text:
Institutional Communications
Principal's Prize
Inappropriate prescriptions sending hospitalized seniors back to the ER
Improving hospital prescribing practices may reduce risk of rehospitalization and death
Published: 5Aug2020
Two in three hospitalized seniors are prescribed drugs that should be avoided by older adults, increasing the risk of injury and adverse drug reactions. Improving hospital prescribing practices can reduce the frequency of inappropriate medications and resulting harm, according to a new study led by McGill University researchers.
Potentially inappropriate medications (PIMs) are drugs that should be avoided by seniors because the risk of harm or injury outweighs the expected benefit, particularly when safer or more effective alternatives are available for the same condition. A new study published in Journal of the American Geriatrics Society is the first to examine the impact of these medications as a result of decisions made by care teams during hospital stays.
As patients recover, they are especially vulnerable in the short-term period after discharge, when the associated risks and adverse reactions can result in emergency department visits, rehospitalization, and even death within 30 days.
Navigating the risks and benefits of medications
To estimate the number of PIMs prescribed to patients, the researchers analyzed data from hospital admissions records, provincial health databases, and filled prescriptions in Quebec between October 2014 and November 2016.
The results they found were surprising. "The risks associated with some of these drugs are well-known to clinicians, yet they are still commonly prescribed," says lead author Daniala Weir, a former doctoral trainee under the supervision of Robyn Tamblyn in the Department of Epidemiology and Biostatistics. "Benzodiazepines, for example, are typically used to treat anxiety and insomnia in adults, but are known to increase the risk of falls for seniors."
Clinicians may judge that the benefit of a PIM outweighs the harm for their patient. "This could be true in specific situations; however, on the whole, these medications still tend to cause more harm than good for the average older adult," says Weir.
The researchers conclude that these findings demonstrate a need not only for further research into the risks associated with continuing PIMs at hospital discharge, but also for better guidance to improve prescribing practices in hospitals for older adults.
"Dr.Weir has identified important and preventable risk factors for avoidable harm in hospitalized seniors. With the introduction of computerized discharge prescriptions, there can be an automated review and alerts for potentially inappropriate medications; interventions that have proven to be effective in reducing PIMs in other jurisdictions," says Professor Tamblyn.
"Both New and Chronic Potentially Inappropriate Medications Continued at Hospital Discharge Are Associated With Increased Risk of Adverse Events" by Daniala L. Weir, Todd C. Lee, Emily G. McDonald, Aude Motulsky, Michal Abrahamowicz, and Steven Morgan, David Buckeridge, and Robyn Tamblyn is published in the Journal of the American Geriatrics Society.
DOI: https://doi.org/10.1111/jgs.16413
About McGill University
Founded in Montreal, Quebec, in 1821, McGill University is Canada's top ranked medical doctoral university. McGill is consistently ranked as one of the top universities, both nationally and internationally. It is a world-renowned institution of higher learning with research activities spanning two campuses, 11 faculties, 13 professional schools, 300 programs of study and over 40,000 students, including more than 10,200 graduate students. McGill attracts students from over 150 countries around the world, its 12,800 international students making up 31% of the student body. Over half of McGill students claim a first language other than English, including approximately 19% of our students who say French is their mother tongue.
http://www.mcgill.ca/newsroom/
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Shirley Cardenas
shirley.cardenas [at] mcgill.ca
Dept. of Epidemiology/Biostatistics/Occupational Health
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Daniala Weir
Department of Epidemiology and Biostatistics
Inappropriate medications
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Robyn Tamblyn
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Sells out on first day of sales on packaging appeal alone without any additional marketing support.
New single malt whisky in progress.
Hearing about our reputation for creating sales success, Radiant Distillers based in Assam asked us to help create a new whisky for the North East and East Indian market. Spotting growth in a more youthful market, we developed the brand concept with Radiant through to naming, packaging and marketing materials.
I would like to take this opportunity to thank each one of you at Pax who have worked on Castle Hill. I take great pleasure in letting you know the great response we have had from customers and trade- it's liquid gold. We hope in it's printing we have done justice to your creative inputs. I once again thankyou from the bottom of my heart.


Kidney stones are mineral crystalizations that are formed from imbalanced mineral ions in the urine. You can prevent kidney stones formation and reduce recurrence by about 70-80% by making simple changes to your diet and lifestyle.
Below is an image of a crystalized kidney stone. Individuals with history of kidney stones will know how excruciatingly painful and crippling it can be, so let's see how we can prevent that from recurring.
Typically, our kidneys are not meant to form stones, but with some people their urine contains more crystal-forming substances—such as calcium and uric acid—and little water in the kidneys to dilute them. Underlying causes may be a metabolic disorder, kidney disease, drugs/medications and other factors that create a favorable environment for stones to form.
Calcium stones: This is the most common type, calcium oxalate. Oxalate is found in some fruits and vegetables such as in dark green vegetables, nuts and chocolate.
Struvite stones: Usually found in women, often formed as a result of urinary tract infections.
Uric acid stones: These are byproduct of protein metabolism, commonly seen with gout, due to certain genetic factors and disorders of your blood-producing tissues.
Cystine stones: These are less common and are a result of a hereditary disorder that causes your kidneys to excrete massive amounts of certain amino acids.
Do Kidney Stones Require Medical Help?
Kidney stones can range in sizes from the size of a grain of sand to larger than a golf ball. Small ones may pass through your urinary tract unnoticed but larger ones can cause excruciating pains, and if a stone fails to pass it may cause permanent damage to the urinary tract as a result.
There are several medical procedures and surgical techniques that can be used to treat kidney stones but are not recommended, unless there's no choice. An advanced option other than surgery, is a treatment such as extracorporeal shockwave lithotripsy. This treatment entails being submerged in a tub of water where sound waves traveling through the liquid will shatter the stones to be passed out over the next few days or weeks.
In most cases, the best way is to let the stones pass on its own. It may take days or weeks by drinking plenty of pure water to reduce the concentration of solids in your urine to allow them to be dissolved. Some of the lifestyle changes below need to be followed to help speed up the dissolution of the stones.
Most often, with lifestyle changes and proper care, it is possible to prevent kidney stones. These steps should be taken especially by those who have a history of kidney stones.
The number one risk factor for kidney stones is not drinking enough water. If you are not drinking enough, your urine will have higher concentrations of substances that can form stones. An easy way to check if you are drinking enough water is to check the color of your urine. It has to be a very light yellow.
Read how much water you should drink daily.
If you happen to be taking any multi-vitamins or B supplements that contain vitamin B2 (riboflavin), the color of your urine will be a very bright, nearly fluorescent yellow. This will mar your ability to take the urine color as a guide to your level of hydration.
When your brain signals the need to empty your bladder, get to the washroom and do it! Regularly holding in urine can damage your kidneys.
When urine remains in the bladder for longer than necessary, it causes bacteria to breed and multiply. This can cause a bladder and urinary tract infection. Retaining urine also increases the pressure on the kidneys and eventually lead to renal failure and incontinence.
Respond to nature call signals and do them!
Magnesium deficiency may be linked to kidney stones as it plays an important role in your body's absorption and assimilation of calcium. Consume too much calcium without adequate magnesium can cause the excess calcium to become toxic and contribute to kidney stones formation.
Magnesium helps prevent calcium from combining with oxalate to form the common type of calcium oxalate stones.
Some excellent sources of magnesium are from green vegetable juice, beans, almonds, pumpkin and sunflower seeds. Avocados are also a good source and are very versatile for using in your dietary.
Have you ever wondered why diabetics have a higher chance of kidney failure? Sugar. An overload of sugar in the blood increases the risk of many diseases, especially of the kidneys.
Studies have shown that drinking a can of soda daily provides a high amount of sugar into the blood system, that causes the body to be acidic. Sugar is an anti-nutrient that depletes your body of nutrients.
Drinking soda excerbates conditions in your urine that lead to formation of calcium oxalate kidney stone problems. It increases kidney size and produce pathological changes in your kidney, such as the formation of kidney stones.
If you have a history of kidney stones, you should avoid eating foods and drinking beverages that are high in sugar. Sugar from fruits can normally be easily assimilated by individuals without kidney issues, but not if you have a kidney issue.
Drinking alcohol in moderation is fine, but drinking excessively and regularly can compromise your kidneys' functions, and impair their ability to regulate the volume and composition of fluid and electrolytes in the body. It dehydrates the body and leads to low blood concentrations of key electrolytes as well as potentially tip the body's pH balance to become more acidic.
Alcoholic beverages causes uric acid to be deposited in renal tubules, leading to tubular obstruction that increases the risk of kidney failure.
It has been found that having more than four drinks a day can double the risk of chronic kidney disease. Heavy drinkers who also smoke increase their chances of developing kidney disease by five times, when compared to people who don't smoke or drink. People who smoke are more likely to have higher levels of protein in the urine—a sign of kidney damage.
Although we now understand that excess calcium and drinking inadequate water may cause calcification of stones, it is nonetheless important to eat foods high in calcium. Calcium from foods are beneficial and necessary as they bind with oxalates (from foods) and prevents both from being absorbed into the blood and later transferred to the kidneys.
Calcium extracts (supplements) and from cow's milk are NOT good sources of calcium. Individuals with kidney stones history are better off avoiding these sources of calcium.
Good sources of calcium that can be made bioavailable to the body, may be obtained from most vegetables, especially "crunchy" ones such as asparagus, celery, broccoli, carrots, kale, cucumber, etc.
When you eat foods high in sodium, your kidneys need to work harder to excrete the excess salt. This in turn can lead to decreased kidney functioning, causing water retention in the body. Water retention can cause a hike in blood pressure and increase the risk of developing kidney disease.
Excess salt is often hidden in processed foods that are labelled by various names. You may be eating a diet high in salt and not know it. Beware your kidneys!
To those who are prone to kidney stones, soy products that contain high levels of oxalates can bind with calcium in the kidney to form kidney stones.
This is one of the reasons why soy products are not health foods. On the other hand, fermented soy products is quite another thing. After a fermentation process, the phytate (which blocks our body's uptake of essential minerals) and and the anti-nutrient levels of soy are reduced, making fermented soy more beneficial to your digestive system.
10. Stop Using OTC Painkillers!
Many people have the habit of taking over-the-counter painkillers to control every little pain instead of using natural remedies. There are pills for fever, for cold, for inflammation—all these can be very damaging to the kidneys. Same goes with long-term medications.
Pharmaceutical drugs reduce blood flow to the kidneys, affecting their functions. Long-term use of these drugs can cause acute renal failure.
A diuretic refers to a drug or food that increases the passing of urine for various reasons, either to increase loss of water and salt from the body, or in cases of infection, to be rid of bacteria in the kidneys or bladder.
This juice recipe helps increase urination that cleanses the kidneys and prevent stones formation. Drinking plenty of water along with drinking this juice is essential for better prevention or for dissolving stones that are already formed.
Another food that is excellent for breaking down kidney stones is watermelon and its rind! Go here to learn more about watermelon rind.
This broth is one of the best for cleansing kidneys and is beneficial for individuals with kidney disease. The key ingredient that heals, is celeriac.
Read more tips on how you can do a complete kidney detox.
This article was originally published in DrMercola.com.


text:
\section{Introduction.}
The  most general class of non-static, perfect fluid solutions of Einstein's
equations that are conformally flat is known as the``Stephani
Universe''\cite{KSMH},\cite{krasbook}. These mathematicaly simple solutions are
characterized (in general) by a shear-free but expanding and accelerating
4-velocity, with perfect fluid FLRW spacetimes being the acceleration free
subcase. The sphericaly symmetric Stephani
Universes\footnote{Unless specificaly stated otherwise, all further mention of
``Stephani Universes'' will refere to sphericaly symmetric Stephani Univeses
excluding their FLRW particular subclass} and some of their subcases have been examined in numerous papers (see
\cite{krasbook} for a comprehensive review). For example,  as star models
\cite{bofa},\cite{TW_1},\cite{TW_2},\cite{nato},\cite{bondi} and
as cosmologies generalizing FLRW\cite{krasbook},\cite{hen},\cite{pdl}. Other
papers have looked at their global properties and
singularities\cite{cook},\cite{kras83},\cite{suss88_1},\cite{suss88_2},
\cite{suss89},\cite{dabr93}, the thermodynamics of their fluid source
\cite{bonacol},\cite{qs},\cite{kqs}, and, more recently, as models with
inhomogeneous pressure in which to test recent astronomical data related to
supernova observations\cite{dabr95},\cite{dabr98}. Although these solutions  are
popular because of their mathematical simplicity, they are not considered
physicaly meaningful because their fluid source seems quite artificial
and excessively restrictive: matter-energy density,
$\rho$, depends only on $t$ while pressure,
$p$, depends on $t$ and $r$ (in comoving coordinates). Another objection
to the Stephani Universes is their incompatibility with a barotropic
relation\cite{mans},\cite{maspar} of the form
$p=p(\rho)$. However, the lack of fulfilment of barotropic
relations is not a strong argument to dismiss a given fluid
solution, since the latter might be compatible with more meaningful
non-barotropic equations of state.

