text:
\section{Introduction.}

Diversity is an important notion in a variety of scientific disciplines.  Historically, the interest stems from ecological applications, where the diversity of species in an ecosystem is a relevant issue.  Other applications include cancer research, where the interest is in the diversity of types of cancer cells in a tumor, and linguistics,  where it is in the diversity of an author's vocabulary.  More generally, in information science, one is interested in the diversity of letters drawn from some alphabet. While there is little controversy on the literal meaning of the word ``diversity'', the best way to quantify this concept is a matter of some dispute. For this reason many indices of diversity have been proposed in the literature.  Perhaps the earliest are Shannon's entropy introduced in Shannon (1948) and Simpson's index introduced in Simpson (1949). Many other popular diversity indices have been proposed, including Emlen's index, the Gini-Simpson index, Hill's diversity number, R\'{e}nyi's entropy, and
Tsallis entropy. These are, respectively, named after the authors of Emlen (1973), Gini (1912), Hill (1973), R\'{e}nyi (1961), and Tsallis (1988). Comprehensive discussions are offered by many, see {\it e.g.}, Magurran (1988) and Marcon (2013). It is quite clear that, when it comes to the question of what constitutes a mathematical diversity index, the ideas are diverse. This paper offers a unified perspective for all of the above mentioned diversity indices based on a re-parameterization and establishes a
general nonparametric estimation procedure for these indices.

In the following section we give a general discussion of diversity indices.  In Section \ref{sec: Entropic Basis} we show that all diversity indices (satisfying very general regularity conditions) are transforms of a relatively simple class of diversity indices, which we call the entropic basis. Further, we show that most diversity indices used in the literature are either linear combinations (possibly infinite) of the entropic basis or equivalent to such indices.  In Section \ref{sec: estimation} we give a general non-parametric approach for estimating such linear diversity indices. We derive an estimator, which, under general conditions, is consistent, asymptotically normal, and has a bias that decays exponentially fast. In Section \ref{sec: estimate Renyi} we show how to extend these results to R\'enyi's entropy, which is not a linear diversity index, but only equivalent to one.

\section{Preliminaries}

Let the species in a population be denoted by letters of a countable alphabet $\mathscr{X}=\{\ell_{k};k\geq 1\}$ and let the abundances of the species be represented by a probability distribution $\mathbf{p}=\{p_{k};k\geq1\}$ associated with these letters. For simplicity of notation, and when there is no risk of ambiguity, let $K=\sum_{k\geq1}1[p_{k}>0]$, where $1[\cdot]$ is the indicator function, be referred to as the cardinality of $\mathscr{X}$, which may be finite or countably infinite.

An index of diversity $\theta=\theta(\mathscr X,\mathbf p)$ is a function taking the alphabet $\mathscr X$ and the probability distribution $\mathbf p$ into the extended real line $[-\infty,\infty]$. What additional conditions $\theta$ must satisfy is a matter of opinion or need depending on the particular population features of interest. The discussion of this paper starts with the following set of axioms:
\beq
    \begin{array}{rcl}\nonumber
    A_{01}:& & \mbox{$\theta$ is fully determined by $\mathbf{p}=\{p_{k};k\ge1\}$.} \\
    A_{02}:& & \mbox{$\theta$ is invariant under any permutation on the index set $\{k;k\geq 1\}$.}
    \end{array}
\eeq
We will refer to these axioms as $\mathscr{A}_{0}=\{A_{01},A_{02}\}$. Note that Axiom $A_{01}$ implies that $\theta=\theta(\mathscr X,\mathbf p)=\theta(\mathbf p)$ does not depend on $\mathscr X$. The axiomatic set $\mathscr{A}_{0}$ is by no means sufficient for a meaningful diversity index, but it provides a minimal constraint for further discussion. Most, if not all, popular diversity indices studied in the literature satisfy $\mathscr{A}_{0}$. These include:
\begin{enumerate}
 \item Simpson's index $\lambda=\sum_{k\geq 1}p_{k}^{2}$,
 \item Gini-Simpson index $1-\lambda=\sum_{k\geq 1}p_{k}(1-p_{k})$,
 \item Shannon's entropy $H=-\sum_{k\geq 1}p_{k}\ln(p_{k})$,
 \item R\'{e}nyi's entropy $H_{r}=(1-r)^{-1}\ln\left(\sum_{k\geq 1}p_{k}^{r}\right)$ for any $r>0, r\ne1$,
 \item Tsallis' entropy $T_{r}=(1-r)^{-1}\left(\sum_{k\geq 1}p_{k}^{r}-1\right)$ for any $r>0, r\ne1$
 \item Hill's diversity number $N_{r}=\left(\sum_{k\geq 1}p_{k}^{r}\right)^{1/(1-r)}$ for any $r>0, r\ne1$,
 \item Emlen's index $D=\sum_{k\ge1}p_ke^{-p_k}$, and
 \item the richness index $K=\sum_{k\geq 1}1[p_{k}>0]$.
\end{enumerate}

Given the abundance of definitions of various diversity indices, let us define a notion of equivalence between two diversity indices.

\begin{definition}\label{defn: equiv} Two diversity indices, $\theta_{1}$ and $\theta_{2}$, are said to be equivalent if and only if there exists a strictly increasing function $g$ such that $\theta_{1}=g(\theta_{2})$. The equivalence is denoted by $\theta_{1}\Leftrightarrow \theta_{2}$.
\end{definition}

Noting that if $g$ is strictly increasing then so is its inverse $g^{-1}$, it is clear that the above definition is symmetric with respect to $\theta_{1}$ and $\theta_{2}$. Two diversity indices are equivalent if they agree on whether one population is more diverse than another, but they may not agree on what the actual difference is.  With this rather trivial notion of equivalence much of the superficial redundance among various indices in the literature can be erased. For example, R\'{e}nyi's entropy $H_r$, Tsallis entropy $T_r$, and Hill's diversity number $N_r$ are equivalent to each other. Further, for $r\in(0,1)$ they are equivalent to a core index
\beq h_r = \sum_{k\geq 1}p_{k}^{r}, \ \ r>0, r\ne1,
\label{renyieqiv}
\eeq
which we will refer to as {\it R\'{e}nyi's equivalent entropy}.  Although these indices are not equivalent to $h_r$ for $r>1$, they are nevertheless continuous transformations of $h_r$. This fact will be useful for statistical estimation, see Section \ref{sec: estimate Renyi} below.

\section{Entropic Basis}\label{sec: Entropic Basis}

In this section we give our first main result.  We introduce a class of diversity indices and show that they are the building blocks from which all diversity indices satisfying the axioms of $\mathscr A_0$ are made. For a given $\mathbf{p}=\{p_{k};k\ge1\}$ and an integer pair $(u\geq 1, v\geq 0)$, let $\zeta_{u,v}=\sum_{k\geq 1}p_{k}^{u}(1-p_{k})^{v}$. We will refer to
\beq
       \boldsymbol\zeta=\{\zeta_{u,v};u\geq 1, v\geq 0\}
\label{zetauv}
\eeq  as the family of the {\it generalized Simpson's diversity indices} after Zhang and Zhou (2010). Furthermore we will refer the sub-family
\beq
\boldsymbol\zeta_{1}=\left\{\zeta_{1,v};v\geq 0\right\}
\label{zeta1v}
\eeq as the {\it entropic basis} with regard to $\mathscr{X}$ and $\mathbf{p}$.

\begin{remark}According to Zhang and Zhou (2010), every member of $\boldsymbol\zeta$ can be expressed as
$\zeta_{u,v}=\sum_{k\geq 1}\left[p_{k}^{u-1}\right]\left[p_{k}(1-p_{k})\right]\left[(1-p_{k})^{v-1}\right]$, which may be regarded as a weighted version of the Gini-Simpson diversity index, with weights $\left[p_{k}^{u-1}\right]\left[(1-p_{k})^{v-1}\right]$ for various choices of $u$ and $v$ at the user's discretion, and hence the term  {\it generalized Simpson's diversity indices}.
\end{remark}

\begin{remark}
Zhang (2012) established an alternative representation of Shannon's entropy $H=\sum_{v=1}^{\infty}v^{-1}\zeta_{1,v}$ provided that $H<\infty$. This is a linear form of $\boldsymbol\zeta_{1}$, hence the term {\it entropic basis}.
\end{remark}

We now show that any diversity index satisfying the axioms of $\mathscr{A}_{0}$ must be a function of $\zeta_{1,v}$, for $v=0,1,\dots$, {\it i.e.} of the members of the entropic basis.

\begin{theorem} Given an alphabet $\mathscr{X}=\{\ell_{k}; k\geq 1\}$ and an associated probability distribution $\mathbf{p}=\{p_{k}; k\geq 1\}$, a diversity index $\theta$ satisfying $\mathscr{A}_{0}$ is fully
determined by the entropic basis $\boldsymbol\zeta_{1}=\left\{\zeta_{1,v};v\geq 0\right\}$.
\label{th1}
\end{theorem}

A proof of Theorem \ref{th1} requires the following lemma due to Zhang and Zhou (2010).

\begin{lemma} The distribution $\mathbf{p} = \{p_k;k\ge1\}$ on $\mathscr{X}$ and the family of generalized Simpson's diversity indices $\boldsymbol\zeta=\left\{\zeta_{u,v};u\geq 1, v\geq 0\right\}$ uniquely determine each other up to a permutation of the index set $\{k;k\geq 1\}$.
\label{lemma1}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{th1}.] By Lemma \ref{lemma1}, $\boldsymbol\zeta$ determines $\mathbf{p}$ up to a permutation, and this fully determines a diversity index $\theta$ satisfying $\mathscr A_{0}$.
It remains to show that every element of $\boldsymbol\zeta$ is fully determined by $\boldsymbol\zeta_{1}$. Toward that end, we note that for any pair of fixed integers $u\geq 2$ and $v\geq0$
\begin{eqnarray*}
    \zeta_{u,v} &=& \sum_{k\geq 1}p_{k}[1-(1-p_{k})]^{u-1}(1-p_{k})^{v} \\
       &=& \sum_{k\geq 1}p_{k}\left[\sum_{i=0}^{u-1}(-1)^{i}\binom{u-1}{i}(1-p_{k})^{i}\right](1-p_{k})^{v} \\
           &=& \sum_{i=0}^{u-1}(-1)^{i}\binom{u-1}{i}\left[\sum_{k\geq 1}p_{k}(1-p_{k})^{v+i}\right] \\
       &=& \sum_{i=0}^{u-1}(-1)^{i}\binom{u-1}{i}\zeta_{1,v+i},
\end{eqnarray*}
which completes the proof.
\end{proof}

The statement of Theorem \ref{th1} holds true for any probability distribution $\mathbf{p}$ regardless of whether $K$ is finite or infinite. Theorem \ref{th1} essentially offers a re-parameterization of $\mathbf p$ (up to a permutation) in terms of $\boldsymbol\zeta_{1}$. This re-parameterization is not just an arbitrary one, it has several statistical implications. First of all, every element of $\boldsymbol\zeta_1$ contains information about the entire distribution and not just one frequency $p_k$. This helps to deal with the problem of estimating probabilities of unobserved species. Second, for a random sample of size $n$, there are very good estimators of $\zeta_{1,v}$ for $v=0,1,2,\dots,n-1$. These are given in Zhang and Zhou (2010) and are discussed below.

