text:
\section{Introduction}

Consider a discrete dynamical system $(D,f)$ on some compact interval $D\subseteq\mathbb{R}$,
called the phase space, given by a function $f:D\to D$, a recursion relation
$x_{n+1}=f(x_n)$ and an initial value $x_0\in D$. The sequence $(x_n)_n$ of
iterates is called the orbit of the dynamical system in phase space corresponding
to the initial value $x_0$. If such a dynamical system is implemented, that is
a computer program is written for calculating a finite initial
segment of the orbit for given $x_0$, care has to be taken in choosing
the appropriate data structure for representing real numbers. Traditionally,
IEEE 754 {\tt double} floating point numbers \cite{ieee87} are used. However,
if the dynamical system shows chaotic behavior, a problem arises. The finite
and constant length of the mantissa of a {\tt double} variable causes round
off errors which are magnified after each iteration step. Only after a few
iterations, the error is so big that the computed values are actually
useless \cite{Mue01}. To put things right, a rigorous method for computations
with real numbers has to be used. In \cite{Bla05}, this issue is addressed
for the logistic map which is also consiedred as a starting point in the next
section. There, the exact real arithmetic in the form of centered intervals
with bounded error terms is used as described in \cite{bl06}. However, the
notation used in \cite{Bla05, bl06} is an algebraic one based on arbitrary
large integers. On the other hand, the aim of the present paper is to keep
the notation as close as possible to the standard in scientific computing
but being precise in the sense of exact real arithmetic. This has as a
consequence first that the basic data type is not an integer as considered
in \cite{Bla05, bl06}, but a floating point number with a definite
mantissa length. Second, the type of error considered here is the
relative error as is standard in floating point arithmetic - in contrast
to the absolute error considered in \cite{Bla05, bl06}. In practice, a
multiple precision floating point library providing floating point
numbers with arbitrary high mantissa length have to be used.
In the following, it is analyzed how the needed
mantissa length behaves in multiple-precision computations of
iterates of discrete dynamical systems. The mantissa length needed
for floating point numbers such that any computed point of the orbit
has a specified and guaranteed accuracy is examined. Therefore,
a precise mathematical framework for floating point computations
has to be established. The main result shows that the
ratio of mantissa length to iteration length in the limit of
iteration length to infinity is related to the Ljapunow exponent.
Comparing, in \cite{Bla05} only the logistic map is
considered explicitly and the connection to the Ljapunow exponent
is not stated, but observed numerically. In the present paper,
this connection is shown mathematically for a general discrete
dynamical system $(D,f)$.
This result also gives some advice for economically designing exact
algorithms simulating one-dimensional discrete dynamical systems.

\section{Roundoff Error, Error Propagation and Dynamic Behavior}

In this section, the discrete dynamical system $(D,f_\mu)$ with $D=[0,1]$ and
$f_\mu:D\to D$, $f_\mu(x):=\mu x(1-x)$ for some control parameter $\mu\in(0,4]$
is investigated. In the literature, the recursion relation $x_{n+1}=f_\mu(x_n)$ is
called the {\em logistic equation} \cite{ce80}. When implementing the logistic
equation on a real computer and demanding to obtain exact values for the
orbit $(x_n)_n$, the analysis of roundoff errors and of error propagation
requires some care. This is due to the fact that for some values of $\mu$
the dynamics is highly chaotic and therefore inaccuracies are magnified
exponentially in time \cite{ce06, hsd04}.

In the following, for a given initial value $x_0$, the true orbit is denoted by
$(x_n)_n$, whereas the really computed orbit, suffering from roundoff errors
and error propagation, is denoted by $(\hat{x}_n)_n$. Note that even
$\hat{x}_0$ may differ form $x_0$ since the conversion to a floating point
number may cause the very first roundoff error. One goal of this section
is to give a rigorous estimation of the total error in dependence of the
iteration step $n$.

Calculating the orbit $(\hat{x}_n)_n$, two types of error are present.
First, error propagation due to the iteration scheme and second the
roundoff error caused by the calculation of $f_\mu$. Now, let $\hat{x}_n$
for some $n\in\mathbb{N}$ be given. Then the true error after one iteration step
is $\hat{x}_{n+1} - x_{n+1}$. Since in reality not $f_\mu(\hat{x}_n)$ is
calculated but some erroneous approximation $\hat{f_\mu}(\hat{x}_n)$,
the true error can be written as
$\hat{x}_{n+1} - x_{n+1} = \hat{f}_\mu(\hat{x}_n)-f_\mu(x_n)$.
Hence, the true error can be written as a sum
\begin{equation}
\label{main_err}
\hat{x}_{n+1} - x_{n+1} = (f_\mu(\hat{x}_n)-f_\mu(x_n)) +
 (\hat{f_\mu}(\hat{x}_n) - f_\mu(\hat{x}_n))
\end{equation}
of two terms. The first term describes solely the error propagation
while the second term gives exactly the newly produced error
due to the approximate calculation of $f_\mu$.

To handle the exact values of both errors computationally, interval
arithmetic can be used \cite{ah83}. Interval arithmetic can be seen
in the setting here as a special case of the computational model
of TTE \cite{wh00}, which gives a precise notion for describing
computations over the real numbers. Another strongly related model,
which in some sense reflects the situation here more adequate is the
Feasible Real RAM model \cite{BH98}. For the sake of simplicity however,
an interval setting is used here. For any time step $n$, let
the phase point $x_n$ together with its error be represented by two
floating point numbers $x^l_n$ and $x^u_n$ ($x^u_n\geq x^l_n$) with
given mantissa length $m_n$ forming an interval $[x^l_n,x^u_n]$. The
interval is an enclosure of the real value $x_n$, that is
$x_n\in[x^l_n,x^u_n]$. It is straightforward to transform
the interval to a floating point value $\hat{x}_n$ of
mantissa length $m_n$ by setting
\begin{equation}
\label{val_def}
\hat{x}_n:=gl\left(\frac{x^l_n + x^u_n}{2}\right)
\end{equation}
where $gl(.)$ performs the rounding to nearest floating point number.
The absolute error $e_n:=|\hat{x}_n - x_n|$ of $\hat{x}_n$ can
be estimated via the interval length $d_n:=x^u_n-x^l_n$ by
\begin{equation}
\label{err_def}
e_n\leq\frac{1}{2}d_n + r_n
\end{equation}
where $r_n$ is an error introduced by the rounding operation $gl(.)$ in
Equation \ref{val_def}. An upper bound on $r_n$ will be discussed later, for
now it suffices to say that in general it is small compared to $d_n$.

For doing an error analysis of the logistic equation analytically, some
idealizing assumptions are made. First, the value of $\mu$ is assumed to be
given with such a high precision that no interval representation is needed. Second,
only the error propagation is considered caused by the initial error due to rounding
$x_0$ to some floating point number of mantissa length $m$. Third, the value
of $r_n$ in Equation \ref{err_def} is neglected. The recursion relation then reads
in natural interval extension
\begin{align*}
x^l_{n+1} &= \mu x^l_n(1-x^u_n)\\
x^u_{n+1} &= \mu x^u_n(1-x^l_n)
\end{align*}
with the interval length $d_n$  given by the recursion relation
\begin{align*}
d_{n+1} &= x^u_{n+1} - x^l_{n+1} = \mu(x^u_n - x^u_n x^l_n - x^l_n + x^u_n x^l_n)\\
 &= \mu d_n
\end{align*}
with the obvious solution $d_n=\mu^ n d_0$. Finally the absolute error $e_n$
of $\hat{x}_n$ according to Equation \ref{val_def} can be bounded from
above by
\begin{equation}
\label{err_bound}
e_n\leq\frac{1}{2}d_n=\frac{1}{2}\mu^n d_0.
\end{equation}
Note that the above analysis only holds if the natural interval extension
for $f_\mu$ is derived from the formula $\mu x(1-x)$. If it is derived
from the formula $f_\mu(x)=\mu(x-x^2)$ or
$f_\mu(x)=\frac{\mu}{4}-\mu(x-\frac{1}{2})^2$, the mathematical analysis
is more difficult. However, the problems described in the following also
appear, but in some different form.

The aim now is to calculate, for given $N\in\mathbb{N}$, $p\in\mathbb{Z}$
and mantissa length $m$, the orbit up to time $N$
with relative error $10^{-p}$. That is, for
$(\hat{x}_n)_{0\leq n\leq N}$ should hold
\begin{equation}
\label{prec_req}
e_n=|\hat{x}_n - x_n|\leq 10^{-p} x_n \leq 10^{-p}.
\end{equation}
The ideal assumptions require the somewhat unreal setting
that the mantissa length has to be set to some finite, but big enough value $m$
for representing $x_0$ and a virtually infinite value $m_\infty$ for doing the
iteration. Finally, some upper bound on $d_0$ is needed. The value of $d_0$
is given as the roundoff error by representing $x_0$ as a floating point number
of mantissa length $m$. For that, the well known estimate
\begin{equation}
\label{round_zero}
d_0\leq 2^{-m+1}x_0\leq 2^{-m+1}
\end{equation}
exists. Combining (\ref{prec_req}), (\ref{err_bound}) and (\ref{round_zero})
gives as a sufficient condition
\begin{equation}
\label{res_cond}
\mu^n\cdot 2^{-m}\leq 10^{-p}
\end{equation}
for $n=0,\dots,N$.

The minimal $m$, fulfilling the precision requirement (\ref{prec_req}) on the
relative error of $x_n$, which depends on $x_0$, $N$ and $p$, is
denoted by $m_{min}(x_0,N,p)$. So, the sufficient condition (\ref{res_cond})
gives an upper bound on $m_{min}(x_0,N,p)$ by
\begin{equation}
\label{ub_req}
m_{min}(x_0,N,p)\leq \lceil p\cdot\mathrm{ld}(10) + N\cdot\max(0, \mathrm{ld}(\mu))\rceil
\end{equation}
where $\mathrm{ld}(.)$ is the logarithm to base $2$. At that stage, a central
quantity of this work is introduced which is a kind of complexity measure.
The {\em loss of significance rate} $\sigma(x,p)$, which may depend on
the initial value $x=x_0$ and the precision $p$ is defined by
\begin{equation*}
\sigma(x,p):=\limsup_{N\to\infty}\frac{m_{min}(x,N,p)}{N}.
\end{equation*}
This quantity describes the limiting amount of significant mantissa length
being lost at each iteration step. Significant means here the part of
the places being exact. A general treatment of this complexity measure
is given in the next section. Roughly speaking,
$\lceil\sigma(x_0,p) N + p\cdot\mathrm{ld}(10)\rceil$ is the mantissa length
for any floating point number needed in an algorithm doing the iteration
starting with $x_0$ and calculating up to $x_N$, if the output should be precise
up to $p$ decimal places. Formula \ref{ub_req} gives an upper
bound for the loss of significance rate by
$\sigma(x,p)\leq\max(0, \mathrm{ld}(\mu))$.

