text:
\section*{Introduction}\label{intro}
Spaces of analytic functions form an essential class of nuclear Fr\'{e}chet spaces. In recent years there have been significant advances in the structure theory of nuclear Fr\'{e}chet spaces  (see \cite{MV}). In attempts to utilise this theory in complex analysis,
one is led to analyse the complex analytic properties shared by Stein manifolds whose
analytic function spaces possess a common linear topological invariant.

In this note, we look into the connection between the existence of
pluricomplex Green functions and certain linear topological properties of
the Fr\'{e}chet space of analytic functions on a Stein manifold; a topic
that was touched upon in \cite{Ay}.

Throughout the note we will denote the space of analytic functions on a
Stein manifold $M$ with compact open topology by $\mathcal{O}(M)$.

After compiling some background material, in section \ref{section1} we
recall a result of \cite{Ay} that relates the existence of pluricomplex Green functions on $M$ to the diametral dimension of $\mathcal{O}(M)$ and look at an example.

In section \ref{section2}, we examine semi-proper pluricomplex Green
functions (see definiton below) and give a characterization of their
existence in terms of a controlled approximation property for analytic
functions on the complex manifold.

We then relate the existence of semi-proper pluricomplex Green functions to
the linear topological properties $\widetilde{\Omega}$ of Vogt \cite{VB}, and the
diametral dimension for $\mathcal{O}(M)$.

In section \ref{section3} we look into pluri-Greenian and locally uniformly pluri-Greenian
complex manifolds introduced by E. Poletsky \cite{P paper}, \cite{P yeni}. We show that a complex manifold is locally uniformly
pluri-Greenian if and only if it is pluri-Greenian and give a
characterization of locally uniformly pluri-Greenian Stein manifolds in
terms of the notions introduced in section 2.

In the last section, we recall some research done for the existence of
pluricomplex Green functions, with the intention of
putting our results in perspective.

The manifolds considered in this note are always assumed to be connected. We
will use the standard terminology and results from functional analysis and
complex potential theory as presented in \cite{MV,K}, respectively.

\section{Pluricomplex Green functions on a Stein manifold $M$ and the diametral dimension of $\mathcal{O}(M)$}\label{section1}

We start by defining the terms appearing in the heading.

Let $\mathcal{X}$ be a Fr\'echet space and $\lbrace \parallel\cdot\parallel_k\rbrace^\infty_k$ an increasing system of semi-norms generating the topology of $\mathcal{X}$. Let $\mathcal{U}_k:=\lbrace x\in \mathcal{X}:\parallel x\parallel_k\leq 1\rbrace$ denote the unit ball of the corresponding local Banach spaces $\mathcal{X}_k$, $k=1,2,\cdots$. For a subspace $L$ of $\mathcal{X}$ and for $k^+>k$, let;
$$\delta(\mathcal{U}_{k^+};\mathcal{U}_k,L):= \inf\lbrace t>0: \mathcal{U}_{k^+}\subseteq t\mathcal{U}_k+L\rbrace.$$
The \emph{$n$-th Kolmogorov diameter} of $\mathcal{U}_{k^+}$ with respect to $\mathcal{U}_k$ is defined as:
$$d_n(\mathcal{U}_{k^+},\mathcal{U}_k):= \inf\lbrace \delta(\mathcal{U}_{k^+};\mathcal{U}_k,L): L \textrm{ subspace of $\mathcal{X}$ of dimension at most } n, n\in \mathbb{Z}^+\rbrace.$$
There is a huge literature on techniques of computing Kolmogorov diameters for specific Banach spaces (see \cite{Pinkus}).

The \emph{diametral dimension} of $\mathcal{X}$ is defined as:
$$\Delta (\mathcal{X}):= \lbrace (t_n)_n\in \mathbb{C}^{\mathbb{N}};\, \forall\, k, \, \exists\, k^+>k; \, t_nd_n(\mathcal{U}_{k^+},\mathcal{U}_k)\to 0\rbrace.$$
The diametral dimension of $\mathcal{X}$ does not depend upon the generating semi-norm system and is a linear topological invariant of the Fr\'echet space $\mathcal{X}$ in the sense that if $\mathcal{X}$ is linear topologically isomorphic to a Fr\'echet space $\mathcal{Y}$, then $\Delta(\mathcal{X})=\Delta(\mathcal{Y})$.

In the case of nuclear Fr\'echet spaces, one can choose a generating system of Hilbertian semi-norms $\lbrace \parallel\cdot\parallel_k\rbrace_k$ for $\mathcal{X}$ and represent the compact linking maps ${i^k_{k^+}:\mathcal{X}_{k^+}\to \mathcal{X}_k}$ as:
$$i_{k^+}^k:= \sum_n\lambda_n \langle \cdot,f_n\rangle_{k^+} e_n,$$
where $\lbrace f_n\rbrace$ and $\lbrace e_n\rbrace$ are orthonormal sequences in $\mathcal{X}_{k^+}$ and $\mathcal{X}_k$, respectively and $(\lambda_n)$ is a sequence of nonnegative numbers converging to $0$.

In this framework, the sequence of Kolmogorov diameters $\lbrace d_n(\mathcal{U}_{k^+},\mathcal{U}_k)\rbrace$ coincides with the non-increasing rearrangement of the sequence $\lbrace \lambda_n\rbrace_n$ of the eigenvalues.

This information is quite useful in computing diametral dimensions. For example, in view of this observation, one can readily verify that
\begin{align*}
	\Delta(\mathcal{O}(\Delta^d))&= \lbrace (\xi_n): \sum_n|\xi_n|^2e^{-\frac{2}{k}n^{1/d}}<\infty, \,\forall\, k=1,2,\cdots \rbrace\\
	\Delta (\mathcal{O}(\mathbb{C}^d))&= \lbrace (\xi_n): \,\exists\, R>1; \, |\xi_n|\leq CR^{n^{1/d}}\rbrace,
\end{align*}
where $\Delta^d$ is the unit polydisc in the $d$ dimensional complex plane $\mathbb{C}^d$, $d=1,2,\cdots$.

For more information about diametral dimension, we refer the reader to \cite{Pietsch,Rolewicz}.

We now recall some background material from Complex Potential Theory.

Let $M$ be a connected complex manifold and $z_0\in M$. Let us denote plurisubharmonic functions on an open $\Omega\subseteq M$ by the symbol $PSH(\Omega)$. The \emph{pluricomplex Green function} $g_M(\cdot;z_0)$ with a pole at $w_0$ is defined as:
$$g_M(z;z_0):=\sup\lbrace u\in PSH(M):u<0\,\textrm{on}\,M, \exists\, c:\, u(z)\leq\ln|z-w_0|+c, z\in U_{z_0}\rbrace,$$
where $U_{z_0}$ is some local neighborhood near $z_0$ (see \cite{K}).

It turns out that either $g_M$ is a plurisubharmonic function that is maximal on $M$ or is $-\infty$. We will say that pluricomplex Green function with a pole at $z_0$ \emph{exists} in case $g_M(\cdot, z_0)\not\equiv-\infty$.

Pluricomplex Green functions are natural analogues in several variables, of the classical Green functions in the theory of open Riemann surfaces and were studied in diverse settings (see \cite{K,Demailly,P paper}).

Our first result, connecting pluricomplex Green functions and diametral dimension comes basically from \cite{Ay}. Indeed Corollary 4.3, Theorem 2.9 and Proposition 4.1 of \cite{Ay} combined yields:

\begin{theorem}\label{thm1}
	Let $M$ be a Stein manifold of dimension $d$, and suppose that $\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\Delta^d))$. Then pluricomplex Green function exists for each point of $M$.
\end{theorem}

However the statement of the above theorem is not an if and only if statement as the example below reveals.

\begin{example}
	Let $M=\Delta\times \mathbb{C}\subseteq \mathbb{C}^2$. Clearly, pluricomplex Green function exists for each point $\zeta_0=(z_0,w_0)$ of $M$, indeed the function:
	$$g((z,w);\zeta_0)=\ln\big|\frac{z-z_0}{1-z\overline{z_0}}\big|$$
	is the pluricomplex Green function for $M$ with a pole at $\zeta_0$.

	We now examine the diametral dimension of $\mathcal{O}(M)$.

	The nuclear Fr\'echet space $\mathcal{O}(M)$ admits a Schauder basis $\lbrace \boldsymbol{z^n}\boldsymbol{w^m}\rbrace$, where $${\boldsymbol{z^n}(z,w)=z^n}, \quad {\boldsymbol{w^m}(z,w)=w^m}, \, n,m=0,1,\cdots.$$

	We choose the exhaustion $D_k=\lbrace (z,w)\in M: |z|\leq e^{-1/k},\, |w|\leq e^k\rbrace$ of $M$, $k=1,2\cdots$, and set $h_k(\theta):= (1-\theta)k -\frac{\theta}{k}$, for $0\leq \theta\leq 1$, $k=1,2,\cdots$.

	Using the $H^2(D_k)$ norms, $k=1,2,\cdots$, to generate the topology of $\mathcal{O}(M)$, we obtain a sequence space representation of $\mathcal{O}(M)$ of the form:
	\begin{eqnarray*}
		\mathcal{O}(M)&\cong &\lbrace (\xi_{n,m})_{n,m=0}^\infty: |(\xi_{n,m})|_k\\
		& &\quad =\big(\sum_{\vec{\eta} =(n,m)} |\xi_{n,m}|^2e^{2|\vec{\eta}|h_k(\frac{n}{n+m})}\big)^{1/2}<\infty,\,\forall\, k=1,2,\cdots \rbrace,
	\end{eqnarray*}
	where, as usual, we used the notation $|\vec{\eta}|:=n+m$ for $\vec{\eta}=(n,m)\in \mathbb{N}\times \mathbb{N}$.

	The corresponding fundamental system of neighborhoods are:
	$$\mathcal{U}_k=\lbrace f=\sum_{n,m}\xi_{n,m}\boldsymbol{z^nw^m}: |(\xi_{n,m})|_k\leq 1,\, k=1,2,\cdots \rbrace.$$
	Fix a $k\in \mathbb{N}$, $k^+ \gg 2k$ and $\varepsilon$, $0<\varepsilon <1$, whose values we will specify later. We will use the usual lexicographical order for $\mathbb{N}\times\mathbb{N}$, and set $\theta(\vec{\eta}):=\frac{n}{n+m}$ for $\overline{\eta}=(n,m)$, $n,m=0,1,\cdots$.

	As was mentioned in the introduction, the sequence of Kolmogorov diameters $\lbrace d_0(\mathcal{U}_{k^+},\mathcal{U}_k),\cdots,d_n(\mathcal{U}_{k^+},\mathcal{U}_k),\cdots\rbrace$ coincides with the non-increasing rearrangement of $$\lbrace e^{2|\vec{\eta}|(h_k(\theta(\vec{\eta})-h_{k^+}(\theta(\vec{\eta}))}\rbrace_{\vec{\eta}\in\mathbb{N}\times\mathbb{N}}.$$

	Evidently, for a fixed $j$;
	\begin{equation}\label{star1}
	j<j+1\leq \boldsymbol{\#}\lbrace \vec{\eta}:|\vec{\eta}|(h_{k^+}(\theta(\vec{\eta}))-h_k(\theta(\vec{\eta})))\leq \varepsilon_j(k^+;k)\rbrace,
	\end{equation}
	where $\#(A)$ denotes the number of elements of the set $A$, and $\varepsilon_j(k^+;k)=-\ln d_j(\mathcal{U}_{k^+}; \mathcal{U}_k)$, $j=0,1,\cdots$.

	On the lattice points $\vec{\eta}$; $\theta(\vec{\eta})\leq 1-\varepsilon$, since $h_{k^+}(\theta(\vec{\eta}))-h_k(\theta(\vec{\eta}))\geq \varepsilon (k^+-k)$, we have:
	\begin{eqnarray*}
	\lefteqn{\#\{\vec{\eta}: \theta(\vec{\eta})\leq 1-\varepsilon,\, |\vec{\eta}|(h_{k^+}(\theta(\vec{\eta}))-h_k(\theta(\vec{\eta})))\leq \varepsilon_j(k^+;k)\}}\\
		&\leq & \# \{\vec{\eta}:|\vec{\eta}|\leq \frac{1}{\varepsilon (k^+-k)}\varepsilon_j(k^+;k)\}\\
		&\leq & \frac{1}{2} \{\big(\frac{1}{\varepsilon(k^+-k)}\varepsilon_j(k^+;k)\big)^2+\frac{1}{\varepsilon(k^+-k)}\varepsilon_j(k^+;k)\}\\
		&\leq &K\varepsilon_j(k^+;k)^2,
	\end{eqnarray*}

	where $K=\frac{1}{2}\big(\frac{1}{\varepsilon^2(k^+-k)}+\frac{1}{\varepsilon(k^+-k)}\big)$.

