text:
\section{Introduction}\label{sec:intro}
Let $a$ and $b$ be positive integers. Bulawa and Lee [1] considered the sequence $\{R_{i}\}_{i \in {\Bbb N} }$ defined by the recurrence relations
\[R_{i+2}=aR_{i+1}+bR_{i}\]
and $R_{0}=0, R_{1}=1$. The generating function is given as
\[R(t)=\frac{t}{1-at-bt^{2}}.\] They established the following necessary and sufficient conditions that should be applied when the rational values in the interval of convergence for the generating function $R(t)$ for the sequence $\{R_{i}\}_{i \in {\Bbb N} }$ are integers:

\vspace{3mm}

\noindent{\bf Theorem 1.1} (Bulawa and Lee [1]).
Let $t$ be a rational number. Let us assume that $b$ divides $a$ and that $t$ lies within the interval of convergence for the generating function $R(t)$. Then, $R(t) \in {\Bbb Z}$ if and only if
\[t \in \bigl\{\frac{R_{2i}}{R_{2i+1}}\bigr\}_{i \in {\Bbb N}}.\]

\vspace{3mm}

This result answers the conjecture developed by Hong [2].

\vspace{3mm}

Independent of Bulawa and Lee, Pongsriiam [3] obtained similar results under the conditions of $a = 1$ and $b = 1$.

However, in this study, their work is expanded for applicability to polynomials.

\vspace{3mm}

Let us define a polynomial sequence $\{F_{i}(x)\}_{i \in {\Bbb N} }$ given by the recurrence relations
\[F_{i+2}(x)=axF_{i+1}(x)+bF_{i}(x)\eqno{(1)}\] and $F_{0}(x)=0, F_{1}(x)=1$. For example, $F_{2}(x)=ax$ and $F_{3}(x)=a^{2}x^{2}+b$.
The generating function is given by
\[F(t)=\sum_{i=0}^{\infty}F_{i}(x)t^{i}=\frac{t}{1-axt-bt^{2}}.\]

Let us also define a polynomial sequence $\{L_{i}(x)\}_{i \in {\Bbb N} }$ given by the recurrence relations
\[L_{i+2}(x)=axL_{i+1}(x)+bL_{i}(x)\eqno{(2)}\] and $L_{0}(x)=2, L_{1}(x)=ax$. For example,  $L_{2}(x)=a^{2}x^{2}+2b, L_{3}(x)=a^{3}x^{3}+3abx$.
The generating function is given by
\[L(t)=\sum_{i=0}^{\infty}L_{i}(x)t^{i}=\frac{2-axt}{1-axt-bt^{2}}.\]

\vspace{2mm}

The main results are as follows:

\vspace{2mm}

\noindent{\bf Theorem 1.2.}
Suppose $b$ can divide $a$, and let $t(x) \in {\Bbb Q}(x)$ be a rational function. For the generating function $F(t)$, $F(t(x)) \in {\Bbb Z}[x]$ if and only if
\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]

\vspace{2mm}

\noindent{\bf Theorem 1.3.}
Suppose $b$ can divide $a$, and let $t(x) \in {\Bbb Q}(x)$ be a rational function. For the generating function $L(t)$, $L(t(x)) \in {\Bbb Z}[x]$ if and only if
\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}, \frac{L_{i}(x)}{L_{i+1}(x)}, -\frac{L_{i+1}(x)}{bL_{i}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]

\vspace{2mm}

Let  $d \in {\Bbb N}$ be a square-free natural number. We define a sequence $\{F_{i}(\sqrt{d})\}_{i \in {\Bbb N} }$ given by recurrence relations
\[F_{i+2}(\sqrt{d})=a\sqrt{d} F_{i+1}(\sqrt{d})+bF_{i}(\sqrt{d})\] and $F_{0}(\sqrt{d})=0, F_{1}(\sqrt{d})=1$.
The generating function is given by
\[F(\sqrt{d})(t)=\frac{t}{1-a\sqrt{d} t-bt^{2}}.\]
We define a sequence $\{L_{i}(\sqrt{d})\}_{i \in {\Bbb N} }$ given by recurrence relations
\[L_{i+2}(\sqrt{d})=a\sqrt{d} L_{i+1}(\sqrt{d})+bL_{i}(\sqrt{d})\] and $L_{0}(\sqrt{d})=2, L_{1}(\sqrt{d})=a\sqrt{d}$.
The generating function is given by
\[L(\sqrt{d})(t)=\frac{2-a\sqrt{d}t}{1-a\sqrt{d} t-bt^{2}}.\]

Furthermore, the convergence radii of these generating functions are all
\[\frac{2}{a\sqrt{d}+\sqrt{a^{2}d+4b}}.\]

Then, we obtain the following as a corollary of Theorem 1.2. (resp.Theorem 1.3.)

\vspace{2mm}

\noindent{\bf Corollary 1.4.} Suppose $b$ can divide $a$. Moreover, we assume that $2b$ can divide $a$ or $d=1$. Let $c \in {\Bbb Q}(\sqrt{d})$. For the generating function $F(\sqrt{d})(t)$, we have $F(\sqrt{d})(c) \in {\Bbb Z}[\sqrt{d}]$ if and only if
\[c \in \bigl\{\frac{F_{i}(\sqrt{d})}{F_{i+1}(\sqrt{d})}\bigr\}_{i \in {\Bbb N}} \text{ or  } c \in \bigl\{-\frac{F_{i+1}(\sqrt{d})}{bF_{i}(\sqrt{d})}\bigr\}_{i \in {\Bbb N}^{+}}.\]

\vspace{2mm}

\noindent{\bf Corollary 1.5.}
Suppose $b$ can divide $a$ and $d=1$, and let $t \in {\Bbb Q}$ be a rational number. For the generating function $L(t)$, $L(t) \in {\Bbb Z}$ if and only if
\[t \in \bigl\{\frac{F_{i}}{F_{i+1}}, \frac{L_{i}}{L_{i+1}}, -\frac{L_{i+1}}{bL_{i}}\bigr\}_{i \in {\Bbb N}} \text{ or  }t \in \bigl\{-\frac{F_{i+1}}{bF_{i}}\bigr\}_{i \in {\Bbb N}^{+}}.\]

\vspace{2mm}

Focusing on the radii of convergence of the generating functions, we have the following results from Corollary 1.4 and Corollary 1.5.

\vspace{2mm}

\noindent{\bf Corollary 1.6}
Under the assumption of corollary 1.4, let $c \in {\Bbb Q}(\sqrt{d})$. We assume that $c$ is in the interval of convergence of the generating function $F(\sqrt{d})(t)$. Then, we have $F(\sqrt{d})(c) \in {\Bbb Z}[\sqrt{d}]$ if and only if

\[c \in \bigl\{\frac{F_{2i}(\sqrt{d})}{F_{2i+1}(\sqrt{d})}\bigr\}_{i \in {\Bbb N}}.\]

\vspace{2mm}

\noindent{\bf Corollary 1.7.}
Under the assumption of corollary 1.5, let $c \in {\Bbb Q}$ be a rational number. We assume that $c$ is in the interval of convergence of the generating function $L(t)$. For the generating function $L(t)$, $L(c) \in {\Bbb Z}$ if and only if
\[c \in \bigl\{\frac{F_{2i}}{F_{2i+1}}, \frac{L_{2i+1}}{L_{2i+2}}\bigr\}_{i \in {\Bbb N}}.\]

\noindent{\bf Remark 1.8.}
Corollary 1.6 and Corollary 1.7 are generalizations of the results obtained by Hong [2], Pongsriiam [3], and Bulawa and Lee [1].

In Corollary 1.4, the statement generally does not hold if the assumption is removed. If $a=1$, $b=1$, and $d=2$, we obtain
\[ F(\sqrt{d})(-2-\sqrt{d})=2-\sqrt{d} \in {\Bbb Z}[\sqrt{d}].\]
But,
\[-2-\sqrt{d} \notin \bigl\{\frac{F_{i}(\sqrt{d})}{F_{i+1}(\sqrt{d})}\bigr\}_{i \in {\Bbb N}} \text{ and  } -2-\sqrt{d} \notin \bigl\{-\frac{F_{i+1}(\sqrt{d})}{bF_{i}(\sqrt{d})}\bigr\}_{i \in {\Bbb N}^{+}}.\]
Also, in Corollary 1.5, the statement generally does not hold if the assumption is removed. If $a=1$, $b=1$, and $d=2$, we obtain\[ L(\sqrt{d})(-2-\sqrt{d})=-4+2\sqrt{d} \in {\Bbb Z}[\sqrt{d}].\]
But,
\[-2-\sqrt{d} \notin \bigl\{\frac{F_{i}(\sqrt{d} )}{F_{i+1}(\sqrt{d} )}, \frac{L_{i}(\sqrt{d} )}{L_{i+1}(\sqrt{d} )}, -\frac{L_{i+1}(\sqrt{d} )}{bL_{i}(\sqrt{d} )}\bigr\}_{i \in {\Bbb N}} \text{ and  }-2-\sqrt{d}  \notin \bigl\{-\frac{F_{i+1}(\sqrt{d} )}{bF_{i}(\sqrt{d} )}\bigr\}_{i \in {\Bbb N}^{+}}.\]
\section{Preliminaries}\label{sec:pre}

Before proceeding to the proof of the main results, the following equations and proposition should be understood.

Let

\[\alpha(x)=\frac{ax+\sqrt{a^{2}x^{2}+4b}}{2},\]

\[
\beta(x)=\frac{ax-\sqrt{a^{2}x^{2}+4b}}{2}
.\]
Then, it follows that
\[F_{n}(x)=\frac{\alpha(x)^{n}-\beta(x)^{n}}{\alpha(x) - \beta(x)}\eqno{(3)}\]
and \[L_{n}(x)=\alpha(x)^{n}+\beta(x)^{n}\eqno{(4)}.\]

Using Equations (3) and (4), the following equations are obtained:

\[F_{n}(x)^{2}-F_{n-1}(x)F_{n+1}(x)=(-b)^{n-1}\eqno{(5)},\]
\[L_{n}(x)^{2}-L_{n-1}(x)L_{n+1}(x)=-(-b)^{n-1}(a^{2}x^{2}+4b)\eqno{(6)},\]
\[F_{2n+1}(x)=L_{n+1}(x)F_{n}(x)+(-b)^{n}\eqno{(7)},\]
\[L_{2n+1}(x)=L_{n+1}(x)L_{n}(x)-(-b)^{n}ax\eqno{(8)},\]
\[L_{2n+1}(x)=(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)+(-b)^{n}ax\eqno{(9)},\]
\[F_{n+1}(x)L_{n}(x)=F_{n}(x)L_{n+1}(x)+2(-b)^{n}\eqno{(10)},\]
\[F_{n+1}(x)=\frac{axF_{n}(x)+L_{n}(x)}{2} \eqno{(11)},\]
\[F_{n}(x)=\frac{-axF_{n+1}(x)+L_{n+1}(x)}{2b} \eqno{(12)},\]
\[L_{n+1}(x)=\frac{axL_{n}(x)+(a^{2}x^{2}+4b)F_{n}(x)}{2} \eqno{(13)},\]
\[L_{n}(x)=\frac{-axL_{n+1}(x)+(a^{2}x^{2}+4b)F_{n+1}(x)}{2b} \eqno{(14)}.\]

The following proposition provides the most robust foundation for the proof of the main results.

