text:
Q: Why doesn't twill show a form? I am trying to access a website using twill: https://cuentas.itesm.mx/IDMProv/portal/cn/GuestContainerPage/ITESMPageForgotPassword
On the website code there is this field that I need to fill personID, inside the form frmChRIndex

<form id="frmChRIndex" name="frmChRIndex" action="validaUsuario.htm" method="POST">

   Usuario (A00000000 / L00000000)(*)

   <input type="text" name="personId" id="personId" value="" maxlength="9"                       onblur="javascript:this.value=this.value.toUpperCase();">

</form>

When I use the command showforms on twill, this form does not appear.
It returns these forms
  Form #1
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     login                    hidden    (None)       login
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#2)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#3)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#4)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#5)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#6)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#7)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Form name=nav__NVL354063pageLinkForm (#8)
## ## __Name__________________ __Type___ __ID________ __Value__________________
1     navUrl                   hidden    (None)       /IDMProv/portal/cn/GuestContainerPag ...
2     uasess                   hidden    (None)       -4301763981694819540

Any ideas on why is this happening?


text:
\section{Introduction to the problem}

Tilings by dominoes, or tiles of dimensions $2\times 1$, and $1\times 1$ square tiles have been widely studied. For example, it is famously true that the number of tilings of a $1\times n$ rectangle with dominoes and squares is equal to the $n+1$-th term of the Fibonacci sequence \cite{BenjaminQuinn}.

Furthermore, in a landmark paper Fisher and Temperley \cite{TemperleyFisher}, and independently in the same year Kastelyn \cite{Kastelyn}, prove an exact formula for the number of tilings of a $2m\times 2n$ rectangle by dominoes.

The main problem that will be discussed in this article is that of tiling a square border with cuisenaire rods of sizes $1$ and $2$. That is, we will use tiles of dimension $1\times 1$ and $2\times 1$ to build a square of size $n\times n$ with a square hole of size $n-2\times n-2$. Below are examples for $n=3$ and $n=4$.

\begin{figure}[h]
\centering
\begin{pspicture}(0,0)(4.5,2)
\psframe[dimen=outer](0,0)(1.5,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=gray](0.5,0.5)(1,1)

\psframe[dimen=outer](0,1)(0.5,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](0.5,1)(1.5,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](1,0)(1.5,1)
\psframe[dimen=outer](0.5,0)(1,0.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](0,0)(0.5,1)

\psframe[dimen=outer](2.5,0)(4.5,2)
\psframe[dimen=outer,fillstyle=solid,fillcolor=gray](3,0.5)(4,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](2.5,1.5)(3.5,2)
\psframe[dimen=outer](3.5,1.5)(4,2)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](4,1)(4.5,2)
\psframe[dimen=outer](4,0.5)(4.5,1)
\psframe[dimen=outer](4,0)(4.5,0.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](3,0)(4,0.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](2.5,0)(3,1)
\psframe[dimen=outer](2.5,1)(3,1.5)

\end{pspicture}
\caption{Two examples of a square border tiling}
\label{fig:tiling0}
\end{figure}

 We explore how to calculate the number of such coverings for any natural number $n$. This problem first came to the author's attention in an article written for the Association of Teachers of Mathematics \cite{Zarzycki}.

\pagebreak

\FloatBarrier

\section{Transition Matrix Approach}

First let us consider a tiling of a square border as a sequence of $1\times 1$ squares from the top left corner of the square clockwise around the border. We can imagine this sequence of squares laid out in a line from beginning to end as in figure \ref{fig:convention} below.

 As one traverses the border our orientation changes through one full revolution. To reflect the one-dimensional nature of the path we are only interested in closure to the front or back of the direction of travel. Hence square $1$ appears to be open to the bottom on the square border, but from the perspective of someone walking from $1$ to $8$, square number $1$ would be open to the rear of the direction of travel, and so appears open to the left when the squares are laid out end-to-end.

\begin{figure}[h]
\centering
\caption{Traversing the square border}
\begin{pspicture}(0,0)(10,2)
\psframe[dimen=outer](0,0)(1.5,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=gray](0.5,0.5)(1,1)

\psframe[dimen=outer,fillstyle=solid,fillcolor=red](0,0.5)(0.5,1.5)
\rput(0.25,1.25){$1$}
\psframe[dimen=outer](0.5,1)(1,1.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](1,0.5)(1.5,1.5)
\psframe[dimen=outer](1,0)(1.5,0.5)
\psframe[dimen=outer,fillstyle=solid,fillcolor=red](0,0)(1,0.5)

\rput(0.25,1.25){$1$}
\rput(0.75,1.25){$2$}
\rput(1.25,1.25){$3$}
\rput(1.25,0.75){$4$}
\rput(1.25,0.25){$5$}
\rput(0.75,0.25){$6$}
\rput(0.25,0.25){$7$}
\rput(0.25,0.75){$8$}

\psline(2.5,0)(3,0)
\psline(3,0)(3,0.5)
\psline(3,0.5)(2.5,0.5)

\psline(3.5,0)(4,0)
\psline(4,0)(4,0.5)
\psline(4,0.5)(3.5,0.5)
\psline(3.5,0)(3.5,0.5)

\psline(4.5,0)(5,0)
\psline(5,0.5)(4.5,0.5)
\psline(4.5,0)(4.5,0.5)

\psline(5.5,0)(6,0)
\psline(6,0)(6,0.5)
\psline(6,0.5)(5.5,0.5)

\psline(6.5,0)(7,0)
\psline(7,0)(7,0.5)
\psline(7,0.5)(6.5,0.5)
\psline(6.5,0)(6.5,0.5)

\psline(7.5,0)(8,0)
\psline(8,0.5)(7.5,0.5)
\psline(7.5,0)(7.5,0.5)

\psline(8.5,0)(9,0)
\psline(9,0)(9,0.5)
\psline(9,0.5)(8.5,0.5)

\psline(9.5,0)(10,0)
\psline(10,0.5)(9.5,0.5)
\psline(9.5,0)(9.5,0.5)

\rput(2.75,0.22){$1$}
\rput(3.75,0.22){$2$}
\rput(4.75,0.22){$3$}
\rput(5.75,0.22){$4$}
\rput(6.75,0.22){$5$}
\rput(7.75,0.22){$6$}
\rput(8.75,0.22){$7$}
\rput(9.75,0.22){$8$}
\end{pspicture}
\label{fig:convention}
\end{figure}

Using this convention one can identify three possible states for each square as you traverse the border. The first state is a closed $1\times 1$ square, the second is open on the left and the third open on the right. In the digraph of figure \ref{fig:transition} below, an arrow indicates that a state on the left of the arrow may precede a state to the right of the arrowhead as one traverses the border using the above convention.

\begin{figure}[h]
\centering
\caption{State transition}
\begin{pspicture}(0,0)(4,5)
\psframe[dimen=outer](0,4)(1,5)

\psline(0,2)(1,2)
\psline(1,2)(1,3)
\psline(1,3)(0,3)

\psline(0,0)(1,0)
\psline(1,1)(0,1)
\psline(0,0)(0,1)

\psframe[dimen=outer](3,4)(4,5)

\psline(3,2)(4,2)
\psline(4,2)(4,3)
\psline(4,3)(3,3)

\psline(3,0)(4,0)
\psline(4,1)(3,1)
\psline(3,0)(3,1)

\psline{->}(1.25,4.75)(2.75,4.75)
\psline{->}(1.25,4.5)(2.75,0.75)

\psline{->}(1.25,2.75)(2.75,4.5)
\psline{->}(1.25,2.5)(2.75,0.5)

\psline{->}(1.25,0.5)(2.75,2.5)

\end{pspicture}

\label{fig:transition}
\end{figure}

From this we can write down the state transition adjacency matrix as follows:

$$M=
\begin{bmatrix}
1 & 0 & 1 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}.$$

Now of course this is only for one step of the journey around the square. In our problem we are looking to enumerate certain paths in the following trellis graph of figure \ref{fig:trellis}.

\begin{figure}[h]
\centering
\caption{Trellis Graph}
\begin{pspicture}(0,0)(10,2.5)
\psframe[dimen=outer](0,2)(0.5,2.5)

\psline(0,1)(0.5,1)
\psline(0.5,1)(0.5,1.5)
\psline(0.5,1.5)(0,1.5)

\psline(0,0)(0.5,0)
\psline(0.5,0.5)(0,0.5)
\psline(0,0)(0,0.5)

\psframe[dimen=outer](1.5,2)(2,2.5)

\psline(1.5,1)(2,1)
\psline(2,1)(2,1.5)
\psline(2,1.5)(1.5,1.5)

\psline(1.5,0)(2,0)
\psline(2,0.5)(1.5,0.5)
\psline(1.5,0)(1.5,0.5)

\psline{->}(0.625,2.375)(1.375,2.375)
\psline{->}(0.625,2.25)(1.375,0.375)

\psline{->}(0.625,1.375)(1.375,2.25)
\psline{->}(0.625,1.25)(1.375,0.25)