While, in general, a diversity index can be any transformation of the entropic basis, in practice, most commonly used indices correspond to transformations of a fairly simple form. Most diversity indices either belong to, or are equivalent to ones that belong to the following class.

\begin{definition}\label{defn: linear div index}
 A diversity index $\theta$ is said to be a linear diversity index if it is a linear combination of the elements of the entropic basis, {\it i.e.}
\begin{eqnarray}\label{eq: lin div index}
\theta = \theta(\mathbf p) = \sum_{v=0}^\infty w_v \zeta_{1,v} = \sum_{v=0}^\infty w_v \sum_{k\ge1} p_k(1-p_k)^v
\end{eqnarray}
for any choice of weights $w_v$ such that, for every $\mathbf p$, the sum either converges or diverges to $\pm\infty$.
\end{definition}
Definition \ref{defn: linear div index} essentially encircles a sub-class of indices among all functions of $\boldsymbol\zeta_{1}$, {\it i.e.}, all diversity indices satisfying $\mathscr{A}_{0}$.
While there are no fundamental reasons why a search of a good diversity index should be restricted to this sub-class, it happens to cover all of the popular indices that we have come across in the literature, up to the equivalence relationship given in Definition \ref{defn: equiv}. These include:
\[ \begin{array}{rrclcl}
    \mbox{Simpson's index:} & \lambda &=& \sum_{k\geq 1}p_{k}^{2}&=&\zeta_{1,0}-\zeta_{1,1} \\ &&& \\
    \mbox{Gini-Simpson index:} & 1-\lambda &=&\sum_{k\geq 1}p_{k}(1-p_{k})&=& \zeta_{1,1} \\ &&& \\
    \mbox{Shannon's entropy:} &  H&=& -\sum_{k\geq 1}p_{k}\ln(p_{k})&=&\sum_{v= 1}^{\infty}\frac{1}{v}\hspace{0.2em}\zeta_{1,v} \\  &&& \\
    \mbox{R\'{e}nyi equiv.\ entropy:} &  h_{r}&=& \sum_{k\geq 1}p_{k}^{r}&=&\zeta_{1,0}+\sum_{v=1}^{\infty}\prod_{i=1}^{v}\left(\frac{i-r}{i}\right)\hspace{0.2em}\zeta_{1,v}, \\&&&\\
    \mbox{Emlen's index:} & D &=&\sum_{k\geq 1}p_ke^{-p_k}&=& \sum_{v=0}^\infty \frac{e^{-1}}{v!}\zeta_{1,v}\\&&&\\
 \mbox{Richness index:} & K&=&\sum_{k\geq 1}1[p_{k}>0] &=&  \sum_{v=0}^\infty \zeta_{1,v}\\&&&\\
    \mbox{Gen.\
    Simpson's index:} &  \zeta_{u,\hspace{0.2em}m} &=& \sum_{k\geq 1}p_k^u(1-p_k)^m&=&\sum_{v=0}^{u-1}(-1)^{v}\binom{u-1}{v}\hspace{0.2em}\zeta_{1,\hspace{0.2em}m+v}.
\end{array}
\]
Note that $\zeta_{1,0}=\sum_{k\ge1}p_k(1-p_k)^0\equiv1$. We also note that Tsallis' entropy is a linear diversity index.  The form of its weights are very similar to those of R\'enyi's equivalent entropy. All of the representations above can be verified using Taylor expansions.  For example, for the richness index, which is the total number of species in a population, we have
\begin{eqnarray*}
    K&=& \sum_{k\geq 1}1[p_{k}>0]\frac{p_{k}}{1-(1-p_{k})}\\
    &=&\sum_{k\geq 1}1[p_{k}>0]p_{k}\sum_{v=0}^{\infty}(1-p_{k})^{v} \\
    &=& \sum_{v=0}^{\infty}\sum_{k\geq 1}1[p_{k}>0]p_{k}(1-p_{k})^{v}=\sum_{v=0}^{\infty}\zeta_{1,v}.
 \end{eqnarray*}

It is not difficult to see that all linear diversity indices discussed above are of the general form
\begin{eqnarray}\label{eq: alt lin div index}
\theta = \sum_{k\ge1} p_k h(p_k),
\end{eqnarray}
where $h$ has a Taylor expansion around $1$ with radius of convergence at least $1$. This is not a coincidence. If $\theta$ satisfies \eqref{eq: lin div index} and $h(t) =  \sum_{v=0}^\infty w_v (1-t)^v$ then $\theta$ is necessarily of the form \eqref{eq: alt lin div index}, and, of course, the converse of this statement holds.

\begin{remark}
 In the literature of diversity indices, it is generally thought that the richness indices, {\it e.g.}, $K$, and the evenness indices, {\it e.g.}, Gini-Simpson's $1-\lambda$, are two qualitatively different types of indices, see {\it e.g.}, Peet (1974) and Heip, Herman, and Soetaert (1998). It may be interesting to note that, in the perspective of the entropic basis, they are both linear diversity indices and merely differ in the weighting scheme $h(p)$ in (\ref{eq: alt lin div index}).
\end{remark}

\section{Estimation of Linear Diversity Indices.}\label{sec: estimation}

In this section we discuss nonparametric estimation of linear diversity indices.  Assume that $X_1,X_2,\dots,X_n$ are independent and identically distributed (iid) from $\mathscr X$ according to $\mathbf p$. We want to estimate
\begin{eqnarray}\label{eq: theta}
\theta=\theta(\mathbf p) = \sum_{v=0}^\infty w_v \zeta_{1,v} = \sum_{v=0}^\infty w_v \sum_{k\ge1} p_k(1-p_k)^v.
\end{eqnarray}
We assume that $\{w_v:v\ge0\}$ has been chosen but that $\mathbf p=\{p_k:k\ge1\}$
is unknown. We will make the following assumptions:
\begin{enumerate}
\item there is an $M>0$ such that $|w_v|\le M$ for all $v\ge0$, and
\item $K<\infty$.
\end{enumerate}
These conditions guarantee that the sum in \eqref{eq: theta} always converges.  Note that the assumption that $|w_n|\le M$ is satisfied by all of the linear diversity indices discussed in Section \ref{sec: Entropic Basis}.
Note further that we are not assuming that $K$ is known, only that it is known that $K<\infty$.  This is realistic in many applications including ecology, where there is a finite (even if very large) number of species.

For simplicity of notation assume the frequencies $p_k$ are ordered such $p_k>0$ for $k=1,2,\dots,K$ and that $p_k=0$ for $k>K$. For $t\in[0,1]$ let
$$
h(t) = \left\{\begin{array}{ll}
  \sum_{v=0}^\infty w_v  (1-t)^v & \mbox{if }t\in(0,1]\\
0& \mbox{if }t=0
\end{array}\right..
$$
In this case $w_v = (-1)^vh^{(v)}(1)/v!$ and
$$
\theta=\sum_{k=1}^K p_k h(p_k).
$$

Let $\{x_k=\sum_{i=1}^n 1[X_i=\ell_k]\}$ be the sequence of observed counts in our sample and let $\{\hat p_k = x_k/n\}$ be the sample proportions. Perhaps the most intuitive estimator of $\theta$ is the plug-in estimator given by
\begin{eqnarray}\label{eq: plug in}
\hat \theta_n = \sum_{v=0}^\infty w_v \sum_{k=1}^K \hat p_k(1-\hat p_k)^v = \sum_{k=1}^K \hat p_k h(\hat p_k).
\end{eqnarray}
However, it is well known that, in many important situations, the plug-in estimator has a bias that decays very slowly. For instance, in the case of Shannon's entropy ({\it i.e.}, when $w_0=0$ and $w_v=1/v$ for $v\ge1$), the bias decays no faster that $\mathcal O(1/n)$, see {\it e.g.}, Paninski (2003). We now propose another estimator, which has a bias that always decays at least exponentially fast. Our approach is influenced by the estimator of Shannon's entropy derived in Zhang (2012).

First note that
\begin{eqnarray}\label{eq: second term}
\theta = w_0+\sum_{v=1}^{n-1} w_v \sum_{k=1}^K p_k (1-p_k)^v+\sum_{v=n}^\infty w_v \sum_{k=1}^K p_k (1-p_k)^v =: \eta_{n}+B_{2,n}.
\end{eqnarray}
From Zhang and Zhou (2010), we know that an unbiased estimator of $\eta_n$ is given by
\begin{eqnarray}\label{eq: estimator}
\hat \theta_n^\sharp &=& w_0+\sum_{v=1}^{n-1} w_v \sum_{k=1}^K \hat p_k \prod_{j=1}^{v}\left(1-\frac{x_k-1}{n-j}\right)\nonumber\\
&=& w_0+\sum_{k=1}^K \hat p_k \sum_{v=1}^{n-x_k} w_v \prod_{j=1}^{v}\left(1-\frac{x_k-1}{n-j}\right) = w_0+\sum_{k=1}^K \hat \theta^\sharp_{n,k},
\end{eqnarray}
where
\begin{eqnarray}\label{eq: estimator k term}
\hat \theta_{n,k}^\sharp = \hat p_k \sum_{v=1}^{n-x_k} w_v \prod_{j=1}^{v}\left(1-\frac{x_k-1}{n-j}\right)= \hat p_k \sum_{v=1}^{\infty} w_v \prod_{j=1}^{v}\left(1-\frac{x_k-1}{n-j}\right).
\end{eqnarray}
It may, at first, appear that one needs to know $K$ in order to evaluate this estimator. However, if a category $k$ is not observed then $\hat p_k=0$ and hence $\hat\theta_{n,k}^\sharp=0$ and does not need to be included in the sum. Thus one does not need to know the value of $K$ in order to evaluate this estimator. A similar comment holds for the plug-in estimator given in \eqref{eq: plug in}.

By construction, the bias of the estimator $\hat \theta_n^\sharp$ is given by $B_{2,n}$, and letting $p_\wedge=\min\{p_k:1\le k\le K\}$ we see that
$$
\left|B_{2,n}\right|\le M K(1-p_{\wedge})^n,
$$
which decays exponentially fast in $n$. We note that in the case of Shannon's entropy this estimator corresponds with the estimator introduced in Zhang (2012) and Zhang (2013).  For that estimator, an approach to further reduce the bias was presented in Zhang and Grabchak (2013).  One can modify that approach for our more general situation.  This will be dealt with in a future work.

Next, we will establish that $\hat \theta^\sharp_n$ is a consistent and asymptotically normal estimator of $\theta$. Along the way, we will show the corresponding results for the plug-in estimator $\hat\theta_n$. Our approach is similar to the one used in Zhang (2013) to prove the asymptotic normality of an estimator of Shannon's entropy.

