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BLACK SABBATH: THE VINYL COLLECTION 1970-1978 AVAILABLE SEPTEMBER 6TH
July 18, 2019Dana 9 Comments
When Ozzy Osbourne, Tony Iommi, Geezer Butler and Bill Ward formed Black Sabbath in 1968, the quartet conjured a primal sound that was heavier, louder and more menacing than anything that had come before. The band recorded a series of metal masterpieces in the '70s that became the blueprint for heavy music and influenced generations of headbangers for years to come.
Rhino's new collection spans the Rock and Roll Hall of Famer's first eight years with a nine-LP boxed set that includes 180-gram vinyl pressings of the band's eight studio albums from the era, along with a new mono singles collection. The set also comes with a 7″ single featuring Evil Woman (Don't Play Your Games With Me) and Wicked World, a single that wasn't originally released in North America. Also included in the set are reproductions of the original tour books from this era.
Black Sabbath: The Vinyl Collection 1970-1978 will be available on September 6th for $249.98. The set will be produced in a limited, numbered edition of 3,000 copies. Pre-order is available here.
The collection brings together some of the most powerful metal albums ever recorded, including the band's eponymous debut (1970); the multi-platinum landmark Paranoid (1970); the platinum albums Master Of Reality (1971), Vol. 4 (1972), and Sabbath Bloody Sabbath (1973); the gold-certified Sabotage (1975); plus Technical Ecstasy (1976) and Never Say Die! (1978)
These albums contain a long list of metal classics, like Black Sabbath, War Pigs, Iron Man, Paranoid, Sweet Leaf, Children Of The Grave, Into The Void,Changes, Supernaut, Sabbath Bloody Sabbath, Never Say Die, and Dirty Woman.
The Vinyl Collection 1970-1978 also comes with Monomania , a new compilation of mono versions for several key songs from the Black Sabbath catalog, including edited versions of Iron Man, Sabbath Bloody Sabbath and a previously unreleased edit of Into The Void. In addition to mono versions of studio tracks, Monomania also features a radio ad released in 1970 to promote Black Sabbath.
Black Sabbath: The Vinyl Collection 1970-1978:
Album Listing:
Black Sabbath (1970)
Paranoid (1971)
Master Of Reality (1971)
Vol 4 (1972)
Sabbath Bloody Sabbath (1974)
Sabotage (1975)
Technical Ecstasy (1976)
Never Say Die! (1978)
Monomania:
1.   Evil Woman (Don't Play Your Games With Me) *
2.   Wicked World
3.   The Wizard
4.   Iron Man (Edited Version)
1.   Paranoid
2.   Into The Void (Edited Version) **
3.   Sabbath, Bloody Sabbath (Edited Version)
4.  It's Alright
5.   Radio Spot For Black Sabbath
7" Single (45rpm)
Evil Woman (Don't Play Your Games With Me)" b/w "Wicked World
* previously unreleased in North America
** previously unreleased
‹ SEBASTIAN BACH REFLECTS ON SKID ROW GETTING KICKED OFF THE BON JOVI TOUR, AND SAYS, "[WE THOUGHT] WE'RE SUCH BADASSES [LIKE KISS AND VAN HALEN] BECAUSE WE GOT KICKED OFF THE TOUR"
7/18: LIVE IN NJ AT STANHOPE HOUSE TOMORROW, RIVAL SONS PODCAST, MORE. ›
Rattlehead on July 19, 2019 at 12:28 am
Gees, how many times and ways can this era of Sabbath be reissued? Time to move on to other eras…..
D.J.H on July 20, 2019 at 10:42 am
Apart from the Ozzy era, the Dio era is the only other 'Sabbath' worth listening too in my opinion. I still consider the Dio stuff to be a different band ('Heaven and Hell' like they were before Ronnie sadly left us), and the Ian Gillan fronted 'Born Again' is a supergroup made to be called Black Sabbath.
As for the Tony Martin stuff……..zzzzzzzzz. I'm sorry, I'm sure he's a nice guy, but I can't stand Tony Martin's voice or lyrics. Even my guitar hero, the great Tony Iommi himself, sounded tired and bored by that point.
Thank god Ozzy (and eventually RJD) came back into the fold.
robert davenport on July 19, 2019 at 6:27 am
I own all the sabbath albums on vinyl plus Ozzys first two- I'm not giving that rotten
B _ _ h Sharon anymore money –
She doesn't manage Black Sabbath, only Ozzy.
But if it's of any interest, I have heard Ozzy on TV tell her: "Sharon, for f**k sake – we don't f**kin need anymore money! We've got more than enough!"
(Reported combined net worth: £120m)
Most of it Ozzy's.
Dana on July 20, 2019 at 12:14 pm
You may want to sit down (lol), but I have to defend Sharon, on this one.
She made plenty of her own money, both being a former judge on the UK version of the, X Factor, and her being a co-hostess on CBS', The Talk, for nine seasons. Starting this September, the show will be in it's tenth season. Plus, I am sure she gets a cut, at least 10 percent, maybe more, of all of Ozzy's earnings, by being his manager. Mrs. O. is both ruthless and very shrewd.
D.J.H on July 20, 2019 at 5:14 pm
She absolutely has; you're right. I certainly wouldn't doubt that for one moment. My comment on most of the money being Ozzy's is because I once saw a break down of the Osbourne's net worth via some media outlet here in England. This was around 2010 I believe.
At the time it was reported to be £120m, of which £100m was Ozzy's and the remaining £20m was Sharon's. But being that that was nine years ago, it has undoubtedly risen on both accounts, particularly Sharon's.
But who knows how true those figures were in the first place? All I know is that I'm not worth that sort of money….and that pi**es me off! :()
(Added note: If I was somehow to become worth such an amount, I wouldn't spend it all on big rock star mansions (Gene Simmons house is the best one I've seen) and fast cars and rock 'n' roll memorabilia; I would actually try and use it to do some good somewhere, somehow……and stay totally out of the limelight and take no credit for it.)
Dana on July 20, 2019 at 7:21 pm
Then you are the type of person worthy of that money. I would do the same, if I had copious amounts of money like that, I would donate to charities that meant something to me like St Jude's Children's Hospital, Veteran's causes, cancer research and building no kill animal shelters.
Instead of spending it all on plastic surgery and Chanel clothing (side eyeing Mrs. O-LOL!!).
D 🙂
D.J.H on July 21, 2019 at 5:12 am
Thank you for your kind comments Dana, and good for you for thinking the same.
I like the Plastic Surgery line by the way, however I have said that I would get my hair done if it decided to leave me! (I wish I had David Coverdale's hair to be honest, but that's just me!). :()
Dana on July 21, 2019 at 11:56 am
As you know, my big crush is John Sykes, so I understand about hair 🙂 . But, I actually think, in 2019, Ozzy has the best hair, as it looks thick and healthy. Back in the day, I thought Sebastian Bach had the most beautiful hair, Sykes circa '84, and Zakk Wylde, especially, in the, No More Tears, video.
As for nips and tucks, I don't fault Sharon for having plastic surgery, as it seems everyone in the "entertainment" industry, including men, have work. But she is obsessed, and addicted, like Cher, and seems to have a new face every couple of years. That is always the risk with plastic surgery, that one can go overboard, and go from attractive, to freakish, in one procedure. I think a few of the Kardashian women are perfect examples of too much of a good thing is not always positive.
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text:
List of key Golden Globe nominees
قائمة بأهم الاعمال الفنية المرشحة لجائزة الجولدين جلوب
Los Angeles, Here are the nominees in key categories for the 77th Golden Globe Awards, which were announced Monday in Beverly Hills.
The awards ceremony will take place on Sunday, January 5:
Best film, drama
Best film, musical or comedy
Best actor, drama
Christian Bale, "Ford v. Ferrari"
Best actress, drama
Best actor, musical or comedy
Leonardo DiCaprio, "Once Upon a Time… in Hollywood"
Best actress, musical or comedy
Ana de Armas, "Knives Out"
Cate Blanchett, "Where'd You Go, Bernadette?"
Beanie Feldstein, "Booksmart"
Annette Bening, "The Report"
Quentin Tarantino, "Once Upon a Time… in Hollywood"
Best musical or comedy series
Best musical or comedy actor
Best musical or comedy actress
Best limited series or TV movie
Best limited series or TV movie actor
Christopher Abbott, "Catch 22"
Best limited series or TV movie actress
FILMS WITH MOST NOMINATIONS
"Marriage Story" — 6
"The Irishman" — 5
"Once Upon a Time… in Hollywood" — 5
"Joker" — 4
"The Two Popes" — 4
The 'Carlos Ghosn challenge' spreading on Japanese social media
Trump lashes out at Apple for refusing to unlock Naval base shooter's iPhones
UN body welcomes milestone in Qatar labour reforms
New facilities for obtaining a Schengen visa starting from February 2020

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
Discretization of space is one of the standard techniques for quantizing fields non-perturbatively.
In case of the gravitational field a computational method, known as the dynamical triangulation(DT)\cite{DT2-KKM, DT2-ADF, DT2-D}, has been popularly tried up to four dimensions.
In $2$-dimension numerical results have satisfactorily reproduced analytical expectations.
For an example, the string susceptibility predicted by the Liouville field theory\cite{LIOUVILLE-P,LIOUVILLE-D,LIOUVILLE-DK} was obtained for  $ S^2 $ and $ T^2 $ topologies with several central charges within statistical errors\cite{DT2DS-OTY}.
Great many numerical efforts have also been poured on higher dimensional cases\cite{Lattice}.
Those investigations have given some insights on the possibility of quantizing the gravitational field in four dimensions.
For an example, the grand-canonical DT simulation\cite{DT4DS-HEY} has shown the existence of second order phase transition when space is coupled to a certain number of matter fields.
However, no conclusive results were drawn at that time.
It may be not only because of lack of the computational ability but also lack of analytical supports for the method in contrast to the lattice QCD.

While the discretization of space has been originally introduced as a technical tool to avoid the ultra-violet problem of quantizing gravity, there exists a possibility to regard the discrete space as the fundamental structure of space-time\cite{Amb,Loop,qc4}.
Based on this idea we have constructed the $d$-dimensional Universe as a complex of $d$-simplices with the $d$-disk topology, where the $ S^{d-1} $ boundary is assigned as space, while the direction perpendicular to the boundary surface as time.
Statistical evolution equation of the Universe has been formulated as the Markov process\cite{QU}, and applied to the $2$-point temperature correlation observed by the WMAP\cite{WMAP-DATA} in $4$-dimension.
In this model the evolution is regarded as a kind of the diffusion limited aggregation of simplices derived by entropic pressure.
For obtaining the distribution function of simplicial complexes we had to employ numerical simulations which have been the unique machinery beyond 2-dimension up to 4-dimension at that time.

As far as the 2-dimensional Universe is concerned mathematical studies of discrete geometry have shown significant results, such as giving analytic forms of distribution functions for various cases of discretization.
These results are elaborated mainly due to the graph counting theory originated by Tutte and his followers\cite{TUTTE,BROWNT,MULLIN}.
As an important application of the results the master equation for evolution of the 2-dimensional Universe has been solved analytically with using asymptotic limits of the exact distribution functions for the force field\cite{TY}.
However, in higher dimensions no analytic form of the distribution function was available, and all the essential properties were gathered only through numerical simulations, for examples\cite{DT3S-BK, DT3S-AM, DT3S-AV, DT3S-HTY, DT3D-WCR,DT3S-T, DT3S-RCK}, in the 3-dimensional case.\footnote[1]{Recently, the colored tensor model has been developed as an analytic model for higher dimensional counter part of the matrix model\cite{GURAU}.
We shall discuss relationships to our model in the last section.}

Recalling numerical efforts without sound theoretical support can conclude only a temporal consolation in the history of lattice quantum gravity, we focus the main aim of this paper to propose an analytical model of  distribution function for the discrete manifold.
We hope it will be able to provide a theoretical support for numerical methods and serve an initial input for the statistical evolution equation of the 3-dimensional Universe.
As one of the first trial we would like to propose an analytic distribution function for the discrete 3-disk which is constructed based on analyses of known exact results in two dimensions.

