text:
\section{Introduction}

\def\S{\mathbb{S}}

Given a topological space $X$, let $C(X)$ be the set of continuous maps defined from  $X$ to $X$, and let
$(f_n)$ be  a sequence of
maps from $C(X)$. A \emph{non--autonomous  dynamical system} is built
by iterating  sequentially the maps  $f_n$, that is, the \emph{orbits} are:
\begin{equation}\label{Eq:NonAut}
x_{n+1}=f_n(x_n), n=0,1,2,\ldots,
\end{equation}
with $x_0\in X$. When the sequence $(f_n)$ is periodic  of period $p$ then the system,
denoted by $[f_0,f_1,\dots,f_{p-1}]$,  is called an \emph{alternating system.}  This type of
dynamical systems as well as the general non-autonomous case have received special attention in the literature due to
their potential applications to
natural and social sciences (for instance, see \cite{CuHen,ElaSa,CaCoSa,Krau}). To this respect, take into account
that the deterministic law can be dependent on the time due, for instance,
to effects of seasonality producing changes in the environment (although we are interested in
discrete systems, for non-autonomous continuous models dealing with the effect of seasonality, the reader
is referred to the survey \cite{Buono}), or due to the appearance of new information or technical aspects
in economical or social features.
Moreover, from the theoretical point of view, many authors have studied the qualitative properties of the orbits generated by these systems,
see for example \cite{kolsnohatro,alsharawiEtAl,canovaslinero06,CanoNonAut,daniellosteele,joao,calisoalternated1,calisoalternated2}.

We focus here on alternating systems generated by 2-periodic sequences of maps on the circle
$\S^1:=\{z\in\mathbb{C}:|z|=1\}$. Namely, $X= \S^1$,  $f,g:\S^1\to \S^1$ are continuous and the orbits generated by $[f,g]$ are sequences $(x_n)$ given by:
$$
x_{n+1}=
\left\{\begin{array}{l}
f(x_n) \textrm{ if $n$ is even,}\\
g(x_n) \textrm{ if $n$ is odd.}
\end{array}
\right.
$$
From the different qualitative aspects of alternating systems, we are interested in the topic
of the periodicity and in the search of periodic structures associated to these systems.
Recall that the sequence
$(x_n)$ and  $x_0$ are \emph{periodic} if there exists $m\in \N:=\{1,2,\ldots\}$ such that $x_{n+m}=x_n$
for all $n=0,1,2,\ldots,$ and the smallest positive
integer $m$ satisfying the previous condition is called the \emph{period}  or \emph{order} of $\sq$ and $x_0$.
In addition, we say that $x_0$, or the sequence $(x_n)$, is \emph{$m$--periodic}.
By $\Per\sisalter$ we understand the set of periods of $\sisalter$, that is,
the set of positive integers $m$ for which there exists $x_0\in X$ such that it is $m$--periodic. Notice that
$\Per\sisalter=\Per[g,f]$.
As usual we will use the following notation for the iterates of $f$: $f^1=f, f^{n} = f\circ f^{n-1}, n\geq 2$. By $f^0$ we mean the identity
on $X$. Observe that when $f=g$, we receive a classical \emph{autonomous dynamical system} and the analogous definitions on periodicity
(for instance, see \cite{alseda}), in this case we will
use the standard notation and $[f,f]$ will be replaced simply by $f$.

In order to manage circle maps, it is necessary to introduce the notion of lifting
(the reader is referred to \cite{alseda} for a deeper development of this topic).
Consider the standard universal
covering $e:\mathbb{R}\rightarrow\mathbb{S}^1$ given by
$e(x)=e^{2\pi ix}.$ If $f\in C(\mathbb{S}^1),$ we find a (non unique) map
$F:[0,1]\rightarrow \mathbb{R}$ such that the diagram
\[
\begin{array}{ccc}
\lbrack 0,1] & \overset{F}{\longrightarrow } & \mathbb{R} \\ e\downarrow &
& \downarrow e \\
\mathbb{S}^{1} & \overset{f}{\longrightarrow } & \mathbb{S}^{1
\end{array
\]
commutes.  Then, $F$ is said to be a  \emph{lifting} of $f.$ Realize that
$e(0)=e(1)=1,$ so $e(F(1))=f(e(1))=f(e(0))=e(F(0))$. Thus, $d:=F(1)-F(0)\in\mathbb{Z}.$
The integer $d$ is said to be the \emph{degree}
of $f,$ and we denote it by $\deg(f).$ Moreover, we can extend
the lifting $F$ from $[0,1]$ to $\mathbb{R}$ by considering
$\widetilde{F}:\mathbb{R}\rightarrow\mathbb{R}$ as $\widetilde{F}(x)=F(x-[x])+[x]\deg(f),$
where $[\cdot]$ is the integer part of a real number $x$. To simplify the notation, in the sequel
we will identify $\widetilde{F}$ with $F$.
It is well known that   $\deg(f\circ g)=\deg(f)\deg(g)$ and then  $\deg(f^m)=(\deg(f))^m$ for all
$m\geq 1$.

In this work we deal with \emph{affine circle maps}. A map $f\in C(\S^1)$ is said to be \emph{affine} if its  lifting is of the type
$F(x)=dx+\alpha$ for some $d\in\mathbb{Z}$ and $\alpha \in \mathbb{R}$. Hence
\begin{equation}\label{Eq:f}
f(z)= z^d e^{2\pi\alpha i}, z\in\mathbb{S}^1.
\end{equation}
The periodic structure of these maps
is  given in \cite[Lemma 2]{calisoalternated1}:

\begin{lemma}\label{periodosrotaciones}
 Let $f\in C(\S^1)$ be  an affine map with associate lifting $F(x)=d x+\alpha$. Then:
\begin{enumerate}
 \item If $|d|\geq 3$ or $d=2$, then $\mathrm{Per} (f)=\N$.
 \item If $d=-2$, then $\mathrm{Per} (f)=\N\menos \{2\}$.
 \item If $d=-1$, then $\mathrm{Per} (f)=\{1,2\}$.
 \item If $d=1$, and  $\alpha=\frac{M}N \in\Q$, with $\gcd(M,N)=1$ and $N>0$,
then $\mathrm{Per} (f)=\{N\}$. Otherwise, $\mathrm{Per} (f)=\emptyset.$
 \item If $d=0$, then $\mathrm{Per}(f)=\{1\}$.
\end{enumerate}
\end{lemma}

Even in the easiest case of alternating systems given by two affine circle maps on the circle,
 the characterization of the set of periods is hard. The present
authors have solved this problem in \cite{calisoalternated1,calisoalternated2}. We sketch the results
in Section~\ref{S:previousresults}. In Section~\ref{S:preliminar} we present some preliminary technical results and we give a
new determination for the odd periods of the system $[f,g]$, whose appearance is only possible whenever $f$ and $g$ commute. This new approach to the calculation of odd periods (based on the iterates of a suitable set $C$ of real numbers and on appropriate properties of congruences of integer numbers) is different from that developed in \cite{calisoalternated1,calisoalternated2} (in terms of a certain discrete map $\phi:\mathbb Z_{\Delta}\rightarrow \mathbb Z_{\Delta}$ in these last cases). Finally, our new approach originates a
direct algorithm, described in Section~\ref{S:algorithm}, to compute odd periods.

\section{Results on $\Per\sisalter$}\label{S:previousresults}

In what follows, $f$ and $g$ are affine   maps on $\S^1$ with respective liftings $F(x)=d_1x+\alpha$ and
$G(x)=d_2x+\beta$. We adopt the notation:
\begin{equation}
 \kk:=\kkdef\label{E:kappa}.
\end{equation}
Next, we introduce the characterization of the set of periods $\Per\sisalter$ through se\-veral results. They
are given depending on whether the degrees of $f$ and $g$ coincide (see Theorem~\ref{MP:samedegree})
or not. In the last case, there are subcases depending on the value $\kk$
(see Theorems~\ref{P:main1} and \ref{T:main}).

\begin{theorem}\label{MP:samedegree}(\cite[Theorem A]{calisoalternated1})
Let $f,g\in C(\S^1)$ be affine maps with liftings $F(x)=dx+\alpha$ and $G(x)=dx+\beta$, respectively,
and such that $\beta-\alpha\notin\mathbb{Z}$. Then:
\begin{itemize}
\item[(1)] If $|d|\geq 2$, then $\mathrm{Per} \sisalter=2\N$ (the set of even positive integers).
\item[(2)] If $|d|=1$, we distinguish the cases
\begin{enumerate}
 \item[(i)] If $\alpha+d\beta	\not\in\Q$  then $\mathrm{Per}\sisalter=\emptyset$.
 \item[(ii)] If $\alpha+d\beta=\frac{M}{N}\in\Q$ with $\textrm{gcd}(M,N)=1$ then $\mathrm{Per}\sisalter=\{2N\}$.
\end{enumerate}
\item[(3)] If $d=0,$ then $\mathrm{Per}\sisalter=\{2\}$.
\end{itemize}
\end{theorem}

When $f$ and $g$ have different degrees we must distinguish two cases depending on whether $\kk$, cf. Eq.~(\ref{E:kappa}),
is or is not an integer number. By $\mathbb{O}$ we denote the set of odd natural numbers.

\begin{theorem}\label{P:main1}(\cite[Theorem B]{calisoalternated1})
Let $f,g\in C(\S^1)$ be affine maps with liftings $F(x)=d_1 x +\alpha$ and $G(x)=d_2 x +\beta$, with $d_1\neq d_2.$
Assume that $\kappa \notin\mathbb{Z}$ and let $\delta:=d_1d_2$. Then:
\begin{enumerate}
\item If $|\delta|\geq 3$ or $\delta=2$, then $\mathrm{Per}[f,g]=2\mathbb{N}$.
\item If $\delta=-2$, then $\mathrm{Per}[f,g]=2\mathbb{N}\setminus\{4\}$.
\item If $\delta=0$, then $\mathrm{Per}[f,g]=\{2\}$.
\item If $\delta=-1$, then $\mathrm{Per}[f,g]=\{2,4\}$.
\end{enumerate}
\end{theorem}

\evt
\begin{theorem}\label{T:main}(\cite[Theorem C]{calisoalternated1})
Let $f,g\in C(\S^1)$ be affine maps with liftings $F(x)=d_1 x + \alpha$ and $G(x)=d_2 x +\beta,$ being $d_1\not=d_2$, $d_2\not=1$ and
$\kappa\in\Z$. Let
$$
A= \left\{
\begin{array}{l}
 \{4\} \textrm{ if $d_1d_2=-2$, }\\
\emptyset \textrm{ otherwise.}
\end{array}
\right.
$$
Let $\Lambda=\mathrm{Per}([f,g])\cap \mathbb{O}$. Then:
\begin{enumerate}
\item  If $d_1\in\{-1, 0, 1\}$, then $\mathrm{Per}([f,g])$ is described by the following table:

\begin{tabular}{|c||c|c|c|c|c|}
\hline
&\rojo $ d_1$&\rojo $d_2$&\rojo $\Lambda$&\rojo $\mathrm{Per}[f,g]$\\ \hline
(a)&0&&&\{1\}\\ \hline
(b)&1&$\notin\{-1,0,1\}$& $\{1\}$&$[2(\N\menos\{1\})\menos A]\cup\{1\}$ \\ \hline
(c)&1&-1&$\{1\}$& $\{1,4\}$\\ \hline
(d)&1&$\notin\{0,1\}$&$\{N\}$& $[2\N\menos A]\cup\{N\}$\\ \hline
(e)&1&-1&$\emptyset$& $\{2,4\}$\\ \hline
(f)&1&$\notin\{-1,0,1\}$&$\emptyset$&$2\N\menos A$\\ \hline
(g)&-1&$\in\{-3,-2\}$&$\{1\}$&$2(\N\menos\{1\})\cup\{1\}$\\ \hline
(h)&-1 & $\notin\{-3,-2,-1,0,1\}$ &$\{1\}$& $[2\N\menos A]\cup\{1\}$\\ \hline
(i)&-1&$\notin\{-1,0,1\}$&$\emptyset$&$2\N\menos A$\\ \hline
\end{tabular}

Here $N$ denotes an odd natural number bigger than $1$. In addition, the role of $d_1$ and $d_2$ in the above table can be changed.

