text:
\section{Introduction}\label{section1}

In elliptic curve cryptosystems, it is important to construct elliptic curves with a required number of points over finite fields. The theory of complex multiplication leads an approach to the construction of elliptic curves over finite fields. When we construct elliptic curves, we often make use of the class polynomial of $J$-invariant function. However, this method has the problem on practical use that the coefficients of the polynomial grow very large.
To overcome the problem, the polynomials with small coefficients have been invented by using Weber functions.
Recently, Enge and Schertz \cite{EN1} gave the method of using the double $\eta$-quotient.
The double $\eta$-quotient is not $\rm{SL}_{2}(\mathbb{Z})$-invariant but $\Gamma^{0}(N)$-invariant. For this reason, they considered $N$-systems and calculated  class polynomials with respect to double $\eta$-quotients.
Furthermore, they can calculate the modular invariants  $J(\mathfrak{a})$, for an ideal $\mathfrak{a}$ of an imaginary quadratic field, by using the modular polynomial from the class polynomials with respect to double $\eta$-quotients. However in their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant.
In this article, we shall give a method to reduce the amount of computation in the process of counting the number of rational points. Thus, if we are in the situation that the modular invariant is a multiple root of the modular polynomial, we may expect that the amount of computation reducers to at most a half of the original one.
In Section 2, we give some basic results and definitions. In Section 3, we determine the relations between class polynomials with respect to double $\eta$-quotients and $N$-systems.
In Section 4 , we give a condition that the modular invariant is a multiple root of the modular polynomial. In section 5, we give an example.

\section{Some basic results and definitions}\label{section2}

For two prime numbers $p_1$ and $p_2$, the double $\eta$-quotient $\mathfrak{w}_{p_{1},p_{2}}(z)$ of level $N=p_1p_2$
is defined by
$$
 \mathfrak{w}_{p_{1},p_{2}}(z)  =  \frac{\eta(z/p_1) \eta(z/p_2)}{\eta(z) \eta(z/p_{1}p_{2})},
$$
where $\eta(z)$ denotes the Dedekind $\eta$-function defined by
$$
 \eta(z) = q^{1/24} \prod_{n=1}^{\infty}(1-q^{n}), \qquad
 q = q(z) = e^{2\pi iz}.
$$
The function $\mathfrak{w}_{p_{1},p_{2}}(z)$ is invariant under the modular group
$$\Gamma^{0}(N) = \left\{
  \left(\left.
   \begin{array}{cc}
    a & b \\
    c & d \\
   \end{array}
  \right)
  \in \rm{SL}_{2}(\mathbb{Z})\right | b\equiv 0\pmod N\right\}$$
(see \cite{MN}).
For a divisor $Q$ of $N$ such that $(Q,\frac{N}{Q})=1$,
set
$$
W_Q =
\left(
 \begin{smallmatrix}
    -Q & N \\
    -y & -Qx \\
 \end{smallmatrix}
\right),
$$
where $x,y \in \mathbb{Z}$ and $\det (W_Q)=Q$. Then we know $W_Q$ is a normalizer of $\Gamma^{0}(N)$. Especially the normalizer $\displaystyle W_N=\begin{pmatrix}0&N\\-1&0\end{pmatrix}$ is called the Atkin-Lehner involution of $\Gamma^{0}(N)$. By Theorem 2 of \cite{EN2}, we know
\begin{equation}\label{eq1}
 \mathfrak{w}_{p_{1},p_{2}}(W_N(z))  =  \mathfrak{w}_{p_{1},p_{2}}(z).
\end{equation}
For $\mathfrak{w}_{p_{1},p_{2}}^{s}(z)$ with $s = 24/\gcd(24,(p_1-1)(p_2-1))$,
Enge and Schertz \cite{EN1} defined the class polynomials $H_\mathfrak{N}(X)$ associated with $N$-systems $\mathfrak{N}$. In the following, we shall recall their results. Let $\mathcal{O}_f$ be the order of conductor $f$ in a imaginary quadratic field $K=\mathbb{Q}(\sqrt{m})$. Let $d_K$ be a discriminant of $K$. Then the discriminant $D$ of $\mathcal{O}_f$ is given by $D=f^2d_k$.
Let $\mathfrak{H}_f$ be the (proper) ideal class group of $\mathcal{O}_f$.
To a proper ideal $\mathfrak{a}=[\beta_1,\beta_2] = \mathbb{Z}\beta_1 + \mathbb{Z}\beta_2$ of $\mathcal{O}_f$, we associate its basis quotient
$$\alpha_{\mathfrak{a}}=\displaystyle{\frac{\beta_1}{\beta_2}} \qquad \left(\alpha_{\mathfrak{a}} \in \mathbb{H} = \{ \tau \in \mathbb{C}~|~ Im(\tau) > 0 \}\right),$$
and a quadratic form $\mathfrak q_{\mathfrak{a}}(X,Y)=N_{K/\mathbb Q}(\beta_1X+\beta_2Y)/N_{K/\mathbb Q}(\mathfrak{a})$. It is noted that the basis quotient $\alpha_\mathfrak{a}$ is determined up to $\rm{SL}_{2}(\mathbb{Z})$-equivalence and the form $\mathfrak{q}_{\mathfrak{a}}(X,Y)$ is a primitive quadratic form with integral coefficients of discriminant $D$. The following results are well known (see \cite{AC}).

Let $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are proper ideals of $O_f$. We write $ \mathfrak{a}_1 \sim \mathfrak{a}_2$ if  $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are in the same ideal class of $\mathcal{O}_f$.
Then
\begin{eqnarray}
  \mathfrak{a}_1 \sim \mathfrak{a}_2
 & \Longleftrightarrow & \alpha_{\mathfrak{a}_1} \; \text{is} \;\rm{SL}_{2}(\mathbb{Z})\text{-equivalent} \; \text{to} \;  \alpha_{\mathfrak{a}_2} \nonumber\\
 & \Longleftrightarrow & \mathfrak q_{\mathfrak{a}_1}(X,Y) \; \text{is proper equivalent to} \; \mathfrak q_{\mathfrak{a}_2}(X,Y).\label{equiv1}
\end{eqnarray}
Further the map $\mathfrak{a}\mapsto \mathfrak q_{\mathfrak{a}}(X,Y)$ gives rise to a bijection between  $\mathfrak{H}_f$ and the proper equivalent classes of quadratic forms of discriminant $D$. Since for an ideal $\mathfrak{a}$  the value $J(\alpha_\mathfrak{a})$ is independent on the choice of basis quotients $\alpha_\mathfrak{a}$, we shall denote by $J(\mathfrak{a})$ the value $J(\alpha_\mathfrak{a})$. Hereafter for a triple $(A,B,C)$ of integers such that $A\ne 0$, we shall denote by $[A,B,C]$ a quadratic form $AX^2+BXY+CY^2$. Furthermore, for a quadratic form $\mathfrak q=[A,B,C]$, we put $\mathfrak{a}_\mathfrak{q}=[A,\frac{-B+\sqrt D}2]$ and $\alpha_\mathfrak{q}=\frac{-B+\sqrt D}{2A}$. Let $h(D)$ be the class number of the order $\mathcal{O}_f$. Then we know  there exist $h(D)$ isomorphic classes of elliptic curves with complex multiplication by $\mathcal{O}_f$. They are represented  by the elliptic curves with $j$-invariants $j(\alpha_{\mathfrak{a_i}})$, where $\mathfrak{a}_i~(i=1,\dots ,h(D))$ are ideals representing all classes of $\mathfrak{H}_f$. Let $K_f$ be the ring class field of $K$ of conductor $f$. Then the theory of complex multiplication shows  $J(\mathfrak{a}_i)$ generates $K_f$ over $K$ for each $i$ and $J(\mathfrak{a}_i),~(i=1,\dots ,h(D))$ are conjugate  to each other over $\mathbb Q$. To calculate the values $J(\mathfrak{a})$, we use the classical class equation for the order $\mathcal{O}_f$ defined by
\[H_D[J](X)=\prod_{i=1}^{h(D)} (X-J(\mathfrak{a}_i)).\]
However, since this polynomial has large integral coefficient, it is hard to compute the polynomial for large $D$'s.  Enge and Schertz  \cite{EN1} devised the method of using the double $\eta$-quotient to obtain a class polynomial with small integral coefficients. Since double $\eta$-quotient is $\Gamma^{0}(N)$-invariant but not $\rm{SL}_{2}(\mathbb{Z})$-invariant, the values $\mathfrak{w}_{p_{1},p_{2}}^s(\alpha_\mathfrak{a})$ depend on the choice of ideals $\mathfrak{a}$ in an ideal class. Therefore to define a class polynomial for $\mathfrak{w}_{p_{1},p_{2}}^s$, we must use the $N$-systems introduced by Schertz \cite{RS}.

\begin{definition}
Let $\mathfrak{N}$ be a set of $h(D)$ primitive quadratic forms $[A_i,B_i,C_i]$ of discriminant $D$. we say
$\mathfrak{N}$ an $N$-system for $D$ if quadratic forms $[A_i,B_i,C_i]$ satisfy the following conditions:
\begin{enumerate}
  \item $\gcd(A_i,N)=1, B_i\equiv B_j\pmod{2N},~N|C_i$ for every $i,j$,
 \item the ideals $[A_i, \frac{-B_i+\sqrt{D}}{2}]$ form a representative system of $\mathfrak{H}_f$.
\end{enumerate}
\end{definition}
The following result is basic for $N$-systems. See Theorems 3.1, 3.2 of \cite{EN1}.
\begin{theorem}\label{theorem2.3}
Assume prime numbers $p_1$ and $p_2$ satisfy the following conditions:
\begin{enumerate}
 \item If $p_1 \neq p_2$, then $\bigl( \frac{D}{p_1} \bigr),\bigl( \frac{D}{p_2} \bigr) \neq -1$,
 \item if $p_1=p_2=p$, then either $\bigl( \frac{D}{p} \bigr)=1$ or $p|f$.
\end{enumerate}
Then there exists an $N$-system $\mathfrak{N}=\{\mathfrak{q}_i\}$ for $D$ . Further $\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_i}) \in K_f$ for every $i$ and  $\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_i})$ are conjugate under $G(K_f/K)$ to each other.
\end{theorem}
Now the class polynomial of $\mathfrak{w}_{p_{1},p_{2}}^{s}$ associated with an $N$-system $\mathfrak{N}$ is defined by
\[
H_\mathfrak{N}(X) = \prod_{i=1}^{h(D)} \left (X-\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_i})\right).
\]
By Theorem~\ref{theorem2.3}, we have $H_\mathfrak{N}(X)\in K[X]$. Further Corollary 3.1 of \cite{EN1} gives
\begin{corollary}\label{Cor2.4}
 Suppose that following conditions hold:
\begin{enumerate}
 \item If $p_1 \neq p_2$, then $\bigl( \frac{D}{p_1} \bigr),\bigl( \frac{D}{p_2} \bigr) \neq -1$, and $p_1,p_2 \nmid f$;
 \item if $p_1=p_2=p \neq 2$, then $\bigl( \frac{D}{p} \bigr)=1$ or $p|f$;
 \item if $p_1=p_2=2$, then $\bigl( \frac{D}{2} \bigr)=1$, or $2|f$, but $D\not\equiv 4\pmod{32}$.
\end{enumerate}
Then $H_\mathfrak{N}(X)\in\mathbb{Z}[X]$.
\end{corollary}
To obtain modular invariants $J(\mathfrak{a}_{\mathfrak{q}})$ from the singular values $\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}})$, we use a modular polynomial $\Phi_{p_1,p_2}$ that associates $\mathfrak{w}_{p_{1},p_{2}}$ with $J$, which is defined as follows.