It is interesting to see whether the upper bound calculated analytically,
which needed strong idealizations, is in the region of the real value.
So, the logistic equation was implemented using a multiple-precision
interval library. For that purpose, the interval library MPFI
\cite{ReRo02} based on the multiple-precision floating point number
library MPFR \cite{Fousse:2007:MMP}, both written in C, was used.
For each control parameter $\mu$ ranging from $0.005$ to $4$ and a
step size of $0.005$, the orbit for initial condition $0.22$ was
calculated up to $N=2000$. For each $\mu$, the minimum
mantissa length $m_{min}$ needed to guarantee $e_n\leq 10^{-6}x_n$
for $n=0,\dots,N$ was searched. Then, $\sigma_{est}:=m_{min}/N$
was calculated. The result shows that $\sigma_{est}$ exceeds the
analytical bound $\max(0,\mathrm{ld}(\mu))$ only slightly. So, the above made
ideal assumptions seem to be valid. In \cite{Mue01}, the logistic
equation was also investigated for $\mu=3.75$ using the exact real
arithmetic package iRRAM based on the Feasible Real RAM model
\cite{BH98}. In the paper, also the precision needed
to guarantee the exactness of the first $6$ decimal places
are reported up to $N=100000$. The values are in full
agreement with the simulation results performed here.

Hence, for $\mu > 1$, the interval length $d_n$
increases exponentially in time $n$. This result should be interpreted
in terms of the dynamical behavior of the logistic equation. So, at
this point is worth having an analytical look at the behavior
of the dynamical system. Despite the fact that these results
are well known \cite{hsd04, de89}, they are reviewed here for the sake of
self containment. First, the equation possesses in the range $D=[0,1]$
exactly one
fixed point $x^o=0$ if $\mu\in(0,1]$ and exactly two fixed points $x^o=0$
and $x^{(\mu)}=1-\frac{1}{\mu}$ if $\mu\in(1,4]$. Since $f'_\mu(0)=\mu$
and $f'_\mu(x^{(\mu)})=2-\mu$, $x^o$ is a stable fixed point (an attractor,
$|f'_\mu(x^o)|<1$) for $\mu\in(0,1)$ and an unstable fixed point
(a repeller, $|f'_\mu(x^o)|>1$) for $\mu\in(1,4]$.
If $\mu=1$, the only fixed point $x^o$ is hyperbolic ($|f'_1(x^o)|=1$)
and a bifurcation occurs at that value of the control parameter $\mu$.
If $\mu\in(1,3)$, $x^o$ becomes unstable
and the newly occurring fixed point $x^{(\mu)}$ is stable. Finally,
$\lim_{n\to\infty}f^n_\mu(x)=x^{(\mu)}$ for $\mu>1$ and
$\lim_{n\to\infty}f^n_\mu(x)=x^o$ if $\mu\leq 1$ holds for all $x\in(0,1)$.
If $\mu\in(0,1)$, this is a direct consequence
of the contraction mapping principle. If $\mu=1$, observe that
$f_1(x)<x$ holds for all $x\in(0,1)$. Hence, any sequence $(f^n_1(x))_n$,
$x\in(0,1)$, is strictly decreasing and bounded from below. So it
converges to the only fixed point $x^o$. For the case $\mu\in(1,3)$,
the interested reader is referred to the literature: \cite{de89},
Proposition 5.3. At $\mu=3$ a second bifurcation occurs
and for $\mu>3$ the system goes into a region of periodic
behavior with period doubling bifurcations. Finally, for some
$\mu<4$, chaotic behavior is reached.

This analysis shows that in the parameter range $\mu\in(0,3)$,
the orbit converges to the stable fixed point for any initial value
$x_0\in(0,1)$. Furthermore, there exists some closed interval $I\subseteq D$,
which depends on $\mu$, containing the stable fixed point such that
$f_\mu(I)\subseteq I$ holds and $f_\mu$ is a contraction on $I$. The interval
computation using a natural interval extension of the recursion
formula $\mu x(1-x)$, on the other hand, is not very compatible with
this picture. While for $\mu\in(0,1)$, the results are
in agreement with the dynamical analysis, the calculations for
$\mu\in(1,3)$ are not handled very well by interval arithmetic since
the interval approach would suggest an exponential divergence of
initially nearby orbits which is not true in reality. The reason is
that the natural interval approach implicitly, due to the dependency problem,
takes account only of the global behavior of $f_\mu$ in the form of the global
Lipschitz constant $\max\{|f'_\mu(x)| : x\in D\}=\mu$. However, the local
Lipschitz constant $\max\{|f'_\mu(x)| : x\in [x^l_n,x^u_n]\}$ governs the
real error propagation at time step $n$ and also describes the dynamic behavior.
This notion can be made precise and finally leads to a more efficient
algorithm for computing orbits.

Let us return to Equation \ref{main_err}. The true error is the sum of
the error propagation (first term) according to the iteration and the
roundoff error due to the computation of $f_\mu$ (second term). The
first term of Equation \ref{main_err} can be handled using the mean
value theorem,
$|f_\mu(\hat{x}_n)-f_\mu(x_n)|=|f'_\mu(y_n)|\cdot |\hat{x}_n - x_n|$
with $y_n\in[\hat{x}_n-e_n,\hat{x}_n+e_n]$. This gives directly the
bound
\begin{equation*}
|f_\mu(\hat{x}_n)-f_\mu(x_n)|\leq
\sup(|f'_\mu([\hat{x}_n-e_n,\hat{x}_n+e_n])|)e_n.
\end{equation*}
The second term can be estimated the following way. As discussed in
\cite{wi63}, the roundoff error produced in calculating $f_\mu$ can be
estimated by
\begin{equation*}
|\hat{f}_\mu(\hat{x})-f_\mu(\hat{x})|\leq 1.06K2^{-m}|f_\mu(\hat{x})|
\end{equation*}
where $K$ is the number of rounding operations performed in computing
$\hat{f}_\mu$ and $m$ is the mantissa length of $\hat{x}$. In
the case considered here, $K=4$ follows since there are 3 arithmetic
operations and the rounding of $\mu$. It is further crucial to
mention that the factor $1.06$ is only valid if $K\leq 0.1\cdot 2^m$
holds so that the mantissa length must not be chosen too small.
Using the fact that $f_\mu(x)\leq\frac{\mu}{4}$ holds and $f_\mu(x)<x$
if $\mu\leq 1$, the unknown value $|f_\mu(\hat{x})|$ can be eliminated.
This calculation shows that there exists a recursive equation
on an upper bound $\overline{e}_n$ on $e_n$ for all $n$:
\begin{equation}
\label{calc_err}
\overline{e}_{n+1}=L(\hat{x}_n,\overline{e}_n)\overline{e}_n+
1.06K2^{-m}E_\mu(\hat{x}_n),\quad \overline{e}_0=2^{-m}
\end{equation}
with $L(x,e):=\sup(|f'_\mu([x-e,x+e])|)$ and
\begin{equation*}
E_\mu(x):=\begin{cases}
x & \text{if } \mu\leq 1\\
\frac{\mu}{4} & \text{if } \mu >1
\end{cases}.
\end{equation*}
The idea is now not to calculate intervals, but pairs of values $\hat{x}_n$
and corresponding guaranteed error bounds $\overline{e}_n$. The difference to
the interval concept is not to compute the errors {\em implicitly},
so that only global behavior can be taken into account, but to compute
them {\em explicitly} and independent of the values of interest. It should be
mentioned that the approach described here is compatible with an interval
approach using special centered forms, namely mean value forms \cite{rr84}.
However, the approach here explicitly devises values and errors, describes
an automated error analysis, whereas an interval approach primarily does not
disclose any error. Furthermore, also the iRRAM package permits a more
elaborate way for computing the iteration, based on a similar algorithm as
described above \cite{mu10}. The rounded
values $\hat{x}_n$ are calculated as usual in floating point arithmetic except
that multiple-precision floats are used. The guaranteed error bounds are also
calculated using floating point according to (\ref{calc_err}), where
interval arithmetic is used for calculating $L$. Only standard
precision is needed for calculating the error bounds.
Implementing this improved
algorithm using MPFR and MPFI, the setting as given in the interval case
produces the following result.
In the parameter range $\mu\in(0,3)$, the dynamic behavior is reflected very
well. Furthermore, in the range $\mu\in[3,4]$, the curve suggests
a relation between the loss of significance rate and the Ljapunow
exponent $\lambda(x)$ for the logistic map (for a curve of the Ljapunow
exponent of the logistic map see \cite{ce80}):
$\sigma(x)=\max(0,\lambda(x))/\ln(2)$ for all $\mu\in(0,4]$. To be
complete, the definition of the Ljapunow exponent reads
\begin{definition}
\label{def:ljap}
Let $(D,f)$ be a dynamical system, $D\subseteq\mathbb{R}$ compact and $f:D\to D$
continuously differentiable on the interior of $D$. Then the
{\em Ljapunow exponent} at $x$ is defined by
\begin{equation}
\label{ljap_def}
\lambda(x):=\lim_{n\to\infty}\frac{1}{n}\sum^{n-1}_{k=0}
\ln\left|f'(f^k(x))\right|
\end{equation}
if the limit exists.
\end{definition}
The Ljapunow exponent may depend on $x$. However, the following
properties hold:
\begin{itemize}
\item[(a)]
If $(D,f)$ has an {\em invariant measure} $\rho$, then the limit in
Equation \ref{ljap_def} exists $\rho$-almost everywhere.
\item[(b)]
Furthermore, if $\rho$ is {\em ergodic} then $\lambda(x)$
is $\rho$-almost everywhere constant and equal to
\begin{equation*}
\int_D \ln\left|f'(x)\right|\,\rho(dx).
\end{equation*}
\end{itemize}
These properties are a direct consequence of the Birkhoff ergodic theorem,
see \cite{kh95}, Theorem 4.1.2 and Corollary 4.1.9.

\section{The General Algorithm and its Complexity}
Let $D$ be a compact real interval and $f:D\to D$ a self mapping. In the following,
$f$ is assumed to be continuous on $D$, continuously differentiable on the interior
of $D$ and $f'$ is bounded. Furthermore, $f$ and $f'$ are assumed to be
computationally feasible. The precise definition of ``computationally feasible''
is given below.

In this section, a general algorithm for computing the iteration
\begin{equation}
\label{main_it}
x_{n+1}=f(x_n),\quad x_0\in D
\end{equation}
is presented. To be more precise, for given $N\in\mathbb{N}$ and $p\in\mathbb{Z}$,
this algorithm computes a finite part of the orbit,
$(x_n)_{0\leq n\leq N}$, exact in the sense that the relative
error at each point $x_n$ does not exceed $10^{-p}$.
The correctness of the algorithm and its computational feasibility is shown.
Finally, its complexity is examined.