\vspace{3mm}

\noindent{\bf Proposition 2.1.}
Let $P(x), Q(x) \in {\Bbb Q}[x]$ be polynomials for which the highest-order coefficient is non-negative. If
\[
P(x)^{2}-(a^2 x^2 +4b)Q(x)^{2}=4(-b)^{r_{0}} \eqno{(*)}
\]
for some $r_{0} \in \{0,1\}$, then there exists a non-negative integer $n$ such that
\[b^{\lfloor \frac{n}{2}\rfloor}P(x)=L_{n}(x), b^{\lfloor \frac{n}{2}\rfloor}Q(x)=F_{n}(x)\]
and $n  \equiv r_{0}$ (mod 2).

\vspace{2mm}

\begin{proof}

To begin, let us consider the case $\mathrm{deg}Q(x) \leq 1.$

If $Q(x)=0$, we have $P(x)=2$ and $r_{0}=0$. Thus, we can obtain
\[b^{\lfloor \frac{0}{2}\rfloor}P(x)=L_{0}(x) \ \text{and} \ b^{\lfloor \frac{0}{2}\rfloor}Q(x)=F_{0}(x).\]

If $Q(x) \neq 0$ and $\mathrm{deg} Q(x) = 0$, then the undetermined coefficients method gives us
$P(x)=ax$, $Q(x)=1$, and $r_{0}=1$.
Thus, we can obtain
\[b^{\lfloor \frac{1}{2}\rfloor}P(x)=L_{1}(x) \ \text{and} \ b^{\lfloor \frac{1}{2}\rfloor}Q(x)=F_{1}(x).\]

If $\mathrm{deg}Q(x) =1$, the method of undetermined coefficients gives us
\[P(x)=\frac{a^{2}x^{2}+2b}{b}, Q(x)=\frac{ax}{b} \ \ \text{and} \ \ r_{0}=0. \]
Thus, we can obtain the following: \[b^{\lfloor \frac{2}{2}\rfloor}P(x)=L_{2}(x) \ \text{and} \ b^{\lfloor \frac{2}{2}\rfloor}Q(x)=F_{2}(x).\]

Next, let us consider the case $\mathrm{deg}Q(x) \geq 2.$

Let \[\widetilde{P(x)}=\frac{-ax(a^{2}x^{2}+4b)Q(x)+(a^{2}x^{2}+2b)P(x)}{2b}\]
and
\[\widetilde{Q(x)}=\frac{(a^{2}x^{2}+2b)Q(x)-axP(x)}{2b}.\]

Then,
\[\widetilde{P(x)}^{2}-(a^{2}x^{2}+4b)\widetilde{Q(x)}^{2}=4(-b)^{r_{0}}.\]
In addition, we have $\mathrm{deg}P(x)=\mathrm{deg}Q(x)+1$ because equation  $(*)$  is satisfied.

Let $N=\mathrm{deg}P(x).$ Since equation $(*)$ is satisfied, $c_{0}, c_{1}, \dots, c_{N}$, $d_{0}, d_{1}, \dots, d_{N-1} \in {\Bbb Q}$ exists
such that $P(x)=c_{0}x^{N}+c_{1}x^{N-1} \dots +c_{N}$
and  $Q(x)=d_{0}x^{N-1}+d_{1}x^{N-2} \dots +d_{N-1}$.

Then, we can obtain
\[c_{0}=ad_{0}, c_{1}=ad_{1}, ac_{2}=a^{2}d_{2}+2bd_{0}\]
and
\[ac_{3}=\begin{cases}a^{2}d_{3}+2bd_{1} &  \text{if} \ \mathrm{deg}Q(x) > 2 \\ 2bd_{1} &  \text{if} \ \mathrm{deg}Q(x)=2 \end{cases}.\]

Thus, we have
\[\mathrm{deg}\widetilde{P(x)} \leq \mathrm{deg} P(x)-2\]
and
\[\mathrm{deg}\widetilde{Q(x)} \leq \mathrm{deg} Q(x)-2.\]

Alternatively, we have
\[P(x)=\frac{ax(a^{2}x^{2}+4b)\widetilde{Q(x)}+(a^{2}x^{2}+2b)\widetilde{P(x)}}{2b}\]
and
\[Q(x)=\frac{(a^{2}x^{2}+2b)\widetilde{Q(x)}+ax\widetilde{P(x)}}{2b}.\]

Therefore, we can see that the highest-order coefficient for $\widetilde{P(x)}$ (resp. $\widetilde{Q(x)}$) is a non-negative rational number.
Moreover, we have
\[\mathrm{deg}\widetilde{P(x)} = \mathrm{deg} P(x)-2\]
and
\[\mathrm{deg}\widetilde{Q(x)} = \mathrm{deg} Q(x)-2.\]

By repeating operation $(P(x),  Q(x)) \mapsto (\widetilde{P(x)},  \widetilde{Q(x)})$ $\lfloor \frac{N-1}{2}\rfloor$ times ,
we see that
\[\mathrm{deg}\widetilde{P(x)} \leq 2.\]
In addition, the operations $(P(x),  Q(x)) \mapsto (\widetilde{P(x)},  \widetilde{Q(x)})$ and $ (\widetilde{P(x)},  \widetilde{Q(x)}) \mapsto (P(x),  Q(x))$ are exact opposites.

Finally, if there exists a non-negative integer $n$ such that

\[b^{\lfloor \frac{n}{2}\rfloor}\widetilde{P(x)}=L_{n}(x), b^{\lfloor \frac{n}{2}\rfloor}\widetilde{Q(x)}=F_{n}(x)\]
and $n  \equiv r_{0}$ (mod 2), then we have

\[b^{\lfloor \frac{n+2}{2}\rfloor}P(x)=L_{n+2}(x), b^{\lfloor \frac{n+2}{2}\rfloor}Q(x)=F_{n+2}(x)\]
and $n+2  \equiv r_{0}$ (mod 2) because
\[L_{m+2}(x)=\frac{ax(a^{2}x^{2}+4b)F_{m}(x)+(a^{2}x^{2}+2b)L_{m}(x)}{2}\]
and
\[F_{m+2}(x)=\frac{(a^{2}x^{2}+2b)F_{m}(x)+axL_{m}(x)}{2}\]
applies to any non-negative integer $m$.
This completes the proof.

\end{proof}

\vspace{3mm}

\noindent{\bf Proposition 2.2.} Suppose $b$ can divide $a$. Moreover, we assume that $2b$ can divide $a$ or $d=1$.
Let $P(\sqrt{d}), Q(\sqrt{d}) \in {\Bbb Q}[\sqrt{d}]_{\geq 0}$. If
\[
P(\sqrt{d})^{2}-(a^2 d +4b)Q(\sqrt{d})^{2}=4(-b) \eqno{(**)}
\]
and
\[
\frac{P(\sqrt{d})+a\sqrt{d}}{a^2 d +4b} \in {\Bbb Z}[\sqrt{d}] \\\\\ \text{or} \\\\\ \frac{P(\sqrt{d})-a\sqrt{d}}{a^2 d +4b} \in {\Bbb Z}[\sqrt{d}]\eqno{(***)},
\] then there exists a non-negative integer $n$ such that
\[b^{\lfloor \frac{2n+1}{2}\rfloor}P(\sqrt{d})=L_{2n+1}(\sqrt{d}), b^{\lfloor \frac{2n+1}{2}\rfloor}Q(\sqrt{d})=F_{2n+1}(\sqrt{d}).\]

\vspace{2mm}

\begin{proof}

If $P(\sqrt{d}) \leq a\sqrt{d}$, then we have
\[P(\sqrt{d})=L_{1}(\sqrt{d}), Q(\sqrt{d})=F_{1}(\sqrt{d}).\]

If $P(\sqrt{d}) > a\sqrt{d}$, $(\widetilde{P(\sqrt{d})},  \widetilde{Q(\sqrt{d})})$ satisfies the above conditions $(**)$ and $(***)$.

Moreover, we obtain
\[{P(\sqrt{d})}^{2}-{\widetilde{P(\sqrt{d})}}^{2} \geq \frac{4b(a^{2}d+2b)^{2}-16b^{3}}{4b^{2}}>1.\]
Therefore, we see $\widetilde{P(\sqrt{d})} \leq a\sqrt{d}$ by some repetitions of the operation $(P(\sqrt{d}),  Q(\sqrt{d})) \mapsto (\widetilde{P(\sqrt{d})},  \widetilde{Q(\sqrt{d})})$. Since the operations  $(P(\sqrt{d}),  Q(\sqrt{d})) \mapsto (\widetilde{P(\sqrt{d})},  \widetilde{Q(\sqrt{d})})$ and  $ (\widetilde{P(\sqrt{d})},  \widetilde{Q(\sqrt{d})}) \mapsto (P(\sqrt{d}),  Q(\sqrt{d}))$ are exact opposites, the proof is completed in the same way as Proposition 2.1.

\end{proof}

\section{Resulting proofs}

Like [4], the main results are demonstrated by applying Proposition 2.1, Proposition 2.2 and $(1)$ to $(14)$.

