text:
\section*{Introduction}

  \ \ \ \ Since 1930s, spheroidal wave functions have made
strong contributions to extensively theoretical and practical
applications in pure mathematics, applied mathematics, physics and
engineering. They appear in the fields, such as wavelet, random
matrix, non-commute geometry, gravitational wave detection,
quantum field theory in curved space-time, black hole stable
problem; 3G mobile and broad band satellite telecommunication;
steady flow of a viscous fluid and so
on\cite{flammer}-\cite{slepian}. Nevertheless, they perhaps are
one of the hardest work for researchers. They welcome the new
thought and methods to deal with them. Since the appearance of the
supersymmetric quantum mechanics (SUSYQM), its great power of
solving the differential equation attracts tremendous attention.

In this paper, we first use SUSYQM to study The spheroidal
differential equations. First, brief introduction to the
spheroidal problems and the ordinary methods to treat them. The
spheroidal differential equations are
\begin{equation} \left[\frac{d}{dx}\left[(1-x^2) \frac{d}{d
x}\right]+E+ \alpha x^2
 -\frac{m^{2}}{1-x^2} \right]\Theta=0, x\in (-1,+1).\label{3}
\end{equation}
With the condition $\Theta $ is finite at the boundaries $x=\pm
1$, they consist of the Sturm-Liouville eigenvalue problem. The
parameter $E $ can only takes the values $E_0,\ E_1,\dots,
E_n,\dots$, which are called the eigenvalues of the problem, and
the corresponding solutions (the eigenfunctions) $\Theta_0,\Theta_1,\dots,
\Theta_n,\dots$   are called the spheroidal wave functions
\cite{flammer}-\cite{li}.

The equations (\ref{3}) have two parameters: $m$, $\alpha$. When
$\alpha=0$, the spheroidal wave functions reduce to the spherical wave
functions, that is, the associated Legendre-functions $P_l^m(x)$.
Though the spheroidal wave equations are extension of the ordinary
spherical wave functions equations, the difference between this
two kinds of wave equations are far greater than their
similarity\cite{flammer}. The spherical wave equations belong to
the case of the confluent super-geometrical equations with one
regular  and one irregular singularities, whereas the spheroidal
wave equations are the confluent Heun equations containing two
regular  and one irregular singularities. The extra singularity
makes it extremely difficult to solve
them\cite{flammer}-\cite{li}, so very little information have been
 obtained  concerning the analytically exact solutions. Therefore
approximate and numerical methods are two main resources to rely
on for the problem.

Traditionally, the three-term recurrence relation methods are used
to evaluate the eigenvalues and eigenfunctions of spheroidal wave
functions: one could solve the transcendental equation in
continued fraction form  or its equivalent or by power series
expansion etc\cite{flammer}-
\cite{li},\cite{caldwell}-\cite{beti}. For the details of these
methods and their advantage and disadvantage, one could see the
reference \cite{beti}. These methods mainly work for the numerical
purpose, and also rely heavily on the numerical method. However,
all previous works concentrate largely on the calculations of the
eigenvalues, and particularly emphasize the small parameter
approximation and large parameter limits form of the eigenvalues.
Little effort has been devoted to the related eigenfunctions due
to the difficulty and complexity.

Here, we give brief review on the eigenfunctions in small
parameter approximation. No matter what method may be used,  all
eigenfunctions come into this kind of form in the end \begin{equation}
\Theta_{n}(x)=P_n^m+\sum_{q=1}^{\infty}\Theta_{nq}(x)\alpha^q\label
{small1}.\end{equation} Though there are many excellent works on the
eigenvalues' approximation of small parameter, no good works exist
for the eigenfunctions approximation $\Theta_{nq}(x)$. Even for the
first eigenfunctions approximation, the existing works are only in
the series' form \begin{equation} \Theta_{n1}(x)=\sum_{r=0,r\ne n
}^{\infty}B_{r}P_{m+r}^m(x)\label{small2}\end{equation} with infinite numbers
$B_r$ needed to evaluate.  Obviously, this series form  does not
reveal much information about the eigen-functions, even about its
1st order approximation eigenfunctions $\Theta_{n1}(x)$ itself.

In recent years, supersymmetric quantum mechanics have attracted
tremendous attention for solvable potential problems. They not
only  provide clear insight into the factorization method of
Infeld and Hull \cite{Infeld}, but also greatly improve the
methods to solve the differential equations. See reference
\cite{Cooper} for review on its development.

The spheroidal eigenvalue problem is treated by the method of
SUSYQM. This is the first time for researchers to use SUSYQM to
study the ground eigenvalue and eigenfunction (that is $\Theta_0$) of
spheroidal wave functions in the small parameter $\alpha$
approximation.

In usual small parameter approximation method, the key concept is
the eigenfunctions and they are expanded in the form
(\ref{small1}). On the contrary, the super-potential is the
central concept in SUSYQM, and is expanded in the series form of
the parameter $\alpha$. This new method is applied to study the
spheroidal equations and unexpected results are obtained: the
ground eigenfunction of the first order is in closed form, this in
turn gives useful information on the eigenfunction and is helpful
for their application. We also get the ground function for higher
order terms in parameter $\alpha$.

\section*{The ground eigenvalue and eigenfunction in the first order }
\ \ \ \

In the following, we will use the new perturbation method in
supersymmetry quantum to resolve the spheroidal eigenfunctions'
problem.

Though the form (\ref{3}) is more familiar for research, the
problem is easier to solve in the original differential equation
than in the equation (\ref{3}). The original form is obtained from
the eq.(\ref{3}) by the transformation \begin{equation} x=\cos\theta, \end{equation} that is,
\begin{equation}
\left[\frac{1}{\sin \theta}\frac{d}{d\theta}\left(\sin \theta
\frac{d}{d \theta}\right)+ \alpha \cos ^2 \theta -\frac{m^{2}}{\sin ^2
\theta}\right]\Theta=-E\Theta\label{2}
\end{equation}
the corresponding  boundary conditions become $\Theta$ is finite at
$\theta=0,\ \pi$,

 One writes the eqn.(\ref{2}) in the form of the
Schr$\ddot{o}$dinger equation by the transformation
\begin{equation}
\Theta =\frac{\Psi}{\sin^{\frac12} \theta}\label{transform}
\end{equation}
the differential equations turn out to be
\begin{eqnarray}
\frac{d^2\Psi}{d\theta^2}+\left[\frac14+ \alpha\cos ^2 \theta
 -\frac{m^{2}-\frac14}{\sin ^2
\theta}+E \right]\Psi=0\label{main eq}
\end{eqnarray}
and the boundary conditions become
\begin{equation}
\Psi|_{\theta=0}=\Psi|_{\theta=\pi}=0
\end{equation}
From the equation (\ref{main eq}), one knows the potential is
\begin{equation}
V(\theta,\alpha, m)=-\frac14- \alpha\cos ^2 \theta
+\frac{m^{2}-\frac14}{\sin ^2 \theta}\label{potential m s b}
\end{equation}
The super-potential $W$ is determined by the potential
$V(\theta,\alpha,m)$ through the Reccita's equation \begin{equation}
W^2-W'=V(\theta,\alpha)-E_0\label{potential and w relation}
\end{equation} where the substraction of the ground energy just
makes the eqn.(\ref{main eq}) factorable. Actually this equation
is the same hard to treat as that in the original form (\ref{main
eq}). The approximate method naturally comes to one's mind. Hence,
when the absolute value of $\alpha$ is small, it is the the
super-potential $W$ that could be expanded as series of the
parameter $\alpha$, that is,
\begin{equation}
W=W_0+\alpha W_1+\alpha ^2 W_2+\alpha ^3 W_3+\ldots .\label{W}
\end{equation}

\begin{eqnarray}
 W^2&-&W'= W_0^2-W'_0+\alpha \left(2W_0W_1-W'_1\right)+\alpha ^2
\left(2W_0W_2+W_1^2-W'_2\right)\nonumber\\
&+& \alpha ^3 \left(2W_0W_3+2W_1W_2-W'_3\right)+\alpha ^4
\left(2W_0W_4+2W_1W_3+W_2^2-W'_4\right)+\ldots .\label{V-W1}
\end{eqnarray}