\subsection{Syntax, Semantics and the Algorithm}
The set of all computationally accessible real numbers are the floating
point numbers of arbitrary mantissa length denoted by $\hat{\mathbb{R}}$.
In the following, by a floating point number any real number
is meant which can be expressed by normalized scientific notation.
Hence, the set $\hat{\mathbb{R}}\subseteq\mathbb{R}$ of all floating point
numbers is countable infinite and therefore a natural basis for standard
computability considerations. Let $\hat{x}\in\hat{\mathbb{R}}$ be some floating
point number, then $\hat{x}$ has as an essential property, its
{\em mantissa length} denoted by $\hat{x}.m$.
Any real number $x$ is represented in an algorithm concerning real
computation by a pair $[x]\in\hat{\mathbb{R}}^2$ consisting of a floating
point number $[x].fl$ approximating $x$ and an upper bound on the
relative error, $[x].err\geq 0$, also being a floating point number.
Furthermore, the inequality $|[x].fl-x|\leq [x].err$ holds. The pair
$[x]$ is called a {\em finite precision representation} of $x$.
Although $[x].err$ has the property mantissa length, it is irrelevant
in what follows. So, the mantissa length of $[x].err$ can be assumed
to be some big enough constant value. Analogously, a function $f:D\to D$,
$D\subseteq\mathbb{R}$, is called {\em computationally feasible} if a pair
$[f]$ exists of a computable (partial) function $[f].fl:\hat{\mathbb{R}}\to\hat{\mathbb{R}}$
approximating $f$ on $D$ and a computable (partial) function
$[f].erf:\hat{\mathbb{R}}^2\to\hat{\mathbb{R}}$ giving an upper bound on the
absolute error of $[f].fl$ in the sense
$|[f].fl([x].fl)-f(x)|\leq [f].erf([x])$. Here, a partial function
$\hat{f}:\hat{\mathbb{R}}\to\hat{\mathbb{R}}$ is called {\em computable}
if $\hat{f}$ is computable as a string function over some finite
alphabet where the floating point numbers are interpreted as
finite strings. Finally, computability over integers, computability
of functions with mixed arguments and computable predicates are
defined in a standard way.

The algorithm with the above described specification reads
\begin{tabbing}
{\tt\ 1}\ \= {\tt Input parameter:} $\hat{x}_0$, $N$, $p$\\
{\tt\ 2}\> {\tt Initialize mantissa length} $m\leftarrow m_0$\\
{\tt\ 3}\> {\tt do} \= \ \\
{\tt\ 4}\> \> {\tt Initialize value and error} $[x]\leftarrow gl(\hat{x}_0,m)$\\
{\tt\ 5}\> \> {\tt for} \= $n=0$ to $N$ {\tt do}\\
{\tt\ 6}\> \> \> {\tt If} \= $prec([x],p)$ {\tt is {\bf true} then}\\
{\tt\ 7}\> \> \> \> {\tt If not printed print } $n$, $[x].fl$, $[x].err$ \\
{\tt\ 8}\> \> \> {\tt else break}\\
{\tt\ 9}\> \> \> $[x]\leftarrow[f]([x])$\\
{\tt 10}\> \> {\tt end for}\\
{\tt 11}\> \> $[x].fl.m\leftarrow [x].fl.m + 1$\\
{\tt 12}\> {\tt while} $prec([x],p)$ {\tt is false}
\end{tabbing}
To initialize $[x]$, a rounding function
$gl:\hat{\mathbb{R}}\times\mathbb{N}\to\hat{\mathbb{R}}^2$ is needed where $gl(\hat{x}_0, m_0).fl$
is a floating point number of mantissa length $m_0$ being
the exactly rounded value of $\hat{x}_0$ to some rounding convention.
Clearly, $gl(\hat{x}_0, m_0).err$ is an upper bound on the absolute
rounding error, e.g.\ $gl(\hat{x}_0, m_0).err=\frac{1}{2}ulp(\hat{x}_0)$ if
the rounding mode is to nearest. The predicate
$prec:\hat{\mathbb{R}}^2\times\mathbb{Z}\to\{{\rm\bf true}, {\rm\bf false}\}$
is a test whether the relative error of $[x]$,
$|[x].fl-x|/|x|$ if $x\neq 0$, is bounded by $10^{-p}$. The
semantics reads: If $[x]\in\hat{\mathbb{R}}^2$ is a finite precision
representation of $x\in\mathbb{R}$ and $prec([x],p) = {\rm\bf true}$
holds, then $|[x].fl - x|\leq 10^{-p}|x|$ follows.

In the following, some abbreviations are used occasionally.
The floating point numbers and functions are indicated by a
hat: $\hat{x}:=[x].fl$
and $\hat{f}:=[f].fl$. An over-bar indicates an error bound:
$\overline{e}:=[x].err$ and $\overline{erf}:=[f].erf$. Hence,
$[x]$ is equivalent to $(\hat{x},\overline{e})$ and $[f]$
is equivalent to $(\hat{f},\overline{erf})$.

Finally a remark on optimization. The algorithm
is not optimized in the performance. Otherwise, in Line {\tt 10}
something like $m\leftarrow 2m$ should be used. Here, the aim is to
find the minimal $m$ to guarantee some given upper bound on the
relative error of $x_n$.

\subsection{Feasibility and Correctness}
It is clear, that the rounding function $gl$ is computationally
feasible. So lets begin with the predicate $prec$.
\begin{proposition}
The computationally feasible formula
\begin{equation}
\label{alg:prec}
prec((\hat{x},\overline{e}),p):=\begin{cases}
{\rm\bf true} & \text{if }
\overline{e}\leq\frac{10^{-p}}{1+10^{-p}}|\hat{x}|\\
{\rm\bf false} & \text{else}
\end{cases}
\end{equation}
fulfills the above described semantics.
\end{proposition}
\begin{proof}
Let $(\hat{x},\overline{e})$ be a finite precision
representation of $x$. So, if $\overline{e}\leq 10^{-p}|x|$
holds, then also $|\hat{x} - x|\leq 10^{-p}|x|$ holds.
If $(\hat{x}-\overline{e})(\hat{x}+\overline{e})\geq 0$, then
$|\hat{x}|-\overline{e}\leq |x|$ holds. Hence, if
$(\hat{x}-\overline{e})(\hat{x}+\overline{e})\geq 0$ and
$\overline{e}\leq 10^{-p}(|\hat{x}|-\overline{e})$ holds,
then also $|\hat{x} - x|\leq 10^{-p}|x|$. Finally, if
$\overline{e}\leq\frac{10^{-p}}{1+10^{-p}}|\hat{x}|$
holds, then also
$(\hat{x}-\overline{e})(\hat{x}+\overline{e})\geq 0$.

Formula \ref{alg:prec} only uses the accessible floating point
values $\hat{x}$ and $\overline{e}$, basic arithmetics and
finite tests. Hence, this formula is computationally feasible.
\end{proof}
Note that the definition of the predicate this way
also gives ${\rm\bf true}$ in the singular case where $\hat{x}=0$
and $\overline{e}=0$ and hence $x=0$.

An algorithm for computing $\hat{f}$ is by assumption possible.
To derive an algorithm for computing $\overline{erf}$ on the absolute
error, return to Equations \ref{main_err} and \ref{calc_err}.
\begin{proposition}
\label{prop:ubound}
Assume that $\hat{f}(\hat{x})$ computes $f(\hat{x})$ up to
a correctly rounded last bit in mantissa according to rounding convention.
Then there exists a constant $K>0$ such that the absolute error of
$f(x)$ of the computation $[f]([x])$ is bounded from above by
\begin{equation}
L(\hat{x},\overline{e})\cdot\overline{e} +
\frac{K2^{-m}}{1-K2^{-m}}|\hat{f}(\hat{x})|
\end{equation}
if $K2^{-m}<1$. Here,
$L(\hat{x},\overline{e}):=\sup(|f'([\hat{x}-\overline{e},\hat{x}+
\overline{e}])|)$ and $m$ is the mantissa length of $\hat{x}$: $\hat{x}.m$.

Furthermore, this bound is computable.
\end{proposition}
\begin{proof}
Using Equation \ref{main_err} and following the calculations leading to
equation \ref{calc_err},
$|\hat{f}(\hat{x})-f(x)|\leq L(\hat{x},\overline{e})\cdot\overline{e}+
|\hat{f}(\hat{x})-f(\hat{x})|$ follows. According to the assumption
on $\hat{f}$,
$|\hat{f}(\hat{x})-f(\hat{x})|\leq K2^{-m}|f(\hat{x})|$ holds, with
a value $K\in\{1,2\}$ depending on the rounding convention. However,
$f(\hat{x})$ is unknown, only $\hat{f}(\hat{x})$ is accessible.
To overcome this, set $\hat{f}(\hat{x})-f(\hat{x})=\delta f(\hat{x})$
with $|\delta|\leq K 2^{-m}$. Since $|\delta|<1$ holds, resolve to
$f(\hat{x})=\frac{1}{1+\delta}\hat{f}(\hat{x})$. Hence,
\begin{equation*}
|\hat{f}(\hat{x})-f(\hat{x})|=\left|\frac{\delta}{1+\delta}\right|\cdot
|\hat{f}(\hat{x})|\leq\frac{K2^{-m}}{1-K2^{-m}}|\hat{f}(\hat{x})|
\end{equation*}
follows. Since an upper bound on $L(\hat{x},\overline{e})$ can be
computed using global optimization techniques, e.g.\ with interval
arithmetic, the above described bound is computable.
\end{proof}