The thermodynamics of the fluid source of the Stephani Universes is given by the
balance  equations of energy, momentum and entropy per particle, as well as by
the particle number conservation and the equilibrium Gibbs equation.
Previous papers have shown that the general Stephani Universe, having as source
a single component perfect fluid, is incompatible with the integrability
conditions of the Gibbs equation\cite{bonacol},\cite{kqs}, so that a ``
thermodynamical scheme'' is not possible (or in other words, a mathematicaly
consistent and closed set of thermodynamical equations of state is incompatible
with the field equations). However, this restriction does not apply to the
sphericaly symmetric case, or to any subcase whose isometry group has orbits
of dimension 2 or larger, hence mathematicaly simple equations of state have
been found\cite{qs} satisfying energy conditions, but without any basis on actual
physical matter models\footnote{The subclass of Stephani Universes studied in ref
\cite{qs} was erroneously reported as lacking isometries when it is sphericaly
symmetric. See \cite{krasbook} and \cite{kqs}}.

Accepting the above mentioned limitations, this paper aims at looking for a
reasonable (or at least, less objectionable) interpretation for the Stephani
Universes. Using the Gibbs equation as a definition of temperature, $T$, it is
posible to determine the four free functions by imposing the constraint:
$p=nk_{_B}T$, where $n$ is particle number density and $k_{_B}$ is Boltzmann's
constant. However, the fulfilment of the relation between $\rho$ and $p$
characteristic of the equation of state of the ideal gas\cite{rkt} (together
with $p=nk_BT$) cannot hold because $\rho=\rho(t)$ while $p=p(t,r)$.  We
consider the equations of state of: (A) a single component classical monatomic
ideal gas, and (B) a mixture of non-relativistic and relativistic ideal gases,
where the internal energy of the former has been ignored (hence, $p$ and $T$ are
the radiation pressure and temperature). By writing up the explicit form of the
state variables for the forms of the free functions that satisfy $p=nk_BT$, we
provide explicitly the conditions for a tight approximation to the full
equations of state (A) and (B) in terms of the convergence of a series of powers
of adimensional products of
$\rho$ and a parameter $\varepsilon_0$, proportional to the energy ratio
$k_B\hat T_0/mc^2$, where $\hat T_0$ is a suitable constant
temperature value and $m$ is the particles mass. This series converges in a
matter dominated low temperature regime characterized by $\varepsilon_0\ll 1$,
for models having spacelike slices (hypersurfaces of constant $t$, orthogonal to
the 4-velocity) with constant positive curvature (like ``closed''
FLRW cosmologies).  Under this approximation, the evolution of the fluid in the
matter dominated regime becomes very close to a dust FLRW cosmology, to such a
degree that observational parameters $H_0,\,\Omega_0,\,q_0$ become practicaly
indistinguishable from those of the FLRW spacetime. The same models for which the
above mentioned series converges are the only ones admiting an evolution range
that avoids unphysical features, such as a ''Finite Density'' singularity and an
``Asymptoticaly deSitter
boundary''(see \cite{suss88_1} to \cite{dabr93}), and so allowing for
a regular initial Cauchy hypersurface. Models whose spacelike slices orthogonal
to the 4-velocity have zero or negative curvature lack a regular Cauchy
hypersurface, while the series governing the approximation to the ideal gas
equations of state (A) and (B) only converge for comoving observers near the
symmetry center
$r=0$.

Considering their qualities and defects, the subclass of Stephani Universes
derived and discussed in the present paper provide a reasonable and theoreticaly
consistent interpretation for this simple and well established class of exact
solutions.

\section{The sphericaly symmetric Stephani Universes.}

The metric in comoving coordinates for the sphericaly symmetric Stephani
Universes is usualy given in such a way as to emphasize its resemblance with
FLRW cosmologies\cite{krasbook}. However, we shall use instead the following
alternative comoving coordinates:

$$ ds^2=-U^2c^2dt^2+ L^2\left[dr^2+f^2\left( {d\theta ^2+\sin ^2\theta \kern
1pt d\varphi ^2}
\right)\right]\eqno(1a)$$
$$U={{{1+\left( K-{R\kern 1pt K_{,R}}
\right)F^2} \over {1+K\kern 1pt F^2}}},\qquad L={{R}\over { {1+K\kern 1pt
F^2} }}\eqno(1b) $$

\noindent where $R=R(t)$, $K=K(t)$ and $K_{,R}=K_{,t}/R_{,t}=d K/dR $, and the
functions $F(r),f(r)$ are given by the three possible combinations
$$\matrix{{f=r,\quad F=r/2}\cr
{f=\sin r,\quad F=\sin (r/2)}\cr
{f=\sinh r,\quad F=\sinh (r/2)}\cr
}\eqno(2)$$

\noindent
The transformation of the radial coordinate $r$, relating (1)-(2) with the usual
radial coordinate for the Stephani Universes, $\bar r$, is given by
$r=\int{d\bar r}/(1+k\bar r^2/4)$, where $k=0,\pm 1$. The matter tensor
associated with (1) is a perfect fluid

$$ T^{a b}=\rho u^a u^b +p h^{a b}\eqno(3)$$
$$u^a=\frac{1}{U} \delta^a_t       \eqno(4)$$

\noindent where $ h^{a b}=c^{-2}u^au^b+g^{a
b}$ and $\rho,\,p$ are matter energy density and equilibrium pressure. The
time coordinate in (1) has been selected so that the expansion
scalar $\Theta=u^a_{;a}$ is given by
$$\frac{\Theta}{3}=\frac{R_{,t}}{R}    \eqno(5)$$

\noindent
The fluid source of the Stephani Universes is shear-free, hence the remaining nonzero kinematic
scalar is the 4-acceleration: $\dot u_a\equiv u_{a;b}u^b$, whose explicit form
for (1) is
$$\dot u_a=\frac{U_{,r}}{U}\delta _a^r=-{{RK_{,R}\,f} \over {2\left[ {1+KF^2}
\right]\left[ {1+\left( {K-RK_{,R}}
\right)F^2} \right]}}\;\delta _a^r\eqno(6)$$

A perfect fluid source like (2) satisfies energy-momentum balance:
$T^{ab}\,_{;b}=0$, particle number conservation $(nu^a)_{;a}=0$ and entropy
balance: $(nsu^a)_{;a}=0$, where $n,s$ are particle number density and entropy
per particle. These balance and conservation laws take the following forms for
(1) and (2)

$$u_aT^{ab}\,_{;b}=\dot \rho +(\rho+p)\Theta=0\eqno(7a)$$
$$h_{ab}T^{bc}\,_{;c}=h_a\,^bp_{,b}+(\rho+p)\dot u_a=0\eqno(7b)$$
$$(nu^a)_{;a}=\dot n+n\Theta=0\eqno(7c)$$
$$ (nsu^a)_{;a}=(\dot n+n\Theta)s+n\dot s=0\quad\Rightarrow\quad \dot s=0
    \eqno(7d)$$

\noindent
where $\dot A=u^a\,A_{,a}=u^t\,A_{,t}$ is the proper time derivative for any
scalar function $A$ and $\Theta,\,\dot u_a$ are given by (5) and (6). The state
variables must also comply with the equilibrium Gibbs equation

$$T\rm{d}s=\rm{d}\left(\frac{\rho}{n}\right)+p\rm{d}\left(\frac{1}{n}\right)\eqno(8)$$

\noindent
where $T$ is absolute temperature.

The field equation $G^t\,_t=(8\pi G/3c^2) T^t\,_t $ for (1), (2) and (3) is

$${{8\pi G} \over {3c^2}}\;\rho =\left( {{{R_{,t}} \over R}}
\right)^2+{{\left( {K+k} \right)c^2} \over {R^2}}\eqno(9)$$

\noindent where:
$$k=\left\{ \matrix{\;\;0,\quad \quad f=r,\quad F=r/2,\hfill\cr
  \;\;1,\quad \quad f=\sin (r),\quad F=\sin (r/2),\hfill\cr
  -1,\quad \quad f=\sinh (r),\quad F=\sinh (r/2),\quad \hfill\cr}
\right.\eqno(10)$$

\noindent while the form of $p$ follows from (7a) as
$$p =-\rho -{R
\over 3}\frac{\rho _{,R}}{U}=-\rho -{R
\over 3}\rho _{,R}\left[ {{{1+K\kern 1pt F^2} \over
{1+\left( {K-R\kern 1pt K_{,R}}
\right)F^2}}} \right]    \eqno(11)$$

\noindent
The momentum ballance (7b) is

$$ p^\prime+{{fR^2\rho _{,R}\kern 1pt K_{,R}} \over
{6\left[ {1+\left( {K-R\kern 1pt K_{,R}} \right)F^2}
\right]^2}}=0\eqno(12)$$

\noindent
where a prime denotes derivative wrt to $r$ and (6) and (11) were used to
eliminate $\dot u_a$ and $\rho+p$. The particle number density is given by
integrating (7c)

$$ n={{N(r)} \over {R^3}}\left[1+KF^2 \right]^3\eqno(13)$$

\noindent
where $N$ is the conserved particle number distribution.  Since the spacial
coordinates in (1) are comoving, the entropy balance (7d) yields simply $s=s(r)$,
and so, with the help of (7a) and (7c), the Gibbs equation (8) reduces to

$$Ts^\prime=(\rho+p)\left(\frac{1}{n}\right)^\prime\eqno(14)$$

\noindent
with $s'\ne 0$, since the fluid of Stephani Universes is not isentropic (or
equivalently, does not admit a barotropic relation $p=p(\rho)$).

\section{Determination of the free parameters.}
The  Stephani Universes contain (in general) two time dependent free functions
($R(t),\,K(t)$ in (1)) whose determination is elusive. The imposition of an
equation of state (ie a constraint relating state variables) should lead to the
determination of these free-parameters, howewer not all relations between the
state variables is compatible with these free functions and with the radial
dependence of the state variables (in particular, the barotropic relation
$p=p(\rho)$ is incompatible, see
\cite{mans},\cite{maspar}). Conversely, if the two free functions are determined
by suitable boundary conditions, or prescribed in order to obtain specific
kinematic effects, a specific constraint among state variables ensues, though
such constraint is a ``formal'' equation of state and might be wholy unphysical.
The usual boundary conditions invoked for determining the free functions are: a
matching with a Schwarzschild vacuum exterior along a comoving boundary and/or a
relation between $p$ and $\rho$ along a symmetry
center\cite{bofa},\cite{TW_1},\cite{TW_2},\cite{nato},\cite{bondi}, or the
desire to obtain specific kinematic efects, like bouncing \cite{bofa} or
acceleration of comoving observers\cite{dabr95},\cite{dabr98}, or a metric
element that is conformal to that of a FLRW
metric\cite{hen},\cite{pdl},\cite{suss89},\cite{dabr95},\cite{dabr98}. Besides
the time dependent free parameters, a proper thermodynamical approach produces
two extra free functions of $r$: the initial particle number distribution,
$N(r)$, and the entropy per particle $s(r)$.

The Stephani
Universes, as exact solutions subjected to the thermodynamical laws (7c), (7d)
and (8), become determined once the four free functions
$R(t),\,K(t),\,N(r),\,s(r)$ are either prescribed, found or obtained. From
equations (1)-(14), once $\rho$ and $K$ are known as functions of
$R$, the field equation (9) becomes the Friedmann equation whose integration
yields $R$ as a function of $t$. However, the functions $N(r),\,s(r)$ cannot be
determined from the geometry but from thermodynamical criteria.
The usual strategies to determine the free functions have lead, in all cases, to
equations of state whose physics has still remained unclear. We propose in
this paper another strategy for the determination of these free functions.