\begin{definition}
$$
\Phi_{p_1,p_2}(X,J) = \prod_{\sigma \in Iso(\mathbb{C}_{\Gamma^{0}(N)}/\mathbb{C}_{\Gamma})}
(X-\sigma(\mathfrak{w}_{p_{1},p_{2}}^{s})),
$$
where $\mathbb{C}_{\Gamma^{0}(N)}$ and $\mathbb{C}_{\Gamma}$ denote the modular function field
of $\Gamma^{0}(N)$ and $\Gamma=\rm{SL}_{2}(\mathbb{Z})$ respectively.
\end{definition}

We know that $\Phi_{p_1,p_2}(X,J) \in \mathbb{Z}[X,J]$ and
$\Phi_{p_1,p_2}(X,J)$ is minimal polynomial of $\mathfrak{w}_{p_{1},p_{2}}^{s}$ over $\mathbb{C}(J)$
by Theorems 7 and 8 by \cite{EN2}. To obtain elliptic curves with complex multiplication by $\mathcal{O}_f$ over finite field $\mathbb{F}_q$ of $q$-elements, the polynomials $H_\mathfrak{N}(X)$ and $\Phi_{p_1,p_2}(X,J)$ are used in the following algorithm.

Assume that $q$ is a prime number which splits completely in $K_f$.

\begin{algorithm}

\begin{enumerate}
 \item Construct an $N$-system $\{\mathfrak{q}_i\}$.
 \item Compute $\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_i})$ and $H_\mathfrak{N}(X)
$.
 \item Compute a roots $\overline{\mathfrak{w}}$ of $H_\mathfrak{N}(X) \bmod q $.
 \item Compute  $\mathbb{F}_q$-rational roots $\overline{J_{k}}$ of $\Phi_{p_1,p_2}(\overline{\mathfrak{w}},J) \bmod q $.
 \item Output the desired $J$-invariant among $\overline{J_{k}}$.
\end{enumerate}

\end{algorithm}

In step 5, it is necessary to count the number of $\mathbb{F}_q$-rational points of each elliptic curve $E_k$ with the j-invariant $\overline{J_{k}}$ to determine which elliptic curve $E_k$ has complex multiplication by $\mathcal{O}_f$. In step 4, if $\Phi_{p_1,p_2}(\overline{\mathfrak{w}},J) \bmod q $ has the degree 2 in $J$ and the multiple root $\overline{J}$ in $J$, then it is not necessary to count the number of rational points. Accordingly, we will consider the condition that the polynomial has the multiple root in Section \ref{section4}.

\section{$N$-systems and class polynomials}\label{section3}

In this section, we study the relation between the class polynomials and $N$-systems.

\
\begin{lemma}\label{lemma1}
Let $\{[A_i,B_i,C_i]\}$ and $\{[A_i',B_i',C_i']\}$ be $N$-systems for $D$.
Suppose that $[A_i,B_i,C_i]$ and $[A_i',B_i',C_i']$ are proper equivalent.
Then $\frac{-B_i+\sqrt{D}}{2A_i}$ is $\Gamma^0(N)$-equivalent to $\frac{-B_i'+\sqrt{D}}{2A_i'}$ if and only if $B_i\equiv B_i'\pmod {2N}$. In particular, if $B_i\equiv B_i'\pmod {2N}$, then
$$
\mathfrak{w}_{p_{1},p_{2}}^{s}(\frac{-B_i+\sqrt{D}}{2A_i})=\mathfrak{w}_{p_{1},p_{2}}^{s}(\frac{-B_i'+\sqrt{D}}{2A_i'}).
$$
\end{lemma}

\begin{proof}
Since $[A_i,B_i,C_i]$ and $[A_i',B_i',C_i']$ are proper equivalent, there exists a matrix
$
 M= \left(
   \begin{array}{cc}
    a & b \\
    c & d \\
   \end{array}
  \right)
  \in \rm{SL}_{2}(\mathbb{Z})
$
such that
\begin{eqnarray}
\frac{-B_i+\sqrt{D}}{2A_i} = \frac{a(\frac{-B_i'+\sqrt{D}}{2A_i'})+b}{c(\frac{-B_i'+\sqrt{D}}{2A_i'})+d}. \label{5}
\end{eqnarray}
We have only to prove
$M \in \Gamma^0(N)$ if and only if $B_i\equiv B_i'\pmod {2N}$.
By (\ref{5}), we have
\begin{eqnarray}
B_iB_i'c-2A_i'B_id+Dc & = &- 2A_iB_i'a+4A_iA_i'b,  \label{6}\\
-B_ic-B_i'c+2A_i'd & = & 2A_ia. \label{7}
\end{eqnarray}

By substituting (\ref{7}) into (\ref{6}), we obtain
\begin{eqnarray}
A_ia(B_i'-B_i)-2A_iC_ic=2A_iA_i'b. \label{7.1}
\end{eqnarray}

Since $\gcd(A_iA_i',N)=1$ and $N|C_i$, we have $a(B_i'-B_i)\equiv 2A_i'b\pmod {2N}$. Therefore $M \in \Gamma^0(N)$ if and only if $B_i\equiv B_i'\pmod {2N}$.
\end{proof}

The following result is deduced from Proposition 3 of \cite{RS}.
\begin{proposition}\label{prop_rs}
Let $[A,B,C]$ be a primitive quadratic form of discriminant $D$ such that $A>0, \gcd(A_,N)=1$ and $N|C$. Then there exists an $N$-system $\mathfrak{N}$ for $D$ containing $[A,B,C]$. In particular, for an integer $B$ such that $B^2\equiv D\pmod {4N}$, there exists an $N$-system $\mathfrak{N}$ for $D$ containing $[1,B,(B^2-D)/4]$.
\end{proposition}

By Proposition~\ref{prop_rs} and Lemma~\ref{lemma1}, we know the class polynomials of double $\eta$-quotient associated with $N$-systems depend only on integers $B$, considered mod $2N$, such that $B^2\equiv D\pmod {4N}$. Thus, hereafter, we shall denote by $H_{B,N}(X)$ the class polynomial $H_\mathfrak{N}(X)$ associated with  an $N$-system $\mathfrak{N}$ containing a form $[1,B,(B^2-D)/4]$. In the following, we shall fix an $N$-system containing $[1,B,(B^2-D)/4]$ and shall denote it by $\mathfrak{N}_B$.
\begin{lemma}\label{lemma2}
Assume that $p_1$ and $p_2$ are odd primes. Let $N(D)$ be the number of integers $B \bmod 2N$ such that $B^2\equiv D \pmod {4N}$.
Then
\begin{eqnarray}
N(D) =
\begin{cases}
4 \quad if \quad {\bigl( \frac{D}{p_1} \bigr)=1 \quad and \quad \bigl( \frac{D}{p_2} \bigr)=1}, \\
2 \quad if \quad {\bigl( \frac{D}{p_1} \bigr)=1 \quad and \quad \bigl( \frac{D}{p_2} \bigr)=0}.
\end{cases}
\end{eqnarray}
\end{lemma}

\begin{proof}
Let us consider the case  $\bigl( \frac{D}{p_1} \bigr)=1$  and  $\bigl( \frac{D}{p_2} \bigr)=1$. Then there exists an integer $\alpha_i$ such that $\alpha_i^2\equiv D \pmod{p_i}$ for $i=1,2$. By Chinese reminder theorem, we see $B^2\equiv D \pmod{4N}$ if and only if
\begin{eqnarray}
 B & \equiv & D \pmod{2}, \\
 B & \equiv & \pm \alpha_i \pmod{p_i}\quad (i=1,2). \label{1}
\end{eqnarray}
This shows $N(D) = 4$. The remaining case can be treated similarly. Thus we omit the details.
\end{proof}
If $N$ is odd, we obtain at most $N(D)$ distinct class polynomials of the double $\eta$-quotient associated with $N$-systems for $D$.
\begin{lemma}\label{lemma3}
Let $B$ be an integer such that $B^2\equiv D \pmod {4N}$. Then $H_{-B,N}(X)=H_{B,N}(X)$.
\end{lemma}

\begin{proof}
We know the $q$-expansion of $\mathfrak{w}_{p_{1},p_{2}}^{s}(z)$ is rational, thus
$$
\mathfrak{w}_{p_{1},p_{2}}^{s}(z) = \sum a_n q^n \qquad (a_n \in \mathbb{Q}, \quad q=e^{2\pi iz})
$$
(see section 3 of \cite{EN2}). Therefore we have
$$
\overline{\mathfrak{w}_{p_{1},p_{2}}^{s}(z)} = \sum a_n \overline{q}^n = \mathfrak{w}_{p_{1},p_{2}}^{s}({\overline{q}})=\mathfrak{w}_{p_{1},p_{2}}^{s}(-\overline z)
$$
and
\[
\overline{\mathfrak{w}_{p_{1},p_{2}}^{s}\Bigl(\frac{-B+\sqrt{D}}{2}\Bigr)} = \mathfrak{w}_{p_{1},p_{2}}^{s}\Bigl(\frac{B+\sqrt{D}}{2}\Bigr).
\]
Since  Corollary~\ref{Cor2.4} shows that $H_{B,D}(X)\in\mathbb Z[X]$, we have $H_{B,D}(X)=\overline{H_{B,N}(X)}=H_{-B,N}(X)$.