\psline{->}(0.625,0.25)(1.375,1.25)

\psframe[dimen=outer](3,2)(3.5,2.5)

\psline(3,1)(3.5,1)
\psline(3.5,1)(3.5,1.5)
\psline(3.5,1.5)(3,1.5)

\psline(3,0)(3.5,0)
\psline(3.5,0.5)(3,0.5)
\psline(3,0)(3,0.5)

\psline{->}(2.125,2.375)(2.875,2.375)
\psline{->}(2.125,2.25)(2.875,0.375)

\psline{->}(2.125,1.375)(2.875,2.25)
\psline{->}(2.125,1.25)(2.875,0.25)

\psline{->}(2.125,0.25)(2.875,1.25)

\psframe[dimen=outer](4.5,2)(5,2.5)

\psline(4.5,1)(5,1)
\psline(5,1)(5,1.5)
\psline(5,1.5)(4.5,1.5)

\psline(4.5,0)(5,0)
\psline(5,0.5)(4.5,0.5)
\psline(4.5,0)(4.5,0.5)

\psline{->}(3.625,2.375)(4.375,2.375)
\psline{->}(3.625,2.25)(4.375,0.375)

\psline{->}(3.625,1.375)(4.375,2.25)
\psline{->}(3.625,1.25)(4.375,0.25)

\psline{->}(3.625,0.25)(4.375,1.25)

\psframe[dimen=outer](6,2)(6.5,2.5)

\psline(6,1)(6.5,1)
\psline(6.5,1)(6.5,1.5)
\psline(6.5,1.5)(6,1.5)

\psline(6,0)(6.5,0)
\psline(6.5,0.5)(6,0.5)
\psline(6,0)(6,0.5)

\psline{->}(5.125,2.375)(5.875,2.375)
\psline{->}(5.125,2.25)(5.875,0.375)

\psline{->}(5.125,1.375)(5.875,2.25)
\psline{->}(5.125,1.25)(5.875,0.25)

\psline{->}(5.125,0.25)(5.875,1.25)

\rput(7.25,1.25){$\cdots$}

\psframe[dimen=outer](8,2)(8.5,2.5)

\psline(8,1)(8.5,1)
\psline(8.5,1)(8.5,1.5)
\psline(8.5,1.5)(8,1.5)

\psline(8,0)(8.5,0)
\psline(8.5,0.5)(8,0.5)
\psline(8,0)(8,0.5)

\psframe[dimen=outer](9.5,2)(10,2.5)

\psline(9.5,1)(10,1)
\psline(10,1)(10,1.5)
\psline(10,1.5)(9.5,1.5)

\psline(9.5,0)(10,0)
\psline(10,0.5)(9.5,0.5)
\psline(9.5,0)(9.5,0.5)

\psline{->}(8.625,2.375)(9.375,2.375)
\psline{->}(8.625,2.25)(9.375,0.375)

\psline{->}(8.625,1.375)(9.375,2.25)
\psline{->}(8.625,1.25)(9.375,0.25)

\psline{->}(8.625,0.25)(9.375,1.25)
\end{pspicture}
\label{fig:trellis}
\end{figure}

A tiling will result from a path; starting in the first state ending in the first or second states; starting in the second state and ending in the third; or starting in the third state and ending in the first or second states.

Our strategy, is to raise the adjacency matrix to the required power (one less than the number of squares traversed) and total the appropriate entries in the resulting matrix.

In the following section we detail some of the calculations required to find the matrix $M$ to any given power.

\FloatBarrier
\section{Linear Algebra Calculations}

In this section we diagonalise the matrix $M$ by solving the characteristic polynomial and changing to a basis of eigenvectors. It is then an elementary matter to raise $M$ to any power.

First we compute the characteristic polynomial.
\begin{align}
| M - xI| &= \begin{vmatrix}1-x & 0 & 1 \\ 1 & -x & 1 \\ 0 & 1 & -x \end{vmatrix} \notag \\
              &= -1 \begin{vmatrix} 1-x & 1 \\ 1 & 1 \end{vmatrix} -x \begin{vmatrix} 1-x & 0 \\ 1 & -x \end{vmatrix} \notag \\
              &= x(-x^{2}+x+1) \notag
\end{align}

The matrix $M$ has $3$ distinct eigenvalues $0$, $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$.

From these we can find the corresponding eigenvectors,

$$ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} \frac{1+\sqrt{5}}{2} \\ \frac{1+\sqrt{5}}{2} \\ 1 \end{pmatrix} ,
 \begin{pmatrix} \frac{1-\sqrt{5}}{2} \\ \frac{1-\sqrt{5}}{2} \\ 1 \end{pmatrix}.$$

Letting,

$$C=
\begin{bmatrix}
1 & \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \\
0 &  \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \\
-1 & 1 & 1
\end{bmatrix},
$$
it is routine to check that,

$$C^{-1}=
\begin{bmatrix}
1 & -1 & 0 \\
\frac{5-\sqrt{5}}{10} & \frac{3\sqrt{5} - 5}{10} & \frac{5-\sqrt{5}}{10} \\
\frac{5+\sqrt{5}}{10} & -\frac{5+3\sqrt{5}}{10} & \frac{5+\sqrt{5}}{10}
\end{bmatrix}.
$$

Further that $D=C^{-1}MC$ is diagonal is a routine consequence. Conjugating by $C$, the last equation becomes $M=CDC^{-1}$.

Now we can compute $M$ to any power easily, as:

$$M^{q} = CD^{q}C^{-1}.$$

In the following section we apply these calculations to prove our main result.

\FloatBarrier
\section{Main Results}

\begin{lem}
\label{lem:count}
The number of $1\times 1$ squares in an $n\times n$ square border is equal to $4(n-1)$.
\end{lem}

\begin{proof}
The number of squares is equal to the difference of the outer square and the inner square.
\begin{align}
 & n^{2}-(n-2)^{2} \notag \\
& = n^{2}-(n^{2}- 4n+4) \notag \\
& = 4(n-1) \notag
\end{align}
\end{proof}

\begin{thm}
The number of tilings of a square border of size $n\times n$, where $n\geq 2$, by tiles of size $1\times 1$, $2\times 1$ or $1\times 2$ is equal to
$$\left( \frac{1+\sqrt{5}}{2} \right)^{4(n-1)}+  \ \left( \frac{1-\sqrt{5}}{2} \right) ^{4(n-1)}$$
\end{thm}

\begin{proof}
Firstly note that the number of state transitions is one less than the number of squares in the border. By lemma \ref{lem:count} means that we need to compute $M^{4(n-1)-1}$. Let $$M^{4(n-1)-1}=:P=(p_{ij}).$$

We will enumerate all paths starting in state $1$ and ending in state $1$ or $2$, those starting in state $2$ and ending in state $3$, or those starting in state $3$ ending in states $1$ or $2$. We will do so by finding the sum,
$$p_{11}+p_{12}+p_{23}+p_{31}+p_{32}.$$

Now we compute the matrix $P$. Recall that $M =CDC^{-1}$, hence:

$$P = CD^{4(n-1)-1}C^{-1}.$$

The latter product equals,

$$ \begin{bmatrix} 1 & \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \\0 &  \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \\-1 & 1 & 1 \end{bmatrix}
\text{diag}\begin{pmatrix} 0, & \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)-1}, & \left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)-1} \end{pmatrix}
\begin{bmatrix}1 & -1 & 0 \\ \frac{5-\sqrt{5}}{10} & \frac{3\sqrt{5} - 5}{10} & \frac{5-\sqrt{5}}{10} \\ \frac{5+\sqrt{5}}{10} & -\frac{5+3\sqrt{5}}{10} & \frac{5+\sqrt{5}}{10}
\end{bmatrix}$$

$$ = \begin{bmatrix} 0 &  \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)} & \left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)} \\ 0 &  \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)} & \left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)} \\ 0 &  \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)-1} & \left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)-1} \end{bmatrix}
\begin{bmatrix}1 & -1 & 0 \\ \left(\frac{5-\sqrt{5}}{10}\right) & \left(\frac{3\sqrt{5} - 5}{10}\right) & \left(\frac{5-\sqrt{5}}{10}\right) \\ \left(\frac{5+\sqrt{5}}{10}\right) & -\left(\frac{5+3\sqrt{5}}{10}\right) & \left(\frac{5+\sqrt{5}}{10}\right)
\end{bmatrix}$$

We now compute the entries of $P$.
\begin{align}
p_{11} &= \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{5-\sqrt{5}}{10}\right)+\left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{5+\sqrt{5}}{10}\right), \notag \\
p_{12} &= \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{3\sqrt{5}-5}{10}\right)-\left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{5+3\sqrt{5}}{10}\right), \notag \\
p_{23} &= \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{5-\sqrt{5}}{10}\right)+\left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)}\left(\frac{5+\sqrt{5}}{10}\right), \notag \\
p_{31} &= \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)-1}\left(\frac{5-\sqrt{5}}{10}\right)+\left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)-1}\left(\frac{5+\sqrt{5}}{10}\right), \notag \\
p_{32} &= \left(\frac{1+\sqrt{5}}{2}\right)^{4(n-1)-1}\left(\frac{3\sqrt{5}-5}{10}\right)-\left(\frac{1-\sqrt{5}}{2}\right)^{4(n-1)-1}\left(\frac{5+3\sqrt{5}}{10}\right). \notag
\end{align}