One can write the perturbation equation as
\begin{equation}
W^2-W'=V(\theta,\alpha,
m)-\sum_{n=0}^{\infty}2E_{0n}\alpha^n=-\frac14+\frac{m^2-\f14}{\sin
^2 \theta}- \alpha\cos ^2 \theta
-\sum_{n=0}^{\infty}2E_{0n}\alpha^n\label{2potential m=s=0 alpha}
\end{equation}
There are two lower indices in the parameter $E_{0n}$ with the
index $0$ refereing to the ground state and the other index $n$
meaning the nth term in parameter $\alpha$. The last term
$\sum_{n=0}^{\infty}2E_{on}\alpha^n$ is subtracted from the above
equation in order to make the ground state energy actually zero
for the application of the theory of SUSYQM. Later, one must add
the term to our calculated eigen-energy.  Comparing the equations
(\ref{V-W1}), (\ref{potential m s b}), and (\ref{2potential m=s=0
alpha}), one could get
\begin{eqnarray}
&&W_0^2-W'_0=-\frac14+\frac{m^2-\f14}{\sin ^2 \theta}-2E_{00}\label{0-term}\\
&&2W_0W_1-W'_1=- \alpha\cos ^2 \theta-2E_{01}\label{1-term}\\
&&2W_0W_2+W_1^2-W'_2=-2E_{02}\label{2-term}\\
&&2W_0W_3+2W_1W_2-W'_3=-2E_{03}\label{3-term}\\
&&2W_0W_4+2W_1W_3+W_2^2-W'_4=-2E_{04}\label{4-term}\\
&&\ \ \ \ \ \ \ \ \vdots
\end{eqnarray}
From the eq.(\ref{0-term}), we get
\begin{equation}
W_0=-\(m+\f12\) \cot\theta,\ 2E_{00}=m(m+1).\label{w-0}
\end{equation}
Then, we can write the other equations more concisely
\begin{eqnarray}
&&W'_1+(2m+1)\cot\theta W_1=\cos ^2 \theta+2E_{01}\label{1-1-term}\\
&&W'_2+(2m+1)\cot\theta W_2= W_1^2+2E_{02}\label{2-1-term}\\
&&W'_3+(2m+1)\cot\theta W_3=2W_1W_2+2E_{03}\label{3-1-term}\\
&&W'_4+(2m+1)\cot\theta W_4=2W_1W_3+W_2^2+2E_{04}\label{4-1-term}\\
&&W'_5+(2m+1)\cot\theta W_5=2W_1W_4+2W_2W_3+2E_{05}\label{5-1-term}\\
&&\ \ \ \ \ \ \ \ \vdots
\end{eqnarray}
After obtaining the zero term $W_0$ for the super-potential $W$,
the first order $W_1$ can be gotten as
\begin{equation}
W_1=\frac{\bar{A}_1}{\sin^{2m+1} \theta}
\end{equation}
with\begin{eqnarray} \frac{d \bar{A_1}}{d \theta}&=& \sin^{2m+1}\theta \(\cos^2\theta+2E_{01}\)\\
\bar{A_1}&=&\int \(\sin^{2m}\theta \cos^2\theta+2E_{01}\sin^{2m}\theta\)\sin\theta
d\theta \end{eqnarray} Suitably changing the independent variable to $x=\cos\theta $
and expanding the term as following \begin{eqnarray} \sin^{2m}\theta
&=&\(1-\cos^2\theta\)^{m}=\(1-x^2\)^{m}=\sum_{k=0}^{m}(-1)^k{m\choose
k}x^{2k}\end{eqnarray} where ${m\choose k}=\frac{m\times (m-1)\times \dots
\times (m-k+1)}{k!}$, then it reaches
\begin{equation}
\bar{A}_1=\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{\cos^{2k+3}\theta}{2k+3}+\frac{2E_{01}\cos^{2k+1}\theta}{2k+1}\].
\end{equation}
 So,
\begin{equation}
W_1=\frac{\bar{A}_1}{\sin^{2m+1}\theta}=
\frac{\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{\cos^{2k+3}\theta}{2k+3}
+\frac{2E_{01}\cos^{2k+1}\theta}{2k+1}\]}{\sin^{2m+1}\theta}\label{w1 in
complex }.
\end{equation}
 The quantity $E_{01}$ needs to be determined by the boundary
 conditions. This in turn require to calculate the ground eigenfunction upon to the first order
  by
\begin{eqnarray} \Psi_0&=&N\exp\[-\int Wd\theta\]\\&=& N\exp\[-\int W_0d\theta\
-\alpha \int
 W_1d\theta\]*\exp{O(\alpha^2)}\\
&=& N\sin^{m+\f12}\theta \exp\[-\alpha \int
 W_1d\theta\]*\exp{O(\alpha^2)}\label{f-w relation}. \end{eqnarray}
Whenever the eigenfunction is obtained, the boundary conditions
$\Psi|_{\theta=0}=\Psi|_{\theta=\pi}=0$ would choose the proper
$E_{01}$. This sounds very easy, but is a tough task in reality.
The complete calculating process is left to the appendix 1. The
results are \begin{eqnarray}
&&2E_{01}=-\f1{2m+3}\label{e1 in simple }\\
&&  W_1=\frac{\sin\theta\cos\theta}{2m+3}\label{good result of w1}.\end{eqnarray}

With the first order term of the super-potential $W_1$, one could
compute the second term $W_2$ by the same process. This is not
easy either. The appendix 2 gives the detail of the calculation.
Nevertheless, the results are elegant: \begin{eqnarray}
&&2E_{02}=-\frac{2m+2}{(2m+3)^3(2m+5)}\label{e2 in simple }\\
&& W_2=
\[\frac{-\sin\theta
\cos\theta}{(2m+3)^3(2m+5)}+\frac{\sin^3\theta \cos\theta}{(2m+3)^2(2m+5)}\]
\label{w2 in complex 1}.\end{eqnarray}

The ground eigenfunction upon to the second order becomes \begin{eqnarray}
\Psi_0&=&N\exp\[-\int Wd\theta\]\\&=& N\exp\[-\int W_0d\theta\
-\alpha \int
 W_1d\theta-\alpha^2\int W_2d\theta\]*\exp{O(\alpha^3)}\\
&=&\left(sin{\theta}\right)^{m+\frac12 } \exp{\[-\frac{\alpha
\sin^2\theta}{4m+ 6}\]}\nonumber\\ &*&  \exp{\[\frac{\alpha^2\sin^2\theta
}{2(2m+3)^3(2m+5)}-\frac{\alpha^2\sin^4\theta
}{4(2m+3)^2(2m+5)}\]}*\exp{O(\alpha^3)} .\end{eqnarray}

When $m=0$, These results (\ref{e1 in simple }),(\ref{good result
of w1}),(\ref{e2 in simple }),(\ref{w2 in complex 1}) reduce
respectively to \begin{eqnarray} && 2E_{01}=-\f13,\ W_{1}=\f13\sin\theta\cos\theta;\\
&& 2E_{02}=-\f2{135},\ W_2=-\f1{135}\sin\theta \cos\theta +\f1{45}\sin^3\theta
\cos\theta. \end{eqnarray}

 They are in complete accordance with that in reference \cite{tian} where
 the spheroidal wave functions are treated by SUSYQM in the case
 $m=0$.

Back  to equation (\ref{3}) with  their relationship
(\ref{transform}), we  could obtain \begin{eqnarray} \Theta_0 &=&
\left(1-x^2\right)^{\frac
m2}\exp{\left(-\frac{\alpha\(1-x^2\)}{4m+6}\right)}\nonumber\\ &*&
\exp{\[\frac{\alpha^2(1-x^2) }{2(2m+3)^3(2m+5)}-\frac{\alpha^2(1-x^2)^2
}{4(2m+3)^2(2m+5)}\]}*\exp{O(\alpha^3)}\label{good ground 1 }.\end{eqnarray}

Expanding the exponential functions in the above equation, the
results elegantly turn out as \begin{eqnarray} \Theta_0 &=&
\left(1-x^2\right)^{\frac m2}\[1-\alpha \frac{1-x^2}{4m+6}+\alpha^2\(\frac{1-x^2
}{2(2m+3)^3(2m+5)}+\frac{(1-x^2)^2 }{8(2m+3)(2m+5)}\)\]\nonumber \\
&& +0(\alpha^3) \label{good ground 2 }.\end{eqnarray}
 The above equation clearly shows how
the ground state function changes as the function of variable $x$
when $\alpha$ is small and could be compared to the non-perturbation
case $P_m^m=(1-x^2)^{\frac m2}$. The result is much better than the
usual result of the series form \begin{equation} \Theta_0(x)=P_m^m+\alpha
\sum_{q=1}^{\infty}B_qP_{m+q}^m(x)+\alpha^2
\sum_{q=1}^{\infty}C_qP_{m+q}^m(x)+O(\alpha^3)\end{equation} with infinite
numbers $B_q,\ q=1,\ 2,\ \dots\ $ need to be determined for the
first order term and $C_q,\ q=1,\ 2,\ \dots\ $ need to be
determined for the second order term. From another point view, the
results in the eqns.(\ref{good ground 1 }), (\ref{good ground 2 })
give the method to determine the infinite numbers $B_q,\ C_q,\
q=1,\ 2,\ \dots\ $.

In conclusion, SUSYQM provides a new opportunity to treat the
spheroidal wave functions and indeed they give new results in the
eqns. (\ref{good ground 1 }),(\ref{good ground 2 }). Further
calculations can be done by the same way, nevertheless, the higher
order term $W_n$ is more complex than the lower one. The maximus
of $W_1$ and $W_2$ satisfy \begin{eqnarray} \max{W_1}=\f16,\
\max{W_2}=\frac{\sqrt{11}}{2160}<\f1{540}<\max{W_1}. \end{eqnarray}
 Though the further calculation is not processed here, the
 reasonable guess is that the higher term $W_n$ is, the smaller
 its maximus. The guess mainly comes from the calculation of the
 quantity $W_1,\ W_2$, see the appendix 1 and appendix 2 for
 details. If the guess is right, \begin{eqnarray}
 W(\theta)=\sum_{n=0}^{\infty}W_{n}\alpha^n \end{eqnarray} might be analytic function
 in all complex plane $\alpha$, the only singularity of the function $W$ as the variable $\alpha$ is $\alpha=\infty$.
  Only the calculation in the appendixes can
 process on and on, could the guess be tested. Might some day the
 computer can do the work, this is the reason that the calculations in
 appendixes are extremely detailed.

\section*{Acknowledgements}
We greatly appreciate the hospitality of the Gravitation Theory
group and the MCFP in UMD . This work was supported in part by the
National Science Foundation of China under grants No.10875018,
No.10773002.