	On the remaining lattice points $\vec{\eta}$; $\theta(\vec{\eta})>1-\varepsilon$, since $k^+\gg 2k$, we estimate:
	\begin{eqnarray*}
	\lefteqn{\#\{\vec{\eta}:\theta(\vec{\eta})>1-\varepsilon, |\vec{\eta}|(h_{k^+}(\theta(\vec{\eta}))-h_k(\theta(\vec{\eta})))\leq \varepsilon_j(k^+;k)\}}\\
		&\leq& \#\{\vec{\eta}:\theta(\vec{\eta})>1-\varepsilon ; |\vec{\eta}|\leq 2k\varepsilon_j(k^+;k)\}.
	\end{eqnarray*}
	A crude estimate on the number of lattice points in the wedge with base an interval in the unit simplex starting from $(0,1)$ with length $\sqrt{2}\varepsilon $ and with norm less than or equal to $2k\varepsilon_j(k^+;k)$ yields:
	$$\#\{\vec{\eta}: \theta(\vec{\eta})>1-\varepsilon ;|\vec{\eta}|\leq 2k \varepsilon_j(k^+;k)\}\leq 1+\frac{\varepsilon}{2}(4k^2+2k):=K'.$$
	Now choose $\varepsilon$ so that $K'\leq 2$ and $k^+$ so that $K<1$. It follows from these choices and inequality (\ref{star1}) that
	$$j\leq 3 \varepsilon_j(k^+;k)^2=3\ln \big(\frac{1}{d_j(k^+,k)}\big)^2,\quad \textrm{or}$$
	\begin{equation}\label{star2}
	d_j(\mathcal{U}_{k^+},\mathcal{U}_k)\big(e^{\frac{1}{\sqrt{3}}\sqrt{j}}\big)_j\leq 1.
	\end{equation}
	Since $j$ was arbitrary, the nuclearity of $\mathcal{O}(M)$ implies that $\big(e^{\frac{1}{\sqrt{3}}{\sqrt{j}}}\big)_j\in \Delta(\mathcal{O}(M))$ (see \cite{Frerick et al}). On the other hand, since $\Delta(\mathcal{O}(\Delta^2))=\{(\xi_j):\sum_j|\xi_j|^2e^{-\frac{2\sqrt{j}}{k}}<\infty, \, \forall\, k=1,2,\cdots\}$, $\big(e^{\frac{1}{\sqrt{3}}{\sqrt{j}}}\big)_j\notin \Delta(\mathcal{O}(\Delta^2))$.
\end{example}
For more information about the diametral dimension of spaces of analytic functions on complete Reinhardt domains in $\mathbb{C}^n$, we refer the reader to \cite{AKT manuscripta,AT Decomposition}.
\section{Semi-proper pluricomplex Green functions}\label{section2}

The sublevel sets of the pluricomplex Green functions of the above example are far from being relatively compact. Moreover the poles of these
	pluricomplex Green functions are not strict in the sense of Poletsky. Recall
	that a function $g$ on a complex manifold is said to have a \textit{%
		strict} \textit{logarithmic} \textit{pole} at a point $w$, in
case there are a coordinate neighborhood $U$ and constants
$c_{1}$ and $c_{2}$ such that
\begin{equation*}
\ln \left\Vert z-w\right\Vert +c_{1}\leq g\left( w\right) \leq \ln
\left\Vert z-w\right\Vert +c_{2}
\end{equation*}%
on $U$ (see \cite{P paper}). One can ask
as to whether these features account for the difference; $%
\Delta (\mathcal{O}(M))\neq \Delta (\mathcal{O}(\Delta ^{2}))$ in the
example above.

In what follows, we will call an extended real valued function $f$ on a topological space $X$ \emph{semi-proper} in case there exists a $c\in \mathbb{R}$ such that the sublevel set $\emptyset\neq \{x\in X:f(x)<c\}$ is relatively compact in $X$.

To answer the question posed above we will enquire into the existence of semi-proper pluricomplex Green functions.
\begin{definition}\label{defn1}
	Let $M$ be a Stein manifold and $\{\parallel\cdot\parallel_k\}_k$ be a generating norm system for $\mathcal{O}(M)$. For $z\in M$ we will say that $M$ has the property $\widetilde{\Omega}_z$ in case there exists a ball with radius $\varepsilon>0$ in some coordinate system around $z$, $B(z;\varepsilon)$, $B(z;\varepsilon)\subset\subset M$ such that
	$$\exists\, k_1,\lambda>0;\,\forall\, k>k_1, \exists\, C>0; \quad \mathcal{U}_{k_1}\subseteq \frac{C}{r}\mathcal{U}_0+r^\lambda \mathcal{U}_k,\, \forall\, r>0,$$
	where $\mathcal{U}_{0}:=\{f\in \mathcal{O}(M):\sup_{w\in B(z;\varepsilon)}|f(w)|\leq 1\}$.
\end{definition}

Clearly, this condition does not depend on the generating norm system of $\mathcal{O}(M)$.

Now we can state the main theorem of this section.

\begin{theorem}\label{thm2}
	Let $M$ be a Stein manifold and $\zeta_0\in M$. The following statements are equivalent:
	\begin{enumerate}
		\item There exists a negative plurisubharmonic function on $M$ that has a relatively compact sublevel set containing $\zeta_0$.
		\item $M$ possesses a semi-proper pluricomplex Green function with a strict logarithmic pole at $\zeta_0$.
		\item $M$ has the property $\widetilde{\Omega}_{\zeta_0}$.
	\end{enumerate}
\end{theorem}

\begin{proof}
\implication{(1)}{(2)}	The proof of this implication is a gluing argument (see e.g. \cite{A D-issue}). We give it in detail since we will utilize the explicit form of some of the functions appearing in the proof in a later discussion.

Let $\sigma_1$ be a negative plurisubharmonic function given in the assumption and assume that $\{z:\sigma_1(z)<-c\}$ is relatively compact for some $c>0$, and contains $\zeta_0$. Choose a ball $B(\zeta_0,\varepsilon)$ in a coordinate neighborhood of $\zeta_0$ such that $$\overline{B(\zeta_0,\varepsilon)}\subset \{z:\sigma_1(z)<-c\}$$ and set $\sigma_2(z):=\sigma_1+\overline{c}$, where $-\overline{c} :=\max_{\xi\in \overline{B}(\zeta_0,\varepsilon)}\sigma_1(\xi)$.

Fix a relatively compact, smooth strictly pseudoconvex domain $D\subset\subset M$ containing $\overline{\{z:\sigma_2(z)<\beta\}}$, $\beta:=\overline{c}-c$, and let $g_D$ denote the pluricomplex Green function on $D$ with a pole at $\zeta_0$. The function $g_D$ is continuous on $D$ and is a proper exhaustion on $D$ (see Theorem 4.3 of \cite{Demailly}, and Theorem 4.3 of \cite{P paper}).

Choose $c_1>0$, $c_2>0$ so that;
\begin{eqnarray*}
\{z:g_D(z)<-c_1\}&\subset& B(\zeta_0,\varepsilon)\subset\subset \{z:\sigma_2(z)<\beta\}\\
&\subset\subset &\{z\in D:g_D(z)<-c_2\}\subset\subset D\subset\subset M.
\end{eqnarray*}

Set $\psi(z):= \big(\frac{c_1-c_2}{\beta}\big)\sigma_2-c_1$, and let
$$U:=\{z\in D:g_D(z)<-c_2\},\quad V:=\overline{\{z\in D:g_D<-c_1\}}^c\cap \{z\in D: g_D<-c_2\}.$$
For any $\xi\in \partial V\cap U$, $\overline{\lim}_{z\to \xi}\psi(z)\leq g_D(\xi)$ by construction. Hence the function $u$ defined by
\begin{equation*}
	u := \left\{
	\begin{array}{cl}
		\max(g_D,\psi) & \text{on } V,\\
		g_D & \text{on } U-V
	\end{array} \right.
\end{equation*}
is a plurisubharmonic function on $U$.

Since on $\overline{\{z\in D:\sigma_2(z)<\beta\}}^c\cap\{z\in D:g_D<-c_2\}$, $\max\{g_D,\psi\}=\psi$, we can extend $u$ to be a bounded plurisubharmonic function on $M$ by setting it equal to $\psi$ off $\{z\in D:g_D(z)<-c_2\}$.

This modification makes $u-\sup_{z\in M}u(z)$ a semi-proper, negative plurisubharmonic function on $M$. Note that $u$
	is equal to $g_{D}$ in a neighborhood of $\zeta _{0}$. The
domain $D$ is hyperconvex in the sense of Poletsky and so $g_{D}$,%
and hence $u$ has a strict logarithmic pole at $%
\zeta _{0}$ according to Theorem 4.3 of \cite{P paper}. Since $g_{M}(\cdot ,\zeta _{0})$ dominates  $%
u-\sup_{z\in M}u(z)$ on $M$, $g_{M}(\cdot ,\zeta _{0})$ also has a
strict logarithmic pole at $\zeta _{0}$.

\implication{(2)}{(3)} To simplify the notation, we will
	denote the pluricomplex Green function with a strict logarithmic pole at $%
\zeta _{0}$ by $p_{M}$. Pick a coordinate chart $U$%
 around $\zeta _{0}$, such that $g_{M}(\cdot ,\zeta
_{0})\geq \ln \parallel \cdot -\zeta _{0}\parallel +c$ for some $c$, on $U$. Choose an $\varepsilon >0$ such that the ball
	around $\zeta _{0}$ with radius $\varepsilon >0$\textbf{, }$%
B(\zeta _{0},\varepsilon )$, in this coordinate system, that
	satisfies $B(\zeta _{0},\varepsilon )\subset \subset U$. Now for $%
\beta \ll \ln \varepsilon -c$ small enough, we can find an $%
0<\varepsilon ^{-}<\varepsilon $ such that $\Omega _{\beta }\cap
B(\zeta _{0},\varepsilon )\subset B(\zeta _{0},\varepsilon ^{-})\subset
\subset B(\zeta _{0},\varepsilon )$, where $\Omega _{\beta
}:=\{z\in M:g_{M}(z,\zeta _{0})<\beta \}$. Since $\Omega _{\beta }$
 is connected, (\cite{PS}), we conclude that $\Omega _{\beta }\subset B(\zeta
_{0},\varepsilon )$ for this choice of $\beta$.  Hence $%
g_{M}(\cdot ,\zeta _{0})$ is a semi-proper negative
	plurisubharmonic function on $M$.

Choose $\varepsilon >0$, $c>0$, $0<c<c^{+}$%
 such that,
  $$\overline{B(\zeta _{0},\varepsilon )}\subset
\{z:p_{M}(z)<-c^{+}\}\subset \{z:p_{M}(z)<-c\}\subset \subset M.$$
	Pick a $C^{\infty }$ strictly plurisubharmonic, proper function $%
\sigma :M\rightarrow \lbrack 0,\infty )$ and set $D_{k}=\{z:\sigma
(z)<k\}$, $k=1,2,\cdots $. For some $k_{1}$ large
	we have:
\begin{equation*}
\overline{B(\zeta _{0},\varepsilon )}\subset \{z:p_{M}(z)<-c^{+}\}\subset
\{z:p_{M}(z)<-c\}\subset \subset D_{k_{1}}.
\end{equation*}

Let $\Omega _{-}:=D_{k_{1}}$, $\Omega _{+}:=\overline{\{z\in M:p_{M}(z)<-c\}}%
^{c}$.

Define a plurisubharmonic function $\rho_t$ on $M$ via $\rho_t(z):=\frac{t}{c^+}p_M(z)+t$, $t>0$.

Clearly;
\begin{enumerate}
	\item $\rho_t(z)\leq 0$ for $z\in B(\zeta_0,\varepsilon)$, $t>0$,
	\item $\rho_t(z)\geq \big(1-\frac{c}{c^+}\big)t$ for $z\in \Omega_+$, $t>0$.
\end{enumerate}
We choose a Hermitian metric on $M$ and denote by $dV$ the volume form coming from this metric.  Using the notation of Lemma 1 of \cite{A manus} we set $d\varepsilon =cd\mu$, where $\mu$ is a measure on $M$ that is equivalent to the volume form, and $c$ is a strictly positive real valued function on $M$.

Fix $f\in \mathcal{O}(\Omega_-)$ with $\big(\int_{D_{k_1}}|f|^2d\varepsilon\big)^{1/2}\leq 1$.

In view of Lemma 1 of \cite{A manus} we can, for each $t>0$, compose $f$ as, $f=f_++f_-$; $f_+\in \mathcal{O}(\Omega_+)$, $f_-\in \mathcal{O}(\Omega_-)$ such that
\begin{eqnarray*}
	\int_{\Omega_+}|f_+|^2e^{-\rho_t}d\varepsilon&\leq &C_1e^{-(1-\frac{c}{c^+})t}\\
	\int_{\Omega_-}|f_-|^2e^{-\rho_t}d\varepsilon&\leq &C_1e^{-(1-\frac{c}{c^+})t}
\end{eqnarray*}
and for some $C_1>0$ which is independent of $f$ and $t$.