Let us define the $(K-1)$-dimensional vectors
$$
\mathbf v=(p_1,\cdots,p_{K-1})^{\tau} \mbox{ and } \hat {\mathbf v}_n=(\hat{p}_{1},\cdots,\hat{p}_{K-1})^{\tau},
$$
and note that $\hat{\mathbf  v}_n\conp \mathbf  v$ as $n\to\infty$. Moreover, by the multivariate normal approximation to the multinomial distribution
\begin{eqnarray}\label{lemma0}
   \sqrt{n}(\hat {\mathbf  v}_n-\mathbf v)\stackrel{L}{\rightarrow} MVN(0, \Sigma(\mathbf  v)),
\end{eqnarray}
where $\Sigma(\mathbf  v)$ is the $(K-1)\times (K-1)$ covariance matrix given by
\begin{eqnarray}\label{sigmamatrix}
\Sigma(\mathbf  v)=\left(\begin{array}{cccc}
                p_{1}(1-p_{1}) & -p_{1}p_{2}&\cdots & -p_{1}p_{K-1}  \\
                -p_{2}p_{1}& p_{2}(1-p_{2})  &\cdots & -p_{2}p_{K-1} \\
                \cdots&\cdots  &\cdots & \cdots \\
                -p_{K-1}p_{1}&- p_{K-1}p_{2}  &\cdots & p_{K-1}(1-p_{K-1})
                \end{array}\right).
\end{eqnarray}
Let
\begin{eqnarray*}
G(\mathbf v) =\sum_{k=1}^{K-1}p_k h(p_k) + \left(1-\sum_{k=1}^{K-1}p_k\right) h\left(1-\sum_{k=1}^{K-1} p_k\right)
\end{eqnarray*}
and
$$
g(\mathbf v):=\nabla G(\mathbf v) = \left(\frac{\partial}{\partial p_{1}}G(\mathbf v),\cdots,\frac{\partial}{\partial p_{K-1}}G(\mathbf v)\right)^{\tau}.
$$
For each $j$, $j=1,\cdots,K-1$, we have
\begin{eqnarray*}
\frac{\partial}{\partial p_j} G(\mathbf v) &=& h(p_j)+p_jh'(p_j)-h\left(1-\sum_{k=1}^{K-1} p_k\right)-\left(1-\sum_{k=1}^{K-1} p_k\right)h'\left(1-\sum_{k=1}^{K-1} p_k\right).
\end{eqnarray*}
The delta method gives the following result.

\begin{proposition} \label{prop: asymp norm MLE}
If $\hat\theta_n$ is the plug-in estimator given by \eqref{eq: plug in} and  $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)>0$ then
\begin{eqnarray*}
\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)\left[ g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)\right]^{-\frac{1}{2}}\stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{proposition}

\begin{remark}\label{remark: simple cond}
It is well-known that $\Sigma(\mathbf v)$ is a positive definite matrix, see e.g.\ Tanabe and Sagae (1992).  For this reason, the condition $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)>0$ is equivalent to the condition that $g(\mathbf v)\ne0$.  The question of when this holds depends of the function $h$. In the case of entropy (when $h(t)=\log t$) and R\'enyi's equivalent entropy (when $h(t)=t^{r-1}$) it is easy to verify that $g(\mathbf v) = 0$ if and only if $p_k=1/K$ for $k=1,2,\dots,K$.
\end{remark}

In order to use Proposition \ref{prop: asymp norm MLE} in applications we need to be able to estimate $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)$. By the continuous mapping theorem, we can estimate $\Sigma(\mathbf v)$ by $\Sigma(\hat {\mathbf v})$. However, $g(\hat {\mathbf v})$ may not be defined when there are species that have not been observed in the sample. To deal with this, for $\mathbf x=(x_1,x_2,\dots,x_{K-1})^\tau\in[0,1]^{K-1}$ with $\sum_{i=1}^{K-1}x_i\in[0,1]$ define
$$
\bar g(\mathbf x) = (\bar g_{1}(\mathbf x), \bar g_2(\mathbf x),\dots,\bar g_{K-1}(\mathbf x))^\tau
$$
where for $j=1,2,\dots,K-1$
$$
\bar g_j(\mathbf x) = \left\{\begin{array}{ll}
   \frac{\partial}{\partial x_j} G(\mathbf x) & \mbox{if } x_j>0\\
0 & \mbox{otherwise}
             \end{array}
\right..
$$
Since $\hat {\mathbf v}_n\conp \mathbf v$, $\bar g$ is continuous for all $\mathbf x\in(0,1]^{K-1}$, and $\mathbf  v \in(0,1]^{K-1}$, the continuous mapping theorem implies that $\bar g(\hat {\mathbf v}_n)$ is a consistent estimator of $g(\mathbf v)$. From this and Slutsky's Theorem we get the following.

\begin{corollary} \label{cor: estim asympt norm MLE}
If $\hat\theta_n$ is the plug-in estimator given by \eqref{eq: plug in} and  $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)>0$ then
\begin{eqnarray*}
 \sqrt{n}\left(\hat{\theta}_{n}-\theta\right)  \left[\bar g^{\tau}(\hat {\mathbf v}_n)\Sigma(\hat {\mathbf v}_n)\bar g(\hat {\mathbf v}_n)\right]^{-\frac{1}{2}}
 \stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{corollary}

Since both $\Sigma(\hat {\mathbf v}_n)$ and $\bar g(\hat {\mathbf v}_n)$ have zeros in locations that correspond to unobserved species, we can pretend that these species do not exist for the purposes of estimating $\bar g^{\tau}(\hat {\mathbf v}_n)\Sigma(\hat {\mathbf v}_n)\bar g(\hat {\mathbf v}_n)$. For this reason, we do not actually need to know $K$ to evaluate this quantity. We now extend our results to the estimator defined in \eqref{eq: estimator}.

\begin{theorem} \label{thetheorem}
If $\hat\theta_n^\sharp$ is the estimator given by \eqref{eq: estimator} and $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)>0$ then
 \begin{eqnarray*}
\sqrt{n}\left(\hat{\theta}_{n}^{\sharp}-\theta\right)\left[g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)\right]^{-\frac{1}{2}}\stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{theorem}

Before giving the proof, we state the following corollary. Its proof is similar to that of Corollary \ref{cor: estim asympt norm MLE}.

\begin{corollary} \label{cor: estim asympt norm sharp}
If $\hat\theta_n^\sharp$ is the estimator given by \eqref{eq: estimator} and $g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)>0$ then
\begin{eqnarray*}
 \sqrt{n}\left(\hat{\theta}_{n}^\sharp-\theta\right)  \left[ \bar g^{\tau}(\hat {\mathbf v}_n)\Sigma(\hat {\mathbf v}_n)\bar g(\hat {\mathbf v}_n)\right]^{-\frac{1}{2}}
 \stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{corollary}

As before, note that we do not need to know $K$ to evaluate $\bar g^{\tau}(\hat {\mathbf v}_n)\Sigma(\hat {\mathbf v}_n)\bar g(\hat {\mathbf v}_n)$. The proof of Theorem \ref{thetheorem} will be based on the following.

\begin{lemma}\label{lemma: rates}
For $p\in[0,1]$ and $n\in\mathbb N$ let
$$
g_n(p) = p \sum_{v=1}^{\lfloor n(1-p)+1\rfloor} w_v \prod_{j=1}^{v}\left(1-\frac{np-1}{n-j}\right)
$$
and let
$$
g(p) =  p \sum_{v=1}^\infty w_v (1-p)^v.
$$
1. If $0<c<d<1$ then
$$
\lim_{n\to\infty}\sup_{p\in[c,d]}\sqrt n|g_n(p)-g(p)|= 0.
$$
2. Let $p_n\in [0,1]$ such that $np_n \in\{0,1,2,\dots,n\}$ then
$$
\sqrt n \left|g_n(p_n)-g(p_n)\right| \le  \sqrt n (n+1) \le 2n^{3/2}.
$$
\end{lemma}

\begin{proof}
Note that
\begin{eqnarray*}
\sqrt n|g_n(p)-g(p)| &\le& M\sqrt{n} p \sum_{v=1}^{\lfloor n(1-p)+1\rfloor}  \left|\prod_{j=1}^{v}\left(1-\frac{np-1}{n-j}\right)-(1-p)^v\right|\\
&& + M\sqrt{n} p \sum_{v=\lfloor n(1-p)+2\rfloor}^{\infty}  (1-p)^v =: M(\Delta_1+\Delta_2).
\end{eqnarray*}
We begin by showing Part 1. Throughout the proof of this part, we assume that $n>2/c$; this ensures that $n(1-p)+1<n-1$ for all $p\in[c,d]$. Fix $v\in\mathbb N$ such that $v\le n(1-p)+1$. Note that
$$
\prod_{j=1}^v\left(1-\frac{np-1}{n-j}\right) = \prod_{j=0}^{v-1}\left(\frac{1-p-\frac{j}{n}}{1-\frac{j+1}{n}}\right)
$$
and thus
\begin{eqnarray*}
\left|\prod_{j=1}^{v}\left(1-\frac{np-1}{n-j}\right)- (1-p)^v\right| = (1-p)^v\left|\prod_{j=0}^{v-1}\left(\frac{1-\frac{j}{n(1-p)}}{1-\frac{j+1}{n}}\right)- 1\right| \le (1-p)^{v-1}\frac{v^2}{n-v},
\end{eqnarray*}
where the inequality follows by the proof of Part 1 of Lemma 2 in Zhang (2013).  Let $V_n = \lfloor n^{1/8}\rfloor$. For large enough $n$, $V_n<\lfloor n(1-d)+1\rfloor$. For such $n$ we have
\begin{eqnarray*}
\Delta_1 &\le& \sqrt{n} p \sum_{v=1}^{\lfloor n(1-p)+1\rfloor}(1-p)^{v-1}\frac{v^2}{n-v}\\
&=& \sqrt{n} p \sum_{v=1}^{V_n}(1-p)^{v-1}\frac{v^2}{n-v}+\sqrt{n} p \sum_{v=V_n+1}^{\lfloor n(1-p)+1\rfloor}(1-p)^{v-1}\frac{v^2}{n-v}=:\Delta_{11}+\Delta_{12}.
\end{eqnarray*}
We have
\begin{eqnarray*}
\Delta_{11} \le \sqrt{n} p \frac{V_n^2}{n-V_n}\sum_{v=1}^\infty (1-p)^{v-1} =  \sqrt{n} \frac{V_n^2}{n-V_n} \le \frac{n^{3/4}}{n-n^{1/8}} \to 0,
\end{eqnarray*}
\begin{eqnarray*}
\Delta_{12} &\le& \sqrt{n} p (1-p)^{V_n}\frac{( n(1-p)+1)^2}{n-( n(1-p)+1)} \sum_{v=1}^{\infty} (1-p)^{v-1}\\
&=& \sqrt{n} (1-p)^{V_n}\frac{[n(1-p)+1]^2}{np-1} \le \sqrt{n} (1-c)^{\lfloor n^{1/8}\rfloor}\frac{[n(1-c)+1]^2}{nc-1}  \to0,
\end{eqnarray*}
and
\begin{eqnarray*}
\Delta_2 \le \sqrt{n} p (1-p)^{\lfloor n(1-p)+2\rfloor} \sum_{v=0}^{\infty}  (1-p)^v \le \sqrt{n} (1-c)^{\lfloor n(1-d)+2\rfloor}\to0.
\end{eqnarray*}