Finally, finding the sum of the entries above proves the claim.
\end{proof}

\begin{rem}
Checking in the $2\times 2$ case gives,
$$\left( \frac{1+\sqrt{5}}{2} \right)^{4(2-1)}+  \ \left( \frac{1-\sqrt{5}}{2} \right) ^{4(2-1)}=7.$$
Checking in the $3\times 3$ case gives,
$$\left( \frac{1+\sqrt{5}}{2} \right)^{4(3-1)}+  \ \left( \frac{1-\sqrt{5}}{2} \right) ^{4(3-1)}=47.$$
Both of these cases can be found in the Appendix in Section \ref{sec:appendix}.
\end{rem}

\begin{thm}
The number of tilings of a rectangular border of size $m \times n$, where $m,n\geq 2$, by tiles of size $1\times 1$, $2\times 1$ or $1\times 2$ is equal to
$$\left( \frac{1+\sqrt{5}}{2} \right)^{2(m+n-2)}+  \ \left( \frac{1-\sqrt{5}}{2} \right) ^{2(m+n-2)}$$
\end{thm}

\begin{proof}
Firstly note that the number of squares in a rectangular border of size $m\times n$ is equal to
\begin{align}
& mn-(m-2)(n-2) \notag \\
& = mn- (mn-2(m+n) + 4) \notag \\
& = 2(m+n-2). \notag
\end{align}
A similar computation with the matrix $D$ raised to the power $$2(m+n-2)-1$$ yields the result.
\end{proof}

\begin{rem}
It would be interesting to find similar formulae for the number of tilings of a square or rectangular border using other cuisenaire rods. A starting point might be to answer how many tilings of a $3\times 3$ square border are there using just $1\times 1$, $2\times 1$ and $3\times 1$ rods?
\end{rem}

\FloatBarrier
\section{Tilings up to symmetry}
 In general the orbits of this group action can be counted efficiently using Burnside's lemma, which states that the number of orbits is equal to the average number of fixed points,

$$|X/G| = \frac{1}{|G|} \sum_{g\in G} |X^{g}|.$$

Firstly we take into account only rotational symmetry in which case $G=C_{4}$, the cyclic group of order $4$.

\begin{thm}
\label{thm:rotational}
The number of tilings of an $n\times n$ square border, where $n\geq 2$ is an integer, by $1\times 1$ and $2\times 1$ cuisenaire rods, distinct up to rotational symmetry is given by the following formula

$$\frac{1}{4} \left[  x_{n}^{4} +(1+4(-1)^{n})x_{n}^{2}+2x_{n}+2(1+(-1)^{n}) \right].$$

Where $x_{n}=\phi^{n-1}+(-1)^{(n-1)}\phi^{-(n-1)}$ and $\phi = \frac{1+\sqrt{5}}{2}$.
\end{thm}

\begin{proof}
We apply Burnside's lemma. The factor $\frac{1}{4}$ comes from the fact that the group has order $4$. The identity fixes all square borders. A $90$-degree rotation, clockwise or anticlockwise, fixes those square borders where we are given choice over a quarter of the squares, which must match up from start to finish so that the count is the same as that for a square border with a quarter of the total number of squares. Finally, a $180$-degree rotation fixes all square borders where half the squares match the other half. One computes,

\begin{align}
 & \frac{1}{4}  \left[ \left( \frac{1+\sqrt{5}}{2} \right)^{4(n-1)}+\left( \frac{1-\sqrt{5}}{2} \right)^{4(n-1)} \right. + \notag \\
& +2\left( \left( \frac{1+\sqrt{5}}{2} \right)^{(n-1)}+\left( \frac{1-\sqrt{5}}{2} \right)^{(n-1)} \right) + \notag \\
&\left. + \left( \frac{1+\sqrt{5}}{2} \right)^{2(n-1)}+\left( \frac{1-\sqrt{5}}{2} \right)^{2(n-1)} \right] \notag
\end{align}

Substitution for $x_{n}$ as given in the statement of the theorem then yields the result.
\end{proof}

\begin{rem}
Theorem \ref{thm:rotational} for even $n$ becomes,

$$\frac{1}{4} \left[  x_{n}^{4} +5x_{n}^{2}+2x_{n}+4 \right] $$

and for odd $n$ becomes,

$$\frac{1}{4} \left[  x_{n}^{4} -3x_{n}^{2}+2x_{n} \right] .$$
\end{rem}

\pagebreak

\begin{thm}
The number of tilings of an $m\times n$ rectangular border, where $m,n\geq 2$ are integers, and $m\neq n$, by $1\times 1$ and $2\times 1$ cuisenaire rods, distinct up to rotational symmetry is given by the following formula

$$\frac{1}{2} \left[  x_{n}^{2} +x_{n} + 2(-1)^{m+n} \right].$$

Where $x_{n}=\phi^{m+n-1}+(-1)^{(m+n-1)}\phi^{-(m+n-1)}$ and $\phi = \frac{1+\sqrt{5}}{2}$.
\end{thm}
\begin{proof}
A similar computation using the rotational symmetry group of the rectangle which is $C_{2}$, the cyclic group of order $2$.
\end{proof}

Finally, taking into account reflections also, the symmetry group $G$ of the square acts on the set $X$ of all square border tilings. Now the group $G$ is the dihedral group of order $8$.

It turns out that the cases of odd and even $n$ are different and we obtain two results.

\begin{thm}
The number of tilings of an $n\times n$ square border, where $n\geq 2$ is an even integer, by $1\times 1$ and $2\times 1$ cuisenaire rods, distinct up to rotations and reflections is given by the following formula

$$\frac{1}{8} \left[ (x_{n}^{2}+2)^{2} -2 + 2x_{n} +3(x_{n}^{2}+2) + \frac{6\sqrt{5}}{5}(\phi^{(n-1)} + \phi^{-(n-1)})x_{n} \right].$$

Where $x_{n}=\phi^{n-1}-\phi^{-(n-1)}$ and $\phi = \frac{1+\sqrt{5}}{2}$.
\end{thm}
\begin{proof} Omitted.
\end{proof}

\begin{thm}
The number of tilings of an $n\times n$ square border, where $n\geq 3$ is an odd integer, by $1\times 1$ and $2\times 1$ cuisenaire rods, distinct up to rotations and reflections is given by the following formula
$$\frac{1}{8} \left[ (x_{n}^{2}-2)^{2} -2 + 2x_{n} +(x_{n}^{2}-2) + \frac{4\sqrt{5}}{5}(\phi^{(n-1)} - \phi^{-(n-1)})x_{n} \right].$$
Where $x_{n}=\phi^{n-1}+\phi^{-(n-1)}$ and $\phi = \frac{1+\sqrt{5}}{2}$.
\end{thm}
\begin{proof} Omitted.
\end{proof}

\pagebreak

\FloatBarrier
\section{Appendix}
\label{sec:appendix}
This appendix features the full list of tilings in the $2\times 2$ and $3 \times 3$ cases. Note that tilings are not considered equivalent up to symmetry.
All possible tilings of a $2\times 2$ square with $1\times 1$ and $2 \times 1$ cuisenaire rods are illustrated in figure \ref{fig:tiling1}. Those tilings of a $3 \times 3$ square border are illustrated in figure \ref{fig:tiling2}.

\begin{figure}[h]
\centering
\caption{The $7$ tilings of a $2\times 2$ square}
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\label{fig:tiling1}
\end{figure}

\begin{figure}[h]
\centering
\caption{The $47$ tilings of a $3\times 3$ square border}
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redpajama_set_name: RedPajamaArXiv


Drop shipping can be tricky precisely because it's not only the seller's nexus that comes into play, but, potentially, the supplier/shipper's as well.
As the seller, you are generally only obligated to collect sales taxes on a sale if you have nexus in the customer's state. However, several states hold a supplier with nexus in that state responsible to collect sales taxes on the sale when the seller lacks nexus, including California, Connecticut, Florida, Hawaii and others. In turn, the supplier's obligation could require you as the seller to either pay sales tax to the supplier or to come up with an exemption certificate so that you don't have to pay tax.
Drop shipping can be a successful solution for small sellers, but it can also add new layers of sales tax complexity. One way for small businesses to make the most out of drop shipping and other innovative solutions is sales tax automation software such as Avalara AvaTax. AvaTax's accuracy is 100% guaranteed, so you can be sure that you're getting sales tax right no matter how your products are getting delivered. Learn more.