Since $\exists\, C_2>0$ such that
$$\int_{\Omega_+\cap\Omega_-}|f|^2e^{-\rho_t}d\varepsilon\leq C_2e^{-(1-\frac{c}{c^+})t},$$
we have;
$$\int_{\Omega_\pm}|f_\pm|^2e^{-\rho_t}d\varepsilon\leq C_1e^{\frac{c}{c^+}t}$$
and
$$\int_{B(\zeta_0,\varepsilon)}|f_-|^2d\varepsilon\leq \int_{B(\zeta_0,\varepsilon)}|f_-|^2e^{-\rho_t}d\varepsilon\leq C_1e^{-(1-\frac{c}{c^+}t)}.$$
Set
\begin{equation*}
	F := \left\{
	\begin{array}{ccl}
		f_+ & \text{on } \Omega_+,\\
		f-f_- & \text{on } \Omega_-.
	\end{array} \right.
\end{equation*}
The analytic function $F\in \mathcal{O}(M)$ satisfies
\begin{equation}\label{2star1}
\exists\, K>0;\quad \int_M|F|^2d\varepsilon \leq Ke^{t(\frac{c}{c^+})},
\end{equation}
$$\int_{B(\zeta_0,\varepsilon)}|F-f|^2d\varepsilon=\int_{B\zeta_0,\varepsilon)}|f_-|^2\leq c_1e^{-(1-\frac{c}{c^+})t}.$$
We set for $f\in \mathcal{O}(M)$;
$$\parallel f\parallel_k:=\big(\int_{D_k}|f|^2d\varepsilon\big)^{1/2},\quad k=1,2,\cdots.$$
Plainly, $\{\parallel\cdot\parallel_k\}_{k=1}^\infty$ forms a fundamental system of norms generating the topology of $\mathcal{O}(M)$, and as usual we denote the unit ball corresponding to $\parallel\cdot\parallel_k$ by $\mathcal{U}_k$, $k=1,2,\cdots$.

Let $B:=\{g\in\mathcal{O}(M): \int_M|g|^2d\varepsilon\leq 1\}$. Observe that $B\subseteq \mathcal{U}_k$ for each $k\in \mathbb{N}$.

The above analysis can be summerized as
\begin{equation}\label{2star2}
\exists\, k_1,\lambda>0;\, \forall\, k>k_1,\exists\, c>0;\, \mathcal{U}_{k_1}\subset \frac{1}{r^\lambda}\mathcal{U}_{B(\zeta_0,\varepsilon^-)}+cr\mathcal{U}_k,\,\forall\, r\geq 1,
\end{equation}
where $0<\varepsilon^-<\varepsilon$ and $\mathcal{U}_{B(\zeta_0,\varepsilon^-)}=\{f\in \mathcal{O}(M):\sup_{\xi\in B(\zeta_0,\varepsilon^-)}|f(\xi)|\leq 1\}$.

Since for $0<r\leq 1$, (\ref{2star2}) is trivial, we conclude that $M$ has the property $\widetilde{\Omega}_{\zeta_0}$.

\implication{(3)}{(1)}	Choose a volume form $dV$ on $M$ and an open\textit{\ strictly pseudoconvex }
exhaustion $\{D_{k}\}_{k=1}^{\infty }$, $\overline{D}_{k}\subset \mathring{D}%
_{k+1}$; $k=1,2,\cdots $, $\bigcup D_{k}=M$, of $M$ with $\zeta _{0}\in
D_{1} $. For $f\in \mathcal{O}(M)$, set $\parallel \cdot \parallel _{k}:=%
\big(\int_{D_{k}}|\cdot |^{2}dV\big)^{1/2}$, $k=1,2,\cdots $. In view of our
assumption $\exists $ $\varepsilon >0$ such that $B(\zeta _{0},\varepsilon
)\subset \subset D_{1}$ and \ref{2star1}
\begin{equation*}
\exists\, k,\beta >0;\,\forall\, k>k_{1}\,\exists\, C_{k}>0\text{ such that }%
\mathcal{U}_{k_{1}}\subset r\mathcal{U}_{k}+\frac{C_{k}}{r^{\beta }}\mathcal{%
	U}_{0},\,\forall\, r>0,
\end{equation*}%
where $\mathcal{U}_{k}$'s are closed unit balls corresponding to $\parallel
\cdot \parallel _{k}$'s and
\begin{equation*}
\mathcal{U}_{0}=\{f\in \mathcal{O}(M):\sup_{\xi \in B(\zeta _{0},\varepsilon
	)}|f(\xi )|\leq 1\}.
\end{equation*}

In the first part of the proof we employ the argument given in 29.16 Lemma
of \cite{MV} to construct a Hilbert space densely imbedded in $\mathcal{O}%
(M) $ and satisfies a certain interpolation condition. We will include this
line of reasoning here for completeness.

In view of nuclearity of $\mathcal{O}(M)$, for each $k=1,2,\cdots$, and $%
0<\varepsilon_k:=\min\{\frac{1}{2^k},\frac{1}{C_{k+2}^{\frac{2}{\beta}+1}}\}$%
, we can find a finite set $M_k\subset \mathcal{U}_{k+1}$ such that
\begin{equation*}
\mathcal{U}_{k+1}\subset \varepsilon_k\mathcal{U}_k+M_k.
\end{equation*}

Since for each $k=1,2,\cdots$, $M_s\subset\mathcal{U}_{k+1}$ for $s>k$, the
set $\Lambda=\bigcup_{s=1}^\infty M_s$ is bounded in $\mathcal{O}(M)$. For
each $g\in \mathcal{O}(M)^*$ we have:
\begin{equation}  \label{2star3}
\parallel g\parallel^*_{k+1}\leq \varepsilon_k \parallel
g\parallel^*_k+\sup_{x\in \Lambda}|g(x)|,
\end{equation}
where $\parallel h\parallel^*_s:=\sup_{x\in \mathcal{U}_s}|h(x)|$ is the
dual norm of $\parallel\cdot\parallel_s$, $s=1,2,\cdots$.

Iterating inequality (\ref{2star3}) we get;
\begin{align}  \label{1}
&\forall\, g\in\mathcal{O}(M)^*, \forall\, n>k_1 \\
\parallel g\parallel^*_n&\leq
(1+\varepsilon_{n-1}+\varepsilon_{n-1}\varepsilon_{n-2}+\cdots+%
\varepsilon_{n-1}\cdots\varepsilon_0)\sup_{x\in
	\Lambda}|g(x)|+\varepsilon_0\cdots\varepsilon_{n-1}\parallel g\parallel^*_0
\notag \\
&\leq 2\sup_{x\in
	\Lambda}|g(x)|+\varepsilon_0\cdots\varepsilon_{n-1}\parallel g\parallel^*_0,
\notag
\end{align}
where $\parallel g\parallel_0^* =\sup_{x\in\mathcal{U}_0}|g(x)|$, and $%
\varepsilon_0=1$.

From (\ref{2star2}) it follows that, $\exists\, k_1,\beta>0$; $\forall\, n>k_1\,
\exists\, C_n$ such that $\forall\, g\in \mathcal{O}(M)^*$;
\begin{equation*}
\parallel g\parallel^*_{k_1}\leq r\parallel g\parallel^*_n+ \frac{C_n}{%
	r^\beta}\parallel g\parallel^*_0,\quad \forall\, r>0.
\end{equation*}
Inserting (\ref{1}) into this expression one gets:
\begin{equation}  \label{2}
\parallel g\parallel^*_{k_1} \leq 2r \sup_{x\in \Lambda}|g(x)|+
(r\varepsilon_0\cdots \varepsilon_{n-1}+\frac{C_n}{r^\beta})\parallel
g\parallel^*_0,\quad \forall\, r, \forall\, g\in\mathcal{O}(M)^*.
\end{equation}
Now for $C_n\leq r^{\beta/2} \leq C_{n+1}$;
\begin{equation*}
r\varepsilon_0\cdots \varepsilon_{n-1}\leq
C_{n+1}^{2/\beta}\varepsilon_0\cdots \varepsilon_{n-1}\leq \frac{%
	C_{n+1}^{2/\beta}}{C_{n+1}^{1/\beta+1}}=\frac{1}{C_{n+1}}\leq \frac{1}{%
	r^{\beta/2}}
\end{equation*}
\begin{equation*}
\frac{C_n}{r^\beta}\leq \frac{r^{\beta/2}}{r^\beta}\leq \frac{1}{r^{\beta/2}}%
.
\end{equation*}
Hence $\exists$ $R_0$;
\begin{equation}  \label{3}
\parallel g\parallel^*_{k_1}\leq 2r\sup_{x\in \Lambda}|g(x)|+\frac{2}{%
	r^{\beta/2}}\parallel g\parallel^*_0,\quad \forall\, r\geq R_0.
\end{equation}
To get a Hilbert ball from $\Lambda$ we first consider the closed absolutely
convex hull of $\Lambda$; call it $\hat{\Lambda}$, and consider the Hilbert
norm $|\cdot|^2:= \sum_{k=1}^\infty \parallel\cdot \parallel^2_k\frac{1}{%
	2^kD_k^2}$, where $D_k=\sup_{\xi\in \Lambda}\parallel \xi\parallel_k$, $%
k=1,2,\cdots$, and let $B:=\{f\in\mathcal{O}(M):|f|\leq 1\}$. We denote by $%
H_B$ the Hilbert space imbedded in $\mathcal{O}(M)$, whose unit ball is $B$.

Taking into account that $\mathcal{O}(M)$ is separable, by enlarging $\Lambda$
if necessary, we can, without loss of generality, assume that $H_B$ is dense in
$\mathcal{O}(M)$.

In view of (\ref{3});
\begin{equation}  \label{4}
\exists\, D>0;\, \forall\, g\in \mathcal{O}(M)^*;\quad \parallel
g\parallel^*_{k_1}\leq r\parallel g\parallel^*_B+\frac{D}{r^{\beta/2}}%
\parallel g\parallel^*_0;\quad \forall\, r>0,
\end{equation}
where $\parallel g\parallel^*_B:=\sup_{x\in B}|g(x)|$.

Taking minimum (over $r$) of the expression (\ref{4}) we obtain an
equivalent condition to $\widetilde{\Omega }_{\zeta _{0}}$, namely:
\begin{align}
\exists\, \varepsilon >0,k_{1},& \,0<\gamma <1,\text{ and a Hilbert space }%
H_{\infty }\text{ densely imbedded in }\mathcal{O}(M)  \label{5} \\
\text{ and }c>0:&  \notag \\
\parallel g\parallel _{k_{1}}^{\ast }& \leq c\big(\parallel g\parallel
_{0}^{\ast }\big)^{\gamma }\big(\parallel g\parallel _{B}^{\ast }\big)%
^{1-\gamma }\quad \forall\, g\in \mathcal{O}(M)^{\ast },  \notag
\end{align}%
\textit{where} $\parallel \cdot \parallel _{0}^{\ast }=\sup_{\mathcal{U}%
	_{0}}|\cdot |$ .

Let $H$ denote closure of $\mathcal{O}(M)$ with respect to the
hilbertian norm $\left\Vert \cdot \right\Vert =\left( \int\limits_{B(\zeta
	_{0},\varepsilon )}\left\vert \cdot \right\vert ^{2}dV\right) ^{\frac{1}{2}%
}. $ Since for some constant $c>0,$ $\parallel g\parallel _{0}^{\ast }\leq
c\parallel g\parallel _{{}}^{\ast }\forall\, g\in \mathcal{O}(M)^{\ast },$ the
above statement holds for the dual norm $\parallel \cdot \parallel
_{{}}^{\ast }$as well.

The imbedding $\imath :H_{\infty }\hookrightarrow H_{{}}$ is
compact since $\mathcal{O}(M)$ is nuclear, and consequently has a
representation of the form:
\begin{equation*}
\imath (x)=\sum_{n=0}^{\infty }\lambda _{n}\langle x,f_{n}\rangle _{\infty
}e_{n},
\end{equation*}%
for an orthonormal basis $\{f_{n}\}$, an orthonormal system $\{e_{n}\}$ for $%
H_{\infty }$ and $H_{{}}$ respectively, for some null sequence $%
\{\lambda _{n}\}$ consisting of strictly positive numbers.

Note that $\parallel f_{n}\parallel =\lambda _{n}$ and $\parallel
f_{n}^{\ast }\parallel _{{}}^{\ast }=\frac{1}{\lambda _{n}}$, where $%
f_{n}^{\ast }(\cdot )=\langle \cdot ,f_{n}\rangle _{\infty }$, $n=0,1,\cdots
$.