\vspace{3mm}
\subsection{Proof of Theorem 1.2.}
First, it is given that
\[F(\frac{F_{i}(x)}{F_{i+1}(x)}) \in {\Bbb Z}[x] \ (i \in {\Bbb N})\]
and
\[F(-\frac{F_{i+1}(x)}{bF_{i}(x)}) \in {\Bbb Z}[x] \ (i \in {\Bbb N}^{+}).\]

If $i=0$, the result is evident. However, if $i > 0$, by using $(1)$ and $(5)$, we can obtain
\[F(\frac{F_{i}(x)}{F_{i+1}(x)})=\frac{F_{i}(x)F_{i+1}(x)}{F_{i+1}(x)^{2}-(axF_{i+1}(x)+bF_{i}(x))F_{i}(x)}\]

\[\overset{(1)}=\frac{F_{i}(x)F_{i+1}(x)}{F_{i+1}(x)^{2}-F_{i+2}(x)F_{i}(x)}\overset{(5)}=\frac{F_{i}(x)F_{i+1}(x)}{(-b)^{i}}\]
\[F(-\frac{F_{i+1}(x)}{bF_{i}(x)})=\frac{-bF_{i}(x)F_{i+1}(x)}{bF_{i}(x)(axF_{i+1}(x)+bF_{i}(x))-bF_{i+1}(x)^{2}}\]

\[\overset{(1)}=\frac{-bF_{i}(x)F_{i+1}(x)}{bF_{i}(x)F_{i+2}(x)-bF_{i+1}(x)^{2}}\overset{(5)}=\frac{F_{i}(x)F_{i+1}(x)}{(-b)^{i}}.\]
Following the recurrence relation described by Equation (1), $F_{i}(x) \in b^{\lfloor \frac{i}{2}\rfloor}{\Bbb Z}[x]$,  $F(\frac{F_{i}(x)}{F_{i+1}(x)})$ exists as $F(-\frac{F_{i+1}(x)}{bF_{i}(x)})$ polynomials over ${\Bbb Z}$.
Next, if $F(t(x))=k(x)$ ($k(x)$ is a polynomial over ${\Bbb Z}$) for some rational function $t(x) \in {\Bbb Q}(t)$, we can show that
\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]
If $k(x)=0$, then
\[\frac{t(x)}{1-axt(x)-bt(x)^2}=0.\]
Hence,
\[t(x)=0=\frac{F_{0}(x)}{F_{1}(x)}\]
If $k(t) \neq 0$, then
\[\frac{t(x)}{1-axt(x)-bt(x)^2}=k(t).\]
Hence,
\[bk(x)t(x)^{2}+(axk(x)+1)t(x)-k(x)=0.\]
Furthermore,
\[t(x)=\frac{-(axk(x)+1)\pm \sqrt{(axk(x)+1)^2+4bk(x)^2}}{2bk(x)}.\]
Here, because $t(x)$ is a rational function over ${\Bbb Q}$, there exists a polynomial $M(x) \in {\Bbb Q}[x]$ for which the highest-order coefficient is non-negative such that
\[(axk(x)+1)^{2}+4bk(x)^{2}=M(x)^{2}.\]
This allows us to obtain
\[\{ (a^{2}x^{2}+4b)k(x)+ax\}^{2}-(a^{2}x^{2}+4b)M(x)^{2}=4(-b).\]
Thus, according to Proposition 2.1, there exists a non-negative integer $n$ such that
\[M(x)=\frac{F_{2n+1}(x)}{b^{n}}, (a^{2}x^{2}+4b)k(x)+ax=\pm \frac{L_{2n+1}(x)}{b^{n}}.\]

From Equation (9), \[(a^{2}x^{2}+4b)k(x)+ax=\pm \frac{(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)+(-b)^{n}ax}{b^n}.\]
Since $k(x) \in {\Bbb Z}[x]$ and $\frac{F_{n}(x)F_{n+1}(x)}{b^{n}} \in {\Bbb Z}[x]$, this means that
\[(a^{2}x^{2}+4b)k(x)+ax=\frac{L_{2n+1}(x)}{(-b)^{n}}\]
for $n \geq 1$ given that $k(x) \neq 0$.
Additionally, by using Equation (9) , \[k(x)=\frac{F_{n}(x)F_{n+1}(x)}{(-b)^{n}}.\]

Consequently, we can obtain
\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}+(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)} \ (n \geq 1) \eqno{(A)}\]
or
\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}-(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)} \ (n \geq 1)\eqno{(B)}.\]

Regarding (A) and (B), by using Equations (7), (10), (11), and (12), we can obtain
\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]

If $n$ is even, then (A) is given as

\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}+(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)}\overset{(7)}=\frac{-axF_{n+1}(x)+L_{n+1}(x)}{2bF_{n+1}(x)}\]
\[\overset{(12)}=\frac{F_{n}(x)}{F_{n+1}(x)},\]

and (B) is given as

\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}-(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)}\overset{(7)(10)}=\frac{-axF_{n}(x)-L_{n}(x)}{2bF_{n}(x)}\]
\[\overset{(11)}=-\frac{F_{n+1}(x)}{bF_{n}(x)}.\]

If $n$ is odd, then (A) is given as

\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}+(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)}\overset{(7)(10)}=\frac{-axF_{n}(x)-L_{n}(x)}{2bF_{n}(x)}\]
\[\overset{(11)}=-\frac{F_{n+1}(x)}{bF_{n}(x)},\]

and (B) is given as

\[t(x)=\frac{-axF_{n}(x)F_{n+1}(x)-(-b)^{n}-(-1)^{n}F_{2n+1}(x)}{2bF_{n}(x)F_{n+1}(x)}\overset{(7)}=\frac{-axF_{n+1}(x)+L_{n+1}(x)}{2bF_{n+1}(x)}\]
\[\overset{(12)}=\frac{F_{n}(x)}{F_{n+1}(x)}.\]

\vspace{3mm}
\subsection{Proof of Theorem 1.3.} It is given that
 \[L(\frac{F_{i}(x)}{F_{i+1}(x)}), L(\frac{L_{i}(x)}{L_{i+1}(x)}), L(-\frac{L_{i+1}(x)}{bL_{i}(x)}) \in {\Bbb Z}[x] \ (i \in {\Bbb N}) \text{ and  }  L(-\frac{F_{i+1}(x)}{bF_{i}(x)}) \in {\Bbb Z}[x] \  (i \in {\Bbb N}^{+}).\]

If $i=0$, the result is evident. However, if $i > 0$, by using Equations $(1)$ and $(5)$, we can obtain
\[L(\frac{F_{i}(x)}{F_{i+1}(x)})=\frac{2F_{i+1}(x)^{2}-axF_{i}(x)F_{i+1}(x)}{F_{i+1}(x)^{2}-F_{i}(x)(axF_{i+1}(x)+bF_{i}(x))}\overset{(1)}=\frac{2F_{i+1}(x)^{2}-axF_{i}(x)F_{i+1}(x)}{F_{i+1}(x)^{2}-F_{i}(x)F_{i+2}(x)}\]
\[\overset{(5)}=\frac{2F_{i+1}(x)^{2}-axF_{i}(x)F_{i+1}(x)}{(-b)^{i}}\]
\[L(\frac{L_{i}(x)}{L_{i+1}(x)})=\frac{2L_{i+1}(x)^{2}-axL_{i}(x)L_{i+1}(x)}{L_{i+1}(x)^{2}-L_{i}(x)(axL_{i+1}(x)+bL_{i}(x))}\overset{(2)}=\frac{2L_{i+1}(x)^{2}-axL_{i}(x)L_{i+1}(x)}{L_{i+1}(x)^{2}-L_{i}(x)L_{i+2}(x)}\]
\[\overset{(6)}=\frac{L_{i+1}(x)(2L_{i+1}(x)-axL_{i}(x))}{-(-b)^{i}(a^{2}x^{2}+4b)}\overset{(13)}=\frac{L_{i+1}(x)F_{i}(x)}{-(-b)^{i}}\]
\[L(-\frac{L_{i+1}(x)}{bL_{i}(x)})=\frac{L_{i}(x)(2bL_{i}(x)+axL_{i+1}(x))}{L_{i}(x)(aL_{i+1}(x)+bL_{i}(x))-L_{i+1}(x)^{2}} \]
\[\overset{(2)}=\frac{L_{i}(x)(2bL_{i}(x)+axL_{i+1}(x))}{L_{i}(x)L_{i+2}(x)-L_{i+1}(x)^{2}}\overset{(6)(14)}=\frac{L_{i}(x)F_{i+1}(x)}{(-b)^{i}}\]
\[L(-\frac{F_{i+1}(x)}{bF_{i}(x)})=\frac{F_{i}(x)(2bF_{i}(x)+axF_{i+1}(x))}{F_{i}(x)(bF_{i}(x)+axF_{i+1}(x))-F_{i+1}(x)^{2}}\]
\[=\frac{F_{i}(x)(2bF_{i}(x)+axF_{i+1}(x))}{F_{i}(x)F_{i+2}(x)-F_{i+1}(x)^{2}}=\frac{F_{i}(x)(2bF_{i}(x)+axF_{i+1}(x))}{-(-b)^{i}}.\]

Given that $F_{i}(x), L_{i}(x) \in b^{\lfloor \frac{i}{2}\rfloor}{\Bbb Z}[x]$ from the recurrence relations described by Equations (1) and (2),
\[L(\frac{F_{i}(x)}{F_{i+1}(x)}), L(\frac{L_{i}(x)}{L_{i+1}(x)}), L(-\frac{L_{i+1}(x)}{bL_{i}(x)}) \in {\Bbb Z}[x] \ (i \in {\Bbb N}) \text{ and  }  L(-\frac{F_{i+1}(x)}{bF_{i}(x)}) \in {\Bbb Z}[x] \  (i \in {\Bbb N}^{+}).\]

Next, given that $k(x)$ is a polynomial over ${\Bbb Z}$, if $L(t(x))=k(x)$ for some rational function $t(x) \in {\Bbb Q}(t)$, we can see that

\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}, \frac{L_{i}(x)}{L_{i+1}(x)}, -\frac{L_{i+1}(x)}{bL_{i}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]

Thus, if $k(x)=0$, then

\[\frac{2-axt(x)}{1-axt(x)-bt(x)^2}=0.\]

Therefore,
\[t(x)=\frac{2}{ax}=\frac{L_{0}(x)}{L_{1}(x)}.\]
Alternatively, if $k(t) \neq 0$, then
\[\frac{2-axt(x)}{1-axt(x)-bt(x)^2}=k(t).\]
Hence,
\[bk(x)t(x)^{2}+ax(k(x)-1)t(x)+2-k(x)=0.\]
Therefore,
\[t(x)=\frac{-ax(k(x)-1) \pm \sqrt{a^{2}x^{2}(k(x)-1)^{2}-4bk(x)(2-k(x))}}{2bk(x)}.\]
Here, because $t(x)$ is a rational function over ${\Bbb Q}$, there exists a polynomial $M(x) \in {\Bbb Q}[x]$ for which the highest-order coefficient is non-negative such that
\[a^{2}x^{2}(k(x)-1)^{2}-4bk(x)(2-k(x))=M(x)^{2}.\]
Then, we can obtain
\[M(x)^{2}-(a^{2}x^{2}+4b)(k(x)-1)^{2}=4(-b).\]
Thus, according to Proposition 2.1, there exists a non-negative integer $n$ such that
\[M(x)=\frac{L_{2n+1}(x)}{b^{n}}, k(x)-1=\pm \frac{F_{2n+1}(x)}{b^{n}}.\]

Hence, we can obtain
\[t(x)=\frac{-axF_{2n+1}(x)+L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})} \ (n \geq 0)\eqno{(C)},\]

\[t(x)=\frac{-axF_{2n+1}(x)-L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})} \ (n \geq 0)\eqno{(D)},\]

\[t(x)=\frac{axF_{2n+1}(x)+L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})} \ (n \geq 1) \eqno{(E)},\]
or
\[t(x)=\frac{axF_{2n+1}(x)-L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})} \ (n \geq 1) \eqno{(F)}.\]

In the cases of (C) to (F), using Equations (7), (9), (10), (13), and (14) gives us
\[t(x) \in \bigl\{\frac{F_{i}(x)}{F_{i+1}(x)}, \frac{L_{i}(x)}{L_{i+1}(x)}, -\frac{L_{i+1}(x)}{bL_{i}(x)}\bigr\}_{i \in {\Bbb N}} \text{ or  }t(x) \in \bigl\{-\frac{F_{i+1}(x)}{bF_{i}(x)}\bigr\}_{i \in {\Bbb N}^{+}}.\]