This paper is organized as follows.
In order to appeal the model to be reasonable, several distribution functions in 2-dimension are reviewed in the following section (Section 2).
In Section 3 a model distribution function for the discrete 3-disk is proposed and its asymptotic limit is calculated.
The predicted distribution function is applied for the $ S^3 $ and $ D^3 $ manifolds to compare with known numerical results obtained by DT simulations in Section 4.
The last section (Section 5) is devoted for discussions and comments on the evolution of the 3-dimensional universe and extension to the 4-dimensional manifold.

\section{Two dimensional distributions}
\subsection{analytical models of distribution functions}
Let us consider a triangulated 2-dimensional plane with the disk topology.
The triangulation is rooted by choosing a vertex on the boundary as the root vertex, and distinguishing the oriented edge on the boundary incident to the root vertex as the root edge.
The root edge is pointing under the right-hand screw rule directing upwards perpendicular to the plane.
Vertices and edges are called external if they lie on the boundary, and the remaining 1- and 2-simplices are internal.
The triangulation with $ n $ internal and $ m+3 $ external vertices is said to be of type $ [n,m] $.
Two triangulations $ T $ and $ T' $ is isomorphic when a one-to-one mapping exists from $ i $-cells($  i= 0,1,2 $) of $ T $ onto $ i $-cells of $ T' $, preserving incidence including root simplices.
Enumeration of the number of isomorphism classes of rooted triangulations has been initiated by Tutte\cite{TUTTE}, and subsequently refined by Brown\cite{BROWNT} and Mullin\cite{MULLIN} and many others for various cases of discretization.

Tutte considered a class of triangulations in which no internal edge is incident with two external vertices ({\it 3-connected}).
The distribution of triangulations of type $ [n,m] $ is shown to be
\begin{eqnarray}
f^T_{n,m} &=& 3 \sum_{j=0}^{min (m,n-1)} (m+j+2)(m-3j)  \nonumber \\
&\times& {4n+3m-j+1 \choose n-j-1,  3n+3m+3}{m+2 \choose j+1 ,  m-j+2}{m-1 \choose j  , m-j} \  (m \neq 0), \nonumber \\
f^T_{n,0} &=& 2 {4n+1 \choose  n+1 , 3n+2}, \label{eq.2.19}
\end{eqnarray}
where the suffix $ T $ represents Tutte.
Here, we have defined the generalized binomial coefficient by
\begin{equation}
{n_1 \choose n_2 , n_3} = {n_1 ! \over n_2 ! n_3 !}. \label{gbc}
\end{equation}

Allowing two end vertices of an internal edge are external ({\it 2-connected})  Brown obtained the distribution function for type $ [n,m] $ to be
\begin{equation}
f^B_{n,m} = 2 {2m+3 \choose m ,  m+2}{4n+2m+1 \choose n ,  3n+2m+3}, \label{eq.2.16}
\end{equation}
where the suffix $ B $ stands for Brown.
These two classes of triangulations are required to satisfy that
(a) no two edges have the same end vertices, and
(b) every edge is incident with two distinct vertices.
Mullin considered a class of triangulations without the constraint (a) ({\it multiple-edges}), and obtain
\begin{equation}
f^M_{n,m} = {2m + 3 \choose m+1,  m +1} 2^{n+1} {3n + 2m+2 \choose n ,  2n + 2m + 4}, \label{eq.2.15}
\end{equation}
where the suffix $ M $ represents Mullin.
Recently, Krikun\cite{KRIKUN} considered the degenerate triangulations without the constraint (a) and (b) ({\it loops}).
The distribution function is shown to be
\begin{equation}
f^K_{n,m}={2m+6 \choose m+2 , m+3} 2^{3n-4} {{3 \over 2} n+m+{1 \over 2} \choose n , {1 \over 2} n+m+{5 \over 2}},\label{eq.2.18}
\end{equation}
where the suffix $ K $ stands for Krikun.
Brown\cite{BROWNQ} also considered the distribution function for the quadrangular dissection of type $ [n,m] $ with $ n $ internal and $ 2 m+4 $ external vertices.
The distribution function is given to be
\begin{equation}
g_{n,m} = 3 {3m+4 \choose m , 2m+3}{3n+3m+2 \choose n , 2n+3m+4}. \label{BQdst}
\end{equation}

To our knowledge they exhaust all the known exact distribution functions of the 2-disk.
Analytical methods for extracting the distribution function from the generating function are often highly technical and sometimes intuitive, and we recommend readers to refer original papers for the details.

It will be essential in our later construction of the distribution function for the discrete 3-disk to keep in mind that distribution functions of the Mullin, Brown, Krikun triangulation,  and the Brown quadrangulation have the product form of two generalized binomial coefficients.
The 2-dimensional manifolds considered in these four cases have a common feature that there exists a 1-dimensional sub-manifold embedded in the 2-dimensional plane.
By extracting $ n=0 $ components of distribution functions of triangulations, we find $ f^{B, M, K}_{0,m}=C_{m+1} $,
where $ C_{m} $ is the Catalan number,
$$
C_m={2m \choose m,m+1},
$$
and the distribution function of quadrangle dissection gives
\begin{equation}
g_{0,m} =3{ 3m+2 \choose m \ 2m+3}. \label{1dsub}
\end{equation}
They all correspond to the distribution functions of branched trees made of either trivalent vertices for triangulations or tetravalent vertices for quadrangulation.
In contrast, the Tutte distribution function does not have such a simple product property, since it prohibits 1-dimensional sub-graphs.
We consider this is the main reason of the product nature of distribution function of maps which possesses the hierarchical structure.

\subsection{asymptotic distributions}
The asymptotic limit of a distribution function is achieved by taking the total number of 2-simplices large.
Writing ${\hat N_i}$ and ${\tilde N_i}$ for $ i=0,1 $ as total numbers of the internal and the external $ i-$simplices, respectively, we have the relationship either $ N_2 = 2{\hat N_0} + {\tilde N_1} -2 $ for triangulations, or $ N_2 = {\hat N_0} + {\tilde N_1}/2 -1 $ for quadrangulations.

Let us consider the asymptotic limit of a distribution function taking the Mullin distribution function as an example.
Other cases are treated exactly in the same manner as this example except the Tutte distribution function, which will be considered separately.
We introduce the new variables $ N = N_2 -1 $, and $ q = m/N $ together with $ p = n/N $.
Then, we have the relationship, $ p = {1 \over 2}(1-q) $, which implies the ranges of these variables as $ q= [0, 1] $, and $p= [0, 1/2] $.
By making use of the Stirling formula the generalized binomial coefficient has the approximate asymptotic form,
\begin{eqnarray}
 {a_1N + b_1 \choose a_2N+b_2, \  a_3N+b_3 } &\rightarrow &
{a_1^{b_1+1/2} \over a_2^{b_2+1/2} a_3^{b_3+1/2} } \left({a_1^{a_1} \over a_2^{a_2} a_3^{a_3}}\right)^N  N^{c-1/2}\\ \nonumber
& \times  & N^{(a_1-a_2-a_3) N} e^{-(a_1-a_2-a_3) N},
\label{afgbc}
\end{eqnarray}
where we write $ c = b_1-b_2-b_3 $, which will represent the critical exponent associating to a map whose distribution is expressed by the generalized binomial coefficient.
Applying the form to the Catalan number, where $a_1=2, a_2=a_3=1, b_1=b_2=0 $ and $ b_3=1 $, the asymptotic limit is written as
$$
C_N \rightarrow N^{-3/2} 2^{2 N}
$$
within an overall constant, where the exponent $-3/2 $ is characteristic to the 1-dimensional map.
We notice that the most diverging factor $ N^{(a_1-a_2-a_3) N} $ disappears in this case.
We shall observe that this diverging factor disappears in all the known distribution functions in the following analyses, and thus, the distribution is bounded exponentially.

The asymptotic limit of the Mullin distribution function expressed in terms of new variables $ (N,q) $ can be written as
\begin{equation}
f_N (q) \rightarrow N^{-2} q^{1/2}  \phi(q) \Phi(q)^N, \label{eq.2.22}
\end{equation}
where
\begin{eqnarray}
\phi(q) & = &{(3+q)^{{5/2}} \over (1-q)^{1/2} (1+q)^{9/2}}, \label{eq.2.23a}\\
\Phi(q) & = & 2^{-1/2} 2^{q/2} {(3+q)^{(3+q)/2} \over (1-q)^{(1-q)/2} (1+q)^{1+q}}. \label{eq.2.23b}
\end{eqnarray}
In the asymptotic limit the distribution is dominated by the term $ \Phi(q)^N $.
Since the second derivative of the entropy density, which is defined by
\begin{equation}
 s_N(q)= -{1 \over N} \log [f_N (q)], \label{entropyD}
\end{equation}
is given asymptotically as
$$
{4 \over (1-q) (1+q) (3+q)},
$$
it is positive in the allowed range of $ q = [0,1] $, and the entropy density is a smooth concave function in this range.
The free-energy density defined by $ -s_N (q) + \mu q $, then, has a peak at $ q_0 $ in $ [0,1) $ for appropriate values of the parameter $ \mu=\mu_0 $, satisfying
$$
\mu_0 = {d s_N(q) \over d q} \Big |_{q_0} ,
$$
$ \Phi(q)^N $ is well-approximated by the Gaussian function for large $ N $ as
\begin{equation}
 \Phi(q)^N  \sim  \Phi(q_0)^N \exp[-{1 \over 2} a_0 N (q-q_0)^2], \label{eq.2.25}
\end{equation}
with
$$
 a_0 = {d^2 s_N(q) \over d q^2} \Big |_{q_0} .
$$
We obtain the asymptotic limit of the Mullin distribution function as
\begin{equation}
f_{N,m} \rightarrow  m^{1 \over 2} N^{-{5 \over 2}}\exp(-{a_0 \over 2 N} m^2+\mu m-\lambda N) \label{adisM}
\end{equation}
within a constant, where $ \mu=a_0 q_0 $, and $ \lambda= a_0 q_0^2/2-\log [\Phi(q_0)]$.
As for the $q$-dependence of the function $ \phi(q) $, which is regular in the range $ q=[0,1) $.
We approximate it by replacing $ q $ to its peak value at $ q_0 $, and absorb in the overall constant.
The exponent $-5/2 $ of the variable $ N $ is characteristic to the 2-dimensional map.

The asymptotic distribution functions of the Brown triangulation, the Brown quadrangulation, and  the Kulikun triangulation can be obtained precisely in the same form as eq.(\ref{adisM}) with the same exponent, but different non-universal constants $ (a_0, \mu, \lambda) $.

