text:
\section{Introduction}
Tameness of Fréchet spaces is an important property frequently used in functional analysis since it brings a kind of control on the otherwise chaotic behaviour of continuous linear endomorphisms. This concept was used effectively in the structure theory of nuclear Fréchet spaces, especially in finding bases in complemented subspaces of certain infinite type power series spaces \cite{DF}. Frequently the Fréchet spaces that arise in practical applications (e.g. in non-linear analysis), enjoy (versions of) the tameness property (\cite{HAMILTON, Z}). In fact, an inquiry about the tameness of the analytic function spaces was conveyed, sometime ago, in this context by D. Zarnadze [private communication]. In this paper we answer this question in a kind of negative way. We show that there are no tame analytic function spaces other than the natural ones (Theorem \ref{theorem:3}).

The organization of the paper is as follows: After establishing the notation and terminology, in section 1, we recall the definition of the linear topological invariants $\underline{DN}$ and $\Omega$ and introduce the technical tool that we will use in the later sections, namely local imbeddings of power series spaces of finite type into Fréchet spaces. After some general results on local imbeddings we establish a link between the approximate diametral dimension of the space and the existence of effective local imbeddings from certain finite type power series spaces (Theorem \ref{theorem:1}).

Section 2 is devoted to Theorem \ref{theorem:2} and its proof which characterizes tame nuclear Fréchet spaces having a finitely nuclear stable exponent sequence and enjoying the properties $\underline{DN}$ and $\Omega$.

In the first subsection of section 3 we show how the local imbeddings can be used to construct Green's functions on complex manifolds. The second subsection of the same section is devoted to diametral dimension considerations of analytic function spaces and the proof of the main theorem Theorem \ref{theorem:3} of this paper.

\subsection*{Notations and Terminology:}
We will use the terminology of \cite{M-V} and refer the reader to this book for the undefined concepts and the standard results of functional analysis that we will use. For the notions from complex potential theory used (especially in section 3) we refer the reader to \cite{KLIMEK}.

Power series sequence spaces play an important role in this paper. Recall that these are Fréchet spaces
\[
    \Lambda_R(\alpha) \doteq \{(\xi_n)^\infty_{n=0}: \  |(\xi_n)|_r \doteq \left( \sum_n |\xi_n|^2 e^{2r\alpha_n} \right)^\frac{1}{2} < \infty, \quad \forall\ -\infty<r<R\}
\]
where $R$ is either $1$ or $\infty$, and $\alpha = (\alpha_n)_n$ is a sequence of complex numbers with $\sup \frac{\ln (n)}{\alpha_n} < +\infty$, called the \emph{exponent sequence} of the space. The grading on these spaces will be the hilbertian grading $\{|\cdot|_r\}_{r<R}$.

These spaces are referred to as \emph{finite type power} series spaces if $R=1$ and \emph{infinite type power} series spaces if $R=\infty$, respectively. We will use the notation $\Lambda_1[\alpha]$ to denote the Hilbert space $\{(\xi_n)^\infty_{n=0}: \ |(\xi_n)|_r \doteq \left( \sum_n |\xi_n|^2 e^{2r\alpha_n}\right)^\frac{1}{2},$ $\forall \ 0<r<1\}$ with the norm $|\cdot|_r$, $r \in \mathbb{R}$.

For a pair of Fréchet spaces $\{\mathcal{X}, \ ^{\|\cdot\|_k}\}_k$ and $\{\mathcal{Y}, \ ^{|\ |_k}\}_k$ the set of all continuous linear operators will be denoted by $\mathcal{L}(\mathcal{X}, \mathcal{Y})$. We will use the symbol
\[
    \|T\|^n_m \doteq \sup_{\|x\|_{n\leq 1}} |T x|_m, \quad n,m \in \mathbb{N}
\]
for a $T \in \mathcal{L}(\mathcal{X}, \mathcal{Y})$. Note that $\|\cdot\|_m^n$ can take the value $+\infty$.

For a Fréchet space, $\{\mathcal{X}, \ ^{\|\cdot\|_k}\}_k$, the local Hilbert spaces corresponding to the norm $\|\ \|_k$ will be denoted by $\mathcal{X}_{k}$, $k=1,2,\dots$. The closed unit ball in $\mathcal{X}$ corresponding to the semi-norm $\|\ \|_k$ will be denoted by $U_k$, $k=1,2,\dots$.

Throughout the paper we will reserve the symbol $\epsilon_n$ to denote the sequence $(0,0,\dots,0,1,0,\dots)$ where $1$ is in the $n^{\text{th}}$ place.

\section{Local Imbeddings of Power Series Spaces of Finite Type into Fréchet Spaces}
In this section we will examine local imbeddings of power series spaces of finite type into nuclear Fréchet spaces with the properties $\underline{DN}$ and $\Omega$.

In subsection \ref{subs:1.1} we recall the definition of the linear topological invariants $\underline{DN}$ and $\Omega$ of Vogt and list some properties of Fréchet spaces with properties $\underline{DN}$ and $\Omega$, that will be used in the sequel. We emphasize again that all the Fréchet spaces that we will deal with in this paper will be assumed to be nuclear and satisfy the properties $\underline{DN}$ and $\Omega$ unless otherwise specified. In subsection \ref{subs:1.2} we will investigate the existence of local imbeddings. In subsection \ref{subs:1.3} we consider Fréchet spaces whose diametral approximative dimension is equal to a finite type power series space and show that this property yields local imbeddings.

\begin{defin} \label{def:1}
A continuous linear operator $T$ from $\Lambda_1(\alpha)$ into a graded Fréchet space $\{\mathcal{X} , \ ^{{\|\cdot\|}_n }\}$ is called an \emph{$(r , k)$-local imbedding} in case;
\[
\exists \  C > 0 : \quad {\|T(x)\|}_k \geq C {|x|}_r , \quad \forall \  x \in \Lambda_1 (\alpha).
\]
We will say that a Fréchet space $\mathcal{X}$ admits an \emph{r-local imbedding} from $\Lambda_1 (\alpha)$, in case there exists a continuous linear operator $T$ from $\Lambda_1 (\alpha)$ in $\mathcal{X}$ with the property;
\begin{align*}
\exists \text{ continuous semi-norm } \ ^{\|\cdot\|} \text{ of } \mathcal{X} \text{ and } C > 0 \\
\text{ such that } \|T(x)\| \geq C |x|_r \quad \forall \ x \in \Lambda_1 (\alpha).
\end{align*}
\end{defin}

\subsection{Linear topological invariants $\underline{DN}$, $\Omega$ and associated exponent sequences} \label{subs:1.1}

\begin{defin} \label{def:2}
A nuclear Fréchet space $\mathcal{X}$ is said to have properties $\underline{DN}$ and $\Omega$ (\cite{Vogt1}, \cite{Vogt-Wagner1}) in case there exists a fundamental system of hilbertian norms $\{\|\ \|_k\}_k$ generating the topology of $\mathcal{X}$ which satisfy
\begin{align*}
(\underline{DN}) : \qquad & \forall k \ \exists \ 0 < \lambda < 1 \text{ and } C>0 \text{ such that } \\
& \quad {\|x\|}_{k+1} \leq C \ {\|x\|}^\lambda_k \  {\|x\|}^{1-\lambda}_{k+2} \quad \forall \ x \in \mathcal{X}  \\
(\Omega) : \qquad & \forall k \ \forall p \  \exists C>0 \text{, }  j \in \mathbb{N} \text{ such that } \\
&\quad U_{k+1} \subset C r^j U_p + \frac{1}{r} U_k \quad \forall \ r>0
\end{align*}
\end{defin}

For more information about these invariants and examples of Fréchet spaces possessing these properties we refer the reader to \cite{M-V}.

Let $U$ and $V$, $V \subseteq U$ be 2 subsets of a Fréchet space $\mathcal{X}$. We will use the notation $\mathcal{E}_n (U,V)$ to denote
\[
    \mathcal{E}_n (U,V) \doteq -\ln d_n (V, U),\quad n=0,1,2
\]
where
\[
    d_n (V,U) \doteq \inf_{L_n} \  \inf \left\{ \lambda : \  \lambda > 0, \ V \subseteq \lambda U + L_n\right\}
\]
(infimum is taken over all n-dimensional subspaces $L_n$ of $\mathcal{X}$) is the $n^{th}$ Kolmogorov diameter of $V$ in $U$.

In the case of a graded Fréchet space $(\mathcal{X} , \ ^{{\|\ \|}_k})$, we simplify this notation by setting
\[
    \mathcal{E}_n (p, k) \doteq \mathcal{E}_n (U_p, U_k), \quad \forall \  p<k.
\]

Let us now fix a nuclear Fréchet space $\mathcal{X}$, with the properties $\underline{DN}$ and $\Omega$, and choose the grading ${\left\{ {\|\ \|}_k \right\}}_{k}$, derived from these properties. In \cite{AKT2} (c.f \cite{AKT1}) we have shown that all the sequences ${\{\mathcal{E}_n(k, k+1)\}}_n$, $k = 0,1,2,\dots$ are equivalent and called this equivalence class the \emph{exponent sequence associated} to $\mathcal{X}$. In this paper we, unless stated otherwise, will use a concrete representation, $\mathcal{E}_n \doteq \mathcal{E}_n (0,1), \ n=0,1,2$, of the associate exponent sequence of $\mathcal{X}$.

We compile (\cite{Terzioglu1}) some properties of these sequences that will be used in the sequel.

\begin{prop} \label{prop:1.1}
{\em Let $(\mathcal{X} , \ ^{{\|\ \|}_k})$ be as above. An $\Omega$-type condition;
\[
    \exists \  0<\lambda<1, D>0 : \quad U_q \subset r^{\frac{1-\lambda}{\lambda}} U_p + \frac{D}{r} U_k , \quad \forall \  r>C \quad (p<q<1)
\]
implies the inequalities:
\begin{enumerate}
\item $\exists \  C>0$; $\mathcal{E}_n (q, k) \leq \lambda \mathcal{E}_n (p;k)+C$,  $\forall \ n$,
\item $\exists \  C>0$; $\mathcal{E}_n (p, k) \leq \frac{1}{1-\lambda} \mathcal{E}_n (p,q)+C$,  $\forall \ n$,
\item $\exists \  C>0$; $\mathcal{E}_n (q, k) \leq \frac{\lambda}{1-\lambda} \mathcal{E}_n (p,q)+C$,  $\forall \ n$,
\end{enumerate}

A $\underline{DN}$-type condition:
\[
    \exists \  0<\lambda<1, D>0 : \quad {\|\cdot\|}_q \leq D {\|\cdot\|}_k^{1-\lambda} {\|\cdot\|}_p^\lambda \quad (p<q<k)
\]
implies the inequalities:
\begin{enumerate}
\item [(1')] $\exists \  C>0$; $\mathcal{E}_n (p, k) \leq \frac{1}{\lambda} \mathcal{E}_n (q;k)+C$,  $n=0,1,\dots$,
\item [(2')] $\exists \  C>0$; $\mathcal{E}_n (p, q) \leq (1-\lambda) \mathcal{E}_n (p;k)+C$,  $n=0,1,\dots$,
\item [(3')] $\exists \  C>0$; $\mathcal{E}_n (p, q) \leq \frac{\lambda}{1-\lambda} \mathcal{E}_n (q,k)+C$,  $n=0,1,\dots$,
\end{enumerate}
In the presence of both $\Omega$ and $DN$-type conditions we have
\begin{enumerate}
\item [(4)] $\exists \  C>0, \ 0<\lambda_1 <1,\ 0<\lambda_2 <1;\ \mathcal{E}_n \leq \frac{\lambda_1 \lambda_2}{(1-\lambda_1)(1-\lambda_2)} \mathcal{E}_n (p,q)+C,\ \forall \ n$
\item [(4')] $\exists \  C>0, \ 0<\lambda_1 <1,\ 0<\lambda_2 <1;\ \mathcal{E}_n (p,q) \leq \frac{\lambda_1 \lambda_2}{(1-\lambda_1)(1-\lambda_2)} \mathcal{E}_n + C,\  n=0,1,...$
\end{enumerate}
}
\end{prop}

\begin{proof}
See \cite{Terzioglu1} (c.f \cite{AKT2}).
\end{proof}

\subsection{Local imbeddings of power series spaces of finite type into nuclear Fréchet spaces with properties $\underline{DN}$ and $\Omega$} \label{subs:1.2}

\

We now turn our attention to the question of existence of local imbeddings into the graded Fréchet space $(\mathcal{X} , \ ^{{\|\ \|}_k})$ fixed in subsection \ref{subs:1.1} above. But we first consider some general considerations concerning local imbeddings.

\begin{prop} \label{prop:1.2}
{\em Let $\Lambda_1 (\alpha)$ be a nuclear power series space of finite type. The following assertions are equivalent;
\begin{enumerate}
\item [(i)] There exists an $(r,k)$-local imbedding from $\Lambda_1 (\alpha)$ into $\mathcal{X}$
\item [(ii)] There exists a sequence $\left\{ g_n \right\}_{n=0}^\infty$ in $\mathcal{X}$ that is orthogonal in $\mathcal{X}_k$ and satisfies;
\[
    \overline{\lim}_n \frac{\ln \|g_n\|_s}{\alpha_n} < 1-r, \quad \forall \  s
\]
\item [(iii)] There exists an isometry from the local Hilbert space $\Lambda_r [\alpha]$ into $H_k$ that induces a continuous linear operator from $\Lambda_1 (\alpha)$ into $\mathcal{X}$
\item [(iv)] There exists a closed, Hilbertian bounded disc $B$ which satisfies;
\[
    \exists \  C>0 \text{ such that } d_n (B, U_k) \geq C e^{-(1-r)\alpha_n}
\]
\end{enumerate}
}
\end{prop}

\begin{proof}
\
\begin{enumerate}

\item [(i)$\Rightarrow$(ii):] Fix an $(r,k)$-local imbedding $T$ with $\|T(x)\|_k \geq C_1 |x|_r$, $\forall \ x \in \mathcal{X}$. Consider
\[
    f_n \doteq \frac{\epsilon_n}{e^{r\alpha_n}},\quad n=0,1,2,\dots
\]
and apply the Gram-Schmidt orthogonalization procedure to the finitely linearly independent sequence $\left\{ T(f_n)\right\}_n$ in $\mathcal{X}_k$ to get a sequence,
\[
    g_n \doteq \sum_{i=0}^n c_i^n T(f_i), \quad n = 1,2,\dots
\]
in $\mathcal{X}$ that is orthogonal in $\mathcal{X}_k$. We estimate;
\[
    1 = \|g_n\|_{k} = \left\|T\left(\sum_{i=0}^{n} c_i^n g_i \right)\right\|_{k} \geq C_1 \left| \sum_{i=0}^{n} e_i^n \frac{\epsilon_i}{e^{r\alpha_i}} \right|_{r} = C_1 \left( \sum |c_i^n|^2\right)^{\frac{1}{2}}.
\]
From continuity of $T$ we get for every $s$, a $r < \sigma(s) < 1$ and $C>0$ such that,
\begin{align*}
    \|g_n\|_s & = \left\| T\left( \sum_{i=0}^{n} c_i^n f_i \right) \right\|_s \\ & \leq C \left| \sum_{i=0}^{n} c_i^n f_i \right|_{\sigma(s)} \\ & \leq C e^{(\sigma(s)-r)_{\alpha n}}\left( \sum_{i=0}^n |c_i|^2 \right)^{\frac{1}{2}} \leq \frac{C}{C_1} e^{(\sigma(s)-r)_{\alpha n}}.
\end{align*}
Hence
\[
    \overline{\lim}_n \frac{\ln \|g_n\|_s}{\alpha_n} \leq \sigma(s)-r < 1-r.
\]

\item [(ii)$\Rightarrow$(iii):] Choose $\{g_n\}_n$ as in (ii) and set
\[
    T\left(\sum_{i=0}^{\infty} c_i \epsilon_i \right) \doteq \sum_{i=0}^{\infty} e^{r\alpha_i} c_i g_i.
\]
This operator is plainly a continuous linear operator from $\Lambda_1(\alpha)$ into $\mathcal{X}$ that extends to an isometry from $\Lambda_r[\alpha_{n}]$ into $\mathcal{X}_k$.