Then, if $n$ is even, (C) is given as

\[t(x)=\frac{-axF_{2n+1}(x)+L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{-axL_{n+1}(x)F_{n}(x)+(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{2bF_{n+1}(x)L_{n}(x)}\]
\[\overset{(14)}=\frac{F_{n}(x)}{F_{n+1}(x)}.\]

If $n$ is odd, (C) is given as

\[t(x)=\frac{-axF_{2n+1}(x)+L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})}\overset{(7)(9)}=\frac{-axL_{n+1}(x)F_{n}(x)+(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{2bL_{n+1}(x)F_{n}(x)}\]
\[\overset{(14)}=\frac{L_{n}(x)}{L_{n+1}(x)}.\]

If $n$ is even, (D) is given as

\[t(x)=\frac{-axF_{2n+1}(x)-L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{-axL_{n}(x)-(a^{2}x^{2}+4b)F_{n}(x)}{2bL_{n}(x)}\]

\[\overset{(13)}=-\frac{L_{n+1}(x)}{bL_{n}(x)}.\]

If $n$ is odd, (D) is given as

\[t(x)=\frac{-axF_{2n+1}(x)-L_{2n+1}(x)}{2b(F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{-axF_{n+1}(x)L_{n}(x)-(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{2bL_{n+1}(x)F_{n}(x)}\]

\[\overset{(13)}=-\frac{F_{n+1}(x)}{bF_{n}(x)}.\]

If $n$ is even, (E) is given as

\[t(x)=\frac{axF_{2n+1}(x)+L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{axF_{n+1}(x)L_{n}(x)+(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{-2bL_{n+1}(x)F_{n}(x)}\]

\[\overset{(13)}=-\frac{F_{n+1}(x)}{bF_{n}(x)}.\]

If $n$ is odd, (E) is given as

\[t(x)=\frac{axF_{2n+1}(x)+L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{axF_{n+1}(x)L_{n}(x)+(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{-2bF_{n+1}(x)L_{n}(x)}\]

\[\overset{(13)}=-\frac{L_{n+1}(x)}{bL_{n}(x)}.\]

Lastly, if $n$ is even, (F) is given as

\[t(x)=\frac{axF_{2n+1}(x)-L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})}\overset{(7)(9)}=\frac{axL_{n+1}(x)F_{n}(x)-(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{-2bL_{n+1}(x)F_{n}(x)}\]

\[\overset{(14)}=\frac{L_{n}(x)}{L_{n+1}(x)}.\]

However, if $n$ is odd, (F) is given as

\[t(x)=\frac{axF_{2n+1}(x)-L_{2n+1}(x)}{2b(-F_{2n+1}(x)+b^{n})}\overset{(7)(9)(10)}=\frac{axL_{n+1}(x)F_{n}(x)-(a^{2}x^{2}+4b)F_{n+1}(x)F_{n}(x)}{-2bF_{n+1}(x)L_{n}(x)}\]

\[\overset{(14)}=\frac{F_{n}(x)}{F_{n+1}(x )}.\]

\vspace{3mm}
\subsection{Proof of Corollary 1.4.}

In the proof of Theorem 1.2, we apply Proposition 2.2 instead of applying Proposition 2.1 and set $x = \sqrt{d}$ to complete the proof.

\vspace{3mm}
\subsection{Proof of Corollary 1.5.}
In the proof of Theorem 1.3, we apply Proposition 2.2 instead of applying Proposition 2.1 and set $x = \sqrt{d}$ to complete the proof.

\section{Acknowledgement}
I thank Editage for the English language editing.

redpajama_set_name: RedPajamaArXiv


text:
thestoryisthething
You didn't go one-on-one with the One-Armed Bandit
Gary Mays never let having one arm stop him from being a power hitter and a catcher base-runners feared.
Originally published in the July 24, 2016 Charleston Gazette-Mail
By Douglas Imbrogno
So, this is a long-ish story about a one-armed guy from No. 1 Holler in rural Kanawha County. A guy who was once recruited by the Harlem Globetrotters and turned them down.
This is a guy who — in a legendary high school game — once flicked sweat from his stump into the eyes of future NBA great Elgin Baylor, stole the ball and swept past him to put the game away.
That's how the story of that game goes, and Gary Mays — aka The One Arm Bandit — is sticking to it, even if the official account of that long-ago night got the game-ending steal and layup all wrong, he said.
And don't even talk about him being handicapped.
"People see you with one arm and they say, 'He's handicapped.' Well, they don't know if I'm handicapped or not!" Mays said.
Long-ish newspaper stories are supposed to be out of fashion in this short-attention-span age. But Gary Mays' story can't be told in tweets or a few paragraphs.
Partly, because he is an 81-year-old man of many, many words. In fact, you can hardly get him to stop telling story after story after story once you get him on the phone from his home in Ft. Washington, Maryland. Fact-checking follow-up calls turn into another avalanche of anecdotes.
Maybe you'll read about some of those stories when he finishes the autobiography he's writing, "The Amazing Gary Mays," which also happens to be the title of a folk song someone released about him.
And the GoFundMe campaign (gofundme.com/garymaysbook) to shoot a documentary and pay for that book's publishing about his colorful life could use a lot more donations if his story is ever to reach a wider audience.
Meanwhile, local readers can get a flavor of the man in person. He is featured in "The Block Speaker Series: A Personal Perspective of African American Life," starting 6 p.m. Thursday at the West Virginia Culture Center Archives, sponsored by the West Virginia Center for African-American Art & Culture.
But for right now — what story to start with?
Maybe that day in 1954 when the Bandit put the kibosh on Rabbit, as Elgin Baylor was known, whose already legendary prowess was shut down by a one-armed nuisance?
Or maybe Mays' career as a high school and sandlot baseball folk wonder as a home-run-hitting baseball catcher with a rifle for an arm? Or how in 1955 he rocked a major league baseball tryout but didn't earn a contract, most probably because his skin was the wrong color?
Perhaps the best place to start is that day at age 5, in Burnwell, on the other side of the mountain where Jerry West was born in Chelyan, when young Garrett Mays lost his left arm.
That was, after all, what was to transform the boy into The One Arm Bandit.
"I was born in 1935 in the middle of the Depression without a father," Mays said. "It was hell."
His mother moved to Washington, D.C., and remarried. Mays lived with his grandmother in Burnwell. His uncles worked in the mines.
The skinny black kid played with the white kids. He was tight with two of them, the brothers Billy and Boo Hudson.
"You know, man, I lived in No. 1 Holler at Burnwell, and I didn't know what color I was until I got ready to go to school," Mays said.
"That was the most disappointing thing that ever happened to me in my life is when I turned 5 in September 1940 and I had to go to school. Boo went up to the white school, and I went to the black school."
There was no segregation in No. 1 Holler, he said. "We didn't know anything about it because Billy and Boo, we did everything together."
The boys had a great time that Halloween. Then, election day rolled around. A local man with a car was supposed to come by to take his grandma to vote. Mays was told to keep an eye out for him.
But one of his uncles had borrowed a friend's new pump-action Winchester to go squirrel hunting. The uncle returned with the gun and some squirrels, which Mays was eager to check out.
"My uncle laid the gun down on the bench on the back porch and the shells rolled off and dropped in the water and my uncle's friend Rufus Harris ran down the steps and shook the porch," Mays said. "The gun fell over and shot my arm off. It hit right in my elbow."
The man who had come to ferry his grandma to vote happened to be out front.
"God was with me because if he hadn't been there in front of the house, I might have died," Mays said.
They drove the boy to a doctor in Mahan, applied a tourniquet to the bloody stump, then hustled him to Charleston General Hospital.
"A red-headed white lady was crying like a baby, and I passed out," he recalled. "The only thing I remember is waking up the next day. And I had one arm. And I said, 'Well, I can deal with this, I guess.'"
He dealt with it. He loved baseball and would make homemade balls out of rolled-up rags, newspaper and black electrical tape.
His Uncle Elijah was a pitcher, his Uncle Charles a catcher, on the segregated sandlot baseball teams of the day. They would bring home a precious thing: an actual stitched baseball.
Elijah would pitch, throwing heat. Charles would catch. Mays would crouch along with his uncle behind the catcher's glove as the balls slammed into his uncle's mitt.
"He just got the fear out of me," Mays said. "He made sure I wasn't afraid of the ball."
By eighth grade, Mays learned to handle another ball. He got so good at basketball playing for the old Cabell Junior High School on Charleston's West Side that he made all-state.
By 1950, he moved to Washington. D.C., for good to live with his mother. He started ninth grade at the city's Armstrong Technical High School, a mecca for black ballplayers in the city's still-segregated schools.
He made the junior varsity team, but he was good enough to scrimmage with the varsity squad.
"And I was stealing the ball from all the varsity guys," Mays said with a chuckle. "Mr. Charles Baltimore, my coach, he said, 'Damn, that's a one-armed bandit!' That's how I got that name."
Bandit ended up being named team captain his senior year as a momentous game approached.
The year was 1954, and segregation was still the law and custom of the land. A tidal wave was coming in May of that year with the Brown vs. The Board of Education of Topeka case, the U.S. Supreme Court decision that declared state laws establishing separate public schools for black and white students to be unconstitutional.
But the wave had not yet hit. So Armstrong was to meet Springarn High School for the basketball title in Division II, home to D.C.'s black teams. (Division 1 was for white teams).
The problem for Armstrong was that the undefeated Springarn had a lucky rabbit. Or rather Elgin "Rabbit" Baylor, who was averaging 37.5 points per game. In two regular season bouts, Baylor — who was friends with Mays from pickup games around the city — had helped to wallop Armstrong, amassing 44 and 45 points in the games.
Both Baylor and Mays were already high school sports icons in D.C., so things got interesting.
Baltimore came up with a "box-in-one" defense. This amounted to four of the Armstrong players jamming things up in the shooting lane, while Bandit went one-on-one with Rabbit everywhere he went.
"Coach said: 'Look, when he goes to the bathroom, I want you to go with him.'" Mays recalled. "I said, 'What you mean?' He said, 'I want you to be that close to him!'"
Mays took the cue. He also added some psych-out mojo to the mix when he got close to Baylor.
"I took my nub and rubbed it right in the crack of his butt. And he was no good after that. He didn't like that!" May said, cackling with amusement.
With about 25 seconds in the game and Armstrong up 48-47, Baylor got the ball. Mays had, so far, held him to 18 points.
Baylor was pissed, said Mays. The fans in the packed gym were on their feet.
"It's you and me," Baylor told Mays, meeting him just past half court.
Mays knew Baylor had a habit of flicking his head. He watched for it.
"When he did that I threw sweat from my nub in his eye, stole the ball and went down his right side and made a layup, and we beat 'em 50-47," he said.
Mays said a sportswriter who wasn't even in the gym that night wrote that a couple of other Armstrong players had made the steal and the score based on what he had been told.
"I never got the right ink. The guy that wrote the article wasn't even in the game. ESPN got it straight," Mays said.
He was referring to a Feb. 28, 2011, profile by ESPN writer Patrick Hruby, a high-profile attempt to tell the tale of a remarkable, trailblazing black athlete who never even went on to a major league career.
Although he came close.
A lot of it had to do with — to borrow an immortal line from that ESPN profile — Mays' one arm. Or as Hruby writes: "Granted, an overstuffed anaconda of a right arm to shame Hulk Hogan …"
Mays' first love was always baseball. At age 13, he shagged balls and was a bat boy for the sandlot team the Georgetown Panthers in D.C., eventually landing a spot on the team. He also made the Armstrong High School baseball team.
Center field was his sweet spot. But at some point, out of need, a coach called him into home base to fill in as a catcher.
A one-armed baseball catcher is a hard thing to envision, much less describe. But the base runners who tried to steal on him certainly knew how the process worked, usually to their dismay.
Preston Estes, a childhood friend from Mays' Charleston days, gave it a go:
"If there was a runner on base that was attempting to steal — which very few were successful at — he would catch the ball, toss the ball in the air, put the glove under his nub on his left side and catch the ball as it came down."
Then, he would throw.
"No one was hardly ever successful in stealing on him," Estes said. "It was phenomenal. A lot of us were amazed at the abilities he had. We were just sort of mesmerized. Couldn't keep our eyes off of him, to tell you the truth."
For all his tale-telling, Mays is a pretty self-effacing guy. But he readily admits that players did not want to try to steal on him on his heyday.
Sure, he had one arm. But about that arm: "Look, if you got a shotgun you have no problem. I had a cannon," he said.
Mays was one of the top baseball prep players in D.C. his senior year, batting .375, yielding zero stolen bases and no errors, according to a Washington Daily News story. The Bandit had also earned profiles in Jet and Ebony magazines.
The Daily News story on Mays came out in 1955, a year after he had graduated. The newspaper was reporting on its two-day Baseball Talent Hunt, which attracted major league scouts and hundreds of boys, ages 16 to 21, with dreams of the big leagues.
The headline on the story: "One Armed Garry Mays Voted Outstanding in Camp."
The story reported that Mays threw out one runner and a few minutes later lofted a 350-foot home run drive over the center fielder's head. (This was not an unusual occurrence for Mays, who had stroked multiple one-arm home runs in games several times.)
The scouts unanimously voted Mays the outstanding player of the camp. The article made a point to note that "the scouts weren't being just sympathetic because of May's handicap."
But the story went on to say that "it is unlikely, however, that any of the scouts will ask him to sign a contract as he remains a catcher," because Mays "would be at a disadvantage with pitched balls thrown at him in the dirt."
To this day, Mays scoffs at the statement. He could dig balls out of the dirt, he said.
Was it racism?
He is matter of fact in his response: "Yeah, sure. It was part of this country."
In another profile about him, he sounded a more philosophical tone: "That is the way it was, and no one ever said life was fair."
In the book he is writing, he has gotten to a section where he talks about the prejudice and racism of those days, which, of course, is still a burning topic in American life and daily news headlines.
"Racism is something," he said. "I get filled up when I hear all this stuff because I lived it. When you live racism, it's terrible. We wasted a lot of time on racism."
Being the multi-sport guy he was, Mays tried to make a basketball career of it, too.
After graduating from Armstrong, Mays followed Baylor — who had always been a friend, despite the nub-rubbing incident — along with some other notable D.C. players, Warren Williams and R.C. Owens, to play at the College of Idaho in Caldwell.
While Mays rode the bench a lot, the team went on to its first undefeated season in Northwest Conference history. The players went their separate ways when the coach was fired after the '54-55 season amid questions about bringing academically unqualified players to the team.
Baylor went on to have a legendary career. Mays, who had little interest in academics, dropped out and headed back to D.C.
He got a couple letters from Abe Saperstein, legendary coach and founder of the Harlem Globetrotters, proposing that Mays join the world-traveling basketball clowns. He'd seen the Bandit play.
Mays didn't pursue it.
"I never liked clowning, no way," he said. "I'd be that one-arm freak show."
He kept playing sandlot baseball for many years. He ran numbers, an under-the-table illegal lottery; opened his own liquor store. He was a cab and limo driver for years in D.C., driving around many a political luminary. He founded his own construction company called Specialized Services.
Most of the jobs he had was him making his own way, a one-armed guy with one whole lot of determination.
"All my life I had to hustle," he said.
Proving himself was something he used as a challenge to make some money. He'd bet people he could tie his shoes faster than you. Or put on his wristwatch faster than you: "Want to bet $20?"
"I don't know how much money I've won off people trying to beat me putting a watch on. That was my hustle," he said, chuckling. "Sometimes for the wristwatch."
His reputation on the court and field precedes him, though. You'll find him listed at the top of the heap on the D.C. Basketball Blog's list of the "D.C. Basketball's Best 55 Schoolboy Players of All Time."
And Dave McKenna, a Washington City Paper writer, who has been a one-man story archivist and champion for the legend of his "hero and friend" Mays, described him in one article as "one of the greatest all-around athletes ever to call the city home."
An old friend and teammate from his Armstrong High days, George Williams, has his own up-close perspective.
"You had to see it to believe, man," said Williams, a lifelong D.C. resident., who played basketball for both Armstrong and Spingarn and has known Mays for 70 years.
Williams said you had to understand the basketball scene in the nation's capital during the 1950s, where the proving ground was the pickup games on the city's playgrounds.
"The basketball in Washington was not played so much in the high school as it was on the playgrounds because you had so many phenomenal players that did not go to high school," he said.
The "One Arm Bandit" brought the crowds.
When word got out that Mays was playing on some playground, you had 400 to 500 people come to see him, recalled Williams. "They said, 'Man, that one-armed guy is playing!' After awhile, they were breathless. He would just shut them up completely with what he could do on the basketball court."
The crowds are all gone now, but the memories are still strong. Google up McKenna's stories on Mays in Washington City Paper and scan through the comments to hear the delighted recollections of D.C. residents who witnessed Mays on the court or the diamond.
Nowadays, Mays enjoys spending time with his two grown children and his wife as he attends daily to the book he is writing about his life while his black Labrador retriever, 'Bandit Garrett Mays,' lounges nearby or barks loud enough to interrupt a phone interview.
Mays credits his grandmother with the advice that set his head straight after he came home from that Charleston hospital so long ago as a one-armed boy.
"My grandmother told me: 'You got one arm. But you don't need but one arm,'" he recalled."'So make do with the one.'"
NOTE: Gary Mays died at age 82 on Monday, Jan. 15, 2018. See Dave McKenna's touching tribute.
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Welcome to a portfolio site of long-form articles and recent writings of freelance writer and multimedia producer Douglas Imbrogno. In this age of teensy tweets and hyper-short newsbits, this site celebrates long-form magazine and essay writing on a wide-range of topics and personality profiles including pieces from my old web magazine and beyond. Subscribe to the site for updates. You can reach me via the Contact page.
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This fabulous roof structure has come up a treat. Tile to Colorbond and metal re-roof transformation in Kincumber, using 0.48bmt Custom Orb and Kliplok sheets. The main roof with Custom Orb sheets are in the colour Shale Grey and the Kliplok flat roof areas are in Surfmist. No gutters were replaced on this property.
See the Kincumber Colorbond Metal Roof here.