\section*{Appendix1: Simplification of the super potential of the first term $W_1$}
\begin{appendix}
\ \ \ \ The calculations are very complex, we rewrite the
eq.(\ref{w1 in complex }) here again for convenience
\begin{equation}
W_1=\frac{\bar{A}_1}{\sin^{2m+1}\theta}=
\frac{\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{\cos^{2k+3}\theta}{2k+3}+\frac{2E_{01}\cos^{2k+1}\theta}{2k+1}\]}{\sin^{2m+1}\theta}\label{w1
in complex 2 }.
\end{equation}
As state before, the quantity $E_{01}$ is determined by the
requirement that the eigenfunction is zero at the boundaries
$\theta=0,\ \pi$; this in turn demands $\int W_1d\theta$ finite at the
boundary.
  Therefore, the calculation of the
term $\int W_1d\theta$ is first processed . By transformation \begin{equation}
\tau=\sin\theta\label{trans2}\end{equation} and denoting $\int W_1d\theta$ by $I$ ,
it reads \begin{eqnarray} I=\int W_1d\theta &=& \int
\frac{\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{(1-\tau^2)^{k+1}}{2k+3}+\frac{2E_{01}(1-\tau^2)^{k}}{2k+1}\]}{\tau^{2m+1}}d\tau\nonumber\end{eqnarray}
Using formula \begin{eqnarray}
&&  \(1-\tau^2\)^{k}=\sum_{l=0}^{k}(-1)^l{k\choose l}\tau^{2l}\\
&&\(1-\tau^2\)^{k+1}=\sum_{l=0}^{k+1}(-1)^l{k+1\choose
l}\tau^{2l}\label{expanding (a+b) },\end{eqnarray} it becomes \begin{eqnarray} I&=&  \int
\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{\sum_{l=0}^{k+1}(-1)^l{k+1\choose l}\tau^{2l-2m-1}}{2k+3}
+\frac{2E_{01}\sum_{l=0}^{k}(-1)^l{k\choose l}\tau^{2l-2m-1  }}{2k+1}\]d\tau\nonumber\\
. &=& \sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\sum_{l=0}^{k+1}\frac{(-1)^l{k+1\choose
l}\tau^{2l-2m}}{(2k+3)(2l-2m)}+\sum_{l=0}^{k}
\frac{2E_{01}(-1)^l{k\choose l}\tau^{2l-2m}}{(2k+1)(2l-2m)}\]\\ &=&
\sum_{k=0}^{m}\sum_{l=0}^{k+1}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose
l}\tau^{2l-2m}}{(2k+3)(2l-2m)}+\sum_{k=0}^{m}\sum_{l=0}^{k}
\frac{2E_{01}(-1)^{k+l+1}{m\choose k}{k\choose l}\tau^{2l-2m
}}{(2k+1)(2l-2m)}\end{eqnarray} In order to exchange the sums order in the
above equations, one must notice the fact that $l\le k+1\
\Rightarrow k\ge l-1$ in the first term
$\sum_{k=0}^{m}\sum_{l=0}^{k+1}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose l}\tau^{2l-2m}}{(2k+3)(2l-2m)}$ and $l\le k\
\Rightarrow k\ge l$ in the second term
$\sum_{k=0}^{m}\sum_{l=0}^{k} \frac{2E_{01}(-1)^{k+l+1}{m\choose
k}{k\choose l}\tau^{2l-2m }}{(2k+1)(2l-2m)}$. So \begin{eqnarray} I &=&
\sum_{l=0}^{m+1}\sum_{k=l-1\ge 0 }^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose
l}\tau^{2l-2m}}{(2k+3)(2l-2m)}+\sum_{l=0}^{m}\sum_{k=l}^{m}
\frac{2E_{01}(-1)^{k+l+1}{m\choose k}{k\choose l}\tau^{2l-2m  }}{(2k+1)(2l-2m)}\\
&=& \sum_{l=0}^{m}\[\sum_{k=l-1\ge 0 }^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose l}}{(2k+3)(2l-2m)}+2E_{01}\sum_{k=l}^{m}
\frac{(-1)^{k+l+1}{m\choose k}{k\choose l}}{(2k+1)(2l-2m)}
\]\tau^{2l-2m}+\frac{\tau^2}{4m+6}\end{eqnarray} where the first sum under the
condition $l=m+1$ becomes $\frac{\tau^2}{4m+6}$.

Defining\begin{eqnarray} N_{1\ l}&=&\sum_{k=l-1\ge 0
}^{m}\frac{(-1)^{k+l+1}{m\choose k}{k+1\choose l}}{(2k+3)(2l-2m)},\ l=0,1,2,\dots,\label{n1}\\
 N_{2\ l}&=& \sum_{k=l}^{m}
\frac{(-1)^{k+l+1}{m\choose k}{k\choose l}}{(2k+1)(2l-2m)},\
l=0,1,2,\dots \label{n2}\end{eqnarray} it gets
 \begin{eqnarray} I&=& \sum_{l=0}^{m}
\[N_{1\ l}+2E_{01}N_{2\ l}\]\tau^{2l-2m}+\frac{\tau^2}{4m+6}\label{sum of I}.\end{eqnarray}

There are terms $\tau^{2l-2m}$ in the above equation. With the
eq.(\ref{trans2}) and the fact $l< m$, these term as
$\sin^{2l-2m}\theta$ become infinite as $\theta \rightarrow 0,\ \pi$. By
eq.(\ref{f-w relation}) and the eigenfunction's boundary condition
at $\theta=0,\ \pi$, the coefficients of those terms must be zero.
There is only one quantity $E_{01}$ unfixed, could  one choose
proper $E_{01}$ to to make the eigenfunction finite at the
boundaries? Actually, one only has one choice to select $E_{01}$
by \begin{eqnarray} N_{1\ 0}+2E_{01}N_{2\ 0}=0,\ \ N_{1\ 0}=\f1{2m}\sum_{k=0
}^{m}\frac{(-1)^{k+2}{m\choose k}}{(2k+3)},\ \
 N_{2\ 0}= \f1{2m}\sum_{k=0}^{m}
\frac{(-1)^{k+2}{m\choose k}}{(2k+1)}\label{choice E01},\end{eqnarray} do the
other terms in eqn.(\ref{sum of I}) automatically become zero
under the condition (\ref{choice E01} ).

Fortunately, this can be done and the following is the proof. The
inductive reasoning is used to give the proof. In order to
determine the quantity $E_{01}$ under the condition (\ref{choice
E01}), one must first simplify $N_{1\ 0},\ N_{2\ 0}$. From the
formula ( see reference \cite{grad} on page 389) \begin{eqnarray} &&
\int_{0}^{1}(1-\tau^2)^md\tau= \int_0^{\theta=\frac{\pi}2}\cos^{2m+1}\theta
d\theta = \frac{(2m)!!}{(2m+1)!!}, \end{eqnarray} and the similarly one\begin{eqnarray}
&& \int_{0}^{1}\tau^2(1-\tau^2)^md\tau=\int_{0}^{1}(1-\tau^2)^md\tau-\int_{0}^{1}(1-\tau^2)^{m+1}d\tau\\
&& =\[\frac{(2m)!!}{(2m+1)!!}-\frac{(2m+2)!!}{(2m+3)!!}\]\end{eqnarray} and also
with the formula (\ref{expanding (a+b) }), it is easy to obtain
\begin{eqnarray} &&
\int_{0}^{1}\tau^2(1-\tau^2)^md\tau=\int_0^1\sum_{k=0}^m{m\choose k}(-1)^k\tau^{2k+2}d\tau=\sum_{k=0}^m\frac{(-1)^{k+2}{m\choose k}}{2k+3}\\
&& \int_{0}^{1}(1-\tau^2)^md\tau=\int_0^1\sum_{k=0}^m{m\choose
k}(-1)^k\tau^{2k}d\tau=\sum_{k=0}^m\frac{(-1)^{k+2}{m\choose
k}}{2k+1}\end{eqnarray} Comparing these equations with that of (\ref{choice
E01}), one reaches \begin{eqnarray} N_{1\ 0}&=&\f1{2m}\sum_{k=0
}^{m}\frac{(-1)^{k+2}{m\choose k}}{2k+3}=\f1{2m}\[\frac{(2m)!!}{(2m+1)!!}-\frac{(2m+2)!!}{(2m+3)!!}\]\nonumber\\
N_{2\ 0}&=& \f1{2m}\sum_{k=0}^{m} \frac{(-1)^{k+1}{m\choose
k}}{2k+1}=\f1{2m}\frac{(2m)!!}{(2m+1)!!}.\end{eqnarray} Consequently, the
quantity $2E_{01}$ is simplified as \begin{eqnarray} 2E_{01}=-\frac{N_{1\
0}}{N_{2\ 0}}=-\f1{2m+3}\label{choice E01-2}.\end{eqnarray}

$N_{1\ 0}+2E_{01}N_{2\ 0}=0$ is guaranteed  by the choice in
eqn.(\ref{choice E01-2}). According to inductive reasoning, one
needs to prove  \begin{eqnarray} M_{1\  l+1}=N_{1\ l+1}+2E_{01}N_{2\ l+1}=0 \end{eqnarray}
under the assumption that \begin{eqnarray} M_{1\  l}=N_{1\ l}+2E_{01}N_{2\
l}=0.\end{eqnarray}