Set $\varepsilon _{n}:=\ln \big(\frac{1}{\lambda _{n}}\big)$. Since $%
\{f_{n}\}$ is a bounded sequence in $\mathcal{O}(M)$, and $\varepsilon
_{n}\rightarrow \infty $, the function
\begin{equation*}
\psi (z):=\overline{\lim }_{\xi \rightarrow z}\overline{\lim }_{n}\frac{\ln
	|f_{n}(\xi )|}{\varepsilon _{n}}
\end{equation*}%
defines a plurisubharmonic function on $M$. Plainly, $\psi (z)\leq 0$ for $%
z\in M$, and since $\parallel f_{n}\parallel =\lambda _{n}$ for $z\in
B(\zeta _{0};\varepsilon ^{-})$ , $\varepsilon ^{-}<\varepsilon $, $\psi
(z)\leq -1$. Choose a $0<\gamma ^{+}<1$ with $\gamma ^{+}>\gamma $.

We will look at the sublevel set
\begin{equation*}
\Omega_{\gamma^+}=\{z:\psi(z)<-\gamma^+\}.
\end{equation*}
This set is non-empty since $B(\zeta_0;\varepsilon)\subseteq
\Omega_{\gamma^+}$.

Choose a point $^{{}}z\in \Omega _{\gamma ^{+}}$. In view of
Hartog's Lemma (p.50 in \cite{K}), $\exists\, c_{{}}=c(z) >0 $
such that
\begin{align}
|f_{k}(z)|& \leq ce^{-\gamma ^{+}\varepsilon _{k}},\,k=1,2,\cdots ,\quad
\label{6} \\
\text{For an }f\text{ }& \in H_{\infty }\text{,\ \ \ \ \ \ \ \ \ \
	\ \ }f\text{ }(z)=\sum_{n}f_{n}^{\ast }(f)f_{n}(z),  \notag
\end{align}%
in view of (\ref{5}) and (\ref{6}) we have;
\begin{eqnarray*}
	|f(z)| &\leq &\sum_{n}|f_{n}^{\ast }(f)||f_{n}(z)|\leq \big(%
	\sum_{n}\parallel f_{n}^{\ast }\parallel _{k_{1}}^{\ast }|f_{n}(z)|\big)%
	\parallel f\parallel _{k_{1}} \\
	&\leq &\tilde{c}\big(\sum_{n}\frac{1}{\lambda _{n}^{\gamma }}e^{-\gamma
		^{+}\varepsilon _{n}}\big)\parallel f\parallel _{k_{1}} \\
	&=&\tilde{c}\big(\sum_{n}e^{(\gamma -\gamma ^{+})\varepsilon _{n}}\big)%
	\parallel f\parallel _{k_{1}}\leq \kappa \parallel f\parallel _{k_{1}}\text{
		\ for some \ }\kappa =\kappa \left( z\right) . \\
	&&\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }
\end{eqnarray*}

In the above estimate we have used the fact that increasing
permutation of the sequence $\left( \varepsilon _{n}\right) _{n}$ dominates
by a constant times the sequence $\left( \frac{1}{d_{n}\left(
	U,U_{k_{1}}\right) }\right)_{n}$, $U$ being the unit ball of $H$, which in turn dominates a constant
times $\left( n^{\frac{1}{d}}\right) _{n}$, where $d=dimM$ (See
Proposition 1.1 of \cite{AKT manuscripta}). Since $H_{\infty }$ is dense in $%
\mathcal{O}(M)$, the above inequality is true for each $f\in \mathcal{O}(M)$%
. Choose a relatively compact holomorphically convex $\hat{D}_{k_{1}}\subset
\subset M$ that contains $\overline{D}_{k_{1}}$. The analysis above yields:
\begin{equation*}
\exists\, C>0:\forall\, f\in \mathcal{O}(M)\quad |f(z)|\leq C\sup_{\xi \in \hat{D%
	}_{k_{1}}}|f(\xi )|,\quad j=1,2,\cdots .
\end{equation*}%
Employing a well known trick of applying this inequality to $f^{n}$
for a given $f\in \mathcal{O}(M)$, taking $n$th root of both sides as $%
n\rightarrow \infty $ we conclude that $z\in \hat{D}_{k_{1}}$.

This finishes the proof of the theorem.

\end{proof}

We now look into the question posed in the beginning of the section.

\begin{theorem}\label{thm3}
	Let $M$ be a Stein manifold of dimension $d$, and suppose it satisfies one of the equivalent conditions of Theorem \ref{thm2} for some $\zeta_0\in M$. Then $\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\Delta^d))$.
\end{theorem}

\begin{proof}
	Choose a $\mathcal{C}^\infty$-strictly plurisubharmonic exhaustion function $p:M\to (-\infty,\infty)$ and let $\Omega_k=\{z:p<k\}$, $k\in \mathbb{Z}^+$ with $p$ choosen such that $\zeta_0\in \Omega_0$. We assume that $M$ has the property $\hat{\Omega}_{\zeta_0}$. Choose $\varepsilon<\varepsilon^+$, $\overline{B(\zeta_0,\varepsilon^+)}\subset \Omega_{k_1}$ and a sequence of integers $\{k_n\}$ with $k_1<k_2<\cdots$ tending to infinity, where $\varepsilon$ and  $k_1$ are as in the definition of $\widetilde{\Omega}_{\zeta_0}$. Using the notation of Theorem \ref{thm2}, \implication{(3)}{(1)}, we set $$\mathcal{V}_0=\{f\in \mathcal{O}(M):\big(\int_{B(\zeta_0,\varepsilon^+)}|f|^2d\varepsilon\big)^{1/2}\leq 1\}\quad \textrm{and}$$
	$$\mathcal{V}_n=\{f\in \mathcal{O}(M):\big(\int_{D_{k_n}}|f|^2d\varepsilon\big)^{1/2}\leq 1\},\quad n=1,2,\cdots.$$
	In view of Proposition 3.8 of \cite{AS} and the condition $\widetilde{\Omega}_{\zeta_0}$ we have:

	\begin{equation}\label{3star1}
	\forall\, n=0,1,\cdots, \forall\, k>n, \, \exists\, j\textrm{ and }C>0\textrm{ with } \mathcal{V}_{k+1}\subset\subset r^j\mathcal{V}_n+\frac{C}{r}\mathcal{V}_k,\quad \forall\, r>0.
	\end{equation}
	Also we have:
	\begin{equation}\label{3star2}
	\forall\, k=0,\cdots,\exists\, 0<\gamma<1 \textrm{ and } C>0:\,\parallel f\parallel_{k+1}\leq C \parallel f\parallel_k^\gamma \parallel f\parallel_{k+2}^{1-\gamma}.
	\end{equation}
	This "two-constants theorem" type condition can be perceived in many ways, one of which is to employ $p$-measures in its proof as in \cite{Ay istanbul} p. 125. Now, from the discussion leading to Proposition 1.1 of \cite{AKT manuscripta} we conclude that:
	$$0<\underline{\lim}\frac{-\ln{d_n(\mathcal{V}_1;\mathcal{V}_0)}}{\alpha_n}\leq \overline{\lim}\frac{-\ln{d_n(\mathcal{V}_1;\mathcal{V}_0)}}{\alpha_n}<\infty,$$
	where $\alpha_n:= n^{1/d}$, $d=\dim M$, is the \emph{associated exponent} sequence of $\mathcal{O}(M)$ (\cite{AKT manuscripta}). Choose $R_0$ large, $1\ll R_0$ such that:
	\begin{equation}\label{3star3}
	\exists\, n_0:\, n\geq n_0 \, \Rightarrow \, \frac{1}{R_0^{\alpha_n}}\leq d_n(\mathcal{V}_1;\mathcal{V}_0).
	\end{equation}
	We estimate $d_n(\mathcal{V}_1;\mathcal{V}_0)$, $n\in \mathbb{N}$ using $\widetilde{\Omega}_{\zeta_0}$ as in \cite{T Karadeniz}. In view of 29.13 Lemma of \cite{MV} and (\ref{3}) of the proof \implication{(2)}{(3)} of Theorem \ref{thm2}, an equivalent form of $\widetilde{\Omega}_{\zeta_0}$ in our notation reads as; $\exists$ bounded set $B$ and $\lambda>0$ such that
	\begin{equation}\label{3star4}
	\mathcal{V}_1\subseteq \frac{1}{r^\lambda}\mathcal{V}_0+CrB;\quad \forall\, r>0.
	\end{equation}

	Now suppose that $\rho>0$ satisfies $B\subseteq \rho\mathcal{V}_0+\mathcal{L}$, for some finite dimensional subspace $\mathcal{L}$ with $\dim(\mathcal{L})\leq n$.

	In view of (\ref{3star4}), this leads to
	$$\mathcal{V}_1\subseteq \big(\frac{1}{r^\lambda}+Cr\rho\big)\mathcal{V}_0+\mathcal{L},\quad \forall\, r>0,$$
	and hence to:
	$$d_n(\mathcal{V}_1;\mathcal{V}_0)\leq \big(\frac{1}{r^\lambda}+Cr\rho\big) \mathcal{V}_0+\mathcal{L},\quad \forall\, r>0.$$

	Taking infimum of $\big(\frac{1}{r^\lambda}+Cr\rho\big)$ over $r$ first, then taking infimum over $\rho$ we get:

	\begin{equation}\label{3star5}
	\exists\, \kappa >0; \quad d_n(\mathcal{V}_1;\mathcal{V}_0)\leq \kappa d_n(B,\mathcal{V}_0)^{\lambda/1+\lambda}.
	\end{equation}

	We claim that
	$$\Delta(\mathcal{O}(M))\neq \Delta(\mathcal{O}(\mathbb{C}^d))=\{(\xi_n):\exists\, R\geq 1\textrm{ and }C>0; |\xi_n|\leq CR^{\alpha_n}\}.$$

	Anticipating a contradiction, we assume that $\Delta(\mathcal{O}(M))= \Delta(\mathcal{O}(\mathbb{C}^d))$. Choose an $R\gg R_0^{1+1/\lambda}$. Since $\{R^{\alpha_n}\}_n\in \Delta(\mathcal{O}(\mathbb{C}^d))$, $\exists\, m\in \mathbb{N}$ such that
	\begin{equation}\label{3star6}
	\lim_nR^{\alpha_n}d_n(\mathcal{V}_m;\mathcal{V}_0)=0.
	\end{equation}

	So by (\ref{3star3}) and (\ref{3star5}):
	\begin{eqnarray*}
	\Big(\frac{R}{R_0^{1+1/\lambda}}\Big)^{\alpha_n}&\leq& R^{\alpha_n}d_n(\mathcal{V}_1;\mathcal{V}_0)^{1+1/\lambda}\leq \big(\kappa^{1+1/\lambda}\big)R^{\alpha_n}d_n(B;\mathcal{V}_0)\\
	&\leq &CR^{\alpha_n}d_n(\mathcal{V}_m;\mathcal{V}_0).
	\end{eqnarray*}

	Hence, (\ref{3star6}) gives us the desired contradiction.

	In view of Theorem 4.4 of \cite{Ay} we conclude that $\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\Delta^d))$. This finishes the proof of the theorem.
\end{proof}

We now look at some immediate corollaries of Theorem \ref{thm3}.

\begin{corollary}\label{cor4}
	Let $M$ be an open Riemann surface. Then the following are equivalent:
	\begin{enumerate}
		\item $M$ is hyperbolic.
		\item $M$ possesses a semi-proper negative subharmonic function.
		\item $\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(B(0,1)))=\{(\xi_n):\sup_n|\xi_n|r^n<+\infty,\,\forall\, r<1\}$.
	\end{enumerate}
\end{corollary}

\begin{proof}
	If $M$ is hyperbolic, it has a non-trivial Green's function $g_M(\cdot,\cdot)$ at each point of $M$, which is harmonic except at its singularity. Fix a point $\zeta\in M$ and a small ball $B(\zeta,\varepsilon)$ in some chart around $\zeta$.

	Consider the sublevel set $\{z: g_M(z;\zeta)<-c\}$ where $-c<\min_{\varepsilon \in \partial B(\zeta,\varepsilon)}g_M(\zeta,\varepsilon)$. Such a $c>0$ exists since $g_M(\zeta,\cdot)$ is continuous. Taking into account that the sublevel sets of the Green functions are connected (\cite{PS}), we conclude that $$\{z:g_M(z,\zeta)<-c\}\subset B(\zeta,\varepsilon).$$

	So $M$ has a semi-proper negative subharmonic function. The other implications are consequences of Theorem \ref{thm3} and Theorem \ref{thm1}, respectively.
\end{proof}

Another consequence of Theorem \ref{thm3} and Theorem \ref{thm1} is:

\begin{corollary}\label{cor5}
	Let $M$ be a Stein manifold. If $M$ possesses a semi-proper complex Green function with a pole at $\zeta_0\in M$, then for each $\zeta\in M$ pluricomplex Green function with a pole at $\zeta$ exists.
\end{corollary}

We would like to close this section by recalling a linear topological invariant introduced by D. Vogt (\cite{VB}) from the structure theory of nuclear Fr\'echet spaces.

\begin{definition}
	Let $\mathcal{X}$ be a Fr\'echet space and $\{\parallel \cdot\parallel_k\}_k$ a fundamental system of seminorms generating the topology. $\mathcal{X}$ is said to have the property $\widetilde{\Omega}$ in case;
	$$\forall\, k_0\,\exists\, k_1,\lambda >0;\,\forall\, k>k_1, \,\exists\, C>0,\quad \mathcal{U}_{k_1}\subseteq \frac{C}{r}\mathcal{U}_{k_0}+r^\lambda \mathcal{U}_k,\quad \forall\, r>0,$$
	where as usual, $\mathcal{U}_s$ denotes the closed unit ball of the local Banach space $\mathcal{X}_s$, $s=1,2,\cdots$.
\end{definition}

This property does not depend upon the generating system $\{\parallel\cdot\parallel_k\}$ and is in fact a linear topological invariant; that is, if $\mathcal{X}$ has $\widetilde{\Omega}$ then any Fr\'echet space isomorphic to $\mathcal{X}$ as a linear topological space, also enjoys this property.