The first thing to do is to find out whether the problem is with your computer's software or the hardware. You can isolate the root of the problem by doing some simple disk checks. The usual signs and symptoms that your hard drive has a problem include whirring noises that come from the computer case or a defragmentation notice that is continually reminding you of disk errors. Also, check the wiring and connections to ensure they are plugged in the right places. You can also download free hard drive diagnostic tools online to help you with the problem. While you are at it, look up sites that can help you with a external hard drive repair. Once done, you can then shut down your computer down. Then, detach the outside cover before searching for the hard drive. It is a thin square case with a disc and random cables are connected to it. You need to unsnap these tentacles before unscrewing it. Do not bend, twist or yank the drive out. Now that you have the feel of it, examine if there are loose nuts and bolts or bent areas. If you can tighten anything, then please do. Put all the pieces back together then try it in another computer or try using another computer's disk drive to troubleshoot the disk.


iCanon 粉絲簽到打卡帖 2018年4月25日-粉絲打卡簽到區-粉絲交流區-iCanon 愛佳攝影俱樂部 - Powered by Discuz!
我在 2018-04-25 00:00 完成簽到,是今天第一個簽到的用戶,獲得隨機獎勵 魔力 1537MP,另外我還額外獲得了 魔力 10MP.
GMT+8, 2019-4-25 22:29 , Processed in 0.608197 second(s), 17 queries , Apc On.


I started using K2 to create my web site but was hacking and creating too many work-arounds for my comfort. I decided to look around and discovered FLEXIContent. I am thrilled that I did! FC does everything that I wanted K2 to do, and more! Too bad I had to manually re-create all my K2 articles for FLEXIcontent!
FC documentation is a bit on light side, but I have found the forum helpful. To be honest, it is pretty intuitive, so I haven't relied too extensively on the forums but there are a few regular that offer good advice.
All in all, this is a great component and should be standard in Joomla!
I installed this after trying many different options to make myself a microblogging type platform for the frontpage.
Not a single component I tried allowed comments to appear expanded on the front page by default - including this one.
I posted this on the FLEXIcontent forum and within a few hours the dev came back to me with a detailed and VERY SIMPLE hack to achieve my goal. I tried it, and hey presto, perfect!
I cannot believe how clearly this is coded, and how attentive the developer is to his customers. Wow!
If I could give 6 stars, I would. Highly recommended. And thank you so much to the dev, who continues to excel himself every time! Simply brilliant.
I was looking for a universal directory component to able to construct (almost) all of my sites.
This has everything I wanted, multiple content types, flexible templating system, very flexible fields, multiple categories per item, etc., and Excellent access control if you combine it with FlexiAccess (I had a little trouble with it but I resolved, please read documentation for FlexiAccess although not complete).
The big bonus is that your content and categories etc are inserted to the standard Joomla tables, only the extra information about fields, etc., is stored in other DB tables.
It is an excellent directory component for the end user, and something a developer would love to use.
I hope I will be able to actively be involved in this component.
Thanks to flexicontent we can archive all requirements from every client.
You name it, FLEXIcontent has it!
And above all, the community is great, fast feedback, great support, attention to detail, and if you have a good idea for FLEXIcontent they will implement it.
Up until now, Joomla is pretty much the go-to-with when asked to build a website. No questions asked.
Now, not only I use Joomla to build pretty much all my site. But FLEXIcontent has become "THE" CCK extension I install to build the sites.
It is so flexible, it's unreal. And unlike other CCK, the templating system is so simple to play with.
I've tried other CCK extensions which can also be powerful, but either the templating system was complicated or the extensions itself was hard to set up.
The FLEXIcontent team is doing an amazing work!
FLEXIcontent has become the second extension with JCE I install right after I install Joomla!
Simple to install, simple to use, and fantastic support.
I've tried many components for creating a directory, but this really works for me.
The forum and support are great and on the odd occasion when I have been stumped have found the solution quickly.


text:
Anil and Cynthia. Photo courtesy Anil Persaud and Cynthia Jairam-Persaud
From Homework to Teamwork
For Anil Persaud and Cynthia Jairam-Persaud, meeting as first-years meant growing up together
September 21, 2015 | By Staff
Anil (BA 1999 UTSC, BEd 2002): We met for the first time when Cynthia asked a mutual friend at U of T Scarborough to introduce us. But I didn't seem to take much notice of Cynthia. I had been engrossed in a book, which she says I was just pretending to read. But then we kept bumping into each other, almost every day for the next week. Once we started talking, we just meshed. It's hard to put into words, but it was very natural. We're both Guyanese, and we hung out at West Indian Students' Connection events, played cards and dominos, studied together and hit it off. We always had fun together. We started talking about marriage after graduation. We were both adamant that we didn't want children, but we now have three so it's funny how that turned out. We're very complementary. She's a hands-on doer, and I'm more of a big-picture procrastinator. I think, over the years, we've figured out how to keep everything in perspective, which has helped us to move along our own paths and grow together.
The Force is strong with the Persaud family! Photo: Philip Dehaney.
Cynthia (BComm 1999 UTSC): We were together for seven years before we got married, so I think we already had a really clear understanding of each other. I think it helped that we were in the same stage of life – our needs and wants matched up. We've also always been good at giving each other space. We've always had our own separate group of friends, instead of just merging everyone together. We've got our own independence and distinct identities. That can be harder now, managing a family with three children, but Anil and I have similar views. We're able to balance each other's strengths and weaknesses. I'm great with organization but he's great with finances. He's very kind and generous and has a great ethical core. He has very strong opinions, yet he doesn't try to sway you. But he's always challenged me to think about things more deeply. He's always there to support me, but he also really makes me think about what I'm doing and why.
Tags: , , The Two of Us, U of T Scarborough
Surviving the 'Shopocalypse'
U of T Scarborough grad Satish Kanwar is helping small businesses thrive in a market that's gone virtual