\item \label{T:mainpartgeneral} If $\{d_1,d_2\}\cap\{-1,0,1\}=\emptyset$ then
 $$\mathrm{Per}[f,g]=[2\mathbb{N}\menos A] \cup \Lambda.$$
\end{enumerate}
\end{theorem}

These results completely characterize the set of even periods in $\Per\sisalter$.
By means of the study of a dynamical system defined  on $\Z_{\Delta}$ for $\Delta=|d_1-d_2|$
it is possible to say  what odd numbers are in $\Per(\sisalter)$.

\subsection{The set $\Lambda$ of odd periods in $\Per\sisalter$}\label{SS:odd}

Let $d_1,d_2$ be integers, $\alpha,\beta$ be real numbers, $\kk=\beta (d_1-1) - \alpha (d_2 -1)$,   $\Delta=|d_1-d_2|$  and define
$\phi_{d_i,\kappa}:\mathbb{Z}_{\Delta }\rightarrow \mathbb{Z
_{\Delta }$,  $i\in\{1,2\}$, by
\begin{equation}
 \label{Eq:phi}
 \phi_{d_i,\kappa}(m):=(d_{i}m+\kappa )\,\modu (\Delta ).
\end{equation}
Since $d_{1}\equiv d_{2}\modu (\Delta )$, we have $\phi_{d_1,\kappa}=\phi_{d_2,\kappa}=:\phi$ and
the following result connects $\Lambda=\mathrm{Per}([f,g])\cap \mathbb{O}$ and $\Per(\phi)$.

\begin{theorem}\label{T:oddPeriods}(\cite[Prop. 15, Th. 17]{calisoalternated1})
Let $f,g\in C(\mathbb{S}^{1})$ be affine maps with associate liftings $F(x)=d_{1}x+\alpha $ and
G(x)=d_{2}x+\beta $ and  $d_{1}\neq d_{2}$. If $\kappa\not\in\Z$ then
$\Lambda=\emptyset$, otherwise  $\kappa \in
\mathbb{Z}$ and   {\rojo $\Lambda=\mathrm{Per}(\phi )\cap \O$.}
\end{theorem}

Then, in order to describe the set $\Lambda=\O\cap\Per\sisalter$ it suffices to study the set $\Per(\phi)$.
Let $p,s\in \N$, and define

$$\sigma(p,s):=\left\{\begin{array}{lcl}
                      1,&\quad & \textrm{ if $p$ is odd or $p=2$ and $s=2$},\\
                      2,&\quad & \textrm{ otherwise}.
                     \end{array}
\right.$$

We introduce the first case, $\Delta=p^s$ with $p$ prime.
\evt
\begin{theorem}\label{main0}(\cite[Theorem A]{calisoalternated2})
 Let $\Delta=p^s$ where $p$ is a prime and $s\geq 1$  and let $\phi_{d,\kappa}:\Z_\Delta\to\Z_\Delta$ be
 defined by $\phi_{d,\kappa}(x)=dx+\kappa$,  $d,\kappa\in\Z_\Delta$. Then $\Per(\phi_{d,\kk})$ is one of the
 following sets:
\begin{enumerate}
 \item
 $\{1\}\cup \{Np^j\}_{j=0}^\alpha$ where $N$ is a divisor of $p-1$ and $\alpha\in\{0,1,\dots,s-\sigma(p,s)\}$.

 \item $\{p^\alpha\}$ for some $\alpha\in\{0,1,\dots,s\}$.

\end{enumerate}

Conversely, let $p$ be a prime, let $\Delta=p^s$ be with $s\geq 1$, and let $\mathcal{P}$ be one
of the above sets $\{1\}\cup \{Np^j\}_{j=0}^\alpha$ or $\{p^\alpha\}$, then there exists
$\phi_{d,\kk}:\Z_\Delta\to\Z_\Delta$ such that $\Per(\phi_{d,\kk})=\mathcal{P}.$

\end{theorem}

Next result helps us to determine exactly $\Per(\phi_{d,k})$ for fixed $d,\kk\in\Z_\Delta.$
\evt

\begin{theorem}\label{main1}(\cite[Prop. 3.1, Th. C]{calisoalternated2})
 Let $\Delta$ be a positive integer, $d,\kappa\in\Z_\Delta$, let $\phi_{d,\kappa}:\Z_\Delta\to\Z_\Delta$ be
 defined by $\phi_{d,\kappa}(x)=dx+\kappa$. Then we distinguish the following cases:

\begin{enumerate}

\item  For any  $\Delta\in\N$ we have  $\Per (\phi_{0,\kappa})=\{1\}$
 and  $\Per (\phi_{1,\kappa})=\{\frac\Delta{\gcd(\Delta,\kappa)}\}$.

\item If $\Delta\geq 3$ is even then $\Per (\phi_{\Delta-1,\kappa})=\{1,2\}$ if $\kappa$ is even and
$\Per (\phi_{\Delta-1,\kappa})=\{2\}$ if $\kappa$ is odd.

\item If $\Delta\geq 3$ is odd then $\Per (\phi_{\Delta-1,\kappa})=\{1,2\}$ .

\item When $\Delta=p^s$, $p$ prime, we have:

\hspace*{-1cm}\begin{minipage}{.3\textwidth}
{\small
\begin{center}
  \begin{tabular}{|c|l|c|}
  \hline
	\multicolumn{2}{|c|}{\rojo Conditions on $d,\Delta,\kappa$}	&\rojo $\Per(\phi_{d,\kappa})$		\\
  \hline
  $\gcd(d,\Delta)=1$&\begin{tabular}{l}
		      $\gcd(d-1,\Delta)=1$\\
		      $d^N\equiv 1\modu(p^\alpha),\,\alpha\geq 1$\\
		      $d^N\not\equiv1\modu(p^{\alpha+1})$\\
		      $N$ is the order of $d$ modulo $p$
		      \end{tabular}		&              $\{1\}\cup N\cdot\{p^j\}_{j=0}^{\max\{0,s-\alpha\}}$\\ \cline{2-3}
  &\begin{tabular}{l}
		      $\gcd(d-1,\Delta)>1$\\
		      $d\equiv1\modu(p^\alpha)$,\,\, $d\not\equiv1\modu(p^{\alpha+1})$\\
		      $\kappa\equiv0\modu(p^\beta)$,\,\, $\kappa\not\equiv0\modu(p^{\beta+1})$\\
		      $1\leq \alpha<s,\;0\leq \beta<s,$\\
		      \textrm{If $p=2$ this only works when $\alpha>1$}
		      \end{tabular}		&
						$\{p^{j}\}_{j=0}^{s-\alpha}$ if $\beta\geq \alpha$\\ \cline{3-3}
 &						&            $\{p^{s-\beta}\}$ if $\beta<\alpha$\\
 \hline
  $\gcd(d,\Delta)>1$&&$\{1\}$\\
  \hline
  \end{tabular}
\end{center}
}
\end{minipage}

\item For $\Delta=2^s\geq 3$, the missing cases  are:

\hspace*{-1cm}{\small
\begin{center}
  \begin{tabular}{|c|l|c|}
  \hline
	\multicolumn{2}{|c|}{\rojo Conditions on $d,\Delta,\kappa$}	&\rojo $\Per(\phi_{d,\kappa})$		\\
  \hline
\begin{tabular}{l}

 $d\equiv 1\modu(2),\,\,\,
d\not\equiv1\modu(2^2)$\\
 $\kappa\equiv 0\modu(2^\beta),\,\,\,\kappa\not\equiv 0\modu(2^{\beta+1})$\\
 $d^2\equiv1\modu(2^\gamma),\,\,\,d^2\not\equiv1\modu(2^{\gamma+1})$\\
		$0\leq\beta<s,\;\gamma\geq 3$\\

\end{tabular}
  &\begin{tabular}{l}
				$\beta=0$
		      \end{tabular}		&
						$\{2\}$ if $s\leq \gamma-1$\\ \cline{3-3}
 &						&$\{2^{s-\gamma+2}\}$ if
 $s> \gamma-1$\\
\cline{2-3}
  &\begin{tabular}{l}
		$\beta\geq 1$
		      \end{tabular}		&
		       $\{2^j\}_{j=0}^{\max\{1,s-\gamma+1\}}$ \\
 \hline
  \end{tabular}
\end{center}
}

\end{enumerate}
\end{theorem}

Finally, in the general case  we have a prime decomposition $\Delta =p_{1}^{s_{1}}p_{2}^{s_{2}}\dots p_k^{s_k}$. Let
$\phi_{d,\kk}:\Z_\Delta\to\Z_\DD$ be defined by $\phi_{d,\kk}(x)=dx+\kk$ and  take
$
d_i\equiv d\modu(p_i^{s_i}),\;
\kk_i\equiv \kk\modu(p_i^{s_i}),\;
\phi_{d_i,\kk_i}:\Zpi\to\Zpi,$ $i=1,2,\ldots,k.$

Using the Chinese Remainder Theorem, it can be proved:

\evt
\begin{theorem}\label{main4}
Let $\Delta =p_{1}^{s_{1}}p_{2}^{s_{2}}...p_{k}^{s_{k}}$ be a decomposition into prime factors.
Then, $n\in \mathrm{Per}(\phi _{d,\kappa })
$ if and only if $n=\mathrm{lcm}(n_{1},n_{2},...,n_{k})$ for $n_{i}\in \mathrm
Per}(\phi _{d,\kappa _{i}})$.
\end{theorem}
This result allows us to obtain a precise description of the set of periods in the general case
$\Delta =p_{1}^{s_{1}}p_{2}^{s_{2}}\dots p_k^{s_k}$. Since the description is something cumbersome,
the reader is referred to \cite[Theorem D]{calisoalternated2} to see all the details of such a general
description of $\mathrm{Per}[f,g]$.

\section{Technical results. Determination of odd periods}\label{S:preliminar}

To study the question of determining the set of odd periods of $[f,g]$ we need  to link periodic points of the
circle maps $f, g$ with ``lifted cycles" of their corresponding liftings $F, G$. In this direction,
we write $[F,G]$ to denote the set of sequences $(x_n)\subset \mathbb{R}$ obtained by applying $F$ and $G$ in an
alternated form.
A point $z\in\S^1$ is said to be a \emph{shared periodic point of $f$ and $g$ of order or period $m$} if and only if it is an $m$-periodic point of $f$ and $g$ whose orbit is shared by both maps,
that is, $f^i(z)=g^i(z)$ for any integer $i\geq 0$. Note that $f^m(z)=g^m(z)=z$ and $f^i(z)=g^i(z)\not=z$ for any $i\in\{1,2,\dots, m-1\}$.
We recall some interesting results we will need later for the determination of odd periods.

\begin{lemma}\label{L:ciclosf_ciclosF}
Let $f\in C(\mathbb{S}^1)$ and let $F:\mathbb{R}\rightarrow\mathbb{R}$ be an associate lifting of $f$. Then
$z\in\mathbb{S}$ is a periodic point of $f$ of order $p\geq 1$ if and only if
the orbit of $x$, with $e(x)=z$, under $F$ is a lifted cycle of period $p$, that is, $F^{p}(x)-x\in\mathbb{Z}$ and
$F^{s}(x)-x\notin\mathbb{Z}$ for all $0<s<p$.
\end{lemma}
\begin{proof}
 See \cite[Sections 3.2 and 3.3]{alseda}.
\end{proof}

By applying the  definition of shared periodic point and the previous lemma we obtain immediately:

\begin{proposition}\label{puntocompartido}
 Let $f,g\in C(\S^1)$ be with liftings $F$ and $G$, respectively. Then $z\in\S^1$ is a shared  periodic point of $f$ and $g$ of period $m$ if and only if for any $w\in e^{-1}(z)$ it holds $F^j(w)-G^j(w)\in\Z$ for all $j\geq1$, $F^m(w)-w\in \Z$ and $F^r(w)-w\not\in\Z$ if $0<r<m.$