Now to show Part 2.  Note that $p_n\ne0$ implies $p_n\ge1/n$, which means that $\frac{np-1}{n-j}\in[0,1]$ when $j\le n(1-p)+1$. Thus, either $p_n=0$ or  $\prod_{j=1}^{v}\left(1-\frac{np_n-1}{n-j}\right)\in[0, 1]$ when $v\le n(1-p)+1$. This implies that
\begin{eqnarray*}
\Delta_1 &=&  \sqrt{n} p_n \sum_{v=1}^{\lfloor n(1-p_n)+1\rfloor}  \left|\prod_{j=1}^{v}\left(1-\frac{np_n-1}{n-j}\right)-(1-p_n)^v\right| 1_{p_n\ne0} \le \sqrt n (n+1).
\end{eqnarray*}
and
\begin{eqnarray*}
\Delta_2 \le \sqrt{n} p_n \sum_{v=0}^{\infty}  (1-p_n)^v 1_{p_n\ne0}\le \sqrt n.
\end{eqnarray*}
This completes the proof.
\end{proof}

\noindent {\it Proof of Theorem \ref{thetheorem}.} Since $\sqrt{n}\left(\hat{\theta}_{n}^{\sharp}-\theta\right)= \sqrt{n}\left(\hat{\theta}_{n}^{\sharp}-\hat{\theta}_{n}\right)+\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)$, by Proposition \ref{prop: asymp norm MLE} and Slutsky's Theorem it suffices to show that $\sqrt{n}\left(\hat{\theta}_{n}^{\sharp}-\hat{\theta}_{n}\right)\stackrel{p}{\rightarrow}0$.  Further, by Slutsky's Theorem it suffices to show that $\sqrt{n}\left(\hat{\theta}_{n,k}^{\sharp}-\hat{\theta}_{n,k}\right) \stackrel{p}{\rightarrow}0$, where $\hat{\theta}_{n,k}^{\sharp}$ is given by \eqref{eq: estimator k term} and
$$
\hat\theta_{n,k} = \hat p_k \sum_{v=1}^\infty w_v (1-\hat p_n)^v.
$$

We can write
\begin{eqnarray*}
\sqrt{n}\left(\hat{\theta}_{n,k}^{\sharp}-\hat{\theta}_{n,k}\right) &=& \sqrt{n}\left(\hat{\theta}_{n,k}^{\sharp}-\hat{\theta}_{n,k}\right)1_{\hat p_k\le p_k/2} + \sqrt{n}\left(\hat{\theta}_{n,k}^{\sharp}-\hat{\theta}_{n,k}\right)1_{\hat p_k \ge (1+p_k)/2}\\
&& + \sqrt{n}\left(\hat{\theta}_{n,k}^{\sharp}-\hat{\theta}_{n,k}\right)1_{p_k/2 < \hat p_k <  (p_k+1)/2} =: \mathcal A_1+\mathcal A_2.
\end{eqnarray*}
By Part 2 of Lemma \ref{lemma: rates} it follows that
\begin{eqnarray*}
\mathrm E\left|\mathcal A_1\right| &\le& 2n^{3/2} \left[\mathrm P\left(\hat p_k > (1+p_k)/2\right)+ \mathrm P\left(\hat p_k\le p_k/2\right)\right] \\
&\le& 2n^{3/2} \left[\mathrm P\left(|\hat p_k-p_k| \ge (1-p_k)/2\right)+ \mathrm P\left(|\hat p_k-p_k|\ge p_k/2\right)\right]\\
&\le& 4n^{3/2} \left[e^{-n(1-p_k)^2/2}+e^{-np_k^2/2}\right]\to0,
\end{eqnarray*}
where the third line follows by Hoeffding's inequality, see Hoeffding (1963). Thus it follows that $\mathcal A_1\conp0$. By Part 1 of Lemma \ref{lemma: rates} it follows that $\mathcal A_2\conp 0$. \qed

\section{Estimation of R\'enyi's Entropy}\label{sec: estimate Renyi}

The only diversity indices that we have discussed that do not belong to the class of linear diversity indices are R\'enyi's entropy and Hill's diversity number.  However,  both are transformation of R\'enyi's equivalent entropy, $h_{r}$. In this section we will extend Theorem \ref{thetheorem} to R\'enyi's entropy.  We can use a similar approach to extend it to Hill's diversity number.

Fix $r>0$ such that $r\ne1$ and let $\psi_r(t) =(1-r)^{-1}\ln t$. Note that
$$
H_r = \frac{\ln h_r}{1-r} = \psi_r(h_r),
$$
where $H_r$ is R\'enyi's entropy and $h_r$ is R\'enyi equivalent entropy. Let
$$
\hat h_r^\sharp(n) = 1+\sum_{k=1}^K \hat p_k \sum_{v=1}^{n-x_k} w_v \prod_{j=1}^{v}\left(1-\frac{x_k-1}{n-j}\right),
$$
where for $v\ge1$
$$
w_v = \prod_{i=1}^{v}\left(\frac{i-r}{i}\right).
$$
This is the estimator of $h_r$ given by \eqref{eq: estimator}. Let
$$
\hat H_r^\sharp(n) = \frac{\ln \hat h^\sharp_r(n)}{1-r}=\psi_r\left(\hat h^\sharp_r(n)\right).
$$
Since $\psi'_r(t) =t^{-1}(1-r)^{-1}$ the delta method together with Theorem \ref{thetheorem}, Remark \ref{remark: simple cond}, and the fact that $\psi'_r(h_r)>0$ implies the following.

\begin{theorem} \label{thetheorem Renyi}
Provided that there exists a $k\in\{1,2,\dots,K\}$ with $p_k\ne1/K$
 \begin{eqnarray*}
\sqrt{n}\left(\hat{H}_{r}^{\sharp}(n)-H_r\right)h_r(1-r)\left[g^{\tau}(\mathbf v)\Sigma(\mathbf v) g(\mathbf v)\right]^{-\frac{1}{2}}\stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{theorem}

We note that in the case when $r=2$ this result is given in Leonenko and Seleznjev (2010). By arguments similar to the proof of Corollary \ref{cor: estim asympt norm MLE} we get the following.

\begin{corollary} \label{cor: estim asympt norm sharp Renyi}
Provided that there exists a $k\in\{1,2,\dots,K\}$ with $p_k\ne1/K$
\begin{eqnarray*}
 \sqrt{n}\left(\hat{H}_{r}^\sharp(n)-\theta\right)  \hat h^\sharp_r(n) (1-r)\left[ \bar g^{\tau}(\hat {\mathbf v}_n)\Sigma(\hat {\mathbf v}_n) \bar g(\hat {\mathbf v}_n)\right]^{-\frac{1}{2}}
 \stackrel{L}{\rightarrow} N(0, 1).
\end{eqnarray*}
\end{corollary}

\section*{acknowledgements}

The authors wish to thank Dr.\ Eric Marcon for correcting a mistake in the formula for $\hat h_r^\sharp(n)$.

redpajama_set_name: RedPajamaArXiv


An analytical procedure involves assessing the viability of financial information to identify any inconsistencies and unexpected relationships. The analytical procedure is critical to the audit process as it gives understanding into the organisation and its environment. The ISA 520 Analytical Procedures standard states the definition and purpose of analytical procedures.
Current and previous year's figures.
Current and budgeted or forecasted figures.
Organisational and industry average figures.
For the purpose of planning the analytical procedure helps in giving understanding into the performance of the organisation over the year and any significant changes to the business such as Sales figure rising by 50% in comparison to last year. In addition, any deviation of previous year figures, budgets or the auditor's knowledge that indicates any misstatement has to be investigated during the final audit process.


The weather has turned cold here in New York, and I'm not in the mood to go outside with the camera hunting for shots while my hands slowly freeze. So I've decided to take my art indoors- first with the shower spirograph, and now with this, the bathtub-food-coloring-painting experiment. Perhaps one day soon I will have completely combined my two obsessions- art, and spending time in the bathroom.
They look like dragons... esp that red on green/blue one [4th from top] - it looks extra menacing.
i love these prints- it's so nice to do something abstract for a change.
very nice. looks like smoke.
These remind me of a type of imaging called "Schlieren Photography" that is used to reveal slight gradations in the density of a liquid or gas.
Have you thought about trying highlighter ink plus blacklight?
tom, as usual you have just blown my mind. of course, i'll most likely be killed by ink inhallation later on tonight. thanks bro!


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text:
The Czech Republic is our friend and strategic partner
| source: Cabinet Office Of The Prime Minister
Viktor Orbán expressed his congratulations to Petr Fiala in a letter on the occasion of his election as Prime Minister of the Czech Republic, head of the Prime Minister's Press Office Bertalan Havasi informed Hungarian news agency MTI.
"We regard the Czech Republic as a friend and a strategic partner", Viktor Orbán underlined in the letter, highlighting: "The challenges of recent years, such as the global pandemic and increasing migration pressure, have highlighted the importance of close cooperation between the countries of Central Europe.
The Hungarian Prime Minister expressed his hope that he and his new Czech counterpart will succeed in further reinforcing the already outstanding bilateral relations between the two countries, and in expanding broad cooperation within the framework of the Visegrád Group (V4).
Viktor Orbán assured Petr Fiala that Hungary's government is ready to enter into cooperation in the interests of a strong Europe based on strong member states, while placing at the forefront stability, competitiveness, and respect for values that are based on Christian tradition.
Never before has the EU been under such migration pressure
Prime Minister Viktor Orbán's press statement after his talks with Prime Minister of the Czech Republic Andrej Babiš
Central Europe is facing a decade offering fantastic opportunities
congratulations Czech Republic expressed partner Petr Fiala strategic
© All right reserved, 2023. | archive | impressum

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Market research and intelligence was front and center in a recent webinar presented by Public Spend Forum (PSF) and NIGP, the National Institute for Government Procurement. Hosts Raj Sharma and Frank McNally of PSF covered key principles and practical tips for conducting effective market research in as little time as possible. Based on extensive research and experience working within procurement across public and private sectors, attendees were treated to insight on asking the right questions early in the acquisition process, knowing where to find information on their product and service needs, and then market research strategies that promote best value outcomes.
Thank you to everyone who attended the webinar and for those who missed it, check out the full webinar below! We appreciate your interest in the topic and the excellent questions you shared with us during the session. Please check out our in-depth recap of this webinar for more information. This is just the first part of a multi-part series we're running here at Public Spend Forum on this very important topic of market research, so stay tuned for more events and webinars from us to help you do your best market research EVER!