\end{proof}

\begin{theorem}\label{theorem2}
Let $N(H_D)$ be the number of distinct class polynomials $H_{B,N}(X)$ associated with $N$-systems.
Then
\begin{eqnarray}
N(H_D) =
\begin{cases}
1,2 \quad &if \quad {\bigl( \frac{D}{p_1} \bigr)=1 \quad and \quad \bigl( \frac{D}{p_2} \bigr)=1}, \\
1 \quad &if \quad {\bigl( \frac{D}{p_1} \bigr)=1 \quad and \quad \bigl( \frac{D}{p_2} \bigr)=0}.
\end{cases}
\end{eqnarray}
\end{theorem}

\begin{proof}
Lemmas \ref{lemma2} and \ref{lemma3} imply that $N(H_D) \le N(D)/2$. Thus we have the assertion.
\end{proof}

In the case $N(H_D)=2$, we have two class polynomials $H_{B,N}(X)$ and $H_{B',N}(X)$,
where $B,B'$ are integers such that $B^2\equiv D\pmod{4N},~B'\equiv B\pmod{p_1},~B'\equiv -B\pmod{p_2}$. We shall show $H_{B',N}(X)$ is obtainable from $H_{B,N}(X)$ by a simple transformation.
We shall use the following transformation formula of the Dedekind $\eta$-function (see Theorem 1 of \cite{EN2}).
\begin{theorem}\label{EN2_Th1}
Let
$
M =
  \left(
   \begin{array}{cc}
    a & b \\
    c & d \\
   \end{array}
  \right)
  \in \rm{SL}_{2}(\mathbb{Z})
$
be normalized such that $c\geq 0$, and $d>0$ if $c=0$.
Write $c=\gamma 2^{\lambda}$ with $\gamma$ odd; by convention, $\gamma=\lambda=1$ if $c=0$. Then
$$
\eta(Mz)  =  \epsilon(M)\sqrt{cz+d} \eta(z)
$$
with
$$
\mathfrak{R}(\sqrt{cz+d})>0 , \quad
\epsilon(M)  = \Bigl( \frac{a}{\gamma} \Bigr) \zeta^{ab+c(d(1-a^2)-a)+3\gamma(a-1)+\frac{3}{2}\lambda(a^2-1)}_{24}.
$$
\end{theorem}

\begin{lemma}\label{lemma_involution}
Let $
W_{p_1} =
  \left(
   \begin{array}{cc}
    -p_1 & N \\
    -y & -p_1x \\
   \end{array}
  \right)
$
with $y < 0,~y\equiv 1\pmod 2$ and $p_1x+p_2y=1$. Then
$$
 \mathfrak{w}_{p_1,p_2}^s(W_{p_1}(z)) = \frac{\Bigl( \frac{p_1}{p_2} \Bigr)^s}{\mathfrak{w}_{p_1,p_2}^{s}(z)}.
$$
\end{lemma}

\begin{proof}

 By Theorem \ref{EN2_Th1},
\begin{eqnarray*}
 \mathfrak{w}_{p_1,p_2}(W_{p_1}(z)) = \mathfrak{w}_{p_1,p_2}(\frac{-p_1z+N}{-yz-p_1x})
 = \frac{\eta(\frac{-p_1z+N}{p_1(-yz-p_1x)})\eta(\frac{-p_1z+N}{p_2(-yz-p_1x)})}{\eta(\frac{-p_1z+N}{-yz-p_1x})\eta(\frac{-p_1z+N}{p_1p_2(-yz-p_1x)})}
 = \frac{\epsilon^{*}}{\mathfrak{w}_{p_1,p_2}(z)}.
\end{eqnarray*}
Here
$$
\epsilon^{*} ={\footnotesize \frac{
\epsilon
  \left(\Bigl(
   \begin{array}{cc}
    -1 & p_2 \\
    -y & -p_1x \\
   \end{array}\Bigr)
  \right)
\epsilon
  \left(\Bigl(
   \begin{array}{cc}
    -p_1 & 1 \\
    -p_2y & -x \\
   \end{array}\Bigr)
  \right)
}{
\epsilon
  \left(\Bigl(
   \begin{array}{cc}
    -p_1 & p_2 \\
    -y & -x \\
   \end{array}\Bigr)
  \right)
\epsilon
  \left(\Bigl(
   \begin{array}{cc}
    -1 & 1 \\
    -p_2y & -p_1x \\
   \end{array}\Bigr)
  \right)
}}
= \frac{\Bigl(\frac{-1}{-y} \Bigr)\zeta^{a}_{24} \Bigl( \frac{-p_1}{-p_2y} \Bigr)\zeta^{b}_{24}}
{\Bigl(\frac{-p_1}{-y} \Bigr)\zeta^{c}_{24} \Bigl( \frac{-1}{-p_2y} \Bigr)\zeta^{d}_{24}}
= \Bigl(\frac{p_1}{p_2} \Bigr)\zeta^{a+b-c-d}_{24}
$$
with
\begin{eqnarray*}
a & = & -p_2-y+3\gamma(-2), \\
b & = & -p_1 -p_2y(-x(1-p_1^2)+p_1)+3p_2\gamma(-p_1-1)+\frac{3}{2}\lambda(p_1^2-1), \\
c & = & -p_1p_2 -y(-x(1-p_1^2)+p_1)+3\gamma(-p_1-1)+\frac{3}{2}\lambda(p_1^2-1), \\
d & = & -1-p_2y+3p_2\gamma(-2).
\end{eqnarray*}
Therefore we see
$$
a+b-c-d = (1-p_2)(1-p_1)(1-y-xy(p_1+1)-3\gamma).
$$
Since $s(p_1-1)(p_2-1) \equiv 0 \pmod{24}$, we have  $(\epsilon^{*})^s= \Bigl(\frac{p_1}{p_2} \Bigr)^s$.

\end{proof}

\begin{proposition}\label{prop5}
Suppose that $\bigl( \frac{D}{p_1} \bigr)=\bigl( \frac{D}{p_2} \bigr)=1$.
Then
$$
H_{B',N}(X)= \frac{X^{h(D)}}{H_{B,D}(0)} H_{B,N}\Bigl( \frac{\Bigl( \frac{p_1}{p_2} \Bigr)^s}{X} \Bigr).
$$
\end{proposition}

\begin{proof}
 Let $\mathfrak{q}=[A,B,C]$ be a form  of the $N$-system $\mathfrak{N}_B$. Put $\alpha_\mathfrak{q}=\frac{-B+\sqrt D}{2A}$. We can write $W_{p_1}(\alpha_\mathfrak{q})=\frac{-B'+\sqrt d}{2A'}$. We see easily that
$B'\equiv B\pmod{p_1}$, $B'\equiv -B\pmod{p_2}$ and $C'=(B'^2-D)/4A'\equiv 0\pmod N$. If $\gcd(A',N)>1$, then we shall show there exists an element $\gamma \in \Gamma^{0}(N)$ such that the first coefficient $A''$ of the quadratic form $[A'',B'',C'']$
corresponding to $\gamma W_{p_1}(\alpha)$ is prime to $N$. It is noted by the proof of Lemma~\ref{lemma1} that $B'\equiv B''\pmod{2N}$ and $N|C''$. Since $\mathfrak{w}_{p_{1},p_{2}}(\gamma W_{p_1}(\alpha_\mathfrak{q})) = \mathfrak{w}_{p_{1},p_{2}}(W_{p_1}(\alpha_\mathfrak{q}))$, we have
\[
H_{B',N}(X)=\prod_{\mathfrak{q}\in\mathfrak{N}_B}(X- \mathfrak{w}_{p_{1},p_{2}}(W_{p_1}(\alpha_\mathfrak{q}))).
\]
By Lemma~\ref{lemma_involution}, we have our result. Thus we have only to prove the existence of the above $\gamma$ in the case $A'$ is not prime to $N$. Set
$\gamma =
  \left(
   \begin{array}{cc}
    r & N \\
    t & 1 \\
   \end{array}
  \right)
$, for integers $r,t$. Then
\begin{eqnarray*}
\gamma W_{p_1}(\alpha) & = & \frac{r \Bigl(\frac{-B'+\sqrt{D}}{2A'}\Bigr)+N}{t\Bigl( \frac{-B'+\sqrt{D}}{2A'}\Bigr) + 1}
= \frac{(-rB'-tB'N+2A'N+2rtC')+\sqrt{D}}{2(-tB'+A'+t^2C')}.
\end{eqnarray*}
Therefore we have $A''= -tB'+A'+t^2C'$. Since $N|C'$, we know $\gcd(A'',N)=1$ if and only if $\gcd(-tB'+A',N)=1$. Assume $N|A'$. Then $\gcd(D,N)=1$ implies $(B',N)=1$. Therefore we can take $r=N+1$, $t=1$. Next assume $p_i|A'$ and $p_j\nmid A'$. Then we have $p_i\nmid B'$. Therefore we can take $r=p_jN+1$, $t=p_j$. This completes our proof.
\end{proof}

\section{Multiple roots of modular equation }\label{section4}
In this section, we assume the conditions in Corollary~\ref{Cor2.4}.
We shall study a condition that for singular values $\alpha$ of double $\eta$-quotients the polynomial $\Phi_{p_1,p_2}(\alpha,J)$ of $J$ has a multiple root.
\begin{proposition}\label{prop1}
The polynomial $\Phi_{p_1,p_2}(X,J)$ has degree $\displaystyle{N \prod_{p|N} \bigl(1+\frac{1}{p}\bigr)}$ as a polynomial of $J$ and has degree $\displaystyle{\frac{s(p_1-1)(p_2-1)}{12}}$ as a polynomial of $X$. For $\tau\in\mathbb H$, the equation $\Phi_{p_1,p_2}(\mathfrak{w}_{p_1,p_2}^s(\tau),J)=0$ has two roots $J(\tau)$ and $J(W_N (\tau))$.
In particular, if $J(\tau)=J(W_N (\tau))$, then the equation has a multiple root.
\end{proposition}

\begin{proof}
The assertion concerning the degree follows from Theorem 9 of \cite{EN2}. The equation $\Phi_{p_1,p_2}(\mathfrak{w}_{p_1,p_2}^s(\tau),J)=0$ obviously has the root $J(\tau)$.
Similarly, $J(W_N(\tau))$ is a root of $\Phi_{p_1,p_2}(\mathfrak{w}_{p_1,p_2}^s(W_N(\tau)),J)=0$. By \eqref{eq1}, we have $\mathfrak{w}_{p_{1},p_{2}}^{s}(W_N(\tau))=\mathfrak{w}_{p_{1},p_{2}}^{s}(\tau)$. Therefore $J(W_N (\tau))$ is a root of $\Phi_{p_1,p_2}(\mathfrak{w}_{p_1,p_2}^s(\tau),J)=0$.
\end{proof}

Let $\mathfrak{q}=[A,B,C]$ be a form of an $N$-system $\mathfrak{N}_B$. Since
\begin{eqnarray*}
W_N(\alpha_\mathfrak{q}) & = & \frac{N}{-\frac{-B+\sqrt{D}}{2A}}
               = \frac{2AN(B+\sqrt{D})}{B^2-D}
               = \frac{B+\sqrt{D}}{2(\frac{C}{N})},
\end{eqnarray*}
the action of $W_N$ on the ideal $\mathfrak{a_\mathfrak{q}}$ is given by $W_N(\mathfrak{a_\mathfrak{q}}) = [\frac{C}{N}, \frac{B+\sqrt{D}}{2}]$.