redpajama_set_name: RedPajamaCommonCrawl


text:
SUCH SWEET SORROW
Toxic Tales from Sandhurst
While our minds immediately go to sugar and fruit flavourings when thinking of confectionery and chocolate, some ingredients were less appealing…
A Bitter End
The sweet flavour of almonds, key in the unique taste of marzipan but also many varieties of hard confectionery, fudges and chocolate liquid centres, has been popular across centuries, particularly in Europe and the Middle East. Sandhurst confectioner, Alexander Henderson, used it to make ratafia, a macaroon-like confection flavoured with the essence.
To this day, confectioners use an essence of bitter almonds in their craft but unlike their domesticated counterpart, the amara variety of the almond prunus dulcis plant produces the toxic amygdalin – hydrogen cyanide, or prussic acid – even when the kernel is crushed and distilled. The oil remaining from the cold press and distilling process is generally resuspended in a substance like ethyl alcohol but some of the acid remains and even contemporarily, it's estimated that as little as 7.5mL of almond oil can be lethal.
A small amount of the toxic essence can lead to difficulty in breathing, dizziness and kidney failure. In the case of Grace Henderson, 41, of Hargreaves Street Bendigo, it was the shock of ingesting a full tablespoon – 14mL – which caused her death on Monday 21 August in 1876.
Alexander and Grace Henderson, a Scottish couple, had first established a confectionery store in Hargreaves Street in 1873; the site is now the vicinity of The Reject Shop and Council multi storey carpark next to Chancery Lane[i]. In 1874 a large sink hole opened up in the footpaths outside the shop, with Alexander writing to the council to describe the 'perfect man and woman trap' and asking that the 'newly-elected mayor or any of his brother councillors' remedy the dangerous problem, as if he filled it himself, he would be 'transgressing municipal bye-laws'[ii].
As well as the confectionery shop, Alex and Grace operated a restaurant in the two shops next door and employed a sizable staff. On Christmas Day in 1875, they gave a picnic for their employees and a group of friends, and in return, on New Year's Day, Alex was presented with a pair of field glasses, and Grace, a diamond ring[iii].
While 1876 started in a positive and collegial manner, by August it had come to light that Grace appeared to have a drinking problem, and was experiencing regular bouts of 'low spirits' and despondency[iv]. Indeed on one occasion, she told cab driver, Robert Newbold, who operated next door to the shop in Hargreaves Street, that she would 'do away with herself':
"Some four or five weeks ago, as I was locking the door, she was standing there; she seemed the worse for liquor and I said, 'What's the matter, Mrs Henderson?' and she asked me if I saw Mr Henderson. I said 'No,' and she replied that he was next door and she wished for me to get him. I said that I had not time as he was going to the 11:15AM train. She then said, 'I will do away with myself,', and I replied 'Don't be so foolish', and went into my place to get my horse and cab. [v]"
On a Monday afternoon, she asked one of their staff, James Anderson, to fetch for her a bottle of almond oil while he was in the storeroom. He indicated that there was none, but she replied, "There is some in the corner". Despite having no label, Anderson located the bottle and took it up to her, which she then tasted from, remarking, "How it burns"[vi]. They spoke for fifteen minutes before he left, locking up the shop and leaving the keys in their usual place. Annie Lovelace, a waitress in the dining room part of the business, arrived at 4PM and was asked by Grace to bring up the keys to the storeroom. She left soon after, just as Grace was about to enter the pantry.
At 4:45PM, Alexander went upstairs to their parlour and found Grace on the floor by the sofa, breathing heavily and near insensible. He asked one of his shop girls if she had given his wife anything to drink and she said that she had not. Alexander explained to the jury that they did not sell spirits in either shop, but a couple of Grace's friends would bring her drink. He had even mentioned to one that Grace may be in need of a stay at Yarra Bend (asylum).
Grace had been sick, and upon tasting the remnants, Alexander determined that his wife had taken either ratafia (an almond-flavoured liqueur) or bitter almond essence oil, and immediately called for Dr Penfold. When he arrived less than ten minutes later, Grace had been moved to a bedroom and was, according to the doctor, 'in the act of death'[vii].
Alexander had found a uncorked bottle of almond essence on the edge of a bench in the storeroom, away from where it normally stood, with an estimated tablespoon, or 'half an inch', of the contents missing[viii]. By this time, Dr Penfold could not find Grace's pulse, her face 'florid but natural looking' and her pupils dilated and fixed. He applied cold water and CPR while waiting for a stomach pump to be brought in[ix].
Bendigo Advertiser, 14 November 1874, p4
Almond prunus dulcis
He injected a drachm (approx. 3g) of a strong ammonia solution into her arm in an attempt to stimulate the heart but finding no improvement, and she was declared deceased. The liquid flushed from her stomach at the house was Alexander showed the bottle to Dr Penfold, who found the culinary essence to contain four times as much pure prussic acid as the preparation that was used in medical preparations.
Dr Penfold also conducted the examination on Grace's body for the inquest which was held on the following day at the Crown Hotel (still 238 Hargreaves Street) – after the jury had viewed her body, still in the bedroom of their house just a few doors up from the Hotel. In his deposition to the jury, he described finding 'half a pint of dark green grumous fluid, also smelling of the same oil', and significant crystallisation of her major organs[x]. In his summary to the foreman, he noted that her liver was 'of a yellow colour, hard and gristly… drawn and puckered in many places'.
Dr Penfold was of the opinion that the state of her liver suggested she 'must have largely indulged in spirit and other stimulants for some time'[xi]. Alexander had recognised that his wife had been very despondent in recent times but thought that her actions were through a need to drink rather than to 'destroy herself', and denied a suggestion that she had thrown herself into the creek on the Christmas Day of their grand picnic[xii].
The jury, which included the Henderson's neighbour, draper Robert Whiteside, unable to determine if the act was one of suicide or accidentally drinking too much essence for the effect of the stimulant, returned a verdict 'that the deceased met her death by taking a large dose of the essential oil of bitter almonds'.
Excerpt from inquest deposition of Alexander Henderson,
Public Record Office Victoria, VA2807 State Coroner's Office, VPRS 24 Inquest Deposition Files, P0 Unit 352, Item 1876/238, Grace Henderson
Read Grace's full inquest here
Excerpt from Dunn's report to council,
Public Record Office Victoria, VA 4862 City of Sandhurst, VPRS 16936 Inward Correspondence, P1 Unit 44, Item 17-30 September 1896
Frederic Dunn had been an analyst for Victorian municipalities since the 1870s, tasked with reviewing the ingredients of a wide variety of foodstuffs, from tea to cheese, milk to flour, butter to meat, and of course confectionery. Locally made and imported lollies, chocolates and liquorices were examined regularly for additives – often used to enhance colour – which included lead, chromate of lead, clay, gypsum, sulphate of baryta, double silicate of alumina and soda, and sulphuret of sodium. The constitution of products was also reviewed – in 1876, several Melbourne liquorice manufacturers submitted samples to Dunn who found that only one third were actually liquorice – other samples were primarily potato starch with a little cane sugar, mineral ash and liquorice extract[iii].
In 1896, Patrick Fahey was the City of Bendigo's Health Inspector; a former sergeant of police at Maryborough, Fahey had been appointed to the role in 1883 as well as holding the positions of Cab Inspector and Inspector of Nuisances. Under the Public Health Act (1890), foodstuffs were required to meet very particular standards, and the role of the Public Health Board was to approve which analysts municipal Health Inspectors were permitted to use.
So it was that Fahey engaged Dunn to undertake a number of examinations in the Spring of that year, focusing on bread and flour but also including confectionery, two brands of which were locally made. Dunn's report to Council indicated that of the three varieties he examined (hokey pokey lollies, rainbow lollies and raspberry drops) all were free of mineral poisons and colouring matter, but did contain small percentages of 'mineral matter'[iv].
The following year, his analysis of Bendigo distributors found all confectionery samples were free from minerals, colours or adulterants. In 1898, Fahey tendered his resignation from the role and John Reid was appointed as municipal health inspector. Reid engaged Dunn to a series of 'raids' on lolly shops, after having investigated milk sellers in the district and by November, the analyst had a report for Council. It showed that that the samples were again all free from 'mineral adulterations or metallic impurity'[v].
At the sitting of council that month, Councillor Harry Marks suggested that the Turkey lolly (fairy floss) vendors should be his next target, having known 'children to be ill after eating the stuff'. Past Mayor, MLA and hotelier, Councillor Alfred Bailes 'quietly hinted' that perhaps Cr Marks' opinion may have been coloured by an over indulgence in his younger days[vi].
Dunn was based in Melbourne, working for various companies including the Industrial & Technological Museum, and this caused some consternation to councillors from the 1890s and right through to war time in the 20th century.
Several times across the late 19th and early 20th century, various councillors proposed that Dunn's contract be terminated in favour of a local analyst through the School of Mines, and additionally offer those services to neighbouring municipalities. Ballaarat had been successful in decentralising analyst services and councillors thought this reflected poorly on Bendigo.
In 1904 Councillor Semmens proposed to terminate the agreement and appoint Boydell of the School of Mines, seconded by Councillor Carolin though Abbott felt that their laboratory's facilities were not sufficient for the job, nor would the cost difference be favourable. The motion was carried regardless but not carried through. The discussion arose again the following year, with the Council going as far as notifying Dunn that his engagement would be terminated in December. This did not occur and Dunn continued as the City's analyst but again in 1916, the Council were again keen to dispense of his services, citing his distance making inspections and testimonials difficult and expensive. Dunn disputed this but by March, James Anderson, director of the School of Mines had been approved by the Public Health Department to be appointed to the role of City Analyst.
[i] Public Record Office of Victoria, VA 4862 City of Sandhurst, VPRS 16267 Rate Books, P0 Unit 22 Year 1878, p89
[ii] 'A man trap', Bendigo Advertiser, 18 Nov 1874, p2
[iii] 'A presentation', Bendigo Advertiser, 5 Jan 1876, p2
[iv] 'Suicide by poison', Bendigo Advertiser, 23 Aug 1876, p2
[v] Public Record Office of Victoria, VA 2807 State Coroner's Office, VPRS 24 Inquest Deposition Files, P0 Unit 352, item 1876/238 Grace Henderson
[vi] Ibid
[vii] Ibid
[viii] Ibid
[ix] Ibid
[x] Ibid
[xi] Ibid
[xii] Ibid
[xiii] 'Confectionery analysis', The Argus, 8 Jul 1876, p9
[xiv] Public Record Office of Victoria, VA 4862 City of Sandhurst, VPRS 16936 Inward Correspondence, P01 Unit 44, Item 17-30 September 1896, Council Report
[xv] 'Raid on lolly shops', Bendigo Independent, 19 Nov 1898, p2
[xvi] 'Raid on lolly shops', Bendigo Independent, 19 Nov 1898, p2
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Blogs are a quick and easy way to learn something new on the fly. Through blogs, you can get inspired, personalize your learning, and connect with other experts.
In one of my earlier blog posts, 6 Popular Classroom Blogging Ideas, you would have read that there are over 58.6 million blogs on Wordpress alone.
How do you sift through them and find what is relevant to you? There is the Google machine, but you still have to skim through them, look for keywords, and determine the author's credibility.
Here is a list of my top 10 blogs, not only for new teachers, but all teachers plus one bonus blog.
Alice Keeler is one of my all-time favorite educators in technology. She is the epitome of what it means to share and give back to educators. She presents at conferences across the country, tweets, and blogs.
She is also a Google Certified Teacher and most of her posts revolve around the G Suite (formerly known as GAFE). So if you are a Google School, you need to know Alice Keeler.
Keeler's blog covers everything Google, from Chrome extensions, G Suite tips and tricks, as well as many math tech uses (TPACK in action!). Not only is she a giver of knowledge, but she also corresponds with us commoners.
She will respond to your comments on her posts as well as any direct tweets. Be sure to scope out the Tag Cloud at the bottom of her pages if you are looking for specific content. You will not be sorry you discovered Alice Keeler; however, you may fall down the rabbit hole of information overload.
Dr. Tim Clark earned his Ed.D. in educational leadership but covers a wide range of educational topics. He's the author of many educational periodicals including his own blog.
Being that he is a Doctor, his posts are a bit more academic and research-based; whereas, Keeler's posts are more application-based. Nonetheless, both are excellent blogs.
Dr. Clark's Blog: BYOT, Bring Your Own Technology (technology has been interchangeable with Device, BYOD), covers topics on educational leadership, collaboration, digital literacy, and fair use.
All of these topics are of value to new teachers, yes, even the leadership piece. He has a rich database of posts, and like Keeler's, use the word cloud to find topics that suit you (are you finding a theme here? Hint: look for the word cloud).
Catlin Tucker is another Google Certified Teacher. She has written several books and also has a wide range of topics covered on her blog. One of my favorite things about her blog is her "Favorite Web Tools" tab.
She covers in a few sentences several must- have web tools with links to access them. She blends the technology seamlessly into her instructional practices (dare I mention TPACK?), including English Lit uses.
If you are an English teacher you should definitely check out her site, but you don't have to be an English teacher to appreciate the tools she shares.
I find this site is a bit more geared to the techie teacher -- someone who has a lot of foundational tech knowledge and wants to advance that with their students is the primary audience here.
There are several posts about coding in the classroom. This is timely because this week (Dec 5-11) is Computer Science Education week, where schools around the world are encouraged to engage in "An Hour of Code".
This is to promote computer science to all students, across all disciplines. If you'd like to join the movement, I suggest checking out this blog to get you started. It covers free and easy sites such as Scratch and Codecademy.
Not only does this site cover computer sciencey stuff, but it covers digital literacy, G Suite, and other digital tools.
Dr. Monica Burns, another teacher turned tech leader, is an Apple Distinguished Educator, conference presenter, and also holds several accolades from leading tech companies around the country.
She has a unique slant on her blog postings from her most recent post, 5 Apps for Student Sports Lovers with Curriculum Connections, Best Technology for School Grant Program, and How to Strengthen Student Listening Skills with Podcasts.
Like Dr. Clark, Dr. Burns connects many of her posts to the theoretical framework that shapes why we do what we do as educators. It is easy to forget about the research side of it as we put the blinders on while planning a lesson.
She'll help remind you that it's always important to focus on the Why first and then the How.
Vicki Davis is the author of this teachers' blog. She is a current educator, presenter, and guest writer for many esteemed publications. She writes this blog as a hobby to share what she's doing with her students and what she finds important.
Her posts are timely, like her latest post on her suggested gift list for techies this year. She has great posts on How to Help Your Subs Make Progress with Your Students and How to Teach Kids to Use Smartphones Safely.
She's been a classroom teacher for over a decade and has a lot of experience and knowledge to pull from. This blog is especially good for beginning teachers because it comes from a veteran.
The Ed Tech Teacher blog is great because it includes a team of educators' posts. Their team is made up of K-12 educators, instructional designers, instructors, etc.
There is a wide perspective posted here to appeal to all teachers. They cover great topics to inspire new ideas in your classrooms, Smashing Apps and harnessing Chromebooks, to name a couple.
If you are looking for something to freshen up your classroom, this is a one-stop-shop.
Michael Fricano II is a technology integration specialist. He has been an educator since 2007 and has some great experience in those emerging technologies like MakerEd, Coding, gaming, and Virtual Reality.
His blog posts lend themselves well to the STEM/STEAM movement if that interests you. He also has a nice library of YouTube tutorials on his channels, when let's face it, sometimes we'd rather watch and listen than read one more thing.
Fricano also has several Apple topic related posts, which I haven't seen much of on the last seven blogs. His posts are easy to integrate into your classroom tomorrow if you are so inclined.
This blog has come into question over its content creation and licensing practices. We have chosen to remove it from the list.
Kasey Bell is another presenter at several conferences, consultant, and writer of educational content. Her blog covers several Google related topics from, becoming a Google Certified Educator and trainer, using the G Suite tools in unique ways, and how to keep up with the constant innovation of Google.
Her posts are short and sweet so they won't take you long to read. So get in that arduous line at the grocery store, pull out your phone and Shake Up your Learning and read what Bell is writing.
We saved the bonus blog for last! Our very own spot on the internet talking about the latest edtech trends, statistics and device news.
Produced by BEYOND Technology Education, The K-12 Daily Dose provides unique content, premium content and features guest bloggers including me! The K-12 daily Dose is always looking for new bloggers.
If you want to try out your writing skills and have something interesting to say to other teachers then give them a shout.
How do you keep track of all these blogs? I like to use a tool called Feedly. It curates all your blogs into one easy place to access. Feedly notifies you of new posts without having to go out to each blog.
Mobile apps are available as well to make your blogs all the more accessible. Information on the go, what's not to love. Now get reading.