As we can expect from the non-product form of the Tutte distribution function, derivation of its asymptotic limit is a little involved comparing to the previous cases.
Nonetheless, the method is basically the same as others.
We replace the intermediate sum over $ j $ by the integral over $ x =  j/m $, regarding the distribution to be a continuous function of $ x $ assuming $ m (=q N) $ is large.
The asymptotic form of the Tutte distribution function is then expressed as
\begin{equation}
f^T_N (q) \rightarrow N^{-{3\over2}} \int_0^{x_{max}} dx \ \phi_T(x,q) \Phi_T(x,q)^N \label{eq.2.28}
\end{equation}
within a constant, where
\begin{eqnarray}
\phi_T(x,q) & = &{ \{2+(1-x) q \}^{3/2} \{1-(1+2 x) q\}^{1/2} \over (1+q)^{7/2} } \nonumber \\
& &\times  { (1+x) (1-3x) \over x^2 (1-x)^3} ,    \label{eq.2.29}
\end{eqnarray}
and where
\begin{eqnarray}
\Phi_T(x,q) & = & {\{ 2+(1-x)q \}^{2+(1-x)q} \over [\{1-(1+2x)q\}/2]^{\{1-(1+2x)q\}/2} \{3 (1+q)/2 \}^{3(1+q)/2}}  \nonumber \\
 & &\times \{x^x (1-x)^{(1-x)}\}^{-2q}.  \label{eq.2.30}
\end{eqnarray}
The upper value of the range of integration is given by
$$
x_{max} = \left\{\begin{array}{ll}
               1 & \mbox{for} \ 0 \leq q \leq {1\over 3} \\
               {1-q \over 2 q} & \mbox{for} \ {1\over 3} \leq q \leq 1 .
\end{array} \right.
$$

We shall first make the Gaussian approximation for $ \Phi_T(x,q)^N $ as a function of $ x $, and obtain
$$
f^T_{N,q} \rightarrow N^{-{3\over2}} \int_0^{x_{max}} dx \phi_T(x,q) \Phi_T(x_0,q)^N e^{ -{N \over 2} a_0 (q) (x-x_0)^2 }.
$$
Here, the peak position $ x_0 $ is determined by solving the equation
$$
{\partial {\hat s}_T(x,q)\over\partial x} = 0 ,
$$
where we define the asymptotic entropy density as $ {\hat s}_T(x,q) = -\log[\Phi_T(x,q)] $, and we obtain $x_0= (1-q)/(3-q) $.
We also write
$$
a_0(q)= {\partial^2 {\hat s}_T(x,q)\over\partial x^2} |_{x=x_0} ,
$$
which turns out to be $ (3-q)^3 (2-q) q/(1-q)^2/6 $.

Since $ x_0 $ is smaller than  $ x_{max} $ for any $ q $ within the range $ [0,1) $, the integration over $ x $ is dominated by the contribution around the peak $ x_0 $.
After carrying out the integration, we repeat the Gaussian approximation for  $ \Phi_T(x_0(q),q)^N $as a function of $ q $, and obtain
$$
f^T_{N,q} \rightarrow N^{-2} q^{1\over2} { (2-q_0)^{1/2}\over  (1+q_0)^{7/2}  (3-q_0)^{1/2} } \Phi_T(x_0,q_0)^N  \exp\{- {N \over 2}A_0 (q-q_0)^2\},
$$
where $ q_0 $ is determined by the equation
$$
 {\partial s_T(x_0(q),q)\over \partial q} = 0 ,
$$
which we solve numerically obtaining $q_0\sim0.2067$, and where we write
$$
A_0= {\partial^2 s_T(x_0(q),q)\over\partial q^2} |_{q=q_0},
$$
which is about $ 2.682 $.
Finally, the asymptotic distribution appears precisely in the same form as other cases:
\begin{equation}
f^T_{N,m} \rightarrow m^{1 \over 2} N^{-{5 \over 2}}\exp(-{A_0 \over 2 N} m^2+\mu m-\lambda N), \label{adisT}
\end{equation}
with $ \mu=A_0 q_0 $, and $ \lambda= A_0 q_0^2/2-\log [\Phi_T(x_0(q_0),q_0)]$.

It is worth noticing that despite the apparent difference of the Tutte distribution function from others, all the distribution functions in asymptotic regions fall in the same universal form with the same critical exponents $ m^{1 \over 2} N^{-{5 \over 2}} $.
This suggests that even a simple product of generalized binomial coefficients, which are supposed to reflect the hierarchical structure of the manifold, can produce the universal form of the distribution function in the asymptotic limit.
We shall rely on this observation for obtaining the 3-dimensional distribution function in the next section.

\section{A mathematical model of the distribution \\ function for the 3-disk}
So far no recursion equation of the generating function is known for three dimensions, and we would like to propose a mathematical model inspired by the knowledge we have gathered in 2-dimensional discrete manifolds.
We shall construct the 3-disk with a simple $ S^2 $ boundary from the 3-sphere.
\subsection{construction of the discrete 3-disk}
The 3-dimensional sphere is prepared as a simplicial complex with the total numbers of $ i $-simplices being $ {\cal N}_i \ ( i = 0,1,2,3) $.
The Euler-Poincare relationship for the $ S^3 $ topology gives
\begin{equation}
{\cal N}_0-{\cal N}_1+{\cal N}_2 -{\cal N}_3= 0, \label{eq.3.0}
\end{equation}
together with the dimensional constraint,
\begin{equation}
{\cal N}_2 = 2 {\cal N}_3 \label{eq.3.1}.
\end{equation}
The 3-dimensional disk may be obtained from the 3-sphere by choosing one vertex among $ {\cal N}_0 $, and removing all the tetrahedra sharing the vertex.
We call the removed complex as the cap, and the remaining disk as the target manifold.
They both have the $ D^3 $ topology with a common $ S^2 $ boundary.
The number of tetrahedra $ x $ of the cap is called the coordination number of the vertex.
Numbers of $ i $-simplices for the cap and the target manifold are written as $ \{ C_i \} $ and $ \{ N_i \} $, respectively.
For the target disk $ {N_i} $ are decomposed into sums of those belonging to the boundary $ \tilde{ N}_i $ and its complement $ \hat{N}_i $ satisfying
$$
N_i = \tilde{ N}_i +\hat{N}_i  \ (i = 0, 1, 2),
$$
which are often called the external and the internal $ i $-simplex, respectively.
Since the original 3-sphere is divided by the common $ S^2 $ boundary into two 3-disks, we have the relationships
\begin{eqnarray}
{\cal N}_i &=& N_i+C_i-{\tilde N}_i   \  (i=0,1,2), \nonumber  \\
{\cal N}_3 &=& N_3+C_3  \label{eq.3.2}.
\end{eqnarray}
According to the construction procedure the cap variables are given to be
\begin{equation}
C_0={\tilde N}_0+1, \
C_1={\tilde N}_0+{\tilde N}_1, \
C_2={\tilde N}_1+{\tilde N}_2, \
C_3={\tilde N}_2. \label{eq.3.3}
\end{equation}
By adding the two topological conditions of the $ S^2 $ boundary:
\begin{equation}
{\tilde N}_0 -{\tilde N}_1+{\tilde N}_2=2, \
2 {\tilde N}_1= 3{\tilde N}_2 , \label{eq.3.4}
\end{equation}
we count altogether 15 independent relationships among 18 variables.

Choosing $ {\hat N}_0, {\hat N}_1 $, and $ {\hat N}_2 $ as the independent variables, other variables representing the target 3-disk are expressed as
\begin{eqnarray}
N_0 &=& {\hat N}_2 - 2{\hat N}_1+3{\hat N}_0 + 4, \nonumber \\
 N_1 &=& 3{\hat N}_2 - 5{\hat N}_1+6{\hat N}_0 + 6, \nonumber \\
N_2 &=& 3{\hat N}_2 - 4{\hat N}_1+4{\hat N}_0 + 4 , \nonumber \\
 N_3 &=& {\hat N}_2 - {\hat N}_1+{\hat N}_0 + 1 ,
\label{ni}
\end{eqnarray}
and
\begin{eqnarray}
{\tilde N}_0 &=& {\hat N}_2 - 2{\hat N}_1+2{\hat N}_0 + 4,  \nonumber \\
{\tilde N}_1 &=& 3{\hat N}_2 - 6{\hat N}_1+6{\hat N}_0 + 6 ,\nonumber \\
{\tilde N}_2 &=& 2 {\hat N}_2 - 4{\hat N}_1+4{\hat N}_0 + 4.
\label{nitilde}
\end{eqnarray}

 Let us consider the practical construction procedure of a possible 3-disk $ ({\hat N}_0,  {\hat N}_1 ,  {\hat N}_2) $.
We start with an elementary tetrahedron (0,0,0), and either attach or detach one tetrahedron to or from the target disk one by one just like a crystal growth through the diffusion limited aggregation.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{fig1a.png}
\end{center}
\caption{$ {\cal M}_1 $-move \label{fig1}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{fig1b.png}
\end{center}
\caption{$ {\cal M}_2 $-move \label{fig2}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{fig1c.png}
\end{center}
\caption {${\cal M}_3 $-move \label{fig3}}
\end{figure}
There are 3 ways to attach a tetrahedron corresponding to the number of neighboring faces to be glued as shown in Figures 1, 2, and 3.
Each Figure shows a pair of circles.
The left circle and the right circle represent the $ S^2 $ boundaries before and after gluing a tetrahedron, respectively.
Solid lines and dotted lines represent the external and the internal edges, respectively, while the dashed lines are also the external edges, which are hidden behind the attached tetrahedron.
These gluing and removing procedures are called {\it moves}, which we represent as $ {\cal M}_i (i=1,2,3) $, and their inverses as $ \bar{\cal{ M}}_i $. They are sometimes called up-moves and down-moves, respectively.
Application of these three up-moves on a target 3-disk changes of variables $ \hat{ N}_i $, and $ N_i $, as listed in the following Table,
$$
\begin{array}{|c||c|c|c|c|c|c|c|c|}\hline
 moves &\Delta {\hat N}_2 & \Delta {\hat N}_1 & \Delta {\hat N}_0  & \Delta N_3 & \Delta N_2 & \Delta N_1 & \Delta N_0 & \Delta {\tilde N}_2 \\ \hline \hline
{\cal M}_1  &  1   &      0   &     0   &     1 &  3  & 3 &  1   &    2 \\ \hline
{\cal M}_2  &   2    &     1   &     0    &    1 &  2  & 1 &  0    &   0 \\ \hline
{\cal M}_3   &  3     &    3    &    1     &   1 &  1 &  0 &  0   &    -2 \\ \hline
\end{array}
$$
where we add $\Delta  {\tilde N}_2 $ in the list for the later use.

Let us first show that any possible 3-disk complex can be constructed by appropriate combinations of these three types of up and down moves.
Writing a net number of the ${\cal M}_i $-move as $ m_i $, {\it i.e.}number of  ${\cal M}_i $-moves minus number of  $ \bar{\cal{ M}}_i $-moves, the target 3-disk acquires the following numbers of simplices:
\begin{eqnarray}
N_0&=&4+ m_1, \nonumber \\
N_1&=&6+3 m_1+ m_2, \nonumber \\
N_2&=&4+3 m_1+2 m_2+m_3, \nonumber \\
N_3&=&1+m_1+m_2+m_3,\label{eq.3.7}
\end{eqnarray}
and
\begin{eqnarray}
{\tilde N}_0&=&4+ m_1- m_3, \nonumber \\
{\tilde N}_1&=&6 +3 m_1- 3 m_3, \nonumber \\
{\tilde N}_2&=&4 +2 m_1 - 2 m_3,\label{eq.3.8}
\end{eqnarray}
also
\begin{eqnarray}
{\hat N}_0&=& m_3, \nonumber \\
{\hat N}_1&=& m_2+3 m_3, \nonumber \\
{\hat N}_2&=& m_1+2 m_2+3 m_3. \label{eq.3.9}
\end{eqnarray}
Inserting $ m_i (i=1,2,3) $ written in terms of $ {\hat N}_0, {\hat N}_1, {\hat N}_2 $ obtained from the last three equations:
$$
m_1={\hat N}_2 - 2{\hat N}_1+3{\hat N}_0, \
m_2 = {\hat N}_1 - 3 {\hat N}_0, \
m_3 = {\hat N}_0,
$$
the remaining 7 equations give the relationships, eq.(\ref{ni}) and (\ref{nitilde}).
This implies that the three types moves are sufficient to construct a simplicial complex with any possible number of variables, $ \{N_i, {\tilde N_i}, {\hat N}_i \} $.
However, this does not necessarily mean any possible complex can be constructed by some combinations of those 3 types of moves.