Like all $\Omega$-type conditions, $\widetilde{\Omega}$ passes to quotient spaces. For more information about the property $\widetilde{\Omega}$, we refer the reader to (\cite{VB,DMV Bulletin France}). Stein manifolds $M$, for which $\mathcal{O}(M)$ possesses the property $\widetilde{\Omega}$ were characterized in \cite{A D-issue}.

Plainly the property $\widetilde{\Omega}$ for $\mathcal{O}(M)$, $M$ a Stein manifold, implies that $M$ possesses the property $\widetilde{\Omega}_\zeta$ for every point $\zeta$ is $M$. Consequently, $\widetilde{\Omega}$ appears as a linear topological invariant of $\mathcal{O}(M)$ that ensures for every $\zeta\in M$ the existence of a semi-proper negative plurisubharmonic on $M$ with a relatively compact sublevel set containing $\zeta$.

On the other hand, possessing $\widetilde{\Omega}_\zeta$ at every point for a Stein manifold $M$ need not imply that $\mathcal{O}(M)$ has the property $\widetilde{\Omega}$. To see this, consider the punctured disc $B(0,1)\setminus\{0\}\subseteq\mathbb{C}$. This domain obviously enjoys the property $\widetilde{\Omega}_\zeta$ for each of its points. Employing Laurent expansion for analytic functions on the punctured disc, it is not difficult to show that $\mathcal{O}(B(0,1)\setminus\{0\})\approx \mathcal{O}(\mathbb{C})\times \mathcal{O}(B(0,1))$ as Fr\'echet spaces (see \cite{Rolewicz}, p. 376). If the space of analytic functions on this punctured disc had the property $\widetilde{\Omega}$, $\mathcal{O}(\mathbb{C})$, being a quotient of $\mathcal{O}(B(0,1)\setminus\{0\})$, would also enjoy this property. But this cannot be, eg., in view of Theorem  \ref{thm3} and the above remarks.

\section{Pluri-Greenian and Locally Uniformly Pluri-Greenian
	Complex Manifolds.}\label{section3}

E. Poletsky in $\left[ 18\right] $ introduced the concepts of \textit{%
	pluri-Greenian }and \textit{locally uniformly pluri-Greenian} complex
manifolds. In this section we look into these notions from our
	point of view.

A complex manifold $M$ is called \textit{pluri-Greenian} if for each $\omega
\in M$, the pluricomplex Green function $g_{M}(\cdot ,\omega )$ has a strict
logarithmic pole at $\omega $.

On pluri-Green manifolds the pluricomplex Green function $g_{M}(\cdot
,\omega )$ satisfies $(dd^{c}g_{M}(\cdot ,\omega ))^{d}=(2\pi )^{d}\delta
_{\omega }$ if $d$ is the dimension of the manifold.

Consequently, in view of Theorem 2.2, a Stein manifold $M$
	is pluri-Greenian if and only if it has the property $\widetilde{\Omega }%
_{z}$ for every $z\in M$.

Recall that a complex manifold $M$ is called \textit{locally} \textit{%
	uniformly} \textit{pluri-Greenian }in case every point $w_{0}\in M$ has
a coordinate neighborhood $U$ with the following property: there is an open
set $W\subseteq U$ containing $w_{0}$ and a constant $c$ such that $%
g_{M}\left( z,w\right) \geq \ln \left\Vert z-w\right\Vert +c$ on $U$
whenever $w\in W$ (see \cite{P paper}, \cite{P yeni}).

Balls in $\mathbb{C}^{N}$ plainly satisfy the required property; hence bounded domains in $\mathbb{C}^{N},$ are \textit{locally} \textit{uniformly} \textit{pluri-Greenian}. It turns out
that existence of a pluricomplex Green function with a strict logarithmic
pole at each point of a complex manifold ensures that it is locally
uniformly pluri-Greenian.

\begin{theorem}
	1) A complex manifold is locally uniformly pluri-Greenian if and only if it is
	pluri-Greenian.

	2) A Stein manifold $M$ is locally uniformly pluri-Greenian if and only if it has
	the property $\widetilde{\Omega}_{z}$ for each $z\in M$
\end{theorem}

\begin{proof}
	Let M be a complex manifold and $w_{0}\in M$. Suppose there exists a
	pluricomplex Green function $g_{M}\left( .,w_{0}\right) $ with a strict
	logarithmic pole at $w_{0}$. That is there exists numbers $A$ and $B$ such
	that%
	\[
	\ln \left\Vert z-w_0\right\Vert +A\leq g_{M}\left( z,w_{0}\right) \leq \ln
	\left\Vert z-w_0\right\Vert +B
	\]%
	on a coordinate neighborhood $U_{0}$ around $w_{0}$. Choose $R>0$ and $\beta
	<<0$ for the time being, so that%
	\begin{eqnarray*}
		\lbrace z:z\in U_{0},\textrm{ }g_{M}\left( z,w_{0}\right) <\beta \rbrace
		&\subset &B\left( w_{0},e^{\beta -A}\right) \textrm{ and} \\
		B\left( w_{0},e^{\beta -A}\right) &\subset \subset &B\left( w_{0},R\right)
		\subset \subset U_{0}\textrm{ \ }
	\end{eqnarray*}%
	where $B\left( w,r\right) $ denotes the ball around $w\in U_{0}$ with
	radius $r>0$ in $U_{0}$. Since the sub-level sets of $g_{M}\left(
	.,w_{0}\right) $ are connected,
	\[
	\Omega _{\beta }=\lbrace z:z\in M\textrm{ },\textrm{ }g_{M}\left(
	z,w_{0}\right) <\beta \rbrace \subset\subset B\left( w_{0},R\right) \subset \subset
	U_{0}\textrm{ }
	\]%
	Choose $R^{-}<<R$ such that $B\left( w_{0},R^{-}\right) \subset \subset $ $%
	\Omega _{\beta }$, and set $\beta _{0}=\sup_{\zeta \in B\left(
		w_{0},R^{-}\right) }g_{M}\left( \zeta ,w_{0}\right)$. Set,%
	\[
	\sigma \left( z\right) =g_{M}\left( z,w_{0}\right) -\beta _{0},\ \ z\in
	M.
	\]

The plurisubharmonic function $\sigma $ is bounded and semi-proper, since%
\[
\lbrace z:\sigma \left( z\right) <\beta -\beta _{0}\rbrace \subset \subset
B\left( w_{0},R\right) .\textrm{ \ }
\]

Let $g$ denote the pluricomplex Green function of $B\left( w_{0},R\right) $
with a pole at $w_{0}$. Choose constants $r_{1}$ and $r_{2}$ , $r_{1}<r_{2},$
such that ;%
\begin{eqnarray*}
	\lbrace z\in B\left( w_{0},R\right) :g\left( z\right) <r_{1}\rbrace
	&\subset \subset &B\left( w_{0},R^{-}\right) \\
	B\left( w_{0},e^{\beta -A}\right) &\subset \subset& \lbrace z\in B\left(
	w_{0},R\right) :g\left( z\right) <r_{2}\rbrace \subset \subset B\left(
	w_{0},R\right) .\textrm{ \ \ \ }
\end{eqnarray*}

Set%
\[
\Phi \left( z\right) =\frac{r_{2}-r_{1}}{\beta -\beta _{0}}\sigma \left(
z\right) +r_{1},\textrm{ }z\in M\textrm{ and }\Omega =\lbrace z\in
B\left( w_{0},R\right) :g\left( z\right) <r_{2}\rbrace .
\]

Define a function $u$ on $\Omega $ , by the formula:
\begin{equation*}
	u\left( z\right) = \left\{
	\begin{array}{cl}
		\max \left( g,\Phi \right)& \textrm{on }\overline{\lbrace z\in
			B\left( w_{0},R\right) :\textrm{ }g\left( z\right) <r_{1}\rbrace }^{c}\cap
		\Omega \\
		g\left( z\right) &\textrm{on }%
		\lbrace z\in B\left( w_{0},R\right) :g\left( z\right) \leq r_{1}\rbrace.
	\end{array} \right.
\end{equation*}
By our construction $u$ becomes a plurisubharmonic function. Note that
outside $B\left( w_{0},e^{\beta -A}\right) $, $g_{M}\left( z,w_{0}\right)
\geq \beta ,$ so $\sigma \geq \beta -\beta _{0}$ outside $B\left(
w_{0},e^{\beta -A}\right) \cap \Omega$. Hence by setting $u=\Phi $ outside $%
\Omega $, we can extend $u$ to $M$ to a bounded plurisubharmonic function on
$M$. Note that on $M,$
\[
u\left( z\right) \leq \sup_{\zeta \in M}\Phi \left( \zeta \right) \leq
\frac{r_{2}-r_{1}}{\beta -\beta _{0}}\left( -\beta _{0}\right) +r_{1}
\]

Setting $u_{M}=\frac{r_{2}-r_{1}}{\beta -\beta _{0}}\left( -\beta
_{0}\right) +r_{1},$ the function $v=u-u_{M}$ becomes a negative
plurisubharmonic function on M that is equal to $g-u_{M}$ on a ball around $%
w_{o}$ where $g$ is the pluricomplex Green function of a ball of radius $R$
with center $w_{0}$ in a coordinate chart around $w_{o}.$

Note that the dependence of $\Phi $ to the function $g$ is through the
constants $r_{1}$ and $r_{2}$ that satisfy the conditions above. If one could
select $r_{1}$ and $r_{2}$ such that the conditions required by these
numbers are fulfilled by all pluricomplex Green functions of $B\left(
w_{0},R\right) $ with poles that lie in a fixed small ball around $w_{0}$;
one can then use the function $\Phi $ defined above and carry out the above
argument simultaneously for all pluricomplex Green functions with poles in
this neighborhood and thereby obtain a negative plurisubharmonic function on
$M,$ for each $x$ in this neighborhood, which equals to $g_{x}+C$, for some $%
C $ that does not depend upon $x,$ on a fixed neighborhood $W$ of $w_{0}$,
 where $g_{x}$ denotes the pluricomplex Green function of $B\left(
w_{0},R\right) $ with a pole at $x.$ We claim that such a choice of
constants is possible.

In other words we wish to show that;
$$\exists \, s_{1}\textrm{ and }s_{2}, \quad s_{1}<s_{2}<0\textrm{ and }\delta >0: $$%
$$B\left( w_{0},\delta \right) \subset \lbrace z\in U_{0}:g_{x}\left(
z\right) <s_{1}\rbrace \subset \subset B\left( w_{0},R^{-}\right) \textrm{ and }$$
$$B\left( w_{0},e^{\beta -A}\right) \subset \subset \lbrace z\in B\left(
w_{0},R\right) :g_{x}\left( z\right) <s_{2}\rbrace \subset \subset B\left(
w_{0},R\right),$$
 for every $x$ in a fixed neighborhood of $w_{0}$.

To this end, choose $r_{1}$ and $r_{2}$ such that the condition is satisfied
for $x=w_{0}$. In view of Lemma 6.2.4 of \cite{K} choose balls around $w_0$, $B$ and $B_{0}$ such that $w_{0}\in B\subset \subset
B_{0}\subset \subset \lbrace z\in B\left( w_{0},R\right)
:g_{w_{0}}\left( z\right) <r_{1}\rbrace ,$%
\[
\left( 1+\varepsilon \right) g_{w_{0}}\left( z\right) \leq g_{x}\left(
z\right) \leq \frac{1}{1+\varepsilon }g_{w_{0}}\left( z\right) ,\ \ \
x\in B,\textrm{ }z\in U_{0}-B_{0},\ \
\]

and $\varepsilon >0$ to be chosen later.