The key idea is to find the connection between  the terms $M_{1\
l}=N_{1\ l}+2E_{01}N_{2\ l}$ and $M_{1\ l+1}=N_{1\
l+1}+2E_{01}N_{2\ l+1}$. By the definitions of $N_{1\ l },\ N_{2\
l},\  l=0,\ 1,\ 2,\ \dots $ in eqns.(\ref{n1})-(\ref{n2}), one has
\begin{eqnarray} N_{1\ l+1}=&=&\sum_{k=l
}^{m}\frac{(-1)^{k+l+2}{m\choose k}{k+1\choose l+1}}{(2k+3)(2l+2-2m)},\\
 N_{2\ l+1}&=& \sum_{k=l+1}^{m}
\frac{(-1)^{k+l+2}{m\choose k}{k\choose l+1}}{(2k+1)(2l+2-2m)}.\end{eqnarray}
Due to the following relation \begin{eqnarray}
 \f1{2k+3}{k+1\choose l+1}&=&\frac{(k+1)!}{(2k+3)(k-l)!(l+1)!}=\frac{(k+1)!}{(2k+3)(k+1-l)!l!}*\frac{k+1-l}{(l+1)}\nonumber \\
&=&\frac{(k+1)!}{(k+1-l)!l!}
*\frac{k+\f32-(l+\f12)}{(l+1)(2k+3)}\nonumber
\\&=&-\frac{l+\f12}{l+1}\times
\f1{2k+3}{k+1\choose l}+\f1{2l+2}{k+1\choose l},\label{choose
relation}\end{eqnarray} it is easy to get \begin{eqnarray} (2l+2-2m)N_{1\
l+1}&=&\sum_{k=l }^{m}\frac{(-1)^{k+l}{m\choose k}{k+1\choose l+1}}{2k+3}\\
&=&\sum_{k=l }^{m}(-1)^{k+l}{m\choose k}\[-\frac{l+\f12}{l+1}\times
\f1{2k+3}{k+1\choose l}+\f1{2l+2}{k+1\choose l}\]\\&=&
\sum_{k=l-1\ge 0 }^{m}\[\frac{l+\f12}{l+1}\times
\f1{2k+3}\](-1)^{k+l+1}{m\choose k}{k+1\choose l}-\frac{{m\choose
l-1}}{2(l+1)}\nonumber\\&+& \sum_{k=l
}^{m}\frac{(-1)^{k+l}}{2l+2}{m\choose k}{k+1\choose l}. \end{eqnarray} Now, the
calculation becomes \begin{eqnarray} (2l+2-2m)N_{1\
l+1}=\frac{l+\f12}{(l+1)}(2l-2m)N_{1\ l}-\frac{{m\choose
l-1}}{2(l+1)}+\sum_{k=l }^{m}(-1)^{k+l}{m\choose
k}\f1{2l+2}{k+1\choose l}\label{relation of n1-1}\end{eqnarray} under with
the help of eqn.(\ref{n1}). Definition of the quantity \begin{eqnarray} Q_{1\
l}&=&-\frac{{m\choose l-1}}{2l+2}+\sum_{k=l
}^{m}\frac{(-1)^{k+l}{m\choose k}{k+1\choose l}}{2l+2}\nonumber\\
&=&\f1{2l+2}\sum_{k=l-1 }^{m}(-1)^{k+l}{m\choose k}{k+1\choose l},
\label{Q1 result}\end{eqnarray} may make the equation (\ref{relation of
n1-1}) simple as \begin{eqnarray} (2l+2-2m)N_{1\
l+1}=\frac{l+\f12}{(l+1)}(2l-2m)N_{1\ l}+Q_{1\ l }\ l=0,\ 1,\ 2,\
\dots,\  m-1.\label{relation of n1}\end{eqnarray}

In completely similar way, one could get the relation between
$N_{2,\ l}$ and $N_{2\ l+1}$. the formula \begin{eqnarray}
\f1{2k+1}{k\choose l+1}&=& \f1{2k+1}\frac{k!}{(k-l-1)!(l+1)!}=\f1{2k+1}\frac{k!}{(k-l)!(l)!}*\frac{k-l}{l+1}\nonumber \\
&=&\frac{k!}{(k-l)!(l)!}*\frac{(k+\f12)-(l+\f12)}{(2k+1)(l+1)}\nonumber \\
&=& -\frac{l+\f12}{l+1}\times \f1{2k+1}{k\choose
l}+\f1{2l+2}{k\choose l}\end{eqnarray} helps to simplify  the quantity $N_{2\
l+1 }$

\begin{eqnarray} (2l+2-2m)N_{2\  l+1}&=&
\sum_{k=l+1}^{m} \frac{(-1)^{k+l}{m\choose k}{k\choose l+1}}{2k+1}\\
&=&\sum_{k=l+1}^{m} (-1)^{k+l}{m\choose k}\[-\frac{l+\f12}{l+1}\times
\f1{2k+1}{k\choose l}+\f1{2l+2}{k\choose l}\]\\&=&\sum_{k=l}^{m}
\frac{l+\f12}{l+1}\times \frac{(-1)^{k+l+1}}{2k+1}{m\choose k}{k\choose
l}-\frac{l+\f12}{l+1}\times
\frac{(-1)^{2l+1}}{2l+1}{m\choose l}{l\choose l} \nonumber\\
&+&\sum_{k=l+1}^{m} \frac{(-1)^{k+l}}{2l+2}{m\choose k}{k\choose
l}.\end{eqnarray} Using the eqn.(\ref{n2}) and the fact
$-\frac{l+\f12}{l+1}\times \frac{(-1)^{2l+1}}{2l+1}{m\choose l}{l\choose
l}=\f1{2l+2}{m\choose l}$, it is easy to obtain \begin{eqnarray} (2l+2-2m)N_{2\
l+1}&=&\frac{(l+\f12)(2l-2m)}{(l+1)}N_{2\ l}+\frac{{m\choose
l}}{2l+2}+\sum_{k=l+1}^{m} \frac{(-1)^{k+l}}{2l+2}{m\choose
k}{k\choose l}\label{relation of n2-1 }.\end{eqnarray} The similar definition
of the quantity \begin{eqnarray} Q_{2\ l}&=&\frac{{m\choose
l}}{2l+2}+\sum_{k=l+1}^{m} \frac{(-1)^{k+l}}{2l+2}{m\choose
k}{k\choose l}\nonumber \\ &=& \frac{1}{2l+2}\sum_{k=l}^{m}
(-1)^{k+l}{m\choose k}{k\choose l}\label{Q2 result} \end{eqnarray} makes the
equation (\ref{relation of n2-1 }) simply become \begin{eqnarray}
(2l+2-2m)N_{2\ l+1}&=&\frac{(l+\f12)(2l-2m)}{(l+1)}N_{2\ l}+Q_{2\
l},\ l=0,\ 1,\ 2,\ \dots,\  m-1.\label{relation of n2 }\end{eqnarray}

The eqns.(\ref{relation of n1}),(\ref{relation of n2 }) tell that
\begin{eqnarray} M_{1\ l+1}&=&N_{1\ l+1}+2E_{01}N_{2\
l+l}\\&=&\frac{(l+\f12)(2l-2m)}{(l+1)(2l-2m+2)}M_{1\  l} +\[Q_{1,\ l
}+2E_{01}Q_{2\ l}\],\ l=0,\ 1,\ 2,\ \dots,\  m-1\label{M relation
}.\end{eqnarray} the relation between $M_{1\ l}$,$M_{1\ l+1}$ would be the
desired result if one could prove that $Q_{1\ l}=Q_{2\ l}=0,\
l=0,\ 1,\ 2,\ \dots,\  m-1 $. This is the following work.

 Using the formula \begin{eqnarray}
(fg)^{(m)}=\sum_{k=0}^{m}{m\choose k}f^{(m-k)}g ^{(k)},\end{eqnarray} under
the special case of $f=\tau,\ g=(1-\tau)^{m}$, one gets \begin{eqnarray}
\[\tau(1-\tau)^m\]^{(l)}&=&\tau\[(1-\tau)^m\]^{(l)}+{l\choose 1}\[(1-\tau)^m\]^{(l-1)}\\
&=&(-1)^lm(m-1)\dots
(m-l+1)\tau(1-\tau)^{m-l}\nonumber\\&+&(-1)^{l-1}l\times
m(m-1)\dots (m-l+2)(1-\tau)^{m-l+1}\label{useful q1-1}.\end{eqnarray}  The
calculation may also go by different way: \begin{eqnarray}
\[\tau(1-\tau)^m\]^{(l)}&=&\[\sum_{k=0}^{m}(-1)^k{m\choose k}\tau^{k+1}\]^{(l)}\\&=& \sum_{k=l-1}^{m}(-1)^k{m\choose k}(k+1)k(k-1)\dots
(k-l+2)\tau^{k-l+1}\\
&=&\f1{(l-1)!}\sum_{k=l-1}^{m}(-1)^k{m\choose k}{k+1\choose
l}\tau^{k-l+1}\label{useful q1-2 }\end{eqnarray}