\item [(iii)$\Rightarrow$(iv):] Choose an operator $T$ which has the properties given in (iii) and set;
\[
    g_n \doteq T \left( \frac{\epsilon_n}{e^{r\alpha_n}} \right), \quad n = 0,1,\dots.
\]
Consider the set
\[
    B \doteq \left\{ x \in \mathcal{X}; \quad \sum_{n=0}^\infty |\langle x, g_n \rangle_k|^2 e^{2(1-r)\alpha_n} \leq 1 \right\},
\]
where $\langle , \rangle_k$ is the inner product corresponding to $\|\ \|_k$.

A direct computation will show that $B$ is the image under $T$ of the compact set $B_0=\{(\xi_n); \quad \sum |\xi_n|^2 e^{2\alpha_n} \leq 1\}$ in $\Lambda_1(\alpha)$. So $B$ is a compact disc that is Hilbertian and it follows that (see Lemma 6.2.2 of \cite{D})
\[
    d_n (B,U_k) \geq d_n(B \cap T(\mathcal{X}), T(\mathcal{X}) \cap U_k) = e^{(r-1)\alpha_n}
\]

\item [(iv)$\Rightarrow$(i):] Let $H_B$ denote the Hilbert space generated by $B$ in $\mathcal{X}$. Since the inclusion $H_B \hookrightarrow \mathcal{X}_k$ is a compact operator we can choose a sequence $\{f_n\}_{n=0}^\infty$ of orthogonal vectors in $H_B$ that are orthogonal in the Hilbert space $\mathcal{X}_k$ and $\|f_n\|_B = \left( d_n(B, U_k) \right)^{-1}, \quad n=0,1,2,\dots$. Set
\[
    T\big((\xi_n)_n\big) \doteq \sum_{n=0}^\infty \xi_n e^{r\alpha_n} f_n, \quad \forall \ (\xi_n) \in \Lambda_1(\alpha).
\]
We fix an $s>k$ and choose, in view of the condition $\underline{DN}$, a $0<\lambda<1$ and $C>0$ such that $\|x\|_s \leq C \|x\|_k^{1-\lambda} \|x\|_B^\lambda, \quad  \forall\ x \in H_B$.

Choose $\lambda^+$ so that, $0<r+\lambda(1-\lambda)<\lambda^+<1$. There exists positive constants $C_1, C_2$ such that;
\begin{align*}
\left\| T\big((\xi_n)\big)_n \right\|_s  & \leq \sum_n |\xi_n| e^{r\alpha_n} \|f_n\|_s \\ & \leq C_1 \sum_n |\xi_n| e^{r\alpha_n} e^{\lambda(1-r)\alpha_n}\\ & \leq C_2\left( \sum_n |\xi_n|^2 e^{2\lambda^+ \alpha_n} \right)^{\frac{1}{2}} = C_2 \left| (\xi_n) \right|_{\lambda^+}.
\end{align*}
It follows that $T$ defines a continuous linear operator from $\Lambda_1(\alpha)$ into $\mathcal{X}$. Moreover
\[
    \|T((\xi_n))\|_k = \left( \sum_n |\xi_n|^2 e^{2r\alpha_n} \right)^{\frac{1}{2}} = |(\xi_n)|_r.
\]
Hence $T$ is a $(r,k)$-local imbedding.
\end{enumerate}
\end{proof}

\begin{xrem} \label{remark:1}
Contrary to the infinite case (\cite{AKT2}), the existence of an $(r,k)$-local imbedding from a finite type power series space into a graded Fréchet space need not imply the existence of an $(r^+, k)$-local imbedding for $r^+>r$. In fact it is not difficult to see that for a nuclear finite type power series space $\mathcal{X}=\Lambda_1(\alpha)$, there exists a $(r_1,r_2)$-local imbedding from $\Lambda_1(\alpha)$ into $\mathcal{X}$ if and only if $r_1 \leq r_2$. Indeed if there is a bounded set $B$ in $\mathcal{X}$ such that $d_n(B,U_{r_2}) \geq C e^{(r_1-1)\alpha_n}$ for some $C>0$, then for any $r_2<s<1$, there exists a $C_1>0$ such that $B \subseteq C_1 U_s$ which implies that
\[
    d_n(B,U_r) \leq C_1 d_n(U_s,U_r) = C e^{(r_2-s)\alpha_n}
\]
for every $n$. Hence $r_1-1 \leq r_2-1$ or $r_1 \leq r_2$.

The condition in (iv) of the above Proposition can also be expressed by using the unit balls of the grading:
\end{xrem}

\begin{prop} \label{prop:1.3}
{\em
There exists a $(r,p)$-local imbedding from $\Lambda_1(\alpha)$ into $\mathcal{X}$ if and only if
\[
    \sup_{k>p} \ \overline{\lim}_n \frac{\mathcal{E}_n(p,k)}{\alpha_n} \leq 1-r.
\]
}
\end{prop}

\begin{proof}
\
\begin{enumerate}
\item [($\Leftarrow$): ] Let $\rho$ and $r$ satisfy the given inequality. For a given $k>p$ in view of Proposition \ref{prop:1.2} (iii) we have;
\[
    \exists \ 0< \rho<1; \quad \overline{\lim}_n \frac{\mathcal{E}_n(p,k)}{\alpha_n} \leq \rho \  \overline{\lim}_n \frac{\mathcal{E}_n(p,k+1)}{\alpha_n} < 1-r.
\]
So for each $k>p$ we can find a $N(k) \in \mathbb{N}$, strictly increasing with respect to $k$, such that
\[
    -\ln d_n(U_k,U_p) < (1-r)\alpha_n.
\]
Let $\delta_1 = 1/2 e^{-(1-r)\alpha_{N(p+1)}}$. Since $U_{p+1}$ is precompact in $\mathcal{X}_p$ there exists a finite set $Z_1^1 \subseteq U_{p+1}$ such that;
\[
    d_n(U_{p+1}, U_{p}) \leq d_n(Z_1^1,U_{p})+\delta_1 \quad \forall \  n
\]
Hence
\[
    d_{N(p+1)}(Z_1^1,U_p) \geq e^{-(1-r)\alpha_{n_{N(p+1)}}} - \delta_1 = 1/2 e^{-(1-r)\alpha_{n_{N(p+1)}}}
\]
For each $n$, with $N(p+1) \leq n < N(p+2)$, we obtain, as above, a finite set $Z^1_n \subseteq U_{p+1}$ satisfying
\[
    d_n(Z_n^1, U_p) \geq \frac{e^{-(1-r)\alpha_n}}{2},
\]
and set $Z^1 \doteq \bigcup_{N(p+1)\leq n < N(p+2)} Z_n^1$.

Continuing in this fashion we get finite sets $Z^s, \ s=1,2,\dots$ with $Z^s \subseteq U_{p+s}$ and $d_n(Z^s,U_p)\geq (e^{-(1-r)\alpha_n})/2$ for $N(p+s)\leq n<N(p+s+1)$. Note that $\bigcup_{s=1}^\infty Z^s$ is a bounded set since $\forall \ k=p+s, Z^{\bar{s}} \subseteq U_k, \bar{s} > s$. We can find a closed Hilbertian disc $B$ in $\mathcal{X}$, containing $\bigcup_{s=1}^\infty Z^s$ (see for example in Lemma 1.2 of \cite{V2}). For this disc and $n>N(p+1)$(say $N(p+s)\leq n<N(p+s+1)$) we have
\[
    d_n(B,U_p)\geq d_n(Z^s,U_p)\geq \frac{e^{-(1-r)\alpha_n}}{2}
\]
So in view of Proposition \ref{prop:1.2} (iv) there is a (r,p)-local imbedding from $\Lambda_1(\alpha)$ into $\mathcal{X}$.

\item [($\Rightarrow$): ] This implication follows immediately from Proposition \ref{prop:1.2} (iv). Indeed in view of Proposition \ref{prop:1.2} (iv) there exists a bounded set $B$ and $C>0$ with $d_n(B,U_p) \geq C e^{-(1-r)\alpha_n},\ n=1,2,\dots$.
Hence $\forall\ k \ \exists \ C_i>0, \ i=1,2$;
\begin{align*}
    \mathcal{E}_n(p,k) & \leq -\ln d_n(B,U_p)+C_1 \\
                       & \leq (1-r)\alpha_n + C_2.
\end{align*}
So;
\[
    \overline{\lim}_n\  \frac{\mathcal{E}_n(p,k)}{\alpha_n} \leq 1-r \quad \forall \ k=p+1,\dots
\]
\end{enumerate}
\end{proof}

\begin{xrem} \label{remark:2}
Although the structural assumption $\underline{DN}$ on $\mathcal{X}$ is put into use in the Proposition above, the implication $(\Rightarrow)$ can be proved without any structural assumptions on $\mathcal{X}$. One can use the argument given in Proposition \ref{prop:1.1} of \cite{AKT1} to give a proof by just using the definition of local imbeddings.
\end{xrem}

\subsection{Approximate diametral dimension and local imbeddings}  \label{subs:1.3}

\

In this subsection we investigate the relationship between approximate diametral dimension of a nuclear Fréchet space $\mathcal{X}$ with properties $\underline{DN}$ and $\Omega$ and the existence of local imbeddings from $\Lambda_1(\mathcal{E}_n)$, the finite type power series space corresponding to the associated exponent sequence of $\mathcal{X}$. Since we wish to use the information obtained in the previous subsections, we are forced to put \emph{an additional assumption} on $\{\mathcal{E}_n\}_n$, namely we will assume that, $\lim_n \ln (n)/\mathcal{E}_n = 0$. This assumption guaranties the nuclearity of $\Lambda_1(\mathcal{E}_n)$ (\cite{D}), which will be needed in the sequel and also for the validity of the results of previous subsections.

\begin{defin} \label{def:3}
\cite{BPR}. \emph{The Approximate Diametral Dimension} of a Fréchet space $Y$, $\delta(Y)$, is defined as;
\begin{align*}
    \delta(Y) & \doteq \bigcup_{U \in \text{ zero neighborhoods of } Y} \  \bigcup_{B \in \text{ bounded subsets of } Y} \left\{ (t_n) : \lim_n \frac{t_n}{d_n(B,U)}=0 \right\} \\
    & = \{ (t_n)_n : \exists \text{ a neighborhood of zero } U \text{ and a bounded set } B \text{ of } Y \\
    & \text{ such that } \lim_n \frac{t_n}{d_n(B,U)}=0 \}.
\end{align*}
Approximate diametral dimension also admits a representation solely in terms of basis for the zero neighborhoods, $\{U_n\}_n$ of the Fréchet space $Y$ as; $\delta(Y)=\{(t_n):\  \exists \  s \ \forall \  k \geq s ; \  t_n / d_n(U_k, U_s) \rightarrow 0\}$ (See \cite{BPR}).