\end{proposition}

In order to investigate what points from the circle generate periodic orbits of odd period, we need the following lemma stated
in \cite[Lemma 6-(6)]{calisoalternated1}:

\begin{lemma}\label{L:Char_lift}\label{lemlinerosalvalifting}
Let $f, g\in C(\mathbb{S}^1)$ be with associate liftings $F,G:\mathbb{R}\rightarrow\mathbb{R}$, respectively. Let $(z_n)\subset \mathbb{S}^1$ be an orbit in the alternating system $[f,g]$,  let $x_0\in\mathbb{R}$ be such that $e(x_0)=z_0$ and let  $m$ be an odd number. Then $(z_n)\subset \mathbb{S}^1$ is a periodic sequence of period $m$ if and only if the following conditions hold:
\begin{enumerate}
 \item
$
(G\circ F)^{m}(x_0)-x_0\in\mathbb{Z},\, (G\circ F)^{s}(x_0)-x_0\notin\mathbb{Z}, \, 0< s <m.
$
\item
$
F^{m}(x_0)-x_0\in\mathbb{Z},\, F^{s}(x_0)-x_0\notin\mathbb{Z},\, 0< s <m.$
\item $G^{m}(x_0)-x_0\in\mathbb{Z},\, G^{s}(x_0)-x_0\notin\mathbb{Z},\, 0< s <m.$
\item
$
F^{i}(x_0)-G^{i}(x_0)\in \mathbb{Z}, \text{ for all } i\geq 1.
$
\end{enumerate}
\end{lemma}

From now on, our continuous circle maps $f$ and $g$ have liftings
$$F(x)=d_1x+\alpha\qquad\textrm{ and } \qquad G(x)=d_2x+\beta,$$ with $d_1\neq d_2$ and we will
be concerned only with $\kk\in\Z$, see Eq~(\ref{E:kappa}),
since it implies that $f$ and $g$ commutes, a necessary condition to obtain odd periods of
$\sisalter$. For more details, the reader is referred to \cite[Section~5]{calisoalternated1}.

In next result  we sketch the key conditions to obtain points generating periodic orbits of odd period,
see \cite[Lemma 16]{calisoalternated1}.

\begin{lemma}[The set $C$]  \label{L:Candi}
Let
\begin{equation}
C:=\left\{\frac{m+\beta-\alpha}{d_1-d_2}: m\in\mathbb{Z}\right\}. \label{conjuntoC}
\end{equation}
Then the  following properties hold:
\begin{enumerate}
 \item $C+\Z=C$.
 \item $F(C)\subseteq C$ and $G(C)\subseteq C$.
 \item If $z\in \S^1$ is a periodic point shared by $f$ and $g$, with odd period, and $z=e(x)$, then
$x=\frac{m+\beta -\alpha}{d_1-d_2}\in C,$ for some $m\in \mathbb Z.$
\item $F^s(x)-G^s(x)\in\mathbb{Z}$ for all $x\in C$ and for all integer $s\geq 1.$
\end{enumerate}
\end{lemma}

It is immediate to realize that

\begin{equation}
e(C):=\{e(x):x\in C\}=\left\{e^{2\pi i x}:x=\frac{m+\beta-\alpha}{d_1-d_2},
0\leq  m< |d_1-d_2|\right\}\label{conjuntoeC}
\end{equation}
is finite, and, by Lemma~\ref{L:Candi},  if $z_0\in e(C)$ and $(z_n)$ is the orbit
generated by $\sisalter$ then $z_n\in e(C).$

\begin{lemma}\label{L:RCandi} Under the above conditions, if $z_0\in e(C)$, then
its corresponding orbit $(z_n)$ in $\sisalter$ is either periodic or eventually periodic, that is, there exist
 $p\in\N$ and $n_0\geq 0$ such that $z_{n+p}=z_n$ for all $n\geq n_0$.
Additionally, $z_{n_0}$ is a shared periodic point of $f$ and $g$ of order or period $p.$
\end{lemma}
 \begin{proof}
The (eventual) periodicity of $z_0$ is guaranteed by the finiteness of $e(C)$ and Lemma~\ref{L:Candi}. Concerning the fact
of being a shared periodic point of $f$ and $g$, the last part of Lemma~\ref{L:Candi} implies that $f^s(z)=g^s(z)$ for all $s\geq 1$.
\end{proof}

Next result establishes an easy way to determine the period of a periodic point in $C$ that will be useful
in the performance of the algorithm. Remember that, given an initial condition $x_0$, by $(x_j)_j$ we denote
the lifted orbit under $[F,G]$ generated by $x_0$ ($x_1=F(x_0), x_2=G(x_1), \ldots$), and that we say that
$x_0$ is a periodic point in $[F,G]$ whenever $x_{n+q}\equiv x_n$ for all $n\geq 0$ and some $q\in\N$.

\begin{lemma}\label{L:eventOrperC}
Let $x_0\in C$ and let $\left(x_n\right)$ be the corresponding orbit in $[F,G]$.
\begin{itemize}
\item[(a)] If $x_n-x_0\in\mathbb Z$ for some $n\geq 1$, then the orbit of $z_0=e(x_0)$ in $[f,g]$ is periodic.
\item[(b)] If $x_0\in C$ is a periodic point in $[F,G]$, then its period is equal to the  smallest positive integer $N$
such that $$x_N-x_0\in \mathbb Z.$$
\end{itemize}
\end{lemma}

\begin{proof} To prove Part (a), take into account that if $x_{n}\equiv x_0,$ since $x_n\in C$ automatically
Lemma~\ref{L:Candi} ensures that $F(x_n)\equiv G(x_n)$ and therefore $F(x_n)\equiv G(x_n)\equiv F(x_0)\equiv G(x_0)$. Then
$x_{n+1}\equiv x_1$ and, in a similar way, $x_{n+j}\equiv x_j$ for all $j\geq 1.$ From here, it is easily seen that $e(x_0)$ is a periodic point in $[f,g]$, and as a direct consequence we also deduce Part (b).
\end{proof}

For a set $A\subset \R$ and $x\in \R$, we denote $x+A=\{x+a:a\in A\}$.

\begin{lemma}\label{L:Iter_mk}
Let $f,g\in C(\S^1)$ be affine maps with liftings $F(x)=d_1x+\alpha$ and $G(x)=d_2x+\beta$, with
$d_1\neq d_2$ and $\kappa \in \Z$. Let $x_0=\frac{m+\beta-\alpha}{d_1-d_2}\in C$, where $m\in\{1,\ldots,|d_1-d_2|\}.$
If $(x_n)$ is the orbit of $x_0$ generated by the alternating system $[F,G]$, then:
\begin{itemize}
 \item  If $d_2\not=1$
$$x_n \in \frac{d_2^n m + \kappa \frac{1-d_2^n}{1-d_2} + \beta -\alpha}{d_1-d_2} + \mathbb{Z} \,\,\,\textrm{ for all } n\geq 0.$$
\item  If $d_2=1$
$$x_n \in \frac{m + \kappa n + \beta -\alpha}{d_1-1} + \mathbb{Z} \,\,\, \textrm{ for all } n\geq 0.$$
\end{itemize}

\end{lemma}

\begin{proof}
By iterating,
	\begin{eqnarray*}
	x_1&=&F(x_0)=d_1 x_0 +\alpha = \frac{d_1 m + d_1 (\beta-\alpha) + \alpha (d_1-d_2)} {d_1-d_2}\\
	&=&\frac{(d_1 - d_2)m + d_2 m+ \beta (d_1 -1) - \alpha (d_2- 1) +\beta -\alpha} {d_1-d_2}\\
	&=& m + \frac{d_2 m+ \kappa +\beta -\alpha} {d_1-d_2} \in \frac{d_2 m + \kappa + \beta -\alpha}{d_1-d_2} + \mathbb{Z},
	\end{eqnarray*}
	and $x_2=G(x_1)\in G(\frac{d_2 m+ \kappa +\beta -\alpha} {d_1-d_2})+\mathbb{Z}.$
	Since \begin{eqnarray*}
	& &G\left(\frac{d_2 m+ \kappa +\beta -\alpha} {d_1-d_2}\right)=
	\frac{d_2^2 m + d_2 \kappa +d_2 (\beta-\alpha)} {d_1-d_2} +\beta  \\
	&=& \frac{d_2^2 m + d_2 \kappa +d_2 (\beta-\alpha) + \beta(d_1-d_2)} {d_1-d_2}
	=  \frac{d_2^2 m + d_2 \kappa +d_1 \beta - d_2 \alpha} {d_1-d_2} \\
	&=& \frac{d_2^2 m + d_2 \kappa +\beta (d_1 -1)- \alpha (d_2 - 1) +\beta - \alpha} {d_1-d_2}
	=  \frac{d_2^2 m + \kappa (d_2 + 1)+\beta - \alpha} {d_1-d_2},
	\end{eqnarray*}
	we deduce that $x_2\in \frac{d_2^2 m + \kappa (d_2 + 1)+\beta - \alpha} {d_1-d_2} + \mathbb{Z}.$

Now, the proof concludes easily by induction.
\end{proof}

Next, we present some properties relative to congruence of integer numbers. Following \cite{Apo}, if $a,b$ are integers,
we take $\gcd(a,b)$ as the non-negative integer $d$ such that $d$ is a common divisor of $a$ and $b$ and
every common divisor divides $d$. Notice that $\gcd(a,b)=0$ if, and only if, $a=b=0$; otherwise, $d\geq 1$. Moreover,
when we write a congruence modulo $q$,$\modu(q)$, it must be understood that $q>0$.

\begin{lemma}\label{L:Congru}
Let $j,q\in\Z\menos\{0\}$, $q>0$. Then  $\mathrm{gcd}(j,q)=1$ if and only if $j^n \equiv 1\modu  (q)$ for some
$n\in\N.$
\end{lemma}
\begin{proof}
Let $d=\gcd(j,q).$ If $d=1$, the well-known Euler-Fermat's theorem establishes that
$j^{\varphi(q)}\equiv 1 \modu (q)$, where $\varphi(\cdot)$ is the Euler function, that is,
$\varphi(n)=\mathrm{Card}\{m\in\N : \gcd(m, n) = 1\}$ (for instance, see \cite{Apo} for a proof; realize that
$\varphi(n)$ is even for all $n\geq 3$, so we can take a general integer $j$ different from $0$).