Secondly, let us show that it is indeed possible by searching a path starting from a certain possible complex, and choosing appropriate down moves to arrive at an elementary tetrahedron.
Then, the inverse path gives one of the actual construction of the desired complex.
We start with a $ D^3 $ complex ($  {\hat N}_0 , {\hat N}_1, {\hat N}_2$), which has $ {\tilde N}_2 $ external faces.
Each external face is sorted to the following 3 types:
a) one of 3 faces of a 3-simplex attached to the main body by sharing one face such as the complex created by an $ {\cal M}_1 $-move.
b) one of 2 faces of a 3-simplex glued to the main body sharing 2 faces such as that created by an $ {\cal M}_2 $-move.
c) the face  of a 3-simplex glued to the main body sharing 3 faces such as that created by an $ {\cal M}_3 $-move.
It is possible any time to remove the 3-simplex whose external face belong to the a-type.
It is not true when the face is either b- or c-type, in case when the down-move results non-allowed triangulation such as creating a pair of 3-simplices which are attached through a vertex or an edge, or creating a vertex or an edge whose neighboring simplices do not form the $ D^3 $, or the semi-$ D^3 $ when it is on the boundary.

We begin with detaching all those simplices whose faces belong to the a-type.
Then the remaining external faces belong either to the b- or the c-type.
We select any one of simplices which are allowed to remove.
After the down-move we again check all the external faces and remove all those simplices whose faces belong to the a-type.
Repeating these procedures $ m_1+m_2+m_3 $ times we eventually arrive to an elementary tetrahedron.
Inverting these moves starting from a tetrahedron we can create the desired complex.

From the above construction procedures we recognize that there exist three hierarchical structures created by combinations of the 3 types of moves: i) applications of the $ {\cal M}_1 $-moves create complexes without any internal vertex and internal edge.
Then, all the internal triangles created by this type of moves have their associated three edges and vertices on the boundary like a branched tree, and thus these complexes belong to one-dimensional sub-manifolds.
ii) Combinations of the $ {\cal M}_1 $- and $ {\cal M}_2 $-moves create complexes having no internal vertices.
In such a complex two end-vertices of all the internal edges must lie on the boundary surface, that constrain the thickness of the 3-disks being the minimum.
It characterizes the manifold to be 2-dimensional like.
iii) Applying all the three types of moves, the resulting 3-disks are the 3-dimensional generic manifolds.

\subsection{the model distribution function}
We assume maps are rooted.
A rooted map of the 3-disk is obtained by specifying an external triangle as the rooted triangle, and selecting one of the three vertices and an arrowed edge pointing to the vertex under the right-hand screw rule directing outward as the rooted vertex and the rooted edge, respectively.
It is sometimes more convenient and physical to choose $ {\tilde N}_2 $ as the independent variable rather than $ {\hat N}_2 $ through eq.(\ref{nitilde}), which we shall employ hereafter.

The distribution function of the rooted 3-disk is expressed by $ f_{n,l,m}^{D^3} $, where we define
\begin{equation}
n={\hat N}_0, \
l={\hat N}_1, \
m={1 \over 2} ({\tilde N}_2-4). \label{eq.3.16}
\end{equation}
By removing a cap with the coordination number $ x $ from the $ S^3 $ manifold , we obtain the target disk with
\begin{eqnarray}
{\hat N}_0&=&{\cal N}_0-{x \over 2}-3, \nonumber \\
{\hat N}_1&=&{\cal N}_0+{\cal N}_3-2x-2, \nonumber  \\
{\tilde N}_2&=& x ,\label{eq.3.10}
\end{eqnarray}
together with
\begin{equation}
N_3={\cal N}_3-x .\label{eq.3.11}
\end{equation}
Therefore, once we obtain the coordination number distribution of the $ S^3 $ complex, $ P_{{\cal N}_0,{\cal N}_3}(x) $, we can calculate the distribution function for the target 3-disk, $ f^{D^3}_{n,l,m} $, as
\begin{equation}
f^{D^3}_{n,l,m}=P_{{\cal N}_0,{\cal N}_3}(x).
\label{eq.3.12}
\end{equation}

Corresponding to the three hierarchical structures there are three types of distribution functions.
When we apply only the $ {\cal M}_1 $-move on the manifold starting with a single tetrahedron, we can produce the restricted distribution function $ f_{0,0,m}^{D^3} $ corresponding to the 1-dimensional sub-manifold.
Maps created by the $ {\cal M}_1 $-moves are expected to be identical to the 2-dimensional quadrangular maps with zero internal vertex, which have the branched tree distribution function $ g_{0,m} $ of eq. (\ref{1dsub}), constructed by tetravalent vertices:
$$
f_{0,0,m}^{D^3}= 3{3m+2 \choose m, \ 2m+3}.
$$
Here, we restrict ourselves considering only spacial degrees of freedom without any matter field coupled ({\it pure gravity}) for simplicity.

By including the $ {\cal M}_2 $-moves we shall obtain maps containing non-zero $ {\hat N}_1 $ and $ {\tilde N}_2 $.
They give the distribution function of the form $ f_{0,l,m}^{D^3} $.
We expect from observations in the previous section that when the maps contain the 1-dimensional sub-manifold, the distribution function will be given by a product of two generalized binomial coefficients similar to $ g_{l,m}  $ eq.(\ref{BQdst}).
By making use of all types of moves we expect to construct a generic 3-dimensional manifold.

Now we give a naive but natural ansatz for the distribution function of the 3-disk, which possesses the double hierarchical structure, by a product of three generalized binomial coefficients,
\begin{eqnarray}
f_{n,l,m}^{D^3} &=&X A^m {a_1m+a_2 \choose m, \ a_3 m+a_4}  B^l {b_1 m+b_2 l+b_3 \choose l, \ b_4 m+b_5 l+ b_6 } \nonumber  \\
& & \times \ C^n {c_1 m+c_2 l+c_3 n+c_4 \choose n, \ c_5 m+c_6 l+c_7 n+ c_8}. \label{eq.3.18}
\end{eqnarray}
In this expression we count total 22 parameters, $A, B, C, X , \{a_i\}, \{b_i\}$, and $ \{c_i\}$, among which we are going to eliminate by imposing various conditions required either from theoretical or numerical reasons.
From the experience in the 2-dimensional exact distribution functions, the parameter sets $ {a_i, b_i, c_i} $
is expected to take either positive integer or half-integer values.
We consider here limiting within integer values for simplicity.

We should remind that the model distribution does not respect details of practical construction rules, such as the $ n $-conectedness, or acceptance of degenerate graphs.
Relying on the observations in the 2-dimensional exact distribution functions, we expect it is only in the asymptotic region the model distribution function reveals the universal form, which is free from constraints put in practical construction procedures.

One of the most basic theoretical requirement, which is not proven mathematically but seems to be true numerically, is the exponential boundedness of the number of independent geometries by the total number of 3-simplices $ N_3 $.
According to the asymptotic limit of the generalized binomial coefficients, the most diverging factor $ N^N $ is required to disappear.
It is satisfied by imposing three conditions,
\begin{eqnarray}
a_1-1-a_3+b_1-b_4+c_1-c_5=0,  \nonumber \\
b_2-1-b_5+c_2-c_6=0, \nonumber \\
c_3-1-c_7=0.
\label{condit1}
\end{eqnarray}

Hierarchical structure of  2-dimensional sub-manifold requires that the distribution function of the 2-dimensional sub-manifold
$$
f_{0,l,m}^{D^3} =X A^m {a_1m+a_2 \choose m, \ a_3 m+a_4} B^l {b_1 m+b_2 l+b_3 \choose l, \  b_4 m+b_5 l+ b_6 }{c_1 m+c_2 l+c_4 \choose 0, \ c_5 m+c_6 l+ c_8},
$$
should be written as a product of two generalized binomial coefficients.
It is achieved by asking
\begin{equation}
b_1=c_5, \ b_2=c_6, \ b_3=c_8,\label{condit2}
\end{equation}
resulting
$$
f_{0,l,m}^{D^3} =X A^m {a_1m+a_2 \choose m, \ a_3 m+a_4} B^l {c_1 m+c_2 l+c_4 \choose l, \  b_4 m+b_5 l+ b_6 }.
$$
When no matter degree of freedom is involved, we ask the distribution function of 2-dimensional sub-manifold has the same critical exponent as the 2-disk distribution function:
\begin{equation}
c_4-b_6=-2. \label{condit3}
\end{equation}

Similarly, the distribution function for the one dimensional sub-manifold,
$$
f_{0,0,m}^{D^3} =X A^m {a_1m+a_2 \choose m, \ a_3 m+a_4}{c_1 m+c_4 \choose 0, \  b_4 m+ b_6 },
$$
is required to be that for branched trees constructed by tetravalent vertices given by $ g_{0,m} $ eq.(\ref{1dsub}).
This gives
\begin{equation}
a_1=b_4, \ a_2=b_6, \label{condit4}
\end{equation}
and
\begin{equation}
X=3, \ A=1, \ c_1=3, \ c_4=2, \  a_3=2, \ a_4=3. \label{condit5}
\end{equation}

At the end we obtain the 3-disk distribution function for pure gravity as
\begin{eqnarray}
f_{n,l,m}^{D^3} &=&3 {a_1m+4 \choose m, \ 2m+3} B^l {b_1 m+b_2 l+5 \choose l, \  a_1 m+b_5 l+4}  \nonumber \\
&\times & C^n {3 m+(b_5+1) l + c_3 n+2 \choose n, \ b_1 m+b_2 l+(c_3-1) n+ 5} , \label{modelD}
\end{eqnarray}
where the parameter $ b_3 $ is set to be $ 5 $, so that $ c_4-c_8=2-b_3 $ being $ -(manifold \\ dimension) $, which has been the case in  $ 1 $ and $ 2 $ dimensions.
In any way $ b_3 $ will not appear explicitly in the asymptotic limit.
We still count 7 free parameters at this moment left undetermined, which may be fixed by comparing to numerical simulations in the following sections.

\subsection{asymptotic distribution}
In order to study the universal behavior of the distribution function which are expected to appear in asymptotic regions, we define new variables similar to the 2-dimensional cases corresponding to the independent variables $ n, l, m $:
\begin{equation}
p=n/N,  \ r=l/N, \ q=m/N, \label{eq.3.237}
\end{equation}
where
\begin{equation}
N=N_3-1. \label{eq.3.28}
\end{equation}
From eq.(\ref{ni}) and (\ref{nitilde}) there is a relationship,
\begin{equation}
q+r-p=1. \label{eq.3.29}
\end{equation}
We shall select $ (N,p,q) $ as the new set of independent variables when we discuss the asymptotic limit.