To summarize, the mathematical iteration (\ref{main_it}) is performed
in the algorithm by iterating a value $\hat{x}_n$ approximating $x_n$
with an upper bound on its absolute error $\overline{e}_n$ according to
\begin{align}
\label{comp_rec1}
\hat{x}_{n+1} & =\hat{f}(\hat{x}_n) & \hat{x}_0 & =gl(x_0,m)\\
\label{comp_rec2}
\overline{e}_{n+1} & =\overline{L}(\hat{x}_n,\overline{e}_n)\overline{e}_n +
\frac{K2^{-m}}{1-K2^{-m}}|\hat{x}_{n+1}| &
 \overline{e}_0 & =\frac{K2^{-m}}{1-K2^{-m}}|\hat{x}_0|
\end{align}
where $\overline{L}(\hat{x}_n,\overline{e}_n)$ is computable
upper bound on $L(\hat{x}_n,\overline{e}_n)$ as described in the
preceding proposition.
This is Line {\tt 9} in the inner {\tt for}-loop of the algorithm
which is executed with successively increasing mantissa length $m$,
controlled by the outer {\tt do-while}-loop. Finally, it has to
be shown that this outer loop eventually terminates. Therefore,
two more propositions are needed.
\begin{proposition}
Let $x\neq 0$ be a real number and $([x]_m)_{m\geq m_0}$
a sequence of finite precision representations of $x$ with
increasing mantissa length $([x]_m).fl.m \geq m$ such that
$\lim_{m\to\infty}([x]_m).err=0$ holds and consequently
$\lim_{m\to\infty}([x]_m).fl=x$. Then
$\lim_{m\to\infty}prec([x]_m,p)={\bf true}$ follows for
all $p\in\mathbb{Z}$.
\end{proposition}
\begin{proof}
Since $x\neq 0$ there
exists some $M\in\mathbb{N}$ such that $0<\frac{1}{2}|x|\leq|([x]_m).fl|$
and $([x]_m).err\leq\frac{10^{-p}}{2(1+10^{-p})}|x|$ holds for all
$m\geq M$. Then, $prec([x]_m,p)={\bf true}$ holds for all
$m\geq M$.
\end{proof}
The next proposition makes the link to Line {\tt 9}
in the algorithm.
\begin{proposition}
\label{prop:alg_halt}
Let $x_n$ be the $n$-th element of the orbit of Equation
\ref{main_it} and $([x_n]_m)_{m\geq m_0}$ a sequence given
according to  the recursion equations (\ref{comp_rec1}) and
(\ref{comp_rec2}) with increasing mantissa length
$([x_n]_m).fl.m \geq m$. Then
$\lim_{m\to\infty}([x_n]_m).err=0$ holds and consequently
$\lim_{m\to\infty}([x_n]_m).fl=x_n$.
\end{proposition}
\begin{proof}
Let  $L:=\sup(f'(D))$ and $\overline{L}\geq L$ be some computationally
accessible value using some global optimization technique. Then
Equation \ref{comp_rec2} leads to
$\overline{e}_{n+1}\leq \overline{L}\overline{e}_n +
\frac{K2^{-m}}{1-K2^{-m}}\overline{M}$ where
$\overline{M}\geq\sup\{|x| : x\in D\}$ such that
$|\hat{x}_n|\leq\overline{M}$ holds for all $n$.
Iteration gives
$\overline{e}_n\leq\overline{L}^n\overline{e}_0+
\frac{K2^{-m}}{1-K2^{-m}}\overline{M}
\sum^{n-1}_{k=0}\overline{L}^k \leq
\frac{K2^{-m}}{1-K2^{-m}}\overline{M}
\sum^n_{k=0}\overline{L}^k$.
Hence, for $n$ fixed, $\lim_{m\to\infty}([x_n]_m).err=0$ follows.
\end{proof}
These two propositions finish the correctness proof of the
algorithm. They show that, if $x_n\neq 0$ for $n=0,\dots,N$,
the outer loop eventually terminates for any $p\in\mathbb{Z}$.

\subsection{Computational Complexity}
After having presented the preliminary work, the main issue of the
paper is addressed - the computational complexity of the presented
algorithm. The complexity measure of interest here is the loss of
significance rate already introduced informally in the last section.
Here is the formal definition.
\begin{definition}
\label{def:losr}
The minimal mantissa length, for which the described algorithm
eventually halts is denoted by $m_{min}(x_0,N,p)$, where $x_0$, $N$
and $p$ are the corresponding input parameters. Then, the
{\em loss of significance rate}
$\sigma:\hat{\mathbb{R}}\cap D\times\mathbb{Z}\to\mathbb{R}$ is defined by
\begin{equation}
\sigma(x,p):=\limsup_{N\to\infty}\frac{m_{min}(x,N,p)}{N}.
\end{equation}
\end{definition}
However, to achieve bounds on the loss of significance rate,
a technical difficulty has to be
circumvented. Therefore, one more assumption on the dynamical system
$(D,f)$, additional to the ones already mentioned in the beginning
of this section, has to be made.
\begin{assumption}
\label{ass:main}
The dynamical system $(D,f)$ is assumed to have the properties
already mentioned in the beginning of this section and
additionally $0\not\in D$.
\end{assumption}
It was already seen in the last subsection that $x_n=0$
makes difficulties such that it cannot be proven that
the algorithm eventually halts. However,
the restriction $0\not\in D$ is no loss of generality. If all
other conditions are fulfilled except that $D$
contains zero, consider the following dynamical system
$(\tilde{D},\tilde{f})$ instead. Choose some $M>\min(D)$ and
set $\tilde{D}:=\{x+M \mid x\in D\}$ as well as
$\tilde{f}(x):=f(x-M)+M$ for all $x\in\tilde{D}$. Then
$(\tilde{D},\tilde{f})$ fulfills all required properties.
Furthermore $\tilde{f}'(x)=f'(x-M)$ holds and therefore there
is no substantial difference in the complexity analysis of the
algorithm between the original system and the modified system.

First, the boundedness of $\sigma(x)$ is shown.
\begin{proposition}
Let $(D,f)$ be as in Assumption \ref{ass:main} and $m_{min}(x_0,N,p)$
as in Definition \ref{def:losr}. Then, for given $p\in\mathbb{Z}$, there exist
some $C_1, C_2\geq 0$, dependent of $f$, such that
$m_{min}(x_0,N,p)\leq C_1 N + C_2$ holds for all
$N\in\mathbb{N}$, $x\in\hat{\mathbb{R}}\cap D$.
\end{proposition}
\begin{proof}
According to the requirements made on $(D,f)$, there are some constants
$L>0$ and $M>0$ such that
$\overline{e}_{n+1}\leq L\overline{e}_n+\frac{K2^{-m}}{1-K2^{-m}} M$
holds for all $n\in\mathbb{N}$ and all mantissa lengths $m$. Without loss
of generality assume $L\neq 1$, otherwise set $L>1$.
Analogous to the treatment in the proof of Proposition
\ref{prop:alg_halt}, iteration gives
$\overline{e}_N\leq \frac{K2^{-m}}{1-K2^{-m}}M\sum^N_{n=0}L^n=
\frac{K2^{-m}}{1-K2^{-m}}M\frac{L^{N+1}-1}{L-1}$.
Since there exists some $B>0$ with $B\leq|\hat{x}_n|$ for all $n$,
$\overline{e}_N/|\hat{x}_N|\leq\overline{e}_N/B\leq C2^{-m}L^{N+1}$
follows with $C:=MK/(B(1-K2^{-m_0})(L-1))$ where $m_0$ is the initial
mantissa length, Line {\tt 2} in the algorithm. Then, if
$C2^{-m}L^{N+1}\leq \frac{10^{-p}}{1+10^{-p}}$ holds,
$prec((\hat{x}_n,\overline{e}_n),p)={\rm\bf true}$ for all
$n=0,\dots,N$. This leads to
$m_{min}(x_0,N,p)\leq\\max(0,ld(L))N +
max(m_0, \mathrm{ld}(L)+\mathrm{ld}(C) + p\cdot\mathrm{ld}(10) + \mathrm{ld}(1+10^{-p}))$.
\end{proof}
\begin{corollary}
Let $(D,f)$ be as in Assumption \ref{ass:main} and $\sigma(x,p)$ the loss
of significance rate. Then, for given $p\in\mathbb{Z}$, there exists some
constant $C\geq 0$ such that $\sigma(x,p)\leq C$ holds for all
$x\in\hat{\mathbb{R}}\cap D$.
\end{corollary}
The treatment has now come to a stage that the main statements of
this paper can be formulated. A lower and an upper bound for the
loss of significance rate is given. Furthermore, these bounds
are strongly related to the Ljapunow exponent $\lambda(x)$ defined
in the previous section.
\begin{theorem}
\label{thm:main}
Let $(D,f)$ be as in Assumption \ref{ass:main}, $\sigma(x,p)$ the loss
of significance rate and $\lambda(x)$ the Ljapunow exponent of
$(D,f)$. Then $\sigma(x,p)\geq\lambda(x)/ln(2)$ holds for all
$x\in\hat{\mathbb{R}}\cap D$, $p\in\mathbb{Z}$ if $\lambda(x)$ exists.
\end{theorem}
\begin{proof}
First there are two constants $B,M > 0$ such that $|\hat{x}_n|\geq B$
and $|\hat{x}_n|\leq M$ holds for all $n$.
According to Equation \ref{comp_rec2} and Proposition
\ref{prop:ubound}, $\overline{e}_{n+1}\geq|f'(x_n)|\overline{e}_n$
holds. Iteration gives
$\overline{e}_N\geq\frac{BK2^{-m}}{1-K2^{-m}}\prod^{N-1}_{n=0}|f'(x_n)|$.
Hence, $\frac{\overline{e}_N}{|\hat{x}_N|}\geq
\frac{BK2^{-m}}{M(1-K2^{-m})}\prod^{N-1}_{n=0}|f'(x_n)|$ follows.
A necessary condition for the algorithm to terminate is therefore
$\frac{BK}{M}2^{-m}\prod^{N-1}_{n=0}|f'(x_n)|\leq
\frac{10^{-p}}{1+10^{-p}}$ which gives
$m_{min}(x_0,N,p)\geq\sum^{N-1}_{n=0}\mathrm{ld}(|f'(x_k)|)+p\cdot\mathrm{ld}(10)+\mathrm{ld}(\frac{BK}{M})+
\mathrm{ld}(1+10^{-p})$. Following the definitions of the loss of significance rate
and the Ljapunow exponent, $\sigma(x_0,p)\geq\lambda(x_0)/\ln(2)$ follows.
\end{proof}

Before a realistic upper bound on the loss of significance rate
can be presented, one more definition is needed.
\begin{definition}
Let $\alpha>0$ then define a function $\eta_\alpha:(0,\infty)\to\mathbb{R}$ by
\begin{equation*}
\eta_\alpha(x):=\begin{cases}
\ln(x) & \text{if } x\geq\alpha\\
\ln(\alpha) & \text{if } x<\alpha
\end{cases}.
\end{equation*}
Furthermore, for any $\alpha>0$ define
\begin{equation*}
\overline{\lambda}_\alpha(x):=\limsup_{n\to\infty}\frac{1}{n}
\sum^{n-1}_{k=0} \eta_\alpha(|f'(f^k(x))|)
\end{equation*}
\end{definition}
\begin{proposition}
For all $\alpha>0$ there exists some constant $C\geq 0$
such that $\overline{\lambda}_\alpha(x)\leq C$
holds for all $x\in D$. Furthermore, if the Ljapunow
exponent $\lambda(x)$ exists,
$\lambda(x)\leq\overline{\lambda}_\alpha(x)$ holds.
\end{proposition}
\begin{proof}
Let $L$ be the Lipschitz constant of $f$ and $\alpha>0$.
Then for all $n\in\mathbb{N}$,
$\frac{1}{n}\sum^{n-1}_{k=0}\eta_\alpha(|f'(f^k(x))|)\leq
\ln(\max(\alpha,L))$ holds. Hence it follows
$\limsup_{n\to\infty}\frac{1}{n}\sum^{n-1}_{k=0}
\eta_\alpha(|f'(f^k(x))|)\leq\ln(\max(\alpha,L))$.
The second assertion follows from the
fact that $\ln(x)\leq\eta_\alpha(x)$ holds for all $x>0$,
$\alpha>0$.
\end{proof}
\begin{proposition}
Let $x\in D$ be given. If $\lambda(x)$ exists, then
also the limit
\begin{equation}
\label{ljap_sup}
\lim_{\alpha\searrow 0}\overline{\lambda}_\alpha(x)=:
\overline{\lambda}(x)
\end{equation}
exists and $\overline{\lambda}(x)\geq\lambda(x)$.
\end{proposition}
\begin{proof}
Since $\ln(x)\leq\eta_\alpha(x)\leq\eta_\beta(x)$ holds
for all $x>0$, $0<\alpha\leq\beta$, also
$\lambda(x)\leq\overline{\lambda}_\alpha(x)\leq
\overline{\lambda}_\beta(x)$ follows. So if $\alpha$
converges in a monotonic decreasing way to $0$,
$\alpha\searrow 0$, the assertion follows.