\begin{lemma}\label{lemma_a_b}
If we set $\mathfrak{a}_B=[N,\frac{-B+\sqrt{D}}{2}]$, then $W_N(\mathfrak{a_\mathfrak{q}}) \sim \mathfrak{a_\mathfrak{q}}\mathfrak{a}_B$.
\end{lemma}

\begin{proof}Since $(A,B,C)=1$,
\begin{eqnarray*}
\overline{\mathfrak{a_\mathfrak{q}}}W_N(\mathfrak{a_\mathfrak{q}})
& = & [A,\frac{B+\sqrt{D}}{2}][\frac{C}{N},\frac{B+\sqrt{D}}{2}] \\
& = & [\frac{AC}{N},\frac{C}{N}(\frac{B+\sqrt{D}}{2}),A(\frac{B+\sqrt{D}}{2}),B(\frac{B+\sqrt{D}}{2})] \\
& = & [\frac{AC}{N},\frac{B+\sqrt{D}}{2}] \\
& = & \frac{1}{N}(\frac{B+\sqrt{D}}{2})[N,\frac{-B+\sqrt{D}}{2}].
\end{eqnarray*}
Thus we have $\overline{\mathfrak{a_\mathfrak{q}}}W_N(\mathfrak{a_\mathfrak{q}})\sim\mathfrak{a_\mathfrak{q}}\mathfrak{a}_B$. Since $\overline{\mathfrak{a_\mathfrak{q}}}\mathfrak{a_\mathfrak{q}}\sim 1$, this proves the assertion.
\end{proof}

\begin{proposition}\label{prop2}
 Let $\mathfrak{q}=[A,B,C]$ be a form of an $N$-system $\mathfrak{N}_B$.  Then
$J(W_N(\alpha_\mathfrak{q}))$ $=J(\alpha_\mathfrak{q})$ if and only if there exist $u,v \in \mathbb{Z}$ such that
\begin{eqnarray}\label{con1}
\begin{cases}
u^2 - Dv^2 = 4N \\
u-Bv \equiv 0 \pmod{2N}.
\end{cases}
\end{eqnarray}
\end{proposition}

\begin{proof}
By \eqref{equiv1} and Lemma~\ref{lemma_a_b}, we have
\begin{eqnarray*}
J(W_N(\mathfrak{a}_\mathfrak{q}))=J(\mathfrak{a}_\mathfrak{q})
\quad & \Leftrightarrow & \quad W_N(\mathfrak{a_\mathfrak{q}}) \sim \mathfrak{a_\mathfrak{q}} \\
\quad & \Leftrightarrow & \quad \mathfrak{a}_B \sim 1.
\end{eqnarray*}
Further we know
the condition $\mathfrak{a}_B \sim 1$ is equivalent to the existence of an element
$
  \left(
   \begin{array}{cc}
    x & y \\
    z & w \\
   \end{array}
  \right)
  \in \rm{SL}_{2}(\mathbb{Z})
$
such that
\begin{eqnarray}\label{con2}
\frac{-B+\sqrt{D}}{2N} = \frac{x(\frac{-B+\sqrt{D}}{2})+y}{z(\frac{-B+\sqrt{D}}{2})+w}.
\end{eqnarray}
Let us assume  \eqref{con2}. Then  we have
\begin{eqnarray}
zB^2+zD-2wB & = & -2xNB+4Ny, \label{8} \\
w & = & xN+zB. \label{9}
\end{eqnarray}
By substituting (\ref{9}) into (\ref{8}), we have $y = -A\frac{C}{N}z$.
Therefore, from $xw - yz = 1$, we obtain $(Bz+2xN)^2-Dz^2=4N$.
Now, we put $u=Bz+2Nx$, $v=z$. Then we have
$u^2-Dv^2=4N$ and $x=\frac{u-Bv}{2N}$. Further, since $x \in \mathbb{Z}$, we have $u-Bv \equiv 0 \pmod{2N}$. Conversely, let $u,v$ be integers satisfying \eqref{con1}. Put $x=\frac{u-Bv}{2N}$, $y= -A\frac{C}{N}v$, $z=v$ and $w=xN+zB$. Then we have $xw-zy=1$ and \eqref{con2}.

\end{proof}
Immediately from  Proposition~\ref{prop2}, we obtain
\begin{corollary}\label{coro4}
If $J(W_N(\mathfrak{a}_\mathfrak{q}))=J(\mathfrak{a}_\mathfrak{q})$, then $D > -4N$.
\end{corollary}

\begin{proof}
The condition \eqref{con1} shows
$
D = \frac{u^2-4N}{v^2} > -4N.
$
\end{proof}
\begin{proposition}\label{prop3}
Assume that there exist integers $u$ and $v$ satisfying \eqref{con1}.
Then the equation $\Phi_{p_1,p_2}(\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_\mathfrak{q}),J)=0$ has
a multiple root $J(\mathfrak{a}_\mathfrak{q})$.
\end{proposition}

\begin{proof}
The assertions is obvious.
\end{proof}

By a similar argument in Proposition~\ref{prop2}, we have:

\begin{proposition}\label{prop5}
Let the notation be as in Proposition~\ref{prop2}. Then
 ${W_N}^2(\mathfrak{a}) \sim \mathfrak{a} $ if and only if there exist integers $X,Y$ such that $Y\neq 0$ and
\begin{eqnarray*}
\begin{cases}
X^2-DY^2=4N^2 \\
X-BY \equiv 0 \pmod{2N} \\
 (\frac{X-BY}{2N})^2 \equiv 1 \pmod{Y}.
\end{cases}
\end{eqnarray*}
\end{proposition}

\begin{theorem}\label{theorem4.7}
Let $\mathfrak{a_{\mathfrak{q}_i}}~(i=1,\dots,h(D))$ be the ideals associated with the quadratic forms $\mathfrak{q}_i=[A_i,B_i,C_i]$ of $\mathfrak{N}_B$.
Then
\begin{eqnarray*}
J(\mathfrak{a_{\mathfrak{q}_1}})=J(W_N(\mathfrak{a_{\mathfrak{q}_1}})) \quad \Leftrightarrow \quad J(\mathfrak{a_{\mathfrak{q}_i}})=J(W_N(\mathfrak{a_{\mathfrak{q}_i}}))
\quad (i=1,\dots,h(D)).
\end{eqnarray*}
\end{theorem}

\begin{proof}
We set $\mathfrak{a}_{B_i}=[N,\frac{-B_i+\sqrt{D}}{2}]$.
Since $B_1  \equiv B_i \pmod{2N}$, we know
$$
\mathfrak{a}_{B_i}=[N,\frac{-B_i+\sqrt{D}}{2}]=\mathfrak{a}_{B_1}.
$$
Consequently, by Lemma~\ref{lemma_a_b}
\begin{eqnarray*}
J(\mathfrak{a}_1)=J(W_N(\mathfrak{a}_1))
 \Leftrightarrow \mathfrak{a}_{1B} \sim (1) \Leftrightarrow \mathfrak{a}_{iB} \sim (1)  \Leftrightarrow J(\mathfrak{a}_i)=J(W_N(\mathfrak{a}_i)).
\end{eqnarray*}
\end{proof}

\begin{corollary}\label{coro5}
Let $\ell$ be a prime number which splits completely in $K_f$. Let $B$ be an integer such that there exist $u$ and $v$
 satisfied with \eqref{con1}. Further let $\mathfrak{N}_B$ be the $N$-system determined by $B$. Then for any $\mathfrak{q} \in \mathfrak{N}_B$,
the polynomial $\Phi_{p_1,p_2}(X,J)$ has a multiple root $\overline{J(\mathfrak{a}_\mathfrak{q})}$ over $\mathbb F_\ell$.
\end{corollary}
\begin{proof}
Our assertion follows from Proposition~\ref{prop2} and Theorem~\ref{theorem4.7}.
\end{proof}

Finally, we give a result for the decomposition of $\Phi_{p_1,p_2}(X,J(\mathfrak{a}))$ for an ideal $\mathfrak{a}$ over finite fields.
\begin{proposition} Assume that $p_1$ and $p_2$ satisfy the condition 1 of Corollary~\ref{Cor2.4}. Let $\ell$ be a prime number which splits completely in $K_f$. Let $\mathfrak{a}$ be an ideal of $\mathcal{O}_f$. Then the polynomial $\Phi_{p_1,p_2}(X,J(\mathfrak{a})) \mod \ell$ has at least four linear factors over $\mathbb F_\ell$.
\end{proposition}
\begin{proof}
By Lemmas~\ref{lemma1} and \ref{lemma2}, we know in the ideal class of $\mathfrak{a}$ there exist four forms $\mathfrak{q}_B$ of $N$-systems $\mathfrak{N}_B$ for four distinct $B \pmod {2N},B^2\equiv D\pmod{4N}~$. By Theorem~\ref{theorem2.3}, we have $\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_B})$ are integers of $K_f$. Therefore we have our assertion.
\end{proof}

\section{Example}\label{section5}
We give an example for the result given in Corollary~\ref{coro5}.
Let $D=-56$ and $N=39$. The integer $B=10$ satisfies $B^2\equiv D\pmod{4N}$. Consider the $N$-system $\mathfrak{N}_B$. For the integer $B$, there exist $u,v \in \mathbb{Z}$ that satisfy \eqref{con1}. For instance, we can take $u=10,v=1$.

The modular equation $\Phi_{3,13}(X,J)$ and the class polynomial $H_{B,N}(X)$ are given as follows.

{\footnotesize
$
\begin{array}{l}
\Phi_{3,13}(X,J) =
X^{56}+(704-J)X^{55}+(168568+39J)X^{54}+(14498520-663J)X^{53} \\
\hspace{62pt} +(187807764+6331J)X^{52}+(744637296-35763J)X^{51} \\
\hspace{62pt} +(-6562036+106392J)X^{50}+(-3840625568-18070J)X^{49} \\
\hspace{62pt} +(1058251610-1082016J)X^{48}+(10302034600+3516903J)X^{47} \\
\hspace{62pt} +(4510900472-1278901J)X^{46}+(-34331690432-18277116J)X^{45} \\
\hspace{62pt} +(-7097865034+40532700J)X^{44}+(84188024320+11574823J)X^{43} \\
\hspace{62pt} +(546780176-161476962J)X^{42}+(-154959173464+168751479J)X^{41} \\
\hspace{62pt} +(-12359340101+230086922J)X^{40}+(327081484064-617987682J)X^{39} \\
\hspace{62pt} +(-49301838300+137626281J)X^{38}+(-576339027576+928366231J)X^{37} \\
\hspace{62pt} +(284363953068-959457720J)X^{36}+(735938431592-477589944J)X^{35} \\
\hspace{62pt} +(-558265224452+1429130144J)X^{34}+(-890017323520-466517064J)X^{33} \\
\hspace{62pt} +(977815434427-963208272J)X^{32}+(966995235128+909996295J)X^{31} \\
\hspace{62pt} +(-1755072840368+158515461J)X^{30}+(-345165085024-607329720J)X^{29} \\
\hspace{62pt} +(2218368968890+197238236J)X^{28}+(-911733108784+179445279J)X^{27} \\
\hspace{62pt} +(-1540031876048-140684622J)X^{26}+(1628026178168-6888479J)X^{25} \\
\end{array}
$
}

{\footnotesize
$
\begin{array}{l}
\hspace{62pt} +(261124933147+37909092J)X^{24}+(-1229692547200-8835450J)X^{23} \\
\hspace{62pt} +(462040501468-4070053J)X^{22}+(441029439032+1885689J)X^{21} \\
\hspace{62pt} +(-422841966612+44928J)X^{20}+(-7261052136-111436J)X^{19} \\
\hspace{62pt} +(163453863300+9516J)X^{18}+(-59787354976+740J)X^{17} \\
\hspace{62pt} +(J^2-1486J-26470898021)X^{16}+(24009911816-49J)X^{15} \\
\hspace{62pt} +(-1731574864+29J)X^{14}+(-3926472080+246J)X^{13} \\
\hspace{62pt} +(1333660406-364J)X^{12}+(158103088-221J)X^{11} \\
\hspace{62pt} +(-172600168+650J)X^{10}+(25597000-221J)X^9 \\
\hspace{62pt} +(5195450-364J)X^8+(-2155088+247J)X^7+(177164+26J)X^6 \\
\hspace{62pt} +(39936-52J)X^5+(-9996+13J)X^4+(600-J)X^3+88X^2-16X+1,
\end{array}
$
}

$
H_{B,N}(X)=X^4-2X^3-X^2+2X-1.
$

 It is noted the degree of the $\Phi_{3,13}(X,J)$ in $J$ is 2. We take a prime number $\ell=3593$, which splits completely in $K_1$. Then $H_{B,N}(X)$ decomposes into linear factors over $\mathbb F_\ell$ as follows.