Consider the following relation associated with the equation of state of an ideal
gas:
$$p=nk_{_B}T\eqno(15)$$
\noindent  where $k_{_B}$ is Boltzmann's constant. Using  $p,\,n$ from (11) and
(13), the fulfilment of (15) implies the constraint

$$k_{_B}T=\frac{k_{_B}(\rho+p)}{s'}\left(\frac{1}{n}\right)^\prime
=\frac{p}{n}\eqno(16a)$$

\noindent
which becomes explicitly
$$\left( {1+KF^2} \right)\left[ {\rho +{{R\rho _{,R}} \over 3}\left(
{1+{{k_BN'} \over {s'N}}} \right)} \right]+RF\left[ {{{2k_BF'} \over {s'}}K\rho
_{,R}-\rho K_{,R}F} \right]=0\eqno(16b)$$
\noindent
The only physicaly reasonable solution of (16) (other solutions yield
negative values of $n$ and $T$) follows if the terms in square parenthesis
vanish independently. Since
$\rho,\,K$ depend on $R$ while $F,\,N,\,s$ depend on $r$, we have
$${{R\rho _{,R}} \over {3\rho }}=-\left( {1+{{k_BN'} \over {s'N}}}
\right)^{-1}=a\eqno(17a)$$
$${{2k_BF'} \over {s'F}}={{K_{,R}/K} \over {\rho _{,R}/\rho }}=b\eqno(17b)$$
\noindent
where $a,\,b$ are arbitrary nonzero real constants. Equations (17) lead to
$$\rho =\rho _0\left( {{{R_0} \over R}} \right)^{3a},\quad  \quad K=k_1\left(
{{\rho  \over {\rho _0}}} \right)^{b}=k_1\left( {{{R_0} \over R}}
\right)^{3ab}\eqno(18a)$$
$$\log N={{2(1-a)} \over b}\log F,\quad \quad s=s_0+{{ak_B}  \over {1-a}}\log
N,\quad \quad a\ne 1\eqno(18b)$$
$$N=N_0,\quad \quad \quad \quad \quad s=s_0+{{2k_B}  \over b}\log F,\quad \quad
\quad a=1\eqno(18c)$$

\noindent
Inserting (18) into (9), (11), (13) and using (15) leads to
$$\left( {{{R_{,t}} \over R}} \right)^2={{8\pi G} \over {c^2}}\rho _0\left(
{{{R_0} \over R}} \right)^{3a}-{{k_1c^2} \over {R^2}}\left( {{{R_0} \over R}}
\right)^{3ab}-{{k_0c^2} \over {R^2}}\eqno(19a)$$
$$p={{(a-1)\rho _0+\left[ {(1-3b)a-1} \right]\rho _0k_1\left( {R_0/R}
\right)^{3ab}F^2} \over {1+(1+3ab)k_1\left( {R_0/R} \right)^{3ab}F^2}}\;\left(
{{{R_0} \over R}} \right)^{3a}\eqno(19b)$$
$$n={N \over {R^3}}\left[ {1+k_1\left( {{{R_0} \over R}} \right)^{3ab}F^2}
\right]^3\eqno(19c)$$
$$k_BT={{R_0^3} \over N}{{(a-1)\rho _0+\left[ {(1-3b)a-1} \right]\rho
_0k_1\left( {R_0/R} \right)^{3ab}F^2} \over {\left[ {1+k_1\left( {R_0/R}
\right)^{3ab}F^2} \right]^3\left[ {1+(1+3ab)k_1\left( {R_0/R} \right)^{3ab}F^2}
\right]}}\;\left( {{{R_0} \over R}} \right)^{3(a-1)}\eqno(19d)$$
From (18d), (18d), (19c) and (19d) it is evident that the choice $a\ne 1$ leads
to a diverging temperature along a symmetry center marked by $F(0)=0$. Hence, we
must consider only $a=1,\,b>0$, but in this case, in order to have positive
$p$ and $T$, we must also assume $k_1<0$. It turns to be convenient to redefine
the constant parameters $b$ and $k_1$ as
$$b=\gamma-1,\quad -k_1=\frac{\varepsilon_0}{3(\gamma-1)}$$
\noindent
where $\varepsilon_0$ is a positive constant. The metric functions in
(1), 4-velocity and 4-acceleration (4) and (6) become
$$U={{1-\varepsilon _0(\gamma _2/\gamma_1)F^2(R_0/R)^{\gamma_1}}  \over
{1-(\varepsilon _0/\gamma _1)F^2(R_0/R)^{\gamma_1}}},\quad L={R \over
{1-(\varepsilon _0/\gamma _1)F^2(R_0/R)^{\gamma_1}}}\eqno(20)$$
$$u^a={1 \over U}\delta ^a_t,\quad \dot u_a=\aleph\delta _a^r  ,\quad
\aleph={{{\textstyle{1 \over 2}}\varepsilon _0(R/R_0)^{\gamma_1}f} \over {\left[
{(R/R_0)^{\gamma_1}-(\varepsilon _0/\gamma _1)F^2} \right]\left[
{(R/R_0)^{\gamma_1}-\varepsilon _0(\gamma _2/\gamma_1)F^2} \right]}}\eqno(21)$$
where
$$ \gamma _1=3(\gamma -1) ,\quad
\gamma _2=3\gamma -2$$
The state variables in (19) take the forms
$$\rho =\rho_0\left( {{{R_0} \over R}} \right)^3\eqno(22a)$$
$$ p={{\rho_0\varepsilon_0\;F^2} \over {1-\varepsilon
_0(\gamma_2/\gamma_1)\left( {\rho /\rho _0} \right)^{\gamma -1}F^2}}\left(
{{\rho
\over {\rho _0}}} \right)^\gamma\eqno(22b)$$
$$ n={{N_0} \over {R_0^3}}\left( {{\rho  \over {\rho _0}}} \right)\left[
{1-\frac{\varepsilon _0}{\gamma_1}\left( {{\rho  \over {\rho _0}}}
\right)^{\gamma -1}F^2} \right]^3\eqno(22c)$$
$$k_BT={{\rho_0\varepsilon_0 R_0^3\;F^2} \over {N_0\left[
{1-(\varepsilon _0/\gamma_1)\left( {\rho /\rho _0} \right)^{\gamma -1}F^2}
\right]^3\left[ {1-\varepsilon _0(\gamma_2/\gamma_1)\left( {\rho /\rho _0}
\right)^{\gamma -1}F^2}
\right]}}\;\left( {{\rho  \over {\rho _0}}} \right)^{\gamma
-1}\eqno(22d)$$
while the Friedmann equation (9) is given by
$$(R_{,t})^2={{8\pi G\rho_0R_0^3} \over
{3c^2R}}+\frac{\varepsilon_0c^2}{\gamma_1}\left( {{{R_0}
\over R}}
\right)^{\gamma_1}-kc^2\eqno(23)$$
This equation must be integrated in order to obtain $R=R(t)$ and have the models
fuly determined.

So far, we have determined the four free parameters of the Stephani Universe by
imposing the constraint (15). From (18c) and (22a), the state variables in
(22b-d) can be all be expressed as functions of two primary
thermodynamical variables $(\rho,s)$, so that rewriting (8) in terms of the
latter and using (15), it is straightforward to prove that the integrability
conditions of this Gibbs equation are fuly satisfied. However, the equations of
state of ideal  gases impose, besides (15), further relations among the
remaining state variables\cite{rkt}. We look at two of these equations of state
in the following section.

\section{Equations of state.}
The fulfilment of the relation $ p=nk_BT$ leads to the consideration of various
possible ideal gas equations of state for the fluid. However, we have:
$\rho=\rho(t)$, while the remaining state variables depend on both $t$ and
$r$, hence these equations of state will not hold exactly, though the constant
parameters $\gamma,\,\varepsilon_0,\,\rho_0$ can be determined in such a way that
the full equation of state holds under suitable approximations.  Two cases will
be examined separately below.

\subsection{The classical monatomic ideal gas}
For this case the equation of state is (15) together with\cite{rkt},\cite{susstr}
$$\rho=mc^2n+\frac{3}{2}n\,k_BT \eqno(24)$$
As mentioned before, an equation like (24) cannot be satisfied exactly. In order
to examine under which conditions it can be satisfied approximately, we use
(15), (22b) and (22c) to write the right hand side of (24) expanded up to third
order in
$\varepsilon_0$
$$mc^2n+{3 \over 2}nk_BT={{mc^2N_0} \over {R_0^3}}\left( {{\rho   \over {\rho
_0}}} \right)+\left[ {{3 \over 2}\rho _0-{{mc^2N_0} \over {(\gamma -1)R_0^3}}}
\right]\left( {{\rho  \over {\rho _0}}} \right)^\gamma F^2\varepsilon _0+\left[
{{{3\gamma -2} \over {2(\gamma -1)}}\rho _0+{{mc^2N_0} \over {3(\gamma
-1)^2R_0^3}}} \right]\left( {{\rho  \over {\rho _0}}} \right)^{2\gamma
-1}F^4\varepsilon _0^2
$$
$$+\left[ {{{(3\gamma -2)^2} \over {6(\gamma
-1)^2}}\rho _0-{{mc^2N_0} \over {27(\gamma -1)^3R_0^3}}} \right]\left( {{\rho
\over {\rho _0}}} \right)^{3\gamma -2}F^6\varepsilon _0^3+O(\varepsilon
_0^4)\qquad\qquad\qquad\qquad\eqno(25)
$$
Since $\rho_0,\,R_0$ are $\rho,\,R$ evaluated at present cosmic time $t=t_0$
when matter dominated conditions are assumed to prevail, then energy density is
overwhelmingly dominated by rest mass energy, and so, looking for the best
possible approximation of (25) to (24), leads to the following identifications
in (25)
$$\rho_0=\frac{mc^2N_0}{R_0\,^3},\qquad
\frac{1}{\gamma-1}=\frac{3}{2}\quad \Rightarrow\quad \gamma=\frac{5}{3}
\eqno(26)$$
Inserting (26) into (25) yields
$$mc^2n+{3 \over 2}nk_BT=\rho \left[ {1+3\left( {{\rho  \over {\rho _0}}}
\right)^{4/3}F^4\varepsilon _0^2+{{13} \over 4}\left( {{\rho  \over {\rho _0}}}
\right)^2F^6\varepsilon _0^3+O(\varepsilon _0^4)} \right]\eqno(27)$$
so that the leading nonzero order term in the series inside the square
brackets is quadratic in $\varepsilon_0$.  Comparing (27) with (24) shows that
(22) and (26) lead to a good approximation to (24) provided the power series
inside the square brackets in (27) converges. A sufficient condition for this
convergence requires that each term (with the exception of the zero order term)
must satisfy
$$\left( {{\rho  \over {\rho _0}}} \right)^{2\xi/3}F^{2\xi}(\varepsilon
_0)^\xi\ll1,\quad
\quad \xi=2,3,..\eqno(28)$$
The form of the adimensional parameter $\varepsilon_0$ follows by
using (22b), (22d) and (26) in order to expand $p$ and $k_BT$ around
$\varepsilon_0$

$$p=\frac{mc^2N_0\,\varepsilon_0\,F^2}{R_0^3}\left(\frac{\rho}
{\rho_0}\right)^{5/3}\left[1+\frac{3}{2}\left(\frac{\rho}
{\rho_0}\right)^{2/3}F^2\varepsilon_0+\frac{9}{4}\left(\frac{\rho}
{\rho_0}\right)^{4/3}F^2\varepsilon_0^2+O(\varepsilon_0^3)\right]
\eqno(29a)
$$
$$k_BT=mc^2\,\varepsilon_0\,F^2\left(\frac{\rho}
{\rho_0}\right)^{2/3}\left[1+3\left(\frac{\rho}
{\rho_0}\right)^{2/3}F^2\varepsilon_0+6\left(\frac{\rho}
{\rho_0}\right)^{4/3}F^2\varepsilon_0^2+O(\varepsilon_0^3)\right]
\eqno(29b)
$$
By looking at the common factor outside the square brackets, it becomes natural
to identify
$$ mc^2\,\varepsilon_0=k_B\hat T_0\qquad \Rightarrow\qquad
\varepsilon_0=\frac{k_B\hat T_0}{mc^2}
\eqno(30)$$
where $\hat T_0$ is a suitable constant temperature value. With the help of
(22), (26) and (30), the exact form of the state variables becomes
$$p={{(N_0k_B\hat T_0/R_0^3)\;F^2} \over {1-{\textstyle{{3} \over
{2}}}\varepsilon _0\left( {\rho /\rho _0} \right)^{2/3}F^2}}\left( {{\rho
\over {\rho _0}}} \right)^{5/3} ={{(N_0k_B\hat T_0/R_0^3)\;F^2} \over
{1-{\textstyle{3 \over {2}}}\varepsilon _0\left( {R_0/R}
\right)^2 F^2}}\left( {{{R_0} \over R}} \right)^5\eqno(31a)$$
$$n={{N_0} \over {R_0^3}}\left( {{\rho  \over {\rho _0}}} \right)\left[
{1-{{\varepsilon _0} \over {2}}\left( {{\rho  \over {\rho _0}}}
\right)^{2/3}F^2} \right]^3={{N_0} \over {R^3}}\left[
{1-{{\varepsilon _0}
\over {2}}\left( {{{R_0} \over R}} \right)^2F^2}
\right]^3\eqno(31b)$$
$$k_BT={{k_B\hat T_0\;F^2} \over {\left[ {1-{\textstyle{{\varepsilon _0}
\over {2}}}\left( {\rho /\rho _0} \right)^{2/3}F^2}
\right]^3\left[ {1-{\textstyle{{3} \over {2}}}\varepsilon _0\left( {\rho /\rho
_0} \right)^{2/3}F^2}
\right]}}\;\left( {{\rho  \over {\rho _0}}} \right)^{2/3}
       ={{k_B\hat T_0\;F^2} \over {\left[ {1-{\textstyle{{\varepsilon _0}
\over {2}}}\left( {R_0/R} \right)^2F^2} \right]^3\left[
{1-{\textstyle{{3} \over {2}}}\varepsilon _0\left( {R_0/R} \right)^2F^2}
\right]}}\;\left( {{{R_0} \over R}} \right)^2\eqno(31c)$$
while the Friedmann equation (22d) is given by
$$(R_{,t})^2={{8\pi GmN_0} \over {3R}}+{{\varepsilon _0c^2} \over {2}}\left(
{{{R_0} \over R}} \right)^2-kc^2\eqno(32)$$

The degree in which these state variables provide a good approximation to the
equation of state (15) (24) strongly depends on the behavior of $F$,
$\rho/\rho_0$ and on the value of $\varepsilon_0$ in (30), a small number for an
enormous range of temperatures (if $m$ is a protonic mass, we have
$\varepsilon_0\approx 10^{-4}$ for as high as $\hat T_0\approx 10^8$ degrees K.).
The fulfilment of (28) for specific energy and temperature ranges of the models
will be discussed in section VII.