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Positioning Theory
Positioning History
Positioning Report
There are two types of categories and two types of strategies.
All categories are not alike. Unless you know what type of category you are dealing with, you may be making a strategic error.
Type No.1 is frequently-bought products which usually have a clear-cut market leader and a strong second-place brand.
Type No.2 is infrequently-bought products which usually have no clear-cut market leader.
(1) Frequently-bought products.
Almost any product or service you buy frequently has two dominant brands: A market leader and a strong second-place brand. Some examples:
Cola . . . . . . . . . . Coca-Cola and Pepsi-Cola.
Energy drinks . . . Red Bull and Monster.
Pizza . . . . . . . . . . Pizza Hut and Domino's.
Coffee shops . . . . Dunkin' Donuts and Starbucks.
Toothpaste . . . . . . Crest and Colgate.
Smartphones . . . . The iPhone and Samsung.
Batteries . . . . . . . Duracell and Energizer.
Parcel delivery . . UPS and FedEx.
Political parties . . Republican and Democratic.
In a duopoly, there is almost no room for a No.3 brand. What is the market share of Royal Crown cola? The brand is lucky if it has more than a one or two percent share.
Everybody knows Red Bull and Monster. But what's the name of the No.3 brand of energy drink?
Everybody knows Crest and Colgate, but what's the name of the No.3 brand of toothpaste?
Everybody knows Duracell and Energizer, but what's the name of the No.3 appliance battery brand?
The No.3 brands are Rockstar, Sensodyne and Rayovac. The latter brand is owned by Spectrum Brands, a company that went bankrupt in 2009. That's not untypical for No.3 brands which are inherently weak.
The dominance of the "top two" brands in a category can be demonstrated by the smartphone category. The iPhone and Samsung have only 36 percent of the global smartphone market, but by one estimate, the two brands account for 90 percent of the category's profits.
A strategy for frequently-bought brands.
Naturally, a company wants its brands to establish leadership positions in every category they compete in. But that's not always possible.
Leadership is usually determined by the first brand in the mind. Coca in cola. Red Bull in energy drinks. Gatorade in sports drinks. The iPhone in smartphones.
But what determines which brand winds up in second place. Often, but not always, the No.2 brand is the "opposite" of the leader.
After the launch of Red Bull, there were more than a thousand energy-drink brands introduced into the American market. Almost all of them in 8.3-oz. cans pioneered by Red Bull. Except Monster, the first major brand introduced in a 16-oz. can. (With a 16-oz. name, too.)
After the launch of Coca-Cola, hundreds of other cola brands flooded the market, including Pepsi-Cola. The difference? Coca-Cola was sold only in 6.5-oz. contour bottles and Pepsi was sold in 12-oz. bottles. Hence, the radio jingle.
Pepsi-Cola hits the spot. Twelve full ounces, that's a lot.
Twice as much for a nickel too. Pepsi-Cola is the drink for you.
And how did Starbucks compete with market leader Dunkin' Donuts? By doubling the price of its coffee.
Strong No.2 brands invariably achieve their positions by strategies that are the exact opposite of the leader's strategy.
If no brand is launched that does not take an "opposite" position in the category, then the leader brand will often achieve what amounts to a monopoly position. Gatorade, for example, outsells the me-too Powerade brand five to one.
(2) Infrequently-bought products.
Take life insurance, which people buy perhaps once in their lifetimes. Which brand is the leader in life insurance?
Northwestern Mutual with a market share of 8.5 percent. Does that surprise you? It should because few people perceive Northwestern Mutual as the leader in the category.
Here are the next ten brands in the category, along with their market shares: New York Life (5.7%), MetLife (5.3%), Lincoln (5.0%), John Hancock (4.2%), Prudential (3.9%), MassMutual (3.8%), State Farm (3.6%), Transamerica (3.6%), AIG (2.9%) and Guardian (2.7%.)
As you can see, there are no strong No.2 brands in the life insurance category either.
Obviously, a life insurance brand can't be the opposite of the leader if no one knows who the leader is. Or what position it owns.
A strategy for infrequently-bought products.
What should a life insurance brand do? It should narrow its focus to one singular attribute. And then hammer that attribute in everything it does, including its name and its strategic position.
Do you know what any of these life insurance brands stand for? Probably not, because they don't stand for much.
Take automobiles which people also buy infrequently. Which brand is the leader in automobiles?
Ford with a market share of 14.2 percent.
And here are the next ten brands in the category, along with their market shares in the year 2016: Toyota (12.0%), Chevrolet (12.0%), Honda (8.4%), Nissan (8.1%), Jeep (5.3%), Hyundai (4.4%), Kia (3.7%), Subaru (3.5%), GMC (3.1%) and Ram (3.1%).
Obviously, an automobile brand can't be the opposite of the leader if no one knows who the leader is. Or what position it owns.
Go back to the year 2000 and look at which automobile brands weren't on the list that year, but are on the 2016 list. Those are the brands that have made the most progress.
There are three: Hyundai, Kia and Subaru. And they all made the list because they focused on something very specific.
But Hyundai and Kia entered the market at the low end, avoiding the competition with mainstream brands like Ford, Toyota and Chevrolet.
Subaru hammered its four-wheel-drive vehicles, the only brand focused exclusively on four-wheel drive. Confidence in motion is Subaru's theme, a relatively weak expression of a powerful idea.
Furthermore, each of these three brands is doing better financially than Ford. Last year, Ford's net profit margin was 3.0 percent compared with Kia (5.2 percent), Hyundai (5.8 percent) and Subaru (13.5 percent.)
Customer versus competitor orientation.
Now you might be thinking, What possible difference does it make? We are customer oriented, not category oriented.
That's a mistake. You win or lose in today's marketplace by being competitor oriented, not customer oriented.
A second mistake is to ignore the category you compete in. The category should also dictate your strategy.
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Bicycles & Coffee is convenient to many wonderful quiet country lanes, adventurous gravel roads as well as the W&OD trail and historic C&O canal. Take a ride to a winery, past sprawling horse farms, battlefields, great BBQ, a local B&B, to the beach or wherever your heart desires. Rentals are new models and professionally maintained. Please contact us for details on available models and hourly, daily and weekend rates.
We are pleased to offer the reliable comfort hybrids in both men's and women's specific models. These bikes are the perfect model for our area, at home on both paved and light-duty unpaved roads (like the C&O canal), it can provide an enjoyable ride where ever your adventure takes you. Lightweight, easy to operate and a comfortable seating position for any rider at all ability levels. Available in S, M, L and XL.
For our customers looking to explore the area on a road bike, we are pleased to offer a limited number of lightweight models. These are the perfect road bike for our area because most can easily conquer the pavement as well as the gravel with equal enthusiasm. They have endurance geometry, designed for a comfortable cockpit for hours in the saddle and are a willing partner for touring, training or centuries. Please confirm availability and sizes.
Sorry, no tandems are available at this time.