$H_{B,N}(X)\equiv (X-607)(X-166)(X-3428)(X-2987) \pmod{3593}$.
By substituting the roots of $H_{B,N}(X) \pmod{3593}$ into  $\Phi_{3,13}(X,J)$, we have
\begin{eqnarray*}
\Phi_{3,13}(607,J) & \equiv & (J-229)^2 \pmod{3593}, \\
\Phi_{3,13}(166,J) & \equiv & (J-2979)^2 \pmod{3593}, \\
\Phi_{3,13}(3428,J) & \equiv & (J-2874)^2 \pmod{3593}, \\
\Phi_{3,13}(2987,J) & \equiv & (J-2696)^2 \pmod{3593}.
\end{eqnarray*}

Let $D(X)$ be the discriminant of $\Phi_{3,13}(X,J)$ as a polynomial in $J$. Then we see $D(X) \equiv 0 \bmod H_{B,N}(X)$. This means that $\Phi_{3,13}(\mathfrak{w}_{p_{1},p_{2}}^{s}(\alpha_{\mathfrak{q}_i}),J)$ has still a multiple root for every form $\mathfrak{q}_i\in\mathfrak{N}_B$. Thus we also have $J(\mathfrak{a_{\mathfrak{q}_i}})=J(W_N(\mathfrak{a_{\mathfrak{q}_i}}))$ for every $i$.

redpajama_set_name: RedPajamaArXiv


text:
Tag: launch
28ft historic wooden boat for sale (or free to someone who will move it quickly)
Fellow musician and boat nut Alan Lamb is looking for someone interested in taking on a Royal Navy launch – he has found that he will be unable to use it for his original purpose.
If you're interested, email me at gmatkin@gmail.com and I'll put you in touch.
The 28ft, 10ft beam double diagonal construction mahogany on oak launch comes with a 68hp diesel engine. The boat was originally the launch for the Royal Fleet Auxiliary salvage vessel Kinbrace. It was built in 1945 and saw service during the Suez Crisis in 1956 and with the UN in Somalia in 1994.
The mother ship was broken up in 2004 and the launch was used for mackerel fishing out of Pool Harbour for some years.
He bought partly as a piece of experimental archaeology: I had moved to a cottage on the banks of the upper Severn near Bridgnorth. Knowing that much larger boats used the river until about 1900 (although they often had to wait till the water was high enough) I intended to restore the boat and see if it was sometimes still possible to use a boat of this size on the river – the boat has a fairly shallow draft and a large engine.
However, as I have got to know the river better over the last few years I realised that the fish weirs that used to be common on the river increased the depth of the pools.These fish weirs and the barge gutters that bypassed them were crucial to the former navigation by large boats, so I had to abandon my plans.
The planking seems sound and in general it seems a strong boat but there is some rot at the top of some of the ribs. A poor quality plywood and fibreglass deck was added at some stage and this would need replacing as would the steel sheathing on the keel timber. When I bought the boat two years ago I was told that it floated and that the engine was in working order but I have not tested either of these claims. The engine is out of the boat at present
The boat is on blocks outside building eight, The Royal Ordnance depot, Weedon, Northants and can be viewed at any time. It has to be moved as soon as possible and in view of this any reasonable or even unreasonable offer will be accepted. The new owner would be responsible for moving the boat and should make their own enquiries about haulage costs, if necessary.
Author Gavin AtkinPosted on July 17, 2017 July 17, 2017 Categories Equipment and boats for sale, Restoration and repair, wooden boat, Working boatsTags launch, navy, restoration project1 Comment on 28ft historic wooden boat for sale (or free to someone who will move it quickly)
BBA student launch, December 2012
This is photographic evidence that this year's winter student launch at the Boat Building Academy really did manage to take place in decent bright, if chilly, weather despite the variable autumn we've had.
Apart from the students and staff, the shots show the launch of an Iain Oughtred-designed Fulmar named Florence after the builder's daughter born a week ago and also a Humble Bee pram dinghy also designed by Oughtred.
Eight boats built by the class of March 2012 hit the water, and details of the boats and the students who built them are here, and there are more photos here.
The BBC and ITV were in attendance, and their reports are here (the launch report starts about 21 minutes in) and here.
Author Gavin AtkinPosted on December 6, 2012 December 7, 2012 Categories Boatbuilders and restorers, Boating, boats, ships and the sea, Locations, Modern boatbuilding, Techniques, wooden boatTags 2012, boat building, boat building academy, boat plans, boatbuilding, iain oughtred, launch, studentLeave a comment on BBA student launch, December 2012
Boat Building Academy students build a good looking Joel White-designed Haven 12 1/2
This lovely-looking shiny new boat is a Joel White-designed Haven 12 1/2 built in epoxy glass and foam with a traditional fit-out and spars. She was made by three Boat Building Academy students, Gary Thompson, Yoann Henric and Nick Lawther (see second photo).
Sadly the Haven 12 1/2 was one of two boats that did not make the water on the students' summer launch day – Lyme's harbour master decided the wind was a bit too frisky.
Gary is at the far left in the second photo; he is to start work with Wessex Resin's technical expert David Johnson; Yoann in the centre of the photo has returned to France to find work near Marseille, and Nick plans to set up a small boatyard in Australia in a few months.
PS – On launch day boat builder, occasional instructor, very good friend of the academy and former Royal Marines sargeant major Roy Gollop donned his bowler hat and did the honours in getting the twelve boats organised and down to the harbour with something approaching military promptness.
Boat builders? Military promptness? It sounds marvellous if slightly implausible, but a bowler-hatted Mr Gollop is probably not someone to argue with!
Author Gavin AtkinPosted on July 7, 2011 July 7, 2011 Categories Boatbuilders and restorers, Boating, boats, ships and the sea, Designs and designers, boat plans,books of plans, Events, Locations, Modern boatbuilding, Sailing boat, Small boats, Suppliers, TechniquesTags boat, boat building academy, epoxy, Gary Thompson, Joel White Haven 12 1/2, launch, Nick Lawther, students, work, Yoann Henric1 Comment on Boat Building Academy students build a good looking Joel White-designed Haven 12 1/2

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction} \label{sec:intro}

Journalistic organisations, such as \textit{Media Bias/Fact Check},\footnote{\url{http://mediabiasfactcheck.com/}} provide reports on news sources highlighting the ones that are propagandistic.
Obviously, such analysis is time-consuming and possibly biased and it cannot be applied to the enormous amount of news that flood social media and the Internet.
Research on detecting propaganda has focused primarily on classifying entire articles as propagandistic/non-propagandistic~\cite{AAAI2019:proppy,BARRONCEDENO20191849,rashkin-EtAl:2017:EMNLP2017}.
Such learning systems are trained using gold labels obtained by transferring the label of the media source, as per \textit{Media Bias/Fact Check} judgment, to each of its articles. Such distant supervision setting inevitably introduces noise in the learning process \cite{Horne2018} and the resulting systems tend to lack explainability.

We argue that in order to study propaganda in a sound and reliable way, we need to rely on high-quality trusted professional annotations and
it is best to do so at the fragment level, targeting specific techniques rather than using a label for an entire document or an entire news outlet.
Therefore, we propose a novel task: identifying specific instances of propaganda techniques used within an article.
In particular, we design a novel multi-granularity neural network, and we show that it outperforms several strong BERT-based baselines.

\noindent Our corpus could enable research in propagandistic and non-objective news, including the development of explainable AI systems. A system that can detect instances of use of specific propagandistic techniques would be able to make it explicit to the users why a given article was predicted to be propagandistic. It could also help train the users to spot the use of such techniques in the news.

\section{Corpus Annotated with Propaganda Techniques} \label{sec:data}

We retrieved 451 news articles from 48 news outlets, both propagandistic and non-propagandistic according to \textit{Media Bias/Fact Check}, which professionals annotators\footnote{\url{http://www.aiidatapro.com}. The company performs professional annotations in the NLP domain, although they were not expert in propaganda techniques before this work.} annotated according to eighteen  persuasion techniques~\cite{Miller},
ranging from leveraging on the emotions of the audience ---such as using \textit{loaded language} or \emph{appeal to authority}~\cite{Goodwin2011} and slogans~\cite{As2015}--- to using logical fallacies ---such as \textit{straw men}~\cite{Walton1996} (misrepresenting someone's opinion), hidden \textit{ad-hominem fallacies}, and \textit{red herring}~\cite[p.~78]{Weston2000} (presenting irrelevant data).\footnote{For a complete list see \url{http://propaganda.qcri.org/annotations/definitions.html}}
Some of these techniques weren studied in tasks such as hate speech detection and computational argumentation \cite{habernal2018}.

The total number of technique instances found in the articles, after the consolidation phase, is $7,485$, out of a total number of $21,230$ sentences (35.2\%).
The distribution of the techniques in the corpus is also uneven: while there are $2,547$ occurrences of  \textit{loaded language}, there are only $15$ instances of \textit{straw man} (more statistics about the corpus can be found in~\cite{EMNLP19DaSanMartino}).
We define two tasks based on the corpus described in Section~\ref{sec:data}: ({\em i})~ \textbf{SLC  (Sentence-level Classification)}, which asks to predict whether a sentence contains at least one propaganda technique, and ({\em ii})~ \textbf{FLC  (Fragment-level classification)}, which asks to identify both the spans and the type of propaganda technique.
Note that these two tasks are of different granularity, $g_1$ and $g_2$, namely tokens for FLC  and sentences for SLC.
We split the corpus into training, development and test, each containing 293, 57, 101 articles and 14,857, 2,108, 4,265 sentences, respectively.

Our task requires specific evaluation measures that give credit for partial overlaps of fragments.
Thus, in our precision and recall versions,
we give partial credit to imperfect matches at the character level, as in plagiarism detection~\citep{Potthast2010a}.

Let $s$ and $t$ be two fragments, i.e., sequences of characters. We measure the overlap of two annotated fragments as
$    C(s,t,h) = \frac{|(s\cap t)|}{h}\delta\left(l(s), l(t) \right)$,
where $h$ is a normalizing factor, $l(a)$ is the labelling of fragment $a$, and $\delta(a,b)=1$ if $a=b$, and $0$ otherwise.