\subsection{Mixture of matter and radiation}
Consider a mixture of two ideal gases, one non-relativistic (superindex ``I'')
and the other ultra-relativistic (superindex ``II''), the equation of state in
this case is
$$\rho=mc^2n^{\rm{I}}+\frac{3}{2}n^{\rm{I}}k_BT^{\rm{I}}
+3n^{\rm{II}}k_BT^{\rm{II}}\eqno(33a)$$
$$p=n^{\rm{I}}k_BT^{\rm{I}}+n^{\rm{II}}k_BT^{\rm{II}}\eqno(33b)$$
If the internal energy of the non-relativistic gas can be neglected:
$n^{\rm{I}}k_BT^{\rm{I}}\ll n^{\rm{II}}k_BT^{\rm{II}}$ but (at least at a
given stage of the evolution) the energy density $n^{\rm{II}}k_BT^{\rm{II}}$ is
not negligible in comparison with the rest mass energy $mc^2n^{\rm{I}}$, then
(33) can be approximated by\cite{susstr},\cite{susspav}
$$\rho= mc^2n^{\rm{I}}+3n^{\rm{II}}k_BT\eqno(34a)$$
$$p=n^{\rm{II}}k_BT\eqno(34b)$$
\noindent
where $T=T^{\rm{II}}$ and the particle number densities independently satisfy
conservation laws like (7c), therefore $n^{\rm{I}},\,n^{\rm{II}}$ are given by
expressions identical to (22c) with $N_0$ replaced by
$N_0^{\rm{I}}$ and $N_0^{\rm{II}}$. The approximation (34a-b) can be especialy
suited for a mixture of photons and barions\cite{susspav} bacause the ratio of
the former to the latter is such a large number ($\approx 10^8$), leading to a
reasonable description of cosmological mixtures of non-relativistic matter and
radiation for energy and temperature ranges prevailing after cosmological
nucleosynthesis, including the radiative era (recombination and decoupling), and
up to the present.

As in the previous subsection, (34a) cannot hold exactly but can be aproximated
expanding its right hand side in terms of $\varepsilon_0$. From the forms of
$n^{\rm{I}}$ and $T=T^{\rm{II}}$\,from (22c) and (22d),
and using (34b) in the form $3n^{\rm{II}}k_BT=3p$ with $p$ given by (22b),\, we
obtain
$$mc^2n^{\rm{I}}+3n^{\rm{II}}k_BT={{mc^2N_0^{\rm{I}}} \over {R_0^3}}\left(
{{\rho   \over {\rho _0}}} \right)+\left[ {3\rho _0-{{mc^2N_0^{\rm{I}}} \over
{(\gamma -1)R_0^3}}}
\right]\left( {{\rho  \over {\rho _0}}} \right)^\gamma F^2\varepsilon _0+\left[
{{{3\gamma -2} \over {(\gamma -1)}}\rho _0+{{mc^2N_0^{\rm{I}}} \over {3(\gamma
-1)^2R_0^3}}} \right]\left( {{\rho  \over {\rho _0}}} \right)^{2\gamma
-1}F^4\varepsilon _0^2
$$
$$+\left[ {{{(3\gamma -2)^2} \over {3(\gamma
-1)^2}}\rho _0-{{mc^2N_0^{\rm{I}}} \over {27(\gamma -1)^3R_0^3}}} \right]\left(
{{\rho
\over {\rho _0}}} \right)^{3\gamma -2}F^6\varepsilon _0^3+O(\varepsilon
_0^4)\qquad\qquad\qquad\qquad\eqno(35)
$$
so that, considering matter dominated conditions and aiming at the best possible
approximation, we identify
$$\rho_0=\frac{mc^2N_0^{\rm{I}}}{R_0\,^3},\qquad
\frac{1}{\gamma-1}=3\quad \Rightarrow\quad \gamma=\frac{4}{3}
\eqno(36)$$
transforming (35) into

$$mc^2n^{\rm{I}}+{3 \over 2}n^{\rm{II}}k_BT=\rho \left[ {1+9\left( {{\rho  \over
{\rho _0}}}
\right)^{2/3}F^4\varepsilon _0^2+11\left( {{\rho  \over {\rho _0}}}
\right)^{4/3}F^6\varepsilon _0^3+O(\varepsilon _0^4)} \right]\eqno(37)$$

\noindent
indicating, as in the previous subsection, that a good approximation to (34a)
requires the convergence of the power series inside the square brackets, leading
to a condition very similar to (28)
$$\left( {{\rho  \over {\rho _0}}} \right)^{\xi/3}F^{2\xi}(\varepsilon
_0)^\xi\ll1,\quad
\quad \xi=2,3,..\eqno(38)$$
The form of $\varepsilon_0$ follows, as in the previous subsection,  by obtaining
with the help of (22b), (22d) and (36) the forms of $p$ and $k_BT$ expanded in
terms of $\varepsilon_0$, leading  to
$$p=\frac{mc^2N_0^{\rm{I}}\,\varepsilon_0\,F^2}{R_0^3}\left(\frac{\rho}
{\rho_0}\right)^{4/3}\left[1+2\left(\frac{\rho}
{\rho_0}\right)^{1/3}F^2\varepsilon_0+4\left(\frac{\rho}
{\rho_0}\right)^{2/3}F^2\varepsilon_0^2+O(\varepsilon_0^3)\right]
\eqno(39a)
$$
$$k_BT=\frac{mc^2N_0^{\rm{I}}\,\varepsilon_0\,F^2}{N_0^{\rm{II}}}
\left(\frac{\rho} {\rho_0}\right)^{1/3}\left[1+5\left(\frac{\rho}
{\rho_0}\right)^{1/3}F^2\varepsilon_0+16\left(\frac{\rho}
{\rho_0}\right)^{2/3}F^2\varepsilon_0^2+O(\varepsilon_0^3)\right]
\eqno(39b)
$$
so that it becomes natural to identify
$$ mc^2N_0^{\rm{I}}\,\varepsilon_0=N_0^{\rm{II}}k_B\hat
T_0,\qquad\Rightarrow\qquad \varepsilon_0=\frac{N_0^{\rm{II}}k_B\hat
T_0}{N_0^{\rm{I}}mc^2}
\eqno(40)$$
where $\hat T_0$ is a suitable constant temperature value for the relativistic
component. Using (22), (23), (36) and (40), the state variables take the
following exact forms
$$p={{(N_0^{\rm{II}}k_B\hat T_0/R_0^3)\;F^2} \over {1-2\varepsilon
_0\left( {\rho /\rho _0} \right)^{1/3}F^2}}\left( {{\rho
\over {\rho _0}}} \right)^{4/3} ={{(N_0^{\rm{II}}k_B\hat T_0/R_0^3)\;F^2} \over
{1-2\varepsilon _0\left( {R_0/R}
\right) F^2}}\left( {{{R_0} \over R}} \right)^4\eqno(41a)$$
$$n={{(N_0^{\rm{I}}+N_0^{\rm{II}})} \over {R_0^3}}\left( {{\rho  \over
{\rho _0}}}\right)\left[ {1-\varepsilon _0F^2\left( {{\rho  \over {\rho _0}}}
\right)^{1/3}} \right]^3={{(N_0^{\rm{I}}+N_0^{\rm{II}})} \over {R^3}}\left[
{1-\varepsilon _0F^2\left( {{{R_0} \over R}} \right)}
\right]^3\eqno(41b)$$
$$k_BT={{k_B\hat T_0\;F^2} \over {\left[ {1-\varepsilon _0F^2\left( {\rho
/\rho _0} \right)^{1/3}}
\right]^3\left[ {1-2\varepsilon _0F^2\left( {\rho /\rho
_0} \right)^{1/3}}
\right]}}\;\left( {{\rho  \over {\rho _0}}} \right)^{1/3}
       ={{k_B\hat T_0\;F^2} \over {\left[ {1-\varepsilon _0F^2\left(
{R_0/R}\right)} \right]^3\left[ {1-2\varepsilon _0F^2\left(
{R_0/R} \right)}
\right]}}\;\left( {{{R_0} \over R}} \right)\eqno(41c)$$
$$(R_{,t})^2={{8\pi GmN_0^{\rm{I}}} \over {3R}}+\varepsilon _0c^2\left( {{{R_0}
\over R}} \right)-kc^2\eqno(42)$$

The fulfilment of (38) with $\varepsilon_0$ given by (40) controls how close
the state variables aproximate (34a). However in the present case the ratio
$N_0^{\rm{II}}/N_0^{\rm{I}}$ must be taken into consideration as well. This
will be discussed in section VII. Independently of the equations of state
presented so far, the evolution range of the models is necessarily restricted by
singularities and other geometric features. We examine these features in the
following section.

\section{Singularities, regularity domain, symmetry centers and FLRW limit}

Looking at $\rho$ $p$ and $T$ in equations (22), (31) and (41), pending the
determination of $R=R(t)$ by integrating (32) and (42), we can identify the
following scalar curvature singularities
$$R=0\eqno(43a)$$
$$1-\frac{3\gamma-2}{3(\gamma-1)}\varepsilon_0F^2
\left(\frac{R_0}{R}\right)^{3(\gamma-1)}=0\eqno(43b)$$
\noindent
The singularity (43a) is marked by a hypersurface $t=\hbox{const}$, all state
variables diverge and is similar to a ``big-bang'' in FLRW spacetimes.
The type (43b) has been identified in previous literature as a ``finite
density'' (FD) singularity\cite{suss88_1},\cite{suss88_2}. It is, in general, a
hypersurface marked by a curve in the
$(t,r)$ or $(R,r)$ plane, though it is an unphysical singularity because $p,\,T$
diverge while $\rho$ remains finite
$$\left[\frac{\rho}{\rho_0}\right]_{\rm{FD}}=\left[\frac{3(\gamma-1)}{(3\gamma-2)
\varepsilon_0F^2}\right]^{1/(\gamma-1)}\eqno(44)
$$
for all comoving observers, except along the center worldline $r=0$ where
$F(0)=0$ and so $[\rho/\rho_0]_{\rm{FD}}$ diverges. Particle number density also
remains finite, as can be verified by inserting (44) into (22c).

Another feature worth remarking is the points in spacetime where
$$1-\frac{\varepsilon_0}{3(\gamma-1)}F^2\left(\frac{R_0}{R}
\right)^{3(\gamma-1)}=0\eqno(45)$$
These points mark a spacetime boundary
characterized by an asymptoticaly inflationary or (asymptoticaly de
Sitter, ``ADS'') behavior\cite{suss88_2}: $p+\rho\to 0$ and
$n\to 0$. Although $k_BT=p/n$ diverges along (45) (because $n$ vanishes for
finite $p$), this is not realy a curvature singularity since all polynomial
curvature scalars are bounded.

Since for the cases we are interested ($\gamma=4/3,\,5/3$), the FD singularity
is marked by larger values of $R/R_0$ than the ADS boundary, and since this
singularity is spacelike \cite{suss88_1}\cite{suss88_2}, we will consider as the
regular evolution domain of the solutions all points of spacetime along the
fluid worldlines lying to the future of the FD singularity. These domain can be
characterized, for every comoving observer, by either one of the following
conditions

$$\frac{R}{R_0}>\left[\frac{(3\gamma-2)\varepsilon_0F^2}{3(\gamma-1)
}\right]^{1/(3\gamma-3)},\qquad
\frac{\rho}{\rho_0}<\left[\frac{3(\gamma-1)}{(3\gamma-2)
\varepsilon_0F^2}\right]^{1/(\gamma-1)}\eqno(46)
$$

From the material presented in previous sections, it is obvious that the
FLRW limit of the Stephani Universes is given by $\varepsilon_0=0\Rightarrow K=
0$ and
$N=N_0$, so that
$\dot u_a=p'=s'=0$. Since $\varepsilon_0=0$ implies $p=T=0$, this
limit leads to a dust FLRW universe, and so a sort of
asymptotic dust FLRW state can be reached as the fluid evolves in
time. Also, for whatever form of $R(t)$ obtained from integrating (32) and
(42), the worldlines marked by $r=r_c$ where $f(r_c)=0$, with $f$
given by (2) and (10), characterize the symmetry centers. Along these worldlines,
the metric function $\sqrt{g_{\theta\theta}}=Lf$, the radius of the orbits of the
rotation group, vanishes regularly. Also, $\dot u_a,\,p'$ vanish, as well the
radial gradients of all other geometric or thermodynamical variables should also
vanish, hence spacetime appears isotropic for observers along symmetry centers.
Depending on the functions in (2) and (10), the Stephani Universes admit one
($r=0$ for $k_0=0,-1$) or two ($r=0,\pi$ for $k_0=1$) symmetry centers.