On $g_{w_{0}}=r_{1}$, $g_{x}\leq \frac{r_{1}}{1+\varepsilon }$, hence $%
g_{x}\left( z\right) <\frac{r_{1}}{1+\varepsilon }$ , on $\left(
z:g_{w_{0}}<r_{1}\right) $ for all $x\in B$. Consequently we can find a $%
\delta >0,$ such that%
\[
B\left( w_{0},\delta \right) \subset \lbrace z\in
U_{0}:g_{w_{0}}<r_{1}\rbrace \subset \lbrace z\in U_{0}:g_{x}\left(
z\right) <\frac{r_{1}}{1+\varepsilon }\rbrace
\]

for all $x\in B.$ On the other hand for $z\notin B_{0},$ and $x\in
B,$ one has:
$$g_{x}\left( z\right) <\frac{r_{1}}{1+\varepsilon }%
\Longrightarrow g_{w_{0}}\left( z\right) <\frac{r_{1}}{\left( 1+\varepsilon
	\right) ^{2}}.$$
Choose $\varepsilon >0$ so small that $\lbrace z:g_{w_{0}}\left( z\right) <\frac{r_{1}}{\left(
	1+\varepsilon \right) ^{2}}\rbrace \subset \subset B\left( w_{0},R^{-}\right) .$
It follows that $\lbrace z\in B\left( w_{0},R\right) :g_{x}\left(
z\right) <\frac{r_{1}}{1+\varepsilon }\rbrace \subset \subset B\left(
w_{0},R^{-}\right) $, $\forall\, x\in B.$

Choose a ball $\Delta $ around $w_{0}$ that is contained compactly in $%
\lbrace z\in B\left( w_{0},R\right) :g_{w_{0}}\left( z\right)
<r_{2}\rbrace $ and contains $\overline{B\left( w_{0},e^{\beta -A}\right) }$.

On the boundary of $\Delta ,$ $g_{x}\leq \frac{r_{2}}{1+\varepsilon }$\ \ \ for
$x\in B$. So;
\[
B\left( w_{0},e^{\beta -A}\right) \subset \subset \lbrace z\in B\left(
w_{0},R\right) :g_{x}\left( z\right) <\frac{r_{2}}{1+\varepsilon }\rbrace .
\]

We put another condition on $\varepsilon $, by requiring that
$$\lbrace z\in B\left( w_{0},R\right) :g_{w_{0}}\left( z\right) <\frac{r_{2}}{\left(
	1+\varepsilon \right) ^{2}}\rbrace \subset \subset B\left( w_{0},R\right) .$$
It now follows that;%
\[
\lbrace z\in B\left( w_{0},R\right) :g_{x}\left( z\right) <\frac{r_{2}}{%
	1+\varepsilon }\rbrace \subset \subset B\left( w_{0},R\right)
\]

By choosing $\varepsilon >0$ satisfying the requirements stated above and
setting $s_{1}=\frac{r_{1}}{1+\varepsilon },$ $s_{2}=\frac{r_{2}}{1+\varepsilon },
$ one proves the claim.

Now since $B\left( w_{0},R\right) $ is locally uniformly pluri-Greenian,
there are open neighborhoods $W$ and $U$ of $w_{0}$, $W\subseteq U,$
contained in $B$ and a constant $c$ such that $g_{x}\left( z\right) \geq
\ln \left\Vert z-x\right\Vert +c$ on $U$ whenever $x\in W$. Combining
this with the analysis above finishes the proof of the first part of the
theorem.

The second part of the theorem follows directly from the first part and Theorem
2.2.
\end{proof}

\section{Concluding Remarks}

\subsection*{\RomanNumeralCaps{1}} We would like to point out that the results  about locally
uniformly pluri-Greenian complex manifolds obtained by E. Poletsky in his
 papers \cite{P paper}, \cite{P yeni} are valid, in view of
Theorem 3.1, for pluri-Greenian complex manifolds. This, in particular
applies to the construction in \cite{P yeni} of \textit{%
	pluri-potential compactifications }for locally uniformly pluri-Greenian
complex manifolds. Poletsky introduced biholomorphically invariant \textit{\
	pluri-potential compactifications }for complex manifolds in \cite{P yeni} as a counterpart to the Martin compactification in the classical
potential theory. Consequently Theorem 3.1 and Theorem 2.1 implies:

\begin{corollary}
	Let $M$ be a Stein manifold and suppose the Fr\'{e}chet space $O\left(
	M\right) $ has the property $\widetilde{\Omega}$. Then $M$ admits a
	pluri-potential compactification in the sense of Poletsky.
\end{corollary}

\subsection*{\RomanNumeralCaps{2}} \ In \cite{HSTI}, Harz, Shcherbina and Tomassini introduced the notion of a
\emph{core} of a complex manifold $M$, $c(M)$, as the set of all points $%
w\in M$ where every smooth global plurisubharmonic function that is bounded
from above fails to be strictly plurisubharmonic near $w$.

In the example of Section \ref{section1}, the core of the manifold is
plainly itself.

Poletsky and Shcherbina in \cite{PS} Corollary 3.3 showed that complex
manifolds possess pluricomplex Green functions with strict logarithmic
singularities at points outside their core. Combining this result with
Theorem \ref{thm2} of Section \ref{section2}, we get;

\begin{corollary}
	A Stein manifold $M$ satisfies $\widetilde{\Omega}_\zeta$, for all points $\zeta\in M$ that lie outside its core.
\end{corollary}

\subsection*{\RomanNumeralCaps{3}} \ Chen and Zhang, in their paper \cite{CZ}, considered Stein manifolds whose pluricomplex Green functions are semi-proper and specified them as manifolds that satisfy the property $(B1)$. They showed, among other things, that Stein manifolds satisfying property $(B1)$ possesses a Bergman metric (\cite{CZ} Theorem 1). This result and Theorem \ref{thm2} of Section \ref{section2} yields:

\begin{corollary}
	A Stein manifold $M$ satisfying $\widetilde{\Omega}_\zeta$ for all its points, possesses a Bergman metric.
\end{corollary}

\section*{Acknowledgement}

The author sincerely thanks Proffessor E. Poletsky for his valuable comments.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


Choose one starter from the following selection.
Fishcake – minced fish, blended with red curry paste, green beans, lime leaves and chilli, served with sweet chilli and ground peanut sauce.
Minced chicken sprinkled with sesame seeds served on toast with sweet chilli sauce.
Deep-fried golden chunks of bean curd, served with sweet chilli and ground peanut sauces.
Choice of chicken or prawns in sweet and sour soup flavored with galangal, lemon grass, lime leaves, light coconut milk, mushroom and spring onion, also available as vegetarian option with mushrooms.
Choice of chicken or beef in Thai masaman curry with coconut milk, onion, peanut, carrot and potato.
Choice of chicken or beef in Thai green curry with coconut milk, fine bean, carrot, bamboo shoots, basil and courgette.
Stir-fried chicken or beef with fresh chilli, spring onion, bamboo shoots, carrot and onion.
Stir-fried chicken, beef, lamb, duck or prawns with cashew nuts, onion, carrot, mushroom, pineapple and spring onion.
Stir-fried noodles with eggs, bean sprouts, carrot, spring onion, ground peanuts and a choice of chicken or prawns.
Mixed vegetables and deep-fried bean curd in a Thai red curry with coconut milk, bell pepper, tomato and pineapple.
Stir-fried mixed vegetables, bean curd, basil and chilli.
Stir-fried noodles with eggs, bean sprouts, mixed vegetables, ground peanuts and bean curd.
All served with either Jasmine Rice or Egg Fried Rice (except no.64 & v26).


I find that when packing an egy if I push the shisha out of the center hole it won't get harsh. Essentially load it like a donut or phunnel style. I find just about every brand will last flavor wise. It could just be your packing method burning out the flavor too quick.
Load lightly just under the rim, keep the middle open for air flow. Like I said every brand will last flavor wise if you pack this way. All the American brands are good starbuzz, fumari, social smoke, tangiers, haze. But the middle eastern brands are great too afzal, armen, al fakher, and nakhla.
Thanks brother for help ... And witch one from American brand giving muchhhhh smoke ?
Personally I get the best clouds from haze and fumari. With haze you must overpack it and push it down under the rim to make it a bit dense. Fumari is the same way I described earlier, pack it under the rim and leave the middle open. With these juicy tobaccos you must stir them up and mix in the juice to get long lasting flavor.
I would also like to add that with these juicer brands it may be wise to invest in a phunnel bowl to get the most of the juicier brands and to save yourself some mess on cleanup.
For egyptians I usually run about 3-4 natural coals on the juicer brands because in most cases they can handle the additional heat to bump up cloud output. I usually take an oyster fork and kind of scoop the shisha into the bowl with a slight overpack then press down a bit to get it at or just under the rim. Careful with how hard you press or you may snuff your coals with boiling molasses as the juiciness will release to the top of the bowl a bit or worse leak down your bowls holes instantly. I usually then take my hold poker and clear out over the center most hole and then just run the poker straight down in a few other spots to make smaller air pockets. Smokes like a charm.
For stickier and or dryer brands like AF I fluff it to under the rim, foil it, and stick to 2-3 coals depending on the bowl size once again.
To really ramp smoke clouds a windcover will do wonders, just take care with less heat hungry brands like AF. In most cases you can use one less coal depending on brand of shisha and coal type and slap a wind cover on and be all set.
Did you get better results now with our advice?


its been some time since the little Miss Lola Jean has been here.
as you can see she's been bulking up. she is hopefully going to become a cyclist like her Daddy used to be…what else would you do w those thighs!
so we're not crawling yet…more of a rolling all around the house which is great for laughs and she picks up dog hair at the same time.
god i love this kid. just uploading these 4 pics brought tears to me eyes. these are right off the camera…LJ doesnt need too much photoshop to be cute.
She looks just like you bro!
Wow Liam does Lola ever look like you now! She's a cutie for sure! I love how much you love her!!
i want to say she looks like Applehead….but, i already made the cute statement.
Awwwww, Liam, what a Dolly!!! I close my eyes and remember what it was like to cuddle her close and kiss those sweet cheeks! My favourite great-niece is a charmer, to be sure. Yes, you're a lucky guy. What's better – you KNOW it! So happy to see these great pix.
i miss her. can't wait to see her in June.


Kernow Classics is committed to quality and service. We have devised our own strict criteria for grading records and their sleeves so that customers can buy with confidence.
Visual checking of records is not in our opinion sufficient to ensure that records are graded accurately - many a 'click' or 'tick' can thereby go unnoticed. Each record Kernow Classics offers in its catalogue has been professionally cleaned, played through in its entirety and listened to attentively. We are therefore confident that the grading of each record within the catalogue reflects accurately its playing condition. Within the Ex and VG grades there are often additional + and - grades. This is a recognition that not all LPs will sound as good as they do on our equipment (Linn Sondek LP12 turntable; Adikt cartridge) which gives phenomenal sound.
VG- or G soft clicks possibly repeating, lasting generally less than thirty seconds or giving occasional intermittent ticks. This category also includes wear commensurate with age; We assume that most fifty-year old LPs will have been played many times, possibly with less than perfect equipment. If this is the case, one will encounter some distortion at climaxes.
Exactly what it says! Very worn sleeve possibly with very poor repairs. Only included if the product is particularly rare or interesting or if the LP is in better condition.