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You are here: Home / News / Burns Night Celebrations in Art at Manchester Airport
Burns Night Celebrations in Art at Manchester Airport
Manchester Airport's international artist in residence latest work is to be exhibited at a special event to celebrate Burns night.
Rob Pointon, who became the airport's artist in residence in 2016, recently visited Inverness and Orkney with Loganair to capture the beauty of Scotland in his paintings. He will be showcasing his work on the 25th of January at a Scottish themed Platform Airport City event at the Manchester Airport interchange.
The Platform Airport City event is free to attend will take place between 11.30am and 2.00pm with traders including 4 Lunch, Somtam and Backs Deli offering a wide range of meals and snacks including some Scottish themed delicacies. Loganair will also be attending and offering one lucky a visitor the chance to win a pair of flights to Stornoway (Isle of Lewis).
St Margaret's Hope, by Rob Pointon, will be exhibited at the Burns Night exhibition at Manchester Airport
Pointon, from Cheshire, will also be on hand to talk about his latest work. Scenes painted include Scapa Flow, Kirkwall Harbour, Aviemore and the Cairngorn Mountains as Rob battled the Scottish winter.
Rob Pointon, said:
"I am enormously grateful to Loganair and Manchester Airport for giving me the opportunity to further explore some far-flung corners of our beautiful country. The colours of Scotland are unique, the clear air can give the great glens and mountains a startling blue appearance. I hope visitors to Manchester Airport enjoy our Burns Night exhibition and are inspired to set off on their own Scottish adventure."
Sam Talbot, Marketing Manager at Loganair said:
"Many of our Scottish destinations could be described as "picture perfect", so we are delighted to partner with Manchester Airport and their artist in residence to illustrate this and inspire those looking for a break closer to home this summer."
"It has never been easier to "get away from it all" without travelling to the ends of the earth: You can explore the dramatic highlands of Inverness, discover ancient archaeological sites in Orkney, spot diverse wildlife in Shetland or stroll the unspoiled white sand beaches of Lewis, all direct from your local airport with Loganair"
Platform Airport City, located within Manchester Airport's Transport Interchange, is a vibrant and interactive space for visitors and employees working at Airport City Manchester to socialise, network and enjoy food and drink from around the world.
Hosted on the last Friday of each month between 11:30am and 2pm, Platform Airport City brings together a variety of pop-up food stalls from specially-picked traders, showcasing cuisines from destinations that Manchester Airport flies to alongside the best of the Greater Manchester community.
For more information visit www.robpointon.co.uk/home or www.manchesterairport.co.uk or to book direct flights from Manchester to Kirkwall, Stornoway or Sumburgh, visit www.loganair.co.uk