On the other hand, the condition $j^n \equiv 1\modu  (q)$ for some positive integer $n$ is equivalent to $j^n -uq =1$ for some $u\in\mathbb{Z}$.
As $\frac1d=\frac{j^n}{d} -\frac{uq}{d}\in\Z$, we conclude that $d=1.$
\end{proof}

\begin{lemma}\label{L:Congru2}
Let $q$ be a positive integer. Let $j,m\in\Z\menos\{0\}.$ Then the following statements are equivalent:
\begin{enumerate}
\item $mj^{n}\equiv m\modu (q)$ for some $n\geq 1$,
\item $j^{n}\equiv 1\modu  ( \frac{q}{\mathrm{gcd}(q,m)})$ for some $n\geq 1,$
\item $\gcd\left(j,\frac{q}{\mathrm{gcd}(q,m)}\right)=1.$
\end{enumerate}
\end{lemma}

\begin{proof}
By Lemma~\ref{L:Congru}, we have that (2) and (3) are equivalent. On the other hand, the congruence $mj^{n}\equiv m\modu (q)$ is equivalent to the equation
$m(j^n-1)=uq$ for some $u\in\mathbb{Z}$. Then $\frac{m}{\gcd(q,m)}(j^n-1)=u\frac{q}{\gcd(q,m)}.$ Taking into account that
$\gcd\left(\frac{m}{\gcd(q,m)}, \frac{q}{\gcd(q,m)}\right)=1,$
we obtain  the equivalence between (1) and (2).
\end{proof}

Next lemma will play an important role in the proof of Proposition \ref{P:orders}.
The reader should have in mind the meaning of the integers $d_1$, $d_2$ and $\kappa$.

\begin{lemma}\label{L:mnew}
Let $d_1,d_2\in \Z$, $d_1\neq d_2$, $d_2\not\in\{0,1\}$. Fix $m\in\{1,\ldots,|d_1-d_2|\}$, with $m(d_2-1)+\kk\not=0.$
 Then:
\begin{enumerate}
\item There exists a minimal value $n_0\geq 0$ such that
\begin{equation}\label{Eq:n0}
m d_2^{n_0}+\kappa \frac{d_2^{n_0}-1}{d_2-1}\equiv m d_2^{n}+\kappa \frac{d_2^{n}-1}{d_2-1}\,\,\modu (|d_1-d_2|)
\end{equation}
for some $n> n_0$.
\item Let $n_{m,\kappa}>n_0$ be the smallest value $n>n_0$ for which (\ref{Eq:n0}) holds. Then $\varepsilon_{m,\kappa}:=n_{m,\kappa}-n_0$ is the smallest positive integer $\ell$ such that
\begin{equation}\label{Eq:nmk}
d_2^{\ell}\equiv 1\modu  \left(\frac{|(d_1-d_2)(d_2-1)|}{\gcd\left((d_1-d_2)(d_2-1),d_2^{n_0}(m[d_2-1]+\kappa)\right)}\right).
\end{equation}
\item The integer $n_0\geq 0$ defined in (1) is the minimal non-negative integer $N$ such that
\begin{equation}\label{Eq:n0Calculo}
\gcd\left(d_2,\frac{(d_1-d_2)(d_2-1)}{\gcd\left((d_1-d_2)(d_2-1),d_2^{N}(m[d_2-1]+\kappa)\right)}\right)=1.
\end{equation}
\end{enumerate}
 \end{lemma}
\begin{proof}
(1) As the set $\left\{m d_2^{r}+\kappa \frac{d_2^{r}-1}{d_2-1} \modu(|d_1-d_2|):r\in \N\right\}$ is finite, then $n_0$ and $n$ exist, and then (1) follows.

(2) Similarly, we prove the existence of $n_{m,\kappa}> n_0$ being the smallest value fulfilling that
$$m d_2^{n_0}+\kappa \frac{d_2^{n_0}-1}{d_2-1}\equiv
m d_2^{n_{m,\kappa}}+\kappa \frac{d_2^{n_{m,\kappa}}-1}{d_2-1}\modu (|d_1-d_2|).$$
Equivalently:
\begin{eqnarray*}
 (d_2^{n_{m,\kappa}}-d_2^{n_0})m+\kappa\frac{d_2^{n_{m,\kappa}}-d_2^{n_0}}{d_2-1}
 \equiv 0\modu (|d_1-d_2|)
 \\
 \Leftrightarrow
 d_2^{n_0}(d_2^{n_{m,\kappa}-n_0}-1)m+\kappa d_2^{n_0}\frac{d_2^{n_{m,\kappa}-n_0}-1}{d_2-1}
 \equiv 0\modu (|d_1-d_2|)
 \\
 \Leftrightarrow
(d_2^{n_{m,\kappa}-n_0}-1)
d_2^{n_0}\left[
m+\kappa \frac{1}{d_2-1}
\right]
 \equiv 0\modu (|d_1-d_2|)
 \\
 \Leftrightarrow
d_2^{n_{m,\kappa}-n_0}
d_2^{n_0}\left[
m (d_2-1)+\kappa
\right]
 \equiv
 d_2^{n_0}\left[
m (d_2-1)+\kappa
\right]
\modu \left(|(d_1-d_2)(d_2-1)|\right).
\end{eqnarray*}

Now, by Lemma~\ref{L:Congru2} (notice that $m(d_2-1 ) +\kappa\neq 0$)
$$
d_2^{\varepsilon_{m,\kappa}}=d_2^{n_{m,\kappa}-n_0}
\equiv
1
\modu
\left(
\frac{|(d_1-d_2)(d_2-1)|}{
\gcd\left(
(d_1-d_2)(d_2-1),
d_2^{n_0}\left[
m (d_2-1)+\kappa
\right]
\right)}
\right)
$$
we prove (2), since if it would exist a positive value $\ell$ smaller than $\varepsilon_{m,\kappa}$
holding $(\ref{Eq:nmk})$, then by the above equivalences $n_\ell:=n_0+\ell<n_{m,\kappa}$ would satisfy $(\ref{Eq:n0})$, in contradiction
with the minimality of $n_{m,\kappa}.$

(3) Finally,  Lemma~\ref{L:Congru} implies that
$$\gcd\left(d_2,
\frac{(d_1-d_2)(d_2-1)}{
\gcd\left(
(d_1-d_2)(d_2-1),
d_2^{n_0}\left[
m (d_2-1)+\kappa
\right]
\right)}
\right)=1,$$ which concludes the proof.
\end{proof}

\begin{definition}[$\varepsilon_{m,\kk}$] Given  $d_1,d_2\in \Z$, $d_1\neq d_2$ with $\kk=\kkdef\in\Z$,
let $m\in\{1,2,\ldots,|d_1-d_2|\}.$ Then:
\label{D:defepsilonmk}
\begin{equation}\label{E:defepsilommk}
 \varepsilon_{m,\kk}:=\left\{
 \begin{array}{lcl}
 1&&\textrm{ if } m(d_2-1)+\kk=0,\\
 1&&\textrm{ if } m(d_2-1)+\kk\not=0,d_2=0,\\
N
&&\textrm{ if } m(d_2-1)+\kk\not=0,d_2=1,\\
  \textrm{the value of Lemma~\ref{L:mnew}-(2)}&&\textrm{otherwise.}
 \end{array}
 \right.
\end{equation}

The integer $N>0$ is the smallest positive integer $n$ satisfying
$$n\kappa\equiv 0\modu\left(|d_1-1|\right).$$
\end{definition}

Next, we give a characterization of periodic orbits shared by two affine circle maps with different degrees.

\begin{proposition}\label{P:orders}
Let $f,g\in C(\S^1)$ be affine maps with liftings $F(x)=d_1 x + \alpha$ and $G(x)=d_2 x +\beta,$ with $d_1\neq d_2$
and such that  $\kappa \in\Z$.
Let $x_0=\frac{m + \beta -\alpha}{d_1-d_2}\in C$, with $m\in\{1,\ldots,|d_1-d_2|\}.$ Then:
\begin{itemize}
\item[(a)] $e(x_0)$ is an eventually periodic point shared by $f$ and $g$.
\item[(b)] If $\ell\in\N$ is the smallest non-negative integer holding that
$f^\ell(e(x_0))=g^\ell(e(x_0))$ is periodic, then its period is
$\lambda=\varepsilon_{m,\kk}$  defined by  Equation~(\ref{E:defepsilommk}).
\end{itemize}
\end{proposition}
\begin{proof}
Let  $(x_n)$ be the orbit of $x_0\in C$ generated by $[F,G]$,
then by Lemma~\ref{L:RCandi} the orbit $(e(x_n))$ in $\mathbb{S}^1$, generated by $\sisalter$, is either periodic or eventually periodic.
Notice that this (eventually) periodic orbit is shared by $f$ and $g$ because,
by Lemma~\ref{L:Candi}, $F^n(x_0)-G^n(x_0)\in\Z$ for all $n\in \N$.
Let $n_0\geq 0$ be the first integer such that $f^{n_0}(e(x_0))$ is periodic.

Now we distinguish these cases.
\begin{description}
\item[(1)] \underline{$m(d_2-1)+\kk=0.$} We have
\begin{eqnarray*}
 x_1&=&
F(x_0)=d_1\left(\frac{m + \beta -\alpha}{d_1-d_2}\right)+\alpha=\frac{md_1+\beta d_1 -\alpha d_2}{d_1 - d_2}\\
&=&\frac{md_1+\kappa +\beta-\alpha}{d_1-d_2}=\frac{md_1 +m(1-d_2)+\beta-\alpha}{d_1-d_2}\\
&=&m+\frac{m + \beta -\alpha}{d_1-d_2} \in x_0 +\Z,
\end{eqnarray*}
 and then $x_1-x_0\in\Z$. Now, Lemma~\ref{L:eventOrperC} implies that the sequence $(x_n)$ generated by $[F,G]$
is a lifted cycle and $z=e(x_0)$ is a cycle of order $\varepsilon_{m,\kk}=1$ of $\sisalter$.

\item[(2)] \underline{$ m(d_2-1)+\kk\not=0 , d_2=0.$}  In this case, for a given $x_0\in C$ the lifted orbit is
$$x_0,x_1=d_1x_0+\alpha,x_2=\beta,x_3=d_1\beta+\alpha,x_4=\beta,x_5=d_1\beta+\alpha,\dots$$
Observe that $\kk=\beta(d_1-1)+\alpha,$ so $d_1\beta +\alpha=\kk +\beta,$ and consequently
$x_n-x_{n-1}=(-1)^n\kk\in\Z$ for any $n\geq 3$. Therefore, the orbit of $e(f^2(x_0))$ is periodic
of period $\varepsilon_{m,\kk}=1$.

\item[(3)] \underline{$ m(d_2-1)+\kk\not=0 , d_2=1.$}
Let $x_0\in C$ and let $(x_n)$ be the orbit generated by $[F,G]$. According to Lemma~\ref{L:RCandi}, let $n_0$ be the smallest
non-negative integer such that $f^{n_0}(e(x_0))$ is a periodic point in $[f,g]$ or, equivalently, such that $x_{n_0}$ generates
a periodic lifted cycle in $[F,G]$. By Lemma~\ref{L:Iter_mk},
 $$x_n\in  \frac{m + \kappa n + \beta -\alpha}{d_1-1} + \mathbb{Z}=x_0 +\frac{\kappa n }{d_1-1}+\mathbb Z$$
 and $x_n-x_{n_0}\in\Z$ implies
$\frac{(n-n_0)\kk}{d_1-1}\in\Z$. By Lemma~\ref{L:eventOrperC} and the last congruence, the period is equal to
the first positive integer $N$ satisfying $N\kk\equiv 0 \modu(|d_1-1|)$, the value of $\varepsilon_{m,\kk}$
in Definition~\ref{D:defepsilonmk}.

\item[(4)]  \underline{$ m(d_2-1)+\kk\not=0 , d_2\notin\{0,1\}.$}
Then, by Lemma~\ref{L:mnew} (cf. Equation~(\ref{E:defepsilommk}) in Definition~\ref{D:defepsilonmk}),  the period of $f^{n_0}(e(x_0))$
is given by $\varepsilon_{m,\kk}$, as described in the statement.

\end{description}

\end{proof}

Thus, we have proved the following result:

\begin{theorem}\label{T:oddLambda}
Let $f,g\in C(\S^1)$ be affine circle maps with liftings $F(x)=d_1 x + \alpha$ and $G(x)=d_2 x +\beta,$ with $d_1\neq d_2$ and such that
  $\kappa=\beta(d_1-1)-\alpha(d_2-1)\in\Z$.
Then the set $\Lambda$ of odd periods from $\mathrm{Per}[f,g]$ is given by
$$\Lambda=\{\lambda=\varepsilon_{m,\kk}: 1\leq m\leq |d_1-d_2|, \lambda \text{ odd }\},$$
where $\varepsilon_{m,\kk}$ is  introduced in Definition~\ref{D:defepsilonmk}.
\end{theorem}

The case $\Lambda=\emptyset$ can occur for degree $-1$, for instance consider the affine liftings $F(x)=-x+\alpha, G(x)=x+\beta$,
with $\beta=\frac{q}{2}$, $q$ odd; here, $d_1=-1, d_2=1$, $\kappa=-2\beta\in\Z$ and $C$ is reduced to two points,
$\frac{1+\beta-\alpha}{-2}$ and $\frac{2+\beta-\alpha}{-2}$; it is easily seen that
both points generate a periodic sequence of order $2$, an even number.