\end{proof}
\begin{theorem}
\label{thm:ubnd}
Let $(D,f)$ be as in Assumption \ref{ass:main}, $\sigma(x,p)$ the loss
of significance rate and $\overline{\lambda}(x)$ as in
(\ref{ljap_sup}). Let $x\in\hat{\mathbb{R}}\cap D$ be given, then for any
$\varepsilon>0$ there is some $p_0\in\mathbb{Z}$ such that for all $p\geq p_0$,
$\sigma(x,p)\leq\overline{\lambda}(x)/ln(2)+\varepsilon$ holds
if $\overline{\lambda}(x)$ exists.
\end{theorem}
The proof is similar to the proof of Theorem \ref{thm:main}
and can be found in the full version of this article \cite{sp10}.

\section{Conclusions}
In this paper, two main issues are addressed. First it is
shown that a mathematically precise treatment
of multiple-precision floating point computability is not hard
to do. Furthermore this
treatment is in a manner which is familiar to people working
in the field of numerical analysis or scientific computing and
also for theoretical computer scientists. Furthermore, the
formalism does not only allow exact answers concerning the
existence of a computationally feasible algorithm, but is also
allows a treatment of its complexity. As a consequence, the
described algorithm is formulated not only in an exact and
guaranteed way, but also enables a motivated reader the real
implementation and gives a practical performance analysis.

Second, the results show that the Ljapunow exponent, a central
quantity in dynamical systems theory, also finds its way into
complexity theory, a branch in theoretical computer science.
In dynamical systems theory, the Ljapunow exponent describes
the rate of divergence of initially infinitesimal nearby
points. For two points having a small but finite initial
separation, the Ljapunow exponent has only relevance for
short time scales \cite{ce06}. The reason is that due to the
boundedness of $D$, any two different orbits cannot separate
arbitrarily far away. However, the loss of significance rate
shows that the Ljapunow exponent has on long time scales
not only an asymptotic significance but also a concrete
practical one.

\subsection*{Acknowledgments}

The author wishes to express his gratitude to Peter Hertling for helpful
discussions and comments.

\bibliographystyle{eptcs}

redpajama_set_name: RedPajamaArXiv


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Wash and Peel the Turnips and Potatoes. Cut them into 6-7 pieces each.
Heat Oil in a pan. Put Heeng powder and Cumin seeds into it. Once the Cumin seeds gets roasted, add Turmeric powder, Green chilly, Ginger and Coriander powder to it. Roast the masala well by continuously stirring it with a scoop. Now put Turnips, Potatoes, Salt and Red chilli powder into the pan and mix them well with the masala. Stir the vegetable for 2-3 minutes to roast it. Pour 2 table spoon water into the pan, cover it with a lid and let it cook for 8-9 minutes on a medium flame.
Potatoes take time to get tender in comparison with Turnips. Open the lid and break the Potatoes to check whether they have become tender. If Potatoes are still raw, then again add 2 table spoon water to it, cover the pan with a lid, and cook it for another 6-7 minutes on a medium flame. This time the Potatoes will get soft. Your Spicy Potato Turnip Recipe (Aloo Shalgam Recipe) is ready.
Sprinkle some Amchoor powder on the vegetable and mix it well. Turn off the stove and take out the vegetable in a serving bowl. Garnish with Coriander leaves and serve it in lunch or dinner with hot crispy paranthas, chapatis or naan.


text:
Home Headline The first House meeting of 2020
The first House meeting of 2020
SourceKory Leslie (freelance writer)
BELIZE CITY, Thurs. Jan. 16, 2020– Parliament will be meeting for the first time in 2020 on Friday, January 17. At the meeting there will be an introduction of one new bill — The Mutual Administrative Assistance in Tax Matters Amendment Bill, which is set to increase international tax transparency and improve access of the Belize Tax Services to information regarding global allocation of the income taxes paid and certain indicators of the location of economic activity among tax jurisdictions in which multinational enterprise groups operate.
The meeting will then proceed with the second reading of: 1) the Social Security Amendment Bill, which will seek to remove offences in relation to insurance stamps; and 2) the Fisheries Bill, which will seek to make provisions in accordance with the international best practices for the management of the aquatic and fisheries resources of Belize with an aim to optimizing present and future benefits through long-term conservation, management, sustainable use, and development within the fisheries waters of Belize. The current Fisheries Act will also be repealed; and there will be a reading of the Hotels Tourist Accommodation Bill, which will seek to amend the Hotels and Tourist Accommodations Act to modernize the minimum registration, licensing and operating requirements for hotels and other tourist accommodations.
Motions on public business will be opened regarding the loan from the OPEC fund for Phase II of the building of the Caracol Road.
Feature photo: Laura Tucker Longsworth, Speaker of the House
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Why Lidl is selling its own alcohol-free 'spirit' for under £10
22nd October, 2019 by Edith Hancock
Following on from the success of brands like Seedlip, challenger grocer Lidl has launched its own alcohol-free 'craft spirit' at less than half the price of the original.
Called CeroCero, the new product, bottled at 0% ABV, contains "all of the flavour of a botanical gin, with absolutely no alcohol," according to the retailer, with notes of juniper, pine citrus, hibiscus and loganberry.
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Expanding category
Low and no ABV brands account for 1.3% of the UK's total beverage alcohol market, according to data from IWSR, but that percentage is growing, as brands look to appeal to a new generation of consumers who are drinking less alcohol.
Research, published in the journal BMC Public Health, found the proportion of 16 to 24-year-olds who do not drink alcohol had increased from 18% in 2005 to 29% in 2015, and now, more than 25% of young people class themselves as "non-drinkers".
However, consumers are wary of appearing different their peers who do drink.
Speaking at the Lo & No Beverage Summit in London last week (15 October), Paul Thompson, innovation manager at Asahi, said that non-drinkers "don't want to stand out or feel excluded from the drinking ritual."
According to Nielsen, like-for-like sales of low- and non-alcoholic alternatives in the UK rose 25 per cent to £116.6m in the year to June 15.
At the same summit, Craig Hutchison, who founded alcohol-free 'alt-gin' brand Ceder's, said he believes, as the category evolves, "there will be alc-free alternatives for all occasions."
In fact, a spokesperson for Pernod Ricard, which distributes Ceder's in the UK, told db the drinks giant is now in "on-going discussions about innovation with the creators of Ceder's and Celtic Soul" to develop new products in the low and no-alcohol category for its own brand portfolio.
Question of cost
Seedlip, which retails at between £22 and £27 in the UK, was bought by spirits group Diageo earlier this year. In the UK, there is no duty paid on beverages with an ABV of 1.2% or lower, so brands like these could be very lucrative for drinks companies and retailers as they do not need to pay tax on them in the same way as a full-strength spirit.
When the deal was announced in August, Javier Gonzalez Lastra, an analyst at Berenberg, said the price of Seedlip could become "unsustainable" if similar products came to market at a fraction of the cost.
Lidl's press release announcing the launch of CeroCero highlights the fact that the spirit is a "fraction of the price of leading alcohol-free spirits on the market", allowing shoppers to enjoy a taste of luxury without breaking the bank."
Head of buying, Paul McQuade, said: "We're always responding to consumer trends, and so we're thrilled to launch CeroCero – our first non-alcoholic premium botanical craft spirit at a market leading price. Available exclusively at Lidl GB, CeroCero is perfect for those who are big on trying something new this party season."
2 Responses to "Why Lidl is selling its own alcohol-free 'spirit' for under £10"
Vinis Mercator says:
When Seedlip tried to sell their products to me a few years ago I thought they were far too expensive – looks like I was right.
This sort of product is basically a luxury version of a cordial so 3 or 4 times the price of a product like Bottle Green is reasonable, 10 times the price is not. Even at £9.99 I bet Lidl are making a good margin.
Looks like Diageo were sold a pup!
Simon Night says:
If just cold pressing some botanicals in a big factory by the 10,000's of litres & mixing that with water, taking all of 10 minutes with little else, sure 10 pounds sounds about right. I'm not sure how Seedlip do things – but I know there are small-batch, artisan non-alcoholic spirit makers doing things at the highest level, taking as much care for their botanicals (where they come from & how they are processed) & creating spirits like, and if not better, than any other premium distillery. The price should reflect this. Sure, not everyone will want to or can't pay the high price but not everyone has a house in the Hamptons & a Ferrari Testarossa either. I understand the tax concern but refute the comparisons to cordial – it's like saying let's compare the price of a bicycle to a motorcycle. It's just not accurate & clearly more education needs to be done within the category.