By making use of the asymptotic formula of the generalized binomial coefficient eq.(\ref{afgbc}), the model distribution function of the 3-disk in asymptotic regions, which we express as $ f^{D^3}_N (p,q) $, is written within an overall constant by
\begin{equation}
f^{D^3}_N (p, q) \sim N^{-5/2} q^{1/2} \phi (p, q) \Phi(p, q)^N, \label{3ddis}
\end{equation}
where
\begin{equation}
\phi(p, q)= { (b_1 q+b_2 r )^{11/2} \over r^{1/2} (a_1 q+b_5 r)^{9/2} } { (3 q+(b_5+1) r+c_3 p)^{5/2} \over p^{1/2} (b_1 q+b_2 r+(c_3-1) p)^{11/2}}, \label{eq.3.30b}
\end{equation}
and where
\begin{eqnarray}
\Phi(p, q) & = & (a_1^{a_1}/4)^q B^r C^p { (b_1 q+b_2 r)^{b_1 q+b_2 r} \over r^r (a_1 q+b_5 r)^{a_1 q+b_5 r}} \nonumber \\
& \times & {\{3 q+(b_5+1) r+c_3 p \}^{3 q+(b_5+1) r+c_3 p} \over p^p \{b_1 q+b_2 r+(c_3-1) p\}^{ b_1 q+b_2 r+(c_3-1) p}} \label{3dPhi}.
\end{eqnarray}

\section{ Phases of the stationary distributions \\ and comparison to numerical results}
We now wish to check our proposition for the distribution function of the discrete 3-disk by comparing to numerical results.
Until now there are number of numerical studies available \cite{DT3S-BK, DT3S-AM,DT3S-AV, DT3S-HTY, DT3D-WCR,DT3S-T} for pure gravity with $ S^3 $ topology, and one result of $ D^3 $ \cite{DT3S-RCK}.
We shall look for appropriate values of parameters left undetermined in the last section from the numerical data.
\subsection{$S^3$ topology}
We can study statistical properties of the discrete $ S^3 $-manifold through the canonical partition function
\begin{eqnarray}
Z_{{\cal N}_3}^{S^3}[\nu] &=& \sum_{{\cal T}(S^3)} e^{\nu {\cal N}_0}\\ \nonumber
&=& \sum_{{\cal N}_0} f^{S^3}_{{\cal N}_0,{\cal N}_3} e^{\nu {\cal N}_0},\label{eq.4.1}
\end{eqnarray}
where sums $ {\cal T}(S^3) $ are taken over all the independent triangulations with $ S^3 $ topology.
The distribution function $ f^{S^3}_{{\cal N}_0,{\cal N}_3}$ can be obtained from the $ D^3 $ distribution function with the least boundary area, which is created from the $ S^3 $ simplicial complex by removing one of $ {\cal N}_3 $ 3-simplices, as
\begin{equation}
 f^{S^3}_{{\cal N}_0,{\cal N}_3} = f^{D^3}_{n,l,0}/{\cal N}_3, \label{eq.4.2}
\end{equation}
where
$$
n={\cal N}_0-4,\ l={\cal N}_0+{\cal N}_3-6.
$$
In the asymptotic limit we get
\begin{equation}
 f^{S^3}_{{\cal N}_0,{\cal N}_3} \rightarrow f^{S^3}_{N}(p) \sim N^{-4} \hat{\phi} (p)  (BD)^N (BCD)^{pN} \hat{\Phi} (p)^N,\label{3sdis}
\end{equation}
where $ N={\cal N}_3-2, p=n/N $, and $ D=b_2^{b_2}/b_5^{b_5} $.
After separating those terms to be included in either the normalization constant or the cosmological constants in $ \phi(p,0) $ and $ \Phi(p,0) $, we have
\begin{eqnarray}
\hat{\phi}(p) &\sim& {1 \over p^{1/2} }, \\ \nonumber
\hat{\Phi}(p) &\sim& { r^{(b_2-b_5-1) r} \over p^p} {\{(b_5+1) r+c_3 p\}^{(b_5+1) r+c_3 p} \over \{b_2 r+(c_3-1) p\}^{b_2 r+(c_3-1) p} }, \label{eq.4.3}
\end{eqnarray}
with $ r=1+p $.
We note the analytic part in $  \phi(p,0) $ is set to a constant because of  $ \hat{\Phi} (p)^N $ factor which dominates the sum over $ {\cal N}_0 $ in the large $ N $ limit.

Asymptotically the partition function can be written in the integral form as
\begin{equation}
Z_{{\cal N}_3}^{S^3}[\nu] \sim N \int_0^{p_{max}} dp \ f^{S^3}_{N}(p) \ e^{\nu p N}.\label{eq.4.4}
\end{equation}
Upper value of the $ p $-integration $ p_{max}$ is determined  from eq.(\ref{eq.3.11}) in the limit $ N \rightarrow \infty $:
\begin{eqnarray}
p &\rightarrow& {\eta-\xi/2 \over 1-\xi},\nonumber \\
q &\rightarrow& {\xi \over 2(1-\xi)},\label{rgpq}
\end{eqnarray}
where $ \xi=x/{\cal N}_3 $ and $\eta= {\cal N}_0 / {\cal N}_3 $.
Since the physical range of $ q $ is $ [0,1) $, $ \xi $ varies within the interval $ [0, 2/3) $, and $ p $ takes the value in $ (3 \eta-1,\eta] $ .
When  $ q \sim 1 \  ({\it i.e.} \ \xi \sim 2/3) $, most of the tetrahedrons have two neighbors to form a linear chain with its two faces attaching to neighboring tetrahedrons, while the remaining two faces left as external faces.
This branched tree structure does not contain internal vertices, and the value $ p=0 $ at this limit fixes the maximum value of $ \eta $ to be $1/3 $, as conjectured by Walkup \cite{WALKUP}.
The upper limit of $ p $ is fixed at the maximum value of $ \eta $:  $ p_{max} = 1/3 $.

One of the important findings of DT simulations is the nature of phase transition, which turned out to be of first order.
Property of the phase transition can be studied through the free energy density defined by
\begin{equation}
s_N^{S^3} (p) =  {1 \over N} \log [f_N^{S^3}(p)]. \label{eq.4.7}
\end{equation}
Asymptotically it converges to
\begin{eqnarray}
s_N^{S^3} (p) &\rightarrow& \log[ {(1+b_5)^{1+b_5} \over b_2^{b_2}}] \ (p\rightarrow0),\nonumber \\
 &\rightarrow&  \log[ {(1+b_5+c_3)^{1+b_5+c_3} \over (b_2-1+c3)^{b_2-1+c_3}}] p \ (p \rightarrow \infty),\label{eq.4.12}
\end{eqnarray}
The second derivative of the free energy density,
\begin{equation}
s_{N}^{(2)}(p) \sim- {c_3 (c_3-1) (1+b_5-c_3) p + b_2 (1+b_5) c_2 r \over p (p+1) ((1+b_5) r + c_3 p)(b_2 r+(c_3-1) p)}, \label{2den}
\end{equation}
behaves as $-1/p $ for small $ p $, and as $ p $ gets larger it increases and cross zero once at $ p= p_1 $ with
\begin{equation}
p_1={1 \over  ({1 \over b_5+1}-{1 \over b_2}) c_3 (c_3-1) -1}.\label{eq.4.13}
\end{equation}
When $ p_1 $ lies below $ 1/3 $, the distribution function has two peaks, one at $1/3 $ and another between $ 0 $ and $ p_1 $, for an appropriate parameter range of $ \nu $.
In this case the system makes 1st order phase transition at the critical $ \nu $, where heights of the two peaks are equal.
Numerical simulation shows that the average $ N_0 $ has a jump at $ \nu=\nu_c \sim 4.0 $, with a small gap of $ \delta <p> \sim 0.03 $ \cite{DT3S-HTY}, and we shall choose the integer parameter triplet $ (b_2, b_5, c_3) $ in the region where $ p_1 $ is less than but close to $ 1/3 $.
We note that the system shows second order phase transition at $ p_1=1/3 $, and it becomes cross over transition as $ p_1 $ gets bigger than $ 1/3 $.

In order to fix parameters from the DT simulation results we need to make direct numerical integrations for the canonical partition function and other expectation values.
Among the unknown parameters there is no observable to determine the values directly except $ B $, and $ C $, which are related to the cosmological constants.
We shall fix the parameter triplet  $ (b_2, b_5, c_3) $ which control the nature of phase transition through $ p_1 $, by asking 1) they are positive integers and choose as small as possible just for simplicity, 2) they give $ p_1 $ reasonably close to $ 1/3 $ in order to make the jump in the average $ N_0 $ small, and 3) they give correct susceptibility peak pattern in the 2-dimensional parameter space of the $ D^3 $ topology, which will be discussed in the next sub-section.
At first we fix $ b_5 $ to be $ 3 $, after checking that $ b_5=1 $, and $ 2 $ do not satisfy the above condition 3.
With the fixed $ b_5 $ values of $ b_2 $ and $ c_3 $ are bounded below by the requirement $ p_1<1/3 $.
Next, we choose $ b_2=6 $, which is the smallest possible value, and lastly $ c_3 $ to be $ 8 $ in order to make $ p_1 $ close to $ 1/3 $.
With these parameters the expectation value $ <p>(\sim {\cal N}_0/{\cal N}_3) $ and the finite size scaling of the $ {\cal N}_0 $-susceptibility are shown in Figures 4 and 5, respectively, which clearly exhibits the characteristics of first order phase transition.

\begin{figure}
\begin{center}
\includegraphics[width=.7\linewidth]{figSavp.png}
\caption{$ <p>(\nu) $. $ N=100 $. $ \nu_c $ is fixed at $ 3.6 $.}
\end{center}
\label{fig4}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.7\linewidth]{figSfss.png}
\caption{ Finite size scaling of $\chi (\nu_N)$ for $ N=100,200,300,400,500 $. $\nu_N $ is chosen at the peak position of $ \chi(\nu) $.The line is for guiding eyes.}
\end{center}
\label{fig5}
\end{figure}

Two constants $ B $ and $ C $ are fixed from the position $ \nu_c $ of the jump in the average value $ < N_0 > $, and the asymptotic entropy density $ \mu_0 $ obtained from the canonical partition function eq.(\ref{eq.4.4}), which we parametrize as
\begin{equation}
{1 \over N} \log (Z_N^{S^3}[0]) \sim  \mu_0 -(3-\gamma) {\log N \over N} \label{eq.4.141} ,
\end{equation}
at a point $ \nu=0 $.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{figSZfit.png}
\caption {Numerical fit of $\log (Z_N^{S^3}[\nu])/N $ by $\mu_0-(3-\gamma) {\log N \over N} $, with $ \mu_0=2.0 $ and $ \gamma=0.35 $ at $ \nu=0 $.}
\end{center}
\label{fig6}
\end{figure}
In Figure 6 we show $\log (Z_N^{S^3}[0])/N $ with a fixed $ \mu_0 \sim 2.0 $, which gives $ \gamma \sim 0.35 $.
This parametrization differs from numerical simulations.
There were two numerical results available for the parameter $ \mu_0 $: in Ref.\cite{DT3S-AV} it was
$$
{1 \over N} \log (Z_N^{S^3}[0]) \sim  \mu_0 -\beta {1 \over N^{\alpha}} ,
$$
with $ \mu_0 \sim 2.06$, $ \alpha \sim 0.32 $ and $\beta \sim 3.90 $ , while in Ref.\cite{DT3S-AM} it was
$$
{1 \over N} \log (Z_N^{S^3}[0]) \sim  \mu_0 -\beta {1 \over \log N}
$$
with $ \mu_0 \sim 1.62$ and $ \beta \sim 1.44 $.
Our parametrization intends to give a meaning to the parameter $ \gamma $ to be the string susceptibility of the discrete $ S^3 $ manifold.
\begin{figure}
\begin{center}
\includegraphics[width=.7\linewidth]{figScc.png}
\caption{$ \mu_0 (\nu)  $ Two lines are $\mu_0= 2 $, and $ 0.55+\nu/3$}
\label{fig7}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.7\linewidth]{figSueg.png}
\caption{  $\gamma (\nu) $}
\label{fig8}
\end{center}
\end{figure}
Numerically $ \mu_0 $ and $ \gamma $ may depend on $ \nu $.
In Figures 7 and 8 we show the $ \nu $ dependence of $ \mu_0 $ and $ \gamma $ obtained by numerical fitting of  $\log (Z_N^{S^3}[\nu])/N $ for $ N $ between $ 1000 $ and $ 10000$, respectively.
They are almost constant up until the transition point $ \nu \sim 4 $, and after that they behave differently.
This is because of the limit in the integration over $ p $ at $ p_{max} =1/3 $.
When $ \nu>\nu_c $, the peak at $ p=1/3 $ dominates in the integration.
As for the constant $ \nu_c $, there exist also two simulation results: $ \sim 4.0 $  determined from the $ S^3 $ DT-simulations\cite{DT3S-AV, DT3S-HTY}, and $ \sim 3.6 $ observed in the susceptibility map of the $ D^3 $ DT-simulation \cite{DT3D-WCR}.
These facts pretend us existence of the strong finite size effect and probably the systematic errors depending on numerical methods.