Now let us fix, for the rest of this subsection, a graded nuclear Fréchet space $\left\{ \mathcal{X}, \ ^{\|\ \|_k} \right\}$ with the properties $\underline{DN}$, $\Omega$, whose associated exponent sequence $(\mathcal{E}_n)$ satisfies $\left(\ln (n)/\mathcal{E}_n  \right)\rightarrow_n 0$, the grading $\{\|\ \|_k\}_k$ that is coming from the definition \ref{def:2}. Given $p<p+1<q$, in view of Proposition \ref{prop:1.1} (1), we have the inequality $d_n(U_q, U_{p+1})^{\frac{1}{d}} \leq C d_n(U_q, U_{p+1})$ for some $C>0$ and $0<\lambda<1$. So if a sequence $(t_n)_n$ satisfies
\[
    \frac{|t_n|}{d_n(U_q,U_p)} \leq C \quad \forall n, \text{ for some } C
\]
then
\[
    \frac{|t_n|}{d_n(U_q,U_{p+1})} \rightarrow 0
\]
Hence under our assumptions, we have
\[
    \delta(\mathcal{X}) = \{ (t_n)_n: \quad \exists \ p; \forall \ q>p \ \sup_n |t_n|e^{\mathcal{E}_n(p,q)} < \infty\}.
\]
It is not difficult to show, by direct computation, that (\cite{BPR})
\[
     \delta(\Lambda_1(\alpha)) = \{(t_n)_n: \quad \overline{\lim}_n \frac{\ln |t_n|}{\alpha_n} < 0\}.
\]
\end{defin}

\begin{prop} \label{prop:1.4}
{\em
    \[
    \delta(\mathcal{X}) \supseteq \delta (\Lambda_1(\alpha)) \Leftrightarrow \inf_p \sup_{q>p} \overline{\lim}_n \frac{\mathcal{E}_n(p,q)}{\alpha_n} = 0
    \]
}
\end{prop}

\begin{proof}
\
\begin{enumerate}
\item [($\Rightarrow$): ] For a fixed pair $p<q$ let \[
    B(p,q) \doteq \{ (t_n) : \quad \sup_n |t_n| e^{\mathcal{E}_n(p,q)} < \infty \}.
\]
This linear space becomes a Banach space under $|(t_n)|_{pq} \doteq \sup_n |t_n| e^{\mathcal{E}_n(p,q)}$. Since $B(p,q^+)\subseteq B(p,q)$ for $p<q<q^+$, the space $B(p) \doteq \bigcap_{q>p} B(p,q)$ with the fundamental generating norms $\left\{|\cdot|_{pq}\right\}$ becomes a Fréchet space. Moreover by means of continuous inclusions $B(\bar{p}) \hookrightarrow B(p), \bar{p}<p$, we can put the inductive limit topology on $\delta(\mathcal{X}) = \bigcup_n B(p)$ and thus view $\delta(\mathcal{X})$ as a locally convex space which is an inductive limit of Fréchet spaces. Let us fix a $r, \  0<r<1$ and define
\[
    S_r \doteq (t_n): \quad \|(t)\| \doteq \sup_n \frac{|t_n|}{r^{\alpha_n}} < \infty.
\]
Clearly $S_r$ is a Banach space with respect to the norm $\|\cdot\|$, and $S_r \subset \delta(\Lambda_1(\alpha))$. By our assumption, $S_r \subset \delta(\mathcal{X})$ and since projections onto coordinates are continuous with respect to the inductive limit topology on $\delta(\mathcal{X})$, the inclusion $S_r \hookrightarrow \delta(\mathcal{X})$ is a sequentially closed linear operator. In view of Grothendieck Factorization theorem (\cite[p.~225]{Kothe1}), there is a $p(r)$ such that for every $q>p(r)$
\[
    \exists \ C>0 : \quad \sup_n |t_n| e^{\mathcal{E}_n(p(r),q)} \leq C \sup_n \frac{|t_n|}{r^{\alpha_n}}
\]
for any $(t_n) \in S_r$. In particular for every $q>p(r)$ there exists a $C>0: \  \mathcal{E}_n(p(r),q)\leq \ln C - \alpha_n \ln r$, which in turn implies that
\[
    \inf_p \sup_{q>p} \overline{\lim}_n \frac{\mathcal{E}_n(p,q)}{\alpha_n} \leq -\ln r.
\]
Since this assertion holds for every $0<r<1$, we have
\[
    \inf_p \sup_{q>p} \overline{\lim}_n \frac{\mathcal{E}_n(p,q)}{\alpha_n} = 0.
\]

\item [($\Leftarrow$): ] Fix an $r<1$ and choose a $(t_n)_n$ which satisfies
\[
    |t_n| \leq C r^{\alpha_n} \text{ for some } C>0 \text{ and } \forall \ n.
\]
In view of our assumption, we choose a $p$ so that $\sup_{q\geq p} \overline{\lim}_n \frac{\mathcal{E}_n(p,q)}{\alpha_n} < - \ln r$. It follows that there exists an $n_0$ such that for $n\geq n_0$;
\[
    e^{\mathcal{E}_n(p,q)} < \frac{1}{r^{\alpha_n}} \text{ for every } q\geq p.
\]
Hence, $\forall\  q>p \ \exists \ C_q>0$; $e^{\mathcal{E}_n(p,q)} \leq C_q/r^{\alpha_n}$; which in turn implies that $\forall \ q>p$; $\sup_n |t_n| e^{\mathcal{E}_n(p,q)} < +\infty$. It follows that:
\[
    \delta\left(\Lambda_1(\alpha)\right) \subseteq \bigcup_p \bigcap_{q>p} B(p,q) = \delta(\mathcal{X}).
\]
\end{enumerate}
\end{proof}

If we focus our attention to the associated exponent sequence $(\mathcal{E}_n)_n$ we have:

\begin{cor}
\[
    \delta(\mathcal{X}) = \delta(\Lambda_1(\mathcal{E})) \Leftrightarrow \inf_p \sup_{q>p} \overline{\lim}_n \frac{\mathcal{E}_n(p,q)}{\mathcal{E}_n} = 0.
\]
\end{cor}

\begin{proof}
Let us fix a $(t_n) \in \delta(\mathcal{X})$, and choose a $p$ such that \[
    \forall \ q>p; \quad \sup_n |t_n| e^{\mathcal{E}_n(p,q)} <+\infty.
\]
In view of Proposition \ref{prop:1.1} (4), there exists constants $C_1>0$, $C_2>0$ such that \[
    \mathcal{E}_n \leq C_1 \mathcal{E}_n(p,p+1)+C_2 \quad \forall \ n.
\]
It follows that;
\[
    \sup_n |t_n|e^{\frac{1}{C_1}\mathcal{E}_n} < \infty.
\]
Hence $\exists \ D>0$;
\[
    \ln |t_n|+\frac{\mathcal{E}_n}{C_1} < D
\]
or \[
    \overline{\lim} \frac{\ln|t_n|}{\mathcal{E}_n} \leq -\frac{1}{C_1} < 0.
\]
Hence we always have $\delta(\mathcal{X}) \subseteq \delta(\Lambda_1(\mathcal{E}))$. Now the corollary follows from Proposition \ref{prop:1.4}.
\end{proof}

We conclude this section with a summary theorem:
\begin{thm} \label{theorem:1}
Let $\mathcal{X}$ be a nuclear Fréchet space with the properties $\underline{DN}$ and $\Omega$. We also assume that the associated exponent sequence $(\mathcal{E}_n)_n$ of $\mathcal{X}$, is a nuclear exponent sequence of finite type (i.e. $\Lambda_1(\mathcal{E}_n)$ is nuclear). Then the following assertions are equivalent:
\begin{enumerate}
\item $\delta(\mathcal{X}) = \delta(\Lambda_1(\mathcal{E}))$
\item $\inf_{1\leq p < \infty} \sup_{q>p} \overline{\lim}_n \frac{\mathcal{E}_n(q,p)}{\mathcal{E}_n} = 0 $
\item There exists an $r$-local imbedding from $\Lambda_1(\mathcal{E})$ into $\mathcal{X}$ for every $0<r<1$.
\end{enumerate}
\end{thm}

\section{Tame Spaces, $\mathcal{X}$, with Properties $\underline{DN}$, $\Omega$ and $\delta(\mathcal{X})=\delta(\Lambda_1(\mathcal{E}))$}

To every continuous linear operator $T$ between two graded F-spaces $\{\mathcal{X}, \ ^{\|\ \|_k}\}_k$ and $\{\mathcal{Y}, \ ^{|\ |_k}\}_k$ one can associate a sequence of natural numbers, $\{\sigma_T(n)\}_n$, called the \emph{characteristic of continuity} of $T$, via
\[
    \sigma_T(n) \doteq \inf \left\{ s : \quad \exists \ C>0; \ |T(x)|_n \leq C \|x\|_s \forall \ x \in \mathcal{X} \right\}
\]

In general the characteristics of continuity of operators between graded Fréchet spaces could be very disorderly. However for certain pairs of Fréchet spaces control over the growth of characteristics of continuity can be obtained. For example, in the space of analytic functions on the unit disc, $\mathcal{O}(\Delta)$, with the grading $
\|f\|_k \doteq \sum_{n=0}^\infty \left|\frac{f^n(0)}{n!}\right|^2 e^{-n/k}, \quad k=1,2,\dots , \  f \in \mathcal{O}(\Delta)$, it is not difficult to see that;
\[
    \forall \ T \in \mathcal{L}(\mathcal{O}(\Delta), \mathcal{O}(\Delta)), \ \exists \ a \in \mathbb{N} \text{ such that } \sigma_T(n) \leq an, \quad \forall \ n=1,2,\dots
\]
Following (\cite{V2.1}, \cite{DF} c.f. \cite{KN}) we specify this property as:
\begin{defin} \label{def:4}
A pair of Fréchet spaces $\mathcal{X}$ and $\mathcal{Y}$ will be called a \emph{tame-pair} in case for a given pair of generating semi-norms $\{\|\ \|_k\}_k$ of $\mathcal{X}$ and $\{|\ |_k\}_k$ of $\mathcal{Y}$ there exists an increasing function $\psi: \mathbb{N} \rightarrow \mathbb{N}$ such that
\begin{align*}
\forall \ T \in \mathcal{L}(\mathcal{X},\mathcal{Y}), \quad \exists \ N \in \mathbb{N};\quad \sigma_T(n) \leq \psi(n),\quad \forall \ n \geq N.
\end{align*}
In case $\mathcal{X}=\mathcal{Y}$ we say that the Fréchet space $\mathcal{X}$ is \emph{tame}.
\end{defin}

We will use the symbol $(\mathcal{X},\mathcal{Y}) \in \mathcal{T}$ in case $(\mathcal{X},\mathcal{Y})$ form a tame-pair.

\begin{xrem} \label{remark:3}
\
\begin{enumerate}
\item The definition does not depend on the choice of semi-norms in the spaces $\mathcal{X}$ and $\mathcal{Y}$.
\item Plainly the definition is equivalent to the existence of a sequence, for a given pair of generating semi-norms $\{\|\ \|_k\}_k$ of $\mathcal{X}$ and $\{|\ |_k\}_k$ of $\mathcal{Y}$, $\{S_K\}_k$ of increasing functions $S_K: \ \mathbb{N} \rightarrow \mathbb{N}$, such that for every $T \in \mathcal{L}(\mathcal{X}, \mathcal{Y})$, there exists a $K$ with $\sigma_T(n) \leq S_K(n)$ \textit{for every} $n=1,2,\dots$.
\item The space $\mathcal{O}(\Delta)$ is tame. More generally every power series space of finite type is tame, as was observed by various authors (see 2.1 on \cite{DF} for details). A proof of this appears in 2.1 of \cite{KN}.
\item If $(\mathcal{X},\mathcal{Y})\in \mathcal{T}$, then the space of all continuous linear operators from $\{\mathcal{X}, \ ^{\|\ \|_k}\}$ into $\{\mathcal{Y}, \ ^{|\ |_k}\}$ admits a representation of the form
\begin{align*}
    \mathcal{L}(\mathcal{X},\mathcal{Y})=\bigcup_{K=1}^\infty \bigcap_{n=1}^\infty \{ & T \in \mathcal{L}(\mathcal{X}, \mathcal{Y}): \\
   & \exists \ C>0 \ \|T(x)\|_n \leq C |x|_{S_K (n)}, \forall \ x \in \mathcal{X} \}
\end{align*}
Using this representation one can put a linear topology on $\mathcal{L}(\mathcal{X},\mathcal{Y})$ making it an LF-space (i.e. inductive limit of Fréchet spaces) by considering the semi-norms (2.1, \cite{DF}) $\|T\|_K^n \doteq \sup_{|x|_{S_K(n)}\leq1} \|T(x)\|_n$ on the space
\[
    \{ T  \in \mathcal{L}(\mathcal{X}, \mathcal{Y}):
    \exists C>0 \ \|T(x)\|_n \leq C |x|_{S_K (n)}, \forall x \in \mathcal{X} \}, K,n \in \mathbb{N}.
\]
therefore .....
\end{enumerate}
\end{xrem}

This structure allows one to use the results of well studied LF-spaces in the study of $\mathcal{L}(\mathcal{X}, \mathcal{Y})$ for $(\mathcal{X}, \mathcal{Y})\in \mathcal{T}$. These ideas are used in the study of nuclear Fréchet spaces $\mathcal{X}$ which form a tame pair with nuclear stable power series spaces of finite or infinite type in \cite{PISZCZEK} where a complete characterization of such spaces in terms of the linear topological invariants of Vogt are obtained.

In this section we will, once again, consider nuclear Fréchet spaces, $\mathcal{X}$ with the properties $\underline{DN}$ and $\Omega$. We will assume that the associated exponent sequence of $\mathcal{X}$, $ \mathcal{E}=(\mathcal{E}_n)_n$ in addition to our usual assumption of being finitely nuclear is also \emph{stable}; i.e. $\sup_n (\mathcal{E}_{2n}/\mathcal{E}_n) <\infty$.

\begin{thm} \label{theorem:2}
Let $\mathcal{X}$ be a nuclear Fréchet space with stable finitely nuclear associated exponent sequence $\mathcal{E} =(\mathcal{E}_n)$ and which has properties $\underline{DN}$ and $\Omega$. Then $\mathcal{X}$ is isomorphic to a power series space of finite type if and only if $\mathcal{X}$ is tame and $\delta(\mathcal{X}) = \delta(\Lambda_1(\mathcal{E}))$.
\end{thm}

\begin{proof}
\
\begin{enumerate}
\item [($\Rightarrow$): ] If $\mathcal{X}$ is isomorphic to a power series space of finite type then $\mathcal{X}$ must be isomorphic to $\Lambda_1(\mathcal{E})$ by Proposition \ref{prop:1.1} of \cite{AKT2}. So $\delta(\mathcal{X}) = \delta(\Lambda_1(\mathcal{E}))$. Moreover $\mathcal{X}$ being a finite type power series space, is tame in view of Remark \ref{remark:3} (4) above.

\item[($\Leftarrow$): ] Let us fix a tame nuclear Fréchet space with stable finitely nuclear associated exponent sequence $\mathcal{E}=(\mathcal{E}_n)$, and which has properties $\underline{DN}$ and $\Omega$. In view of \cite{V2.1}, we can imbed $\mathcal{X}$ into $\Lambda_1(\mathcal{E})$ as a closed subspace. We will do so and consider the grading on $\mathcal{X}$ induced by $(\Lambda_1(\mathcal{E}), \ ^{|\ |_r})$. Throughout this proof we will, as usual, use the notation $\epsilon_n$ to denote the sequence $(0,\dots,0,1,0,\dots)$ where $1$ sits in the $n^{\text{th}}$ place, $n=1,2,\dots$. In view of Theorem \ref{theorem:1} above there exists an $r$-local imbedding from $\Lambda_1(\mathcal{E})$ into $\mathcal{X}$ for every $0\leq r < 1$. Let us fix an $r_0 <1$ and a corresponding $(r_0,r_{k_0})$, $r_{k_0} < 1$, local imbedding $T$, from $\Lambda_1(\mathcal{E})$ into $\mathcal{X}$. Say $|Tx|_{r_{k_0}} \geq C_0 |x|_{r_0}, \ \forall \ x \in \Lambda_1(\mathcal{E})$. Let $e_n \doteq (\epsilon_n/e^{r_0 \mathcal{E}_n}), \ n=1,2,\dots$ be the canonical  orthonormal basis of $\Lambda_{r_0}[\mathcal{E}_n]$, and set $g_n \doteq T(e_n), \ n=1,2,\dots$.