The singularities (43a-b) and regular boundary (45) have been
examined extensively elsewhere, and so we refer the reader to the appropriate
references (see \cite{suss88_1} to \cite{dabr93}). Considering (46) for
the cases $k=0,-1$ of $F$ in (2) and (10), it is evident that
these cases do not allow for the existence of a hypersurface
$t=\hbox{const}$ in the regularity domain that is also a regular Cauchy surface.
In the case $k=1$ we have two symmetry centers marked by $r=0$ and
$r=\pi$, and so the full domain of the radial coordinate is $0\le r\le \pi$.
Since in this case we have $0\le F=\sin(r/2)\le 1$, it follows from (46) that a
regular Cauchy hypersurface $t=t_i$ can be found in the regularity domain, so
that a regular evolution takes place in the range $t_i<t<t_M$, where
$t_M$ marks the hypersurface of maximal expansion characterized by $\Theta=0$.
In this case, the regular hypersurfaces $t=\hbox{const}>t_i$ will be
diffeomorphic to the invariant spacelike slices of a ``closed'' FRLW spacetime.

\section{Observational features}
As shown in section IV, the fluid sources of the models approximately
satisfy the ideal gas equations of state (15),(24) and (34), provided conditions
such as (26), (28), (36) and (38) hold. The evolution of the models towards the
present time ($t\to t_0$, $R\to R_0$) approaches that of a FLRW dust cosmology,
characterizing a homogeneous and isotropic matter dominated universe. It is not
surprising that (15), (24) and (34) approach dust sources, the latter being realy
a convenient approximation of ideal gases in a low temperature and pressure
limits. However, the ``nearly dust FLRW'' approximation of the models must be
examined by verifying how the observational parameters approach as $t\to
t_0$ those associated with FLRW cosmologies.

The resemblance of the field equation (9) with the Friedmann equation of a FLRW
spacetime, together with the fact that $\rho=\rho(t),\,\Theta=\Theta(t)$, leads
to an invariant definition of observational parameters $H=\Theta/3=\dot
L/L=R_{,t}/R$ and $\Omega=8\pi G/(3c^2H^2)$ that is also very similar to their
FLRW equivalents. Evaluating along $R=R_0$, we have

$$H_0^2=\frac{8\pi
G}{3c^2}\rho_0+\left[\frac{\varepsilon_0}{\gamma_1}
-k\right]\frac{c^2}{R_0^2}\eqno(47a)
$$
$$\Omega_0\equiv \frac{8\pi G \rho_0}{3c^2H_0^2}=\frac{8\pi G\rho_0
R_0^2/3c^4}{8\pi G\rho_0 R_0^2/3c^4+(\varepsilon_0/\gamma_1)-k}\eqno(47b)
$$
$$
\Omega_0-1=\left(k-\frac{\varepsilon_0}{\gamma_1}\right)\frac{c^2}{(H_0R_0)^2}
\eqno(47c)
$$
generalizing the FLRW parameters
$$\tilde H_0^2 =\frac{8\pi
G}{3c^2}\rho_0-k\frac{c^2}{R_0^2},\qquad \tilde\Omega_0=\frac{8\pi G
\rho_0}{c^2\tilde H_0^2}=\frac{8\pi G\rho_0 R_0^2/3c^4}{8\pi G\rho_0
R_0^2/3c^4-k}\eqno(48)$$
where $\gamma_1=3(\gamma-1)$ and all quantities with tildes will henceforth be
FLRW quantities. Since we are interested in comparing observations in the
Stephani Universes and in FLRW dust cosmologies at $R=R_0$, it is useful to
expand $H_0$ and $\Omega_0$ in terms of $\tilde H_0,\,\tilde\Omega_0 $ up to
first order in $\varepsilon_0$

$$H_0\approx\tilde H_0\left[1+\frac{c^2}{\gamma_1(\tilde H_0
R_0)^2}\varepsilon_0+O(\varepsilon_0^2)\right],\qquad \Omega_0\approx
\tilde\Omega_0\left[1-\frac{c^2}{\gamma_1(\tilde H_0
R_0)^2}\varepsilon_0+O(\varepsilon_0^2)\right],\qquad
\tilde\Omega_0-1=\frac{kc^2}{(\tilde H_0 R_0)^2}\eqno(49)
$$
showing that, in general $H_0>\tilde H_0 $ and $\Omega_0<\tilde\Omega_0$, and
that
$H_0,\,\Omega_0$ are roughly equal to $\tilde H_0,\, \tilde\Omega_0$ plus
corrections of the order
$\approx\varepsilon_0\tilde\Omega_0\approx\varepsilon_0$. Another important
observational quantity is the deceleration parameter
$\tilde q$, defined for FLRW cosmologies as
$$\tilde q_0\equiv -\left[\frac{R_{,tt}R}{R_{,t}^2}\right]_0=
\frac{1}{2}\tilde\Omega_0$$
We can compute, with the help of $H$ given by (9) for the Stephani Universes, an
equivalent quantity $q_0$ and expand it in terms of $\tilde q_0$
$$q_0=-\left[\frac{H_{,t}+H^2}{H^2}\right]_0=\frac{1}{2}\frac{8\pi G\rho_0
R_0^2/3c^4+\varepsilon_0}{8\pi
G\rho_0R_0^2/3c^4+\varepsilon_0/\gamma_1-k}=\frac{\Omega_0}{2}
\frac{2+\varepsilon_0}{2-\varepsilon_0}-\frac{\varepsilon_0}{2-\varepsilon_0}
\eqno(50a)$$
$$q_0\approx\tilde
q_0+\frac{1}{2}\left[1-\frac{\tilde\Omega_0}{\gamma_1}\right]\frac{c^2}
{(\tilde H_0R_0)^2}\,\varepsilon_0+O(\varepsilon_0^2)\eqno(50b) $$
However the definition of $q$, unlike $H,\,\Omega$, is coordinate dependent. In
order to derive the appropriate equivalent of $\tilde q_0$ for the models based
on the Stephani Universes, we need to derive the general form of the
relation between red shift and luminosity distance for local
observations along
$t=t_0,\,r=r_0$.

Consider the simplest form of a radial null
vector
$k^a=(k^t,k^r)$  for the metric (1) with $U$ and $L$ given by (20)

$$k^t=\frac{cdt}{dv}=\frac{L}{U}=\frac{R}{1-\varepsilon _0\gamma
_2F^2(R_0/R)^{3(\gamma -1)}},\qquad k^r=\frac{dr}{dv}=\pm
1,\qquad u_ak^a=-L\eqno(51)$$
The effects of inhomogeneity and anisotropy on red shift observations in $t=t_0$
of nearby sources ($z\ll 1$) can be appreciated from the relation between
luminosity distance vs red shift given for the radial light rays (51) by
\cite{elismac}
$$z=\left[ {u_{a;b}\hat k^a\hat k^b} \right]_0\Delta \ell  _0+{1 \over 2}\left[
{u_{a;bc}\hat k^a\hat k^b\hat k^c} \right]_0\Delta \ell _0^2+O(\Delta \ell
_0^3)\eqno(52a)$$
$$\hat k^a=\left( {u_bk^b} \right)^{-1}k^a,\quad \hat k^t=-U^{-1}=-u^t,\quad
\hat k^r=\mp L,\qquad \Delta \ell _0=-\left[ {u_bk^b} \right]_0\Delta
\lambda=L_0 \Delta \lambda\eqno(52b)$$
where $\Delta \lambda$ is the geodesic parameter of an observed gallaxy. For an
expanding, accelerating but shear-free 4-velocity, the quantities involved in
(52) are
$$u_{a;b}\hat k^a\hat k^b={\Theta  \over 3}-\dot u_a\hat k^a\eqno(53a)$$
$$u_{a;bc}\hat k^a\hat k^b\hat k^c={{\Theta _{,c}} \over 3}\hat k^c+{{2\Theta
^2} \over 9}-\Theta \dot u_a\hat k^a-\dot u_{b;c}\hat k^b\hat k^c+\dot u_a\dot
u_b\hat k^b\hat k^c\eqno(53b)$$
Evaluating these expressions for (20), (21), (48b) and (47), expanding up to
first order in $\varepsilon_0$ and evaluating at $(t_0,r_0)$ yields
$$z\approx{{H_0} \over c}\left[ {1\mp {{f_0c} \over {2H_0R_0}}\varepsilon _0}
\right]\Delta \ell _0+{{H_0^2} \over {2c^2}}\left\{ {3+q_0+\varepsilon _0\left[ {\mp (1+q_0)F_0^2+
{{(\gamma _1+5)f_0c} \over {2H_0R_0}}\mp {{(1-2kF_0^2)c^2} \over {2(H_0R_0)^2}}}
\right]} \right\}\Delta \ell _0^2+O(\Delta \ell _0^3)\eqno(54)
$$
where $H_0,\,q_0$ are given by (47a) and (50). As shown by (54), the red shift
observed from a given source depends in general on the position of the observer
(its radial coordinate $r_0$ determining $f_0,F_0$). This is expected, since the
metric given by (1) and (20) has an explicit radial dependence contained in the
functions $f=4FF'$, and so the red shift distribution is anisotropic because of
the inhomogeneity of the cosmological model. In particular, an
accelerating and shear-free fluid leads to a monopole plus dipole term in (53a),
since the direction of the 4-acceleration ($\partial/\partial r$) is a
privileged direction. Hence, for a connecting vector in the radial
direction $\Delta\ell_0$ has a maximal value, while along other directions it
must be multiplied by a factor $\cos\Psi$, where $\Psi$ is the telescopic
angle\cite{elismac}. Observations along the symmetry center
$r=0$ detect an isotropic distribution

$$z|_{r=0}=\frac{H_0}{c}\Delta\ell_0+\frac{H_0^2}{2c^2}\left[3+q_0\mp
\varepsilon_0\frac{c^2}{2(H_0R_0)^2}\right]\Delta\ell_0^2+O(\Delta\ell_0^3)
\eqno(55)$$
Equation (54) provides then the maximal
magnitude of the corrections to local red shift isotropy that inhomogeneity of
spacetime introduces into to the red shift vs luminosity distance.  From (47c)
and since $f_0\approx F_0$, these corrections are of the
order $\varepsilon_0\Omega_0F_0\approx
\varepsilon_0\bar\Omega_0F_0\approx \varepsilon_0F_0$. For these corrections to
be small, we need $\varepsilon_0F_0\ll 1$, which for the models considered in
sections III and IV, means that either $k=1$ (where $0\le r\le \pi$ and $f,F$
are finite), or if $k=0,\,-1$, the domain $r$ of $f,F$ must be necessarily
restricted. However, as discussed in section V, the cases
$k=0,-1$ do not allow for regular hypersurfaces $t=\hbox{const}$ nor a regular
initial Cauchy surface at $t=t_0$. For the case
$k=1$, if $\varepsilon_0\ll 1$ the corrections due to inhomogeneity are small
for all $r_0$ compared with the isotropic zero order terms $H_0,\,q_0$, which
means that local observations are overwhelmingly determined by $H_0,\,q_0$,
leading to a distribution of red shifts that is almost isotropic but whose
observational parameters slightly differ from the FLRW parameters $\tilde
H_0,\,\tilde q_0$. However, because of (49) and (50), $H_0,\,q_0$ differ from
$\tilde H_0,\,\tilde q_0$ by corrections of order $\varepsilon_0$, and so for
sufficiently small
$\varepsilon_0$, compatible with (28) and (38), this difference should be much
smaller than the 10-15 percent or so observational uncertainty\cite{peacock} in
the most accurate possible local measurements of $\tilde H_0$. This will be
discussed in the following section.

\section{Discussion and conclussion.}
We have presented in previous sections important general features of the
subclass of Stephani Universes derived in sections III and IV. We still need
to prove that the approximation to ideal gas equations of state (conditions (28)
and (38)), as well as the approximation to isotropic local observations, are
actualy met for physicaly reasonable numerical values of the parameters
involved. We also need to integrate the Friedmann equations (32) and (42) in
order to obtain a fuly determined dynamical evolution of the models, allowing us
to verify their prediction of the ``age of the universe today'' in comparison
with the prediction of standard cosmology (dust FLRW universes). Each case,
$\gamma=4/3$ and $\gamma=5/3$, is examined separately below.