\subsection{$ D^3 $ topology}
Since the current study is intended for a construction of the 3-dimensional Universe, which stands alone in the non-framed 'space', there is no meaning to specify a root in the map, and the distribution should be counted for non-rooted maps.
Transformation of distribution functions from rooted to non-rooted $ D^3 $-maps is approximately performed through dividing the $ D^3 $-distribution function by  the number of possible root simplices, which equals to the number of boundary $2$-simplex $ {\tilde N}_{2}$.
If a map has spacial symmetries, such as rotational, reflectional, etc. this simple rule cannot be applied.
However, such symmetric maps are rare especially when $ N $ is large, and we expect they will not give much effects on the distribution function in the asymptotic limit.
We shall write the approximate distribution function of non-rooted maps as
\begin{equation}
{\hat f}_{n,l,m}^{D^3}={1 \over m} f_{n,l,m}^{D^3}. \label{eq.4.142}
\end{equation}

Statistical properties of the discrete $ D^3 $-manifold are studied through the canonical partition function
\begin{equation}
Z_N^{D^3}[\mu,\nu] = \sum_{{\cal T}(D^3)} e^{-S[\mu,\nu]},\label{eq.4.151}
\end{equation}
where sums ${\cal T}(D^3)$ are taken over all the non-rooted maps with the $ D^3 $ topology.
Here the action for the $ D^3 $ topology is given by
\begin{eqnarray}
S[\mu, \nu]=\mu {\tilde N}_2 -\nu N_0 ,\label{eq.4.152}
\end{eqnarray}
In the asymptotic limit the partition function is written in the integral form as
\begin{equation}
Z_N^{D^3}[\mu,\nu] = N^2 \int_0^{1/3} dp \int_0^{q_{max}(p)} dq \ {\hat f}^{D^3}_N (p,q) e^{-\{(2 \mu-\nu) q-\nu p \} N},\label{eq.4.16}
\end{equation}
where $ p=n/N $ and $ q=m/N $ as before, and where
$$
{\hat f}_{N}^{D^3}(p,q)={\hat f}_{n,l,m}^{D^3}. \label{eq.4.17}
$$
The distribution function ${\hat f}^{D^3}_{N}(p,q)$ is defined from the original distribution function  ${\hat f}^{D^3}_{n,l,m}$by replacing $ l $ to $ N $ using the topological relationship $ N=l-n+m $.
For a fixed $ p $, eq.(\ref{rgpq}) gives $ q_{max}(p)=1-3p $.

The model distribution function for non-rooted $ D^3 $-maps in the large $ N $ limit is written as,
\begin{equation}
{\hat f}^{D^3}_N (p, q) \sim N^{-7/2} q^{-1/2} {\hat \phi} (p, q) B^N (BC)^{pN} (a_1^{a_1}/4/B)^{qN}   {\hat \Phi} (p, q)^N, \label{eq.4.17a}
\end{equation}
where
\begin{equation}
{\hat \phi}(p, q)= {1 \over r^{1/2} p^{1/2}},\label{eq.4.17b}
\end{equation}
and
\begin{equation}
{\hat \Phi}(p, q) = {q^{(a_1-3) q} \over r^r p^p} { (b_1 q+b_2 r)^{ b_1 q+b_2 r} \over  (a_1 q+b_5 r)^{ b_4 q+b_5 r}} {(3 q+(b_5+1) r+c_3 p)^{3 q+(b_5+1) r+c_3 p} \over (b_1 q+b_2 r+(c_3-1) p )^{ b_1 q+b_2 r+(c_3-1) p }}. \label{eq.4.76c}
\end{equation}
In the above equations $ r $ should be replaced by $ 1+p-q $.
Analytic part of $ \phi(p,q) $ is set to a constant in the same reason as the $ S^3 $ case, and also in order to suppress the $ N $ dependence in numerical integrations with finite $ N $.
We need to make the double integrations for the patition function and other expectation values, which is time consuming.
For getting a better convergence and to save computer time we set $ N $ as small as possible, and choose $ N=30 $ in the following calculations for the susceptibility landscapes in the two parameter space.

To carry out the numerical integrations, we need to fix additional two integers for $ a_2 $ and $ b_1 $ for $ \hat{\Phi}(p,q) $.
Starting with $ a_1=1 $ and $ b_1=1 $, we find $ a_1 $ should be larger than $ 3 $ under the similar stuation as the condition on $ b_5 $.
After fixing $ a_1=3 $ we search appropriate $ b_1 $ starting from $ 1 $.
Until $ b_1 $ reaching to $ 5 $ the susceptibility peak range lying vertically is tilted, and we finally select $ (a_1, b_1)=(3,5) $.
We need to adjust the parameters $ B $ and $ C $ in order to fit the horizontal range of susceptibility peaks lying along $ \nu \sim 3.6 $, instead of the $ S^3 $-sumulation value of $ 4.0 $, and also $ \mu_c = 1.95 $, which lyes between two numerical results $ 1.6 $ and $ 2.0 $.
The agreement between the numerical result\cite{DT3D-WCR} and our model turned out to be unexpectedly good as we see in Figure 9 for the average $ N_0 $, and Figure 10 for the $ N_0 $-susceptibility.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{figDavN0.png}
\caption{$ \ <N_0>(\mu, \nu)/N $. $ N=30 $.}
\end{center}
\label{fig9}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{figD368.png}
\caption{$ N_0 $-susceptibility. Red dots are taken from \cite{DT3D-WCR} with dashhed lines for guiding eyes. }
\end{center}
\label{fig10}
\end{figure}

We also calculate the average values of $ p \sim {\hat N}_0/N $, and $ q \sim {\tilde N}_2/N $ as shown in Figures 11 and 12, respectively.
They show clearly the existence of three phases reflecting the hierarchical structure of the $ D^3 $-manifold:
 i)  The platou in $ <p> \sim 1/3 $(blight area) correspond to the 3-dimensional generic manifold.
 %
ii) The blight area in $ <q>\sim1 $ represents the 1D sub-manifold (branched tree) phase, and iii) the remaining area, which is the dark areas in both $ <p> $ and $ <q> $ is considered to be the 2-dimensional sub-manifold ($ <r> \sim 1 $).
These three areas are surrounded by susceptibility peaks determined by the DT-simulations.

\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{figDavp.png}
\caption{$<p>(\mu,\nu)$. Hights of the blight platau is $ \sim 1/3$ and the dark bottom $\sim 0 $.}
\end{center}
\label{fig11}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{figDavq.png}
\caption{$ <q>(\mu,\nu) $. Hights of the blight platau is $ \sim 1$ and the dark bottom $\sim 0 $. }
\end{center}
\label{fig12}
\end{figure}
\section{Discussions and Summary}
For predicting the distribution function of the discrete 3-disk we have made use of the common structural property in the exact forms of certain 2-dimensional disks.
We rely on the following two observations.
Firstly, when a discrete 2-disk manifold contains 1-dimensional sub-manifold the distribution function is written in a simple product of two generalized binomial coefficients.
Secondly, even if a manifold does not belong to above mentioned class, such as the Tutte distribution function, the distribution function will show asymptotically the same universal form with the same critical exponent as the product type.
In the 3-dimensional case the true distribution function of a manifold constructed by the DT may not be written in a simple product form.
However, we know that the discrete 3-disk constructed by the DT method possesses hierarchical substructure which persists in the asymptotic regions, and we expect a simple product distribution function of generalized binomial coefficients will show the same universal asymptotic behavior as that obtained by the DT.

In oder to check above prediction we compare our formula of the discrete 3-disk distribution function with numerical results obtained by the DT simulations.
We choose a set of parameters which fulfill exponential boundedness of the state volume, and show the hierarchical structure of sub-manifolds and 1st order phase transition, which is one of the significant feature of DT  simulation in $ S^3 $ manifold.
In general, our predicted distribution function shows the possibility of three types of transitions, either crossover, 1st order, or 2nd  order, depending on the choice of the parameter set.
Since 2nd order phase transition appears at a point on a line in the parameter space where phase changes from crossover to 1st order, we understand 1st order transition has more probability of appearance.
There will be a possibility to obtain  2nd order phase transition if we put appropriate matter fields on 3-dimensional discrete space.

Phase structure of the 3-dimensional manifolds is another important point to check.
It can be visualized by the expectation values of $ p $ and $ q $ in the two dimensional parameter space $ (\mu,\nu) $.
We observe three characteristic regions: i) for large $ \nu $ with small $ \mu $,  a plateau with the hight $ <p> \sim 1/3 $, which  corresponds to generic 3-dimensional manifold,
 ii) for large $ \mu $, a plateau with $ <q> \sim 1 $ which exhibits the 1-dimensional sub-manifold property,
 and iii) small $ \nu $ and $ \mu $ where $ <p>, <q> \sim 0 $, which is expected for the 2-dimensional sub-manifold.
Three phases are well-separated by two walls in the susceptibilities: one in $ \chi_p $ which separate the generic 3-dimensional disk and the 2-dimensional sub-manifold, and another in $ \chi_q $ which divides the 1-dimensional sub manifold from others.
Over all view of phase structure is quite similar to the numerical result \cite{DT3D-WCR}.

Our primary task of achieving an analytical form for the distribution function of 3-disk is to apply it for the statistical evolution equation of the 3-dimensional Universe.
The 'free energy' $ \log [f] $ acts as the force field in the master equation for the probability of the 3-dimensional Universe with the volume $ V=v_0 N_3 $, the surface area $ A=a_0 {\tilde N}_2 $, and the 'dilatation' $ D=d_0 {\hat N}_0 $ along the 'Markov-time' $ \tau $.
When we define the physical time $ t $ by
\begin{equation}
t=\int^{\tau} {1 \over A(\tau) } {dV(\tau) \over d\tau} d \tau,
\end{equation}
so that it satisfy the rate of change of 3-volume to be $ dV(t)=A(t) dt $, the relationship between the physical time and the Markov time is given by
$$
t \sim \left\{\begin{array}{ll}
               {1 \over 1-\kappa} \tau^{1-\kappa}  & \mbox{if} \ f\neq 1 \\
               \log \tau & \mbox{if} \ f= 1
\end{array} \right.
$$
for a solution of the master equation with the asymptotic behavior, $ V \sim \tau $ and $ A \sim \tau^\kappa $.
If we ask the physical time to posses the physical dimension $ 1 $, then the Markov time $\tau$ will have a dimension $ 1/(1-\kappa) $, and thus the body (V) and the face (A) carry the dimension $1/(1-\kappa)$ and $\kappa/(1-\kappa)$, for the case $ \kappa \neq 1$, respectively, while for the case $ \kappa=1 $ the space dimensions are infinity.
We recall in the 2-dimensional Universe the statistical evolution equation had three asymptotic solutions, $ \kappa = (1, 1/2, 0) $, and thus the face dimension varies $ (\infty, 1, 0) $.
We expect for the 3-dimensional Universe there will be a $\kappa=2/3 $ solution in the master equation, so that the dimensions $ V $ and $ A $ are 3 and 2, respectively.
We are now underway to obtain solutions for three dimensional evolution equation.
We also focus our investigation on the effect of phase transition in the evolution of the Universe.
Extension to the 4-dimensional case is straight forward, and it will soon appear in separate articles.