We note that $\{g_n\}_{n=1}^\infty$ is a finitely linearly independent sequence since $T$ is a local imbedding. We choose a sequence $\{f_n\}$ in $\mathcal{X}\subseteq\Lambda_1(\mathcal{E})$ satisfying;
\begin{enumerate}
\item [(1)] $f_n \in \text{span} \{g_1,\dots,g_{2n}\}, \ n=1,2,\dots$.
\item [(2)] $\langle f_n, f_s \rangle_{r_{k_0}} = 0, \ s=1,\dots, n-1; \ n=1,2,\dots$.
\item [(3)] $\langle f_n, \epsilon_k \rangle_{r_{k_0}} = 0, \ k=1,\dots, n; \ n=1,2,\dots$.
\item [(4)] $\langle f_n, f_n \rangle_{r_{k_0}} = 1, \ n=1,2,\dots$.
\end{enumerate}
where $\langle$, $\rangle_{r_{k_0}}$ is the inner product in $\Lambda_{r_{k_0}}[\epsilon]$.

Such a sequence exists and can be selected by induction since for each $n \in \mathbb{N}$, the space $\{g_1, \dots, g_{2n}\}$ is $2n$-dimensional and we impose $2n-1$ conditions for $f_n$.

Hence $f_n = \sum_{i=1}^{2n} c_i^n g_i$ for some scalars $\{c_i^n\}_i, \ n=1,2,\dots$ . We have;
\[
    1 = |f_n|_{r_{k_0}} \geq C_0 \left| \sum_{i=1}^{2n} c_i^n e_i \right|_{r_{0}} = C_0 \left( \sum_{i=1}^{2n} |c_i^n|^2 \right)^{\frac{1}{2}}, \   n=1,\dots
\]
So;
\begin{align}
    \sum_{i=1}^{2n} |c_i^n|^2 \leq \frac{1}{C_0}, \ \forall \ n \in \mathbb{N}.
\end{align}

Now fix an $s$, $0<s<1$, and estimate $|f_n|_s, \ n\in \mathbb{N}$ as:

\begin{align}
    |f_n|_s & = \left| \sum_{i=1}^{2n} c_i^n g_i \right|_s \leq \sum_{i=1}^{2n} |c_i^n| |g_i|_s \leq C_1 \sum_{i=1}^{2n} |c_i| |e_n|_{\sigma_T(s)} \nonumber \\
    & = C_1 \sum_{i=1}^{2n} |c_i^n| \left| \frac{\epsilon_i}{e^{r_0 \mathcal{E}_i}} \right|_{\sigma_T(s)} = C_1 \sum_{i=1}^{2n} |c_i^n| e^{(\sigma_T(s)-r_0)\mathcal{E}_i}
\end{align}
for some $C_1>0$, where $\sigma_T$ is the characteristic of continuity of $T$ with respect to the connonical gradings of $\Lambda_1(\mathcal{E}).$

Choose a $K(s)$, with $\max \{\sigma_T(s), r_0\} < K(s) < 1$, and proceed with the estimate (2.2),

\begin{align*}
    \sum_{i=1}^{2n} |c_i^n| \  & e^{(\sigma_T(s)-r_0)\mathcal{E}_i + K(s)\mathcal{E}_i - K(s)\mathcal{E}_i} \\
    \leq \  &  e^{(K(s)-r_0)\mathcal{E}_{2n}} \left( \sum_{i=1}^{2n} |c_i^n|^2 \right)^{\frac{1}{2}} \left( \sum_{i=1}^{2n} e^{2(\sigma_T(s)-K(s))\mathcal{E}_i} \right)^{\frac{1}{2}} \\
    \leq \  &  C e^{(K(s)-r_0)\mathcal{E}_{2n}},
\end{align*}
to obtain $\forall s<1$, $\exists C=C(S,T)$ and $K(s) < 1$ such that

\begin{align}
    |f_n|_s \leq C e^{(K(s)-r_0)\mathcal{E}_{2n}}, \quad n=1,2,\dots
\end{align}

We choose an increasing sequence $\{ K^+(s)\}_s$ with $K(s)<K^+(s)<1, \ \forall s<1$.

On the other hand we also have, for each $n \in \mathbb{N}$, a representation of $f_n$ as;
\[
    f_n = \sum_{k=n+1}^\infty \beta_k^n \epsilon_k
\]
for some sequence $\{\beta_k^n\}_k$. For an $-\infty<r<r_{k_0}$ we estimate;
\begin{align}
    |f_n|_r^2 & = \sum_{s=n+1}^\infty |\beta_s^n|^2 e^{2r\mathcal{E}_s} = \sum_{s=n+1}^\infty |\beta_s^n|^2 e^{2r_{k_0}\mathcal{E}_s} e^{2(r-r_{k_0})\mathcal{E}_s} \nonumber \\
    & \leq e^{2(r-r_{k_0})\mathcal{E}_{n+1}} |f_n|^2_{r_{k_0}} = e^{2(r-r_{k_0})\mathcal{E}_{n+1}}, \quad n=1,2,\dots.
\end{align}

In view of stability of $\{\mathcal{E}_n\}$ there exists a $C_0>0$ such that $C_0 \mathcal{E}_{2n} \leq \mathcal{E}_{n+1}, \ \forall n=1,2,\dots$. Hence proceeding with the estimate (2.4) we have;

\begin{align}
    |f_n|_r^2 \leq e^{2C_0(r-r_{k_0})\alpha_n} \quad \forall n \in \mathbb{N} \text{ and } -\infty<r<r_{k_0}.
\end{align}

We now choose and fix an $s_0$ with $-\infty < s_0 < -2/c_0$. In view of (2.5) we have;
\begin{align}
    |f_n|_{s_0} \leq e^{(C_0 s_0 - C_0 r_{k_0} + r_0)\mathcal{E}_{2n}} e^{-r_0\mathcal{E}_{2n}} \leq e^{-\mathcal{E}_{2n}} e^{-r_0\mathcal{E}_{2n}} \quad n=1,2,\dots.
\end{align}

We stress that $s_0$ depends only on the associated exponent sequence $\mathcal{E}$. Now for a sequence $\{\lambda_n\}_{n=1}^\infty$ and $s<1$, (2.3) above gives:
\[
    \sum_{i=1}^\infty |\lambda_i| |f_i|_s e^{r_0 \mathcal{E}_{2n}} \leq C \left( \sum |\lambda_i|^2 e^{2K^+(s)\mathcal{E}_{2n}} \right)^{\frac{1}{2}}
\]
for some $C=C(s)$.

It follows that the assignment that sends $\epsilon_n$ to $f_n e^{r_0 \mathcal{E}_{2n}}, \ n=1,2,\dots$, defines a continuous linear operator $\widehat{T}$ from $\Lambda_1(\mathcal{E}_{2n})$ into $\mathcal{X}$ that satisfies;

\begin{align*}
    |\widehat{T}(x)|_{r_{k_0}} = \left|\widehat{T}\left( \sum^\infty_{i=1} x_i \epsilon_i \right)\right|_{r_{k_0}} & = \left|\sum^\infty_{i=1} x_i f_i e^{r_0 \mathcal{E}_{2i}}\right|_{r_{k_0}} \\
    & = \left( \sum^\infty_{i=1} |x_i|^2 e^{2 r_0 \mathcal{E}_{2i}} \right)^\frac{1}{2} \\
    & = |x|_{r_0}, \quad \forall x \in \Lambda_1(\mathcal{E}_{2n}).
\end{align*}

Hence $\widehat{T}$ is an $(r_0, r_{k_0})$-local isomorphism from $\Lambda_1(\mathcal{E}_{2n})$ into $\mathcal{X}$.

Moreover $\widehat{T}$ has the additional property that it extends to be continuous from $\Lambda_0[(\mathcal{E}_{2n})]$ into $\Lambda_{s_0}[(\mathcal{E}_{n})]$. Indeed in view of (2.6), for $x \in \Lambda_{0}[(\mathcal{E}_{n})]$ we have;

\begin{align*}
    |\widehat{T}(x)|_{s_0} = \left|\widehat{T}\left( \sum_i x_i \epsilon_i \right)\right|_{s_0} & = \left| \sum^\infty_{i=1} x_i f_i e^{r_0 \mathcal{E}_{2i}} \right|_{s_0} \\
    & \leq \sum^\infty_{i=1} |x_i| |f_i|_{s_0} e^{r_0 \mathcal{E}_{2i}} \leq \sum^\infty_{i=1} |x_i| e^{-\mathcal{E}_{2i}} \\
    & \leq \left( \sum^\infty_{i=1} e^{-2 \mathcal{E}_{2i}} \right)^\frac{1}{2} |x|_0.
\end{align*}

We now vary $r<1$, and obtain a family $\{\widehat{T}_r\}$ of $(r,r_{k_r})$-local imbeddings from $\Lambda_1\big((\mathcal{E}_{2n})\big)$ into $\mathcal{X}$ with the additional property mentioned above.

Using the elementary inequality
\[
    |\ |_{t_2} \leq |\ |_{t_1}^{\frac{t_3-t_2}{t_3-t_1}} |\ |_{t_3}^{\frac{t_2-t_1}{t_3-t_1}} ,
\]
for a given $t_1<t_2<t_3$, which is valid in any power series space, for any $s_0<s<1$ we choose a $\rho(s)<1$ such that
\begin{align}
    \forall \ 0\leq r < 1 \ \exists C > 0: \quad |\widehat{T}_r(x)|_s \leq C |x|_{\rho(s)}
\end{align}
After these preparations we now proceed to show that $\Big(\mathcal{X}, \Lambda_1\big((\mathcal{E}_{2n})\big)\Big) \in \mathcal{T}$.

According to our assumption, there exists a sequence $\{S_\alpha\}_{\alpha=1}^\infty$ of increasing functions from $\mathbb{N}$ into $\mathbb{N}$, such that $\forall T \in \mathcal{L}(\mathcal{X}, \mathcal{X})$ and $\exists \alpha \in \mathbb{N}$ such that $\sigma(T)\leq S_\alpha$.

Let now $S$ be a given continuous linear operator from $\mathcal{X}$ into $\Lambda_1\big(\mathcal{E}_{2n})\big)$. In view of (2.7) above, the family, $\{\widehat{T}_r \circ S\}_{r<1}$ of continuous linear operators from $\mathcal{X}$ into $\mathcal{X}$ satisfy;
\[
    \sigma_{\widehat{T}_r \circ S} \leq \sigma_S \circ \rho.
\]
Hence this family is in $F \doteq \{U \in \mathcal{L}(\mathcal{X}, \mathcal{X}): \quad \sigma_U \leq \sigma_{S} \circ \rho \}$. On $F$ we consider the topology coming from the semi-norms $\{\|\cdot\|_s^{\sigma_{S} \circ \rho(s)}\}_{s=1}^\infty$, and on $\mathcal{L}(\mathcal{X}, \mathcal{X}) = \bigcup_{\alpha=1}^\infty \bigcap_{s=1}^\infty \{U \in \mathcal{L}(\mathcal{X}, \mathcal{X}): \ \|U\|_s^{S_\alpha(s)} < \infty \}$, we consider the LF-space structure as explained in Remark \ref{remark:3} (4) above. Since evaluation at points of $\mathcal{X}$ are continuous in both $F$ and $\mathcal{L}(\mathcal{X}, \mathcal{X})$, the inclusion $F \subseteq \mathcal{L}(\mathcal{X}, \mathcal{X})$ has a sequentially closed graph. It follows that there exists $\alpha$ such that $F \subseteq \bigcap_{s=1}^\infty \{U \in \mathcal{L}(\mathcal{X}, \mathcal{X}): \ \|U\|_s^{S_{\alpha(s)}} < \infty\}$ in view of Grothendieck factorization theorem \cite[p.~68]{Kothe2}. It follows that there exists an $\alpha \in \mathbb{N}$ such that;
\[
    \sigma_{\widehat{T}_r \circ S} \leq S_\alpha \quad \forall r<1.
\]
In particular for each $r<1$ there exists a $\widehat{C}>0$ such that;
\begin{align}
    |S x|_r \leq C \left|\widehat{T}_r(Sx)\right|_{r_{k_r}} \leq \widehat{C} |x|_{S_\alpha (r_{k_r})}.
\end{align}
Now if we set $\widehat{S}_\alpha (r) \doteq S_\alpha(r_{k_r})$, $r<1$, $\alpha \in \mathbb{N}$, the analysis above shows that $\forall S \in \mathcal{L}\Big(\mathcal{X}, \Lambda_1\big((\mathcal{E}_{2n})\big)\Big) \  \exists \ n \in \mathbb{N}$ such that $\sigma_S \leq \widehat{S}_\alpha$. Hence it follows that $\Big(\mathcal{X}, \Lambda_1\big((\mathcal{E}_{2n})\big)\Big) \in \mathcal{T}$.

Now Theorem 11 of \cite{PISZCZEK} implies that $\mathcal{X}$ satisfies the strong $\Omega$ condition, $\bar{\Omega}$, of Vogt. This together with our assumption that, $\mathcal{X}$ has the property $\underline{DN}$ allows us to conclude that $\mathcal{X}$ is isomorphic to a finite type power series space (Proposition 2.9.18 of \cite{M-V}). Hence $\mathcal{X} \cong \Lambda_1 (\mathcal{E})$ (Proposition 1.1 of \cite{AKT2}).
\end{enumerate}
\end{proof}

\vspace{0.5cm}

\section{Spaces of Analytic Functions}

In this section we will focus our attention to a particular class of nuclear Fréchet spaces with the properties $\underline{DN}$ and $\Omega$, namely the spaces of analytic functions on Stein manifolds. Stein manifolds, being closed connected submanifolds of complex euclidean spaces $\mathbb{C}^N$, possess a rich supply of analytic functions. These spaces, with the usual topology of uniform convergence on compact subsets, form an important subclass of Fréchet spaces with the properties $\underline{DN}$ and $\Omega$. The linear topological properties of $\mathcal{O}(M)$, the Fréchet space of analytical functions on a Stein manifold $M$, and the complex analytic properties of $M$ that are reflected on $M$ by the type of $\mathcal{O}(M)$, have been studied by several authors (see \cite{AYI} \cite{Zah1} and the references therein). In this context, we show, in subsection \ref{subs:3.1} that local imbedding of finite type power series spaces into $\mathcal{O}(M)$ can be used to construct Green's functions in $M$. In subsection \ref{subs:3.2} we classify Stein manifolds $M$, for which $\mathcal{O}(M)$ is tame. Some results and concepts from Pluripotential Theory will be used in the sequel. For undefined terminology and background in these matters we refer the reader to \cite{KLIMEK}.