Nevertheless,  under the conditions of Theorem~\ref{T:oddLambda}:
\begin{corollary}
Let $d_1=-1$ and assume that $\Lambda\neq\emptyset$. Then $\Lambda=\{1\}.$
\end{corollary}

\begin{proof}
Let $m\in\{1,2,\ldots,|d_1-d_2|\}.$ If $d_1=-1$, then $\kappa=-2\beta - \alpha (d_2-1)\in\mathbb Z$. Fix $m$.
Now, we apply Definition~\ref{E:defepsilommk} and Theorem~\ref{T:oddLambda}. We distinguish, according to the referred definition,
the following cases:

-- If $m(d_2-1)+\kappa=0$ or  $m(d_2-1)+\kappa\neq 0, d_2=0$, then $\varepsilon_{m,\kappa}=1$.

-- If $m(d_2-1)+\kappa\neq 0, d_2=1$, then $\kappa=-2\beta\in\mathbb{Z}$, and we know that, from Definition~\ref{E:defepsilommk},
we have to add the period $N$, the smallest positive integer such that $n\kappa\equiv 0 \modu(|d_1-1|)$, that is, $-2n\beta\equiv 0\modu(|d_1-1|).$
Thus, either $N=1$ or $N=2$. If $N=1$, we add it to $\Lambda$. If $N=2$, this case does not originate a value of $\Lambda$.

-- If $m(d_2-1)+\kappa\neq 0, d_2\notin\{0,1\}, d_2\neq d_1$, by Theorem~\ref{T:oddLambda} and its previous Lemma~\ref{L:mnew},
firstly we have to compute the smallest non-negative integer value $n_0$ such that
$$\gcd\left(d_2,\frac{(d_1-d_2)(d_2-1)}{\gcd\left((d_1-d_2)(d_2-1),d_2^{N}(m[d_2-1]+\kappa)\right)}\right)=1.$$
We short the notation by setting $d=d_2$. Since $d_1=-1$, the above condition is written as
\begin{equation}\label{Eq:cn0}
\gcd\left(d,\frac{1-d^2 }{\gcd\left( 1-d^2, d^{N}(m[d -1]+\kappa)\right)}\right)=1.
\end{equation}
Notice that $1=\gcd(d,d+1)=\gcd(d,d-1)$ if $|d|\geq 2$. From here,
if we put $w:=\frac{1-d^2 }{\gcd\left( 1-d^2, d^{N}(m[d -1]+\kappa)\right)}$, obviously $\gcd(d,w)=1$ since each prime factor of $w$
is a divisor of $1-d^2$ and we know that $\gcd(d,1-d^2)=1.$ Therefore, (\ref{Eq:cn0}) is satisfied by all the non-negative integers and
consequently $n_0=0$. Now, in order to evaluate $\varepsilon_{m,\kappa}=n_{m,\kappa}-n_0=n_{m,\kappa},$ we have to obtain the smallest positive
integer $N$ holding $d^N\equiv 1\modu(\frac{|1-d^2|}{\gcd\left(1-d^2, m(d-1)+\kappa\right)}).$ Taking into account
that $d^2 -1 = \gcd\left(1-d^2, m(d-1)+\kappa\right)\frac{d^2-1}{\gcd\left(1-d^2, m(d-1)+\kappa\right)}$, we find that either $N=1$ or $N=2$.
If $N=2$, this case does not provide a value of $\Lambda$. If $N=1$, we add it to $\Lambda$.

Summarizing, with the study of the above cases, we have seen that the unique odd period we can obtain in $\Lambda$ is precisely $1$.
\end{proof}

\begin{center}
\begin{table}[ht]
\begin{tabular}{|c|c|c|c|c|c|c
c|}
\hline
$d_1$&$d_2$&$\alpha$&$\beta$&$|d_1-d_2|$& $\kappa$&$\Lambda
\\ \hline
38&7&0&0&31& 0&$\{1,15\}
\\ \hline
46&16&0&1/45&30& 1&$\{15\}
\\ \hline
16&6&0&1/15&10& 1&$\{5\}
\\ \hline
39&16&0&0&23& 0&$\{1,11\}
\\ \hline
5&36&0&0&31& 0&$\{1,3\}
\\ \hline
10&4&0&1/9&6& 1&$\{3\}
\\ \hline
31&256&2&7&225& -300&$\{1,3,5,15\}
\\ \hline
10&2&0&1/9&8& 1&$\{1\}
\\
\hline
31&1&23&1/5&30&6&$\{5\}
\\ \hline
-1&9&1/4&1&10&-4&$\{1\}
\\ \hline
\end{tabular}
\caption{Examples of odd periods of alternating systems after applying the proposed algorithm.}
\label{Table:algoritmo}
\end{table}
\end{center}

\section{The algorithm}\label{S:algorithm}

Although Theorem~\ref{T:oddLambda}  provides the characterization of odd periods for an alternating system, often it
is necessary to do a certain number of hard computations to obtain it. Hence, we are going to give
an algorithm to do these calculations with a computer. The program introduced in next subsection has been implemented in \emph{Maxima}, which is a free software of symbolic calculation.

The last function of the program, ComputeLambda, must be called with four arguments ($d_1$, $d_2$, $\alpha$ and  $\beta$)
and returns the set $\Lambda$ of odd periods. This function calls another one called Epsilonmkappa which distinguishes in what case of Definition~\ref{D:defepsilonmk} we are and computes the value of $\varepsilon_{m,\kappa}$. This value is easy to find  when we are in the two first cases of the definition of $\varepsilon_{m,\kappa}$, however  when we are in the third and fourth cases   stronger calculations are needed and made respectively by  the functions Epsilon3 and Epsilon4.

Epsilon3 is an easy function that uses a loop in order to find the least integer $N$ satisfying $N\kappa\equiv 0\modu\left(|d_1-1|\right)$. In exchange, in order to call Epsilon4 from the function Epsilonmkappa we need previously to know the value of $n_0$ satisfying the conditions of Lemma~\ref{L:mnew}-(i). The function Computen0 searches this number recursively, next we call the function Epsilon4 with  $n_0$ and, by means of a loop for, we find the smallest integer $n>n_0$ satisfying the condition in Lemma~\ref{L:mnew}-(ii). This completes the computation of $\varepsilon_{m,\kk}$.

ComputeLambda  works by means of a loop for between the numbers 0 and $|d_1-d_2|$.
 It checks if the  calculated value by Epsilonmkappa is odd and in this case it adds the number to the set $\Lambda$.

\newcommand{\comentariomaxima}[1]{\textrm{$/*$ #1 $*/$}}
\newcommand{\fincomandomaxima}{\$}
\newcommand{\espaciovertical}{\vspace*{.25cm}}

\subsection*{The algorithm implemented in Maxima.\newline}
\hfill

\noindent\comentariomaxima{In the following definition we compute, by means of a recursive function, the value $n_0$ given by Lemma~\ref{L:mnew}--(1) and (3). This
value is needed to obtain $\varepsilon_{m,\kappa}$ in the fourth alternative of Definition~\ref{D:defepsilonmk}
which we implement in function Epsilon4 below. We call the function with $n=0$, it  checks if the greatest common divisor in Proposition~\ref{P:orders} is
1 and in this case $n_0=0$; if not, the function calls itself with $n=1$, do the test for $n=1$ and so on

\espaciovertical

\noindent\texttt{
\noindent{}Computen0(d1,d2,m,k,n):=
\newline\indent if gcd(d2,(d1-d2)*(d2-1)/gcd((d1-d2)*(d2-1),d2\^{}n*(m*(d2-1)+k)))=1
\newline\indent \indent then n
\newline\indent \indent else
Computen0(d1,d2,m,k,n+1)\fincomandomaxima
\espaciovertical
}

\noindent\comentariomaxima{The following function, Epsilon4, computes the value of $\varepsilon_{m,\kappa}$ when the values of $d_1,d_2,\alpha$ and $\beta$ are in the condition of the fourth alternative in Definition~\ref{D:defepsilonmk}. This function will be called by Epsilonmkappa}

\espaciovertical
\noindent
\texttt{Epsilon4(d1,d2,k,n0,m):=block(
\newline\indent l:1,
\comentariomaxima{l is the variable used to obtain the value of  $\ell$}
\newline\indent
for n:1 thru abs(d1-d2) while abs(remainder(
\newline\indent\indent d2\^{}n-1,(d1-d2)*(d2-1)/gcd((d1-d2)*(d2-1),d2\^{}n0*(m*(d2-1)+k))))$>$0
\newline\indent\indent do
l:l+1,
\newline\indent
l \comentariomaxima{Epsilon4 returns the value of $\ell$}
\newline)\fincomandomaxima\comentariomaxima{End of definition of Epsilon4}
}
\espaciovertical

\noindent\comentariomaxima{Function Epsilon3, defined below, computes the value of $\varepsilon_{m,\kappa}$ when the values of $d_1,d_2,\alpha$ and $\beta$ are in third alternative in Definition~\ref{D:defepsilonmk}. This function will be called by Epsilonmkappa}

\espaciovertical
\noindent
\texttt{Epsilon3(d1,k):=block(
\newline\indent
N:1, \comentariomaxima{N is the variable used to obtain the value of  $N$}
\newline\indent
modcong:abs(d1-1),
\newline\indent
    for n:1 thru modcong while
    \newline\indent \indent
    abs(remainder(n*k,modcong))$>$0
    \newline\indent \indent
    do N:N+1,
    \newline\indent
    N  \comentariomaxima{Epsilon3 gives back the value of $N$}
    \newline
    )\fincomandomaxima \comentariomaxima{End of definition of Epsilon3}
}
\espaciovertical

\noindent\comentariomaxima{Function Epsilonmkappa computes the value of $\varepsilon_{m,\kappa}$ distinguishing the
four possibilities given in Definition~\ref{D:defepsilonmk}}

\espaciovertical
\noindent
\texttt{
Epsilonmkappa(d1,d2,alpha,beta,m):=block(
\newline\indent     emk:0,
\newline\indent         k:beta*(d1-1)-alpha*(d2-1), \comentariomaxima{the value of $\kappa$}
    \newline\indent     condition:m*(d2-1)+k, \comentariomaxima{the value needed to distinguish the cases of Definition~\ref{D:defepsilonmk}}
    \newline\indent     if condition=0 then emk:1 else(
        \newline\indent\indent     if d2=0 then emk:1 else(
         \newline\indent\indent\indent            if d2=1 then emk:Epsilon3(d1,k) else(
                    \newline\indent\indent\indent\indent n0:Computen0(d1,d2,m,k,0),
                    \newline\indent\indent\indent\indent  emk:Epsilon4(d1,d2,k,n0,m)
                \newline\indent\indent\indent)
  \newline\indent\indent          )
   \newline\indent ),
   \newline\indent
emk \comentariomaxima{we return the value of $\varepsilon_{m,\kappa}$}
\newline)\fincomandomaxima \comentariomaxima{End of definition of Epsilonmkappa}
}
\espaciovertical

\noindent\comentariomaxima{Function ComputeLambda calculates the set $\Lambda$
}

\espaciovertical
\noindent
\texttt{
ComputeLambda(d1,d2,alpha,beta):=block(
\newline\indent difference:abs(d1-d2),
\newline\indent k:beta*(d1-1)-alpha*(d2-1),
\newline\indent Lambda:[], \comentariomaxima{the vector that will contain the odd period}
\newline\indent if integerp(k) and not(d1=d2) then ( \comentariomaxima{ we check if k is integer}
\newline\indent \indent    for m:0 thru difference do(
\newline\indent \indent\indent        emk:Epsilonmkappa(d1,d2,alpha,beta,m),
\newline\indent \indent\indent        candidate:emk,
\newline\indent \indent\indent        if oddp(candidate) then Lambda:append(Lambda,[candidate])
\newline\indent \indent\indent \comentariomaxima{we only add candidate to Lambda when it is odd}
\newline\indent \indent    )
\newline\indent ) else Lambda:[],
\newline\indent setify(Lambda) \comentariomaxima{the function returns the set made of the components of the components from Lambda}
\newline )\fincomandomaxima \comentariomaxima{End of definition of the function ComputeLambda}
}
\espaciovertical

\noindent\comentariomaxima{  Finally, the computation of the odd periods of an alternating affine system with values $d_1,d_2,\alpha,\beta$ will be obtained by executing the order:}

\espaciovertical
\noindent
\texttt{
ComputeLambda(d1,d2,alpha,beta)
\fincomandomaxima
}

\espaciovertical

\newcommand{\datos}[8]{#1&#2&#3&#4&#5&#6&#7&$#8$
\hline
\\}

We apply this algorithm to some alternating systems and we obtain some additional examples presented in Table~\ref{Table:algoritmo}.