Comparing the results of eqns. (\ref{useful q1-1}), (\ref{useful
q1-2 }) and taking the function $\[\tau(1-\tau)^m\]^{(l)}$ at
$\tau=1$ under the condition $l<m$, the good result reaches \begin{eqnarray}
\[\sum_{k=l-1}^{m}(-1)^k{m\choose k}{k+1\choose l}\tau^{k-l+1}\]_{\tau=1}=\sum_{k=l-1}^{m}(-1)^k{m\choose k}{k+1\choose l}=0\label{useful
q1-3 }.\end{eqnarray} By eqn.(\ref{Q1 result}), it is easy to get \begin{eqnarray}
Q_{1l}=0,\ l=0,\ 1,\ 2,\ \dots,\  m-1 .\label{q1=0}\end{eqnarray} Similarly,
one also could prove that \begin{eqnarray} Q_{2l}=0,\ l=0,\ 1,\ 2,\ \dots,\ m-1
.\label{q2=0}\end{eqnarray} Therefore, the eqns.(\ref{M relation }),
(\ref{q1=0}), (\ref{q2=0}) imply that\begin{eqnarray} M_{1\  l+1}=0,\ l=0,\ 1,\
2,\ \dots,\  m-1 \end{eqnarray} under the condition \begin{eqnarray} M_l=0;\end{eqnarray} by
induction one gets \begin{eqnarray} M_{n}=0,\ n=1,2,\dots ,  m.\end{eqnarray} Hence, the
boundary conditions could be satisfied by just selecting the only
one quantity $2E_{01}=-\f1{2m+3}$. With the good results $M_{1\
l}=N_{1\ l}+2E_{01}N_{2\ l}=0, l=0,\ 1,\ 2,\ \dots,\ m$ , one can
greatly simplify the first order super-potential $W_1$ in the
eqn.(\ref{w1 in complex 2 }). By similar method as before, rewrite
$W_1$ by changing the independent variable to \begin{equation} \tau=\sin\theta \end{equation}
and expanding terms $(1-\tau^2)^{k+1},\ (1-\tau^2)^k$, that is \begin{eqnarray}
&&W_1=\frac{\sum_{k=0}^{m}(-1)^{k+1}{m\choose k}\[\frac{(1-\tau^2)^{k+\f32}}{2k+3}+\frac{2E_{01}(1-\tau^2)^{k+\f12}}{2k+1}\]}{\tau^{2m+1}}\\
&=&(1-\tau^2)^{\f12} \sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}\[\frac{\sum_{l=0}^{k+1}(-1)^l{k+1\choose l}}{2k+3}
+\frac{2E_{01}\sum_{l=0}^{k}(-1)^l{k\choose
l}}{2k+1}\]\tau^{2l-2m-1}\end{eqnarray} then changing the order of the sums
just as before and dividing $W_1$ by $(1-\tau^2)^{\f12}$ for good
looking in the formula, it reads \begin{eqnarray} \frac{W_1}{(1-\tau^2)^{\f12}}&=&
\sum_{l=0}^{m+1}\sum_{k=l-1\ge 0
}^{m}\frac{(-1)^{k+l+1}{m\choose k}{k+1\choose l}\tau^{2l-2m-1}}{(2k+3)}\nonumber\\
&+&\sum_{l=0}^{m}\sum_{k=l}^{m}
\frac{2E_{01}(-1)^{k+l+1}{m\choose k}{k\choose l}\tau^{2l-2m-1  }}{(2k+1)}\\
&=& \sum_{l=0}^{m}\sum_{k=l-1\ge 0 }^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose l}}{(2k+3)}\tau^{2l-2m-1}\nonumber\\
&+&\[\frac{(-1)^{k+l+1}{m\choose k}{k+1\choose
l}}{(2k+3)}\tau^{2l-2m-1}\]_{{l=m+1}\atop{k=m}}+2E_{01}\sum_{k=l}^{m}
\frac{(-1)^{k+l+1}{m\choose k}{k\choose l}}{(2k+1)} \tau^{2l-2m-1}\\
&=& \sum_{l=0}^{m}\[\sum_{k=l-1\ge 0 }^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k+1\choose l}}{(2k+3)}+ 2E_{01}\sum_{k=l}^{m}
\frac{(-1)^{k+l+1}{m\choose k}{k\choose l}}{(2k+1)}\] \tau^{2l-2m-1}\nonumber\\&+&\frac{\tau}{2m+3}\\
&=&
\[N_{1\ l}+2E_{01}N_{2\ l}\]*(2l-2m)\tau^{2l-2m-1}+\frac{\tau}{2m+3}\\&=& \frac{\tau}{2m+3}\end{eqnarray}
So the quantity $W_1$ could be written tidily as \begin{eqnarray}
W_1=\frac{\tau(1-\tau^2)^{\f12}}{2m+3}=\frac{\sin\theta \cos\theta}{2m+3}.\end{eqnarray}

\section*{Appendix2: Simplification of the super potential of the first term $W_2$}
\ \ \ \ Though the process of the calculation of $W_2$ repeats
that of $W_1$ in the appendix 1, there still are some needs to
write it down. Perhaps this may be used for further computation,
even by computer.

\begin{eqnarray} W_2&=& \frac{\bar{A_2}}{\sin^{2m+1}\theta}\end{eqnarray} where \begin{eqnarray}
\bar{A_2}&=& \int (W_1^2 +2E_{02})\sin^{2m+1}\theta d\theta\\
&=&\int \[\f1{(2m+3)^2}\sin^{2m+3}\theta \cos^2\theta+2E_{02}\sin^{2m+1}\theta\] d\theta\\
&=&-\int  \[\f1{(2m+3)^2}(1-x^2)^{m+1}
x^2+2E_{02}(1-x^2)^{m}\]dx\ \ \ \dots, x=\cos\theta \\
&=&\int  \[\f1{(2m+3)^2}\sum_{k=0}^{m+1}(-1)^{k+1}{m+1\choose
k}x^{2k+2}+2E_{02}\sum_{k=0}^{m}(-1)^{k+1}{m\choose
k}x^{2k}\]dx \\
&=& \sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}x^{2k+3}}{(2m+3)^2(2k+3)}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}x^{2k+1}}{2k+1} \\&=&\sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}\cos^{2k+3}\theta}{(2m+3)^2(2k+3)}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}\cos^{2k+1}\theta}{2k+1}\end{eqnarray} The other important term is \begin{eqnarray}
&&II=\int W_2d\theta\\&&=\int
\[\sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}\cos^{2k+3}\theta}{(2m+3)^2(2k+3)\sin^{2m+1}\theta}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}\cos^{2k+1}\theta}{(2k+1)\sin^{2m+1}\theta}\]d\theta. \end{eqnarray} By the
transformation $\tau=\sin\theta$, one may have \begin{eqnarray} &&\int W_2d\theta =
\int
\[\sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}(1-\tau^2)^{k+1}}{(2m+3)^2(2k+3)\tau^{2m+1}}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}(1-\tau^2)^{k}}{(2k+1)\tau^{2m+1}}\]d\tau
\\&&=\sum_{k=0}^{m+1}\sum_{l=0}^{k+1}\frac{(-1)^{k+l+1}{m+1\choose
k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}+2E_{02}\sum_{k=0}^{m}\sum_{l=0}^{k}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l }\tau^{2l-2m}}{(2k+1)(2l-2m)}\end{eqnarray}

Exchanging the sum order, it reads \begin{eqnarray} &&
II=\sum_{l=0}^{m+2}\sum_{k=l-1\ge 0
}^{k+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}+2E_{02}\sum_{l=0}^{m}\sum_{k=l}^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l }\tau^{2l-2m}}{(2k+1)(2l-2m)}\\
&&= \sum_{l=0}^{m}\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}+2E_{02}\sum_{l=0}^{m}\sum_{k=l}^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l
}\tau^{2l-2m}}{(2k+1)(2l-2m)}\\&&+\sum_{l=m+1}^{m+2}\sum_{k=l-1\ge
0 }^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}\\ &&
=\sum_{l=0}^{m}\[\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}+2E_{02}\sum_{k=l}^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l
}\tau^{2l-2m}}{(2k+1)(2l-2m)}\]\\&&+\sum_{l=m+1}^{m+2}\sum_{k=l-1\ge
0 }^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}\end{eqnarray}

It is better to define the following quantities \begin{eqnarray} N_{3, \
l}=\sum_{k=l-1\ge 0 }^{m+1}\frac{(-1)^{k+l+1}{m+1\choose
k}{k+1\choose l }}{(2m+3)^2(2k+3)(2l-2m)},\ l=0,\ 1,\ 2,\ \dots,\
m\label{n3}\end{eqnarray} \begin{eqnarray} \ N_{4\
l}=\sum_{k=l}^{m}\frac{(-1)^{k+l+1}{m\choose k}{k\choose l
}}{(2k+1)(2l-2m)}, \ l=0,\ 1,\ 2,\ \dots,\ m\label{n4}\end{eqnarray} and
simplify the last term \begin{eqnarray} && \sum_{l=m+1}^{m+2}\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}\tau^{2l-2m}}{(2m+3)^2(2k+3)(2l-2m)}\\&&=-\frac{(2m+6)\tau}{(2m+3)^3(2m+5)}+\frac{\tau^3}{3(2m+3)^2(2m+7)}.\end{eqnarray}
Then the express becomes \begin{eqnarray} II=\sum_{l=0}^{m}\[N_{3\
l}+2E_{02}N_{4\
l}\]\tau^{2l-2m}-\frac{(2m+6)\tau}{(2m+3)^3(2m+5)}+\frac{\tau^3}{3(2m+3)^2(2m+7)}\end{eqnarray}
Just as done before, one needs to select proper $E_{02}$ to make
all $N_{3\ l}+2E_{02}N_{4\ l}$ zero. By comparing the quantities
$N_{1\ l},N_{2\ l},N_{3\ l},N_{4\ l}$, the good relations among
them could be revealed: \begin{eqnarray} N_{4\ l}=N_{2\ l},\ l=0,\ 1,\ 2,\
\dots,\ m\end{eqnarray} and the great similarity between $N_{1\ l},N_{3\ l}$.
The transformation forms between $N_{1\ l+1},N_{1\ l}$, $N_{3\
l+1},N_{3\ l}$ are similar.

\begin{eqnarray} N_{3, \ l+1}=\sum_{k=l\ge 0 }^{m+1}\frac{(-1)^{k+l+2}{m+1\choose
k}{k+1\choose l+1 }}{(2m+3)^2(2k+3)(2l-2m+2)}\end{eqnarray} By the relation
(\ref{choose relation}) \begin{eqnarray}
 \f1{2k+3}{k+1\choose l+1}=-\frac{l+\f12}{l+1}\times
\f1{2k+3}{k+1\choose l}+\f1{2l+2}{k+1\choose l}\end{eqnarray}