\subsection{Local imbeddings of finite type power series spaces into $\mathcal{O}(M)$ and Green's functions} \label{subs:3.1}

\

Let $M$ be a complex manifold and fix a point $z_0 \in M$. We will use the symbol $PSH(M)$ to denote the set of all plurisubharmonic functions on $M$. Employing norms in a local chart centered at $z_0$ we consider
\[
    \mathcal{L}_{z_0} \doteq \{u \in PSH(M), \ u \leq 0 \text{ and } u(z)-\ln\|z-z_0\| \text{ is bounded near } z_0\}
\]
and set
\[
    g(\xi, z_0) \doteq \sup\{u(\xi): \ u \in \mathcal{L}_{z_0}\} .
\]
This assignment, if not $\equiv - \infty$, defines a plurisubharmonic function on $M$. We will call $g(\cdot z_0)$ the \emph{Green's function} of $M$ with pole at $z_0$, and say that Green's function with pole at $z_0$ exists in case $g(\cdot, z_0)$ is not identically equal to $-\infty$. If $M$ is parabolic i.e. if it does not possess a nonconstant bounded plurisubharmonic function then of course, no Green's function exist. In one variable nonparabolicity characterizes existence of Green's functions, \cite{S-N}, however in several complex variables there is no such general result. The difficulty seems in constructing a negative plurisubharmonic function with a pole at the given point from the class of negative plurisubharmonic functions.

\begin{prop} \label{prop:3.1}
{\em
Let $M$ be a Stein manifold of dimension $d$ and let $z_0$ be a point in $M$. If there exists a local imbedding from $\Lambda_1(\alpha)$ into $\mathcal{O}(M)$, for some finitely nuclear exponent sequence $\{\alpha_n\}_n$, with $\underline{\lim} \  n^{1/d}/\alpha_n > 0$, then Green's function with a pole at $z_0$ exists.
}
\end{prop}

\begin{proof}
Let us fix $M$ and the point $z_0$. In view of \cite{S-F} there exists a local biomorphism $\Phi: \ \Delta^d_e \twoheadrightarrow M$, from the polydisc with centre $0$ and radius $e$ of $\mathbb{C}^d$ onto $M$ such that $\Phi(0)=z_0$. We will use $\Phi$ to imbed $\mathcal{O}(M)$ into $\mathcal{O}(\Delta^d_e)$ via the composition operator $f \mapsto f\circ \Phi$, $f \in \mathcal{O}(M)$ (\cite{AT1}). We will choose a bijection $\rho: \mathbb{N} \rightarrow \mathbb{N}^d$ such that $|\rho(n)|$ is strictly increasing with $n$ and is ordered lexicographic on the level sets of $|\rho(\cdot)|$, and use it to define an isomorphism between $\Lambda_1(n^{1/d})$ and $\mathcal{O}(\Delta^d_e)$ by the correspondence $\epsilon_i \rightarrow z^{p(i)} = z_1^{p_1(i)} \dots z_d^{p_d(i)}, \ i=1,2,\dots$. We note that there exists constants $\beta_1 > 0$ and $\beta_2>0$ such that $\beta_1(\rho(n)) \leq n^{1/d} \leq \beta_2(\rho(n)), \ n=1,2,\dots$ (see e.g. \cite{R}).

Let us fix an $(r_0, k_0)$-local imbedding, $T$, from $\Lambda_1(\alpha_n)$ into $\Lambda_1 (n^{1/d})$, $T(\Lambda_1(\alpha_n)) \subseteq \mathcal{O}(M) \subseteq \Lambda_1(n^{1/d})$, which exists in view of our assumptions. Let
\[
f_n \doteq T \left(\frac{\epsilon_n}{e^{r_0\alpha_n}}\right), \ n=1,2,\dots.
\]
The sequence $\{f_n\}_n$ is finitely linearly independent, so we can, choose a sequence $\{g_n\}^\infty_{n=1}$ of elements of $\mathcal{O}(M)$ with
\begin{enumerate}
\item $g_n \in \text{span} \{f_1,\dots,f_n\}, \ n=1,2,\dots$
\item $\langle g_n, \epsilon_i \rangle_{k_0}$, $i=1,\dots,n-1$ where $\langle$, $\rangle_{k_0}$ is the inner product in $\Lambda_{k_0}[n^{1/d}] \  \forall \ n=1,2,\dots$
\item $|g_n|_{k_0} = 1$, $n=1,2,\dots$
\end{enumerate}
Note that if $g_n = \sum^n_{k=1} c_k^n f_k$ then we estimate, as in the above arguments;
\begin{align*}
    \sum^n_{k=1} |c_k^n|^2 = \sum^n_{k=1} \left| \frac{c_k^n}{e^{2r_0\alpha_k}} \right|^2 e^{2r_0\alpha_k} = \left| \sum^n_{k=1} c_k^n \frac{\epsilon_k}{e^{2r_0\alpha_k}}\right|^2_{r_0} & \leq C \left| T \left( \sum^n_{k=1} c_k^n \frac{\epsilon_k}{e^{2r_0\alpha_k}}\right) \right|^2_{k_0} \\
    & = C |g_n|^2_{k_0} \leq C,
\end{align*}
for some $C>0$ and for all $n=1,2,\dots$.

Hence, for a given $r<1$;
\begin{align} \label{eqn:3.4}
|g_n|_r & = \left| \sum_{i=1}^n c_i^n f_i \right|_r \leq C_1 \sum_{i=1}^n |c_i^n| \left| \frac{\epsilon_i}{e^{r_0\alpha_n}} \right|_{\sigma_T(r)} \nonumber \\
& = \sum_{i=1}^n |c_i^n| e^{(\sigma_T(r)-r_0)2n} \leq C_2 e^{\rho(r)\alpha_n}, \quad n=1,2,\dots
\end{align}
and for some constants $C_1>0$, $C_2>0$ where $\sigma_T(r)-r_0 \leq \rho(r) < 1$ is a chosen number that depends on $r$ (and $T$).

On the other hand in view of equation \eqref{eqn:3.2} above, each $g_n,\ n=1,2,\dots$ has an expansion as \[
    g_n = \sum_{i \geq n} d_i^n \epsilon_i \leftrightarrow \sum_{i \geq n} d_i^n z^{\rho(i)}
\]
in $\Lambda_1(n^{1/d})$ and in $\mathcal{O}(\Delta^d_e)$, respectively. By abuse of notation we will think of $g_n \in \mathcal{O}(M) \subseteq \mathcal{O}(\Delta_e^d), \ n=1,2,\dots$ as an analytic function on $\Delta_e^d$ with Taylor series;
\begin{align} \label{eqn:3.5}
    g_n(z) = \sum_{|t|\geq \frac{1}{\beta_2} n^{1/d}} \alpha_t^n z_1^{t_1} \dots z_d^{t_d}, \quad n=1,2,\dots.
\end{align}

Choose $h_n \in \mathcal{O}(M)$ such that $g_n = h_n \circ \Phi, \ n=1,2,\dots$. For a given compact set $K \subseteq M$ choose $r<1$ so that $\Phi(\Delta^d_{e^r}) \supseteq K$. So in view of equation \eqref{eqn:3.4};
\[
    \sup_{z \in K} |h_n(z)| \leq \sup_{\xi \in \Delta_{e^r}^d} |g_n(\xi)| \leq C |g|_{r^+} \leq C C_2 e^{\rho(r^+)\alpha_n}
\]
for some constant $C>0$ and a choice of $r<r^+<1$.

Hence the function
\[
    u(z) \doteq \overline{\lim_{\xi \rightarrow z}}\ \overline{\lim}_n \frac{\ln |h_n(\xi)|}{\alpha_n},\quad z \in M
\]
defines a plurisubharmonic function on $M$ that is bounded by $1$. This plurisubharmonic function is not $\equiv -\infty$. To see this let us assume, with the anticipation of getting a contradiction that it is identically equal to $-\infty$. Fix large $k_0<r<1$ such that $|x|_{k_0} \leq C \sup_{z\in \overline{\Delta}_{e^r}^d} |x(z)|, \ \forall x \in \mathcal{O}(\Delta^d_e)$ and choose a compact set $K \subset M$ that contains $\Phi(\overline{\Delta}_{e^r}^d)$.

Our assumption and Hartogs theorem (\cite[p.~70]{KLIMEK}) gives for each $N\in \mathbb{N}$, an $n_0 \in \mathbb{N}$ such that:
\[
    \sup_{z \in K} |h_n(z)| \leq e^{-N\alpha_n}, \quad n\geq n_0.
\]

This in turn gives that for some $C>0$
\[
    1 = |g_n|_{r_0} \leq C e^{-N\alpha_n}, \quad \text{ for } n \geq n_0.
\]
So $u$ is not identically $-\infty$.

Now consider a $z \in M$ near $z_0$, say with $\|z-z_0\| = e^r$ for some very large and negative $r$. Since $\Phi$ is a local biholomorphism, there exists a $C_0>0$, independent of $r$, and $\xi \in \Delta_e^d$ with $\|\xi\| < C_0 e^r$ such that $\phi(\xi) = Z$. Using \eqref{eqn:3.5} we estimate, with, $k_0^- < k_0$ and $C_1>0$,
\begin{align*}
    |g_n(\xi)| & \leq \sum_{|t| \geq C n^{1/d}} |\alpha_t^n| C_0^{|t|} e^{r|t|} e^{-k_0^-} |t| e^{k_0^- |t|} \\
    & \leq C_0 e^{C(r-k_0^-)n^{1/d}} \left(\sum_{|t|} |\alpha_t^n|^2 e^{2k_0 |t|}\right)^{1/2} \leq C_1 e^{C(r-k_0^-)n^{1/d}}, \quad n=1,2,\dots
\end{align*}

Hence our assumption on $(\alpha_n)$ implies $\exists \  C_2>0$, $C_3>0$:
\[
    \overline{\lim}_n \frac{\ln |h_n(z)|}{\alpha_n} \leq \overline{\lim}_n C(r-k_0^-) \frac{n^{1/d}}{\alpha_n} \leq C_2 \ln \|z-z_0\|+C_3.
\]
So $\frac{u}{C_2}$ has a logarithmic singularity at $z_0$, and is a bounded plurisubharmonic function. It follows that $g_m(\cdot,z_0)$ is not identically $-\infty$.
\end{proof}

\subsection{Diametral dimensions of analytic function spaces} \label{subs:3.2}

\

In this subsection we will investigate the diametral dimension of the spaces of analytic functions on Stein manifolds. These invariants for Fréchet spaces are in a sense dual to approximate diametral dimensions, however they are more extensively studied.

For a nuclear Fréchet space $\mathcal{X}$ with a neighborhood basis of $0$, $\{U_p\}_{p=1}^\infty$, consisting of discs, $\Delta(\mathcal{X})$, is defined by (in the notation of Section 1):
\begin{align*}
    \Delta(\mathcal{X}) & = \{(t_n);\ \forall p \ \exists q:\quad t_n d_n(U_q,U_p)\rightarrow 0\}. \\
                        & = \{(t_n);\ \forall p \ \exists q:\quad t_n e^{-\mathcal{E}_n(p,q)}\rightarrow 0\}.
\end{align*}
As the notation suggests, it is easy to see that, this sequence space does not depend upon the neighborhood basis chosen and is an invariant of the Fréchet space $\mathcal{X}$.

In this subsection we will use a spacial generating norm-system for the analytic function spaces, unless stated otherwise. To describe these norms, suppose a Stein manifold $M$ of dimension $d$ is given. We choose a strictly plurisubharmonic $C^\infty$ exhaustion function, $\rho$, of $M$ and consider a sequence $r_p \uparrow \infty$ such that the sub-level sets $D_p \doteq \{z \in M; \ \rho(z)<r_p\},\ p=1,2,\dots$, are strictly pseudoconvex. We set $K_p\doteq \bar{D}_p, \ p=1,2,\dots$. Following (\cite{Zah2}, c.f. \cite{Zah1}) we choose for each $p$, a Hilbert space, $H_p$, that satisfies;
\[
    A(K_p) \hookrightarrow H_p \hookrightarrow AC(K_p) \hookrightarrow \mathcal{O}(D_p)
\]
where $A(K_p)$ denotes the germs of analytic functions on the compact set $K_p$ with the inductive limit topology, $AC(K_p)$ denotes the Banach space that is the closure of $A(K_p)$ in $C(K_p)$, the Banach space of continuous functions on $K_p$ with the sup-norm, and the $\hookrightarrow$'s are imbeddings with dense range, $p=1,2,\dots$. The norms that we will use to generate the topology $\mathcal{O}(M)$ will be the hilbertian norms of $H_p$'s, $\{\|\ \|_p\}_p$. Their corresponding unit balls in $\mathcal{O}(M)$ will be denoted by $U_p$, $p=1,2,\dots$. The primary reason of our usage of these semi-norms is because of the beautiful formula of Nivoche, Poletsky, Zaharyuta:

\begin{align} \label{eqn:3.1}
\forall p<q \quad \lim_n \frac{\mathcal{E}_n(p,q)}{\left(\frac{2\pi d! n}{\tilde{C}(K_p,D_q)}\right)^\frac{1}{d}} = 1
\end{align}
where $\tilde{C}(K_p,D_q)=\sup\{\int_{K_p} (d d^c u)^n:\ u \in PSH(D_q)-1\leq u \leq 0\}$ (Proposition 4.6, 4.8, 4.12 on \cite{Zah2}). We refer the reader to \cite{Zah2} and \cite{Zah1} for a proof and a discussion about the history of this formula.