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Home Portfolio Item Terrahawks
By Jamie Anderson
Terrahawks first aired in 1983, and was the first Gerry Anderson puppet series produced since The Secret Service had ended in 1969. Moreover, due to a noticeable downturn in his fortunes, it was the first television series he'd made at all since production had ended on the second season of Space:1999 at the end of 1976. The puppets that would star in Terrahawks however would not be the marionettes he was best known for working with, but instead latex hand puppets capable of a greater freedom of movement than their predecessors. Supermarionation was no more, and instead Terrahawks would be produced in Supermacromation! (And whatever Hudsoncolor was.)
The story of Terrahawks starts in the year 2020, as an alien space armada led by the android Zelda, Queen of the planet Guk, obliterates a human colony on Mars before occupying the planet and declaring war on Earth. Mankind's only hope for survival are the Terrahawks, an elite fighting force with a secret base hidden beneath the White House in South America and led by the stern but fair Doctor Tiger Ninestein. As his name implies, Ninestein is one of nine clones, with the remaining eight quietly living out their lives elsewhere on Earth ready and waiting to have their minds wiped and replaced with a backup copy of Tiger's brain patterns should the worst occur.
The less nightmarish side of the Terrahawks organisation includes second-in-command Captain Mary Falconer, Captain Kate Kestrel and her co-pilot Lt Hawkeye aboard the Hawkwing fighter, and Lt Hiro who mans the Terrahawks first line of defence, Spacehawk, and has about as much luck repelling alien intruders as the SHADO interceptors used to.
From Hawknest, the Terrahawks are able to launch an array of fighting machines and equipment to combat Zelda's continual assaults, but the android Queen is a formidable opponent – and she isn't alone. Also residing in her Martian base are her slovenly son Yung-star, and her airheaded fashion obsessed sister Cy-star, while somewhere in the base is a cryogenic store carried dozens of alien exiles and refugees who have aligned with Zelda to destroy humanity. These included Sram, a demon drum player whose roar could bring down mountains, Yuri the telekinetic Space Bear, Lord Tempo the Time Lord, and MOID, the Master of Infinite Disguise, plus an army of robotic Cubes.
Aside from Zelda, whom any older unpopular woman in the public eye is always at risk of being compared to, perhaps the most memorable characters in the show were the Zeroids, the Terrahawks' army of spherical robot soldiers led by Sergeant Major Zero. Essentially reprising his role of Sergeant Major Williams from BBC sitcom, Windsor Davies excelled in the role of Zero, the loyal and lovable robot permanently locked in a battle of wits with Doctor Ninestein. Other Zeroids under his command included the moustachioed French Zeroid 18, rhyming Zeroid 55, and stuttering Zeroid 21, while the Zeroid contingent on Spacehawk was led by the fussy and increasingly camp Space Sergeant 101.
Fans expecting a long-awaited triumphant return to the glory days of Thunderbirds were to be disappointed when Terrahawks first aired in 1983. The show was produced on a significantly lower budget than those of the ITC era, and the earliest episodes showed that the stiff and serious nature of its the stories clashed with the cartoony look of the puppets and pantomime behaviour of the villains. Realising this, Terrahawks very quickly embraced its own shortcomings and used them to its advantage, becoming a madcap and at times deeply subversive comedy. This had the effect of lending greater weight to the dramatic stakes on the rare occasions that the show tried to tell a more serious story, while also providing more of a focus on the characters and their relationships than had been evident in the earliest episodes.
The first season of 26 episodes ended with a cliffhanger that saw Cy-star make a startling announcement, and as the second season opened a devious new character joined the Martian hordes. It-Star possessed two personalities; a timid little girl, and a male mad scientist with a German accent. Even for a series that was already very weird It-star's addition to the cast plays like something out of a fever dream, and the scene of his birth was so inappropriate for a child audience that it was edited out on first broadcast.
Adding to the strangeness was Kate Kestrel's side career as an international pop superstar, regularly taking a break from her Terrahawks duties to record songs at Anderburr Records, under the supervision of studio manager Chic King and permanently stressed recording engineer Stew Dapples. In real life, singer Moya Griffiths provided Kate's singing voice and made public appearances to promote the show, and even released a 7 inch vinyl featuring two songs from the show.
Despite proving popular with younger viewers, Terrahawks came to an end after 39 episodes, but as it turned out this was to be more of an extended hiatus than a cancellation. In 2014 the series returned in audio form from Big Finish Productions, featuring three of the original cast – Denise Bryer, Jeremy Hitchen and Robbie Stevens, plus newcomer Beth Chalmers inheriting the late Anne Ridler's characters.
The new audio episodes brought back all the classic characters, vehicles and settings of the original television series, while bringing a uniquely 21st-century attitude to the stories. In addition to new regular characters and vehicles, the audio episodes of Terrahawks also succeeded in bringing a greater emotional core to the series, while addressing some of the moral and ethical questions regarding the clones that the television show had largely ignored. The result was 26 brand new episodes produced over the next three years.
Despite being one of the more controversial Gerry Anderson series, particularly among older viewers, Terrahawks has remained a fond and enduring memory for its many fans. Its lovable characters and offbeat sense of humour have ensured it continues to find new viewers on DVD and blu-ray, and its recent critically-acclaimed audio revival proved that the series still had a wealth of untapped potential yet to be explored. More than 35 years after it first appeared on our screens, Terrahawks still continues to "Stroll on!"
Anderson Burr Pictures/LWT – 1983
39 episodes x 25 mins
Producers: Gerry Anderson & Christopher Burr
Associate Producer: Bob Bell
Directors: Alan Pattillo, Desmond Saunders, Tony Bell & Tony Lenny
Lighting Cameramen: Harry Oakes BSC & Paddy Seale
Art Director: Gary Tomkins
Special Effects Director: Steven Begg
Music Composed & Performed by Richard Harvey
Additional Music Composed by Gerry Anderson & Christopher Burr
Voice Cast:
Jeremy Hitchen, Anne Ridler, Denise Bryer, Ben Stevens, Windsor Davies
Want to know more about the Terrahawks audio series from Big Finish? Read more here.
Terrahawks Episode List:
Expect The Unexpected Part 2
Thunder-Roar
Close Call
From Here To Infinity
Space Samurai
The Sporilla
Happy Madeday
Gunfight At Oaky's Corral
The Ugliest Monster Of All
Thunder Path
Mind Monster
To Catch A Tiger
The Midas Touch
Operation S.A.S.
Ten Top Pop
Unseen Menace
Play It Again, Sram
My Kingdom For A ZEAF
Zero' Finest Hour
The Ultimate Menace
Ma's Monsters
Jolly Roger One
Terratomb
Cry UFO
Space Cyclops
Timewarp
Space Giant
Cold Finger
Operation Zero
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Gerry Anderson was born on 14 April 1929, and through a combination of skill and an incredible level of determination earned his first credit as a producer when he was still in his 20s. Years before the word 'brand' was popularly applied to television, Gerry's name came to represent an inimitable style of entertainment that proved hugely successful around the world. From Fireball XL5 to Space: 1999, Captain Scarlet to Space Precinct, Supercar to Thunderbirds - he has become the UK's equivalent of Walt Disney. During the 1960s Gerry used puppetry to realise epic science fiction scenarios that would have been impossible to achieve in anything other than miniature scale. The names of the shows he produced during this era are familiar to millions, and continue to entertain audiences today. An incredible feat unsurpassed by any other British TV and film producer. As the optimism of the 1960s faded, Gerry Anderson created darker scenarios for the new decade. Live-action series UFO and Space: 1999 anticipated the accomplishments of Star Wars and are still regarded by many as his finest work. He was never content to stand still, and his programmes became increasingly diverse in the 1980s. He made a welcome return to puppetry for Terrahawks and brought the animated private detective Dick Spanner to the screen, while developing a parallel career as an award-winning director of sophisticated television commercials. In the 1990s he created and produced the lavish live-action series Space Precinct, before returning to children's entertainment with the enchanting Lavender Castle. In 2001 he was awarded an MBE for services to the British film industry. The astonishing New Captain Scarlet premièred in 2005, by which time Gerry (then 76) was long past the age when most people consider retiring. His enthusiasm for the next project, and his fascination for the latest technology, remained undimmed through much of his final illness. He passed away on 26 December 2012. Despite being relatively shy, and never really spending much time looking back and reminiscing over his achievements Gerry Anderson came to appreciate that for many people his shows represented a special time in their lives. And he was proud that the ingenuity and optimism of his characters galvanised some viewers who went on to distinguished careers within film, television and other fields. Thunderbirds has inspired movies, stage shows and a vast array of merchandise, all reflecting its phenomenal impact on our popular culture, and together with his other shows is cherished by many generations of viewers.
Our Top 10 Gerry Anderson Title Sequences
Jamie Anderson
Finding Straker's Car
Shane Rimmer has Died
Thunderbirds: The Origins Of International Rescue
FAB Facts: Doctor Who guest stars who were extras on Space:1999
All the way with KLA! The inspiration for the Thunderbirds story Ricochet
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Ha! Bet I caught your attention with that title! There is something mysterious about gold other then its allure as a wealth asset. Sure you can make jewelry out of it, coins and a little more useful items like circuit boards, but are there other, a little more hidden, capabilities of the stuff?
Some people think so. If you search the net, you'll find a huge collection of postings that reference lectures given by the late David Hudson or Laurence Gardner. But it's pretty rare to find people that have taken their information (teachings) and applied them in order to take it to the next level.
I guess there are doers and watchers in this world just like there are readers and TV viewers. From looking at the ratio between people who watch TV verses those that don't, I would be willing to bet that a huge majority of the people are watchers rather than doers.
So, I decided to take a little walk using Google to see if there was anyone in the blog-o-sphere that is actively reproducing high spin M-state elements from sea water. I Googled "ORMUS making process -Sunwarrior" on Google blogs and started looking. (as a side note, I like the fact that Google allows you to look for postings that don't contain words. Sunwarrior has a product called ORMUS greens that has a big following.) After a few hours and poking around, following links and scanning through to see how they referenced the word "ORMUS", I have to conclude that no one is actively blogging about making ORMUS.
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That's just a shame. There are lots of products where people make versions of White Gold Powder and they must have learned how to perform this operation from someone somewhere. You'd think the process would be echoed on the net.
Maybe you'll find it useful and maybe I'll want to take a deeper look at some of the sites that Google found someday soon. Thus, what follows is a little outline of what I found.
1) This website, White Powder Gold lab is a blog that talks a lot about Ormus and monatomic elements, but doesn't seem to provide any new content. They do have a good selection (store) of books. Looks to me like they're just trying to sell books. Maybe they're making enough money selling books that they don't need to investigate ORMUS any more.
2) Another site called The Bamboo jungle has a link to EMR Labs, LLC where someone outlines how to make ORMUS. But the recipe is like all the others that simply precipitate minerals from sea water. There is no mention of anything that would lead me to believe they precipitated anything unusual. There is no mention of high-spin state or metaphysical (or any) interactions with the stuff.
3) Here is another blog, LabVirus that simply echo's old David Hudson stuff. The topics they cover sure don't make me feel good. Didn't like the name either and almost didn't visit figuring I'd just stumble across another virus or something.
4) Here is another, Ormus Living. Looks like the blog is a single article. Ha. I believe I've seen this signature in Ormus forums – Denis Of Oz. Unfortunately, doesn't look like he liked blogging. Maybe I'm just confusing the name with someone else.
However I think I am open minded enough to look at the content within the container and to discern for myself what is real and was isn't.
It totally made me smile. I wonder if that container is half empty or half full?
6) Here is a site that I'd hoped to find something at, yet, just more scare tactics about heavy metals and toxins. I always find it a little funny when someone assumes that lye is always toxic to the body. Its just sodium hydroxide. That stuff is used to make soap. People have been using it on their skin for generations, and what you put on your skin is absorbed in the body. Also, these people are precipitating elements found in edible salt. If there were toxic compounds in these salts, we would have found out a long time ago!
7) Here is a blogger that's caught my eye; RawketScience. This person has done a really nice write up on Ormus. It's short and to the point. I like the fact that she's referenced FOOD as an Ormus source. (Might touch on that in a later posting…) and I think I could spend some time on this blog.
So through my m-state works i found a lot of the techniques had some flaws to them. So i did some work on them. I don't mind sharing this information if anyone is interested. If you find anything wrong with this work or ways to simplify please speak up. If you would like to post your methods, please do so.
There are a good few different methods i will post here. Some of these methods are from others work. So i surely do not claim them to be of my own.
How to make M State gold from Dead sea salt.
2 Add 2 cups of distilled water.
3 Now have sea water from dead sea salt.
4 Pour sea water through a coffee filter to filter out scum.
5 Pour the sea water into a stainless steel pot. Slowly, drop-by-drop,add the food grade lye solution WHILE STIRRING. Every ten drops or so, test the pH. You might want to take at least 3 to 5 samples from different regions of the liquid.
6 bring the pH up to 9.5, then stop to be on the safe side. If you are using a pHmeter, stop just before you get to pH 10.78.
7 A white precipitate which includes m-state elements will form.
heavy metals. It is alleged that the Dead Sea salt water does not produce any Gilcrest precipitate. This has not been proven and should not be assumed.
9 Once you are at the correct pH, stop.
10 Pour the solution into a clean glass jar or test tube.
13 Add distilled water to the precipitate (filling the jar), stir thoroughly, and let it settle again for at least 4 to 5 hours, preferably overnight. Put the jar inside of a metal cookie jar inside of another metal jar . Steal wool between the jars will keep the solution away from magnetic fields.
14 Do 12 and 13 4 times to wash the material well. Using distilled water each time.
15 This step use to be to boil the solution for 5 minutes. Boiling is probably not the best idea unless you are using straight sea water and not using the dried sea salt reconstituted in water.
16 Take ph to 12 .
17 Filter material and keep liquid.
18 Use vinegar to bring the ph to 10.7 . Jello like precipitate will form.
19 To further filter the material to dry material m state elements.
This procedure removes the Mg(OH)2 by dissolving it below pH 9. First get some HCl (or muriatic acid) and coffee filters. A safer alternative to HCl is distilled white vinegar.
one or two hours. This forms a dry powder.
water. For example, one liter of water for one cup of powder.
the pH to 5 or 6.
9) The Book of ThoTH has an interesting historical mix of information about "ORMUS", but for some reason, I think they're over complicating things. Someone did a bunch of reading, but did they make ORMUS? I doubt it.
In the end, maybe I'll find a community on the net that doesn't hide behind what could be, but publishes what is.
Meanwhile, I've got some new books to read and another batch of sea salt to mix up. Hopefully this time, I'll get to the 'Jello' that is mentioned in the above directions.
ORMUS, Is this the wet method?
A couple years ago I played around with trying to distill M-State material from Dead Sea Salt and Celtic Sea Salt. I'd read the David Hudson lectures a number of times and searched the web for any sources that might have reproduced what David discovered.
One of the sources I came across was the website SubtleEnergies.com. After digging around a bit there, I found the Wet-Method Procedure for isolating M-State from these salts. The procedure can be found here.
After reading a bunch of sites and seeing that the directions were all very similar, I figured that the directions that I found were 'good enough' for me to get the job done.
Then, I set out to follow the instructions explicitly.
To sum up a couple months worth of testing, I never got to what I felt was M-State material. I remember coming across a statement from David Hudson that the material was like cow sperm (he had a farming background) or a gelatinous concoction. All I came up with was a milky white substance.
It left me a little discouraged. So, I packed up the chemistry set and figured I'd wait until the time was right and keep my eyes open.
Well, it just so turns out that I stumbled across a video on YouTube where it looks like HealingMindN is able to get to the point that I wanted to get to. Here is his video (also shown below). If you've been looking for a good video on the wet-method M-State distillation process, this might be it.
Special note: when watching the video, he adds music about 2:23 in. Fortunately, he's funneled it to the one speaker and his voice is on the other. If you want to hear him without the interference, just turn off the speaker playing the music and turn up the other.
Here are the steps as outlined by the video. I've made a few screen shots with the idea of lining up the wet-method with what it is that HealingMindN shows.
Mineral water. Turns out he's bought a German import, Apollinaris water. A quick search on the net finds an Amazon source. Search Amazon for product number: 030494881229.
So, that part holds up. In any case, he claims that 2 cups distilled water plus 1 teaspoon of Epsom salts may be substituted if the water is not found (video 00.15).
Half cup Celtic Sea Salt. That stuff looks like this (above). Search Amazon for product number: B000SWVPV8.
And, again, HealingMindN suggests that other sea salts may be substituted such as Lima Atlantic or Dead Sea Salt.
I believe he's saying that he's adding 450 milliliters to the beaker too which he adds the salt. After stirring for a moment, it sounds like he says he's going to let it sit overnight.
Great. That is a step that I do not remember from the Wet-Method procedure written up at SubtleEnergies. Yet, maybe it doesn't have to sit overnight.
Looks like the starting pH of the salt water is 5.9.
And now for his setup – food grade lye water. Looks like he's using Tropics Lye Water. Turns out that Amazon also shows this product.
He claims it's about 12.7 pH. Anyone should be able to validate that pretty easily. It looks like at second 2:10 in the video, he's got 140 milliliters of Lye Water that he'll be using.
Looks like he continues to add the lye solution until the Sea Water hits 10.5. He points out that it is point 1 to go from stop. Thus, he's looking for 10.6 as the stopping point for this pH swing.
…For the processes detailed in the book, a 1:8 strength works very, very well. That is ½ cup of lye (NaOH) crystals mixed as described in the book into 1 quart of distilled water. This would be about 125 grams per liter of water in the metric system.
At 3:17 in the video, The participate has formed, it's mainly Magnesium Hydroxide Lye … and a little M-State. He is now at pH 10.6.
Overall, he's added about 140 milliliters of lye just like he did last time and he's going to let it sit overnight. So that it settles.
He makes a comment to not use anything but glass so as to not disrupt the M-State material.
Here, he shows his Faraday cage that he uses to place the M-State material in overnight. His faraday cage is two cookie containers where the center one is lined with steel wool where there is a brown paper bag inside of that. His participate container has cellophane with a rubber band holding it on.
It might be work investigating this a little more. If M-State is sensitive to electromagnetic energy, shielding it might be a good idea. Yet, I'm not sure how well his box is working, for one of the key elements of a Faraday cage is grounding. Yet, let's continue.
He shows about 300 milliliters of precipitate and the covering liquid has a pH of about 10.4.
He comments that he's going to wash the precipitate three times.
Here he shows the plastic tubing that he's going to use to remove the liquid on top of the precipitate.
Here we can almost see the brand of distilled water that he's going to use to water the precipitate. He fills the container up to about 600 milliliters again so that he can let it settle again overnight.
At this point I have to assume that he goes through the washing process all three times before coming back to the video. Thus, at this point, he's just about doubled the water content in his beaker three times and removed the substrate covering the precipitate as he outlined. If he lets it settle overnight each time, a few days have passed.
He then jumps straight into the boiling process. We can see from his text that he boils for 5 minutes.
Here he shows the precipitate after boiling and he's going to take it to pH 12. He stated that last time it took nearly 50 milliliters of Lye water to perform that task.
Here he shows that he's raised the pH to 12.0. At this point, the M-State will be re-dissolved in solution. Shortly after showing the pH, he comments that he should have drained off more substrate (the clear liquid on top) for he recognizes that he has more volume than what he was expecting.
Here we see the filter that he's going to use to remove the precipitate leaving just the clear liquid holding the M-State material.
At this point, I wish I knew what that filter was. I've searched around on some different chemistry supply websites, but when you don't know what you're looking for, it makes things a little harder to find. Hopefully, I'll find a picture that matches what he's using.
Here is a good picture showing his flask with the rubber stopper holding the funnel filter. What gathers in the flask (the filtrate) is what he's looking for.
Here he shows the pH of the filtrate. He comments that it's gone down a little to 11.8.
Now he goes to bring the solution down to 10.78 pH. He uses acetic acid. The form of the acid is white distilled vinegar that you'd find in the grocery store. During this process, the 'coopers pairs' should start forming.
After watching this video, it seems pretty simple. I just might have to give it another try.
As a point of reference, I've included the process form SubtleEnergies.com below. The steps are nearly identical. The only real difference comes at the end when HealingMindN mentions that it's a feeling out process from here on out.
1. First, you might want to pour the sea water through a coffee filter to remove any scum.
2. If the starting material does not contain magnesium hydroxide (sea water does contain magnesium hydroxide), add some, or add a teaspoon of Epsom salts per gallon of water.
3. Pour the sea water into a stainless steel pot. Slowly, drop-by-drop, add the lye solution WHILE STIRRING. Every ten drops or so, test the pH. You might want to take at least 3 to 5 samples from different regions of the liquid. If you are using pH paper, the goal is to bring the pH up to 9.5, then stop to be on the safe side. If you are using a pH meter, stop just before you get to pH 10.78.
A white precipitate which includes m-state elements will form.
CAUTION: You must proceed slowly and patiently so that you do not exceed pH 10.78 with a meter or pH 9.5 with pH paper. If you go higher than pH 10.78, you might get a "Gilcrest precipitate" of toxic heavy metals. It is alleged that the Dead Sea salt water does not produce any Gilcrest precipitate. This has not been proven and should not be assumed.
4. Once you are at the correct pH, stop.
5. Pour the solution into a clean glass jar or test tube.
6. The white precipitate (slurry) slowly settles on the bottom of the jar. Let the slurry settle overnight. If metals or other toxins have been ruled out by prior testing of your starting material, the slurry is probably mostly calcium hydroxide, Mg(OH)2, lye, and a small amount of m-state.
You can speed this settling process with a centrifuge, which forces the precipitate to settle rapidly. Inexpensive second-hand centrifuges may be found at American Science and Surplus, http://www.sciplus.com.
7. Using a large syringe (or siphon), remove the liquid above the slurry.
8. Add distilled water to the precipitate (filling the jar), stir thoroughly, and let it settle again for at least 4 to 5 hours, preferably overnight.
9. Repeat steps 7 and 8 at least three times to thoroughly wash the precipitate. This should remove almost all of the lye. The remaining lye can be neutralized with HCl or distilled white vinegar as well. Washing three times is intended to reduce the dissolved "impurities" (like salt, for example) by 87.5%. Four washes would provide a 93.75% reduction, five washes a 96.875% reduction, and so on.
At this point, the precipitate is likely to contain some m-state, milk of magnesia Mg(OH)2, calcium, and perhaps some impurities.
Pour the precipitate and water into a stainless steel pot on a stove burner. A gas burner is preferred over electric because any magnetic fields from the electric burner may drive off some of the m-state material. Cover the pot with a lid to contain the m-state, and boil the solution for 5 minutes to sterilize it. Be careful not to spill the hot solution! Let it cool back to room temperature and recheck the pH to make sure it hasn't exceeded pH 9.
1. Boil before adding lye solution.
PROS: Faster reaction, faster precipitation. CONS: You may spill the hot lye solution. You may inhale fumes.
2. Boil while adding lye solution.
PROS: Faster reaction, faster precipitation. CONS: You may spill the hot lye solution. You may inhale fumes. Danger of lye spurting out of pot. Not recommended.
3. Boil and cool after adding lye solution.
PROS: No danger of inhaling fumes. Little danger of spilling hot lye solution. CONS: Slower reaction, slower precipitation.
4. Boil the washed precipitate (recommended).
PROS: No danger of inhaling fumes. No danger of spilling hot lye solution. pH is unlikely to change after boiling because the reaction has already taken place. CONS: Slower reaction, slower precipitation. If safety is the main issue, this seems to be the best method.
Caution: If you boil the solution on an electric burner, the magnetic field in the burner may "blow off" some of the m-state materials, resulting in a small yield. This can be minimized by adding a source of sodium (such as sodium hydroxide or salt) to the solution before boiling.
Since sea water contains sodium in salt, none of the boiling methods will be a problem with sea water. However, if you are starting with low-sodium fresh water, add a sodium source (such as table salt or lye solution) before boiling.
Once the precipitate and water have been sterilized, the next step is required to concentrate the m-state.
1. Suppose you just made a precipitate by adding lye solution to sea water. The precipitate is m-state mixed with Mg(OH) 2.
Superconductivity in the human body?
A few years ago I came across a very interesting video from a man talking about some mysterious white powder gold that seemed to ring true (at some level) with me. I don't remember the original video (and thus I don't have a link to it), but here is a link to the first of many YouTube clips that make up the lecture I heard.
I have to admit that this lecture sparked my curiosity and a couple years later I'm still researching aspects of what he talked about. If you have the time, well, if you're here you probably have the time, I'd recommend giving it a listen or reading a transcript or two.
If you're looking for a transcript, the website asc-alchemy has a bunch of them all strung together. You can find them here.
We've seen what light can do to molecules. Specific wavelengths of light (electromagnetic energy) can elevate electrons into higher orbits that allow for the formation of molecules like sugar. What David Hudson seems to be saying is that a superconducting atom (or atom pair) have electrons that stop acting like normal electrons, but acting like light – or a source of light.
Did you know that the way cells communicate with each other is by superconductivity? That the U.S. Naval Research Facility knows that the way cells communicate with each other is by superconductivity? That they have actually measured it using SQUIDS? Superconducting Quantum Interference Devices with a superconducting ring around the body. And they have seen by this procedure that literally light flowed between cell to cell to cell to cell. Did you know that your nerve impulses are not electricity that they travel closer to the speed of sound than they do to the speed of light? And electricity travels closer to the speed of light? Do you know what speed the superconducting wave travels? The speed of sound. This, in fact, is what is in your body that we call the consciousness. It's what separates you from a computer.
This last little bit opens the doors for investigation. The idea that the body could produce a non polar magnetic field, which is the aura, is a bit different then what I'd think it was based upon the body's ability to break down sugar with releases light (electromagnetic energy) – which might be seen by the correct sensitive equipment.
What's more interesting is that if these superconductive atoms are in our bodies, how might our bodies use them? And, probably more important, if these atoms are light (electromagnetic energy) sources, might some functions within our bodies need this type of electromagnetic energy to function properly?
He also makes mention of magnetic fields, electricity and how superconductors can store energy (and release energy). There is mention that to get electricity through a wire, you need voltage. Yet, to get it into a superconductor, you have to apply voltage at just the right frequency. Well, this makes me think just a bit, if superconductors can take or give light and if your body can break down molecules of sugar to release light, might these superconductive atoms act like batteries storing the light?
It's worth thinking about, thus it's worth posting. I'll think about this some more to see if there are other ideas I can post about this.