If you are looking to sell your home at any point in the near future then you are obviously going to want to make a profit from it. Over the years there would have been some wear and tear on your property from the elements and it's imperative that you keep on top of them and repair/renovate everything before you get your house evaluated.
However, if you want to really up the value of your house before you sell it, then you are going to have to invest in making some wholesale changes to your home. Having a look at the local market and asking experts what the trends are is a great way to start. It is also widely known that eco-friendly houses are fetching much better prices than those that are ill-equipped for looking after the environment.
Another way to add value to your house is obviously to add space – most real estate agents will tell you that an extra bedroom and bathroom will help the value of your property to rise. This also goes for extra living spaces and the easiest way to achieve extra rooms in the house is to do simple conversions. There are a number of different spots in the house that you can convert to ensure your house price increases including loft, garage, shed, basement, conservancy and garden conversions.
One of the easiest places to add another bedroom and bathroom is the loft. It is also probably the least disruptive conversion and it won't have much of an effect on the everyday life going on in the house below. You have to make sure that the loft is easy to reach and safely accessible before you work on it. You will most likely have to strengthen the floor in the loft, which could result in a raised ceiling level – so just make sure that you are left with enough room to stand in. According to many estate agents - loft conversions can increase your house value by 12.5%.
If you don't own a car, like most of Britain's urban residents, and you have a garage – then you are sitting on wasted space. Garage space is ample, and more often than not, used as storage space rather than somewhere to keep a car. Use your garage to add value to your house by converting it into another living space. It can be made into a downstairs lounge or bar, children's play area or a spare bedroom if you convert it correctly.
This conversion is definitely one that can't be taken lightly, and a whole lot of consideration need to happen before you decide to add a basement to your home. It's one of the most expensive conversions that you can do to your house, you will most definitely have to move out for a while, there is lots of digging and it will most likely have some sort of impact on the overall structure of your house – so it's a massive undertaking. But, it can add massive value to your house, and in some cases you can almost double the amount of living space that you have in your home, without your garden or other space being compromised.
If you have the space outside, then adding a conservatory can do great things for the value of your house. Ensure that it complements the style of the rest of the house by using the same decorations and flooring that you find inside. Consistency is keen when adding a conservatory to your home, because even though it was added afterwards – you certainly don't want it to look like it was just added as an afterthought. The conservatory makes a nice sanctuary that can be used as a sunroom, a reading room or a play room for children and it can add around 7% to the value of your home.
If you live in an urban area, chances are that you there is very little parking available on the street. This means that if you have somewhere to park your car – then your property is going to be worth more than those next door that don't. While it is lovely to have a garden, the upkeep can be pretty hard work and the benefits of converting it are huge. You may need to get planning permission first, but then get to work on turning your garden into a drive way or parking space and you could add a whopping 50 000 quid onto your property.
There are a great number of other changes you can make to your property to add value, such as kitchen and bathroom renovations, repainting, redecorating and up-to-date heating and insulation.
Mr H wanted to sell a house fast in Bristol, so that he could release some capital and downsize to a more suitable house.


American author Napoleon Hill once wisely remarked, "No man can succeed in a line of endeavor which he does not like".
Every year, millions of students around the globe find themselves at a stage of life where they have to choose a career path. A small fraction of these young men and women go to a career professional for counseling and advice. An even smaller percentage of people find themselves at the right career direction at a certain point in their life.
A recent study suggests that 65% of Americans are not happy with their career choice. This is slightly higher than their level of happiness when it comes to choosing a life partner.
Unfortunately in the developing world, career counseling has not yet been accepted as an important thing to do in life. People choose their career either because of their parents' pressure or by simply following the footsteps of some other person in the society.
Career counseling is as important as choosing a life partner. If you end up choosing the wrong career, the feeling would not be any lesser than how it might feel to choose a wrong life partner.
Our aim is to introduce career counseling as an important part of life and that at a very early stage of life. We want to make it interesting and adaptable by both parents and kids through interactive and innovative means.
Imagine not being able to choose a career path at the right time and spending rest of your life doing something you don't like. It would hurt even more thinking about how different life would have been if the career, that you have always dreamed about, was pursued instead of going the wrong direction.
We believe that the counseling needs to be started at a very early age and this is why we came up with Chooza.
Chooza aims to engage students, starting grade 5th onward, in exploring their abilities through gamification which will help them choose their career path by identifying their hidden talent.
The system would include a web and mobile app built on basis of Holland's Theory of Career Choice.
The system will be intelligent enough to give basic suggestions but the user can always be linked to a professional career counselor through an online interface.
An important part of the system is a separate section for parents to oversee their child's profile before they turn 18.


text:
namespace TraktApiSharp.Objects.Get.Users.Json.Writer
{
    using Basic.Json.Writer;
    using Newtonsoft.Json;
    using Objects.Json;
    using System;
    using System.Threading;
    using System.Threading.Tasks;

    internal class UserSettingsObjectJsonWriter : AObjectJsonWriter<ITraktUserSettings>
    {
        public override async Task WriteObjectAsync(JsonTextWriter jsonWriter, ITraktUserSettings obj, CancellationToken cancellationToken = default)
        {
            if (jsonWriter == null)
                throw new ArgumentNullException(nameof(jsonWriter));

            await jsonWriter.WriteStartObjectAsync(cancellationToken).ConfigureAwait(false);

            if (obj.User != null)
            {
                var userObjectJsonWriter = new UserObjectJsonWriter();
                await jsonWriter.WritePropertyNameAsync(JsonProperties.USER_SETTINGS_PROPERTY_NAME_USER, cancellationToken).ConfigureAwait(false);
                await userObjectJsonWriter.WriteObjectAsync(jsonWriter, obj.User, cancellationToken).ConfigureAwait(false);
            }

            if (obj.Account != null)
            {
                var accountSettingsObjectJsonWriter = new AccountSettingsObjectJsonWriter();
                await jsonWriter.WritePropertyNameAsync(JsonProperties.USER_SETTINGS_PROPERTY_NAME_ACCOUNT, cancellationToken).ConfigureAwait(false);
                await accountSettingsObjectJsonWriter.WriteObjectAsync(jsonWriter, obj.Account, cancellationToken).ConfigureAwait(false);
            }

            if (obj.Connections != null)
            {
                var sharingObjectJsonWriter = new SharingObjectJsonWriter();
                await jsonWriter.WritePropertyNameAsync(JsonProperties.USER_SETTINGS_PROPERTY_NAME_CONNECTIONS, cancellationToken).ConfigureAwait(false);
                await sharingObjectJsonWriter.WriteObjectAsync(jsonWriter, obj.Connections, cancellationToken).ConfigureAwait(false);
            }

            if (obj.SharingText != null)
            {
                var sharingTextObjectJsonWriter = new SharingTextObjectJsonWriter();
                await jsonWriter.WritePropertyNameAsync(JsonProperties.USER_SETTINGS_PROPERTY_NAME_SHARING_TEXT, cancellationToken).ConfigureAwait(false);
                await sharingTextObjectJsonWriter.WriteObjectAsync(jsonWriter, obj.SharingText, cancellationToken).ConfigureAwait(false);
            }

            await jsonWriter.WriteEndObjectAsync(cancellationToken).ConfigureAwait(false);
        }
    }
}

redpajama_set_name: RedPajamaGithub


I hope you find the enjoyment in reading my novels that I did in writing them! It is my intention to make and create loyal fans whom I can serve with future novels that I write based on your suggestions. Please let me know if there is something you like, or something that you don't like. I have a wide range of ideas that I can put in print and look forward to getting them down on paper. Thank you for your support!


text:
High-Throughput Cell Motion Detector
Courtesy of ReifyHigh-content imaging systems for cell-based assays have proliferated in the past year, but so far none of the systems available allows users to directly quantify dynamic processes such as muscle contraction or cellular migration. To analyze these events, users generally have to monitor the cells throughout the course of the experiment. But a new imaging system developed by Cambridge, Mass.-based Reify http://www.reifycorp.com promises to change all that."The approaches that are
Aileen Constans
Courtesy of Reify
High-content imaging systems for cell-based assays have proliferated in the past year, but so far none of the systems available allows users to directly quantify dynamic processes such as muscle contraction or cellular migration. To analyze these events, users generally have to monitor the cells throughout the course of the experiment. But a new imaging system developed by Cambridge, Mass.-based Reify http://www.reifycorp.com promises to change all that.
"The approaches that are in market today are really focused on how you can take a single picture and quantify what is occurring in that single picture," says Andrew Hack, Reify's director of life sciences. The company's Visible Discovery system, on the other hand, "is focused on the ability to quantify events as they occur over time."
The system, encompassing both instrumentation and analysis, enables unsupervised real-time, high-content detection and measurement of dynamic cellular processes using a high-resolution video microscopy...
Revolutionary Crystallographer Ned Seeman Dies at 75
Hydrogen Fueled Life's Origins: Study
Book Club Discussion of Venomous by Christie Wilcox
2021 Top 10 Innovations


Did Israeli FM's Comments Encourage PKK Attacks Against Turkey?
Following up with their claim of responsibility for the recent bombing attacks in Ankara and Kemer, the Kurdistan Workers Party (PKK) wing the Kurdistan Freedom Falcons (TAK) is now threatening to launch additional attacks against Turkish targets.
"Big Turkish cities will be our main targets," the statement warned. This is a significant change in tactics, as well as aggression, from an organization that in the past has mostly centered its attacks on ambushing Turkish soldiers and fleeing across the border to neighboring Iraq.
The growing aggression by the PKK comes just in the two weeks after Israeli Foreign Minister Avigdor Lieberman was reported to be pushing for the Israeli government to arm the militant faction to "punish" Turkey.


This two bedroom villa, close to shopping centers and the beach, has been tastefully renovated and given a modern feel.
Wooden flooring in the main living area, two well sized bedrooms with built in wardrobes and main with ceiling fan and double sized wardrobe. The modern bathroom has also been renovated to reflect the modern style. The large kitchen with plenty of storage and bench space features timber bench tops and quality electric appliances, quality fridge, living area with built in wall unit make this property perfect for entertaining! Laundry also including a washing machine.
Moving outside the outdoor deck over looking the fully fenced court yard with landscaped gardens and single car space complete the package along with dual entry. Sorry, no pets!