An immediate consequence of equation \eqref{eqn:3.1} is;

\begin{prop} \label{prop:3.2}
Let $M$ be a Stein manifold of dimension $d$. Then
\[
    \Delta(\mathcal{O}(M)) \equiv \{(t_n):\ \forall p \ \exists q \text{ such that } \sup_n |t_n| e^{-\alpha_n/c(p,q)} < \infty\}
\]
where for $p<q$, $c(p,q) \doteq \tilde{C}(K_p,D_q)^\frac{1}{d}$, and $\alpha_n \doteq (2\pi d!)^\frac{1}{d} n^\frac{1}{d}, \ n=1,2,\dots$.
\end{prop}

\begin{proof}
\
\begin{enumerate}
\item [($\subseteq$): ] Choose a $(t_n)_n \in \Delta(\mathcal{O}(M))$. Fix $p$ and choose $q$ such that
\[
    \sup_n |t_n| e^{-\mathcal{E}_n(p,q)} < +\infty.
\]
Now in view of the (2') of Proposition \ref{prop:1.1} there exists a $q^+$, $C>0$ and $0<\lambda<1$ such that;
\[
    \mathcal{E}_n(p,q) \leq (1-\lambda) \mathcal{E}_n (p,q^+)+C.
\]

For $\varepsilon>0$ such that $(1+\varepsilon) < \frac{1}{1-\lambda}$ in view of \eqref{eqn:3.1} there exists an $N$ such that
\[
    (1-\varepsilon)\frac{\alpha_n}{C(p,q^+)} \leq \mathcal{E}_n(p,q^+) \leq (1+\varepsilon)\frac{\alpha_n}{C(p,q^+)}, \quad n \geq N.
\]
Hence we have;
\begin{align*}
    |t_n| \leq C_1 e^{\mathcal{E}_n(p,q)} & \leq C_2 e^{(1-\lambda)\mathcal{E}_n(p,q^+)} \\ & \leq e^{(1-\lambda)(1+\varepsilon)\frac{\alpha_n}{C(p,q^+)}} \leq e^\frac{\alpha_n}{C(p,p^+)}, \quad n \geq N.
\end{align*}

\item [($\supseteq$): ] Choose a sequence $(t_n)$ from the right hand side. For a fixed $p$, choose a $q$ such that $\sup_n |t_n| e^{-\alpha_n/c(p,q)} < \infty$. By (2') of Proposition \ref{prop:1.1} choose $k$, $0<\rho<1$ and $C>0$ so that $\mathcal{E}_n(p,q) \leq (1-\rho)\mathcal{E}_n (p,k)+C$. Let $\varepsilon>0$ be so that $1-\rho < 1-\varepsilon$ and choose an $N$ (from \eqref{eqn:3.1})
\[
    \frac{-(1+\varepsilon)\alpha_n}{C(p,q)} \leq -\mathcal{E}_n(p,q) \leq -\frac{(1-\varepsilon)\alpha_n}{C(p,q)}, \quad n \geq N
\]
Hence
\[
    |t_n|^{(1-\varepsilon)} e^{-\mathcal{E}_n(p,q)} \leq \left( |t_n| e^{\frac{-\alpha_n}{C(p,q)}} \right)^{1-\varepsilon} \leq Q < \infty.
\]
for some $Q>0$. Moreover since $-\mathcal{E}_n(p,q) \geq -(1-\varepsilon)\mathcal{E}_n(p,k)-C$ we have;
\[
    \sup_n\left( |t_n| e^{-\mathcal{E}_n(p,k)} \right)^{1-\varepsilon} \leq Q e^C \Rightarrow \sup_n |t_n| d_n(U_k,U_p) < \infty.
\]
It follows that $(t_n) \in \Delta(\mathcal{O}(M))$.
\end{enumerate}
\end{proof}

\begin{cor} \label{corollary:3.2}
Let $M$ be a Stein manifold of dimension $d$. Then \[
    \Delta(\mathcal{O}(M)) = \Delta(\Lambda_1(n^\frac{1}{d})) \Leftrightarrow \inf_p \sup_{q\geq p} \frac{1}{C(p,q)} = 0 \Leftrightarrow \delta(\mathcal{O}(M)) = \delta(\Lambda_1(n^\frac{1}{d})).
\]
\end{cor}

\begin{proof}
Suppose that $\Delta(\mathcal{O}(M)) = \Delta(\Lambda_1(n^\frac{1}{d}))$. Suppose that there is an $\varepsilon>0$ such that $\inf_p \sup_{q\geq p} \frac{1}{c(p,q)} > \varepsilon > 0$. In view of equation \eqref{eqn:3.1} this implies,
\[
    \inf_p \sup_{q\geq p} \lim_n \frac{\mathcal{E}_n(p,q)}{n^\frac{1}{d}} > \varepsilon(d!2\pi)^\frac{1}{d}
\]
Setting $\kappa \doteq \varepsilon(d!2\pi)^\frac{1}{d}$, we have;
\[
    \forall p \ \exists q \text{ and } N \text{ such that } \frac{1}{d_n(U_q,U_p)} \geq e^{\kappa n^\frac{1}{d}}, \quad n \geq N.
\]
Hence,
\[
    e^{\kappa n^\frac{1}{d}} \in \Delta(\mathcal{O}(M)) = \Delta(\Lambda_1(n^\frac{1}{d})) = \{(\xi_n):\ \forall\ r<1, \quad \lim_n |\xi_n| r^{n^{1/d}} = 0\}.
\]
This contradiction shows that $\inf_p \sup_{q\geq p} 1/c(p,q) = 0$.

On the other hand if $\inf_p \sup_{q\geq p} 1/c(p,q) = 0$, and $r<1$ is given, choose a $p$ such that, in the notation of Proposition \ref{prop:3.1},
\[
    \sup_{q\geq p} \frac{\alpha_n}{c(p,q)n^\frac{1}{d}} \leq -\ln r \quad \forall n.
\]
In view of Proposition \ref{prop:3.1}, for $p$ there is a $q>>p$ such that $\sup_n |t_n| e^{\frac{-\alpha_n}{c(p,q)}} < \infty$. Hence $\sup_n |t_n| r^{n^\frac{1}{d}} < \infty$. This shows that
\[
(t_n) \in \{(\xi_n):\ \forall r<1, \ \lim_n|\xi_n|r^{n^\frac{1}{d}} = 0\} = \Delta(\Lambda_1(n^\frac{1}{d})).
\]
So $\Delta(\mathcal{O}(M)) \subseteq \Delta(\Lambda_1(n^\frac{1}{d}))$.

On the other hand, $(n^\frac{1}{d})_n$, being the associated exponent sequence of $\mathcal{O}(M)$, always satisfies $\Delta(\Lambda_1(n^\frac{1}{d})) \subseteq \Delta(\mathcal{O}(M))$ (Proposition 1.1, \cite{AKT2}). So we have $\Delta(\Lambda_1(n^\frac{1}{d})) =\Delta(\mathcal{O}(M))$. Since $\mathcal{O}(M)$ is isomorphic to a closed subspace of $\Lambda_1(n^{1/d})$, the other equivalence directly follows from Proposition \ref{prop:1.4} and the fact that approximate diametral dimension of a nuclear Fréchet space is greater than the approximate diametral dimension of its subspaces (\cite{BPR}).
\end{proof}

\begin{thm} \label{theorem:3}
Let $M$ be a Stein manifold of dimension $d$. Then either $\Delta(\mathcal{O}(M)) = \Delta(\mathcal{O}(\Delta^d))$ or $\Delta(\mathcal{O}(M)) = \Delta(\mathcal{O}(\mathbb{C}^d))$.
\end{thm}

\begin{proof}
    Let us fix a Stein manifold $M$ with dimension $d$. We will use, in what follows, the grading on $\mathcal{O}(M)$ described in the beginning of the section. Using the notation above, for a given $p\geq 1$, we set, $c(p) \doteq \lim_{q>p} c(p,q) = \inf_{q>p} c(p,q)$. Since $c(\cdot,\cdot)$ is increasing in the first variable, the sequence $\{c(p)\}$ is an increasing sequence. We have two cases:
\begin{enumerate}
\item [1.] $c(p)$ is zero for all $p\geq 1$.
    Suppose this is the case. Let us choose a $(t_n)_n \in \Delta(\mathcal{O}(\mathbb{C}^d))=\Delta(\Lambda_\infty(n^\frac{1}{d}))$, and choose an $R>1$ and $C>0$, so that $|t_n| \leq C R^{n^\frac{1}{d}}, \ \forall n$. For a given $p$ let us choose a $q$ such that $c(p,q) \leq \frac{1}{\ln (\tilde{R})}$ for some $\tilde{R}$ with $R^{1/(2\pi d!)^\frac{1}{d}} << \tilde{R}$. Then
\[
    |t_n| e^{-\frac{\alpha_n}{c(p,q)}} \leq |t_n| e^{-\alpha_n \ln \tilde{R}+ n^{1/d}\ln R -n^{1/d}\ln R} \leq C.
\]
So $(t_n)_n \in \Delta(\mathcal{O}(M))$ in view of Proposition \ref{prop:3.1}. Hence
\[
\Delta(\Lambda_\infty(n^{1/d})) \subseteq \Delta(\mathcal{O}(M)).
\]
However, $\{n^\frac{1}{d}\}_n$, being the associated exponent sequence of $\mathcal{O}(M)$, the inclusion $\Delta(\mathcal{O}(M)) \subseteq \Delta(\Lambda_\infty(n^{1/d}))$ is always true. It follows that in this case $\Delta(\mathcal{O}(M)) = \Delta(\mathcal{O}(\mathbb{C}^d))$.

\item [2.] The sequence $\{c(p)\}$ increases to a non-zero number $c$.
Suppose this is the case. We have two possibilities either $c\in \mathbb{R}$ or $c=\infty$.

Let us suppose that $c>0$ is a real number. Fix natural numbers $\alpha<\gamma<\beta$. For given natural numbers $n$ and $m$, plainly $d_{n+m}(U_\beta,U_\alpha) \leq d_n(U_\beta,U_\gamma)d_m(U_\gamma,U_\alpha)$, where $U_t$'s, $1\leq t < \infty$, are the unit balls corresponding to our grading. In particular we have
\begin{align} \label{eqn:3.2}
\mathcal{E}_{n+m}(\alpha,\beta) \geq \mathcal{E}_n(\gamma,\beta) + \mathcal{E}_m(\alpha,\gamma); \quad n,m\geq 1
\end{align}

Fix an $\varepsilon >0$ and using \eqref{eqn:3.1}, choose $N=N(\alpha,\gamma,\beta)$ such that
\begin{align*}
    (1-\varepsilon) \frac{\alpha_{n+m}}{c(\alpha,\beta)} & \leq \mathcal{E}_{n+m}(\alpha,\beta) \leq (1+\varepsilon) \frac{\alpha_{n+m}}{c(\alpha,\beta)} \\
    (1-\varepsilon) \frac{\alpha_{n}}{c(\gamma,\beta)} & \leq \mathcal{E}_{n}(\gamma,\beta) \leq (1+\varepsilon) \frac{\alpha_{n}}{c(\gamma,\beta)} \\
    (1-\varepsilon) \frac{\alpha_{m}}{c(\alpha,\gamma)} & \leq \mathcal{E}_{m}(\alpha,\gamma) \leq (1+\varepsilon) \frac{\alpha_{m}}{c(\alpha,\gamma)}, \text{ for } n,m \geq N
\end{align*}
Hence we have
\[
    (1-\varepsilon) \frac{m^\frac{1}{d}}{c(\alpha,\gamma)} + (1-\varepsilon) \frac{n^\frac{1}{d}}{c(\gamma,\beta)} \leq (1+\varepsilon) \frac{(n+m)^\frac{1}{d}}{c(\alpha,\beta)}
\]
for $n,m \geq N$, taking $m=n$, and $n>N$, we get, after cancellation:
\[
     \frac{(1-\mathcal{E})}{c(\alpha,\gamma)} +  \frac{(1-\mathcal{E})}{c(\gamma,\beta)} \leq  \frac{(1+\mathcal{E}) 2^{1/d}}{c(\alpha,\beta)}
\]
which upon letting first $\beta$ then $\gamma$ and $\alpha$ to go to infinity, respectively, gives $2 \leq 2^{1/d}$. Hence for $d>1$, $c$ must be $\infty$.

For $d=1$, we will use the exhaustion given in \cite[p.~145]{AS} and an associated fundamental hilbertian norm system as explained in the beginning of the section. In this context we will use the \emph{modulus inequality} as given in \cite[p.~14]{S-N}, which in our notation states that for $1\leq p<q<s<\infty$,
\begin{align} \label{eqn:3.3}
    \frac{1}{c(p,s)} \geq \frac{1}{c(p,q)} + \frac{1}{c(q,s)}
\end{align}

First letting $s\rightarrow \infty$ and then $q \rightarrow \infty$ we see that $1/(c(q))$ approaches to $0$. Hence in this case we conclude that $c=\infty$, as well. So the first possibility does not occur and we conclude that
\[
    \lim_p \frac{1}{c(p)} = \inf_p \sup_{q \geq p} \frac{1}{c(p,q)} = 0.
\]
The theorem now follows from Corollary \ref{corollary:3.2}.
\end{enumerate}
\end{proof}

Now we turn our attention to tameness in the spaces of analytic functions. Recall that a Stein manifold $M$ is called \emph{hyperconvex} in case it possesses a bounded plurisubharmonic exhaustion function. We refer the reader to \cite{KLIMEK} and the references therein for an account of hyperconvex manifolds. From a functional analysis point of view, hyperconvex Stein manifolds $M$ are precisely those Stein manifolds that satisfy, $\mathcal{O}(M) \approx \mathcal{O}(\Delta^d)$, $d=\text{dim}M$, (\cite{ZahH,AytM}). Hence for hyperconvex manifold $M$, $\mathcal{O}(M)$ is a tame Fréchet space. Our next and final result tells us that they are the only ones having this property:

\begin{mainthm} \label{theorem:3.3}
    Let $M$ be a Stein manifold. Then $\mathcal{O}(M)$ is tame if and only if $M$ is hyperconvex.
\end{mainthm}

\begin{proof}
Let us fix a Stein manifold $M$ with dimension $d$. In view of the remarks preceding the theorem, it suffices to show that if $\mathcal{O}(M)$ is tame then $\mathcal{O}(M) \approx \mathcal{O}(\Delta^d)$. In view of Theorem \ref{theorem:3} either $\Delta(\mathcal{O}(M)) = \Delta(\mathcal{O}(\mathbb{C}^d))$ or $\Delta(\mathcal{O}(M)) = \Delta(\mathcal{O}(\Delta^d))$. The first case cannot occur. To see this first observe that the assumption on the diametral dimension implies that $\mathcal{O}(M)$ contains a complemented copy of $\mathcal{O}(\mathbb{C}^d)$ by Theorem 1.3 of \cite{AKT1}. Since plainly tameness passes to complemented subspaces and $\mathcal{O}(\mathbb{C}^d)=\Lambda_\infty(n^{1/d})$ is not tame \cite{DF}, indeed this case cannot occur. Now the theorem follows from Corollary \ref{corollary:3.2} and Theorem \ref{theorem:2}.
\end{proof}