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Retired Eckerd College Professor Dr. David Hastings Promotes Banning of Single-Use Plastics in Local Communities
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GAINESVILLE, FL / ACCESSWIRE / October 22, 2020 / Dr. David Hastings retired Eckerd College professor, weighs in on the debate surrounding the banning of single-use plastics within communities in Florida.
With single-use plastic bans in Florida being rescinded almost as quickly as they are announced and implemented, it's a fight that retired Eckerd College professor Dr. David Hastings simply isn't prepared to lose. With a focus on the devastating and long-lasting impact of single-use and other plastics on the ocean environment, the marine science expert weighs in on the debate and explains more about the issue.
"Calling for an end to single-use plastics in Florida is not new, but it's an ongoing struggle," suggests retired Eckerd College professor Dr. David Hastings, speaking from his home in Gainesville, Florida, "and one which continues to face almost constant opposition."
According to marine science professor Dr. David Hastings, previously implemented single-use plastic bans in Surfside, Palm Beach, and elsewhere in Florida have already been overturned.
Even a ban in the retired Eckerd College professor's home city of Gainesville seemingly met with such opposition that it was rescinded not long after it was implemented, the Alachua County-based expert reports. "Alachua County very quickly called for an end to a single-use plastics ban which had not long been in place, after which Gainesville simply followed suit," reveals Dr. Hastings.
Of some solace to Dr. Hastings, however, he says, is the fact that municipalities themselves aren't simply choosing to overturn rules aimed at reducing plastic waste and protecting the environment; they are backtracking on policy decisions to avoid legal action. "The Florida Retail Federation has pushed back," explains theretired Eckerd College professor, "stating that single-use plastic bans violate state laws."
Threats to sue local governments, Dr. Hastings says, made by the controversial trade association-which represents retailers such as Publix, Walmart, and Target-have proved to be an early death knell for what is, he's keen to stress, an extremely important cause.
The Florida Retail Federation has, it's reported, cited a law that prevents local governments from regulating certain plastics determined to be of a single-use nature, including disposable plastic bags and so-called auxiliary containers.
"What the Florida Retail Federation and the corporations that it represents are failing to acknowledge, however, is the catastrophic rate at which single-use and other plastics are already destroying delicate marine environments, both in the U.S. and elsewhere around the world," stresses expert and retired marine science professor, Dr. David Hastings.
Retired Eckerd College professor Dr. David Hastings is, therefore, calling, again, for bans on single-use plastic items in Florida and its communities to be implemented and upheld.
"We must act now if we are to turn the tide on plastic pollution in our oceans," concludes Dr. David Hastings, "or we risk catastrophic damage to marine environments which may well prove irreversible."
Dr. David Hastings is a retired college professor who landed on the shores of Tampa Bay, Florida, in 2000 to teach marine science and chemistry at Eckerd College. Eckerd College is a prestigious liberal arts institution located in nearby St. Petersburg. Passionate about conservation, Dr. David Hastings is, though retired, still extremely keen to champion the importance of experts actively engaging in policy conversations in order to ensure the future of the planet and its oceans, and in the imperative nature of taking direct action to combat further damage to the world's precious marine environments as a result of plastic pollution.
Caroline Hunter
Web Presence, LLC
SOURCE: David Hastings
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MICHAEL FROM THE EAST Drops Sweet RnB Cut THAT'S IT
NEON DREAMS Drop New Single TURN BACK NOW
Tugan Shows Up With NEED IT
Perttu Is Finding His DAMAGE CONTROL
NOVA HALL Shares COLLATERAL
FM Presents | CHRIS SCHOLAR
Celebrities Who Are Hooked On Online Gaming
1st November 2019 No Comments
Online gaming is a past time which has got the whole world hooked, and that includes celebrities. One of the great things about online gaming is that there is something to suit everybody, with each person having their own preferences. So, let's delve into the world of celebrities and see who loves playing online and what they're enjoying.
Drake, real name Aubrey Graham, is a man of many talents and interests. The rapping megastar is said to find time between recording sessions to kick back and play Fortnite and was even seen on a live stream on Twitch with gaming legend Ninja. However, while everyone seems to be getting stuck into battle royale titles, Drake offered a fantastic insight into his life as a gamer when he mentioned League of Legends, which makes him hardcore in our opinion.
Ben Affleck has always had a love for card games, and he took his passion to the tables of many a table at land-based casinos to play both blackjack and poker. However, as we all know, there are now several online casinos which offer an exciting and comprehensive service such as PartyCasino, who also have Live Dealer tables as well. Therefore, it's not that shocking to find out that after spending so much of his free time, and money, at land-based venues, Affleck, knowing all the rules of blackjack, has taken his love for the game into the online world. So, you might be lucky and end up sitting next to him in the virtual world!
There's more to Vin Diesel than meets the eye when it comes to gaming, with the Fast and the Furious star owning his own video game development studio. Tigon Studios produced The Chronicles of Riddick: Escape from Butcher Bay, ranked as one of the top movie to game ports of all time. Vin Diesel, as you would expect, is a big gamer himself, and has played multiple titles on console and PC, with the Call of Duty series being one of his favourites.
Pamela Anderson came to the fore when she starred in Baywatch, and since then she's become known for being a headstrong woman who does what she wants when she wants. Anderson, therefore, is always looking to have a good time, and casinos became venues she would visit regular, both offline and online. The Baywatch star married professional poker player Rick Salomon back in 07, with many suggesting she did so after losing a game of poker to him.
Mila Kunis, the voice behind Meg Griffin on Family Guy since 1999, is often ranked as one of the most glamorous women on the planet, but she's also a bit of a nerd when it comes to gaming. Several years ago, Kunis recalled her love for Blizzard's World of Warcraft game (WoW), a title often associated with hardcore gamers. Kunis even went as far as saying she would regularly play all-day alongside her friends, as they went around wiping raids.
Gaming is a hobby which everyone will likely enjoy at some point in their life, especially celebrities, when they have free time on their hands. While most celebs are into online gaming in the form of playing poker, blackjack and roulette, there are plenty who enjoy video gaming on console and PC.
GAMESINTERNETSHOWBIZ