\section{Conclusions}
The algorithm given in this manuscript is a valuable instrument for computing the odd periods in
$\mathrm{Per}(\sisalter)$ when considering affine circle maps, $f$ and $g$,  and it finishes the  problem of characterizing $\mathrm{Per}(\sisalter)$. However we think that in  this line of research a lot of work can be done in the future. We mention some valuable problems to analyze.

Admittedly the problem of characterizing  $\mathrm{Per}(\sisalter)$  for continuous circle maps $f,g$ seems too much ambitious.
We propose to make a deep analysis of alternating systems $\sisalter$ for circle homeomorphisms since the sets of periods for these maps are simple and depend on the rotation number of the homeomorphisms when the degree is 1. However,  the triviality of the sets of  periods of homeomorphisms does not guarantee an easy control of the set  $\Per(\sisalter)$ since the periods of $f\circ g$ strongly  depends on the rationality of its rotation number. In this point it is interesting to remark that the composition of two   degree 1 homeomorphisms with irrational numbers  (and then with empty set of periods) can give a degree 1
homeomorphism with rational number (and then with nonempty set of periods), see \cite[Chapter 1, Section 4]{demelovanstrien}.

In the setting of affine circle maps we still propose to make a deep research by considering alternating systems of more than two maps.

\def\cprime{$'$}

\section*{Acknowledgments}
Authors have been partially supported by the Grants MTM2014-52920-P and MTM2017-84079-P from Agencia Estatal de Investigaci\'on (AEI) y Fondo Europeo de Desarrollo Regional (FEDER). We acknowledge the referees for their suggestions which allowed us to improve the reading of the paper.

redpajama_set_name: RedPajamaArXiv


Performing artist and educator, is a Flint Michigan native, Mr. Bradley is the founder of Tapology Tap Festival for Youth, The Flintstone Hoofers, now Tapology Youth Ensemble, and co-founder of Creative Expressions Dance Studio. He is also the promotional voice of Genesee County. Bradley is a graduate of Alabama State University where he earned a Bachelor of Arts degree and began his theatrical training. Upon returning to the Flint area after college he became actively involved with the McCree Theatre as a performing artist. As a result of his work through the McCree Theatre he was given the opportunity to work professionally in the hit Off Broadway musical "One Mo Time" in the lead role as Papa Du which resulted in an extended run at The Village Gate in Toronto, Ontario.
Photo by Rick Smith, Courtesy of the C.S. Mott Foundation.


text:
Засенчваща бобина e вид реле с намотка на късо съединение, чиято част от полюса на се активизира от променлив ток, чрез магнит за намаляване на магнитния поток. Индуцираните токове в намотката са използвани за самопуск на еднофазни двигатели.

Свойството на свойството позволява затварянето на арматурните пера, когато основният поток спада до нула (например 100 или 120 пъти на секунда с 50 (или 60) цикъла AC).

Без този засенчващ пръстен, арматурата би имала тенденция да се отваря всеки път, когато основният поток премине през нула и създава шум, топлина и механични повреди по лицевите повърхности на магнита (например намаляване на подскачащите контакти на релето или захранването и трептенето.

Източници
 Riba, J. R.; Espinosa, A. G.; Cusido, J.; Ortega, J. A.; Romeral, L. (2008). Design of Shading Coils for Minimizing the Contact Bouncing of AC Contactors. 2008 Proceedings of the 54th IEEE Holm Conference on Electrical Contacts. pp. 130–136. doi:10.1109/HOLM.2008.ECP.34. ISBN 978-1-4244-1901-2.

Релейни защити

redpajama_set_name: RedPajamaWikipedia


We require TWO WEEKS notice in order to perfect every detail of your custom graduation cake. Pricing begins at $5 per serving and goes up with fondant and other design details. We have several designs to choose from and can also with your vision. We look forward to creating your masterpiece!
Email your budget, event date, school name and colors, plus the number of servings needed, to customersvc@dessertgallery.com.


text:
The Story of "When Water Moves"
Published on September 7th, 2021 by FLOW Editor - Blog Posts, News
Editor's note: This article served as a preview for FLOW's live-streamed premier of the film "When Water Moves" on Sunday, September 12, at 5:00 p.m. EDT. Photo by Tyler Franz.
By Anne-Marie Oomen
Ari Mokdad (left) and Anne-Marie Oomen (right) read from Seiche Ways aboard the Nauti-Cat in 2018.
It begins for me with the question that haunts my days: How can I use my art, my words, my one small gift which brings me joy, to make a difference in those causes I consider most critical to supporting eco-vigor? In recent years, that attempt to "make a difference" has been focused on water. Thank goodness for FLOW, because through their Art Meets Water program, that impulse found a home. Their program seeks to connect their mission with artistic resources to "develop this natural creative synergy into an intentional and inspiring outreach platform that motivates action and change." In other words, they welcome and support interaction with artists who are thinking about water.
In 2018, FLOW had graciously supported the book launch of The Lake Michigan Mermaid: A Tale in Poems (written with Linda Nemec Foster, illustrated by Meridith Ridl). Then too through the publication (with Ari Mokdad) of a pocket anthology, Seiche Ways, a gentle fundraiser in which Ari and I collected water poems from our regional poets as a gratitude gift for FLOW donors. Ari is a poet, dancer and choreographer. She also happens to be married to the captain of the Nauti-Cat, and though her roots are in Detroit, she had also enthusiastically fallen in love with these more northern waters. I enjoyed our interaction immensely, and in August of 2019, one of her beautifully choreographed dances, "Water Studies" was accepted in the Detroit Dance City Festival, incorporating my poem "Water Gratitude" into the work. Inspired by the pleasure of those collaborations, we looked at each other and asked, "What's next?"
With FLOW, we began to build a program for the 50th anniversary of Earth Day, in March 2020, to address the ongoing water crisis in the state. We hoped to celebrate and heighten the water spirit permeating the work of water activists, including the water protectors of the tribal communities. We planned a multi-dimensional project that would include dance, poetry, community, and water. All was canceled due to COVID-19. For a while we floundered, but then like so many artists, we became inspired by the "creative synergy" of the challenge and refocused the grant on video presentation—motivated by a more precise question: How could we communicate both the aesthetic and spiritual narrative of our beloved lakes, and simultaneously heighten the sense of urgency for water issues?
We extended the original concept of "When Water Moves," to a dance/text/image/performance video centered on an original water parable that incorporates water issues. In the Michigan Council for Arts and Cultural Affairs (MCACA) grant application, we wrote: "As Michigan-born artists with deep ties to water culturally, our work coalesces around both the movement of water, represented through traditional Anishinaabe dance and new contemporary choreography, and through text (poetry and storytelling) that uses water themes. We believe that water activism can successfully build from arts-based collaborations centered on the Great Lakes and current water issues." We also hoped that this water-art collaboration might initiate deeper thinking about how we belong to the lakes instead of the other way around. We hoped that the aesthetics of story, of poetry and dance, and the imagery of video might move viewers to think about a more comprehensive and heartfelt water protection and preservation ethic.
"When Water Moves" evolved as a video performance poem about a water woman who becomes a lake being, embodying in spirit the love of water and assuring its longevity. The story also describes how a water woman faces what happens when we fail to respect this ultimately life-giving resource. I had the pleasure of writing and narrating the poem, but the real pleasures were the contributing voices from the Northern Michigan and greater community. The story is inspired by both Sierra Clark's Anishinaabe stories and tales from Ari Mokdad's Lebanese heritage. Joe VanderMeulen stepped in to bring it all together—videotaping Ari's lovely choreography, incorporating the voices of water protectors and the Anishinaabe jingle dress dance for healing.
Thanks to Joe's skilled work, friends who view the short film will have the pleasure of watching a narration that holds both threads: aesthetically beautiful rendering of the concept and reflections on water work. The video offers hope for healing the relationships of humans to water. And also, just perhaps, offers a model for how artistic collaboration can build water consciousness, both for the artists and for the audiences who participate in open follow-up discussions.
In addition, Ari and I are grateful that MCAC and the Northwest Michigan Arts and Culture Network were comfortable awarding support for this project to a grant-seeker (FLOW) that is not categorized as an arts organization. FLOW's history of meaningful relationships with artists gave the project a solid track-record to stand on. To that end, Ari and I wrote the following with heartfelt conviction, "As artists we strive to foster conversations… and shift our perspective towards a culture that values, honors, and celebrates water and water equity. This conversation and development of art provides a starting point to create a healthier water culture and a more joyful participation in water consciousness and preservation. That said, we do this work because we share an ethos with FLOW as artists to preserve our greatest resource in Michigan and continue to advocate for sustainable changes for water equity and justice. This work is integral to our beings as water is a part of what makes our culture and very psyches whole."
Tags: Anne-Marie oomen, ari mokdad, art meets water, Joe vandermeulen, when water moves

redpajama_set_name: RedPajamaCommonCrawl


The no. 1 afro house label MoBlack Records offers you four timeless tracks for their third sampler e.p. First up, Italian duo Armonica and South African songstress Toshi have their incredible track 'Ngeke' remixed into a deeply hypnotising, glitched out roller by Andhim. Armonica then take on Dele Sosimi's 'Turbulent Times' reworking the afrobeat flavours into a swelling, synthesised house number filled to the brim with emotion.
On the B side FatouMata & Juan remix Mabiisi's 'King' into a deep and brooding techno track with a darker tinge to it. The final slot on this MoBlack sampler comes in the form of Christos Fourkis' – 'Doumasi' that captivates with its use of African vocal samples behind percussive rhythms, mesmerising marimbas and sweeping pads.


text:
Oregon and the 30x30 Conservation Initiative
Oregon is known for its wild forests, rivers, estuaries, mountains, and deserts that stretch across the state. These landscapes purify the air we breathe, filter the water we drink, and support the salmon and other wildlife that are part of our region's identity. But Oregon's communities and wildlife are facing unprecedented environmental challenges in the face of climate change and ongoing logging, grazing, and development. Given these impacts, the need to conserve the natural world for both people and wildlife has become even more apparent.
Webcast: Hiking the Oregon Coast Trail
From wide beaches and lush forests to windswept bluffs and dramatic sea stacks, the stunning wild coast of Oregon is emerging as the next great long-distance hiking experience. This world-class hiking trail right in Oregon's backyard is an increasingly popular destination, and now it finally has a dedicated guidebook!
The Forest Service Doth Protest Too Much
Wed, 10/27/2021 - 14:39 | rob
Not long ago, we asked our members and supporters to speak up for Eastern Oregon's old-growth forests. As expected, Trump's Forest Service didn't listen. Now the Biden administration is defending a grievous mistake.
Worse yet, in a recent email many of our members received, the Forest Service claimed they had strengthened protections. It's simply not true, and we want to set the record straight.
Webcast: Unpacking the Status of Oregon's Wolves
Wolves once had a range that covered a vast majority of North America. A concentrated killing campaign drove wolves to the brink, and it is only through hard-fought conservation efforts that these native animals have started to re-establish across their range. Wolves are still slowly returning to the places their ancestors once howled and roamed. Unfortunately, short-sighted politicians have resumed the last century's war-on-wolves, threatening to undo decades of recovery. This webcast provided a thorough overview of the status of wolf protections.
Shadow of the Condor: Act 4
Global Warming Commission Says it's Time to Put Oregon's Forests to Work Fighting Climate Change
Mon, 10/11/2021 - 10:19 | la@oregonwild.org
It's no secret that Oregon needs to do a better job of stewarding it's amazing wildlands and waters — this is especially true for its forests. While the overall forest area has remained relatively steady in our state, the same cannot be said for the quality of those forests. They have been logged extensively, and some estimates show that as little as 10 percent of old growth forests remain. This poor management has led to degraded watersheds, impacted fish and wildlife, and millions of metric tons of carbon dioxide released into the atmosphere.
October 11th is Indigenous Peoples Day!
Sun, 10/10/2021 - 10:06 | Ally
Learn more about what this day means and why it is so important.