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Pakistan-India Cooperation: A Way Forward
« Security Threats and Challenges to Pakistan: A Way Forward
Pakistan – Romania Economic Relations: A Way Forward »
Islamabad Institute of Conflict Resolution organized one-day seminar on May 29, 2014, in collaboration with Department of Political science, Lahore College for Women University, Lahore, titled as "Pakistan-India Cooperation: A Way Forward". The key speakers were: Dr. Hassan Askari Rizvi, renowned Political Scientist and Defence Analyst and Dr. Shabbir Ahmed, Assistant Professor in Department of Political Science, Punjab University, Lahore. Prof. Dr. Sabiha Mansoor, Vice Chancellor Lahore College for Women University, Lahore welcomed the guests. Ms. Sabah Aslam, Executive Director, Islamabad Institute of Conflict Resolution (IICR) concluded the seminar.
The seminar was well-attended by students of Political Science, International Relations, and Pakistan Studies, faculty members, scholars, young researchers, and media persons. They patiently listened to the speakers and actively participated in question answer session. The speakers gave a fresh input on the on-going debate of Pakistan-India relations, especially after the political change in India through May 2014 elections in which Mr. Narender Modi's politically right and religiously oriented party BJP, has emerged as victorious. With the emergence of Mr. Modi, various questions are raised regarding nature of future relationship between Pakistan and India.
The guest speakers, while keeping in view global and regional imperatives, were optimistic about the future cooperation between India and Pakistan and agreed that it is high time two nations learned from past and never allowed it to damage present and the future cooperation between the two.
The key objective of the seminar was to address those questions that arise out of new situation developed because of domestic political change in the two countries and the changing regional and global dynamics. It was aimed to make the participants develop a rational discourse about the changing circumstances in international relations and especially about the on-going debates about the future of Pakistan-India Relations. The seminar also aimed to engage students to apply their knowledge in understanding new situations in international affairs.
Prof. Dr. Sabiha Mansoor:
The Vice Chancellor of LCWU, Prof. Dr. Sabiha Mansoor welcomed the guests and appreciated the IICR Team for organizing a seminar on the right occasion and topic. She said nobody wants war, everybody is pacifist and our new generation has more inclination towards peace and cooperation. There is a greater need for enhancing cooperation between Pakistan and India. Two nations have already drained their resources on defeating each other militarily. There is urgent need to turn the pages of history and write a new chapter of cooperation.
Dr. Hassan Askari Rizvi:
The key-note speaker at the seminar Dr. Hassan Askari Rizvi, a renowned Political Scientist and Defence Analyst, said that the relations between the two nations have always been shaped by domestic politics. Dr. Askari said, whenever there is debate on the subject of Pakistan India Cooperation, it give rise to so many controversies since the subject is part of the domestic politics. And the recent visit of Prime Minister Nawaz Sharif to India for witnessing oath-taking ceremony of Mr. Modi, has received mixed reactions in different quarters including Media. We need to keep emotions aside and try to understand the nature and delicacy of the problem. You have to deal with India, BJP, and Modi without negative reaction, you have to acknowledge the success of Modi, whether it is good or bad decision of electorate.
The emergence of political right and religiously oriented party in India has generated much reaction in Pakistan. It is believed in some quarters of media that BJP has won the elections on the basis of hatred against Pakistan. Mr. Askari, while rejecting such discourse, explained that it was internal task of economic welfare and poor performance of congress that have contributed in the success of Narender Modi. He said, "BJP's Past has nothing to do with elections". He agreed that there are extremist elements in in India i.e RSS, but on other hand there is much support of corporate sector to Mr. Modi, which will not want any war/conflict. Mr. Askari stressed that, Mr. Modi has got image problem, it is in his interest to develop good relations with Pakistan; and to recover that he invited SAARC leaders including Mr. Sharif in oath-taking ceremony. To Mr. Askari, this was a positive development which has "unfrozen the frozen dialogue process". However, there is a diplomatic challenge for them in finding a middle ground
when both states follow a different approach towards resolution of dispute. Another challenge is to make the resumed dialogue more sustainable.
There is a challenge for Mr. Modi to develop his positive image and creating balance in internal ethnically and religiously diverse India.
There are two ways to deal with the problem between India and Pakistan. First is delving into past; but problem is that the discussion on both sides is similar. Both the nations carry with themselves the baggage of historical grievances, with a charge sheet against each other. To Mr. Askari, this is ineffective way and is the major cause of stalemate. The second approach, which Mr. Askari advocated is to learn from past and move forward. "You cannot to be the prisoners of past", said Dr. Askari, "You have to draw lessons from history and take steps cautiously and lead the situation in positive direction"
Dr. Askari underscored that the effectiveness of foreign policy depends on the internal strength which comes through firstly having political and societal harmony; that is what the social and political nature of conflict is, such as terrorism which has social and political dimension, and secondly having sound economy. Dr. Askari stressed that without sound economy you cannot make foreign policy effective. He believe that independent economy automatically widens the foreign policy options, and wide options in foreign policy matter a lot in securing national interest, which are achieved through internal strength. During cold war era some major power came to help you as part of global politics but today there is no such situation where some external power may come and rescue you.
Mr. Askari suggested Pakistan needs to diversify its foreign policy and do away with dependence on external powers through building internal social, political and economic strength. Pakistan's challenge is internal, that is of economy, terrorism and other social problems; if it deals with them successfully, it can deal India as well. Modi's challenge is his image and Indian global aspirations, which would be achieved when he develops good relations with Pakistan
He suggested Pakistan should continue its policy of dialogue. He pointed out that both nations want to resolve their problems but unfortunately there is a difference of approach. Indian approach to the dialogue based on single issue and Pakistan's on multiple issues is the real challenge for diplomacy. He warned that single issue relationship would not work for dialogue. If multiple issues are taken into account like Kashmir, Sir Creek, water, movement of people, and economy, there will be likely progress on some issues and some issues will further thinking and deliberations.
He suggested that carry on those issues which some amount of progress. Through diplomacy the difficult and frozen situations are resolved, and there is a diplomatic challenge to harmonize Pakistan India Relations. He said Pakistan needs to wait for few months as the new government in India fully establishes and hope that has developed out of Pakistani PM's visit to India to take a concrete shape. However, to Mr. Askari, there are challenges attached to this hope.
Answering a question on MFN, he said giving access to Pakistani Market is separate from giving India corridor to Afghanistan. Pakistan will have a right to stop goods from India. On
another question, Mr. Askari suggested that, Pakistan needs to understand Indian Political System and design foreign policy keeping in view internal contradictions of India
Mr. Shabbir Ahmed:
Another speaker Mr. Shabbir Ahmed, Assistant Professor in Punjab University, stressing on future he said he is seeing a good future of Pakistan-India cooperation. Emphatically, Mr. Shabbir warned, "If you look at the past, you can't move forward, you have to bury the past." He pointed out imperatives that lead to future cooperation between two nations. Firstly, issue of terrorism brings two nations closer because both have suffered at the hands of terrorist. There is coincidence of stance on it. Secondly, both are now democratic nations, and the democratic principle both sides make them less feasible to go for war. Thirdly, global politics is supporting cooperation; it is moving from traditional security towards non-traditional security; it is focusing on socio-economic development. Fourthly, Mr. Modi and Mr. Sharif share the same vision, both represent corporate sector. Fifthly, war is not an option for both to resolve the issues, probably it was in1960's, but now it is not since both are nuclear states. Pakistan is an important regional state, and regional politics would impose certain concerns. Finally, vision of founding fathers, especially Quai-i-Azam, is based on peace and friendly relations with neighbors. He envisaged the relations between India and Pakistan same like that of US and Canada. Mr. Shabbir set European history of conflicts and its transformation into cooperation and as an exemplary point.
Ms. Sabah Aslam:
While concluding the session, the Executive Director of Islamabad Institute of Conflict Resolution (IICR), Ms. Sabah Aslam, said that the relationship between India and Pakistan are marred by the deep scars dating back to Partition of sub-continent with animosity embedded in the memory lane of both nations. What started from the skewed Radcliff award has developed into wide gulf furthered each time by military show downs, proxy, media and lately water wars.
More so, she pointed out that we have tried traditional military and non-military means already which appear to have not worked so far. Media efforts, cultural exchanges, enhanced business, relaxed trade and sports etc. must continue, as these help in defusing tension and increase interdependence which is an important step in enhancing prospects of reconciliation. However, it must be realised that such efforts cannot progress well, if these are one sided. The euphoria and efforts of years must not be allowed to die down by imprudent acts. If Pakistan's water is stopped, interference by India in Baluchistan and support to anti – federation forces continues, and if Pakistan interferes in India, how can the relations between both improve?
To be fare both need to sit and listen to each other's' grievances, and take steps to address the genuine ones, she suggested. Both the countries have paid enough of human and economic costs. As said by the PM Nawaz Sharif, one and a half billion people's lives, their future and progression are dependent upon the progress both leaders make on the peace process. She stressed that there is a need to detach the process from past and make a fresh beginning. Words, Jargons and terminologies are not important. What is important is to move forward and not look back to the dark alleys. Neither India and nor Pakistan are in a position to solve the problems by military or economic war, which would at best bring stale mate and devastation.
While highlighting the considerable potential for trade in the entire South Asian region, Ms. Sabah said that regional integration is only possible when the two nuclear powers would be together as trade partners. It would open new venues of cooperation for South Asia. In fact, regional integration is a key for economic growth and intra-regional trade and would also produce considerable economic gains for South Asia.
Hence future of both Nations lies in resolving the bilateral disputes through negotiations, while staying within the broad parameters of partition and taking care of each other's legitimate concerns. Both the leaders owe this to their people.
She thanked all the participants for taking keen interest in the seminar and active involvement in the discussions. She said she had been greatly indebted to the gratitude of whole Department of Political Science and wished great future to young researchers and students.