one could get \begin{eqnarray} && (2m+3)^2(2l+2-2m)N_{3\
l+1}=\sum_{k=l }^{m+1}\frac{(-1)^{k+l}{m+1\choose k}{k+1\choose l+1}}{2k+3}\\
&=&\sum_{k=l }^{m+1}(-1)^{k+l}{m+1\choose
k}\[-\frac{l+\f12}{l+1}\times \f1{2k+3}{k+1\choose
l}+\f1{2l+2}{k+1\choose l}\]\\&=& \sum_{k=l-1\ge 0
}^{m+1}\[\frac{l+\f12}{l+1}\times \f1{2k+3}\](-1)^{k+l+1}{m+1\choose
k}{k+1\choose l}-\frac{{m+1\choose l-1}}{2(l+1)}\nonumber\\&+&
\sum_{k=l }^{m+1}\frac{(-1)^{k+l}}{2l+2}{m+1\choose k}{k+1\choose l};
\end{eqnarray} with the help of eqn.(\ref{n3}), the calculation becomes \begin{eqnarray}
N_{3\ l+1}=\frac{(l+\f12)(2l-2m)}{(l+1)(2l+2-2m)}N_{3\
l}+\frac{\[-\frac{{m+1\choose l-1}}{2(l+1)}+\sum_{k=l
}^{m+1}(-1)^{k+l}\frac{{m+1\choose k}{k+1\choose
l}}{2l+2}\]}{(2m+3)^2(2l+2-2m)}.\label{relation of n3-1}\end{eqnarray} The
definition of the quantity \begin{eqnarray} Q_{3\ l}&=&-\frac{{m+1\choose
l-1}}{2l+2}+\sum_{k=l
}^{m+1}\frac{(-1)^{k+l}{m+1\choose k}{k+1\choose l}}{2l+2}\nonumber\\
&=&\f1{2l+2}\sum_{k=l-1 }^{m+1}(-1)^{k+l}{m+1\choose k}{k+1\choose
l}\label{Q3 result}\end{eqnarray} makes the above equation (\ref{relation of
n3-1}) become \begin{eqnarray} N_{3\
l+1}=\frac{(l+\f12)(2l-2m)}{(l+1)(2l+2-2m)}N_{3\ l}+\frac{Q_{3\
l}}{(2m+3)^2(2l+2-2m)},\ l=0,\ 1,\ 2,\,\dots,\ m-1.\label{relation
of n3}.\end{eqnarray} Another useful relation is \begin{eqnarray} N_{4\ l+1}&=&N_{2\
l+1}=\frac{(l+\f12)(2l-2m)}{(l+1)(2l+2-2m)}N_{2\
l}\\&=&\frac{(l+\f12)(2l-2m)}{(l+1)(2l+2-2m)}N_{4\ l}\end{eqnarray} by the use
of $Q_{2\ l}=0$. Now the quantities $M_{2\ l}$ is defined as  \begin{eqnarray}
M_{2\ l}=N_{3\ l}+2E_{02}N_{4\ l},\ l=0,\ 1,\ 2,\ \dots,\ m\end{eqnarray}
then the relation between $M_{2\ l}$ and $M_{2\ l+1}$ is \begin{eqnarray} M_{2\
l+1} =\frac{(l+\f12)(2l-2m)}{(l+1)(2l-2m+2)}M_{2\ l}+\frac{Q_{3\
l}}{(2m+3)^2(2l+2-2m)},l=0,1, \dots, m-1\label{M2 relation }.\end{eqnarray}

In the following, it will be proven that $Q_{3\ l}=0$. With the
help \begin{eqnarray} (fg)^{(m)}=\sum_{k=0}^{m}{m\choose k}f^{(m-k)}g ^{(k)}\end{eqnarray}
under the special case $f=\tau,\ g=(1-\tau)^{m+1}$, one gets \begin{eqnarray}
\[\tau(1-\tau)^{m+1}\]^{(l)}&=&\tau\[(1-\tau)^{m+1}\]^{(l)}+{l\choose 1}\[(1-\tau)^{m+1}\]^{(l-1)}\\
&=&(-1)^lm(m+1)\dots
(m-l+2)\tau(1-\tau)^{m-l+1}\nonumber\\&+&(-1)^{l-1}l\times
m(m+1)\dots (m-l+2)(1-\tau)^{m-l+2}\label{useful q3-1}.\end{eqnarray} The
alternative way to compute is \begin{eqnarray}
\[\tau(1-\tau)^{m+1}\]^{(l)}&=&\[\sum_{k=0}^{m+1}(-1)^k{m+1\choose k}\tau^{k+1}\]^{(l)}\\&=& \sum_{k=l-1}^{m+1}(-1)^k{m+1\choose k}(k+1)k(k-1)\dots
(k-l+2)\tau^{k-l+1}\\
&=&\f1{(l-1)!}\sum_{k=l-1}^{m+1}(-1)^k{m+1\choose k}{k+1\choose
l}\tau^{k-l+1}.\label{useful q3-2 }\end{eqnarray}

The valuation of  $\[\tau(1-\tau)^{m+1}\]^{(l)}$ at $\tau=1$ under
the condition $l<m+1$ is  \begin{eqnarray}
\[\sum_{k=l-1}^{m+1}(-1)^k{m+1\choose k}{k+1\choose l}\tau^{k-l+1}\]_{\tau=1}=\sum_{k=l-1}^{m+1}(-1)^k{m+1\choose k}{k+1\choose l}=0\label{useful
q3-3 }.\end{eqnarray} From the eqn.(\ref{Q3 result}), it is easy to get \begin{eqnarray}
Q_{3\ l}=0\ l=0,\ 1,\ 2,\,\dots,\ m-1.\label{q3=0}\end{eqnarray}

Finally, one gets the good result \begin{eqnarray} M_{2\ l+1}
&=&\frac{(l+\f12)(2l-2m)}{(l+1)(2l-2m+2)}M_{2\ l},\ l=0,\ 1,\
2,\,\dots,\ m-1. \label{M2 relation2 }\end{eqnarray} So the choice of $M_{2\
0}=0$ guarantees $M_{2,\ l}=1,\ 2,\,\dots,\ m$. The quantity
$E_{02}$ is obtained by \begin{eqnarray} E_{02}=-\frac{N_{3\ 0}}{N_{4\ 0}}.\end{eqnarray} As
in the case $N_{1\ 0}$, \begin{eqnarray} N_{3\ 0}&=&\f1{2m}\sum_{k=0
}^{m+1}\frac{(-1)^{k+2}{m+1\choose k}}{(2m+3)^2(2k+3)}=\f1{2m(2m+3)^2}\[\frac{(2m+2)!!}{(2m+3)!!}-\frac{(2m+4)!!}{(2m+5)!!}\]\nonumber\\
N_{4\ 0}&=&N_{2\ 0}= \f1{2m}\sum_{k=0}^{m} \frac{(-1)^{k+2}{m\choose
k}}{2k+1}=\f1{2m}\frac{(2m)!!}{(2m+1)!!}.\end{eqnarray} Hence \begin{eqnarray}
E_{02}=-\frac{N_{3\ 0}}{N_{4\ 0}}=-\frac{2m+2}{(2m+3)^3(2m+5)}.\end{eqnarray}  The
expression  \begin{eqnarray} W_2=\frac{\sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}\cos^{2k+3}\theta}{(2m+3)^2(2k+3)}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}\cos^{2k+1}\theta}{2k+1}}{\sin^{2m+1}\theta}\end{eqnarray} could greatly simplified
by $\tau=\sin\theta$ and the use of the elegant formula $M_{2\ l}=0$:
\begin{eqnarray} && \frac{W_2}{\cos\theta}=\sum_{k=0}^{m+1}\frac{(-1)^{k+1}{m+1\choose
k}(1-\tau^2)^{k+1}}{(2m+3)^2(2k+3)\tau^{2m+1}}+2E_{02}\sum_{k=0}^{m}\frac{(-1)^{k+1}{m\choose
k}(1-\tau^2)^{k}}{(2k+1)\tau^{2m+1}}\\&&=\sum_{k=0}^{m+1}\sum_{l=0}^{k+1}\frac{(-1)^{k+l+1}{m+1\choose
k}{k+1\choose l
}}{(2m+3)^2(2k+3)\tau^{2m-2l+1}}+2E_{02}\sum_{k=0}^{m}\sum_{l=0}^{k}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l}}{(2k+1)\tau^{2m-2l+1}}.\end{eqnarray} Exchanging the sums
order, it is easy to get \begin{eqnarray} &&
\frac{W_2}{\cos\theta}=\sum_{l=0}^{m+2}\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}}{(2m+3)^2(2k+3)\tau^{2m-2l+1}}+2E_{02}\sum_{l=0}^{m}\sum_{k=l}^{m}\frac{(-1)^{k+l+1}{m\choose
k}{k\choose l}}{(2k+1)\tau^{2m-2l+1}},\end{eqnarray} comparing the
eqns.(\ref{n3}) and (\ref{n4}), one may have \begin{eqnarray} &&
\frac{W_2}{\cos\theta}=\frac{\sum_{l=0}^{m}\[N_{3\ l}+2E_{02}N_{4\
l}\]}{\tau^{2m-2l+1}}+\sum_{l=m+1}^{m+2}\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}}{(2m+3)^2(2k+3)\tau^{2m-2l+1}}\\
&& =\sum_{l=m+1}^{m+2}\sum_{k=l-1\ge 0
}^{m+1}\frac{(-1)^{k+l+1}{m+1\choose k}{k+1\choose l
}}{(2m+3)^2(2k+3)\tau^{2m-2l+1}}\\ &&=
\frac{-\tau}{(2m+3)^3(2m+5)}+\frac{\tau^3}{(2m+3)^2(2m+5)}\end{eqnarray} by the use
of $M_{2\ l}=N_{3\ l}+2E_{02}N_{4\ l}=0$. The elegant form of
$W_2$ is \begin{eqnarray}
W_{2}=\[\frac{-1}{(2m+3)^3(2m+5)}+\frac{\sin^2\theta}{(2m+3)^2(2m+5)}\]\sin\theta
\cos\theta .\end{eqnarray} This ends the appendix 2.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


1 Department of Physics, West Virginia University, Morgantown, WV 26506, USA.
2 National Radio Astronomy Observatory, Green Bank, WV 24944, USA.
3 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia.
4 Department of Physics and Astronomy, Franklin and Marshall College, Lancaster, PA 17604, USA.
A rapid and powerful burst of radio waves is found through an analysis of archival pulsar data, suggestive of a new class of radio bursts, perhaps from a supernova.