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"Ancient [and Modern] Gower": Presenting Shakespeare's Pericles
Dymkowski, C., 2007, The Narrator, the Expositor and the Prompter in European Medieval Theatre. Butterworth, P. (ed.). Turnhout: Brepols, p. 235-64
"And I was struck still by time": Contemporary British Theatre and the Metaphysical Imagination
Megson, C., 2013, Contemporary British Theatre: Breaking New Ground. Angelaki, V. (ed.). Basingstoke: Palgrave Macmillan, p. 32-56
"And The Wind Wheezing Through That Organ Once In A While": Voice, Narrative, Film
Gibson, A., 2001, In: New Literary History. p. 639-57
"Anglia libera": commonwealth politics in the early years of George I
Champion, J., 2005, Cultures of Whiggism. Womersley, D. (ed.). University of Delaware Press, p. 86-107
"Approaches to Intellectual History: the Case of Pre-Marxian Socialism"
Claeys, G., 2014, In: History of European Ideas. 40, p. 893-904 11 p.
"Appropriating primal indeterminacy": language, landscape and postmodern poetics in Susan Howe's Thorow
Montgomery, W., 2006, In: Textual Practice. 20, 4
"Approval Voting and Scoring Rules with Common Values
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"Aunt Anna's Report": the Buxton women and the Aborigines Select Committee, 1835-1837
Laidlaw, Z., May 2004, In: The Journal of Imperial and Commonwealth History. 32, 2, p. 1-28 28 p.
"Auteurs grecs", "Gloire", "Imagination", "Présomption", "Vanité", "Xénophon"
O'Brien, J. & Desan, P. (ed.), 2004, Dictionnaire de Michel de Montaigne. Paris: Champion
"Autheur de merveilleux poids": Montaigne et Xénophon
O'Brien, J., 2005, In: Montaigne Studies. 17, p. 17-34 18 p.
"Back story brought to centre stage": Review of Stephen Greenblatt, Will in the World: How Shakespeare Became Shakespeare
Ryan, K., 24 Dec 2004, Times Higher Education Supplement.
"Beauty and Promise": Sonic Narratives and the Politics of Freedom in the Literary Imagination of Lovelace
Ramnarine, T. K., 2008, Caribbean Literature After Independence: The Case of Earl Lovelace. Schwartz, B. (ed.). London: Institute for the Study of the Americas, p. 175-190
"Beauty Like a Tightened Bow": Analysing the Juxtaposition of Drama and Lyricism in Schubert's Paratactic Sonata-Form Movements
Hyland, A., 2013, (Accepted/In press) Irish Musical Studies. Four Courts Press, (Irish Musical Studies; vol. 11).
"Behold! The long-awaited new Hungarian opera has been born!" Discourses of denial and Petrovics' C'est la guerre
Beckles Willson, R., 2003, In: Central Europe. I, 2, p. 133-145
"Black Muslim Outreach and Interfaith Activism in Louis Farrakhan's Nation of Islam"
Gibson, D-M., 5 Apr 2021, (Submitted) In: Journal on Muslim Philanthropy and Civil Society.
"Blindness Gain" as "Worldmaking": audio description as a new "partage du sensible"
Thompson, H. & Chottin, M., 1 May 2021, (Submitted) In: L'Esprit Créateur.
"Bonnie Scotland, Eh?": Scottish Cinema, the Working Class and the Films of Ken Loach
Hill, J., 2009, Scottish Cinema Now. Murray, J. (ed.). Newcastle: Cambridge Scholars, p. 88-104 17 p.
"Book Review: The Charismatic Chameleon: The Actor as Creative Artist by Leslie O'Dell"
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"Boys talk facts, girls talk feelings? Questioning gendered consumption discourse in online communities of consumption"
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"Broken Down and Fast Breaking Up": Style, Technique and Vision in the "Eumaeus" Episode in Ulysses
Gibson, A., 1984, In: Southern Review. 17, p. 22-39
"Buenos seamos, que Dios nos ve": La madre naturaleza and Foucault's Panopticon
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"But the Image Wants Danger": Georges Bataille, Werner Herzog, and Poetical Response to Paleoart
Lee, N. & Dicker, B., Mar 2012, In: Time and Mind: The Journal of Archaeology, Consciousness and Culture. 5, 1
"But the Image Wants Danger": Georges Bataille, Werner Herzog, and Poetical Responses to Paleoart
Lee, N. & Dicker, B., 1 Mar 2012, In: Time and Mind: The Journal of Archaeology, Consciousness and Culture. 5, 1, p. 33-52 19 p.
"by the best masters…": advertising commercial concerts in London 1660–1750: 23rd Annual Conference on Music in 18th-century Britain
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"Can the Odyssey ever be tragic? Historical perspectives on the theatrical realization of Greek epic"
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Live It Again 1947
Author: Annie's
Publisher: Annie's
"Featuring the best of The Saturday Evening Post"--Cover.
I'd Live it Again
Author: J. M. Blakiston Houston
Play it Again
Cover Songs in Popular Music
Author: George Plasketes
Publisher: Ashgate Publishing, Ltd.
The cover phenomenon in popular culture may be viewed as a postmodern manifestation in music as artists revisit, reinterpret and re-examine a significant cross section of musical styles, periods, genres, individual records, and other artists and their catalogues of works. The cover complex, with its multiple variations, issues, contexts, and re-contextualizations comprises an important and rich popular culture text. These re-recordings represent artifacts which embody artistic, social, cultural, historical, commercial, biographical, and novel meanings. Through homage, allusion, apprenticeship, and parody, among other modes, these diverse musical quotations express, preserve, and distribute popular culture, popular music and their intersecting historical narratives. Play it Again represents the first collection of critical perspectives on the many facets of cover songs in popular music.
The Diary of Anaïs Nin, 1944–1947
Publisher: HMH
Category: Biography & Autobiography
The fourth volume of "one of the most remarkable diaries in the history of letters" (Los Angeles Times). The renowned diarist continues her record of her personal, professional, and artistic life, recounting her experiences in Greenwich Village for several years in the late 1940s, where she defends young writers against the Establishment—and her trip across the country in an old Ford to California and Mexico. "[Nin is] one of the most extraordinary and unconventional writers of [the twentieth] century." —The New York Times Book Review Edited and with a preface by Gunther Stuhlmann
The "Jew" in Cinema
From The Golem to Don't Touch My Holocaust
Author: Omer Bartov
Publisher: Indiana University Press
Analyzes the image of the Jew in flims from the 1920s through the 1990s, in a number of countries (the USSR, Poland, Hungary, Czechoslovakia, East and West Germany, France, Italy, the USA, and Israel). Highlights the influence of stereotypes not only in antisemitic films, but also in pro-Jewish and anti-antisemitic ones, revealing how latent antisemitic images have been projected even in the latter two types of films. Contains a broad range of films, including Nazi documentaries and Hollywood films with Jewish or "Jewish" heroes (e.g. Paul Newman in "Exodus"). The sections deal with the "Jew" as perpetrator, victim, hero, and anti-hero. The analyses of many different films explore issues related to antisemitism and the Shoah. Celluloid images of the "Jew" have both reflected history and helped to mold it.
The Dream, 1947
Author: Winston S. Churchill
Publisher: Rosetta Books
A powerful, emotional short story from the prime minister and Pulitzer Prize–winning author, detailing a conversation with the ghost of his beloved father. Legendary politician and military strategist Winston S. Churchill was a master not only of the battlefield, but of the page and the podium. Over the course of forty books and countless speeches, broadcasts, news items and more, he addressed a country at war and at peace, thrilling with victory but uneasy with its shifting role in global politics. In 1953, he was awarded the Nobel Prize for Literature for "his mastery of historical and biographical description as well as for brilliant oratory in defending exalted human values." During his lifetime, he enthralled readers and brought crowds roaring to their feet; in the years since his death, his skilled writing has inspired generations of eager history buffs. In this rare work of fiction, Churchill imagines a visit from the ghost of his father, Randolph. Churchill reveals to his father all that has happened in the world since his death in 1895, leaving out one crucial detail: his own critical role in determining the unfolding of world events. His yearning for his late father shines through his terse, careful prose, lending emotional weight and nostalgia to this unusual foray into fiction.
The Shaping of Indian Science: 1914-1947
Publisher: Universities Press
Category: Indian Science Congress
This is a compendium of the speeches of the Presidents of the Indian Science Congress Association (ISCA) from 1914-2003. Through the years, these Presidents have inspired the Congress by their speeches-some of them visionary, some impassioned in their plea for Science, but all of them with a message that Science must be used for the good of the human race.
Author: Edmund Morris
Publisher: Modern Library
Selected by the Modern Library as one of the 100 best nonfiction books of all time • WINNER OF THE PULITZER PRIZE • WINNER OF THE NATIONAL BOOK AWARD Although Theodore Rex fully recounts TR's years in the White House (1901–1909), The Rise of Theodore Roosevelt begins with a brilliant Prologue describing the President at the apex of his international prestige. That was on New Year's Day, 1907, when TR, who had just won the Nobel Peace Prize, threw open the doors of the White House to the American people and shook 8,150 hands, more than any man before him. Morris re-creates the reception with such authentic detail that the reader gets almost as vivid an impression of TR as those who attended. One visitor remarked afterward, "You go to the White House, you shake hands with Roosevelt and hear him talk—and then you go home to wring the personality out of your clothes." The rest of this book tells the story of TR's irresistible rise to power. (He himself compared his trajectory to that of a rocket.) It is, in effect, the biography of seven men—a naturalist, a writer, a lover, a hunter, a ranchman, a soldier, and a politician—who merged at age forty-two to become the youngest President in our history. Rarely has any public figure exercised such a charismatic hold on the popular imagination. Edith Wharton likened TR's vitality to radium. H. G. Wells said that he was "a very symbol of the creative will in man." Walter Lippmann characterized him simply as our only "lovable" chief executive. During the years 1858–1901, Theodore Roosevelt, the son of a wealthy Yankee father and a plantation-bred southern belle, transformed himself from a frail, asthmatic boy into a full-blooded man. Fresh out of Harvard, he simultaneously published a distinguished work of naval history and became the fist-swinging leader of a Republican insurgency in the New York State Assembly. He had a youthful romance as lyrical—and tragic—as any in Victorian fiction. He chased thieves across the Badlands of North Dakota with a copy of Anna Karenina in one hand and a Winchester rifle in the other. Married to his childhood sweetheart in 1886, he became the country squire of Sagamore Hill on Long Island, a flamboyant civil service reformer in Washington, D.C., and a night-stalking police commissioner in New York City. As assistant secretary of the navy under President McKinley, he almost single-handedly brought about the Spanish-American War. After leading "Roosevelt's Rough Riders" in the famous charge up San Juan Hill, Cuba, he returned home a military hero, and was rewarded with the governorship of New York. In what he called his "spare hours" he fathered six children and wrote fourteen books. By 1901, the man Senator Mark Hanna called "that damned cowboy" was vice president of the United States. Seven months later, an assassin's bullet gave TR the national leadership he had always craved. His is a story so prodigal in its variety, so surprising in its turns of fate, that previous biographers have treated it as a series of haphazard episodes. This book, the only full study of TR's pre-presidential years, shows that he was an inevitable chief executive, and recognized as such in his early teens. His apparently random adventures were precipitated and linked by various aspects of his character, not least an overwhelming will. "It was as if he were subconsciously aware that he was a man of many selves," the author writes, "and set about developing each one in turn, knowing that one day he would be President of all the people."
The Religion of the Tempasuk Dusuns of North Borneo
Author: I. H. N. Evans
Category: Social Science
This 1953 volume describes the Dusun people of the Tempasuk region in what was formerly North Borneo.
Muslims in India Since 1947
Islamic Perspectives on Inter-Faith Relations
Author: Yoginder Sikand
Muslims in India today are responding to the challenge of religious pluralism in a variety of ways. This book explores the attempts being made by scholar-activists and Muslim organisations to develop new understandings of Islam to relate to people of other faiths and to the modern nation-state, and to deal with issues such as democracy and secularism. It examines how a common predicament, characterised by a sense of siege and the perception of being an oppressed minority, is producing new expressions of Islam, some of which seek to relate to non-Muslims in terms of confrontation, and others which call for dialogue, reconciliation and inter-faith harmony.
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How to get American Netflix on Roku? You can change your Netflix region on your Roku. Thus, you can unblock and watch more content than what is available in UK, Canada, Australia, Germany, France. VPN and Smart DNS proxy help you bypass regional restrictions. Read this help guide on how to unblock American Netflix on Roku using VPN or DNS proxy.
Unblock American Netflix on Roku – Compatible Models.
The guides below work on all Roku models.
American Netflix on Roku – VPN.
In order to unblock US Netflix on Roku, you have to block Google's public DNS on your router, too. That's because Google's DNS servers are hard-coded on the Roku. There are many tutorials on how this can be done. Since you cannot setup VPN directly on your Roku, you have to install it on a VPN-enabled router. The VPN providers I've listed below all have easy-to-follow guides to setup VPN on your Roku.
American Netflix on Roku – Smart DNS.
As with VPN, you need router access to unblock US Netflix on your Roku using Smart DNS. In this case, you need to block Google public DNS too. Once done, add the DNS server addresses you've obtained from a Smart DNS proxy service to your router. Click here fore a free Unlocator account. You'll need their guides to finish the setup. In addition to Netflix, there are many American channels you can add to your Roku.
Given the vast superiority of US Netflix over other regions, switching to American Netflix is a no-brainer. It's possible to unblock American Netflix on Roku thanks to VPN and Smart DNS.