We've got our goal of season. True, it's only the second week of August, but we have it. Oh, and it's probably the best goalkeeper goal of all time, as well.
It'd be one thing if Zlatan pulled this off… but a goalkeeper? Down 2-1 against the league's reigning champs, PSV? On the final play of the game?


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This Runaways exclusive spotlights the team's new boomer superhero mentor
All images: Marvel Comics
Runaways has always had a tenuous relationship with its superhero identity. The book has always been best when it downplays superhero elements to focus on the challenges faced by a group of teens and young adults on the run from authority figures, and writer Rainbow Rowell fully understands this strength. But it's still a comic published by Marvel, set in a superhero universe, and starring superpowered characters, so eventually it's going to dive back into the genre. The latest arc of Rowell's outstanding run leans into the superhero element harder than the series ever has before, bringing the group under the wing of a Los Angeles superhero looking to revive his team of young costumed crusaders from the '90s: Doc Justice And The J-Team. This story is a fascinating journey through teen superhero team history, and it's a particularly strong showcase for Gertrude, who is forced to sit on the sidelines because none of the J-Team's old costumes fit her plus-sized body.
This exclusive preview of next week's Runaways #29 offers some more background on the team's new mentor as Gertrude bonds with Doc Justice's long-time assistant, Matthew. Artist Andrés Genolet and colorist Dee Cunniffe kick off the issue with a burst of energy as they show the J-Team in action, showing quick snapshots of excitement that contrast with the drabness of Gertrude and Matthew monitoring the team. The first issue of this arc featured the return of Rowell's original Runaways collaborator Kris Anka, who came back to debut the costumes he designed for the J-Team, and it highlighted just how well Genolet has held on to the bright, stylish aesthetic that Anka brought to this series. The character acting in these pages is full of expression, and the detail put into the emotional storytelling strengthens the bond between Gertrude and Matthew, who share more than just a bag of 0% gluten "Worry-Free Cookies."
Cover by Kris Anka
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The collection of elements situated in the body, as well as me who dwells in the body, are weakened by ignorant people. Know them to be of devilish resolves.
Let's say it is the first day of the new year, and we make a resolution to lose some weight. The very next day, we join a gym for the very first time, and spend one hour running on the treadmill. This will not be a good idea. Instead of helping our body, we may harm it. Weight loss, like any other system, requires a combination of knowledge and action. Unless we understand the techniques involved in weight loss, we should avoid starting an exercise regimen.
A similar state is found in some people who practice terrible austerities in the name of worship. Shri Krishna explains their fate here. He says that those who torture the body and mind in the name of austerity are doing nothing but weakening the elements, meaning the organs, in their body. Nowhere in the scriptures does it ask us to harm our body. He goes one step further by asserting that such people are also hurting Ishvara himself. As we saw in the ninth chapter, Ishvara dwells in all of us as the saakshi, the witness of our thoughts and actions.
Shri Krishna calls such people achetesaaha which means ignorant, or even unconscious. They chose to ignore Ishvara's teaching in the form of scriptures. They do not have faith in any ideal except their own preservation. They are driven only by egoism and desire. It is no wonder that their plans, their resolves, will be aasuric or devilish, as explained in the prior chapter. So the warning to us is clear. As we are conducting the analysis of our faith, we should in no way emulate these people. We should stay away from them.


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Gena Branscombe Project
Pilgrims of Destiny
Articles, Speeches & Interviews
Branscombe Choral
Marian Anderson String Quarter
Gena Branscombe
Branscombe Family Photos
Life! Love! Song!
TheGenaBranscombeProject.com
The Gena Branscombe Project
"The life of Gena Branscombe is an inspiring record of a woman of great ability who carved...a viable musical career…when American composers were just coming into their own in this country." (Laurine Elkins-Marlow, Gena Branscombe: American Composer and Conductor, A Study of Her Life and Works; Doctoral Dissertation - 1980). In today's language, Branscombe was a woman who did and had it all, establishing a career in music before marriage and afterwards successfully balancing work with an active family life.
For over twenty years I have worked on the life and music of Gena Branscombe. What began as a recording of her songs and piano works progressed to co-authoring and performing a one-woman show, Life! Love! Song! A Visit with Gena Branscombe. We told her story with first-person dialogue and 15 of her melodic songs. Captivated audiences continually asked why they had never heard of this woman composer and why had her music disappeared. Their questions propelled me to work on bringing Miss Branscombe to the attention of present day musicians and audiences.
Through my blog Music Meets Drama I wrote about Miss Branscombe, her life and music. What intrigued me most was her humanity - her passion for life, her kindness to colleagues, her honest and forthright emotions expressed in her compositions, her drive to promote American women composers and her love of family.
Her work ethic inspired me to seek out performance opportunities for her music. By reconstructing her dramatic oratorio's orchestral score and for the first time in over 70 years, Pilgrims of Destiny was performed at Clark University in 2019. With colleagues, we created The Gena Branscombe Project not-for-profit. We will be sponsoring concerts of Miss Branscombe's music as well as some of her colleagues. We award yearly scholarships to a student composer, conductor and arts administrator.
The woman composer who motivates our project was born in Picton, Ontario in 1881. Miss Branscombe attended the Chicago Musical College for her advanced music training as a piano major with an emphasis in composition. She furthered her musical experiences when in 1909 Gena departed for Berlin, Germany for a year of intensive piano study with Rudolf Ganz and composition with Englebert Humperdinck.
Upon her return from Berlin, she married John Ferguson Tenney on October 5, 1910 in Ontario. They immediately moved to New York City to pursue their individual careers. Gena's professional life quickly flourished. Throughout her life she credited her husband for his support of her work and his constant help with home and their four daughters: Gena born in 1911, Vivian Allison in 1913, Betty in 1916 and Beatrice in 1919.
Gena founded her women's chorus, Branscombe Choral in 1934. For 20 years she served as its conductor/composer/organizer and fund-raiser. The Choral performed extensively in New York City with their radio broadcasts reaching audiences far beyond the concert hall.
Ahead of her time, Gena Branscombe's life was filled with family and music. Influenced early by the late German romantic style, she helped develop a 20th century American/Victorian musical voice with a body of work encompassing 150 art songs, piano and chamber music, as well as choral works, all of which were readily available during her life. She was a renowned composer and conductor. Upon her death in July 1977 her richly melodic music seemed lost to history, despite its outstanding beauty.
Miss Branscombe was an important "bridge" for American women composers. Her professional training and perseverance, continuous support of American musical organizations and her many contemporary associations helped lay an enduring foundation for modern women in music. I introduce her now to the 21st century.


Speaker Phat Jaaye lyrics in Hindi from movie Total Dhamaal, sung by Harrdy Sandhu, Abuzar Akhtar, Aditi Singh Sharma, Jonita Gandhi. lyrics penned by Kumaar, music composed by Gourov-Roshin. Starring Ajay Devgn, Riteish Deshmukh, Arshad Warsi, Javed Jaffrey, Madhuri Dixit and Anil Kapoor. Music label Saregama India Ltd.
Song Label: Saregama India Ltd.