\makeatletter
\renewcommand\@biblabel[1]{#1.}
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redpajama_set_name: RedPajamaArXiv


text:
1924-07 Watch Word July 1924
July, 1924	THE WATCH WORD	31
Helen DeMien left June 20 to be one of the many June brides. She was presented with a lovely gift by her co-workers.
Henrietta Peters became the bride of V. H. Gutschow June 7. She was presented with a magnificent gift by her many friends in the department.
Florence Darlison and Edna Rudinger were enjoying their annual vacations at this writing.
Mabel Adams, of the office force, has become economical and has started a new style in line with her new policy of economy. She wears the belt to her sweater as a tie. Some style, Mabel!
Katherine Erdan, Alice Keenan, Elsie Garner and Dorothy Young are among the new ones enrolled in this department recently.
Mrs. John Dahlstrom returned to work June 23, following a two weeks honeymoon, spent motoring through the east. She found her bench beautifully decorated and was presented with a bridge lamp by her co-workers. Mrs. Dahlstrom was formerly Miss Madeline Swahberg.
Alfrieda Duewel and Estelene Skelley are new in this department.
Mr. and Mrs. Harold McLaughlin and Mr. and Mrs. Martin Seybold motored to Griswold Lake June 22.
Here's one heard in the Escape: "Say, Bill, when you got married, how many cigars did you buy and how much did they cost?" Bill�"I dunno, Heine. Why?"
Miss Florence Dahlstrom entertained a group of friends Friday evening, June 27 in honor of Miss Gertrude Bjork, sister of Helen Bjork, of the Escape department office. Miss Bjork is a missionary in China, home on a furlough of eighteen months. The evening was spent enjoying games and music. A dainty two-course luncheon was served by the hostess.
Mr. and Mrs. Herron Sullivan, of Benton, Ill., spent a week with Mr. and Mrs. Sydney Sullivan.
Ruth Shaver passed cream puffs and candy June 17 in honor of her birthday.
Ernest Volmar, grandfather of Leona Zerneck, passed away Saturday, June 28. Sympathy of the department is extended to Leona in her bereavement.
Mr. and Mrs. Sydney Sullivan motored to DeKalb one Sunday.
Mr. and Mrs. Frank Davis were entertaining David Burzell and son, Clifford Burzell, and the latter's wife from California at this writing. The Messrs. Burzell formerly resided in Elgin.
Marjorie Mink spent a week-end visiting a friend at the Northern Illinois Teachers' College in DeKalb.
Frank Davis left Thursday, July 3 for Sparta, Wis., to spend the Fourth with his parents.
Mayme Schreiber was absent be- cause of illness at this writing.
Elsie Beck is a new bobbed head on Loechner's job.
Leona Zerneck spent the Fourth of July vacation in Chicago.
Mr. and Mrs. Sydney Sullivan, Marjorie Mink and Councilman Joe Day spent the Fourth of July vaca-
�	tion at the Dells of Wisconsin.
Mrs. Fern Palmer is a new arrival on Loechner's job.
Clara Myhre was on the absent list at this writing because of illness.
Marguerite Danielson and husband motored to Ziegler, Ill., and visited Mr. and Mrs. Elmer Swanson. Mrs.
WHICH IS WHICH?
We'll bet you're trying to tell which is the boy. Well, we'll tell you. The boy is the youngster standing, and he is none other than our own Al Radloff, of the Assembling, as he appeared when he wore pretty curls and a hair ribbon. The baby with him is his sister, Helen. Al doesn't know where we got this one.
Swanson was formerly Hazel Fagens, of the Gilding.
Marie D. Pundt is a new arrival on Anderson's job.
To all workers in the Gilding: We are about to spill to you a very interesting fact regarding our popular friend, Alma Brunneman, who works in our office. She has a very nice cottage in Trout Park, where she lives during the summer time. Of course, the people up there have no city water, so they use nice, cold spring water for drinking purposes�which water can't be beat. One day, however, the water was gone from the spring, which fact naturally puzzled everyone. An investigation was started and the facts gathered, so now we have the extreme pleasure of announcing that, while withdrawing a pail of the clear, cold fluid, Alma slipped, falling backwards into the spring. She splashed all the water out of the spring, and the spring was, of course, dry for a time. Now a word to Alma: We have a new swim- ming tank at Wing Park, free to the public, and it is deep enough for you to swim in. So why try to swim in the spring at Trout Park? It may be large enough, but it is not nearly deep enough.
Ruth Berggren was on the absent list at this writing, her teeth causing her trouble.
Mr. and Mrs. Donald Beverly spent the Memorial holidays at the Dells of Wisconsin. They reported a wonderful time.
Margaret M. Johnson and George Johnsen, formerly of the Stewart-Warner speedometer, are new in the Jewel.
Alvina Reed has been transferred to the Plate department.
Lydia Steffen was enjoying a week's visit with her sister and friends in Pulaski, Ia., at this writing.
Esther Sanwald has left the shop. She planned to return to the farm.
The girls certainly enjoyed a wonderful picnic supper at the home of Mrs. Anna Kerin, in honor of Lydia Krueger, Margaret Perrelet and Sylvia Borgh, who left the shop recently. Sylvia was very entertaining with her vocal selections. Dancing on the spacious porch was also enjoyed.
Elizabeth Lang, Bertha Reichert, Nora V. Foss, Verrella D. Bryant and Vera Fellows are beginners in the Jewel, the latter having been transferred from the Plate department.
Belle Fritch thought it would be an easy matter to keep a secret. But you don't think so now, do you, Belle? She became Mrs. W. M. Eberly Saturday evening, June 21 in Wheaton. Lots of good luck, Belle. Belle was presented with a lovely gift of glassware by her co-workers.
Dick Liebau, of the die job, says he plays in a fine band, but the boys don't agree with him at all.
Fowler has a friend who caught a three-pound silver bass, so he (Fowler) wins the hardware this month for the "best" story on fishing.
Even if the first of July didn't feel like summer, the die job supported a lot of summer suits that they had to give some wear this year.
The punch and die makers are a jealous lot. One evening two of the bunch were standing in the Square and when a third member of the job drove through for about the tenth time within a half-hour one of the standing die makers remarks to the other: "They ought to pinch that guy for a public pest." Moral: Sour grapes.
Mort Anderson finally received his radio set only eight or ten months after placing his order.
Walt Johnson had the dope on the New Buick all right. Just like all these "inside tips," it was all wrong.
Milton Robbins broke a looking glass on Friday, the thirteenth, and
Title 1924-07 Watch Word July 1924
Description The July 1924 issue of the Watch Word, a magazine for and about the employees of the National Watch Company
FullText July, 1924	THE WATCH WORD	31 Helen DeMien left June 20 to be one of the many June brides. She was presented with a lovely gift by her co-workers. Henrietta Peters became the bride of V. H. Gutschow June 7. She was presented with a magnificent gift by her many friends in the department. Florence Darlison and Edna Rudinger were enjoying their annual vacations at this writing. Mabel Adams, of the office force, has become economical and has started a new style in line with her new policy of economy. She wears the belt to her sweater as a tie. Some style, Mabel! Katherine Erdan, Alice Keenan, Elsie Garner and Dorothy Young are among the new ones enrolled in this department recently. Mrs. John Dahlstrom returned to work June 23, following a two weeks honeymoon, spent motoring through the east. She found her bench beautifully decorated and was presented with a bridge lamp by her co-workers. Mrs. Dahlstrom was formerly Miss Madeline Swahberg. Alfrieda Duewel and Estelene Skelley are new in this department. Mr. and Mrs. Harold McLaughlin and Mr. and Mrs. Martin Seybold motored to Griswold Lake June 22. Here's one heard in the Escape: "Say, Bill, when you got married, how many cigars did you buy and how much did they cost?" Bill�"I dunno, Heine. Why?" Miss Florence Dahlstrom entertained a group of friends Friday evening, June 27 in honor of Miss Gertrude Bjork, sister of Helen Bjork, of the Escape department office. Miss Bjork is a missionary in China, home on a furlough of eighteen months. The evening was spent enjoying games and music. A dainty two-course luncheon was served by the hostess. GILDING Mr. and Mrs. Herron Sullivan, of Benton, Ill., spent a week with Mr. and Mrs. Sydney Sullivan. Ruth Shaver passed cream puffs and candy June 17 in honor of her birthday. Ernest Volmar, grandfather of Leona Zerneck, passed away Saturday, June 28. Sympathy of the department is extended to Leona in her bereavement. Mr. and Mrs. Sydney Sullivan motored to DeKalb one Sunday. Mr. and Mrs. Frank Davis were entertaining David Burzell and son, Clifford Burzell, and the latter's wife from California at this writing. The Messrs. Burzell formerly resided in Elgin. Marjorie Mink spent a week-end visiting a friend at the Northern Illinois Teachers' College in DeKalb. Frank Davis left Thursday, July 3 for Sparta, Wis., to spend the Fourth with his parents. Mayme Schreiber was absent be- cause of illness at this writing. Elsie Beck is a new bobbed head on Loechner's job. Leona Zerneck spent the Fourth of July vacation in Chicago. Mr. and Mrs. Sydney Sullivan, Marjorie Mink and Councilman Joe Day spent the Fourth of July vaca- �	tion at the Dells of Wisconsin. Mrs. Fern Palmer is a new arrival on Loechner's job. Clara Myhre was on the absent list at this writing because of illness. Marguerite Danielson and husband motored to Ziegler, Ill., and visited Mr. and Mrs. Elmer Swanson. Mrs. WHICH IS WHICH? We'll bet you're trying to tell which is the boy. Well, we'll tell you. The boy is the youngster standing, and he is none other than our own Al Radloff, of the Assembling, as he appeared when he wore pretty curls and a hair ribbon. The baby with him is his sister, Helen. Al doesn't know where we got this one. Swanson was formerly Hazel Fagens, of the Gilding. Marie D. Pundt is a new arrival on Anderson's job. To all workers in the Gilding: We are about to spill to you a very interesting fact regarding our popular friend, Alma Brunneman, who works in our office. She has a very nice cottage in Trout Park, where she lives during the summer time. Of course, the people up there have no city water, so they use nice, cold spring water for drinking purposes�which water can't be beat. One day, however, the water was gone from the spring, which fact naturally puzzled everyone. An investigation was started and the facts gathered, so now we have the extreme pleasure of announcing that, while withdrawing a pail of the clear, cold fluid, Alma slipped, falling backwards into the spring. She splashed all the water out of the spring, and the spring was, of course, dry for a time. Now a word to Alma: We have a new swim- ming tank at Wing Park, free to the public, and it is deep enough for you to swim in. So why try to swim in the spring at Trout Park? It may be large enough, but it is not nearly deep enough. JEWEL Ruth Berggren was on the absent list at this writing, her teeth causing her trouble. Mr. and Mrs. Donald Beverly spent the Memorial holidays at the Dells of Wisconsin. They reported a wonderful time. Margaret M. Johnson and George Johnsen, formerly of the Stewart-Warner speedometer, are new in the Jewel. Alvina Reed has been transferred to the Plate department. Lydia Steffen was enjoying a week's visit with her sister and friends in Pulaski, Ia., at this writing. Esther Sanwald has left the shop. She planned to return to the farm. The girls certainly enjoyed a wonderful picnic supper at the home of Mrs. Anna Kerin, in honor of Lydia Krueger, Margaret Perrelet and Sylvia Borgh, who left the shop recently. Sylvia was very entertaining with her vocal selections. Dancing on the spacious porch was also enjoyed. Elizabeth Lang, Bertha Reichert, Nora V. Foss, Verrella D. Bryant and Vera Fellows are beginners in the Jewel, the latter having been transferred from the Plate department. Belle Fritch thought it would be an easy matter to keep a secret. But you don't think so now, do you, Belle? She became Mrs. W. M. Eberly Saturday evening, June 21 in Wheaton. Lots of good luck, Belle. Belle was presented with a lovely gift of glassware by her co-workers. MACHINE Dick Liebau, of the die job, says he plays in a fine band, but the boys don't agree with him at all. Fowler has a friend who caught a three-pound silver bass, so he (Fowler) wins the hardware this month for the "best" story on fishing. Even if the first of July didn't feel like summer, the die job supported a lot of summer suits that they had to give some wear this year. The punch and die makers are a jealous lot. One evening two of the bunch were standing in the Square and when a third member of the job drove through for about the tenth time within a half-hour one of the standing die makers remarks to the other: "They ought to pinch that guy for a public pest." Moral: Sour grapes. Mort Anderson finally received his radio set only eight or ten months after placing his order. Walt Johnson had the dope on the New Buick all right. Just like all these "inside tips," it was all wrong. Milton Robbins broke a looking glass on Friday, the thirteenth, and

redpajama_set_name: RedPajamaCommonCrawl


text:
MNN.com > Tech > Research & Innovations
Students solve water woes by mimicking nature
A system designed by students at the University of Toronto was inspired by fish gills, tadpoles and human arteries.
John Platt
Photo: "Nature Inspired Removal of Air Entrained in Water," the winner of the Biomimicry Student Design Challenge.
What do human arteries, fish gills and newt tadpoles have to do with safe drinking water? A team of graduate students from the University of Toronto looked to those three things in nature to come up with a novel water pipe system that could be used to decrease water-borne illnesses while improving water delivery efficiency and lowering wastewater operation costs. Their system won them first place and a $2,500 prize in the first round of the 2013 Biomimicry Student Design Challenge, which asked students to study and imitate nature's best ideas to solve human problems.
The contest was sponsored by the Biomimicry 3.8 Institute, a nonprofit founded to study nature's designs. "We asked students to look at how nature deals with water access and management and apply that knowledge to solve a human design problem," Megan Schuknecht, director of university education Institute, said in a prepared release announcing the results.
Nearly 70 teams from 14 U.S. states and 18 countries entered this first round of the competition. Check out some of the finalists below:
How do you tap the heavy monsoon-season rain of the Sonoran desert to solve problems of rainwater scarcity in other seasons? A team from the University of Arizona looked at the root structure and cortex of the saguaro cactus to create an efficient rainwater collection system.
A tropical plant and spider webs inspired a project from Universidad Autónoma de Yucatán in Mexico, which was designed to catch and store water in rural communities. Their Chaac Ha' water system collector received the competition's Autodesk Sustainability Award.
Can fog be a good source of water for agriculture? A team from Pontificia Universidad Católica de Chile created a dynamic system that would "farm" water from fog. They were inspired by a plant called Tillandsia landbeckii that gets its water and nutrients from the air. The team won third prize.
Although it did not place as a finalist, the AquaBloom developed by a team from Saint Louis University was one of the competition's most visually striking entries. The system was inspired by bromeliads, which use overlapping leaves to absorb and release water. The team used this idea to develop a rainwater collection system that could be hooked up to an underground reservoir.
Honeybees inspired a team from Artesis University College in Belgium to create an evaporation cooling system to keep fruits and vegetables from spoiling and going to waste at minimal cost. Bees in a hive flap their wings to spread water and increase humidity, dropping the temperature in the process. The team mimicked this by flowing a fan over a wet cloth. Their idea was good enough to earn them second place.
Ten finalists from this first round and several other teams will now also enter Round 2, where they will receive startup mentoring and compete for $11,000 in prizes.
Also on MNN:
Biomimicry: Science inspired by nature could feed the hungry, reduce impact of technology
7 amazing examples of biomimicry
Climate change researcher, biomimicry expert among annual Heinz Award winners
Biomimicry a hot trend in renewable energy technology
Related topics: Nature, Waste, Water Conservation
What are dog whiskers for?
10 surprising facts about Neanderthals
Firefighters save grove of trees in Australia that predate the dinosaurs


text:
<?php

namespace Process\Controller;

use Zend\Mvc\Controller\AbstractActionController;
use Zend\View\Model\ViewModel;
use Process\Model\Entity\Modelo;

class ListaController extends AbstractActionController
{

	private $modelo;
	public function __construct (){
		$this->modelo = new modelo();
	}

    public function indexAction()
    {
    	$consulta=array();
        $elementos=array('nombreInterseccion'=> 1, 'horaInicioConteo'=> 1,'horaFinalConteo'=>1,'codificador'=> 1,'fechaCodificacion'=> 1);
    	$informacion = array('status'=>'empty');
    	$intersecciones = $this->modelo->find($consulta, $elementos);