A novel readout procedure for a mercury sensor has been proposed. The sensor is based on the surface adsorption. Mercury vapor is adsorbed at the surface of a thin gold layer patterned into four meanders which are connected into a Wheatstone bridge configuration. Sensing is achieved by the resistivity change of the gold film during mercury adsorption. Direct output from the sensor is electric voltage. It takes an additional procedure to convert voltage to mercury concentration. This type of sensor is improved by introducing continuous readout procedure for the voltage–concentration conversion. By using this readout procedure it is possible to monitor mercury vapor concentration as a function of time. Readout is based on the Langmuir time–dependent adsorption theory. This paper illustrates a practical implementation of the new readout procedure.
This work was supported by the Ministry of Education and Science of Republic of Serbia, Contract No. TR 32008 of 2011.


text:
X-Men Blue: Generation Gap
The original Generation X comes calling for the old school X-Men!
by Sarah Cooke
In the upcoming X-MEN: BLUE #18 by writer Cullen Bunn and artist R.B. Silva coming December 27, we'll see the original Generation X crossing paths with the original five team. So how will it all go down? We chatted with writer Cullen Bunn about what we can expect when these mutants all go head-to-head.
Marvel.com: We've got the original Generation X team coming in, but we're not quite sure whether they'll help the Blue team, or start a fight. Could you give us a couple motivations they might have for each?
Cullen Bunn: The X-Men Blue team has discovered that time is breaking down and the past, present, and future are changing all around them. So, when they meet up with Generation X, they meet a group of mutant kids who have been dealing with circumstances that are a little different than what you might expect. The Generation X team is on edge and on guard, so when the Blue team appears before them, they don't trust them at all. In fact, the original five X-Men are seen as enemies for reasons we have yet to explain.
Marvel.com: If it does come to blows, can you weigh the odds for us? What advantages does each side have?
Cullen Bunn: Well, Generation X has the X-Men Blue team outnumbered just slightly, but I'd say that at this point in time, the X-Men are much more seasoned. That said, Monet and Chamber are wild powerhouses and Synch and Husk both throw in such a wild card factor that Generation X is not to be underestimated. If it comes down to a fight, it could go either way.
Oh, who am I kidding? They're gonna fight.
And the matchups between Penance and Jimmy Hudson, Monet and Bloodstorm, and Husk and Cyclops are all exciting for me.
Marvel.com: In recent X-Men history, there we've seen a lot going on with the timestream. Like younger versions of Jean, Scott, etc. coming into the present. So this seems like a really interesting time for the X-Men to do Marvel Legacy stories. How does the timing of all of this influence this arc?
Cullen Bunn: Anyone who reads my [X-Men] stories knows that I'm almost always looking to the legacy of the creators who came before me. I grew up reading UNCANNY X-MEN. The X-Men are very important to me. Stories like this one—and the previous "Mojo Worldwide" story—allow me to dive into that love for the X-adventures of the past in a big way. I would have wanted to tell this tale no matter what, but the Legacy angle was a happy accident.
Marvel.com: I think a lot of us—including myself!—consider Jubilee a favorite. Can you tease a little about the role she'll have to play in issue #18?
Cullen Bunn: I was so happy to be writing the Jubilee from this era. For me, she represents everything fun about the X-Men. This chapter of the X-Men Blue story certainly has some dark moments, but Jubilee reminds us that we're meant to have a blast reading this story. Also, there's conversation between Jubilee and Bloodstorm that I really enjoyed writing. It's fun and maybe a little meta, but Jubilee also sums up what I think should be the ever-present philosophy for X-Men tales.
Marvel.com: Would you like to mention anything else?
Cullen Bunn: Don't forget that a story featuring Generation X from this timeline also means Banshee and the White Queen have a role to play. In particular, Emma plays an interesting role. In the present, the X-Men see her as an enemy. But when they encounter her in the past, they see a different side of her. Maybe—just maybe—this will inform how they interact with her when—[and] if—they make it back home.
Traverse the timestream with Cullen Bunn and R.B. Silva in X-MEN: BLUE #18, due out December 27!
In this article: X-Men
Revisiting 'Daken: Dark Wolverine' with Writer Rob Williams
Look back on a significant chapter of Daken's deep history with Marvel Unlimited!
'X of Swords' So Far
Catch up with the latest chapters on Marvel Unlimited, and recap what's happened in the opening arc!
Family Trouble Strikes Second 'Xavier Institute' Prose Novel, 'First Team'
Your first look at Robbie MacNiven's upcoming novel, hitting shelves on March 2!
Writer Si Spurrier Previews the New X-Men Series, 'Way of X'
Get the superstar creator's insights on Nightcrawler, the X-Men, mutant life, and more!


Receive the most current information available on a used motorcycle.
Note: Cyclechex does not have reports available for all motorcycles. Please enter your VIN to see if a report is available.
Purchasing a pre-owned motorcycle in today's market can be scary!
Knowledge = Power and a Cyclechex Motorcycle History Report gives you both!


text:
GONG SHOW: LOVE THE TRADE WINDS…
Rhonda Annesley, Editor-in-Chief
I have a couple of things I would like to Gong off my mind… just a little rant to top off my two week return to the Rock in the middle of the Pacifika sami… love the trade winds… but not standing down wind.
My question: How can ASG ask for more money, from the private sector, when it won't or can't even support the main source of our economy's revenue stream in the private sector, our one and only cannery?
StarKist Samoa is shutting down production once again due to lack of fish supply. How long is this going to go on? The cannery wants to build more freezer storage, and the government needs to meet its request because right now we have no other horse pulling the cart (economy) — despite the government's best efforts (and I'm being kind…).
As it stands, other than the government itself, which is planning to support itself by taxing us more, there are currently no other irons in the fire — a lot in the clouds maybe, but 'would-a, could-a, should-a,' don't pay the bills.
What's worse is the government even admits that its FY 2018 budget proposal is predicated on "the continued maintenance of StarKist's current production and employment level."
So far this year, we have seen at least 3 extra weeks of down time due to fish-stock shortage. And there's more coming down the pipeline unless the fish storage problem is resolved — never mind other issues. Samoa News understands an additional 5 weeks is on the down time for the only cannery left in the Territory, and this does not have to do with fish-stock shortage.
And then there is that budget for FY 2018 proposal, which begs the question: How do you propose a budget in two separate documents?
Well according to the governor's cover letter on the FY 2018 budget proposal, it's been determined that revenue collections in FY 2018 will be at 90% of FY 2017 levels and these projected revenues will be sufficient to finance only 90% of FY 2018 planned expenditures.
Therefore, a directive was issued for agencies to prepare budgets at the 90% level (of the FY 2017 budget).
Governor Lolo continued, "Since many of the agencies could not fully cover the cost of their respective operations, the second directive was issued calling for the preparation of supplemental budgets and for a majority of the agencies, the supplemental budget threshold was set at 10% of the FY 2017 budget."
As a result, for FY 2018, two separate budgets have been submitted — the Basic Budget, which is 90% of the approved FY 2017 budget, and the Supplemental, which is 10% of FY 2017, said Lolo — adding that the two budget proposals are submitted concurrently to facilitate for simultaneous consideration and approval by the Fono.
He also said that approval of the Supplemental Budget is dependent on passage of the five revenue measures now before the Fono. However, if the measures are not approved, the supplemental is thereby invalidated.
Okay then… I would GONG this alone on — huh?
Here goes an explanation:
If I understand what's being said:
The Basic budget document only shows revenues supporting 90% of what the government says it needs in FY 2018; and, it can't give you a Basic budget with a 100% of its needs for FY 2018, because its revenue collections don't support it — they're 10% short.
So… it's submitting the Supplemental budget, which lives or dies on whether or not those new proposed revenue measures pass or not — and that would then allow ASG to meet its FY 2018 needs (which the Basic budget does not meet being 10% short).
A bit convoluted, but okay, let's take a look at the numbers the government is trying to cover for FY 2018, according to them:
First, according to ASG, they are currently short by around $3Mil a month to cover its FY 2017 budget — creating a hole of about $36Mil at the end of the FY 2017.
To counteract this shortfall, the Administration is saying first: its Basic budget is 10% less of its FY2017 local budget that stands at $108.16Mil (in FY 2017) — which is around $10.8Mil.
However, the Supplemental budget then adds back the 10% it took out to adjust for the FY 2017 budget shortfall, but it does so with the understanding that the newly passed revenue measures will fund these expenses.
However, there are a couple of problems with this:
It's estimated that for 2018, the new revenue measures will raise $21.6Mil, which doesn't cover the balance of the $36Mil it's touting is its annual shortfall estimate; and in 2019, it's estimating $32.3Mil from the revenue measures, which is still short of the $36Mil.
This is dependent of course on the belief that the numbers the administration is saying is their shortfall are actually correct.
I think we need to keep in mind here that these numbers being touted by the Lolo & Lemanu Admin are "unaudited".
The Single Audit Report for ASG for FY 2016 prepared by Moss Adams LLP, states in part that its total deficit was only around $3.1Mil. So what changed? How did the 2017 deficit estimate end up at $3Mil monthly or possible $36Mil from the $3.1Mil for last year?
Is it the bond issue payment? Is it a bloated payroll? Are there expenses happening that are political payouts? Or is the government raising more money to fund its 'cloud projects? Is it all of the above?
And don't forget, the governor said, it's "important to remember and to take note" that the forecast revenues for FY 2018 "are predicted on the continued maintenance of StarKist's current production and employment level."
So, until we get some audited numbers, I don't believe we should support these revenue measures proposed by the administration, which we all know ultimately will be paid by the households in American Samoa. Businesses pass on expenses, period. The latest estimate is $2,000 per household, and that's a huge hit for households.
After all, that's at least 4 faalavelaves or faifeau donations…GONG!
At this stage, I think a cut in expenses funded with local funds — worth around the admin's $3Mil figure per month is a quick solution, until an audited financial can be done and a 'true figure' obtained.
Guess what government entities are 100% locally funded… yep… the Fono, the Office of Samoan Affairs, certain Governor's Office Special Projects… GONG!


text:
Q: Find column Vbscript PowerDesigner I would like to find column in tables.
Why I have mistake when I run VBscript in Power Designer?
Dim table, column
for each table in activeModel.tables
for each column in table.columns
output column.name
next
next

Mistake is Object doesn't support this property or method: 'columns'.

redpajama_set_name: RedPajamaStackExchange


text:
Police searching for 8 suspects in motel robbery
Two men allegedly robbed at La Quinta Inn
Updated: 7:12 AM CST Dec 16, 2015
Police are searching for eight suspects in an an early morning robbery.They were called to La Quinta Inn near Interstate 680 and Maple Street at about 12:30 a.m. Wednesday.Officers said a group of eight people - three women and five men - got in an argument about a wallet with two men staying at the motel.Officers also said the group was allegedly ransacking the room, then someone pulled out a gun.Police said evidence of drug use was left and the group got away with cash and cell phones.They used two getaway vehicles, one of which police said was stolen.More details on this developing situation on KETV NewsWatch 7 and KETV.com as they become available.
OMAHA, Neb. —
Police are searching for eight suspects in an an early morning robbery.
They were called to La Quinta Inn near Interstate 680 and Maple Street at about 12:30 a.m. Wednesday.
Officers said a group of eight people - three women and five men - got in an argument about a wallet with two men staying at the motel.
Officers also said the group was allegedly ransacking the room, then someone pulled out a gun.
Police said evidence of drug use was left and the group got away with cash and cell phones.
They used two getaway vehicles, one of which police said was stolen.
More details on this developing situation on KETV NewsWatch 7 and KETV.com as they become available.


text:
Larcidiocesi di Antsiranana (in latino: Archidioecesis Antsirananensis) è una sede metropolitana della Chiesa cattolica in Madagascar. Nel 2020 contava 823.000 battezzati su 2.005.300 abitanti. È retta dall'arcivescovo Benjamin Marc Ramaroson, C.M.

Territorio
L'arcidiocesi comprende la città di Antsiranana, dove si trova la cattedrale di San Matteo.

Il territorio si estende su 37.924 km² ed è suddiviso in 29 parrocchie.