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Amazon is turning 25 – here's a look back at how it changed the world
Venkatesh Shankar, Texas A&M University
Amazon has redefined retail over the last 25 years.
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A quarter of a century ago, on July 5, 1994, a company, which shared a name with the world's largest river, was incorporated. It sold books to customers who got to its website through a dial-up modem.
It wasn't the first bookstore to sell online. (Books.com launched in 1992.) But it behaved like a local store, whose shopkeeper knew customers by name – a bell even rang in the company's Seattle headquarters every time an order was placed.
Amazon's founder, Jeff Bezos, set his sights on making it an "everything store." The company would go on to become not just an everything store, but an "everything company."
Today, 25 years later, Amazon has reshaped retailing permanently. It is one of the top three most valuable companies in the world, with a market capitalization hovering around US$1 trillion, greater than the GDP of nearly 200 countries.
If you had bought $100 worth of its IPO shares in 1997, it would be worth about $120,000 today.
Redefining retail
Amazon continually took shopping convenience to newer levels.
Before 1994, shoppers had to travel to stores to discover and buy things. Shopping used to be hard work – wandering down multiple aisles in search of a desired item, dealing with crying and nagging kids, and waiting in long checkout lines. Today, stores try to reach out to shoppers anywhere, anytime and through multiple channels and devices.
After first experiencing two-day free shipping from Amazon's Prime membership program, shoppers started expecting no less from every online retailer. About 100 million shoppers worldwide have Amazon Prime.
The company made shopping more convenient through features like one-click ordering; personalized recommendations; package pickup at Amazon hubs and lockers; ordering products with the single touch of a Dash button; and in-home delivery with Amazon Key.
Shoppers can also search for and order items through a simple voice command to an Echo or by clicking an Instagram or a Pinterest image. Amazon now even has a cashierless "Go" store in Seattle.
Amazon has been a factor in the rising closures of brick-and-mortar stores that can't keep pace with the changes in retail. In the first 15 weeks of 2019 alone, there were about 6,000 store closures in the U.S. overall, higher than the number of closures in all of 2018. Analysts fear a coming "retail apocalypse."
A major employer
Amazon's impact extends to other industries, including smart consumer devices (Alexa), cloud service (Amazon Web Services), and technology products and services (drones).
Such is Amazon's impact that industry players and observers use the term "Amazoned" to describe their business model and operations being disrupted by Amazon.
Today, Amazon is the largest tech employer by far. It employs more people than the next five tech companies combined. No wonder Amazon created such a buzz last year before selecting a location for its second headquarters.
Amazon's work culture is intense. It has a reputation as a cutthroat environment with a high employee burnout rate. It is automating as many jobs as possible, mostly in warehousing.
At the same time, after criticism from policymakers, Amazon stepped up in October 2018 by raising minimum wage for all U.S. employees to $15 per hour.
Faced with growing criticisms about the mounting impact of Amazon's boxes and other packaging material on the environment, Amazon has pledged to disclose more information about its environmental impact at the end of 2019.
What's in store for Amazon's future?
The Amazon Echo Plus.
ClassyPictures/shutterstock.com
Bezos has said that Amazon's efforts focus on preventing it from dying. As he noted at a 2018 all-hands meeting, "Amazon is not too big to fail."
As a professor of marketing, having conducted research on online retailing and analyzed hundreds of cases, I believe that Amazon's future, like shoppers' and the society's future, is inextricably linked to the rise of artificial intelligence. Starting with Alexa, the company's virtual assistant, Amazon is betting on AI.
In fact, Amazon is testing anticipatory shipping, a practice in which it anticipates what shoppers need and mails them the items without shoppers ordering them. Shoppers could keep the items they like and return those they don't want at no charge.
A diagram in Amazon's patent for anticipatory shipping.
Google Patents
More immediate questions relate to Amazon's entry in two stodgy yet critical industries: health care and financial services.
Although Amazon has disrupted many industries, these two are heavily regulated industries in which the company hasn't had much experience.
Amazon is considering becoming a major player in the pharmaceutical and the health insurance markets. In May, it acquired online drugstore PillPack for $1 billion to crack the $500 billion market for prescription drugs and has formed a joint health venture with Berkshire Hathaway and JP Morgan Chase.
Meanwhile, with 310 million customer accounts, Amazon is building a suite of high-tech financial services, such as Amazon Cash, a way to add cash to your online balance, and Amazon Pay, an online payment service. These programs are aimed at developing markets such as India, which has a huge population that does not use banks.
Amazon has reshaped retailing permanently in the last 25 years. In the next 25, it might totally redefine how the world shops.
Venkatesh Shankar, Professor of Marketing; Director of Research, Center for Retailing Studies, Texas A&M University
TCU scores high in 'Most Beautiful Campus' ranking
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Commentary: O.K. Carter: Kennedale finds itself in middle of challenges – and opportunities
Richard Connor: Did you hear the one about the fox and the henhouse?
National Cowgirl Museum announces 2019 inductees
Stepping out on faith and passion
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Superintendent for Grand Prairie ISD dies in Arizona accident
316-unit apartment in Fort Worth sold
Fort Worth M.D. school begins and city hopes to reap benefits
Richard Connor: Did you hear the one about the fox and the henhouse? (5)
Panther Island review delayed for staff briefings; Mayor calls for public release (2)
As Panther Island review nears release, White Settlement business owners hear from contractor (2)
Texas Health Resources making way for $300M expansion in Fort Worth (2)
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Commentary: O.K. Carter: Kennedale finds itself in middle of challenges – and opportunities (2)
Council Report: Botanic Garden accessibility plans (2)
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Fort Worth Business Press July 8 – 14, 2019
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Number 8;- L/Cpl. J.W. Ray. Civil police, Baghdad.
Number 9;- No name or date.
The last one was on the Maud bridge, Baghdad.
What a find YT, and how wonderful it would be to find the decendants of those photographed.
Certainly need to contact the owner of the tree on Ancestry.
The album seems to have belonged to L/Cpl. J.W.Ray.