text:
Like the Jetsons' microwave oven and wristwatch sized TV screens first seen worn by Dick Tracy, "autonomous" or self-driving cars appear ready to make the leap from science-fiction to everyday reality. However, while the technology has been pushed hard by the automotive industry, both well-publicized glitches and the Covid-19 pandemic have slowed the transformation from the test track to Main Street, USA. As recently as 2018, driverless taxis were predicted to be deployed in large numbers by 2021. Currently, a single fleet of Chrysler Pacifica vans operated by the ride-share company "Waymo" are in experimental use in Phoenix, AZ.
Florida, hoping to lure developers of autonomous technology from California, passed Florida Statute 316.85 in 2019 which prohibits constraints on the development and use of fully autonomous technology in vehicles, stating: " a fully autonomous vehicle may operate in this state regardless of whether a human operator is physically present". The law also makes provisions for special traffic lanes for such vehicles, prohibits local authorities from imposing additional taxes and fees. It also makes texting and hand held phones lawful for the occupants of fully automated vehicles.
The SAE (Society of Automotive Engineers) has developed standards for the different degrees of motor vehicle automation. The scale is as follows:
0: no automation/ all driver controlled;
1: adaptive cruise control
2: partial traffic assist (lane change warning; auto braking)
3: conditional automation (asks for driver assist with navigation)
4: high automation ( limited to certain roads)
5: full automation ( no driver involvement)
Given the negative publicity generated from a number of crashes (and one death of a pedestrian), the introduction of fully autonomous vehicles remains a few years in the future. That being said, pressure is being placed on both Congress and state legislatures by the automotive industry to restrict the public's legal remedies should they be injured by a self-driving vehicle. Their lobbyists have proposed that legislation should be passed by Congress to establish a nationwide procedure. This would pre-empt the individual states' ability to fashion their own laws. It would also most likely eliminate the right to a jury trial, which ARE currently afforded in motor vehicle negligence cases. At the state level, the push has been to force litigants to submit to arbitration, further limiting an injured party's right to a jury trial.
Both victims' advocacy groups and plaintiffs' lawyers associations are vigorously attacking these suggestions as unconstitutional and overly protective of an industry which needs no such protection. Expect this debate to continue until such time as the Uber vehicle of the near future shows up (minus a steering wheel) to pick up its passengers.

redpajama_set_name: RedPajamaCommonCrawl


New age Fly Fishing !
We are forever on the look out, for anything new and innovative, Fly Fishing in particular, of course.
As a convert to Czech Nymphing some time ago, instructed by one of the Worlds Best (and still learning) was recently introduced to Sun Ray Lines for Czech nymphing and Nymph fishing, the concept is micro thin lines which work amazingly well, so well in fact the angler can feel the fish thru the line, the most delicate of takes, the line colour is black and very thin reducing drag, splash, shadows, line flash, we hope to have these in stock end of September.
Couple this line with a HANAK Alpen or Champion Rod one has the perfect package.


Today uptrend of Gold should ended around $1206, because on yesterday Gold was moved upward due to weakness of dollar. So today correctional movement are expected near to support level $1194 towards $1191. Today some high importance fundamental event create positive or negative impact on gold, if data shows negative impact then Gold again keep the average range near $1200 and if data shows positive impact then $1185 confirm.


Men's lacrosse (3–5, 0–2 Ivy) defeated University of Denver (5–3) 14–13 in a nail-biter on Tuesday evening. The Tigers pulled ahead early in the game at Sherrerd Field, fell behind, and came back to secure victory against the Pioneers.
Denver's head coach, Bill Tierney, won an impressive six national championships while he was Princeton's head men's lacrosse coach from 1988–2009. Despite a pleasant ceremony before the first whistle, Tuesday night's homecoming did not end well for Tierney's new team.
Princeton got off to a hot start with two goals in the first minute. Senior attacker Emmet Cordrey scored the Tigers' first goal only 45 seconds into the game after the team dominated possession in their offensive half. First-year midfielder Jake Stevens scored off of the resulting face-off, bringing Princeton two goals ahead of Denver.
The Pioneers responded two minutes later with two goals in quick succession. They scored at 12:46 and converted quickly on the face-off at 12:33, tying the game 2–2. They pulled ahead in the 10th minute with another goal and increased their lead with six minutes left in the first quarter after a defensive clear. Princeton's sophomore attacker Chris Brown scored with less than two minutes to go and the quarter finished 4–3 in Denver's favor.
Denver opened the second quarter with a goal two and a half minutes after the starting whistle. Junior attacker Michael Sowers responded with his first goal of the game, although the Pioneers capitalized on the resulting face-off to bring the game to 6–4. Another goal by Sowers at 7:44 and Brown's second of the game 20 seconds later brought the Tigers even with their opponents. Both teams scored once more before the half finished 7–7, although Denver had eight shots while the Tigers had one.
Sophomore middie Jamie Atkinson was the first to score in the third quarter, giving Princeton its first lead since the opening minutes of the game. The two teams traded points for the remainder of the half. After Atkinson made it 8–7, Sowers scored with five minutes to go and assisted Brown and junior attacker Phillip Robertson in their goals. The third quarter finished 11–10 in the Tigers' favor with no team completely dominating play.
Robertson scored first in the fourth quarter, bringing Princeton two ahead. Although Denver responded with a minute later with a goal, first-year midfielder Beau Pederson scored with seven minutes to go, bringing the score to 13–11. Cordrey's goal with three minutes remaining increased the Tigers' point buffer, but two goals in the last minute from Denver brought everyone on edge. Princeton held out, and the game ended 14–13.
The Tigers resume Ivy League play this Saturday when they will take on Brown University (3–5, 1–0 Ivy).
"We absolutely are ready for Ivy play," said Coach Madalon.


Milan and New York spring 2013.
19 Greek Street has launch this limited edition of Re-imagined chairs giving using end of line fabrics from Marc BY Marc Jacobs' spring collections. This range is an evolution of Re-Imagine's philosophy of adapting WASTED elements and ALLOWING them with a useful second life.


text:
Dispatch from Eastpoint: Requiem for an oyster and a way of life
By Richard Bickel Guest Columnist
(Editor's note: this article originally appeared earlier this month in The Tallahassee Democrat)
There is, perhaps, no town in America that so profoundly reveals the human toll from a loss of a fishery than Eastpoint.
Perched on the edge of Apalachicola Bay, whose waters once provided 90 percent of Florida's annual oyster harvest and 10 percent of the nation's, tiny Eastpoint, population 2,000, is literally built on a foundation of oyster shells. Indeed, dig anywhere along the town wharf where fish houses have for over a century shucked and packed literally tons of the bivalves, shipping them to lands far and wide, and your shovel will hit nothing but oyster shells from five to 10 feet down.
Eastpoint is the historical landing and processing port of most of Apalachicola Bay's oyster crop. In its heyday, the town wharf was chock-a-block with shucking houses, oyster skiffs and docks, and pulsing with an energy that one finds wherever men and women ply a wage from a fertile sea.
I arrived in Eastpoint on assignment for a national travel magazine in 1995 at daybreak, and immediately recalled epic waterfront scenes I've photographed on Burma's Irrawaddy River, Mali's Niger River and other bustling ports. Outboard engine smoke and the smell of fish guts thickened dank air and shores were clogged with an armada of wooden boats and fishermen preparing for a day at sea. It was at once raw and beautiful.
On that morning I met and photographed several Eastpoint oystermen. One was James Polous, who fed and clothed his family and built his house on oysters. James offered to take me aboard his plywood skiff to see up close the art of tonging oysters. We motored to Cat Point, a productive oyster reef near the St. George Island Bridge. Lowering his oyster tongs and pausing from the brutality of his work, the strapping oysterman surveyed the dawn horizon.
"Jesus is in these waters," Polous shared, shaking off a sweat-soaked cap. "Look at what he's provided." And then offering, "This here's a drive-in bank -- drop an anchor and get what you want. Make it a day or tong enough for a six-pack."
James was emblematic of the many oystermen I've been privileged to call friends: proud, soft-spoken, hard-working and never, ever complaining about the harshness of his trade.
Today, the Eastpoint wharf is but a ghostly whisper of that past. Abandoned, half-submerged oyster boat bows and decaying dock pilings poke out of the steely Bay water like a grotesque sunken forest while remnants of shuttered, hurricane-ravaged shucking houses scatter the banks with twisted boards and chunks of mortar.
The oystermen have vanished. As have the armies of oyster shuckers – mostly women-- who once earned a good paycheck processing oysters for out-of-state export in the dozens of Eastpoint shucking houses. Some of the fishers and shuckers are now cleaning tourist rental houses on nearby St. George Island; some are finding odd jobs such as cutting grass and others, by virtue of their fierce independence, cannot easily adjust to a structured workplace and have yet to find gainful employment.
Yet, there is newly-emerging life on the Eastpoint waterfront. Tourist-catering bars and restaurants are opening and Eastpoint even boasts its own brewery. This is Florida with a water view, and tourism, the probable future of the dying oyster port. But sadly for many out-of-work oystermen, the new Eastpoint holds few opportunities to put bread on the family table.
Tammy Boone, wife of Eastpoint oysterman Michael Boone, succinctly stated the mindset of many members of the Eastpoint fishing community. "This is becoming a town for rich tourists. It's my home but I don't belong anymore."
Add an oil spill, a fire and a hurricane to the life of an Eastpoint oysterman and there is little left to go wrong. The Deepwater Horizon spill closed the Bay to harvesting for nearly a half-year in 2010. In 2018, a controlled burn by the state of Florida erupted into an uncontrolled inferno, leveling 40 homes in the fishermen's quarters. (To date, fire victims have received no compensation). Later that year, epic Hurricane Michael struck, again closing the Bay and putting oystermen out of work for months.
"People say you can't live on nothing," says Tammy Boone, "but we do." The Boone's home was a total loss from the fire and they now live in a confined camper. Tammy works part time in a nearby convenience store; Michael gets occasional work repairing small engines. In the Boone's front yard is a barrel heaped with discarded aluminum cans. "A guy comes around every week and gives us ten dollars a barrel," says Tammy. "We do what we can."
Factors propelling the Apalachicola oyster to near-extinction are many: Agricultural run-off up-stream; oyster over-harvesting; Army Corp of Engineers river dredging, silting the Bay and River to smother oysters and other aquatic life. The slicing of St. George Island in two (in 1954, to give fishing boats direct access to the Gulf from Apalachicola) created a channel that sweeps much of the oysters' food source -- dissolved leaf litter -- out of the Bay and into the open Gulf.
But Atlanta and its monstrous thirst is the core of the problem. The entire Apalachicola Bay ecosystem, once one of the most prolific cradles of life on the planet, is struggling as the megalopolis, with its exponential growth, syphons the Bay's life-giving waters through a series of dams with impunity. (At the Florida-Georgia border, the Apalachicola River changes its name to the Chattahoochee which reaches to metropolitan Atlanta). Thus, there has been a drastic alteration of Bay water chemistry. Blue crabs are vanishing, fin fish are far fewer and the Bay floor, once a living carpet of oysters, is a vast expanse of bottom mud.
With the oyster's demise, so have imploded the livelihoods of hundreds of generational Apalachicola Bay fishermen who have asked for little but the right to board a plywood boat to take an honest wage from the sea. Oystering is a profession not only of brutal work but one of dignity and honor. That we as a society have deemed these people and the waters that have nourished their way of life expendable, even worthless, is wrong and patently obscene.
There has been an ethereal connection between man and the sea since the beginning of time. This bond was never so deeply revealed to me than by oysterman friend, James Cain. James was 75, and in his 55th year working on the Bay when I drove him to Tallahassee to visit his hospitalized wife. We reached a highway fork: to the left was the land route to the hospital; to the right, the Gulf route.
I signaled for the quicker land route. But James tapped my shoulder and said quietly. "Let's go right. I don't want to lose sight of the water."
Richard Bickel is a photojournalist and an Apalachicola resident who has documented the fishermen of Apalachicola Bay and their way of life for two decades. He encourages readers to support the environmental organization, ApalachicolaRiverkeeper.org, which is aggressively fighting for the River and Bay and those who depend on its waters. More of Bickel's work may be seen at www.richardbickelphotography.com
In August of 2020, the State of Florida banned wild oyster harvest in Apalachcola Bay for five years to allow oysters to repopulate. The ban had the support of environmental groups, The Franklin County Seafood Workers Association and most, but not all, oyster harvesters. But the closure was largely symbolic; the oysters were already gone as were most of the fishers who harvested them.
Meanwhile, Atlanta's massive thirst continues with no viable plan for conservation. The issue of Apalachicola River water allotment has become the East's first water war and is now in the United States Supreme Court awaiting a ruling.