We now define variants of precision and recall able to account for the imbalance in the corpus:
\begin{equation}
P(S,T) = \frac{1}{|S|}\!\sum_{\begin{minipage}[t]{0.97cm}\footnotesize $s\in S,\\$ $t\in T$\end{minipage}}\!\!\! C(s,t,|s|), \hspace{1cm}
R(S,T) = \frac{1}{|T|}\!\sum_{\begin{minipage}[t]{0.97cm}\footnotesize $s\in S,\\$ $t\in T$\end{minipage}}\!\!\! C(s,t,|t|),
\label{eq:precisiontask3}
\end{equation}
In eq.~\eqref{eq:precisiontask3}, we define $P(S,T)$ to be zero if $|S|=0$ and $R(S,T)$ to be zero if $|T|=0$.
Finally, we compute the harmonic mean of precision and recall in Eq.~\eqref{eq:precisiontask3} and we obtain an F$_1$-measure.
Having a separate function $C$ for comparing two annotations gives us additional flexibility compared to standard NER measures that operate at the token/character level, e.g.,~we can change the factor that gives credit for partial overlaps and be more forgiving when only a few characters are wrong.

\section{Models}
\label{sec:baselines}

We depart from BERT~\citep{devlin2018bert}, and we design three baselines.

\textbf{BERT.} We add a linear layer on top of BERT and we fine-tune it, as suggested in~\cite{devlin2018bert}. For the FLC  task, we feed the final hidden representation for each token to a layer $L_{g_2}$ that makes a 19-way classification: does this token belong to one of the eighteen propaganda techniques or to none of them (cf.\ Figure~\ref{fig:arch}-a).
For the SLC  task, we feed the final hidden representation for the special \verb![CLS]! token, which BERT uses to represent the full sentence, to a two-dimensional layer $L_{g_1}$ to make a binary classification.

\textbf{BERT-Joint.} We use the layers for both tasks in the BERT baseline, $L_{g_1}$ and $L_{g_2}$, and we train for both FLC  and SLC  jointly (cf.\ Figure~\ref{fig:arch}-b).

\textbf{BERT-Granularity.}
We modify BERT-Joint to transfer information from SLC  directly to FLC. Instead of using only the $L_{g_2}$ layer for FLC, we concatenate $L_{g_1}$ and $L_{g_2}$, and we add an extra 19-dimensional classification layer $L_{g_{1,2}}$ on top of that concatenation to perform the prediction for FLC  (cf.\ Figure~\ref{fig:arch}-c).

\textbf{Multi-Granularity Network.}
We propose a model that can drive the higher-granularity task (FLC) on the basis of the lower-granularity information (SLC), rather than simply using low-granularity information directly. Figure~\ref{fig:arch}-d shows the architecture of this model.

More generally, suppose there are $k$ tasks of increasing granularity, e.g., document-level, paragraph-level, sentence-level, word-level, subword-level, character-level.
Each task has a separate classification layer $L_{g_k}$ that receives the feature representation of the specific level of granularity $g_k$ and outputs $\boldsymbol{o}_{g_k}$.
The dimension of the representation depends on the embedding layer, while the dimension of the output depends on the number of classes in the task. The output $\boldsymbol{o}_{g_k}$ is used to generate a weight for the next granularity task $g_{k+1}$ through a trainable gate $f$:

\begin{equation}
w_{g_{k}} = f(\boldsymbol{o}_{g_k})
\end{equation}
The gate $f$ consists of a projection layer to one dimension and an activation function. The resulting weight is multiplied by each element of the output of layer $L_{g_{k+1}}$ to produce the output for task $g_{k+1}$:

\begin{equation}
\boldsymbol{o}_{g_{k+1}} = w_{g_{k}} * \boldsymbol{o}_{g_{k+1}}
\end{equation}
If $w_{g_{k}}=0$ for a given example, the output of the next granularity task $\boldsymbol{o}_{g_{k+1}}$ would be 0 as well. In our setting, this means that, if the sentence-level classifier is confident that the sentence does not contain propaganda, i.e.,~$w_{g_{k}}=0$, then $\boldsymbol{o}_{g_{k+1}}=0$ and there would be no propagandistic technique predicted for any span within that sentence.
Similarly, when back-propagating the error,
if $w_{g_{k}}=0$ for a given example, the final entropy loss would become zero, i.e.,~the model would not get any information from that example. As a result, only examples strongly classified as negative in a lower-granularity task would be ignored in the high-granularity task.
Having the lower-granularity as the main task means that higher-granularity information can be selectively used as additional information to improve the performance, but only if the example is not considered as highly negative.

\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{overview}
\caption{The architecture of the baseline models (a-c), and of our multi-granularity network (d).}
\label{fig:arch}
\end{figure}

For the loss function, we use a cross-entropy loss with sigmoid activation for every layer, except for the highest-granularity layer $L_{g_K}$, which uses a cross-entropy loss with softmax activation. Unlike softmax, which normalizes over all dimensions, the sigmoid allows each output component of layer $L_{g_k}$ to be independent from the rest. Thus, the output of the sigmoid for the positive class increases the degree of freedom by not affecting the negative class, and vice versa. As we have two tasks, we use sigmoid activation for $L_{g_1}$ and softmax activation for $L_{g_2}$. Moreover, we use a weighted sum of losses with a hyper-parameter $\alpha$:

\begin{equation}
\mathcal{L}_\mathcal{J} = \mathcal{L}_{g_1} * \alpha + \mathcal{L}_{g_2} * (1-\alpha)
\end{equation}

Again, we use BERT~\cite{devlin2018bert} for the contextualized embedding layer and we place the multi-granularity network on top of it.

\section{Experiments and Evaluation}
\label{sec:experiments}

We used the PyTorch\footnote{\url{http://pytorch.org}} framework and the pretrained BERT model,\footnote{\url{http://github.com/huggingface/pytorch-pretrained-BERT}} which we fine-tuned for our tasks.\footnote{Our source code together with the dataset are available in GitHub: \url{http://anonymous.for.review}.}
To deal with class imbalance, we give weight to the binary cross-entropy according to the proportion of positive samples. For the $\alpha$ in the joint loss function, we use 0.9 for sentence classification, and 0.1 for word-level classification.
In order to reduce the effect of random fluctuations for BERT, all the reported numbers are the average of three experimental runs with different random seeds.
As it is standard, we tune our models on the dev partition and we report results on the test partition.

The left side of Table~\ref{Table:task3} shows the performance for the three baselines and for our multi-granularity network on the FLC  task. For the latter, we vary the degree to which the gate function is applied: using ReLU is more aggressive compared to using the Sigmoid, as the ReLU outputs zero for a negative input.
Table~\ref{Table:task3} (right) shows that using additional information from the sentence-level for the token-level classification (BERT-Granularity) yields small improvements.
The multi-granularity models outperform all baselines thanks to their higher precision.
This shows the effect of the model excluding sentences that it determined to be non-propagandistic from being considered for token-level classification.

\begin{table}[t]
\centering
\begin{tabular}{@{}l@{\hspace{1mm}} c @{\hspace{1mm}}c@{\hspace{1mm}}c c@{\hspace{1mm}}c@{\hspace{1mm}}c@{}}
\toprule
 \multirow{2}{*}{Model} & \multicolumn{3}{c}{Task SLC} & \multicolumn{3}{c}{Task FLC} \\
 & P & R & F$_1$ & P & R & F$_1$ \\
 \midrule
 All-Propaganda  & 23.92 & 100.0 & 38.61 & - & -  & - \\
 BERT & \bf{63.20} & 53.16 & 57.74 & 21.48  & \bf 21.39 & 21.39\\
\,\,\, Joint      &  62.84 & 55.46 & 58.91 & 20.11 & 19.74 & 19.92\\
\,\,\, Granu      &  62.80 & 55.24 & 58.76   & 23.85 & 20.14 & 21.80\\
\multicolumn{2}{l}{\hspace{-3mm}Multi-Granularity} \\

\,\,\, ReLU    &  60.41 & \textbf{61.58} & \textbf{60.98} & 23.98 & 20.33 & 21.82\\
\,\,\, Sigmoid & 62.27 & 59.56 & 60.71 & \bf 24.42 & 21.05 & \bf 22.58\\
\bottomrule
\end{tabular}
\caption{Sentence-level (left) and fragment-level experiments (right).  \textit{All-propaganda} is a baseline that always output the propaganda class.\label{Table:task3}}
\end{table}

The right side of Table~\ref{Table:task3} shows the results for the SLC  task. We apply our multi-granularity network model to the sentence-level classification task to see its effect on low granularity when we train the model with a high granularity task. Interestingly, it yields huge performance improvements on the sentence-level classification result. Compared to the BERT baseline, it increases the recall by 8.42\%, resulting in a 3.24\% increase of the F$_1$ score. In this case, the result of token-level classification is used as additional information for the sentence-level task, and it helps to find more positive samples. This shows the opposite effect of our model compared to the FLC  task.

\section{Conclusions\label{sec:conclusions}}

We have argued for a new way to study propaganda in news media: by focusing on identifying the instances of use of specific propaganda techniques. Going at this fine-grained level can yield more reliable systems and it also makes it possible to explain to the user why an article was judged as propagandistic by an automatic system.

We experimented with a number of BERT-based models and devised a novel architecture which outperforms standard BERT-based baselines.
Our fine-grained task can complement document-level judgments, both to come out with an aggregated decision and to explain why a  document ---or an entire news outlet--- has been flagged as potentially propagandistic by an automatic system.

In future work, we plan to include more media sources, especially from non-English-speaking media and regions. We further want to extend the tool to support other propaganda techniques.

\section{Acknowledgements}

This research is part of the Propaganda Analysis Project,\footnote{\url{http://propaganda.qcri.org}} which is framed within the Tanbih project.\footnote{\url{http://tanbih.qcri.org}}
The Tanbih project aims to limit the effect of ``fake news'', propaganda, and media bias by making users aware of what they are reading, thus promoting media literacy and critical thinking. The project is developed in collaboration between the Qatar Computing Research Institute (QCRI), HBKU and the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL).


The Thinking Schools Academy Trust is shortlisted for Employer of the Year at the 2018 Schools Awards, run by education magazine TES.
Thinking Schools, which runs primary and secondary academies in the South-East, has been recognised for the progressive teacher pay structure it has developed alongside the National Education Union (NEU), transforming the way teachers are paid and helping to tackle the national challenge of teacher recruitment and retention.
The Trust will find out whether they have won the top prize at an awards ceremony on Friday 22nd June at the Grosvenor House Hotel, in London, due to be attended by more than 1,000 VIP guests, including Government ministers and the great and good from the education world.
Stuart Gardner, Chief Executive of the Thinking Schools Academy Trust, said: "We put a huge amount of work into supporting our fantastic staff, and central to this has been our development of a new pay structure to transform the way they are paid. It will properly reward teachers for their hard work, and incentivise them stay in the profession and with us – a simple yet powerful aim.
"In fostering a financially secure and motivating environment for teachers, our new pay model directly benefits students through the excellent standard of teaching they will receive. And that is ultimately what our model is all about: ensuring students enjoy a classroom experience which enables them to achieve their best.
The awards, now in their 10th year, celebrate the extraordinary commitment, quality, and innovation shown by teachers and support staff across the UK.