The NZ Transport Agency says many roads have closed or remain closed in the Bay of Plenty and Waikato after the severe weather event.
The eastern Bay of Plenty has several closures because of fallen power lines and slips. State Highway 29 the Kaimai Range is also now closed after a large slip.
All of the closures will stay in place overnight and contractors will be out on the state highway network at first light assessing any damage and beginning work where they can.
People can report any further damage to 0800 4 HIGHWAYS (0800 44 44 49).
Check our website for the latest (including if roads have opened): www.nzta.govt.nz/traffic(external link).
State Highway 2 between Ohiwa Beach Road and Opotiki is closed because of slips.
State Highway 34 is closed at Otakiri between Edgecumbe and Te Teko because of fallen power lines.
State Highway 30 between Thornton Rd and Awakeri is closed because of fallen power lines.
State Highway 2 near Pikowai camping ground on the Matata straight is closed because of fallen tree and powerlines. It will remain closed tonight.
The roads closed during Cyclone Debbie or previous projects (SH2 Edgecumbe, SH2 at Waimana Gorge, SH2 Pekatahi Bridge) in the Eastern Bay of Plenty remain closed.
State Highway 29 Kaimai Range is also currently closed because of a large slip at the bottom of the Waikato side. Not expected to open tonight.
State Highway 29 Kaimai Range is currently closed because of a large slip at the bottom of the Waikato side. Not expected to open tonight.
State Highway 25 – Thames-Coast Rd is closed from Tararu to Preece Point because of several slips and won't be opening in the near future.


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Home » Tech » Report suggests games market might see decline in 2022
Report suggests games market might see decline in 2022
Interested in learning what's next for the gaming industry? Join gaming executives to discuss emerging parts of the industry this October at GamesBeat Summit Next. Learn more.
A new report from Ampere Analysis suggests the global games market might not be as "recession-proof" as it might have seen. The report predicts the market will see a decline in 2022 — perhaps not by a dramatic amount, but a notable one.
The report's graphs show that the market expanded rapidly during the plantation, adding $39 billion in spending from 2019 to 2021. The market performance of 2021 was $191 billion, while Ampere predicts the 2022 market will reach $188 billion. Ampere suggests that the decline has multiple sources, including hardware supply chain issues and game delays, as well as the more general effects of inflation.
Mobile gaming might also see a slight decline, thanks to privacy disputes on mobile platforms. The other platforms might also see either decline or no change, except for cloud gaming. That said, the report also predicts 2023 will be a more stable year, with fewer hardware shortages and delayed games finally launching.
Piers Harding Rolls, Ampere's research director, said in a statement, "After two years of huge expansion, the games market is poised to hand back a bit of that growth in 2022 as multiple factors combine to undermine performance. Even so, the year will end well ahead of pre-pandemic performance, and the outlook for the sector as a whole remains positive, with growth forecast to return in 2023."
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