text:
Reviews in Brief
To Live And Defy In La: How Gangsta Rap Changed America
Home » To Live And Defy In La: How Gangsta Rap Changed America
Author: Felicia Angeja Viator
Publisher: Cambridge, MA: Harvard University Press, 2020. 352p.
Reviewer: Nicholas Stoia | May 2021
In 1984, the Los Angeles Police Department acquired two tank-like, six-ton armored personnel carriers that the US Army had used in the Vietnam War, and repurposed them for their fight against two connected and growing problems in the city: gangs and rock cocaine. The police were frustrated by their inability to break through the increasingly sophisticated security of "rock houses" before suspects inside could destroy evidence. So, they attached steel battering rams to the carriers, painted them blue, wrote "L.A.P.D. Rescue Vehicle" on the sides, and anticipated a new tactical edge in the war on drugs. On 6 February 1985, Chief Daryl Gates christened one of the rams with a bottle of Thunderbird wine and rode along as it rolled through the city streets and then smashed through the front wall of a suspected rock house. The gaping hole gave police quick and easy access, but inside they found only a small group of terrified citizens, including small children, sitting in the living room eating ice cream. Among the responses to this debacle were lawsuits, community outrage, and attempts to decommission the ram. Another was the song "Batterram," in which Compton rapper Toddy Tee described the unstoppable power of the police, as embodied by the ram. The track quickly became a local sensation, because it resonated with so many young Black men in South Los Angeles who resented the LAPD for the militarized policing of their communities.
The story of "Batterram" is one of many in To Live and Defy in LA: How Gangsta Rap Changed America, in which Felicia Angeja Viator shows how LA rap was, from its beginning, an artistic response to police power. This persistent focus on the police distinguished the regional style from the New York rap that dominated the genre early on and earned it the label "gangsta rap." As it became increasingly influential within hip-hop, its attributes infused the genre as a whole; and when rap came to dominate popular music, gangsta rap changed the culture nationwide.
The backdrop of To Live and Defy in LA is the acute plight in the cities of Southern California after the "tax revolt" of the late 1970s and early 1980s, and the slashing of public services. These shifts "coincided with, and resulted in, the further expansion of local street gangs, the rise of violent crime, and the spread of illicit industries, including guns and narcotics trafficking." The police, along with judges, prosecutors, and others, conflated full-fledged gang members with the much larger number of young Black Angelinos who merely "'affiliated' or 'associated' themselves with local gang sets." Most "managed generally to abide by the law," but "accepted gang members in their communities as if they were the troublesome relatives in an extended family." Eventually, this expanding view of gang membership "meant that all young black Angelinos, whether involved with local gangs or not, bore the racialized stigma of the gangster." (They often had a reciprocal perception of the LAPD, whom they feared "as if it were just as ruthless as any street gang, and more sinister.")
Mobile dance parties were essential to the emerging regional genre, but dependent on the participation of gang members: "The fact is that gun-toting gangbangers, and the thousands of young people who were peripherally affiliated with them, were integral to the core mobile dance demographic that so many young strivers depended on." Early LA rappers like Toddy Tee and his partner Mixmaster Spade sold their homemade "street tapes" to the very entities that the police and community leaders despised, and locals loved the music because it "was a rare reflection of their lived reality."
Although New York derided LA rap, it was the Queens group Run-DMC that galvanized the nascent West Coast scene. They had severed rap from disco and honored hip-hop's breakdancing origins with "the sneakers, jeans, T-shirts, bomber jackets, and black hats" that were the "uniform of the inner city." In 1986, on tour to prove the nationwide potential of hip-hop, pandemonium broke out at their Los Angeles concert. Following the "rap riot," Run-DMC vacillated between defending hip-hop from moral crusaders and joining them, blaming the "gangbangers" who had "sabotaged" the show. (Viator astutely contextualizes rap among contemporaneous genres like punk and heavy metal that likewise stoked concerns about violence, noting that only rap led to debates centered on race that cast all fans as gangsters.) Modern gangsta rap emerged in the wake of the concert, which "presented a window of opportunity for LA's own rap hopefuls, who recognized the value of the limelight, even when it was harsh."
The creation and ascension of NWA is central to Viator's account. She relates the early days in detail: the entrepreneurial Eric Wright (Eazy-E), a drug dealer seeking a way out, who launched the small label Ruthless Records in 1986; his poaching of Andre Young (Dr. Dre), the talented DJ and audio engineer, from Kru-Cut Records, where he was frustrated by lack of creative control and "tired of crafting electro-ballads and performing dance moves in medical scrubs." She describes O'Shea Jackson (Ice Cube) cutting his teeth at a skating rink, where he delivered verses that were "totally street, totally dirty," including obscene parodies—"My Penis," for instance, to the tune of Run-DMC's "My Adidas"—to the delight of Compton's toughest audiences; and Dre's subsequent recruitment of Ice Cube from Kru-Cut, where the latter felt restrained and unfulfilled, "unable to do what he did best: write authentic street tales about the homeboys from around the way." At Ruthless, they were unrestrained, encouraged to be "as hardcore as you want to be." The label was "rooted in the reality that those same toughs and gangsters" that Run-DMC had denounced "were LA's most loyal rap fans," and in "the counterintuitive belief that their stories were marketable."
With the 1987 release of the single "The Boyz-N-The-Hood," Ruthless set the new standard for LA rap. The lyrics "were plot-driven narratives, in contrast to the loosely constructed rap braggadocio that characterized most hip-hop records." The song pursued the realism that had been growing in rap since the 1982 release of "The Message," a disco-inspired track by Grandmaster Flash and the Furious Five; but it was also—like "Batteram"—highly localized, "written in the LA vernacular of drugs, classic cars, and police abuse." The record combined the localized narratives descending from Toddy Tee and others with the modern, disco-free style of Run-DMC.
Early gangsta rap hinged on its resonance with those caught up in gang violence and the LAPD's powerful new mandate for "hard-nosed measures." After violence spilled over into the city's posh neighborhoods, engendering a flood of media attention, the state passed laws including mandatory sentencing rules for felonies committed by people just marginally associated with gang organizations, and in LA Chief Gates—possibly the most central character in this book after Eazy-E et al.—announced "Operation Hammer," which would round up juvenile delinquents in "gang sweeps," and "'make war' on all the 'rotten little cowards' terrorizing the city." It was in this climate that NWA released their groundbreaking Straight Outta Compton. They were responding in part to the growing attention to gang violence in popular culture, exemplified by the 1988 Dennis Hopper film Colors, but they told the story from the receiving end of "Operation Hammer," rather than from the perspective of the film's two white LAPD officers. As Ice Cube put it, "We're trying to show how it is on the other side, what it's like to have to deal with that kind of asshole cop on a day-to-day basis." They represented themselves as "underground street reporters" and claimed a kind of authenticity missing from both Colors and New York rap.
Viator scrutinizes the emergence of MTV, its early exclusion of many Black artists, and its eventual reliance on them in order to remain relevant. Their show Yo! MTV Raps was instrumental in gaining broader exposure for hip-hop and capturing a wider national audience, especially the white teenage boys who would become such devoted fans. Unlike most established rockers (but like their new wave contemporaries) hip-hop artists immediately recognized the opportunities music videos offered for cultivating their personas. NWA's first music video, "Straight Outta Compton," depicted a gang sweep from the target's point of view. Despite the bluster and braggadocio of the song, the video casts it as "an expression of powerlessness. Only the police wield actual deadly weapons, and they use them, zealously, to round up NWA and their friends." But MTV rejected the video because it "glorified violence." "The cable outlet that had always seemed willing to rile censors with rock videos now proved uncomfortable with a dramatization of the sort of police harassment considered routine in black Los Angeles." NWA was, naturally, deeply resentful. For Ice Cube, the only "problem" with the video was that it told the truth and forced viewers to reckon with injustice. But, perhaps predictably, the network's refusal led to a boost for competing channels that agreed to play the video and generated a cascade of sales and publicity for the group. The whole affair ultimately played into NWA's marketing strategy, in which controversy was currency.
NWA's most inflammatory song was "Fuck Tha Police." Law enforcement was not a fan of the track, and the Fraternal Order of Police advised members not to provide security during the group's 1989 national tour, in an effort to shut down shows. Even the FBI chimed in, scolding the group's label and pontificating on the role of music in society. But NWA used the criticism as a "springboard for more exposure and public sympathy." (The FOP and FBI apparently failed to anticipate this brilliant counterstroke.) Both the album and the song had a huge influence outside of LA. New York rapper Jay-Z found more in the album than just the "nod in his direction" that he got from prominent New York groups; "he heard his own voice." In Philadelphia, the sixteen-year-old ?uestlove was "thunderstruck" after a friend showed him "Fuck tha Police": "There were certain things that artists didn't do…You just didn't say, 'Fuck the police.'" He ditched chamber orchestra so that he could listen to rest of the album. NWA helped make 1988 the watershed year when hip-hop broke out of its New York mold and the California "gangster" archetype took hold within the genre.
Viator doesn't gloss over NWA's faults: she notes the battery charges against Dr. Dre for an assault on Denise "Dee" Barnes; his and Eazy-E's tasteless jokes about the brutal police beating of Rodney King; the publication of "luridly misogynistic" tracks like "To Kill a Hooker," "Findum, Fuckum, & Flee," and "One Less Bitch"; and she includes incisive criticisms from various perspectives. Some in the industry "argued that LA rappers had abused their public platform by promoting an apolitical version of black militancy, even violent anarchy." A critic told the Los Angeles Times that NWA made "black exploitation records," while another at the Sentinel, the long-established African American-owned newspaper, called them "the ultimate sell-outs." LA activist Leon Watkins worried that NWA and Ice Cube were "normalizing gang violence and urban plight" and exploiting a "volatile situation."
The book ends with the 1992 Los Angeles uprisings in response to the Rodney King verdict, which appeared to confirm gangsta rap as an "honest response to a real American crisis." Multiple rappers, including Ice-T and Tupac Shakur, argued that they had been warning about the explosive conditions for years. "Anything you want to know about the riots," observed Ice Cube, "was in the records before the riots."
To Live and Defy chronicles the emergence of gangsta rap and its transformation of hip-hop and popular culture. As Viator notes in her closing paragraph, "The lasting legacy of Los Angeles gangsta rap is not a hip-hop genre 'co-opted by the mainstream' but rather a twenty-first century pop culture landscape co-opted by hip-hop." But the book is also a thorough and timely study of an important intersection between music and social conditions, because the ascendance of gangsta rap since the 1980s reflects the rise of militarized policing over that same span, and we've seen all too clearly in recent years why it continues to resonate.
Nicholas Stoia is Assistant Professor of Music at Duke University
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I still don't fully understand what Bitcoin is all about to this day, so there's no way I'm going to understand the latest virtual currency springing up, Coinye West. That's right, it's a currency based on Kanye West, using a cartoon version of his face. The first coins were issued last night ahead of its initial release date. It's all fine and well, but I think we all know by now that Kanye has zero sense of humour, so of course his lawyers have filed a cease and resist to the anonymous folks behind the new currency.
The legal document, dated Jan. 6, includes an image of Coinye – a cartoon representation of West on a gold medallion. West's lawyer argues trademark infringement.
"Given Mr. West's wide-ranging entrepreneurial accomplishments, consumers are likely to mistakenly believe that Mr. West is the source of your services," wrote Brad Rose, a partner at Pryor Cashman LLP, which has previously worked for West.
LOL, I mean, I doubt any of this will last very long, but who knows. I don't think it's right if it takes off that the people behind Coinye will profit off of Kanye's image, but that's a conversation to have if it ever makes it that far, you know?
Actually they were sort of bothered.. they changed the name from coinye west to just coinye and changed the logo to an unknown man with a fishtail thing. Which they had to do because Kanye definitely has justiciable rights.
while they had zero legally, i love them for yanking kanye's chain like that. the tweet is my everything.


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At the heart of the RX0 II sits a 1.0-type stacked 15.3-megapixeli Exmor RS™ CMOS image sensor and an advanced BIONZ X™ image processing engine that offer enhanced[i]color reproduction, including beautiful skin tones. This power packed combination has been optimized to deliver lightning fast performance and exquisite image quality for both stills and movie shooting across a wide sensitivity range of ISO 80[ii]-12800. The ZEISS® Tessar T* 24mmiii F4.0 fixed wide-angle lens has a shortened minimum focusing distance of 20cm, making it perfect for selfies or table-top photos.
The RX0 II is truly a worry-free camera that can be taken into places that no other camera can go. Measuring just 59mm x 40.5mm x 35mm[iii] and weighing just 132g,[iv] the RX0 II fitseasily into a pocket and is ready for whatever the elements can throw at it. It is waterproofii up to 10 meters deep, dustproofii, shockproofvii up to 2 meters and crushproofviii up to 200KG force.
The RX0 II offers 4K 30p internal movie recordingiv with full pixel readout and no pixel binning to collect approximately 1.7 times the amount of data required for 4K video. By oversampling this data, the appearance of moiré and jaggies is reduced to deliver smooth, high-quality 4K footage with exceptional detail and depth. Using the recently introduced Sony "Imaging Edge™"mobile applications, this footage can be transferred to a smartphone[v], edited and shared easily across social networks.
The RX0 II introduces in-body electronic stabilization[vi] for steady footage, even when shot handheld. This can be enhanced even further when footage is exported to a smartphone or tablet running the "Movie Edit add-on"[vii] application, where the additional information captured during filming can be processed to produce a video with gimbal-like smoothness.[viii] An additional new feature that can be accessed via Sony "Movie Edit add-on" is Intelligent Framing, where the selected subject is kept in the center of the frame and image distortion is corrected in a final edit. Depending on where the video will be shared, a variety of aspect ratios can then be selected.
Additional movie features of the RX0 II include Super Slow Motion recording at up to 1,000 fps[ix], uncompressed 4K HDMI output and simultaneous proxy movie recording. Users can utilize Picture Profile, S-Log2 and Time Code / User Bit functions to ensure a final result that exactly matches their creative vision.
The versatility of the RX0 II means thatit offers a host of still photography features in addition to its video capabilities. It boasts an Anti-distortion Shutter of up to 1/32,000 sec. and can shoot at up to 16 fpsv to capture any fleeting moment of emotion that passes across the subject's face. In an upgrade from the original RX0, color reproduction has been enhanced to reproduce natural and vivid color of human skin, and the optional Soft Skin Effect can be applied to cover minor blemishes and wrinkles.Rating and Protect functions can be applied in camera and a grouped display of continuously shot images can be viewed on the camera.
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[xi] Not available during 4K video, FHD 120p (NTSC)/100p (PAL) video or interval recording. DSC-RX0 and DSC-RX0M2 models cannot be used with radio control at the same time. Depending on the specifications of the access point, smartphone, and tablet to be used, and the radio wave environment of the installation location, the number of devices that can be connected may change.