\subsection{The classical monatomic ideal gas source.}

For this model the state variables $\rho,p,n,T$ are given by (22a) and (31a-c),
satisfying (15) and providing (if (28) holds) a close approximation to (24).
Consider matter dominated conditions along $t=t_0$, so that $\rho_0$ is
matter-energy density today. In order to examine the fulfilment of (28),
suppose that $m$ is a proton mass and (say) $\hat T_0\approx 100$ degrees K,
leading to $\varepsilon_0\approx 10^{-10}$ in (30). The terms of the form (28) in
the series in (27) have the form

$$\left(\frac{\rho}{\rho_0}\right)^{2\xi/3}F^\xi\,\varepsilon_0^\xi=F^\xi\,
\left(\frac{\rho} {\rho_0}\right)^{2\xi/3}
\left(\frac{k_B\hat T_0}{mc^2}\right)^\xi\approx
\left(10^{-10}\right)^\xi F^\xi\,\left(\frac{\rho} {\rho_0}\right)^{2\xi/3}\ll
1,\qquad \xi=2,3,..\eqno(56)$$

\noindent controling the goodness of the approximation of (27) to (24) along the
directions of increasing $r$ (increasing $F$) and $t$ (decreasing $\rho$).
For the cases $k=0,-1$, the function $F$ increses monotonously as $r$ increases,
so that even for very small values of $\varepsilon_0$ and $\rho/\rho_0$ we might
have a diverging series in (27), since $F^\xi<F^{\xi+1}$ for $r>2$. Hence in the
cases $k=0,-1$, a necessary (but not sufficient) condition for (27) to converge
into (24) is $r<2$, restricting the domain of regularity to comoving observers
near the center worldline $r=0$. In the case $k=1$, the function $F$ is bounded
and satisfies $F^\xi>F^{\xi+1}$ for all values of $\xi$ and $0\le r\le \pi$.
Therefore, in this case the convergence of the series in (27) and the fulfilment
of (56) only depends on the comparison of $\varepsilon_0$ with ratio
$\rho/\rho_0$ for the leading term of nonzero order ($\xi=2$) in (27). From (46)
and using the value
$\varepsilon_0\approx 10^{-10}$, the allowed values of $\rho/\rho_0$ (with the
maximal value at the FD density) is given by

$$\frac{\rho}{\rho_0}<\left[\frac{2}{3\varepsilon_0F^2}\right]^{3/2}\approx
\frac{10^{15}}{\sin^3(r/2)}
\eqno(57a)
$$
Considering (56) with $\xi=2$ for $k=1$, the conditions for a good
approximation of (24) become
$$\varepsilon_0^2\sin^2(r/2)\left(\frac{\rho}{\rho_0}\right)^{4/3}\approx10^{-20}
\left(\frac{\rho}{\rho_0}\right)^{4/3}\ll 1,\qquad  \Rightarrow\qquad
10^{-15}\frac{\rho}{\rho_0}\ll 1\eqno(57b)
$$
Thus, comparing (57a) and (57b), we see that the approximation to (24) is a good
one as long as one excludes very early evolution stages close to the FD density.
However, from (29b) and (31c), a ratio $\rho/\rho_0\approx 10^{15}$ can be
associated with to $T/\hat T_0\approx 10^{10}$, corresponding to
relativistic temperatures far above the range validity of the classical equation
of state (15), (24). Hence, the condition (57b) is perfectly reasonable and
allows for a large and physicaly justified range of validity for the
approximation of (27) to (24), based on the state variables (31). A convenient
limit of this validity range can be set for $\rho/\rho_0<10^6$, corresponding
approximately to $T/\hat T_0 <10^4$, roughly (and slightly larger than) the
range of post-decoupling temperatures for which the evolution a classical ideal
gas without interacting with radiation can be justified. The reasoning applied
above for the case $k=1$ is valid  for $k=0,-1$, as long as we consider only
spacetime sections with $r<2$.

The range of validity of the ideal gas equation of state affects also the
question, raised in the previous section, on whether local observations on the
models are significantly different from those performed in dust FLRW
cosmologies. As far as the parameters $H_0,\,\Omega_0,\,q_0$ are concerned, the
small value $\varepsilon_0\approx 10^{-10}$ is sufficient to make them (via
(49), (50b)) practicaly identical to the equivalent FLRW parameters.
However, the red shift vs luminosity distance (54) contains products of
the form $\varepsilon_0F_0^2$ and $\varepsilon_0f_0^2$, and so for the
cases $k=0,-1$, these terms can be large even for such a small
$\varepsilon_0$. Hence, the same reasoning applies: for observations to
be close (up to order $\varepsilon_0$) to FLRW spacetimes, either we assume $k=1$
or, if $k=0,-1$, then we must only consider comoving observers near
$r=0$.

The dynamical evolution of the models, especificaly the
model $k=1$, follows from the integral of the Friedmann equation (32), leading
to

$$H_0(t-t_i)={{1} \over {\sqrt{\nu_0}}}\int\limits_{A_i}^A  {{{\bar Ad\bar A}
\over {\left[ {\varepsilon _0/2+\mu _0\bar A-\bar A^2} \right]^{1/2}}}}={{1}
\over {\sqrt
{\nu _0}}}\left[ {\mu _0\arcsin \left( {{{2\bar A-\mu _0} \over {\sqrt {\mu
_0^2+2\varepsilon _0}}}} \right)-\left[ {\varepsilon _0/2+\mu _0\bar A-\bar A^2}
\right]^{1/2}} \right]_{A_i}^A\eqno(58a)$$
where
$$A\equiv \frac{R}{R_0}\qquad\nu _0={{c^2} \over
{(R_0H_0)^2}}={{2(\Omega _0-1)} \over  {2-\varepsilon _0}}={{2(2q_0-1)}
\over {4-\varepsilon
_0}},\quad \quad
\mu _0={{8\pi G}
\over {3c^4}}\rho _0R_0^2={{(2-\varepsilon _0)\Omega _0} \over {2(\Omega
_0-1)}}={{(2-\varepsilon _0)q_0+\varepsilon _0} \over {2q_0-1}}\eqno(58b)
$$
and $A_i$ is the value of $A$ associated with a suitable initial surface
marked by $t=t_i$, for example the surface characterized by $T/\hat
T_0=10^4$ corresponding (from (31c)) approximately to $R/R_0\approx 10^{-2}$.
The regularity domain, starting (say) at $A_i=R_i/R_0= 10^{-2}$, must exclude
unphysical features like the FD singularity and ADS boundary,
marked by the coordinate values

$$\left[\frac{R}{R_0}\right]_{_{\rm{FD}}}=A_{_{\rm{FD}}}=
\sqrt{\frac{3\varepsilon_0}{2}}\sin(r/2)\le
\sqrt{\frac{3\varepsilon_0}{2}},\qquad
\hbox{FD singularity}\eqno(59a)
$$
$$\left[\frac{R}{R_0}\right]_{_{\rm{ADS}}}=A_{_{\rm{ADS}}}=
\sqrt{\frac{\varepsilon_0}{2}}\sin(r/2)\le
\sqrt{\frac{\varepsilon_0}{2}},\qquad
\hbox{ADS boundary}\eqno(59b)
$$

\noindent corresponding to very early stages of the evolution ($R/R_0\approx
10^{-5}$), well outside the stage where the initial value surface $R/R_0=
10^{-2}$ has been suggested. This surface is a regular Cauchy surface and the
evolution range of the models is given by

$$A_{_{\rm{FD}}}\ll A_i=10^{-2} \le A \le A_{_{\rm{MAX}}}=\frac{\mu_0}{2}+
\sqrt{\left(\frac{\mu_0}{2}\right)^2+
\frac{\varepsilon_0}{2}}\approx \mu_0\eqno(60)
$$
where $A_{_{\rm{MAX}}}$ marks the surface of ``maximal expansion'' characterized
by $H=\Theta/3=0$, after which the fluid recollapses following a time symmetric
pattern. From (58b) and (60), and bearing in mind that $\varepsilon_0\approx
10^{-10}$, we have $A_{_{\rm{MAX}}}\approx \Omega_0/(\Omega_0-1)$, a very close
fit to the value of $A_{_{\rm{MAX}}}$ obtained for a FLRW dust universe with
$k=1$.

The time required for the fluid to expand up to $A=1$ (the ``age of
the universe today'') and up to $A_{_{\rm{MAX}}}$ follows directly from (58) and
can be expressed as a function of either $(\Omega_0,\varepsilon_0) $ or
$(q_0,\varepsilon_0) $. As shown by figure 1, the curve relating $H_0(t_0-t_i)$
obtained from (58a-b) vs $\Omega_0>1$ for $\varepsilon_0=10^{-10}$ is almost
identical with the curve relating these same quantities for a FLRW dust universe
with $k=1$. As shown in Figure 2, for any given $\Omega_0 >1$ the value of
$H_0(t_0-t_i)$  obtained from a dust FLRW universe with $k=1$ is slightly larger
than this same quantity obtained from (58). In fact, the ratio of the
difference of these values of $H_0(t_0-t_i)$ to the FLRW value is of the order
of $10^{-4}$ of the latter value. Therefore, the age of the universe today
predicted by this model is practicaly the same as that predicted by a dust FLRW
universe with $k=1$, with $H_0(t_0-t_i)\to 2/3$  as $\Omega_0\to 1$, thus
sharing the same ``age problem'' associated with standard cosmology. Depending
on the value of $H_0$, in the range  $50-70\, \hbox{km}\,(\hbox{mpc s})^{-1}$,
standard cosmology leads to a range of age values $t_0-t_i\approx 10^{9-12}$
years that barely meets accepted age values obtained from globular clusters and
from nuclear and geological estimates\cite{peacock}. The same result follows for
models with $k=-1$ (assuming $r\ll 1 $), the ``age of the universe'' is
almost the same as that predicted by standard cosmology with $k=-1$. The case
$k=0$ (also assuming $r\ll 1$) is slightly different, since in the FLRW models
$k=0$ necessarily implies $\Omega_0=1$, but for the $k=0$ Stephani case the
value $k=0$ corresponds to $\Omega_0<1$. We can only have $\Omega_0=1$ in the
Stephani models if $k=1$ and $\varepsilon_0=3(\gamma-1)$, a possible but
undesirable combination of values since $\varepsilon_0\approx 10^{-10}$ while
$\gamma=4/3,\,5/3$.

From equations (31) for $k=1$ we have $p(t,0)=T(t,0)=0$ along the symmetry
center $r=0$, while both $p$ and $T$ have their maximal value along any surface
$t=\hbox{const}$ along the other symmetry center $r=\pi$ (see subsection C for
discussion on these points), that is, there is  a ``cold center'' along $r=0$ and
a ``hot center'' along $r=\pi$. However, if (28) and (56) hold and for matter
dominated conditions, the variation of $p$ and $T$ at any surface
$t=\hbox{const}$ is minimal (of the order $\varepsilon_0\approx 10^{-10}$) as
illustrated by the approximated forms of state variables at $t=t_0$

$$\rho_0=\frac{mc^2N_0}{R_0^3},\quad p_0\approx \frac{N_0k_B\hat
T_0}{R_0^3}\sin^2(r/2)\left[1+O(\varepsilon_0)\right],\quad n_0\approx
\frac{N_0}{R_0^3}\left[1+O(\varepsilon_0)\right],\quad T_0\approx \hat
T_0\sin^2(r/2)\left[1+O(\varepsilon_0)\right]\eqno(61)
$$
clearly indicating a matter dominated regime $p_0/\rho_0\approx \varepsilon_0 \ll
1$, with very low pressure and temperature values providing an
excelent approximation to dust-like conditions. The models with
$\gamma=5/3$ might not be realistic becouse they do not allow  for the
description of another gas (CBMW radiation or other relique gases) as parts of a
decoupled mixture accompanying the monatomic ideal gas. However, if the matter
content of the present universe is dominated overwhelmingly by non-barionic CDM,
and if we can justify this CDM to satisfy the equation of state  of a classical
monatomic ideal gas ((15) and (24)), then the case
$\gamma=5/3$ could provide a nice toy model of a CDM universe in its latter
stages approaching near dust conditions. Obviuosly, it would be necessary in
this case to infere the mass of the particles forming the CDM ideal gas, so
that an appropriate value of $\varepsilon_0$ be given.

\subsection{The matter-radiation mixture.}

The state variables for the case $\gamma=4/3$ follow from (41a-c). This case is
perhaps more interesting than the case examined before because of the
possibility of accomodating a joint description of non-relativistic matter and a
relativistic relic gas. Considering the non-relativistic and ultra-relativistic
particles to be barions and photons, the ratio $N_0^{\rm{II}}/N_0^{\rm{I}}$ can
be associated with the ratio of photons to baryons, and so we have
$N_0^{\rm{II}}/N_0^{\rm{I}}\approx 10^8$ (considering $h\approx .6$ and
$\Omega_B\approx \Omega_0$\cite{kotu}). Assuming $m$ to be a proton mass and
$\hat T_0\approx 3$ degrees K., a good numerical estimate in (40) is
$\varepsilon_0\approx 3\times 10^{-5}$, a value five orders of magnitude larger
than
$\varepsilon_0$ in (30). However $\varepsilon_0$ is still sufficiently small so
that (38) might hold and (37) might provide a good approximation for (34) over a
wide temperature and density ranges. Since $T$ under (34a-b) is the  temperature
of the photon gas (so that $\hat T_0\approx 3$ degrees K), we can set again
$\rho_0$ to be matter-energy density today (overwhelmingly rest mass density).

The arguments discussed in the previous subsection regarding the convergence
of the  series in (27) for cases $k=0,-1$ apply also for the series in (37).
Hence, we consider only the case $k=1$, though the results can be applied to the
cases $k=0,-1$ for comoving observers near $r=0$. As before, we use look at
(46), now with the value $\varepsilon_0\approx 10^{-5}$, obtaining the allowed
values of $\rho/\rho_0$ (with the maximal value at the FD density) given by

$$\frac{\rho}{\rho_0}<\left[\frac{1}{2\varepsilon_0F^2}\right]^3\approx
\frac{10^{15}}{\sin^6(r/2)}
\eqno(62a)
$$
Considering (38) with $\xi=2$ for $k=1$, the conditions for a good
approximation to (34) become
$$\varepsilon_0^2\sin^2(r/2)\left(\frac{\rho}{\rho_0}\right)^{2/3}\approx10^{-10}
\left(\frac{\rho}{\rho_0}\right)^{2/3}\ll 1,\qquad  \Rightarrow\qquad
10^{-15}\frac{\rho}{\rho_0}\ll 1\eqno(62b)
$$
Hence, from (62a) and (62b), a good approximation of (37) to (34a) requires
the exclusion of evolution stages close to the FD density.
From (39b) and (41c), the ratio $\rho/\rho_0\approx 10^{15}$ corresponds to
$T/\hat T_0\approx 10^{5}$, a higher value than temperatures of
the matter-radiation decoupling era, thus the range of
validity for the approximation of (37) to (34) exceeds that of a decoupled
mixture.

