season of 1998, the experiment was conducted on the breeding density of Common magpie Pica pica, western jackdaw Corvus monedula and hooded crow Corvus cornix in towns in north-eastern Slovenia. In each town I randomly selected plots.They measured 99 ha in Celje, 65.5 ha and 52.3 ha in Maribor, 46.0 ha in Ptuj, 73.4 in Slovenska Bistrica, 11.4 ha in Slovenske Konjice and 96 ha in žalec. The most common species was common magpie which was present in all seven towns and the rarest western jackdaw which bred only in two towns. Common magpie density in Slovenian towns was approximately the same as in other similar towns in Central Europe and smaller than in other bigger towns. Interestingly,densities of common magpie in towns under study are even higher than density in rural areas in Slovenia and elsewhere. The main reasons for higher densities in towns are probably a result of food availability throughout the year, and lack of predators. Relationship between density of common magpie and number of inhabitants in towns was not significant (r= -0.30, P>0.05, n = 5). Due to lower densities of common magpie in smaller towns I predict that colonization was recent. Based on historic data I predict that Celje was colonized by common magpie about 20 years ago, Maribor some years earlier, whereas colonization of Slovenj Gradec, Slovenska Bistrica and žalec was in recent years. Since densities of hooded crow is lower then common magpie, colonization of this species is probably even later.
Milan Vogrin , 2003. Common Magpie Pica pica, Western Jackdaw Corvus monedula and Hooded Crow Corvus cornix in Some Towns in North-eastern Slovenia (Central Europe). Journal of Biological Sciences, 3: 688-693.


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Mongolian Cashmere Headband – Le Bandeau, made in Mongolia. Shown here in Burnt Orange, a statement in glamorous headwear.
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Founder of 'Solana Killer' Aptos (APT) Reveals Plans for 2023, Says Leaps in Innovation Coming
The co-founder of Aptos (APT) says a lot of innovation is coming to the 'Solana killer' blockchain project in 2023.
In a new interview with crypto influencer Scott Melker, Aptos CEO and cofounder Mo Shaikh says that the blockchain will see significant development next year, especially catering to decentralized finance (DeFi).
"There's some really cool DeFi stuff that's going to be coming live soon that takes advantage of not only things like Move, but also parallel transaction processing. So, you know, central limit order books, AMMs (automated market makers), DEXes (decentralized exchanges) – all of these things will be really cool things to keep an eye on Aptos. That'll be step changes in innovation relative to everything that we've seen in the previous generation of blockchain. So we're looking forward to all of that."
The Aptos chain uses a programming language called Move, which was originally created for Diem, Meta's since abandoned crypto project. Aptos aims to advance Diem's original goal of creating a fast and scalable blockchain.
Shaikh also says that Aptos was built with billions of users in mind, which appeals to larger entities.
"The other thing that we were focused on was, well, it was kind of frustrating to hear the large companies talk about Web3, but not really do anything. And partly that was because, [they thought] 'well, we're going to do a little pilot and do some testing here, but we know that this stuff doesn't really work for millions of people. And so we're not risking bringing all our users to these layer twos or layer ones.'
Back in our early days of Meta, we were building principally for billions of people. So we had taken that approach to protect three-plus billion users. When we started having conversations with folks like Google and Npixel, which is a triple A gaming studio in Korea that has over 10 million users, those folks saw what Aptos can do for their users and actually innovate inside of their organization.
So all those folks have started building on top of Aptos. Web3 users are building new innovative products on Aptos, things that they could not do on the blockchains that were out before. And large entities and enterprises that were… excited about Web3, but weren't comfortable are now building on Web3 in a meaningful way, and we're excited to see all those use cases now come to life and in 2023."
At time of writing, Aptos is changing hands at $3.24, down more than 40% from its month's high of $5.43.
O Featured Image: Shutterstock/Mia Stendal
2022: A Year Of Grassroots Bitcoin Adoption For Africa →
← FTX-backed Projects Plunge as 'Sam Coins' Underperform