Provincia ecclesiastica
La provincia ecclesiastica di Antsiranana, istituita nel 1958, comprende 3 suffraganee:
diocesi di Ambanja
diocesi di Mahajanga
diocesi di Port-Bergé

Storia
Il vicariato apostolico del Madagascar settentrionale fu eretto il 5 luglio 1898 con il breve Universi fidelium di papa Leone XIII, ricavandone il territorio dal precedente vicariato apostolico del Madagascar settentrionale, che contestualmente assunse il nome di vicariato apostolico del Madagascar centrale (oggi arcidiocesi di Antananarivo).

La nuova circoscrizione ecclesiastica, che comprendeva tutto il territorio a nord del 18º grado di latitudine sud, fu affidata ai missionari della Congregazione dello Spirito Santo, noti come "missionari spiritani".

Il 20 maggio 1913 in virtù del decreto Cum in generalibus della Sacra Congregazione per la Propagazione della Fede  assunse il nome di vicariato apostolico di Diégo Suarez.

Il 15 marzo 1923 cedette una porzione del suo territorio a vantaggio dell'erezione del vicariato apostolico di Majunga (oggi diocesi di Mahajanga).

Il 2 febbraio 1932 cedette una porzione di territorio alla prefettura apostolica delle Isole di Mayotte, Nossi-bé e Comore (oggi diocesi di Ambanja).

Il 14 settembre 1955 con la bolla Dum tantis di papa Pio XII il vicariato apostolico fu elevato a diocesi, con il nome di diocesi di Diégo Suarez, suffraganea dell'arcidiocesi di Tananarive (oggi arcidiocesi di Antananarivo).

L'11 dicembre 1958 è stata ancora elevata al rango di arcidiocesi metropolitana con la bolla Qui benignissima di papa Giovanni XXIII.

Il 21 maggio 1959 cedette un'altra porzione di territorio a vantaggio dell'erezione della diocesi di Ambatondrazaka.

Il 28 ottobre 1989 ha assunto il nome attuale per effetto del decreto Apostolicis sub plumbo della Congregazione per l'evangelizzazione dei popoli..

Il 30 ottobre 2000 cedette un'ulteriore porzione di territorio a vantaggio dell'erezione della diocesi di Fenoarivo Atsinanana.

Cronotassi dei vescovi
Si omettono i periodi di sede vacante non superiori ai 2 anni o non storicamente accertati.
 François-Xavier Corbet, C.S.Sp. † (5 luglio 1898 - 25 luglio 1914 deceduto)
 Auguste Julien Pierre Fortineau, C.S.Sp. † (25 luglio 1914 succeduto - aprile 1946 dimesso)
 Edmond-Marie-Jean Wolff, C.S.Sp. † (13 febbraio 1947 - 13 aprile 1967 dimesso)
 Albert Joseph Tsiahoana † (13 aprile 1967 - 14 novembre 1998 dimesso)
 Michel Malo, Ist. del Prado (28 novembre 1998 - 27 novembre 2013 ritirato)
 Benjamin Marc Ramaroson, C.M., dal 27 novembre 2013

Statistiche
L'arcidiocesi nel 2020 su una popolazione di 2.005.300 persone contava 823.000 battezzati, corrispondenti al 41,0% del totale.

|-
| 1949 || 36.948 || 541.334 || 6,8 || 48 || 7 || 41 || 769 ||  || 41 || 64 ||
|-
| 1970 || 76.898 || 813.227 || 9,5 || 42 || 1 || 41 || 1.830 ||  || 64 || 87 || 3
|-
| 1980 || 104.232 || 1.110.000 || 9,4 || 50 || 21 || 29 || 2.084 ||  || 49 || 94 || 20
|-
| 1990 || 194.355 || 1.665.000 || 11,7 || 48 || 23 || 25 || 4.049 ||  || 68 || 111 || 21
|-
| 1999 || 340.776 || 2.468.466 || 13,8 || 59 || 44 || 15 || 5.775 ||  || 52 || 148 || 22
|-
| 2000 || 231.000 || 1.780.040 || 13,0 || 63 || 33 || 30 || 3.666 ||  || 65 || 144 || 16
|-
| 2001 || 231.000 || 1.780.040 || 13,0 || 64 || 50 || 14 || 3.609 ||  || 42 || 185 || 16
|-
| 2002 || 310.133 || 1.566.162 || 19,8 || 39 || 29 || 10 || 7.952 ||  || 42 || 116 || 17
|-
| 2003 || 312.140 || 2.010.736 || 15,5 || 50 || 38 || 12 || 6.242 ||  || 35 || 117 || 17
|-
| 2004 || 343.354 || 2.211.809 || 15,5 || 46 || 36 || 10 || 7.464 ||  || 40 || 119 || 17
|-
| 2006 || 369.757 || 1.209.600 || 30,6 || 44 || 33 || 11 || 8.403 ||  || 44 || 150 || 23
|-
| 2012 || 590.796 || 1.431.000 || 41,3 || 65 || 45 || 20 || 9.089 ||  || 42 || 139 || 25
|-
| 2015 || 682.304 || 1.536.326 || 44,4 || 80 || 56 || 24 || 8.528 ||  || 58 || 163 || 28
|-
| 2018 || 718.225 || 1.666.415 || 43,1 || 68 || 53 || 15 || 10.562 ||  || 48 || 232 || 30
|-
| 2020 || 823.000 || 2.005.300 || 41,0 || 72 || 57 || 15 || 11.430 ||  || 46 || 229 || 29
|}

Note

Bibliografia
 Breve Universi fidelium, in Le canoniste contemporain, 23º anno, Paris, 1900, pp. 99–100
 Decreto Cum in generalibus, AAS 5 (1913), p. 274
 Bolla Dum tantis, AAS 48 (1956), p. 113

Altri progetti

Collegamenti esterni
Annuario pontificio del 2021 e precedenti, in
 Sito ufficiale della diocesi

Antsiranana
Antsiranana

redpajama_set_name: RedPajamaWikipedia


Staff Development is the continuing education of teachers other staff members. It is an essential ingredient to keep educators abreast of current issues and trends that are important to the well being of students at educational facilities.
This project is designed to give educators a different perspective of topics they may think they know a great deal about. This activity will have three to four senior "Principles of Communication" students research an area of concern to high school teenagers today. Students will add their perspectives as teenagers to their research findings. The culminating activity will be a day in which the students present their findings and opinions to the faculty in a small group setting.
Students should form groups of three to four.
The groups then need to brainstorm various areas of research that they see as "crisis" areas for today's teenagers. Some examples topics could be: Cancer, Death, Divorce, Asthma, Teen Pregnancy, Teen Alcohol Use, or Teen Drug Use. Remember these are just some example areas. Certainly you may feel free to explore other areas of concern since that what this project is ultimately trying to garner-A teen's perspective!
Groups should then submit their research topics to the teacher. Each group will have a different area of research.
d. Interviews with students or adults that have insights into your topic. Example: Your topic is teens with Cancer. You interview a student who has cancer. Another example would be: Your topic is "A teen's perspective of living with a family member that has cancer." You interview the student or the family member that is afflicted with this disease.
d. Facts, Opinions, Visuals, Handouts, and other forms of information needed to present the topic.
f. A group designed teacher feedback form. This form should include a few questions for the faculty to answer, which provides feedback to the group. The form should also include an area where each attending faculty member assigns the group an overall grade from 60 to 100.
The teacher will read all term papers and grade them on a 100 point system. The main areas of focus will be ability to follow directions, grammar, and usage.
Each class member will fill out the survey sheet that the group has created for the staff development session. Each class member will assign the group a grade between 60 and 100. The teacher will add together all students' grades and divide those grades by the total number of students present in class to determine the group's Class Presentation grade.
Staff Development Presentation-100 Points maximum.
Each member of the staff that attends the group's presentation will be given a survey sheet at the conclusion of the presentation. One question that should appear on each group's survey sheet should be a place where each staff member assigns the group a grade from 60 to 100. The teacher will forewarn the faculty about the survey sheets and ask the staff to be honest about each group's presentation.
The teacher will divide the total number of staff members attending the staff development day by the total number of groups presenting topics. For example: If 100 staff members were attending and 10 different groups were presenting, I would limit the faculty to 10 staff members per topic sign up sheet. This will ensure balance of attendance and equity in grading to all groups.
The teacher will post a list of all the different group topics in the teachers' workroom. Each group topic will be assigned a corresponding room number. Each member of the staff will sign up for one of the groups' topics. There will be a previously determined, equitable number of staff members per topic. This will provide student groups with approximately the same number of staff members.
Student groups should prepare the classroom they will be presenting in immediately following the dismissal of school at 1:00. It is very important for groups to reserve any pieces of technology they will be using ahead of time for the staff development day. See the teacher ahead of time if there are any problems with this.
At the conclusion of the group's presentation, the group members should have the staff members fill out their survey forms. The teacher will supply a large envelope for those completed forms to be placed in. One member of each group should take the initiative of collecting the surveys, putting the surveys into the envelope, and giving the envelope to the teacher. Please be sure to put your group members' names and your group's topic on the outside of the envelope. The teacher will add together the total number of surveys grades per envelope and divide by the total number of surveys completed. The groups will get to review the surveys during the next Principles of Communications class period.
This is the newest lesson that I have created. I hope to experiment with this lesson for the fist time this year during a staff development day.


Avoid picture games. There is nothing more boring than a hangman style game with a footballer pictured and guessing letters to unveil who it is. These games have been done to death, and that reminds me..
Avoid too easy. The one problem many football apps do have is that they are not made by people with a good knowledge of football and hence it all tends to be too easy with questions galore on the same narrow subjects.
The type of games that are really working at the moment in the world of app quizzing are head to head games…Quiz Up, Quiz Planet etc. So invest time and money and making it a more competitive element to it, where rival fans can play and friends can show off their knowledge. The quiz games of just playing for your own personal score are long dead and player interaction is the way forward!
Invest in a quality writer. As an experienced quizzer and writer myself, I see so many awful apps with questions that have been cobbled together. The quality of questions will keep players coming back and if your app is advert reliant then that is what you want. Spend time finding a good writer, get questions updated all the time and always upload them. 100,000 questions should be the very basic base line you should be looking at when launching an app about football.
So, don't waste your time reinventing the while, get a good interactive format mixed with solid questions and you will be laughing all the way to the bank! Also check out Best Football Apps for all the best football related apps out there.
Previous PostPrevious Ole Gunnar Solksjaer – How can he not….?


To promote outstanding attendance, St Michael's CE Primary School has a number initiatives and celebrations throughout the year.
Class attendance is tracked weekly and a certificate given to the class with the best attendance during celebration assembly. In Key Stage 2, the class with the highest attendance for the week also receives a trophy, in Key Stage 1 and EYFS they receive the 'attendance bear'. We also give out individual attendance certificates every term for those children who have been to school every day and therefore have 100% attendance. At the end of the year children with 100% attendance receive a medal.
Congratulations to the following classes who had best attendance for the term.


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