    	$tabla = $this->modelo->getTablaCodificadores($intersecciones);
    	return new viewModel(array('tabla' =>$tabla));
    }

}

redpajama_set_name: RedPajamaGithub


text:
Marathon Runner Fights MS
Forty-year-old Cheryl Hile runs marathons, bicycles and holds a demanding full-time job—impressive accomplishments but, on the surface, not especially remarkable in this day and age.
Cheryl, however, is remarkable. She engages in these strenuous activities despite a dropped foot, weakened thigh muscles, fatigue and a "creepy-crawly" feeling in her body—all symptoms of multiple sclerosis (MS). MS is an autoimmune disease in which the body's immune system destroys myelin, the protective sheath that covers nerve fibers. This affects the brain and spinal cord, leading to pain and disability.
"I was so depressed after receiving a diagnosis of MS," she says. "I didn't know anything about the disease except that I had a co-worker in a wheelchair and thought that was my future. But I kept on running and exercising because that was my stress reliever, where I could be myself."
In keeping with her take-charge personality, Cheryl did extensive research on the disease and the best places for treatment and research. She soon discovered Virginia Mason Medical Center and Benaroya Research Institute at Virginia Mason (BRI). In partnership with Virginia Mason, BRI's team of expert scientists is focused on identifying causes and cures for MS and other devastating autoimmune diseases.
Cheryl and her husband moved to Seattle so that she could become a patient of Mariko Kita, MD, BRI clinical investigator, director of the Virginia Mason Multiple Sclerosis Center and head of the Department of Neurology at Virginia Mason. From their first meeting, Cheryl knew Dr. Kita was right for her.
"Dr. Kita is very compassionate and doesn't talk down to you," Cheryl says. "She gives me information so that I can make my own choices and doesn't tell me what to do. And she is very willing to help keep me active and running."
BIOREPOSITORY PARTICIPANT
Even though she has moved back to San Diego, Cheryl continues to see Dr. Kita. She also participates in BRI's MS biorepository. A biorepository consists of blood and tissue samples linked to medical and demographic information collected from people with a specific disease or condition. Researchers use this information to better understand progression of diseases and possible therapies, accelerating the translation of discoveries from the lab to the patient.
"Through the biorepository, researchers at BRI discovered a biomarker for MS that can be used as a target for new therapies," she says. "What I hope for is better treatments for MS and I would really like to see a cure, so if it means regularly donating a small amount of blood, then I will gladly extend my arm!"
BRI and Virginia Mason have participated in more than 30 clinical studies that have advanced MS treatment including FDA approval of new therapies shown on this chart. One of the most promising new areas of research lies in remyelination—reversing damage and growing back nerves. BRI and VM are now testing a new molecule for the first time in humans that has been shown to repair and reduce lesions.
Originally published in BRING IT ON newsletter - Fall 2015
Patient Stories, Dr. Mariko Kita, Virginia Mason, Cheryl Hile, Marathon, Multiple Sclerosis, Biorepository
BRI Biorepositories


So, I'm here with my second write-up for Vulnhub – Kioptrix Level 2 challenge. So, we usually start by doing some enumeration on services. but before that we have to find out the IP Address of our machine.
netdiscover will scan for all devices connected on your network or you can use arp-scan your choice.
Now we have our target IP Address let's take a look which services are running on that server.
These are the services running on targeted machine.
Remote System Administration Login, Username and Password field let's take a look at page source might we find something interesting.
Nothing special but since we do not have any security checks on username and password field we can try SQLi.
Below is the request which we're sending and check the response we got.
So, it was simply SQL Injection (1′ or '1' = '1).
We have tested manual SQL Injection. Let's test SQLmap now.
It generated the payload for us!
There's an another field to run a ping command and tested SQL Injection vulnerabilities again and found a simple ';' semicolon can be used to bypass this.
And i ran "nc -lvp 1337″ on my machine to get reverse shell.
Now we're going after root access.
By running LinEnum.sh script we couldn't find anything interesting but you take a look at kernel version we have some exploits available for that kernel version.
After doing Google searches found that this exploit might work so i gave it a try and it actually worked.


text:
package cofh.api.block;

import java.util.ArrayList;

import net.minecraft.entity.player.EntityPlayer;
import net.minecraft.item.ItemStack;
import net.minecraft.world.World;

/**
 * Implemented on Blocks which have some method of being instantly dismantled.
 *
 * @author King Lemming
 *
 */
public interface IDismantleable {

	/**
	 * Dismantles the block. If returnDrops is true, the drop(s) should be placed into the player's inventory.
	 */
	ArrayList<ItemStack> dismantleBlock(EntityPlayer player, World world, int x, int y, int z, boolean returnDrops);

	/**
	 * Return true if the block can be dismantled. The criteria for this is entirely up to the block.
	 */
	boolean canDismantle(EntityPlayer player, World world, int x, int y, int z);

}


text:
Lost in Brexit chaos - abortion rights in Northern Ireland
On the back of Ireland's successful repeal campaign, the spotlight has now turned to Northern Ireland and there is an impetus amongst activists to see these draconian laws repealed and abortion decriminalised (Photo: William Murphy)
By Emma Rainey
BRUSSELS, 13. Dec 2018, 13:59
On the 26 May 2018, the day that Ireland voted 'yes' to repeal the eighth amendment from its constitution, I sat with other pro-choice activists in one of the few Irish pubs in Brussels as we watched the results roll in.
As the only person from Northern Ireland in the group, my emotions that day could best be described as bittersweet.
Unlike the rest of the UK, abortion care is still illegal in Northern Ireland as the 1967 Abortion Act was never extended to the region.
Northern Irish abortion legislation is based on the 1861 Offences against the Person Act, and the 1945 Criminal Justice Act (NI), making abortion illegal in almost every circumstance and the law is one of the most restrictive in the world.
Between 2016/17, only 13 abortions took place in Northern Ireland.
On the back of Ireland's repeal campaign, the spotlight has now turned to Northern Ireland and there is an impetus amongst activists to see these draconian laws repealed and abortion decriminalised.
However, unlike Ireland's binding referendum, the political landscape in Northern Ireland and the wider UK has made the process for change much more difficult.
Three factors contribute to this difficult environment.
Firstly, the collapse of Stormont which houses Northern Ireland's devolved government and assembly which has been in suspension since January 2017; prime minister Theresa May's reliance on Northern Ireland's Democratic Unionist Party (DUP) to secure a majority government; and lastly (and always) – Brexit.
Westminster and the Brexit blockade
Devolution enables Northern Ireland, Scotland and Wales to self-govern on a number of regional matters, healthcare being one of them.
However, with no functioning devolved government in place, it means that abortion reform from within Northern Ireland is currently not possible.
This has meant that pro-choice activists have had to take the fight for abortion rights to Westminster (which has ultimate constitutional sovereignty over all regions) in order to navigate around the constitutional devolved barriers that are being instrumentalised by the Tory government to appease the anti-abortion DUP.
The DUP maintain that the region is as "British as Finchley [a north London suburb]" and must not diverge from the rest of the UK when it comes to Brexit and the border backstop - whilst also maintaining divergence on many other issues like abortion.
The irony is not lost on anyone in Northern Ireland.
The complexity of the challenges facing pro-choice activists is not helped by the fact that the clock is rapidly counting down to the UK's departure from the EU on the 29 March 2019.
The vast majority of legislation going through Westminster is Brexit-related; therefore, domestic issues – especially those from a distant province across the water – are sitting on the backburner.
In the meantime, it is estimated that three women a day travel from Northern Ireland to England for an abortion.
By deploying a campaign strategy to prevent devolution from being used as a shield for inaction, activists are aiming to create a legal environment to put further pressure on the UK government.
The 2018 UN report by the Committee on the Elimination of Discrimination against Women, found that the UK is in violation of women's rights in Northern Ireland by restricting access to abortion.
On the back of this report, long-term campaigner Sarah Ewart is set to challenge Northern Ireland's abortion laws as a breach of human rights in the High Court for cases of fatal-foetal abnormalities.
Labour MP Diana Johnson has brought a private members bill to Westminster that proposes to decriminalise abortion in the whole of the UK, which means that, if successfully passed, current provisions for Northern Ireland will also be repealed.
This would leave a legal void around abortion.
Medical practitioners would still not be able to issue abortion pills; however, women who seek to procure the pills online would no longer have to fear prosecution.
While this avenue would not provide a sustainable solution, it would require the UK government to act.
While Brexiteers rampage on in their perceived battle for Britain's sovereignty, freedom of movement has enabled me to gain back sovereignty over my own body.
Living in Belgium has allowed me to access rights I would not have back home in Northern Ireland.
Like so many other aspects of the UK at present, uncertainty will continue to prevail for abortion rights in Northern Ireland unless something fundamental changes.
However, the one certainty that can be guaranteed is that activists will continue to remind the UK government that human rights are not a devolved matter.
Emma Rainey is a committee member of Repeal Brussels, a campaign for abortion rights for the island of Ireland
EU court adds to knife-edge Brexit drama
May on whistle-stop EU tour to seek new backstop pledges
Norway plays politics with abortion laws
EU's 'old men' must pressure Poland on abortion rights
Why 60 Romanian hospitals are refusing abortions
The Abortion Exodus - more Poles and Croats going abroad
EU judges have granted the UK the right to unilaterally stop Brexit - amid question marks on whether Tuesday's crunch vote in London will take place.
The British prime minister dramatically delayed a parliamentary vote on the Brexit deal at the last minute, as she faced defeat. Theresa May will now speed-tour EU capitals to try to secure further political guarantees.
Norway's conservative prime minister, Erna Solberg, has proposed tightening the country's abortion laws, in a political gambit that goes against Europe's liberal trend.
Despite fresh crackdowns on Poland's already restrictive abortion laws, EU commission president Juncker did not raise the issue with the new Polish PM Morawiecki - perhaps because it was an all-male event?
Some 60 of Romania's public hospitals are refusing to provide women with legal abortions - as a result of doctors invoking a "conscience clause".
Why are more women from Poland and Croatia seeking pregnancy terminations abroad?


text:
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Richard Lawson
Richard Lawson, namesake of the new Richard and Patricia Lawson Computer Science Building received an honorary doctorate on May 14, 2006 at the 9:30 a.m. Purdue University commencement ceremony. Richard was accompanied to graduation by his wife, Patricia. Richard's business associate, whom he credits as being instrumental in his decision to study computer science at Purdue, Dr. Peter Patton, also attended the ceremony with the Lawsons.
Richard Lawson graduated from Purdue with an MS in Computer Science in 1968. He came to Purdue with a BS in Computer Science from Oklahoma Christian University. He credits Purdue with teaching him the know how to think about computer science and adjust to the many changes in the technology over the years. Currently, Richard is on the Board of Trustees at Oklahoma Christian University and the chairman of Lawson Software, Inc., which he co-founded with his brother and one other partner in 1975, seven years after receiving his MS from Purdue.
Lawson Software
Lawson Software began as a consulting business, and in the past thirty years has flourished into a $364 million company providing business software to large and mid-sized organizations in healthcare, retail, professional services, the public sector, financial services, and other strategic markets. Lawson Software also serves more than 140 government and education clients. The growth of Lawson Software can be attributed to Richard's strong balance of computer science knowledge and business acumen.
The company's stellar achievements represent the innovative, humble leadership style of Richard Lawson. "Technology is a tool for business," Lawson said. "Because of our experience with companies around the world, Lawson can offer the applications and experience to help businesses make best use of their technology – and get maximum returns on these investments."
In 1995, Lawson Software received the IBM Star Stream Award in recognition of its outstanding achievements and leadership in the AS/400 Application Modernization Category. In September 1998, Red Herring named Lawson Software "the strongest private ERP company" and listed Lawson among the world's top 50 private companies. Lawson Software was also selected to the Red Herring 100 in 1999.
Richard Lawson's Success
Richard Lawson's success has made him a sought after speaker for many information technology events. In 1997, he spoke at the Forbes CEO Forum: The Next Millennium where he addressed 250 chief executives. His initiative, innovation, pursuit in better technology, and his ability to always keeping the customer in mind also made him the right candidate to speak at the 2001 ASP European Summit in Rome.
Richard Lawson and his wife, Pat, are the namesakes and lead donors of the new Computer Science building at Purdue. Philanthropy is a way of life for the Lawsons. They also made a gift of stock to the Oklahoma Christian University valued at more than $33 million, and a $5.7 million contribution to the university for operational funds and to help fund capital and endowment needs.
In 2002, he was named the Purdue Computer Science Distinguished Alumnus of the Year.
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Welcome to CPR Test Center. CPR Test Center certifications have been issued to and accepted by people and organizations around the world including members of the US Army, US Navy, state and federal government agencies, public school districts, universities, hospitals, security companies, hotels, sports & fitness groups, massage & health centers, students, moms & dads. Knowing when and how to perform proper CPR can mean the difference between life and death. When cardiac arrest strikes someone close to you, knowing CPR can greatly increase the chances of survival. You can learn CPR right now in Enid from your smartphone or computer.
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