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Machine Builders
Dilbert Cartoon
Replace manual gauges with new INTUO software for Equator
Renishaw plc
The Renishaw Equator flexible gauge is now offered with the new INTUOTM gauging software, combined with a range of industry standard Renishaw probes. INTUO simplifies and automates the gauging of a wide variety of parts, removing dependence on the skill of manual gauge users. Reducing gauging inconsistency reduces scrap, whilst ensuring parts that are out of tolerance are always identified.
Renishaw is also launching the Equator Button Interface (EBI), with simple push-button controls for shop-floor operators, removing the need for a mouse and keyboard.
INTUO with Equator is a useful alternative to multiple manual devices such as vernier or digital callipers, micrometres and plug gauges. Equator is highly repeatable and runs fast automated routines, meaning significantly reduced labour costs. With many workshops using hundreds of manual gauges this cost saving can be considerable.
INTUO, with minimal training, uses intelligent functions to help the engineer set up gauging of parts in a matter of minutes. Shop-floor operators then select and run those programs using the user-friendly Renishaw Organiser front-end software.
INTUO allows the programmer to create gauging routines using just a part with an engineering drawing. With the Feature Predict function activated, the programmer uses the joystick to take points on each feature while INTUO does most of the work, by predicting the type of feature, the nominal value and a possible tolerance band. It is then easy to adjust the program to change the number of points taken or to space them evenly, and match nominals and tolerances to the engineering drawing.
It is a simple task to input the actual values for features on the master part, which does not need to be a 'perfect part'. In fact it is preferable that a production part manufactured with the same method and material is used. First the features required are measured on a certified device, like a calibrated manual gauge or any co-ordinate measuring machine (CMM). The values from this master part inspection report can be manually added to a table in Organiser or to the corresponding field in INTUO – changing one will update the other.
INTUO has gone through extensive development to make it as simple to use as possible, which means that programmers only need a training course lasting a few hours. Training is 'hands-on', using the joystick to take points while interacting with icon based menus and a graphical workspace to rapidly build up the features to be gauged. In just a few hours programmers will learn the four stage INTUO process of Align – Measure – Dimension – Report, and be ready to inspect production parts.
In many shop-floor environments simple operator controls are essential. Customer feedback has shown that an alternative to the keyboard and mouse is required, so Renishaw has introduced the Equator Button Interface (EBI), which works directly with the Organiser software.
EBI uses large push buttons which can be operated by operators wearing gloves. It is not affected by shop-floor contamination and can be mounted on the front of the Equator or on the framework of a table or enclosure. At the most basic level it can be used to start gauging programs, but can also be used to stop those programs mid-cycle, select other programs and navigate the Organiser environment. Equator can be moved using the jog keys if necessary and there are buttons for initial alignment, switching between master/measure modes, changing view screens and easy error recovery.
Equator has various reporting options, including simple pass/fail, a more detailed view in Process Monitor (a process monitoring window for the user interface), or automatic data capture for updating machine tool offsets.
Process Monitor provides a chart showing the history of feature measurements, and an easy graphical view of proportion of tolerance for each feature. Being able to view the inspection data history of a part is an invaluable function for controlling manufacturing processes. This is not possible when using a Go/no-go gauge, which can only indicate the state of each feature.
Process Monitor also enables shop floor management of the mastering process according to temperature, time or number of parts measured. Use of a calibrated master part allows the system to be re-zeroed whenever shop-floor conditions change, meaning Equator can cope with a rapidly changing thermal environment.
Reflecting the international nature of many modern manufacturing organisations, Equator customers have been able to take advantage of Renishaw's extensive network of Equator support engineers. Projects initiated in one country or region can be easily transferred and locally supported in multiple locations. Renishaw now has over 70 regional support offices and multiple approved partners, covering every industrialised area of the world.
Equator is unique in its design and method of operation, and has already changed the thinking of thousands of production engineers, making it their new gauge of choice. The versatility and repeatability Equator offers is re-defining the world of gauging, and now with INTUO and the Equator Button Interface an even wider range of capabilities is offered to manufacturers globally.
For further information on gauging, the Renishaw Equator flexible gauge and the new INTUOTM gauging software, visit www.renishaw.com/gauging.
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© Copyright 2006-14 Damte Ltd.


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Home / torture
New report 'details UK involvement in rendition to torture'
A new report presents, for the first time, a comprehensive overview of the Central Intelligence Agency's black sites network, using new data derived from an unprecedented analysis of the Senate Intelligence Committee's 2014 study of CIA detention.
Northern Ireland Legacy consultation responses published
The UK Government has published a summary of responses to the consultation on the legacy of the Troubles in Northern Ireland.
Federal court blocks Trump administration policy of arbitrarily jailing asylum-seekers
A federal court has blocked a Trump administration policy that denies bond hearings to asylum seekers.
BMA 'gravely concerned' over findings of China Tribunal on organ harvesting
The China Tribunal is sure beyond reasonable doubt that in China forced organ harvesting from prisoners of conscience has been practised for a substantial period of time involving a very substantial number of victims.
MPs urge Jeremy Hunt to press Bahrain to end 'punitive measures' against female detainees
Sixteen cross-party MPs and Peers have delivered a letter to the British Foreign Secretary Jeremy Hunt urging him to press the Bahraini government to end the abuse of female political prisoners.
Call for immediate investiigation into death of former Egyptian president Mohamed Morsi
Mohamed Morsi's death in court is deeply shocking and raises serious questions about his treatment in custody, says Amnesty International.
UN expert says 'collective persecution' of Julian Assange must end
A UN expert who visited Julian Assange in a London prison says he fears his human rights could be seriously violated if he is extradited to the United States, and condemned the "deliberate and concerted abuse inflicted for years" on the Wikileaks co-founder.
Head of UK intelligence services watchdog to step down
The UK's intelligence services watchdog has announced that the commissioner, Sir Adrian Fulford, will be stepping down in October.
Surge in suicide attempts on Manus Island 'illustrates crisis of Australia's refugee policy'
The newly-elected Australian government has promised to repeal legislation which allows severely ill people to be transferred to Australia for medical treatment.
UN urges investigation following death of detainee in Lebanon
Hassan Toufic Dika's death occurred despite numerous interventions by a variety of UN entities, including the UN Human Rights Office, with the Lebanese authorities after he was allegedly tortured in prison following his arrest in November 2018.