We would like to add some comments on the colored tensor model  which has been claimed to be a generalization of the matrix model for higher dimensions.
It is then regarded to be an analytic realization of the dynamical triangulation of space-dimensions higher than two.
In the model the dominant contributions come from graphs constructed with an elementary block called melon, which is formed by a pair of 3-simplices glued along 3 faces.
The generating function of rooted melon graphs is known to be
$$
T(\lambda)=1+\lambda^2 T(\lambda)^4
$$
where $ \lambda $ is the vertex coupling constant in the dual graph of the 3-dimensional triangulation.
This equation can be interpreted as the generating function of branching trees made by 5-nary vertices, which is known to give the distribution function,
$$
T_m={4 m \choose m , 3 m +1},
$$
where $ m=(\tilde{N}_2 -4)/2 = N $.
Asymptotically the number of graphs is  $ T_N \sim N^{-3/2} e^{\mu_0 N} $ with $ \mu_0 =(1/2) \log(256/27) $, which is about $ 1.12 $.
On the other hand in the DT simulations it is either $ 1.6 $ or $ 2.0 $, which considerably surpasses the melonic graphs.
We also remark that in the standard DT simulations the melon graph is excluded as degenerate graphs.
From the distribution functions $ T_m $ we can observe that the melonic graphs belong to the one dimensional universality class.
Recently, a new type of tensor model is introduced\cite{BENEDETTI-GURAU} with the critical property showing the 2-dimensional character in the double scaling limit.
We expect that another new tensor model which exhibit a genuine 3-dimensional phase transition will appear, so that the transition properties can be compared with our phenomenological model quantitatively.

redpajama_set_name: RedPajamaArXiv


…. and sometimes a trait required by the most effective of health and patient advocates.
And more importantly, do you know how and when to use it?
I ask this because I think there are appropriate times, and inappropriate times, when an advocate needs to showcase his or her chutzpah. Lately I have experienced both, and I wonder how that translates to its use for clients. I know some of you experience it, too.
Having chutzpah, and being able to use it, means you have enough confidence in what you are asking for that you believe you are entitled to, or possibly deserving of the outcome you request.
An example: your elderly client is in the hospital, sharing a room, and a single room comes open across the hall. You know your client would be much happier in that single room, so you ask to have him moved. Of course, they have already determined that someone else who is private pay is going to get that room, but you make the case (just enough of a stink) and ultimately your client finds himself alone in that single room. Yes, you demonstrated your chutzpah. Now your client is happier.
That's chutzpah! But – does it cross a line? Is it more like bribery? or blackmail? or…. ?
That's the big question. Where is the line? And is it appropriate to cross it?
I think it boils down to that word bolded below: entitled.
There is such a thing as true entitlement, and then there is the other side of that coin – non-entitled. But yes – there is also a gray area – where someone is possibly deserving, too.
As an advocate, working on behalf of your client, you need to be aware of which area a request or a need falls under, and approach it appropriately. If your client is truly entitled, then of course, go for it. If your client is possibly deserving, then go for it too!
But if your client is clearly not entitled, then you run the risk of creating bigger problems for yourself and your client down the road. Because making the decision-maker, the person who holds the winning cards, mad enough that they dig in their heals, may make it impossible for you to successfully get your way in the future.
Recently one of our APHA PACE members (who paid $49 for her membership) asked me if she could have access to the client contract we offer Premium members (who pay as much as $289 for membership.) I politely declined her request, but invited her instead to upgrade her membership so she could have the access she needed.
I gave her some credit! I didn't blame her for asking. The worst thing I could say was no, and that would leave her exactly where she was. She felt she was possibly deserving based on another situation that had transpired during her membership. It required some chutzpah on her part to ask – and I actually admire that. Why not ask?
A threat! THAT crossed the line!
She was never entitled, and now, in my opinion, she was no longer possibly deserving. My response to her was to watch for her refund on the balance of her membership dues, and good luck to her in the future.
She's miffed – very upset with me. I'm beyond miffed. She has also stymied her own chances to benefit from membership in the Alliance. She will not be invited to return.
Yes, crossing the line isn't just about pushing too far, it's about the penalties that may result.
As you work with your clients and identify their needs, I strongly encourage you to demonstrate your chutzpah when its required, or even just when it will improve the situation for your client. That's part of what your client pays you for and will appreciate.
But know where the line is, and be sure that you (and your client) are at least possibly deserving of the outcome you've requested. To step over that line puts your client, and your ability to do your job in jeopardy.
Long live chutzpah! …. to a point.
That woman is simply horri as my daughter would say, and I would add, unethical, after you said no, that should have been it.
Here, in europe's "socialized medicine", people do not feel a lot of entitlement, so my battle is the other way round. I have to fight for them to understand that they have far more rights than they believe!
having said that, I would not change the universal healthcare system for any other, it feels wrong that if you have more money you can get better treatment…sure, we do have private medicine here, but that gives you exactly that, private rooms, nicer looking nurses, and a bit less waiting time for surgeries.
But the outstanding doctors are in public medicine, where you have the challenging cases and the workload.
I really believe every human being is entitled to good medicine, without taking into account something that I only heard in the USA , a "person's net worth", what is that? people have net worth???
sorry if I have opened a can of worms……….
Well said!! I believe that one essential trait for a patient advocate must be assertiveness (which can often be analogous to chutzpah). But, as with chutzpah, you need to know how and when to use it. Assertiveness is important, but when it approaches aggressiveness or confrontative-ness, it is no longer effective and does not benefit the client .
The big thing for me here is credibility. If you make a stink over everything, people will just start pinching their noses and moving away. To be taken seriously as an advocate, decisions makers need to believe in you so that if you fight for something, they respect you enough to step back and examine the situation. Don't fight because you don't like the system or the rules, fight to give your client the best healthcare at a fair cost.
It also moves beyond the individual client. It only takes a couple of experiences of advocate/navigators as "complainers" to turn a provider off to the idea of working with them. If instead they are seen as people who improve outcomes and patient satisfaction, then they will be granted greater access and respect.
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text:
Silence is Death: The Life and Work of Tahar Djaout
"Šukys … has written a beautiful but unsentimental book about her search for Algerian poet Tahar Djaout, who was killed in 1993 at the height of Islamic terrorism against intellectuals. No work of dreary lit crit, this is creative nonfiction at its best. Beginning and ending in the Iowa city of Elkader (which was named after a young Algerian hero), the book reveals Djaout's extraordinary eloquence on every page. … Highly recommended." — miriam cooke, Choice Reviews Online
"One gets a sense of the man, the artist, and Ms. Šukys herself. With a lyrical nomadism, she combines her study with detours into memoir and even fiction on themes of bones and memory." — Nina C. Ayoub, The Chronicle of Higher Education
"The sheer range is ambitious." — Lydia Wilson, Times Literary Supplement
"Šukys' text isn't straightforward literary biography, and much of the book concerns itself with the question of how to present the man, memory, and writings. … The book works best as an example of how we create our own stories in any book we read, how facts don't necessarily provide more than the fictions we spin around them. … Šukys has created something that's of greater interest and appeal than the usual author-monograph." — The Complete Review
About Silence is Death
On May 26, 1993, the Algerian novelist and poet Tahar Djaout was gunned down in an attack attributed to Islamist extremists. An outspoken critic of the extremism roiling his nation, Djaout, in his death, became a powerful symbol for the "murder of Algerian culture," as scores of journalists, writers, and scholars were targeted in a swelling wave of violence.
The author of twelve books of fiction and poetry, Djaout was murdered at a critical point in his career, just as his literary voice was maturing. His death was a great loss not only for Algeria and for Francophone literature but also for world literature. Rage at the news of his slaying was explosive but did nothing to quell the increasing bloodshed.
Silence Is Death considers the life and work of Djaout in light of his murder and his role in the conflict that raged between Islamist terrorist cells and Algeria's military regime in the 1990s. The result is an innovative meditation on death, authorship, and the political role of intellectuals. By collapsing the genres of history, biography, personal memoir, fiction, and cultural analysis, Julija Šukys investigates notions of authorial neutrality as well as the relationship between reader and writer in life and in death. Her work offers a view of reading as an encounter across time and place and opens the possibility of a relationship between different cultures under peaceful terms.
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Chapters Indigo


text:
Gobiodon ist eine Fischgattung aus der Familie der Grundeln. Die Arten dieser Gattung leben in den Korallenriffen des Indopazifik versteckt in ästigen Steinkorallen (Scleractinia) der Gattung Acropora und der Familie Pocilloporidae, oft vergesellschaftet mit Pelzgroppen und Porzellankrebsen. Die schlechten Schwimmer leben ständig zwischen den Ästen und verteidigen die Korallen gegen Fressfeinde wie Falterfische und Feilenfische. Zum Laichen nagen sie ein kleines Stück Korallengewebe ab und kleben ihre Eier an die kahle Stelle. Die Haut der Korallengrundeln ist schuppenlos und mit einem giftigen Schleim bedeckt, der sie vor Fressfeinden schützt. Sie werden 2 bis 6,5 Zentimeter groß. Korallengrundeln leben von Zooplankton, auch von solchem das die Korallenpolypen schon zum eigenen Verzehr eingefangen haben.

Arten
 Gobiodon acicularis
 Schwarzweiße Korallengrundel (Gobiodon albofasciatus)
 Gobiodon albolineatus
 Gobiodon aoyagii Shibukawa et al., 2013
 Gobiodon ater
 Gobiodon atrangulatus
 Gobiodon axillaris
 Gobiodon bilineatus
 Gobiodon brochus
 Gobiodon ceramensis
 Zitronengrundel (Gobiodon citrinus)
 Gobiodon erythrospilus
 Gobiodon fulvus
 Gobiodon fuscoruber
 Gobiodon heterospilos
 Blaupunkt-Korallengrundel (Gobiodon histrio)
 Gobiodon irregularis
 Gobiodon micropus
 Gobiodon multilineatus
 Gobiodon oculolineatus
 Okinawagrundel, Gelbe Korallengrundel (Gobiodon okinawae)
 Fünfstreifen-Grundel, Braune Korallengrundel (Gobiodon quinquestrigatus)
 Gobiodon reticulatus
 Rotstreifen-Grundel (Gobiodon rivulatus)
 Gobiodon spilophthalmus
 Schwarzstreifen Korallengrundel (Gobiodon unicolor)

Nahe verwandt ist die Gattung Paragobiodon, deren Kopf mit kleinen Stacheln bedeckt ist und die auch die gleiche Lebensweise hat.

Literatur
 Atakan Sver: Korallengrundeln im Riffaquarium, in KORALLE, Meerwasseraquaristik-Fachmagazin, Nr. 6 Dezember/Januar 2000/2001, Natur und Tier Verlag Münster,