The dynamical evolution for the case $k=1$ follows from the following integral of
the Friedmann equation (42)

$$H_0(t-t_i)=\sqrt {1-\varepsilon _0}\int\limits_{A_i}^A  {{{\sqrt {\bar A}d\bar
A} \over {\sqrt {\Omega _0-\varepsilon _0-(\Omega _0-1)\bar A}}}}=\sqrt
{{{1-\varepsilon _0} \over {\Omega _0-1}}}\left[ {\xi _0\arcsin \left( {{{\bar
A} \over {\xi _0}}-1} \right)-\bar A^{1/2}\left( {2\xi _0-\bar A} \right)^{1/2}}
\right]_{\bar A=A_i}^{\bar A=A}\eqno(63)
$$
$$\hbox{where:}\qquad A\equiv \frac{R}{R_0},\qquad \xi _0\equiv {{\Omega
_0-\varepsilon _0}
\over {2(\Omega _0-1)}}
$$
The regularity domain, starting at $\rho/\rho_0\approx 10^{15}$ or
$R/R_0\approx10^{-5}\approx\varepsilon_0$ must exclude the FD singularity and ADS
boundary  marked by
$$
\left[\frac{R}{R_0}\right]_{_{\rm{FD}}}=A_{_{\rm{FD}}}=
2\varepsilon_0\sin^2(r/2)\le
2\varepsilon_0,\qquad
\hbox{FD singularity}\eqno(64a)
$$
$$\left[\frac{R}{R_0}\right]_{_{\rm{ADS}}}=A_{_{\rm{ADS}}}=
\varepsilon_0\sin^2(r/2)\le
\varepsilon_0,\qquad
\hbox{ADS boundary}\eqno(64b)
$$
indicating that we can set $A_i=A(t_i)$ to a value greater than
$2\varepsilon_0\approx 10^{-5}$ that would correspond to the decoupling era ($
T/\hat T_0\approx 10^3$), for example: $A_i=10^{-3}$. The regular evolution
range of the model extends from $A_i$ up to the value $A_{_{\rm{MAX}}}$, marking
the maximal expansion $\Theta=0$, where the fluid bounces and begins to collapse
in a time symmetric pattern. Evaluating
$A_{_{\rm{MAX}}}$ from (63) yields

$$A_{_{\rm{MAX}}}=2\xi_0={{\Omega
_0-\varepsilon _0}
\over {\Omega _0-1}} \approx \frac{\Omega_0}{\Omega_0-1}
\eqno(65)$$
a value almost equal to that obtained for a dust FLRW universe with $k=1$.
The ``age of the universe today'' can be evaluated by setting $A=1$ in (63).
Ploting the resulting value $H_0(t_0-t_i)$ vs $\Omega_0>1$ and comparing with
the FLRW spacetime leads to curves almost identical to those shown in figure 1.
However, the ratio of differences of $H_0(t_0-t_i)$ to the FLRW value
yields an even closer fit to the FLRW expression (see figure 2). Hence, as
in the case $\gamma=5/3$, this model presents the same ``age problem'' of
standard cosmology mentioned before. If we assume that the non-relativistic
component  includes cold dark matter (assuming also that it can be described as
an ideal gas), we would have a value
$N_0^{\rm{I}}$ larger by two orders of magnitude, modifying (40) to
$\varepsilon_0\approx 10^{-7}$. However, this alternative value of
$\varepsilon_0$ does not change the close fit to the dust FLRW universe, in
fact, it makes this fit even closer, since the only way to obtain values of
$H_0(t_0-t_i)$ significantly different from the dust FLRW values is to have
a larger $\varepsilon_0\approx 1$, though the resulting values of
$H_0(t_0-t_i)$ are then {\it significantly smaller} than those of the FLRW
model. As in the case $\gamma=5/3$, this hopeless situation also occurs
for the cases with $k=0,-1$.

As in the previous case, from (39) and (41), we also have a ``cold'' symmetry
center $r=0$ where $p(t,0)=T(t,0)=0$ (see subsection C for discussion on this
aspect), and a ``hot'' center at $r=\pi$ where both $p$ and $T$ have their
maximal value along any surface $t=\hbox{const}$. At $t=t_0$ the state variables
take the approximated forms

$$\rho_0=\frac{mc^2N_0^{\rm{I}}}{R_0^3},\quad p_0\approx
\frac{N_0^{\rm{II}}k_B\hat
T_0}{R_0^3}\sin^2(r/2)\left[1+O(\varepsilon_0)\right],\quad n_0\approx
\frac{(N_0^{\rm{I}}+N_0^{\rm{II}})}{R_0^3}\left[1+O(\varepsilon_0)\right],\quad
T_0\approx
\hat T_0\sin^2(r/2)\left[1+O(\varepsilon_0)\right]\eqno(66)
$$
so that matter dominated dust-like conditions do prevail at $t=t_0$, with the
variation of $p$ and $T$ of the order $\varepsilon_0\approx 10^{-5}$. Also, as
with the case $\gamma=5/3$, as long as $F_0$ and $f_0$ in (54) are bonded,
observation parameters are almost identical to their FLRW equivalents. However,
the pressure and temperature in this case correspond to the pressure and
temperature of the MWCB (with $\hat T_0=3$ degrees K.). Hence, the ratio
$p_0/\rho_0\approx \varepsilon_0\approx 10^{-5}$ from (66), is roughly the
accepted value of ratio of the energy density of the CMBR ($\approx 3p_0$) to
that of visible non-relativistic matter. Regarding the temperature, a deviation
of order $\varepsilon_0\approx 10^{-5}$ gives an excellent fit with the observed
deviations of the isotropy of the temperature of the MWCB\cite{peacock},
\cite{kotu}. We might consider including CDM in the matter content of the
non-relativistic component so that $\varepsilon_0\approx 10^{-7}$.
This might sound reasonable, but CDM is most possibly non-barionic, and so, as
in the case $\gamma=5/3$, we would have to justify that it satisfies an equation
of state like (34). With or without CDM, the case $\gamma=4/3$ could be a
nice toy model of a matter dominated universe approaching near dust
conditions, though it would be a better model than that provided by $\gamma=5/3$
as it would include the MWCB.

\subsection{The drawbacks.}
 It emerges from the discussion in the previous subsections that the $k=1$
models, whether $\gamma=4/3$ or $\gamma=5/3$, share two unappealing features:
(A) The maximal and minimal values of the rest mass density ($mc^2n$ or
$mc^2n^{\rm{I}}$) respectively occur along the symmetry centers $r=0$ and
$r=\pi$, while the maximal and minimal values of the internal energy density
($\rho-mc^2n\approx 3p/2$ or $\rho-mc^2n^{\rm{I}}\approx 3p$) occur along $r=\pi$
and
$r=0$.  This means that along $r=0$ all energy density is rest mass energy
density and there is a higher density of particles but zero temperature and
pressure for all times (see discussion further ahead), while at $r=\pi$ there
are less particles by volume but a higher temperature and pressure. (B) The fact
that pressure and temperature vanish along $r=0$ for all evolution times (forced
by the specific spacial dependence of the metric contained in the function $F$)
is a very different situation from $p,\,T$ tending to zero asymptoticaly as the
fluid expands and cools. This situation leads to the entropy per particle (18c)
diverging as $r\to 0$, so that the left hand side of the Gibbs equation (14)
($T s'$), remains finite (as the right hand side does). Although this feature is
problematic, it is consistent with the fact that an absolute zero temperature
(if $T$ is stricly zero instead of $T\to 0$ asymptoticaly) is impossible to
attain in equilibrium classical thermodynamics\cite{huang}. This
feature (appearing also for the cases $k=0,-1$) is perhaps the strongest
drawback on the physical plausibility of the models.

\subsection{Final remarks.}
The models derived and presented in this paper are mathematicaly
simple, theoreticaly consistent and satisfy physicaly motivated equations of
state. Whether one considers it a virtue or a defect, the models bear an
extremely close resemblance (observationaly and dynamicaly) to FLRW
cosmologies in the matter dominated regime with a dust source, therefore they
 share the possible failure of matter dominated standard cosmology in
dealing with recent observational evidence\cite{peacock}. However, even if hard
core cosmologists might dismiss them as not sufficiently realistic, the
solutions can at least be applied as self-consistent cosmological toy models or
as useful educational tools in teaching cosmology. Hopefuly the results of the
present paper, besides rescueing the Stephani Universes from oblivion, will
motivate and improve further research based on this nice and elegant class of
solutions.

\acknowledgements
This work is supported by grant DGAPA-IN122498. The wise meowing and feline
wisdom of Mimi, Tontina, Ni\~na, Chata, Pintita, Martin, Feucho, Bonito, Ulises,
Vitorio and Pocholo provided the author with enough inspiration as to make the
writing of this paper an enjoyable venture.

redpajama_set_name: RedPajamaArXiv


text:
Q: Using the C preprocessor to effectively rename variables I'm writing a few very tight loops and the outermost loop will run for over a month. It's my understanding that the less local variables a function has, the better the compiler can optimize it.  In one of the loops, I need a few flags, only one of which is used at a time. If you were the proverbial homicidal maniac that knows where I live, would you rather have the flag named flag and used as such throughout or would you prefer something like
unsigned int flag;

while (condition) {

#define found_flag flag
  found_flag = 0;
  for (i = 0; i<n; i++) {
    if (found_condition) {
      found_flag = 1;
      break;
    }
  }
  if (!found_flag) {
     /* not found action */
  }

/* other code leading up to the next loop with flag */
#define next_flag flag
  next_flag = 0;
/* ... /*
}

This provides the benefit of allowing descriptive names for each flag without adding a new variable but seems a little unorthodox. I'm a new C programmer so I'm not sure what to do here.

A: I completely agree with dreamlax: the compiler will be smart enough for you to ignore this issue entirely, but I'd like to mention that you neglected a third option, which is rather more readable:
while (something) {
  /* setup per-loop preconditions */
  {
    int flag1;
    while (anotherthing) {
      /* ... */
    }
    /* deal with flag found or not-found here */
  }

  /* possibly some other preconditions */
  {
    int flag2;
    while (stillanotherthing) {
      /* ... */
    }
  }
}

which would tell a dumb compiler explicitly when you are done with each flag. Note that you will need to take care about where you declare variables that need to live beyond the flag-scope blocks.

A: Your trick would only be useful on very old, very simple, or buggy compilers that aren't capable of correct register (re)allocation and scheduling (sometimes, that's what one is stuck with for various or ancient embedded processors).  gcc, and most modern compilers, when optimizations are turned on, would reallocate any register or local memory resources used for local variables until they are almost hard to find when debugging at the machine code level.  So you might as well make your code readable and not spend brain power on this type of premature optimization.

A: Don't bother doing this, just use a new variable for each flag. The compiler will be able to determine where each one is first and last used and optimise the actual amount of space used accordingly. If none of the usage of the flag variables overlap, then the compiler may end up using the same space for all flag variables anyway.
Code for readability first and foremost.

redpajama_set_name: RedPajamaStackExchange


text:
Meilenstein ist eine österreichische Partyschlagerband.

Bandgeschichte
Gegründet wurde die Band 2002 von Helmut "Heli" Brunner, Charly Höfer, Stefan Konrad und Martin "Rick" Rosenberger.

Der bekannteste Song der aktuell fünfköpfigen Band ist bis heute Christin, der 2008 als Single beim  Ötztaler Label VM Records erschien und über MCP Sound & Media mit Sitz in  Mittenwald vertrieben wurde. Der Titel, der auf beiden Alben der Band vertreten ist, erschien 2011 beim Stammlabel HomeRun Music in einer weiteren Singlefassung und ist auch auf Schlagersamplern enthalten. Das bei Major Babies 2011 veröffentlichte Debütalbum mit dem Titel Gib mir deine Liebe stieg im Juni 2011 in die österreichischen Albumcharts auf Platz 10 ein.

2014 wurde die Single Hey Leute das ist schön! (HomeRun Music) aufgenommen, welche beim Wolfsberger AC bei einem Ball in den Maschen erklingt.

Noch bei der Aufnahme der Single Wovon ich träume im Jahr 2015 spielte die Band mit Sigi Obereder am zweiten Keyboard in Sechserbesetzung. Das 2017 veröffentlichte zweite Album Na na verlorene Liebe entstand in Fünferbesetzung und stieg im August 2017 auf Platz 15 in die österreichischen Albumcharts ein, wo es eine Woche lang auf Platz 14 seine Bestplatzierung hatte.