text:
Essential Retail Magazine Home » Analysis » Is Ocado putting its infrastructure up for tender?
1 Jul 2014 Analysis
Is Ocado putting its infrastructure up for tender?
Online supermarket Ocado gave further indication this week that it views itself as a technology business as well as an internet grocer, signalling its intent to create further partnerships like its recent tie-up with Morrisons.
In a half-year report, which showed that the company recorded profit before tax of £7.5 million in the 24 weeks ending 18 May 2014, the business said it is making progress with the re-platforming of its IT systems, which it hopes will "enable faster replication and roll out of our technology".
At the same time, CEO Tim Steiner said that the online supermarket group continues to receive interest from potential partners in international markets, with discussions reportedly centring on how Ocado "might assist them in introducing or improving online business in their own markets".
Ocado launched its partnership with the UK's fourth largest grocer, Morrisons, earlier this year – and the deal, which sees the online company provide the backbone to the Yorkshire-based supermarket's online grocery operations, could prove to be a template for further collaborations within the retail sector.
Steiner described January's Morrisons.com launch as "successful" and said it "paves the way for future agreements to commercialise the value of our intellectual property". It was also suggested that the e-tailer's new modular, scaleable fulfilment hubs will allow the business to offer potential partners "a lower risk, less capital intensive platform solution for online retailing".
"We expect the model for these services to be weighted towards ongoing annual usage fees rather than the high upfront fees typical of the technology licensing market, which we believe will make it a compelling proposition to potential partners," Steiner continued.
Stephen Springham, senior retail analyst at industry intelligence group Planet Retail, argued that Steiner's comments on international partnerships were "a major signal of intent", in relation to licensing its model to other retailers.
"Ocado is not putting itself up for sale, but is effectively putting its infrastructure and know-how up for tender," he told Essential Retail.
"Who may take them up on this is still subject to speculation, but the business is clearly courting international operators. If suitable partners are successfully found, the oft-debated issue of whether the business can be profitable may ultimately be put to bed."
Analysts have suggested for some time now that licensing its fulfilment assets could be the future primary profit stream for Ocado, as opposed to simply online retailing, but there is no denying that the company is operating in a growing market where there is still significant growth potential for the businesses that manage to get e-grocery right.
Research group IGD published new data on Monday, which estimated that the UK grocery market will be worth £203 billion by 2019 – an increase of 16% from its current value of £175 billion.
Discounters, online and convenience is expected to account for over 40% of the market in five years' time, with e-tail set to be the fastest growing part of the market, more than doubling in value to £17 billion over that period. IGD says it expects online to represent 8% of the market, boosted by a surge in usage of home delivery and click & collect services.
Ocado reported a 20% increase in its top line for the six-month period, with revenue at £429.7 million, while active customers showed a year-on-year increase from 360,000 to 396,000 and the average basket size dipped slightly to £114.43.
The improved sales come at a hugely challenging and competitive time for the grocery market, with all the major players experimenting with new fulfilment options. Ocado's sourcing partner Waitrose, meanwhile, has also quickly developed a popular eCommerce operation in its own right over the last few years.
"Ocado's core business model remains unproven despite the welcome move into the black in the first half," Springham explained.
"Stripping out the fillip from Morrisons.com – for which Ocado has been overpaid by Morrisons in its desperate efforts to play catch-up – the level of profitability would have been much lower."
1 Aug 2014 Comment: How online grocery will grow in the next 5 years
Online grocery retailing is still at a nascent stage and we can expect plenty of exciting development ahead in ordering, delivery and mobile, argues IGD's Lisa Byfield-Green.
29 Sep 2015 Five online grocery trends to watch
10 Sep 2014 Comment: Grocers and suppliers must combine to boost online growth
With Tesco recruiting its new CEO from Unilever, IGD's Lisa Byfield-Green looks at how grocers and suppliers can better work together to maximise eCommerce growth.
17 Jul 2014 NetSuite acquires tech firm Venda as European expansion continues
23 Jun 2017 Over 25% of shoppers open to conversational commerce, says IGD
Over a quarter of shoppers are showing an interest in technology using voice and conversational messaging to make their grocery shopping easier.


This is a placeholder page for Brad Houser, which means this person is not currently on this site. We do suggest using the tools below to find Brad Houser.
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Here LA designers prepared the conceptual design for a 6-Acre nature preserve atop the bluffs of Rancho Palos Verdes. Visions for the rehabilitated site included trail planning and rationalization of site circulation. Project completed while at previous firm.


The list of the best telescopic step ladder, available on this page, is based with the help of customer reviews found at the marketplace. Additionally, we compared telescopic step ladder with the highest users score to pick the very best models. Considering the above, you can be sure that every telescopic step ladder listed below offers a great price/quality ratio.
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The 2006 BMW RT P 1200 Police Motorcycle for Sale is a 2006 BMW RT1200 Police model bike. If you are not familiar with the Police model, when you do a little research you will quickly note that it has so many more features than the stock BMW RT.. much more!
The RTP has only 36,550 miles and its 36,000 mile maintenance service as been completed which is a $1000 service fee right there alone! All service procedures were performed by MAX.BMW and I have receipts for all.
LED emergency lighting, with programmable flashing lights and sirens and yes... the front and rear flashing lights and siren work.
an AUX fan for the oil cooler that is not available on the stock BMW R1200.
I have a clean NY State title in hand plus the owner will receive an extra key as well as the owners maintenance and shop repair manuals.
in the amount of $500).
The motorcycle is currently being used as a daily rider so the mileage number will increase. This bike is excellent condition and you are welcome to make an appointment to come see for yourself. (No test rides) using the space below that has been provided by Strike Lightning's GoGocycles GoGoCycles Used Motorcycles for Sale (and PLEASE do not neglect to include your contact information in your email or to provide a phone number or email address!) so that I can respond with answers to your questions or set up an appointment as quickly as possible.
In my opinion there is not a better deal on a used BMW police motorcycle for sale in Syracuse New York NY, on Craigslist or in an Ebay auction as this one so hurry and make an offer before it gets away from you!
Strike Lightning would also encourage bikers to submit a review of the 2006 BMW RT P 1200 Police Motorcycle for Sale to make the biker community aware of any problems that are commonly associated with the 2006 model or to highlight its positive features and help potential buyers make informed decisions.
Get in touch today to see it in person and don't forget to mention that you found this ad on Strike Lightning's Motorcycle Classifieds www.gogocycles.com in your email! Have a wonderful day and RIDE safe, my friends!
I want to sell the bike so I am taking best offers.
I would like to get $8500 - but you can make an offer.
Tim... I'm in Brooklyn. If this bike is still available I'm interested. Thanks.
Is this RTP still for sale? I am in Brooklyn, and I am interested.


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I Was a Gay Pastor's Beard
My adoration was a convenient coverup—until it all fell apart.
As a pastor-in-training at my church, Colin* was smart, dedicated, and kind of a snob—the exact qualities I was attracted to in my early 30s. A lifelong member of the Evangelical Lutheran Church in America, I liked the rigor of devout religion, from the march through the major and minor holidays of the year to the repetitious demands of weekly worship. One of my church's mottos was that we were always being made new, and I loved that idea.
Colin seemed to feel the same way, and since he was almost a pastor, I trusted him easily. He was a few years younger than me, but so earnest and serious that he seemed older. Soon, I had a crush on him, and, over coffee one day, I confessed my feelings.
"Hmmm," he said. "Let's think about this like a book. We've only turned the first page. We don't know what the other pages hold." He went on to explain he could be kicked out of seminary if he was romantically involved with a member of the congregation he was serving. He'd have more freedom after he wrapped up his time at my church in a few months.
"Can we wait and see?" he asked. I eagerly agreed, thrilled that someone I liked and respected so much might reciprocate my interest.
I referred to our future together, making plans, and he didn't correct me.
What I didn't know was that Colin was already in a relationship. And, even more unfortunately for me, it was with a man. Although my particular congregation was proudly inclusive, with a senior pastor who rebelliously married same-sex couples long before it was accepted by the national church, the fact of the matter was that the ELCA did not yet accept gay people as clergy.
But I couldn't allow myself to really see it. I shrugged off any personality trait of Colin's which could be perceived as stereotypically gay. Sure, he was fond of classical music and loved fancy cheeses, but so do plenty of straight Lutheran pastors (and humans). Besides, as a playwright, I was used to working in an industry where gender and sexuality were understood to be fluid. Plenty of people who identified as gay still found themselves in straight, romantic relationships, and vice versa. So even if he was, maybe, potentially gay...that didn't mean he wasn't romantically interested in me. Wasn't that the true meaning of acceptance?
I was blind to the fact that in order to be ordained in the Lutheran Church, Colin needed to go back into the closet. Until the Church changed, Colin needed to pretend to be celibate, or at least a heterosexual man in a committed relationship with a woman. While he may not have gone looking for a beard, apparently I fit the role of one perfectly.
His ambiguous answer when I confessed my feelings had felt less like a rejection and more like a promise. He couldn't (and didn't) tell me his true feelings, but I believed that all would be well soon. In the meantime, he welcomed my devoted help with his work. I stayed up all night writing a series of devotions with him, and ran an important committee at the church at his request.
Before long, we had a social life together outside church. The idea of socializing with the clergy may seem bizarre to those raised to revere them, but the Lutheran church is generally more casual about such things, since part of our beliefs is that we're all of equal standing. Pastors at that church went to the symphony or to dinner with members of the congregation; some of them even vacationed together. Far from finding anything troubling about our relationship, everyone at the church seemed to be delighted that we were becoming an item, and often commented about how well we complemented each other.
I enjoyed playing the role of charming companion to a spiritually wise counselor, standing beside him at church functions and the brunch our church held after every Sunday morning service. I listened to Colin's sermon drafts and suggested jokes here and there. We went to concerts and museums together. He joined me on a trip to Long Island to see one of my plays in performance. I referred to our future together, making plans, and he didn't correct me.
At some point, though, I found myself having a twisted version of The Talk. What are we?! Months after we'd started our relationship, I confronted him and asked if he planned to date me when his internship ended. That's when he said he would never be interested in me. I sobbed. He didn't say that he was gay, and I still didn't believe that could be the reason—I thought it must be something else. Maybe it was me.
It felt like a break-up, of course, and I was unbelievably hurt. I couldn't find the grace or love he preached about in his actions. Eventually, he was forced to apologize, after the senior pastor asked me why he had seen me run by his office, sobbing, that day. He understood that Colin was responsible, even if only by sins of omission.
What I remember most about the apology was how Colin seethed at being forced to say it. He gave a litany of the ways he was embarrassed, too—how he found my affection disgusting but needed to push through for his cover story. At least, I thought to myself, he was finally acknowledging the truth.
Every person who found genuine solace in Colin's ministry irritated me, which was unfair to us all.
A few weeks after he apologized, I ran into another member of the congregation. Unaware of what had gone on between us, the man, out and proud himself, told me about welcoming Colin and his boyfriend to dinner. He teased me that he had wondered if Colin and I were a "thing," but wasn't surprised that it turned out I was helping him "pass." I didn't say a word about what had really happened—mostly to protect myself—but was hurt even further. Even if Colin was out to those he could be sure were sympathetic, I realized sadly, he still had never shared it with me directly, as a friend.
I tried to stay in that congregation, but I couldn't, especially after Colin was ordained and took a full-time job there. Every person who found genuine solace in Colin's ministry irritated me, which was unfair to us all. I was bitterly angry at myself for being duped. I was angry, too, at people like the friend I ran into on the train, who knew the truth but never realized what Colin was doing to me in front of them. I couldn't find any comfort in the rigor and litany of religion.
For a while, I wandered around, trying out different churches. Nothing suited me. Finally, I stumbled into a Lutheran congregation in a new town, one with a lovely lady pastor, who I loved–platonically!–on sight. At home again in the Church I had never wanted to leave, I felt renewed.
It was a difficult, painful way for grace to do its business, but I couldn't deny that it had succeeded. As with so many other tough situations, I'm a better person for experiencing it: less dreamy-eyed, less passive, more focused, more discerning.
As of 2010, the ELCA officially, happily ordains LGBT ministers. Reverend Asher O'Callaghan of Denver became the first known transgender person to be ordained by the ELCA earlier this year. At his ordination ceremony, he received the following blessing: May God bless you with discomfort at easy answers, half truths and superficial relationships, so that you may live deep within your heart.
*Name has been changed.
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UKIP deselects leading anti-Muslim activist Anne Marie Waters
David Lawrence - 29 04 17
The National Executive Committee (NEC) made the decision after HOPE not hate and other media sources put pressure on the party to disavow the candidate. Waters, the former Pegida UK leader, founder of Sharia Watch UK, and organiser of a controversial "Muhammad Cartoons" exhibition, has also recently developed links to the extreme far right.
UKIP leader Paul Nuttall, who himself has a long history of making discriminatory remarks about Islam, has been condemned by both the press and leading members of his own party over the explicitly anti-Muslim direction in which he has led the party. After a hum of negative press around the candidate, Nuttall stated on BBC 5Live yesterday morning that Waters' anti-Islam rants "make me feel a bit uncomfortable".
Waters was deselected despite the social media campaign of former British National Party (BNP) star Jack Buckby, now of the extreme anti-Muslim organisation Liberty GB, who encouraged supporters to put pressure on the NEC. Worryingly, Buckby's campaign was picked up by UKIP NEC member Anish Patel, who vowed to "do [his] best" to keep Waters as a candidate.
Patel has previously displayed his own troubling beliefs about Islam, writing publicly on his personal Facebook account that "these people will not stop until the whole of the west has completely changed and our western culture is a minority". Also among the 16 members of UKIP's NEC, who make up the principal management and administrative authority of UKIP, is Katie Fanning, who we previously revealed held racist and extreme anti-Muslim views.
While Waters' deselection is a step in the right direction, people with extreme views remain at the very heart of the party, and the radical anti-Islam direction in which Nuttall has led UKIP will continue to empower them.
HOPE not hate will continue to challenge UKIP to disavow these prejudiced policies and calls upon UKIP to take the principled stance against discrimination that we should expect from any party with mainstream aspirations.
Southend West By-Election: Candidate Briefing
Old Bexley and Sidcup roundup