Swing into action with this awesome Marvel comic book framed poster which features your friendly neighbourhood Spider-Man. A cool compilation of retro comic book cartoons. Complete with a 61cm x 91.5cm black wooden frame, fronted with transparent Perspex and ready to hang in your home.
High quality, framed Black Wooden Framed Retro Spiderman Compilation, Marvel Comics framed poster with a bespoke frame by our custom framing service.


There is no end to the adventures an 18 year old boy can have at Walt Disney World. Take him to meet Mickey Mouse at the Town Square Theater on Main Street U.S.A. Let him fly over California on Soarin'. Take him to outer space on Star Tours 3D. Thrill him on Space Mountain. By him a Dole Whip.Teach him at the Hall of Presidents. Give him the chills on Haunted Mansion. Be a kid again on 'it's a small world.' Show him the world at World Pavilion. While there eat in a foreign country. Listen to some great live music in Downtown Disney. Eat a churro. Run around Tom Sawyer's island. Remind him on Spaceship Earth. Have him taste the Beverly at Club Cool. Watch Wishes. Have some Tonga Toast. Purchase him some Mickey Mouse Ears. Pin trade. Ride the Monorail loop. Stop at Disney's Contemporary Resort and spot the 5 legged goat on the mural by Mary Blair. Have a swim at Typhoon Lagoon. Take a boat to Epcot. Have a meal with Disney Characters. Eat at House of Blues. Take a ride on Summit Plummet. Take the Kilimanjaro Safari Tour. Eat at the Yak & Yeti. Watch Illuminations under the stars at Epcot. Watch the film in Norway. As you can see, I could give him great tips all day. Use these, add new ones and have the time of your life at the Happiest Place on Earth.


Link to Liturgy :: 13th Sunday :: Fast. Free. Faithful.
Jarius and the woman afflicted with hemorrhages show us the steps of Faith and how to pray, trusting in both the words and works of Christ. The faith of these two people allow life to be lived to the full. On the other hand there are those in this Gospel that ridicule Christ. This packet also includes a lesson on atheism and the cause and effect of those that have ridiculed Christ in modern history.


text:
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redpajama_set_name: RedPajamaGithub


text:
Q: Why do inline-block divs get screwed up by internal content? I dont understand why inline-block divs are losing their position due to the addition of internal content.
My CSS:
#header {
    height: 80px;
    width: 66%;
    min-width: 1024px;
    top: 0px;
    margin: 0 auto;
    background-color: rgb(128,128,128);
}
#header-left {
    position: relative;
    display: inline-block;
    height: 80px;
    width: 25%;
    left: 0px;
    top: 0px;
    background-color: rgb(128,0,0);
}
#header-center {
    position: relative;
    display: inline-block;
    height: 80px;
    width: 50%;
    background-color: rgb(0,128,0);
    text-align: center;
}
#header-right {
    position: relative;
    display: inline-block;
    height: 80px;
    width: 25%;
    background-color: rgb(0,0,128);
}

My HTML:

Hello

See Fiddle: https://jsfiddle.net/ude04hf4/
I want to add internal content and not have it affect its parent's position. The only way I have found to do this is to add float: left; to every internal element's style. There has to be a better way?

A: You need to change the vertical-align property value of the element.
By default, the vertical-align property's value is baseline, which is why the center element's text is being aligned to the previous element's baseline (notice that this issue doesn't occur when the element doesn't contain text).
If you change the vertical-align property value to something like top, it will work as expected:
Updated Example
#header-center {
    position: relative;
    display: inline-block;
    vertical-align: top;
    height: 80px;
    width: 50%;
    background-color: rgb(0,128,0);
    text-align: center;
}

redpajama_set_name: RedPajamaStackExchange


text:
Job posting: MWC Development Executive
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Young people lead barrier crossing
Posted: November 15, 2022 | Category: Testimonies
GKMI Petra-Depok, Indonesia, celebrated being a new creation on Peace Sunday , 18 September 2022, with a special guest. Sadanand Hembrom joined the service from India by video. For the last few years, GKMI Petra-Depok in greater Jakarta has been... more/más/suite
Raúl O. García
Posted: November 14, 2022 | Category: News
Raúl O. García, Mennonite World Conference president 1990-1997 died 30 October 2022. A long-term pastor, teacher and professor, he was born in Argentina 26 August 1930. more/más/suite
We speak the same language
"I cannot be grateful enough that even though we are a large and diverse group, we speak the same language: the language of love for Christ and his people," says Daniel Nugroho. He was part of the team that made it possible for all to understand. Up... more/más/suite
How can Christian values be embodied?
Posted: November 3, 2022 | Category: From Our Leaders
My encounters with Anabaptist-Mennonite brothers and sisters in South Korea shaped me. They demonstrate what it means to be a Christ follower. These believers transform their own trauma into a Christ-like life. They bear passion, peace, love and a... more/más/suite
Sermon advances mission
Posted: October 31, 2022 | Category: News
With sermons from Honduras, Canada, Indonesia and more, "the competition was great this time," says Fernando Enns, jury member for the Menno Simons Sermon Prize, an initiative of Hamburg-Altona Mennonite Church in Germany and the Centre for Peace... more/más/suite
A conversation about Scripture
"I was inspired by the radical faith of early Anabaptists, but I don't want us to be stuck in the 16th century… Anabaptism now has a culturally diverse body," says Hyejung Yum, co-founder of Sowing for Peace, an intercultural peace ministry based in... more/más/suite
"Thank you for carrying us in your hearts," says Siaka Traore, MWC regional representative for Central and West Africa. Mennonite World Conference members are invited to gather together Online Prayer Hour, 18 November 2022. This event will briefly... more/más/suite
God's mission in Anabaptist communities
Posted: October 15, 2022 | Category: From Our Leaders
"The relationship of Anabaptism and mission is a hot topic, and the field continues to expand to include a number of disciplines and sub-disciplines emerging which attempt to integrate a vision that is both missional and faithful to the Anabaptist... more/más/suite
A school of love and service
"MWC was a school for me. It really helped me to know Anabaptist doctrine and understand how the church operates," says Francisca Ibanda. Known around the world as "Maman Cisca", Francisa Ibanda, along with Barbara Nkala, recently completed her... more/más/suite
Mesach Krisetya
Mesach Krisetya, Mennonite World Conference (MWC) president 1997-2003, died 30 September 2022. He was born in Jepara, Indonesia, 9 September 1939. A theologian, professor and academic administrator, author and counsellor, Mesach Krisetya was beloved... more/más/suite
Ministry partner update: ICOMB – October 2022
Posted: October 6, 2022 | Category: News
We ask for your prayers dear brothers, that God may allow us to continue growing in the ministry in which we are currently working, that God may always provide what is necessary in terms of economic needs and also that in this month of October 2022... more/más/suite
A journey of understanding
Posted: October 5, 2022 | Category: From Our Leaders
It was with great anticipation that I went to attend a Brethren In Christ Church (BICC) conference in Mozambique, in July 2016. This was my first trip as a regional representative for Mennonite World Conference. It was held in a small town called... more/más/suite
"We felt the connection"
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"I think this type of 'hybrid' connectedness has great potential for strengthening the communion of Mennonite-related churches around the world," says Ray Brubacher. more/más/suite
Believers bring water for drought baptism
In the drought-prone Borena region of Ethiopia, believers saved water to hold a baptism for 120 new believers. The area in Southern Ethiopia currently has a severe water shortage. Believers excavated land and lined it with plastic to preserve... more/más/suite
Watch for it!
"It was nice to hear something from Indonesia and take part a little bit," says Erwin Röthlisberger of Evangelische Mennoniten-Gemeinde Bern, Switzerland. He attended Assembly from his home in Europe. Videos from MWC's Assembly Indonesia 2022 will... more/más/suite
An opportunity to cross barriers
"Jesus is our hope: even if we go through the valley of the shadow of death, he is by our side." In the Anabaptist World Fellowship Sunday materials for 2023, meet the Christian believer in Burkina Faso who declares this faith despite challenges –... more/más/suite
Ministry partner update: ICOMB – September 2022
IEU has 16 local churches. Serving the churches, due to the distance between them, is a bit difficult for the conference. For this reason, the conference in Panama works by region. They have five regions and each region has one or two coordinators... more/más/suite
Fostering relationships
Posted: September 8, 2022 | Category: News
"We want our regional representatives to develop deep relationships with each member church in their region," says Arli Klassen, regional representatives coordinator. MWC's regional representatives are part-time volunteers who develop and support... more/más/suite
Time to correct the page
Posted: August 29, 2022 | Category: From Our Leaders
"Indigenous solidarity hits the heart of what we do in the Philippines with Coffee for Peace," says Joji Pantoja, chair of the Peace Commission (2015-2022). The Peace Commission drafted a Statement of Solidarity with Indigenous Peoples that was... more/más/suite
Jesus' message of resilience and liberation
Posted: August 26, 2022 | Category: Testimonies
Reading: Matthew 5:3-20 In June 1981, our family moved to Cochabamba, Bolivia, where my parents were to teach in a Baptist seminary that wanted more Anabaptist input. We arrived at a particularly raucous point in Bolivian history. In July 1980, Luis... more/más/suite