These materials are extracted from the roasted pods of Theobroma cacao, called the cacao or cocoa tree, a small evergreen tree in the family Malvaceae native to the deep tropical regions of Central and South America. These are, of course, the exact same seeds pods that give us chocolate. Used in perfumery for their deep, rich, luscious aroma, they serve a growing interest in gourmand perfumes and blend well with tonka, coumarin notes, coffee, hazelnut CO2, cherry CO2. banana CO2, vanilla, honey and can also provide interesting effects when used with florals. Did you know there are about 25 key ingredients which give chocolate its smell, but not one of them has a characteristic chocolatey scent?


There's a fair bit of citrus to harvest but the south side of the yard continues to die back. No sun. I've slowed down feeding the fish. There's very few eggs from the chooks but the quails are still doing well.
Well, it's finally getting cooler, and oddly enough, wetter. Along with the cooler temperatures, comes increasing darkness.
Annuals on the south side of the yard are starting to fail, even the cherry tomatoes. The water spinach and shallots are still okay, but not for a lot longer. They need to be eaten while it's warm in the day. Even the birds are going into hiatus. It's not just about the cooler temperatures. It's mostly about the decrease in sunlight hours as winter draws closer.
This is when the north side of the yard comes into its own, as tiny as it might be. The elephant garlic needs to go back in over there. I should still get tomatoes from there and silver beet will provide me with greens and celery-like branches. Chitting potatoes will be shoved into hanging buckets over there as well. I want to get some cabbage in too and maybe broccoli.
Luckily, both sides of the yard are well stocked with citrus which will develop over the next month or three. Nature provides, so I'll have oranges, lemons, limes and mandarin to give us vitamin C. The colder months are also time for hearty meat dishes. I have an excess of male quails that are doing little more now than trying to injure each other. It is a necessary evil to cull them over the next few months. The bigger perch will also be eaten. The fingerlings that I added in November have grown big enough to provide sufficient waste to keep the aquaponics bed cycling as well as it can in the darker months.
Even tiny farm homesteading means you need to learn to store excess crops. We did well with beans over summer. Jeez I have ample dried beans! LOL. I still have lots of chili powder and lentils (pigeon pea to be accurate). Ginger and turmeric are getting low. For some reason, I also didn't make mango chutney over summer.
I love the way the year turns and my diet changes. I love when "snuggle weather" comes in. Life is good.
This year I am doing well with eggs. I'm getting a dozen quail eggs every second day (and almost 20 of them aren't old enough to lay yet!) I'm also freezing processed male quails now and currently have 5 vacuumed sealed ready to cook. I might even have a crack at smoking one. The chickens are still producing 2 or 3 eggs a day. If I could keep a rooster, my poultry could go "off grid."
My fruit have also finally (for the most part) started to be old enough to produce well. The "ball tree" (named by my toddler granddaughter) has produced around 25 lemonade fruit now. We are also getting ruby grapefruit, passion fruit, mandarins, the odd lemon and have bananas on the lady finger.
I have cabbages and lettuce growing under a rat proof cover. Otherwise, when the big aquaponics bed had to be removed, my supply of greens dried up. I'm working on fixing that - around my oh-so-many career obligations.
After all these years, I still haven't replaced my fingerlings. I do however have some really big fish out in the backyard. One will feed four adults at Easter! Tiny city life is pretty good!
Thought I'd do a non-video update for a change (based on March 2015 which is below).
I still have a lot of spinach type greens. I've been drying them, making pesto and all sorts of things. However, and this is a big HOWEVER, none are as crispy as lettuce. I really want to find a summer green that is crispy like lettuce but won't just bolt in the heat. I'm happy to use all my other greens for lots of cooking, but in a salad, you really need crispy. Anyone got any ideas?
There is probably double the citrus production as last year which sounds good, but is really only 5 or so fruit per tree.
I'm still eating lots of eggs and loving it. The Quails add some painful but fun touches to things. Quail eggs taste great but have a very thick skin under the shell making peeling and cracking a bugger. I know that my eggs will slow down to a trickle pretty soon. I'm okay with that because we've had quite a bounty. Talking of Quail, I have 5 babies and 20 more eggs in the incubator. If I am lucky, I'll get another lot of eggs in before it starts to get cold. I think I've also located a male to breed with Big Bertha (the Plymouth Rock). Unfortunately, she's decided to moult and has stopped laying. I'm wondering if I'll get a breeding in before winter with her now.
I'l be eating Perch for Easter, and over winter (in replacement of the eggs). I'm running out of time to pick up some fingerlings to start to grow as replacements for the ones I eat. They are really good eating and great when I don't have as many eggs so I need to get an order in.
My final thoughts for this blog: I am getting some degree of self-sufficiency. I manage eggs, spinach greens, turmeric, chilli, passion fruit and fruit based beer/wine. My composting has increased lots and I have begun to actively manage my raised bed with organic soil additives like blood and bone, trace elements, seaweed etc.
I've been busy doing some careers work, so there's been a delay in blogging. Sorry about that.
Things at the Manor haven't gone so badly. We don't buy greens like lettuce or spinach any more. I also realised that I might have over focussed on greens. Man, I got greens!
We also don't buy eggs, but I do keep a close eye on the supply, in case I need to cut back on usage.
On the bright side, the fish are also growing like crazy. I still intend to leave them until they are huge, until they are eaten.
I have suferred greatly trying to be patient, watching the fruit trees grow. They are mostly still around 12 months old, so they aren't going to produce at this stage. I just need to learn patience until they get bigger. Aaargh. They are averaging between 1 - 3 fruit each, at this stage. The temptation is sending me crazy.
I am also growing Mignonette, and another one the type with frilly leaves that you can pick the leaves as you want them. My Cos bolted but I think it was due to the exceptional heat we had a couple of weeks ago. I still eat the leaves though.
I agree with the choice of mignonette Susan. In the past, I have found so much lettuce goes to waste (worms or chooks aren't really waste), if I pick the whole plant. I have also had cos lettuce reward me with much longer life, by picking leaf by leaf off a few plants. Lettuce in an AP growbed recover quickly when stripped of leaves.
Thanks folks. I'll give a few a go (he says, we move towards autumn... I hope).
Hi guys, I grow mignonette lettuces. If you pick them (and I pick them leaf by leaf) and wash them in the lettuce spinner in fridge, I find they are quite crispy. My kids don't know the difference. They do go to seed quicker in summer but I can pick for 3 weeks off one set of lettuce so I don't find that too bad and even if they do start shooting up, doing the above sweetens them back up and you can get another week or two out of them. Never had a problem with this type of lettuce.
What about Brazilian Spinach, I grow that year round and it is nice and crispy. Also Endive is another one always crispy and if you don't like your greens a little bitter, make one of your homemade dressings to put on it. Bitter Greens are very good for you. I sometimes buy the Little Gem Lettuce and Elaine is right they stay crisp for ages. Also other Selections could be Upton Cress, Lambs Lettuce, Lettuce Romaine, Mizuna, Some Chards and Rocket.
Radish! Yes, although they thrive in the cooler months :-( There's nothing quite like thinly-sliced Radish with some salt. Yummm.
I know how you feel Andy, my family will eat all the different type of greens I'm able to grow as a novelty but then they want to know where the real lettuce is, so I revert to buying it.
I love the way you are raising your own Perch, you are certainly further along the self-sustainable trail than I am. A nice piece of fish, vegies, a few herbs and a glass of wine sounds heavenly to me.
Not quite the replacement you might have had in mind, Andy … we buy Little Gem lettuces from Woolies (gasp!). Yup horror and not organic either. Fair chance they are hydroponic or at least greenhouse although I don't know for sure. They are not cheap either, around $1.50 each. Available all year round which makes me think they are hydroponic. Crisp as and keep very well.
You can buy Little Gem seeds, btw.


In order to ensure the success of the Paralympic enterprise, more than 8.000 volunteers were selected from amongst associations, federations and sports clubs, secondary schools and universities, associations for the disabled, etc. Very often, their tasks were similar to those carried out during the Olympic Games.
Amongst the body of Paralympic volunteers the task of one group, the hosts, was that of attending to the needs of the Paralympic Family. According to the various specific required to give logistic support, provide information to the athletes about the city or the Games, or accompany them to different places.
Other tasks, however, made more specific demands of the Paralympic volunteers. These included the tasks to be perfomed at the Prothesis Repair Centre or at the Classification Centre. Besides this, volunteers at the Barcelona'92 IX Paralympic Games also fulfilled a very important social function: that of contributing to the diffusion of personal values and attitudes based on normal, natural behaviour towards persons with physical or sensorial disabilities.
Programa de voluntarios de los Juegos Paralímpicos : IX Juegos Paralímpicos Barcelona'92 : Mayo 1991 / COOB'92. Divisió de Paralímpics. [Barcelona : COOB'92], 1991.
Els Voluntaris del Jocs Paralímpics = Los Voluntarios de los Juegos Paralímpicos / COOB'92. Divisió de Paralímpics. Barcelona : COOB'92, D.L. 1991.


Released in the USA on July 19 of 1996, "Trainspotting" is a film written by John Hodge (screenplay) . Ewen Bremner and Jonny Lee Miller are starring, alongside Kevin McKidd, Robert Carlyle, Kelly Macdonald and Peter Mullan.
Choose life. Choose a job. Choose a starter home. Choose dental insurance, leisure wear and matching luggage. Choose your future. But why would anyone want to do a thing like that?
Trainspotting was first released in the United States on Friday, July 19, 1996. Also released on this day were Fled (dir. Kevin Hooks), The Frighteners (dir. Peter Jackson), and Bye-Bye (dir. Karim Dridi). This was twenty-two (22) years ago.
It has been released on Friday, February 23, 1996 in the United Kingdom; Friday, July 26, 1996 in Canada; Wednesday, June 19, 1996 in France; Thursday, August 15, 1996 in Germany; Friday, October 04, 1996 in Italy; Monday, September 23, 1996 in Spain; Thursday, May 23, 1996 in Australia and on Saturday, November 30, 1996 in Japan.
Irvine Welsh and Ewen Bremner, Kevin McKidd can be seen in The Acid House, released in 1998.
Jonny Lee Miller, Kevin McKidd and Eileen Nicholas also appeared together in Regeneration, released in 1997.
Peter Mullan, Keith Allen and Billy Riddoch also worked together in Shallow Grave, released in 1994.
Kevin McKidd, Peter Mullan and James Cosmo also worked together in The Last Legion, released in 2007.
Mark Renton (McGregor) and Spud (Bremner) are running down Princes Street pursued by security guards. Renton's circle of friends are introduced: amoral con artist Sick Boy (Miller), clean-cut athlete Tommy (McKidd), simple-minded, good-natured Spud, and violent sociopath Begbie (Carlyle). Sick Boy, Spud and Renton are all heroin addicts and spend their time shooting up at the flat of their drug dealer "Mother Superior" Swanney (Mullan).
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Michael Fassbender to be in Trance?
Resources: Channel Four Films, Figment Films, Polygram Filmed Entertainment, The Noel Gay Motion Picture Company, Wikipedia and contributions from movie fans. Anyone can submit additionnal information and corrections, you can post a message into the forum, or you can contact me by e-mail. Information and materials are submitted by users and thus may not always contain up-to-date and correct information, so do not hesite to report mistakes, and submit corrections.Thank you!. Direct editing is no longer available, sorry.


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Q: Expected value conditional on a function Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$?
I have been trying to Google or look in books but I'm having trouble even articulating this properly. Any insight would be greatly appreciated.

A: $P(Y=y \& g(X)=g(a)) =P(Y=y | g(X)=g(a))P(g(X)=g(a))\\
\begin{align} &=P(Y=y \& g(X)=g(a) \& X=a)+ P(Y=y \& g(X)=g(a) \& X \neq a)
\\&=P(Y=y \&  X=a)+ P(Y=y \& g(X)=g(a) \& X \neq a)
\\&=P(Y=y |  X=a) P(X=a)+ P(Y=y | g(X)=g(a) \& X \neq a)P(g(X)=g(a) \& X \neq a)
\end{align}$
And we get this when $P(g(X)=g(a)) \neq 0$. :
$P(Y=y | g(X)=g(a))=P(Y=y |  X=a) r +P(Y=y | g(X)=g(a) \& X \neq a)(1-r)$
where $r= P(X=a)/P(g(X)=g(a))$.
$\int y f_{Y|g(X)=g(a)} dy= r \int y f_{Y|X=a} dy+ (1-r) \int y f_{Y|g(X)=g(a) \& X \neq a}dy$
$E[Y|g(X)=g(a)]= r E[Y|X=a] +(1-r)E[Y|g(X)=g(a) \& X \neq a]$
When g is a one-to-one function, r would be 1 so they are equal.
I guess this equation could be related to Bayesian statistics.
I posted misleading answer before and now I improved it and post it again. Any advice and crrections are always welcome. I am also a studying student.

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