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Ventilators and COVID-19: what you need to know
by Yale University
The COVID-19 pandemic has cast a spotlight on ventilators—but few know much about what they do or how they work.
A ventilator pumps air—usually with extra oxygen—into patients' airways when they are unable to breathe adequately on their own. If lung function has been severely impaired—due to injury or an illness such as COVID-19—patients may need a ventilator. It is also used to support breathing during surgery.
Ventilators, also known as life-support machines, won't cure an illness, but they can keep patients alive while they fight an infection or their body heals from an injury.
Yale Medicine's Lauren Ferrante, MD, MHS, a pulmonary and critical care specialist, explains how ventilators work and why they are sometimes necessary for battling a COVID-19 infection.
How does COVID-19 affect the lungs?
There is much researchers still don't understand about COVID-19, but we do know that many who are infected with the novel coronavirus get a fever, cough, and sore throat, among other symptoms.
If the body's immune system does not fight off the infection, it can travel to the lungs and cause a potentially fatal condition called acute respiratory distress syndrome (ARDS). In ARDS, the alveoli (tiny air sacs that allow oxygen to reach the blood stream and remove carbon dioxide) fill with fluid, which diminishes the lungs' ability to provide vital organs with enough oxygen.
"ARDS entails severe inflammation of the lungs, but the main problem is that it makes portions of the lungs unusable," Dr. Ferrante explains. "It can be very serious, and many of these patients will need to be on a ventilator."
How does a ventilator work?
Normally, when someone takes a breath, their chest wall expands, which creates negative pressure (i.e., a vacuum) inside the lungs that draws air in. When a person is sick and weak and can't pull the breaths in on their own, a ventilator creates positive pressure that forces air into the lungs.
A ventilator is typically used in a hospital's intensive care unit (ICU), though those who need it for a longer period of time may be in a different part of the hospital, at a rehabilitation facility, or even at home.
The first step in putting a patient on a ventilator is general anesthesia. Then, a medical professional will place a tube into the mouth or nose and snake it into the windpipe. This is called intubation.
The tube is connected to an external machine that blows air and oxygen into the lungs. The machine can help do all or just some of the breathing, depending on the patient's condition. The ventilator can also help hold the lungs open so that the air sacs do not collapse.
While patients are on a ventilator, doctors will monitor their heart and respiratory rates, blood pressure, and oxygen saturation. Other tests, such as X-rays and blood draws, may be done to measure oxygen and carbon dioxide levels (sometimes called blood gases).
What is it like to be on a ventilator?
The goal is for patients to be awake and calm while they are on a ventilator, but that can sometimes be difficult; many require light sedation for comfort, Dr. Ferrante says.
"Sometimes, patients develop delirium, or an acute state of confusion. And when patients become confused, they might try to pull out their endotracheal tube, which connects them to the ventilator," she says. "Patients with delirium can be lucid one moment and confused the next. Although we try to avoid sedation as much as possible, particularly in delirious patients, we may have to give some sedation to prevent people from causing self-harm, like pulling out the breathing tube."
Furthermore, patients with ARDS often feel a natural instinct to take in very big breaths, Dr. Ferrante adds. "Very large breaths can be harmful to an ARDS patient's lungs, so we try to have their breath size match what we have set on the ventilator," she says.
Typically, most patients on a ventilator are somewhere between awake and lightly sedated. However, Dr. Ferrante notes that ARDS patients in the ICU with COVID-19 may need more heavy sedation so they can protect their lungs, allowing them to heal.
What is prone positioning?
As doctors have gained more experience treating patients with COVID-19, they've found that many can avoid ventilation—or do better while on ventilators—when they are turned over to lie on their stomachs. This is called prone positioning, or proning, Dr. Ferrante says.
"Instead of lying on your back, we have you lie on your belly. Because of how the lungs are positioned, this lets you use parts of your lungs that aren't being used when you are on your back," she explains, adding that it reduces pressure from the heart and diaphragm on the lungs. "Often, we see oxygenation improve quickly. We are using this a lot for COVID patients on a ventilator, and for those who are in the hospital on oxygen. It can help COVID patients from needing the ventilator."
How long does someone typically stay on a ventilator?
Some people may need to be on a ventilator for a few hours, while others may require one, two, or three weeks. If a person needs to be on a ventilator for a longer period of time, a tracheostomy may be required. During this procedure, a surgeon makes a hole in the front of the neck and inserts a tube into the trachea. The tube is connected to the ventilator.
The tracheostomy tube is inserted below the vocal cords, making it difficult to talk. As patients are weaned from the ventilator, they can start to talk again, using a device called a speaking valve.
Most tracheostomies are not permanent; they are often used to help wean a patient off a ventilator after long-term use, Dr. Ferrante says. However, not everyone will be able to come off a ventilator and breathe successfully on their own—and that reality can prompt important discussions for families, Dr. Ferrante says.
"Many people may be okay with being on the ventilator for a few weeks, trying to get better from an acute illness, but they may not be willing to stay on a ventilator permanently," she says. "Many find that unacceptable. This is why it is good for patients and their families to have advance care planning discussions."
How does someone come off a ventilator?
A patient can be weaned off a ventilator when they've recovered enough to resume breathing on their own. Weaning begins gradually, meaning they stay connected to the ventilator but are given the opportunity to try to breathe on their own.
"When someone is on a ventilator, especially with COVID-induced ARDS, they are often on very high levels of support," Dr. Ferrante explains. "As you improve, the support comes down to what we call 'minimal vent settings,' meaning you don't need a lot of oxygen through the ventilator, and you don't need higher pressures."
When a certain threshold is reached, doctors will have patients try daily spontaneous breathing trials. "The way we test is by having you breathe for 30 minutes on your own while still connected to the ventilator," she says. "There are certain numbers we track to let us know if you have passed the spontaneous breathing trial. You also have to be awake and, ideally, interacting with us."
When those milestones are achieved, the doctors may decide to try taking the patient off the ventilator for a trial. If it's not successful, weaning can be attempted another time.
For patients with acute respiratory or cardiopulmonary failure, another therapy called ECMO (extracorporeal membrane oxygenation), may be necessary. ECMO is a highly specialized form of life support that can take over the work of the heart and lungs, allowing them to rest and heal.
What are the risks of being on a ventilator?
There are risks associated with ventilator use. "It's not natural to have positive pressure forcing air into your lungs," Dr. Ferrante notes. "But a big part of our training as critical care physicians is on the proper use of a ventilator, so that we're giving a patient as much benefit as possible while also minimizing harm."
Infection is one potential risk associated with being on a ventilator; the breathing tube in the airway can allow bacteria to enter the lungs, which can lead to pneumonia. A ventilator can also damage the lungs, either from too much pressure or excessive oxygen levels, which can be toxic to the lungs.
Delirium is another concern, and fits in with what is called post-ICU syndrome (PICS), a collection of problems that can present—and linger—after a critical illness.
With a critical illness, and particularly with ventilator use, "the three domains we worry about are impairments in physical function, cognitive function, and mental health," Dr. Ferrante says, adding that the lack of movement during hospitalization can present other challenges after a patient is discharged.
"It can take months to recover," she explains. "Continuing physical therapy and occupational therapy after you go home is very important." (At Yale New Haven Hospital, an ICU-based mobility program has physical and occupational therapists working with patients to get them moving, even while they are on a ventilator.)
Dr. Ferrante says that older patients, in particular, are likeliest to experience a decline in their physical and cognitive function. "ICU survivors may feel like their thinking and processing isn't as quick as it was before they were in the ICU," she says. "When you take someone out of their home environment, put them in an unfamiliar place, and give them medications they don't normally take, it can put them at a higher risk for delirium. And if they experienced delirium or needed sedatives in the ICU, that may lead to cognitive problems after an ICU stay. Patients may also experience mental health issues, such as PTSD [post-traumatic stress disorder]."
Nonetheless, ventilators can be life-saving and, indeed, many of those who've survived severe cases of COVID-19 would be unlikely to have made it without one.
Follow the latest news on the coronavirus (COVID-19) outbreak
Provided by Yale University
Citation: Ventilators and COVID-19: what you need to know (2020, June 4) retrieved 20 January 2022 from https://medicalxpress.com/news/2020-06-ventilators-covid-.html
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Q: WAnt to Replace EANF from much Extract iMacros Bellow is iMacros iim code and i want to change EXTRACT result #EANF# to be null or '' or change maybe to "BRAVO"
here is still not work any tipe EVAL code, i'm still learn iMacros.
And the problem is, total EXTRACT=TXT is more then 12 whic is cannot put in to !VAR1 and so on. Helps please.
TAB T=2
TAG POS=1 TYPE=DIV ATTR=CLASS:addr&&TXT:* EXTRACT=TXT
SET EXTRACT EVAL("'{{!EXTRACT}}'.replace(/[\'#EANF#\']/,'');")
TAG POS=1 TYPE=SELECT FORM=ID:resetForm ATTR=ID:PortalCode EXTRACT=TXT
SET EXTRACT EVAL("'{{!EXTRACT}}'.replace(/[\"#EANF#\"]/,'');")
TAG POS=1 TYPE=DIV ATTR=CLASS:addr&&TXT:*<SP>* EXTRACT=TXT
SET EXTRACT EVAL("'{{!EXTRACT}}'.replace(/['#EANF#']/,'');")
TAG POS=1 TYPE=DIV ATTR=ID:info&&TXT:* EXTRACT=TXT
SET EXTRACT EVAL("'{{!EXTRACT}}'.replace(/[\"#EANF#\"]/,'');")
TAG POS=1 TYPE=DIV ATTR=CLASS:STYLE:padding-top:* EXTRACT=TXT
SET EXTRACT EVAL("'{{!EXTRACT}}'.replace(/[\"#EANF#\"]/,'');")

A: Try something like this:
TAB T=2
TAG POS=1 TYPE=DIV ATTR=CLASS:addr&&TXT:* EXTRACT=TXT
TAG POS=1 TYPE=SELECT FORM=ID:resetForm ATTR=ID:PortalCode EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=CLASS:addr&&TXT:*<SP>* EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=ID:info&&TXT:* EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=CLASS:STYLE:padding-top:* EXTRACT=TXT
SET !EXTRACT EVAL("'{{!EXTRACT}}'.replace(/#EANF#/g, '');")

A: You can try this code.
SET !TIMEOUT_STEP 0
SET !ERRORIGNORE YES
SET !EXTRACT_TEST_POPUP NO

TAB T=2
TAG POS=1 TYPE=DIV ATTR=CLASS:addr EXTRACT=TXT
TAG POS=1 TYPE=SELECT FORM=ID:resetForm ATTR=ID:PortalCode EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=CLASS:addr&&TXT:*<SP>* EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=ID:info EXTRACT=TXT
TAG POS=1 TYPE=DIV ATTR=STYLE:padding-top* EXTRACT=TXT

SET !EXTRACT EVAL("\"{{!EXTRACT}}\".replace(\"#EANF#\", \"\");")

redpajama_set_name: RedPajamaStackExchange


Annual sales figures for the Chevrolet Tracker in the US.
The Geo Tracker was renamed Chevrolet Tracker when General Motors discontinued the Geo brand in 1998. General Motors had introduced the Geo (sub-)brand in 1989 for Japanese engineered small cars to be sold by Chevrolet dealers. Models include the Geo Metro, Geo Spectrum, Geo Prizm, Geo Storm, most of which were also sold as Chevrolet during their lifetime.
Related model: Suzuki Vitara/XL7, Suzuki Grand Vitara (Europe).