In Unit 3, students move from analyzing challenges faced by others, to learning challenges that they face, specifically with reading. This is framed with the book More Than Anything Else by Marie Bradby, which describes the reading challenges Booker T. Washington faced. Students hear the whole text read aloud and analyze in detail an excerpt of text that is rich in figurative language and describes the challenges Booker faced in detail. For a mid-unit assessment, students demonstrate their writing skills by writing an informative paragraph recounting Booker's story from More Than Anything Else and the lesson they learned through the challenges faced and how those challenges were overcome.
How can I overcome learning challenges?
I can overcome learning challenges by being an effective learner: taking initiative and responsibility, persevering, and collaborating.
Task: Students write an informative reading contract identifying two personal reading challenges and two strategies for overcoming each of those challenges. They then create a reading strategies bookmark outlining the information in their reading contract.
The ability to read and comprehend text is the heart of literacy instruction. Comprehension is taught, reinforced, and assessed across both components of this curriculum: module lessons and the Additional Language and Literacy block. Refer to the 3M1 Module Overview for additional information.
Prioritizing lessons for classrooms with many ELLs: Consider prioritizing and expanding instruction in Lessons 1 and 2 to support students in preparation for the mid-unit assessment. Students may benefit from an additional review of informational essay structure. The content in More Than Anything Else includes serious and challenging subject matter, which may require additional time as well. Be sure to complete the Language Dives in Lessons 2 and 5. Consider placing less focus and condensing instruction in Lessons 11, 13, and 14.
Language Dives: All students participate in their first full Language Dive in Lesson 2, embedded in a close read. ELLs participate in Language Dives in Lessons 2 and 5. Most lessons also offer optional Mini Language Dives for ELLs. Language Dives are guided conversations about the meaning of a sentence from the central texts, models, or learning targets. The conversation invites students to unpack complex syntax, or "academic phrases," as a necessary component of building both literacy and habits of mind. Students then apply their understanding of language structure as they work toward the assessments and performance task. All Language Dives follow a Deconstruct-Reconstruct-Practice routine, in which students discuss and play with the meaning and purpose of the sentence and each chunk of the sentence; put the chunks back together into the original order and any possible variations; and practice using the chunks in their own speaking and writing. To maximize language practice and accommodate time, consider dividing or reviewing each Language Dive over multiple lessons. A consistent Language Dive routine is critical in helping all students learn how to decipher complex sentences and write their own. In addition, Language Dive conversations can hasten overall English language development for ELLs. Avoid using the Language Dive Guide to lecture about grammar; the Guide is designed to prompt students as they grapple with the meaning and purpose of the chunks and the sentence. Consider providing students with a Language Dive log inside a folder to track Language Dive sentences and structures and collate Language Dive note-catchers. Assure students that this log will not be graded; however, consider inviting students to use their log and note-catchers to gauge the progress of their speaking and writing skills. For more information on Language Dives and supporting English language learners, please see the Tools page.
Goal 2 Conversation Cues: Encourage productive and equitable conversation with Conversation Cues, which are questions teachers can ask students to help achieve four goals: (Goal 1) encourage all students to talk and be understood; (Goal 2) listen carefully to one another and seek to understand; (Goal 3) deepen thinking; and (Goal 4) think with others to expand the conversation (adapted from Michaels, Sarah and O'Connor, Cathy. Talk Science Primer. Cambridge, MA: TERC, 2012. Based on Chapin, S., O'Connor, C., and Anderson, N. . Classroom Discussions: Using Math Talk to Help Students Learn, Grades K–6. Second Edition. Sausalito, CA: Math Solutions Publications). Refer to the Tools page for the complete set of cues. Goal 2 Conversation Cues are introduced in Lesson 1. Heightened language processing and development is a primary potential benefit for ELLs.
Diversity and inclusion: Investigate the routines, practices, rituals, beliefs, norms, and experiences that are important to ELLs and their families. Integrate this background into the classroom as students explore informational texts taking place across different cultures and countries. Because this unit explores issues related to slavery, oppression, and learning challenges, it is important to be particularly sensitive to students' experiences. Some may have similar challenges as the characters in the texts, such as low self-confidence or exhaustion from demanding family responsibilities. It is also possible that some students have been enslaved. Remain inclusive and create space for students to express their feelings about sensitive issues, while being aware that they may experience some trauma or social stigma. Consult with a guidance counselor, school social worker, or ESL teacher for further investigation of diversity and inclusion concerns.
Ensure families are aware of the sensitive content of More Than Anything Else by Marie Bradby and prepare students who may be affected by this context in advance.
Ensure you have access to the painting materials required for The Painted Essay® in Lesson 5.
Prepare technology necessary to record the audiobooks in in Lesson 10 (e.g., a device to record audio or video).
Consider enlisting an art or technology teacher to support work on the reading strategies bookmark in Lessons 11 and 13.
Gather craft materials for students to create bookmarks in Lesson 13 (e.g., colored cardstock, ribbons, paint, colored paper, glue, ribbon, stickers, etc.).
Speech to Text - To create writing by speaking: Students create written work by speaking using Speech to Text. Dragon Dictation. Many newer devices already have this capability; there are also free apps for this purpose, including Dragon Dictation.
Additional Work with Complex Text: Students read and analyze the Writing Contract: Teacher Model used in module lessons to prepare them to write their own reading contract.
Reading and Speaking Fluency/GUM: Students reread the excerpt of More Than Anything Else closely read in module lessons for fluency and accuracy.
Writing Fluency: Students build writing stamina by either free writing or by responding to a prompt.
Word Study and Vocabulary: Students use their vocabulary logs to analyze two words relevant to the module content.


text:
Septoria slaptoniensis är en svampart som beskrevs av D. Hawksw. & Punith. 1973. Septoria slaptoniensis ingår i släktet Septoria och familjen Mycosphaerellaceae.   Inga underarter finns listade i Catalogue of Life.

Källor

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slaptoniensis

redpajama_set_name: RedPajamaWikipedia


text:
Solution Focused Brief Therapy (SFBT): A Complete Guide
Clinically Reviewed by Elizabeth Keohan, LCSW-C
Written byWendy Wisner
Updated on 12/9/2021
When many of us think of therapy, we think of sitting on a therapist's couch, pouring our heart out, digging deep into our past, and trying to understand our motivations and emotional coping mechanisms. We think of therapy as a process that will take many months, if not years.
Yet not all therapy is like that, and one method in particular encourages us to think more about the future than that past, and isn't meant to drag on for years at time. Enter solution focused therapy (SFT), or solution -focused brief therapy (SFBT session).
What is Solution Focused Therapy?
So what exactly is solution focused brief therapy, or the SFBT approach, and how can it help you lead a happier, more balanced life?
Effective solutions to your problems are the bedrock of solution focused therapy. It's a goal-oriented therapy that focuses on what needs to change in your life in order for you to live more peacefully and with fewer stressors. Your problems are addressed, but the problems themselves aren't deeply scrutinized; rather, forward-thinking methods of improvement are discussed.
Besides taking a more goal-oriented approach, solution focused therapy assumes that you know yourself best, that the possible solution to your problems lies within, and simply needs to be teased out by a skillful therapist. Solution based therapy offers a much-needed breath of hope and positivity for many people seeking care.
The History Of Solution Focused Therapy
Solution focused therapy has been around since the last 1970s and early 1980s. The method was developed by two psychotherapists, Insoo Kim Berg and Steve deShazer, along with their colleagues at the Brief Family Therapy Center in Milwaukee, Wisconsin. The team was looking for short-term, effective solutions to their patient's problems, with a practical, solution-based approach.
What Does The Research Say About Solution Focused Therapy?
Since its inception, solution focused therapy has been studied extensively, and has proven to be effective. For example, there have been 150 randomized clinical trials studying solution focused therapy. Overall, it was found to be effective, and produced similar results as other popular therapy methods, including Cognitive-Behavioral Therapy and Interpersonal Psychotherapy.
Is Solution Focused Therapy Effective?
How Does Solution Focused Therapy Work?
Solution focused therapy is similar to other therapy methods in that you generally sit with a therapist one-on-one to do the work. These days, virtual and online options are also available for solution focused therapy. The difference between solution focused therapy and other methods is where the focus lies, and what is discussed during a typical therapy session.
Who Should Seek Solution Based Therapy?
Someone who is experiencing a serious mental health issue, such as anxiety, depression, or bipolar disorder, should not necessarily seek solution based therapy alone. Solution based therapists do not generally diagnose mental health issues and cannot cure these bigger underlying problems. However, when used in conjunction with other therapies, solution based therapy can be helpful for anyone who is experiencing stress in life.
People who may benefit from solution focused therapy include:
Children experiencing behavior problems
Children experiencing problems at school
Families with dysfunctional dynamics
People who are battling addiction
Couples who are having relationship issues
What Issues Are Treated With Solution Based Therapy?
Serious ongoing mental health struggles should not be treated solely with solution based therapy methods. But solution based therapy can address many common issues that people struggle with, including relationship issues, issues at school, issues in the family, and workplace issues.
In general, if you have a hunch of what the solution to your problem might be, but you aren't sure — or if you feel you need someone to help you sort out your options — solution based therapy might be the right choice for you. A therapist who practices solution based therapy techniques will help you become the "expert" on your own life, and help you determine solutions to your problems hidden within yourself.
"In solution focused brief therapy you can work toward your goals by focusing on what is going well and applying that to other concerns in your life. Being able to examine what is working helps move you forward in a practical way while at the same time addressing any barriers to achieving your goals."
Talkspace therapist Jill Daino, LCSW-R, BC-TMH
Often, a solution based therapy approach will be used in conjunction with other methods — such as cognitive behavioral therapy, or traditional psychotherapy — though it can be used on its own as well.
What Techniques Do Solution-Focused Therapists Use?
A solution -focused therapy session will likely begin with a series of questions. These questions may be about what you are struggling with, what you currently see as possible solutions to your problems, and what has worked for you in the past as you have solved other problems.
Your therapist will not tell you what to do, but will help you organize your thoughts on what solutions may work for you and guide you in feeling confident about your own instincts. You and your therapist will work to zero in on = your specific goal to solve the problem. At times, you and your therapist will make concrete plans for how your specific goal will be accomplished.
What To Look For In A Solution Focused Therapist
Many therapists have training in solution focused techniques, along with other therapy modalities. When looking for a solution focused therapists, you can ask about how they incorporate solution focused therapy into their practice, what therapy will look like, and how long you may take to complete the therapy.
In addition, you will want to look for the features of any reputable therapist, including licensure in your state, experience, education, and a warm, nurturing approach. You will be discussing personal issues with this therapist, so you want to select someone you feel comfortable opening up to. Sometimes it can take a few tries to find a therapist who "clicks" with you — that's normal and expected.
"Within any therapy relationship it is important to know that you can speak up about your needs and goals especially as they may change over time. Many therapists are trained in several modalities and can incorporate them into their work with you, don't hesitate to ask."
Traditional psychotherapy, or psychodynamic therapy, where you spend years delving into your childhood, trying to uncover and understand your hidden pain and fears, most definitely has a place, and can have real value for many of us. But sometimes you need a more practical, forward-thinking approach for dealing with life's struggles and stressors.
Solution focused therapy can offer a positive, hopeful approach to solving problems. With its emphasis on validating your own instincts, and coming up with effective solutions that work for your lifestyle and sensibilities, it might just be the breath of fresh air you need to make real and lasting changes in your life.
Murray H. Solution focused brief therapy (SFBT). Solution Focused Brief Therapy (SFBT) | Simply Psychology. https://www.simplypsychology.org/solution-focused-therapy.html. Published January 25, 2021. Accessed November 30, 2021.
What is solution-focused therapy? SolutionFocused Therapy Institute. https://solutionfocused.net/what-is-solution-focused-therapy/. Published May 5, 2021. Accessed November 30, 2021.
What does the research say about solution-focused brief therapy? SolutionFocused Therapy Institute. https://solutionfocused.net/research-in-solution-focused-therapy/. Published May 5, 2021. Accessed November 30, 2021.
Murray H. Solution focused brief therapy (SFBT). Solution Focused Brief Therapy (SFBT) | Simply Psychology. https://www.simplypsychology.org/solution-focused-therapy.html#:~:text=In%20a%20solution%2Dfocused%20therapy,might%20not%20have%20noticed%20before. Published January 25, 2021. Accessed November 30, 2021.
Wendy Wisner
Wendy Wisner is a freelance writer and lactation consultant (IBCLC) whose work has appeared on/in The Washington Post, Family Circle, ELLE, ABC News, Parents Magazine, Scary Mommy, Babble, Fit Pregnancy, Brain Child Magazine, Lilith Magazine, and elsewhere. Find her at www.wendywisner.com.
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