are you desperate for zero time dilemma??? need something to refresh your memory??? want to have a little fun before you suffer in hell for the rest of eternity???????
3 protagonists. 23 routes. 46 endings.
sudoki is a high school au(?) dating sim(?) rated T for Teen. includes some swearing, creative threats, mild violence, pda, brief emeto mention, discussion of food, spoilers for 999 and vlr, and my very heart and soul.


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Transgender protections now law after Christie signs dozens of bills
Catherine Carrera
After talking baseball and O.J. Simpson's parole on sports radio Friday, Gov. Chris Christie made time to sign dozens of new laws, including one that protects transgender students' privacy rights, and permits them to use bathrooms and dress according to their gender identity.
Christie also signed legislation designed to combat the state's heroin crisis and another named after a slain Hillsdale girl.
Fourteen bills were vetoed, including one that would have benefited anti-smoking initiatives by dedicating 1 percent of the tax revenues from the sale of cigarettes and tobacco products to those programs. He also agreed to raise the minimum age to buy tobacco products from 19 to 21.
The transgender rights law requires the commissioner of education to develop guidelines for school districts to address an array of transgender students' needs. Christie, a Republican who had sought his party's nomination for president after making some appeals to conservatives, signed the law cleared by the state's Democratic-controlled Legislature without comment.
"These guidelines are needed to ensure that transgender students can safely be themselves without fear of being persecuted," said Assemblywomen Valerie Vainieri Huttle, D-Englewood, one of the bill's sponsors, in a statement.
Some school districts have already implemented similar policies, allowing transgender students to use bathrooms and locker rooms that align with their gender identities.
Schools are also required to use the name or pronoun that corresponds to a student's gender identity, and a student's school ID should be in the name corresponding to the student's gender identity. Transgender students are also given equal opportunities to participate in physical education and in gender-segregated school activities.
Fighting heroin epidemic
In the final year of his second term, Christie's agenda has aimed to focus on fighting the state's heroin epidemic by expanding treatment and resources.
Statistics released this week by the Bergen County Prosecutor's Office show that this year could be Bergen County's most deadly year on record, with 244 overdoses, including 57 fatalities.
The new legislation requires the state's Department of Human Services to start a daily report, which would be posted on their website, that includes information on open beds available in facilities across the state for those in need of mental health or substance-use disorder treatment.
Further, the Department of Education is tasked with developing an educational fact sheet, targeted specifically for parents of student athletes and cheerleaders, about the use and misuse of prescription opioids.
Hospice programs, under the new laws, are permitted to accept unused prescription medication from their patients for safe disposal. Those programs are also required to use sensitive terminology when referring to people with substance use disorders or certain disabilities.
Joan's Law
The latest amendment to Joan's Law, a bill named after Hillsdale's 7-year-old Joan D'Alessandro, who was raped and killed by her neighbor 44 years ago, will require life in prison without parole for anyone convicted of murdering and sexually assaulting a person under the age of 18.
The previous bill required the same punishment for those convicted of that crime to minors under 14.
"This day is a tremendous victory," said Rosemarie D'Alessandro, Joan's mother and leading advocate of the law, which was first enacted 20 years ago.
D'Alessandro was hoping for a public signing of the bill, similar to the one former Gov. Christine Whitman did when signing the bill in 1997. She had been advocating for this latest amendment to the bill for seven years.
"They will never have to go through parole hearings and appeals like our family had to for through for years," D'Alessandro said. She was motivated to get Joan's Law established when her daughter's murderer, Joseph McGowan, became eligible for parole 14 years after receiving a life sentence. He has been denied parole several times.
Other bills signed
Christie signed five bills aimed to benefit the state's military veterans, including one that establishes a Gold Star Family Counseling Program in the Department of Military and Veterans Affairs. Another bill allows all disabled veterans to receive parking privileges reserved for people with disabilities.
Among the bills signed that are aimed to preserve and protect the environment was legislation appropriating $71.7 million from the corporate business tax revenue and Green Acres funds to the Department of Environment Protection for local government open space acquisitions and park development projects.
Christie began his day once again filling in on the WFAN-660 morning show, "Boomer & Carton."
"Sounds like a great day for me," Christie said as he kicked off the show. Christie began auditioning this month for a permanent hosting gig at the CBS sports radio station.
But, Friday's 20-minute segment was not part of the governor's series of auditions, said co-host Boomer Esiason.
The radio appearance came ahead of Christie's trip to Colorado next week for a meeting with the Republican Governors Association.
Email: carrera@northjersey.com


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(director: Clint Eastwood; screenwriters: Paul Haggis/based on stories from "Rope Burns: Stories From the Corner" by F.X. Toole; cinematographer: Tom Stern; editor: Joel Cox; music: Clint Eastwood; cast: Clint Eastwood (Frankie Dunn), Hilary Swank (Maggie Fitzgerald), Morgan Freeman (Eddie Scrap-Iron Dupris), Brian O'Byrne (Father Horvak), Jay Baruchel (Danger), Lucia Rijker (Billie "The Blue Bear"), Mike Colter (Big Willie Little); Runtime: 137; MPAA Rating: PG-13; producers: Clint Eastwood/Paul Haggis/Tom Rosenberg/Albert S. Ruddy; Warner Brothers; 2004)
"Leaves its mark not with a knockout blow, but with round after round of jabs that hit their mark until there's nothing more left to save except one's dignity."
The 74-year-old Clint Eastwood ("Play Misty For Me"-1971, the first film he directed) directs, stars, produces, and writes the score for this brilliant drama about boxing, unlikely friendships, and redemption. In its simple aim, it's about a 31-year-old hillbilly woman from a dysfunctional family, Maggie Fitzgerald (Hilary Swank), who works as a waitress but aspires to be a boxer. On a more complicated level it's about three characters in search of survival: Maggie, who needs boxing to make something of herself and gain respect; the elderly grizzled has-been fight manager and "cut man" (someone who stops the bleeding), Frankie Dunn (Clint Eastwood), who also runs a dilapidated gym in the shabby part of Los Angeles and still dreams the dream–but lives it through others; and, the elderly ex-boxer Eddie Scrap-Iron Dupris (Morgan Freeman), who fought one fight too many under Frankie's tutelage without becoming a champion and lost an eye and now works for Frankie as a janitor in his Hit Pit gym. Each of the three characters never got out of life what they wanted. Maggie and Frankie still have fanciful dreams and aspirations, despite the odds against them ever succeeding. Scrap, who narrates the story (so everything is told from his viewpoint), doesn't ask for much out of life and is just trying to get by. "Baby" is also about searching for family, trying to correct past mistakes, and regrets about former relationships. In the third act the film transcends even these personal aims–becoming a boxing film unlike other bleakly told great ones such as Body and Soul, The Setup, The Harder They Fall, Fat City, and Raging Bull. It's one that has more to say about living and dying than it lets on at first–raising social issues that play for real life-and-death stakes.
The film was adapted by Canadian Paul Haggis from Rope Burns: Stories From the Corner. There are six short stories and the writer and Eastwood picked out the one that was most appealing. The book was written by Jerry Boyd, writing under the pseudonym F.X. Toole. When the book was finally published after being rejected for 40 years, the 70-year-old died a few years later and never had the pleasure of seeing his story made into a film. Boyd had been a fight manager and corner man, and this was the first book he ever published.
Big Willie Little (Mike Colter), an ambitious black fighter with a family, is Frankie's most promising prospect to win a championship. But the cautious Frankie thinks he needs a few more fights before he takes his one shot at the title. Willie is hungry for the fight now and leaves Frankie for a more prominent manager with business connections. Willie then wins the championship benefiting from all that Frankie taught him.
Frankie is used to disappointments, and after that setback turns his attention back to the gym and the odd characters and mostly minorities who come there dreaming of reaching it to the top. The crusty Frankie has not gone along with the recent trend to manage women boxers but relents after badgering from the earnest wannabe boxer Maggie, who blurts out that boxing is the only thing she "feels good doing." Scrap adds some more subtle and not so subtle persuasions to influence Frankie, seeing in her all the hopes and passion he once had to be "somebody."
Most of what we know about the three characters is drawn from what happens in the Hit Pit. The three performers are so good that they make all the boxing movie clichés it uncovers seem fresh. Their story always seems genuine and is accomplished in a relaxed pace giving it a very satisfying special flavor. The little we learn about them is handed out like nuggets of gold. We know that Frankie has an estranged daughter he writes to every week, but the letters are returned unread. There's no mention about her mother or what might have gone wrong in the relationship. That Frankie, a Catholic and regular attender of Mass, has a confrontational relationship with a priest–questioning the rhetoric of the church and asking for counsel on his dilemmas. Frankie is also studying Gaelic and reading Yeats, evidently brushing up on his Irish heritage. We eventually meet Maggie's horrible white trash family; but, know nothing about her sex life or how she functioned all these years, except she earns little as a waitress and has to take home leftover scraps from her patrons to survive. In turn, we learn of how Scrap lost his eye and that Frankie never forgave himself for not stopping the fight.
Everything inside the gym is grim. No apologies are made for the sport, nor is it romanticized–just that life is brutal and so is boxing, that's just the way it is. The characters are trapped in a bleak and despairing world that doesn't seem natural, but they can't change it. These characters live on the periphery of society and the odds are overwhelmingly against them becoming a "somebody." The only time they smile is over little things: a puppy dog Maggie sees in another girl's car, Frankie finding a diner that sells real homemade lemon pie, and Scrap bemused by a retarded white boy harmlessly punching the air and dreaming of becoming a champ. It's that kind of a film where there's no religion, doctors, or family to help out the unfortunate, the injured, and the lonely. It leaves its mark not with a knockout blow, but with round after round of jabs that hit their mark until there's nothing more left to save except one's dignity.
REVIEWED ON 2/1/2005 GRADE: A


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