using the smart tools provided by the Ministry of Education to efficiently manage and enhance school property.
the Innovative Learning Environments (ILE) assessment tool.
Check your Property occupancy document to ensure you have the core principles of school property management, use and development sorted.
Make sure school policies and procedures take account of your resourcing, school environment and the application of the leadership dimensions of the property smart tools to every phase of property decision-making and action.
Reflect on how well your property knowledge and skills prepare you to meet the expectations of the New Zealand School Property Strategy 2011-2021 and your school's achievement goals.
The property pages on the Ministry of Education's website are an essential reference for all new principals and board members.
The focus is on state school property, but there is some information for integrated schools.
The pages clearly explain all the regulations and requirements for state schools and describe the processes involved in each stage of the property management process.
The Property portal, which replaced the Property Management Information System in 2017, provides information on your school's annual property management grant and allows you to work out whether your school has the space it is entitled to.
Pay particular attention to the resources that support planning and preparedness for use in times of emergency.
Each board manages school property using systematic processes that meet the requirements and responsibilities defined in the Property Occupancy Document.
General maintenance from the operational funding.
Ministry of Education general support, advice and, in special cases, funding in times of emergency.
Community support and funding where locally generated funds provide for district facilities on the school site. For example for a swimming pool or hall.
Health and safety procedures to meet the standards required for all aspects of school property use and maintenance including, for most schools, meeting the requirements for building warrants of fitness. Rural schools are likely to have procedures to ensure water supplies are safe for consumption.
The principal plays a key role in building strong working relationships with the Ministry of Education, property consultant and property project managers. In this way you learn quickly about property matters but do not get buried in the details. Your goal is to make sure their expertise directs property processes to strongly support school achievement goals.
Some large schools have one or two board members who undertake much of the policy and implementation work for property.
Many boards will expect their principal to lead the processes and step in to take action as required.
Some rural schools have pooled resources so they can employ a part-time person to oversee correct application of Ministry of Education policies and attend to tenders, contracts with builders and liaison with the property consultant.
Most property projects of any size now require the employment of a professional project manager.
Make well-defined school achievement goals and plans the basis for your property decision-making and strategy. The building Conditions Assessment methodology and the Innovative Learning Environments (ILE) assessment tool help you do this.
Implementing a regular pattern of prioritising spending and actions to maintain and add to a school environment so that it provides a pleasant and encouraging learning place for both staff and pupils will strongly support school achievement goals.
Keep in mind that property-related actions can be at the maintenance, refurbishment or new capital work levels.
Engage in "open to learning conversations" about school property and its use.
Discuss the capacity of classrooms to provide the setting for meeting achievement goals with the teachers and students. Do this in their space. If there is a worst classroom in the school go there first!
Involve staff and students in hazard identification and minimisation and keeping up to date with maintenance. Supplement recording of and acting on hazards with minor maintenance actions. Many eyes help identify property needs and speed up the process.
Regularly observe and analyse student and staff travel patterns around, in and out of the school. Identifying and alleviating bottlenecks and crowding may reduce student conflict and bullying issues. Staff at the perimeter of a large school may feel isolated and find the long trek to the staffroom an issue. Evolving traffic patterns at the beginning and end of the school day can create new needs and hazards.
View the school property after extreme weather conditions. Heavy rain and drainage issues may have a significant impact on student and staff well-being.
Look for what needs maintaining, what has been improved by student or staff action and what would benefit from a re-think about use and appearance.
Make sure the school has an operative maintenance plan. Do not put off maintenance work. An active preventative maintenance regime will avoid serious and expensive problems occurring later.
Encourage the display of evidence of student achievement in public spaces and all classrooms. Living reminders of the community and its culture are important.
You are not expected to be a property 'expert'. Your skill is recognising when you need help and knowing where to go to get it.
Take advantage of the expertise of your property consultant to gather guidance on how to work through the complexity of property issues.
Use the expertise of property project managers to develop solutions for property issues and the needs of achieving classrooms.
Engaging in these two groups of open to learning conversations develops two-way learning and helps everyone recognise that there are limits and constraints that force you to prioritise.
Ask for help – Seek the advice and support of the Ministry of Education when questions and issues with property arise. You are leading the guardianship of school property for achievements by learners in the year ahead.
Anne Kenneally has decided to create a radically different learning space for her learners from the one they are used to.


text:
Building Chargebee
Adobe Fully Adopts Subscription Model With Creative Cloud
John Solomon
~ 3 min read | June 9
Adobe, one of the largest creative software services providers, is now fully pursuing a subscription business model. The company's flagship product, Adobe Creative Suite, has been evolved to be accessible on the cloud on a subscription basis. What does this mean for creatives and other software services competitors?
For over three decades, creatives have been paying a license fee to use Adobe design software. However, as the subscription business evolves, large software companies are not being left behind. Adobe Creative Cloud was introduced in late 2012 as a subscription option and so far, everything indicates the company got it right. In the first 9 months, over 450,000 premium users and more than 2 million free members had subscribe to the subscription package.
Adobe's Creative Cloud Pricing
Creative Cloud was initially introduced to individual members and later to teams. The response has been positive so far. Users can access a single Adobe product for a monthly fee of $29.99 or the entire Creative Suite for a monthly fee of $49.99.
As consumer software consumption moves from perpetual licensing (where you purchase a software to install to your PC) to a SaaS model, Adobe has decided to go all-in to offer a SaaS service with its Creative Cloud. While the company will continue selling the current versions of its products, no further feature updates will be made for them
Together with its cloud storage service, Adobe is providing creative professionals with all they need to successfully execute projects. According to Adobe senior executives, the shift to a subscription based model is a major but necessary step for the company's future sustainability. Users can now expect more stable versions and quicker patches, fixes and updates with the Creative Cloud.
While the subscription model will reduce the instance of cracked and patched Adobe products, the company says this way not their major drive into the subscription model. For example, it cites Poland, a country where piracy levels are traditionally high, to have more than average number of registered users. The point is that creatives are willing to pay for software if they are given a pricing option that they can afford. Instead of the large upfront cost typical of the perpetual licensing model, creative professionals can pay affordable monthly subscription fees to access the products they need.
Is Adobe Getting it Right?
Design software are typically RAM intensive and require PCs with robust performance features. How does Adobe Creative Cloud augur in performance?
Application performance is one of the issues that Adobe has to deal with when designing the cloud offering. Unlike most other SaaS software, Adobe has to deal with the reality of performance issues with its products. Creatives need robust PCs to optimally run products like Photoshop and After Effects. As a result, the company decided to offer the application offline but users need to log into the Internet at least once a month to verify their subscription.
Several shots have been fired at Adobe from different quarters, including Microsoft, for abandoning the packaged software services to focus on cloud services. Unlike Adobe, Microsoft does not plan to stop its packaged software services even as it pushes for its subscription based Office365.
There has been an outcry on the internet from creative professionals who see they will be affected by the subscription based services. Over 5000 creative professionals have signed a petition at Change.org asking Adobe to keep its perpetual licensing products. Some consumers are wary of having to be locked up in subscription pricing plans. However, others are happy with the new move, citing the lower cost.
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Experience : At least 1-year experience of working with the Government/N onGovernment organizations.
Interested and eligible candidates can download in the prescribed form from website jkicds.com and send their application form on or before 11-02-2019. The Prescribed format along with the photocopy of all relevant documents should be submitted to the address mentioned below.


text:
Tunnel visions
by Brian Hudgins
Ten months before he embarked on US Air Force pilot training, David Thirtyacre ('87 Mech. Eng.) forged a bond with aerodynamics in Washington State University wind tunnels.
Recently graduated from WSU and awaiting his trip to Laughlin Air Force Base in Del Rio, Texas, he worked as a research assistant on several projects that centered around using tiny beads instead of smoke to visualize air flow.
David Thirtyacre (Courtesy David Thirtyacre)
"A photo would show the movement of the bead during the exposure as a vector which could be measured," Thirtyacre explains. "Another experiment involved using white noise injected into the wind tunnel to increase the energy of the airflow and improve wing aerodynamics."
The experience with WSU's wind tunnels helped propel Thirtyacre through 26 years of active duty as a fighter pilot and gradual transition to his role as chair of the department of flight at the College of Aeronautics of the Daytona Beach, Florida-based Embry-Riddle Aeronautical University (ERAU).
During his Air Force career, Thirtyacre served as an assistant weapons officer for four years at Ramstein Air Base in Germany before being assigned as chief of weapons and tactics at Aviano Air Base in Italy. From 1997 to 2014, he served at Nellis Air Force Base in Nevada.
He finished a master's degree at ERAU in the early 2000s and taught at Nellis AFB as an adjunct professor. Those experiences as both student and educator gave Thirtyacre a path to return home to Washington, where ERAU has five campuses. "Back here, we have horses, chickens, and everything on a farm," says Thirtyacre, who grew up in University Place, just south of Tacoma, and now lives in Eatonville.
The ERAU department of flight is dedicated solely to drones, and one of Thirtyacre's top concerns is drone operational safety. With the growth of small unmanned aircraft systems, Thirtyacre says, "The degrees offered have to keep course content current, which is difficult in a fast-moving industry."
The ERAU curriculum for unmanned systems instruction has grown to over 30 courses—everything from how to pass the Federal Aviation Administration exam to industrial inspection courses and several courses for public safety agencies. Even for experienced pilots, there is a learning curve. "Every one of the faculty is a pilot," Thirtyacre says. "Students are trained to not only fly, but to do it precisely and accurately. Going from manned aviation to a drone, it is totally different."
Once the mechanics are mastered, practical uses for drones span far and wide. Last year, ERAU assisted marine biologists by using drones to monitor injured mammals, such as seals who need medication or help getting disentangled from objects. Through a partnership with Verizon and use of a drone platform, the drones deliver data without increasing the mammals' distress or disturbing the surrounding environment.
"We set up about 10 different aircraft with a test area for them," Thirtyacre says. "We came up with some good rules of thumb for what aircraft to use, how close to fly, and how to approach wildlife."
Now, Thirtyacre teaches undergraduate and graduate aerodynamics. And he has a home-built wind tunnel in his garage, a passion that was ignited nearly 35 years ago at WSU.


text:
Dum is a 2003 Indian Tamil-language action-romantic comedy film directed by A. Venkatesh and written by Puri Jagannadh. The film stars Silambarasan and Rakshitha (in her Tamil debut) while Ashish Vidyarthi and S. S. Rajendran play pivotal roles. It is the Tamil remake of 2002 Kannada film Appu.

Plot
Satya (Silambarasan) is the son of a police constable. He gets into fights and ends up in jail where his own father gets him out on bail. While he is returning home from a party drunk, a group of college students beat him up. That is when Suchitra (Rakshitha) comes and takes him to the hospital and gives blood. She is the daughter of a police commissioner. Later, Satya falls in love with Suchitra. That leads to several problems which are faced bravely by Satya in the later part of the film. Finally, all goes well, and Satya also receives the letter confirming his selection for IPS.

Cast

 Silambarasan as Sathya
 Rakshitha as Suchitra
 Ashish Vidyarthi as Thilak
 S. S. Rajendran as IGP
 Delhi Ganesh as Chokkalingam
 Livingston as Boopathi
 Radha Ravi as Inspector
 Sumitra
 Neelima Rani
 Besant Ravi
 Madhan Bob
 Cool Suresh

Production
The film was initially set to be titled Idiot after the Telugu version, but the title was later changed. A. Venkatesh remade the film from Puri Jagannadh's 2002 Kannada film Appu, which was also remade in Telugu in 2002 as Idiot. Venkatesh was keen to cast Kiran Rathod, but later selected Rakshitha, who appeared in all three versions of the film. During the making of the film, Silambarasan did his own stunts including a risky jump from the fifth floor of a building.

Release
Dum was an average hit at the box office.

Soundtrack
There are eight songs composed by Deva, while Sabesh–Murali handled the film's background score. The songs Chanakya, Polladha Padava, Manase are reused from the original Telugu movie Idiot. The song "Chanakya Chanakya" was sampled by Bulgarian pop-folk singer Emilia in her song "Ti si mi". Lyrics written by Pa. Vijay and Kabilan.

References

2003 films
Tamil remakes of Kannada films
2000s Tamil-language films
2000s masala films
Indian action comedy films
2003 action comedy films
Films directed by A. Venkatesh (director)
2003 comedy films

redpajama_set_name: RedPajamaWikipedia