Meilenstein treten neben kleineren Volksfesten auch auf bekannteren Veranstaltungen auf und waren beispielsweise bei der Wiener Wiesn, als Headliner beim Donauinselfest und beim Schweizer Musi Open Air zu sehen. In der ORF-Reihe Wenn die Musi spielt hatten sie bislang 2011, 2012 und 2015 Auftritte.

Diskografie
Alben
 2011: Gib mir deine Liebe (Major Babies)
 2017: Na na verlorene Liebe (HomeRun Music)

Singles
 2008: Christin (VM Records/MCP Sound & Media)
 2011: Gib mir deine Liebe (HomeRun Music)
 2011: Christin (HomeRun Music)
 2012: Es gibt keinen (HomeRun Music)
 2012: Weil ich Dich liebe (HomeRun Music)
 2013: Männer wie wir (HomeRun Music)
 2014: Hey Leute das ist schön! (HomeRun Music)
 2014: So wie Du bist (HomeRun Music)
 2015: Wovon ich träume (HomeRun Music)
 2017: Hey, ich lieb dich doch (HomeRun Music)
 2017: Was hab ich davon? (HomeRun Music)

Kompilationsbeiträge
 2008: Christin auf Tollhaus Joker 09 (Universal Music Austria)
 2009: Christin auf Schlager Fox Party (Eurotrend (MCP Sound & Media))
 2011: Weil ich Dich liebe auf Tollhaus Joker 21 (Universal Music Austria)
 2013: Oberkörper frei auf Tollhaus Joker 24 (Universal Music Austria)
 2015: Christin auf 50 Hits Schlager Party (Eurotrend (MCP Sound & Media))
 2015: Christin auf Schlager Disco Fox – 20 Hits im Discofox (MCP Sound & Media)

Weblinks

 Bandwebsite
 Meilenstein beim HomeRun Verlag

Einzelnachweise

Schlagerband
Österreichische Band

redpajama_set_name: RedPajamaWikipedia


A common treatment for Staph infection is the use of an antibiotic either topical or oral. Pus may also be drained (only by doctor!). Dr. Clay, what are the common types/names of antibiotics prescribed for Staph infection? And if an antibiotic cream is prescribed, what is the frequency and duration of topical application? Should a parent consult the doctor again if there's no improvement within the first two days (for possibility of different strain of bacteria involved)? Can a parent apply the same cream the next time a similar-looking rash occurs on the child's skin?
Dr. Clay: There are a number of antibiotics that are effective against Staph, but unfortunately, many strains of Staph are now resistant to the antibiotics that have been used for years. The most commonly used antibiotics are in the penicillin family, and one of the best is methicillin. Cephalosporins, such as cephalexin, are also good for sensitive strains. Topical antibiotics include bacitracin, Neosporin and mupirocin (Bactroban). These are usually applied two to three times a day. It usually takes several days for improvement to be noted, and there may be redness that lasts for several more days than that. There should be no further spreading of the process, and the pain and redness should start to resolve in two to three days.
What if there is no improvement after treatment?
If there is no improvement, it should be checked by a doctor as it is possible that it is being caused by another rare organism or fungus, or it may be a sign that the patient's immune response is too weak to help eradicate it with a topical antibiotic alone. In general, if the patient is doing well and develops a similar process, the family can try to treat topically at first. If it does not improve in a few days, the doctor should be notified.
A worrying trend is that there is a strain of Staph bacteria, namely the MRSA, which has developed resistance against the commonly prescribed antibiotics noted above. As explained on Dermnet NZ, there is a mec gene in the bacterium that "alters the site at which the methicillin binds to kill the bacteria." As such, the binding is rendered ineffective, and the MRSA cannot be killed by antibiotics such as methicillin, floxacillin, lactamase-resistant pencillin and amoxicillin.
What are the antibiotics that can be given to a child that can kill the MRSA?
I read that Vancomycin is one such antibiotic, but there are also strains of Staph bacteria that have become resistant to it. What about Bactrim DS (double-strength sulfamethoxazole trimethoprim)?
Dr. Clay: Yes, vancomycin is a commonly used antibiotic that is effective against MRSA, but it cannot be taken orally. If the infection is relatively mild, we generally use Bactrim DS, as it can be taken by mouth and is effective. It is also important to supplement this with measures to try to eradicate it from the skin such as the use of bleach-containing cleansers, like CLn® BodyWash.
MarcieMom: Thank you Dr Cockerell, next week we shall learn about prevention – definitely beats having to treat a staph infection!
I would highly suggest purchasing CLn. We've found that it is worth every penny! It is so easy and hassle free to use also! The benefits can't be beat in my opinion. I'm not sure what would be going on with my 4 year old had we not found this product!
Thanks again for sharing this link with me! I hope other people can be blessed by CLn like we have.
We use CLn body wash every single day in our family! We have 4 kids and all of them have varying skin conditions.
Our most severe case was in our daughter. She had cellulitis twice in a couple months. We were fortunate to be included on the CLn study. It changed our lives. We believe that it is a miracle solution! Our daughter had been on oral and topical steroids over and over and nothing would help. After a week on CLn we retired all steroids and used CLn ONLY. We began using CLn 2 years ago and we have never looked back!
We will only use CLn on all family members. Our kids ask for it even if they just get itchy at all even from just playing outside.
I cannot wait to see the success of this product as it has been a great addition to our family!
Thanks Jessica for your sharing and really glad to know that your kids' skin got so much better! We chlorhexidine whenever our girl's skin is itchy, has the same effect 🙂 A little more troublesome though, cos it's an extra step to take.


This is the Freakland tribute to the best stop motion leyend ever, creating one of his amazing monster figures.
Dimensions : 8,6 in - 9 in aprox ( 22 - 23 cm ) including base.
Materials : Rigid figure made with polyurethane resin.


text:
❤
Do-or-die Chazz with Zillions-of-Games
This page is written by the game's inventor, DavidMoeser.
"DO-OR-DIE" CHAZZ
invented by david moeser
ASCII DIAGRAM:  INITIAL POSITION
+---+---+---+---+---+---+---+---+
5   |   |   |   |   | k |   |   |   |
4   | p | p | p | p | p | p | p | p |
3   |   |   |   |   |   |   |   |   |
a   b   c   d   e   f   g   h
"Do-Or-Die" Chazz combines the features of Chazz and "Do-Or-Die
Chess."  Both are chess variants invented by David Moeser of Cincin-
nati, Ohio, USA.
Like "Do-Or-Die Chess," D-O-D Chazz is played on a 40-square board
consisting of five ranks and eight files.  Like Chazz, the only pieces
in the initial position are Kings and Pawns.  D-O-D Chazz was invented
on April 18, 1999, for submission in the 40-square Chess Invention
Contest hosted by Hans Bodlaender's chess variants web site.
In D-O-D Chazz each player starts the game with a King on the 'e'
file and eight Pawns on the second rank.  In the initial position all
other squares on the board are vacant.
D-O-D Chazz is played like regular chess (a.k.a. "the FIDE vari-
ant") except for three special provisions dealing with Pawn movement:
(1) Pawns are allowed to move backwards.  They may not capture
backwards (either diagonally or orthogonally), but may move backwards
on the file.  Moving back to the first rank is legal.
(2) Promotion occurs on the last rank but can only be to a Rook,
Bishop, or Knight.  Promotion to Queens is not allowed.
(3) No double jumping is allowed.  All Pawn moves, forwards or
backwards, can only be one square at a time.
All other rules are the same as regular chess.
DO-OR-DIE CHAZZ:  ADDITIONAL RULES
========= =====   ========== =====
(1) No stalemates.  The player who can't move loses that turn.
(As in backgammon, the opponent would keep making moves until the
"staled" player can move.)
(2) A player can move a pawn backwards one rank from the first
rank.  That pawn is then off the board and out of the game --
permanently!  This isn't a requirement; it's merely an option.  No
player is ever "required" to self-destruct pawns in this manner, but
it's something that can be done if the player believes it to be
advantageous.
A regular chess set can be used for this game.  A regular chess
board can be used by covering over three ranks.  Over-the-board Chazz
is supposed to be played with clocks set at five minutes.
Chazz was first revealed in THE PARKWAY POST issue of May 1992, and
has been included in the NEUE CHESS book series.  It also appeared in
THE ENCYCLOPEDIA OF CHESS VARIANTS, published by David Pritchard in
The author of this variation may be contacted at this e-mail
address: (email removed contact us for address) ou.com.  The address of the chess variants web
site is: http://www.chessvariants.com/index.html.
Written by David Moeser.
This is a submission to the contest to design a chess variant on a board with 40 squares.
WWW page created: July 7, 1999. Last modified: November 1, 1999.
2dSmall8 x 5
Do-or-die Chazz. Pawns that can move backward and kings only on 5 by 8 board. (8x5, Cells: 40)
https://www.chessvariants.org/40.dir/dodiechaz.html
First Created: Tuesday, November 30th, -0001
Last Modified: Monday, March 26th, 2001
File Timestamp: Wed, 14 Jul 2021 17:37:27 -0400

redpajama_set_name: RedPajamaCommonCrawl


text:
Penguins' Evgeni Malkin to meet Flyers for first time since stick-swinging suspension | TribLIVE.com
Penguins/NHL
Penguins' Evgeni Malkin to meet Flyers for first time since stick-swinging suspension
Jonathan Bombulie
Fri., February 22, 2019 8:50 p.m. | Friday, February 22, 2019 8:50 p.m.
Pittsburgh Penguins' Evgeni Malkin (71) leans on the boards in front of his bench after Philadelphia Flyers' Jakub Voracek scored an empty-net goal during the third period to cap a 4-2 Flyers win in an NHL hockey game in Pittsburgh, Saturday, Dec. 1, 2018.
PHILADELPHIA – In the old days of the rivalry between the Pittsburgh Penguins and Philadelphia Flyers, there's little doubt Evgeni Malkin would have been starting off Saturday's Stadium Series game with a fight.
These days, it's no sure thing.
The last time the teams met Feb. 11 at Wells Fargo Center, Malkin received a match penalty for swinging his stick at the head of Flyers forward Michael Raffl. Malkin, who acted in retaliation after Raffl punched him in the back of the head, received a one-game suspension.
With the NHL's revenge culture growing less pronounced in recent years, bad blood might carry over, but it might not.
"I think it's like that whenever we play these guys," winger Jake Guentzel said. "You know it's going to be physical. It's a big two points. We both need these points right now. It's going to be that type of game, I think."
Malkin didn't do interviews after Friday's practice.
Follow the Pittsburgh Penguins all season long.
Jonathan Bombulie is a Tribune-Review staff writer. You can contact Jonathan by email at [email protected] or via Twitter .
Categories: Sports | Penguins
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Tim Benz: Can Penguins coach Mike Sullivan be empowered by his contract extension?


dance, poetry, physical and psychological theories to unravel the mystery.
first prize from he National Press Association for best Photography.
Stage at Reston as well as in part supported by the D.C. Commission on the Arts and Humanities.


Recently I did a tad of shopping (nothing new there), so today I thought I would run you through the things I picked up and my first impressions. First things first, there is a new Kiko shop both in Sheffield and in Birmingham which are the two cities I live in so I took it as a sign and popped in. Lucky for my bank account I only picked up one thing and that was the Long Lasting Eyeshadow Stick in shade 363 which word on the street is meant to be a dupe for a much more expensive product, the By Terry Obre BlackStar in Misty Rock. So can't wait to give that a go and I'll report back whether it is a good cheaper alternative.
Whilst we are talking about dupes I'll mention the lip pencil currently on my lips. It's the Rimmel Exaggerate in Eastend Snob which is meant to be a good alternative to the rather pricy Charlotte Tilbury lip liner in Pillow Talk. So far I've got a lot of use out of it and it looks great under a few of my different pinky lipsticks.
I saw lots of people posting about a free nail varnish that came with Elle Magazine back in October but I never managed to get my hands on it unfortunately. However I picked up a Models Own shade from their metallic range which is very different to my normal neutral nails but I have to say I am a fan, especially for this festive season. I got Rose Gold which is as you'd expect a rose gold/ pinky shade, might have to look at more in the range because I like Models Own nail varnishes and they have a great colour range.
Final thing to mention is the Origins Out of Trouble mask. I am a huge fan or Origins, especially their masks, as this is the third or fourth I've tried from them and I love it. I'll admit the smell isn't great, it's a bit medicinal and strange but the mask is a good quick fix for those spottier days.


The names listed below are alumni who have been searched for on this site from Community Education Partners in Houston, Texas. If you see your name among the Community Education Partners graduates, someone is looking for you! Register to let other graduates of Community Education Partners find and contact you. We realize that a Community Education Partners year book may be hard to find so registering to complete your school's Virtual Yearbook may be the next best thing to finding Community Education Partners yearbooks. The list of names is also useful for getting together a Community Education Partners reunion if you don't have a complete list of the students from your graduating class of Community Education Partners.


Topic: How to control another player?
Re: How to control another player?
I know how to change the player in edit-mode, but once I try playing with it, I can't. Is there a wau to "unlock" other players for me?
Yes, it would be only a matter of adding this "revolutionary city" feature. Do you think I could add this editing some rulseset, or I need to edit the source code?