Weblinks

 Bericht über die Fortpflanzung von Grundeln bei Spiegel online

Grundeln
Gobiidae

redpajama_set_name: RedPajamaWikipedia


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Executors' obligations after obtaining a grant of probate in the BVI
The powers and obligations of an executor change after a grant has been obtained and also depend on whether the "administration period" is ongoing or whether it has ended.
The administration period starts from the moment of death of the deceased and finishes once the executor has both obtained a grant and paid any debts or other liabilities payable from the deceased's BVI assets (if any). Since BVI estates of persons domiciled outside the BVI are often very simple, with few debts or liabilities other than legal fees, the administration period will essentially end soon after a grant has been obtained.
Prior to obtaining a grant, the executor's powers are quite limited. For example, the executor is unable to transfer any shares held by the deceased to the beneficiaries of the estate until a grant has been obtained.
We explain below the powers and corresponding obligations of executors after a grant has been obtained.
Powers of an executor
Once an executor has obtained a grant, the executor has a variety of powers. The executor can essentially act as if they are the deceased ie as the absolute owner of the deceased's property. By way of illustration the Executor can:
collect assets;
pay debts;
run companies; and
instruct nominees.
Powers in relation to companies
Where the deceased owned shares in a company, an executor is able to exercise the deceased's rights as shareholder. These rights may allow the executor to replace the director(s) of the company, including by appointing the executor as a director. If appointed as director, the executor can exercise all the usual powers of a director according to the company's memorandum and articles of association. Such powers may, depending on the provisions of the company's memorandum and articles of association, include commencing Court proceedings on behalf of the company, representing the company in conversations with banks, or changing the registered agent of the company, amongst other things.
However, BVI law also allows an executor to transfer shares in a company directly to the beneficiaries, without an intermediate transfer to the executor and in most estates this is what happens.
Powers where assets were held under a nomineeship
Depending on the terms of any nominee agreement, an executor may instruct the nominee to substitute the executor as the beneficial owner of the nominee assets and thereafter exercise the rights of the beneficial owner, such as to replace the nominee. However, the executor may also instruct the nominee to substitute the beneficiaries of the estate as the beneficial owners and, in circumstances where the estate is a simple one (ie without debts and liabilities), the executor should do so.
Communications with beneficiaries
There is no strict duty to inform beneficiaries of their entitlement during the administration period, and many executors do not consider it is necessary to inform those beneficiaries who are only entitled to a small sum of their entitlement.
Likewise, executors are not strictly required to consult with beneficiaries regarding any act carried out during the administration period.
If there is a risk of litigation by beneficiaries in relation to the estate, executors will often consult with major beneficiaries before taking substantial steps so as to minimise the risk of a claim by a beneficiary later on.
Accounts of executorship
In making an application for a grant of probate in the BVI, an executor must undertake to "render a just and true account of my executorship whenever required by law to do so". In practice, such accounts are rarely required. Nonetheless, the executor must be in a position to account accurately for their administration of the assets if called upon to do so.
Accounting for an executorship would require a valuation of the BVI assets as at the date of the deceased's death and a valuation of the assets as at the end of the administration period.
An executor is not permitted to charge for their personal time unless authorised by the deceased's will or the Court. However, an executor may seek payment from the estate for costs incurred by them in relation to their duties, including:
taking the opinion of legal counsel concerning any matter related to the duties of the executor; and
employing agents in any part of the world including lawyers or accountants to transact any business in connection with the executor's duties.
The executor's costs are taken from the estate before the residue is divided among the beneficiaries. It is not necessary to confirm the quantum of the costs with beneficiaries before paying those costs out of the estate.
If there is a risk that the beneficiaries may later challenge the costs incurred (eg as unreasonably incurred or unreasonable in quantum), it may be advisable for an executor to seek the beneficiaries' agreement to costs to avoid a dispute later on.
Executor liability
An executor may become personally liable or accountable from their own assets if they violate or neglect their duties in respect of the estate. They owe duties to preserve the assets, to deal with them properly, and to apply them in the due course of administration for the benefit of those interested.
If they breach any of those duties and cause loss to the estate, they are guilty of a "devastavit" and may be personally liable to make good the loss caused to the estate. An executor can be guilty of a devastivit through:
deliberate abuse (eg using estate assets to pay their personal debts);
negligence (eg needlessly delaying the payment of a debt and incurring interest, or paying a claim they were not legally required to satisfy).
An executor will not be liable for the actions of another person that causes loss to the estate if the executor was unable to prevent those actions.
An executor has an "executor's year" from the date of death to administer the estate. After that year, they must be able to show a valid reason for the delay in realising the estate.
Of course in practice most international estates take longer than a year to administer because the executor may need to obtain grants in multiple jurisdictions, and so Courts tend to be lenient towards executors who can show they have acted reasonably, particularly in complex estates. Further, if a beneficiary later sought compensation on the basis of any delay, that beneficiary would need to demonstrate that the delay in fact caused the loss before the executor will be personally liable.
Rights and obligations after the administration period
Once the Executor has obtained the grant and has paid any debts or other liabilities payable from the BVI assets (if any), they are ready to transfer the estate to the beneficiaries (or in the case of a nominee agreement, instruct the nominee to transfer the beneficial ownership of the assets). At this point the administration period ends.
The powers and obligations of the executor after the administration period are governed by the laws of the place of the deceased's domicile. For example, if the deceased died domiciled in Hong Kong, Hong Kong law would govern the executors' powers and obligations. Therefore, matters such as to whom the executor must distribute the BVI assets, and in what proportions, would also be governed by Hong Kong law.
Henry Mander
Nicola Roberts
Lucy Hickmet
Matthew Howson
Guide to the BVI Probate Rules for BVI Company Owners


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Columbus Crew stunned by rookie Kirk Urso's death
By Steve DavisAug 5, 2012, 12:47 PM EDT
What terrible news much of Major League Soccer woke up to this morning.
Columbus Crew rookie Kirk Urso has died. He was 22.
According to the Columbus dispatch: "Uniformed police officers were dispatched to Park Street Patio at 533 Park St. at 12:50 a.m. Urso was transported from there to Grant Hospital, where he was pronounced dead at 1:51 a.m. The coroner's office is handling the investigation."
The Crew later issued this brief statement:
The Columbus Crew has learned that midfielder Kirk Urso has passed away on Saturday night.
While the circumstances of Urso's passing are still being determined, no further comments will be made at this time.
The thoughts and prayers of the entire Columbus Crew and Hunt Sports Group are with the Urso family in this time of need.
Urso was a promising talent who started the club's first five games this year.
He had previsouly been a member of the United States Under 17 program and played with the U.S. side during the 2007 FIFA U-17 World Cup in South Korea. His game-winning goal propelled the United States into the second round.
Later, Urso was a captain for North Carolina's NCAA championship team.
Here's what Columbus Crew coach Robert Warzycha said about Urso after pre-season training, when the 10th selection in January's MLS supplemental draft had so impressed coaches and supporters: "He's just works. He works very hard and he has a good shot from distance. He just settled down in the midfield with Mirosevic and Tchani. He's one of those players that can change the game for you."


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Mike Watson - The Fundamentally Sound Super Star
He's ranked number four on the Canadian all-time money list, has almost $6.4 million in live tournament earnings and a WPT, two Aussie Millions and a €50,000 High Roller title. These victories are just the tip of the iceberg, as the man known as SirWatts has 11 six-figure cashes and two of which are more than $1.3 million. Online SirWatts has been a High Stakes MTT regular since 2006 and he granted iGaming.org time to dig into who he is, what drives him and how he became so successful on the highest level. Watson is currently ranked fifth in the GPI Player of the Year race and has not been outside their overall Top 10 since October of last year. Watson seems focused more than ever, but it started all for him back in 2008 when he had a breakout win in the WPT Bellagio Cup for almost $1.7 million.
Front page photo credit: Neil Stoddart for PokerNews.com during the 2013 PCA High Roller final table.
Does win tilt exist, that when you win a lot you can become complacent about your game?
Yes, absolutely. I definitely think that it exists. Back in 2008 I had an amazing year and I was winning both live and online. I was crushing. In 2009 and 2010 I didn't really do much at all, and I definitely got a bit complacent. I was running bad but I also think I got a bit unlucky as well. A couple of bad years in a row I was just not putting in the work. That is a big part of it I think, of having a couple of bad years. Recently I've been getting back into studying more, and trying to really work on my game. I think that it's a combination of, both me starting to run better again, and playing better. I've been winning some crucial coin flips deep in high rollers over the last year or so. It's also just about playing a lot better and patching up a lot of leaks, I think that's what happened. The last year and a half I played a lot better and ran a lot better at the same time, and all of a sudden I'm on a pretty big run.
What is the point where you start realizing that you're getting complacent, and how did you approach the game from that point on?
It's hard, and I think that at some point I realized that I wasn't putting in as much effort to work on the game as I used to. I also thought that it didn't matter that much because I thought I was good enough to win, playing the way I did. I'd had a lot of success in poker and at that point I was trying to balance other aspects of my life a bit more, and poker had become a lesser priority. It was still pretty important to me, but not the all-consuming game that it was for the first few years when I started grinding and playing for a living. It can just take over and become your whole life. I felt like in terms of me being happy I was a lot better off that way, but definitely my poker slipped a bit. It's hard to be really honest with yourself about something like that. Even for poker players, who are supposed to be the most open and honest with themselves, who require being almost ruthlessly objective, struggle with that sometimes. I don't know if there's a secret to make it easy, but I think you ideally need to have some friends who are looking out for you; who are always keeping you on your toes and bouncing new ideas off on you. You need to trust to talk to them and be objective with them.
Has someone ever told you, "Mike, you're playing really bad. How about you step it up?"
Definitely. I credit a lot of my success recently to talking a lot more about the game, mostly with Dan Smith. He's probably the guy I'm closest to and I'm talking the most poker to. There are also other people in that group that I've lived with in Vegas and talked a lot of poker with like LuckyChewy (Andrew Lichtenberger), aejones (Aaron Jones) and all kinds of bosses. I started talking poker a lot more with them and they helped me to get back up to speed in a pretty short amount of time. I think I always had the fundamentals there, but there was some stuff I was missing out on. It helps a lot when people kick you in the ass and say, "You need to step your game up. The way you were playing in 2008 is not going to get it done anymore."
Do you think fundamentals are less important nowadays because there are so many players now with very sound fundamentals, especially in the High Roller events, and it's more about adjusting?
I think fundamentals are still very important, but it's more about to which degree. Obviously in a lot of common spots in terms of open-raising or re-raising versus calling, I still feel like some people are not perfect. Especially in terms of which hands they choose to re-raise in certain situations. A lot of it is adjustment based, depending on type of player you're up against. You do want to have a little bit of a different strategy and I don't think people necessary fully understand how to adjust. Then also, every tournament you play you're going to see someone open a hand in a position where it's just unacceptable to open that hand. I see it all the time, and people try to use the excuse, "It's a soft table." While there's obviously some validity to that, I've seen them raise hands 20% away from being an open. When it's a spot where they should be able to get away with it by opening 35% of the hands, they are opening something like a 60th percentile hand or something insane. There are always people who are making very basic fundamental mistakes in that regard. Blind versus blind I feel like has become a thing that you need to play super accurately. Every little blind battle matters, and I think that there are people whose strategies in that situation are not the best.
You're known as a tight player, does that mean you rely more heavily on your fundamentals or on other player's mistakes?
That's not a one-or-the-other situation. I think my fundamentals are one of the strengths of my game, and staying close to a more theoretically correct strategy is important. Especially since tournaments are getting tougher. At the same time I'm aware of what's going on around me. I'm definitely going to open my game up if I feel like it's the right situation to do that. I'm also going to tighten up when I'm at a particularly tight table, or when there's someone extremely crazy on my left. I'm going to adjust either way, and of course everything you're doing is based on taking advantage of your opponent's mistakes. Even when you're playing more theoretically sound.
Bad player's fundamentals are worse. They are going to make mistakes and that's where you're going to profit. Specifically when I see somebody make a certain type of mistake a lot, I'm going to adjust in a way to try to exploit that. I'm not always just going to stick to a game theory-ish type approach. I realize that it's obviously a lot more profitable to exploit something that they're doing wrong.
Remko Rinkema - Igaming.org Interviewer
Remko Rinkema has covered the biggest poker tournaments in the world since 2008, including many WSOP, EPT, Aussie Millions, APPT, MCOP and Unibet Open events. As an in-depth interview and story enthusiast he tries to do things a little differently. Besides the usual writings Rinkema grabs every chance to appear on podcasts, live streams and in the occasional video.
Connect with Remko Rinkema via: Google+ | Twitter
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Winning Poker Network (WPN)
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Equity Poker Network
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World Series of Poker Europe (WSOPE)
World Series of Poker Circuit
Latin America Poker Tour (LAPT)
Aussie Millions
World Series of Poker Asia Pacific
UK & Ireland Poker Tour (UKIPT)
Grosvenor UK Poker Tour (GUKPT)
Australia New Zealand Poker Tour (ANZPT)
Asian Poker Tour (APT)
Asia Pacific Poker Tour (APPT)
The Grand Live
Hold'em Manager (HEM)
Easy Seat Poker
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PokerOffice
PokerTableStats
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Tracking Sites
Playerscope
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OfficialPokerRankings (OPR)
TheHendonMob
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