text:
Google News - Australia
Reading View
One-atom switch supercharges fluorescent dyes
miragenews.com 09/10/2019 00:03:37 Mirage News
Having trouble reading this article? Read the Original Article »
It only took the replacement of one atom for Rice University scientists to give new powers to biocompatible fluorescent molecules.
The Rice lab of chemist Han Xiao reported in the Journal of the American Chemical Society it has developed a single-atom switch to turn fluorescent dyes used in biological imaging on and off at will. The technique will enable high-resolution imaging and dynamic tracking of biological processes in living cells, tissues and animals.
The Rice lab developed a minimally modified probe that can be triggered by a broad range of visible light. The patented process could replace existing photoactivatable fluorophores that may only be activated with ultraviolet light or require toxic chemicals to turn on the fluorescence, characteristics that limit their usefulness.
The researchers took advantage of a phenomenon known as photo-induced electron transfer (PET), which was already known to quench fluorescent signals.
Rice University chemist Han Xiao and his colleagues have discovered a simple method to turn fluorescent tags on and off with visible light by switching one atom. Photo by Jeff Fitlow
They put fluorophores in cages of thiocarbonyl, the moeity responsible for quenching. With one-step organic synthesis, they replaced an oxygen atom in the cage with one of sulfur. That enabled them to induce the PET effect to quench fluorescence.
Triggering the complex again with visible light near the fluorescent molecule's preferred absorbance oxidized the cage in turn. That knocked out the sulfur and replaced it with an oxygen atom, restoring fluorescence.
"All it takes to make these is a little chemistry and one step," said Xiao, who joined Rice in 2017 with funding from the Cancer Prevention and Research Institute of Texas (CPRIT). "We demonstrated in the paper that it works the same for a range of fluorescent dyes. Basically, one reaction solves a lot of problems."
Researchers worldwide use fluorescent molecules to tag and track cells or elements within cells. Activating the tags with low-powered visible light rather than ultraviolet is much less damaging to the cells being studied, Xiao said, and makes the long exposures of living cells required by super-resolution imaging possible. Super-resolution experiments by Theodore Wensel, the Robert A. Welch Chair in Chemistry at Baylor College of Medicine, and his team confirmed their abilities, he said.
"We feel this will be a really good probe for living-cell imaging," Xiao said. "People also use photoactivatable dye to track the dynamics of proteins, to see where and how far and how fast they travel. Our work was to provide a simple, general way to generate this dye."
At top, a sequence shows the design of thio-caged dyes designed at Rice University to be triggered by visible light. At bottom, confocal and super-resolution imaging of a lipid droplet in living adipocytes incubated with BODIPY (green), SNile Red (red) and Hoechst 33342 (blue), followed by photoactivation using a 561-nanometer laser. Scale bar: 10 µm. Scale bar for super-resolution image of lipid droplet labeled with SNile Red, bottom right: 1 µm. Courtesy of the Xiao Lab
The researchers found their technique worked on a wide range of common fluorescent tags and could even be mixed for multicolor imaging of targeted molecules in a single cell.
Rice postdoctoral researcher Juan Tang is lead author of the paper. Co-authors are Rice graduate students Kuan-Lin Wu and Jingqi Pei; postdoctoral fellow Michael Robichaux of Baylor; and graduate student Nhung Nguyen and Yubin Zhou, an assistant professor at the Center for Translational Cancer Research at Texas A&M University. Xiao is the Norman Hackerman-Welch Young Investigator and an assistant professor of chemistry, biosciences, and bioengineering.
CPRIT, the Robert A. Welch Foundation, a Hamill Innovation Award, a John S. Dunn Foundation Collaborative Research Award and the National Institutes of Health supported the research.
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text:
Q: What is the difference between colors and logColors in configuration field I have a project based on angularjs and written with coffeescript. I am working to add unit tests to my project with karma and jasmine
Building my configuration file I found two way to specify colors in logging.
colors:true

and
logColors:true

What is the difference between colors and logColors?
How do I have to specify log level?

A: You can specify:

*

*colors

Type: Boolean
Default: true
Description: Enable or disable colors in the output (reporters and logs).

*

*logLevel

Type: Constant
Default: config.LOG_INFO
Possible values:

*

*config.LOG_DISABLE

*config.LOG_ERROR

*config.LOG_WARN

*config.LOG_INFO

*config.LOG_DEBUG

Description: Level of logging.

From https://karma-runner.github.io/0.12/config/configuration-file.html

redpajama_set_name: RedPajamaStackExchange


text:
Q: Error in dumbbell plot with facet grid, unable to create a side by side plot I tried plotting a dumbbell plot side by side with a facet grid but got some interesting errors.
Error: At least one layer must contain all faceting variables: Trial_type, Session..

*

*Plot is missing Trial_type, Session.

*Layer 1 is missing Session.

*Layer 2 is missing Session.

*Layer 3 is missing Session.

*Layer 4 is missing Session.

*Layer 5 is missing Trial_type, Session.

*Layer 6 is missing Session.
Run rlang::last_error() to see where the error occurred.
In addition: Warning messages:
1: In min(x) : no non-missing arguments to min; returning Inf
2: In max(x) : no non-missing arguments to max; returning -Inf
3: In min(x) : no non-missing arguments to min; returning Inf
4: In max(x) : no non-missing arguments to max; returning -Inf

data10A <- structure(list(Session = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2), conditon2 = structure(c(10L,
4L, 12L, 6L, 11L, 5L, 1L, 7L, 3L, 9L, 2L, 8L, 10L, 4L, 12L, 6L,
11L, 5L, 1L, 7L, 3L, 9L, 2L, 8L), .Label = c("CEN_LLL", "CTL_LLL",
"IPS_LLL", "CEN_RRR", "CTL_RRR", "IPS_RRR", "CEN_RLR", "CTL_RLR",
"IPS_RLR", "CEN_LRL", "CTL_LRL", "IPS_LRL"), class = "factor"),
    Trial_type = structure(c(2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L,
    1L, 2L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 2L, 1L,
    2L), .Label = c("Retention", "Transfer"), class = "factor"),
    NormalizedJerk_102 = c(1270.168699, 2099.703957, 3259.268053,
    1152.257445, 3810.890123, 4601.918336, 1792.371775, 1288.768888,
    2699.08162, 1650.968794, 2018.457394, 6159.567785, 931.350429,
    1053.84252, 1611.673955, 1034.363607, 5352.195367, 2499.83996,
    1560.678962, 915.3845866, 1948.757464, 1341.815274, 2113.732859,
    2051.140838), NormalizedJerk_104 = c(853.7034116, 924.8554548,
    2268.966702, 675.7160839, 2442.874632, 1603.954653, 1010.111276,
    794.1752256, 1313.813984, 1197.638788, 1039.577947, 3125.131019,
    561.2311988, 767.7541159, 1019.744071, 769.6067294, 2232.404471,
    1292.509181, 884.8343164, 663.0273865, 1230.369444, 717.8466364,
    1536.027898, 1027.358586), key = c("Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk", "Change in normalized jerk",
    "Change in normalized jerk")), row.names = c(NA, -24L), class = c("tbl_df",
"tbl", "data.frame"))

  library(tidyverse)
  library(dumbbell)
 dumbbell::dumbbell(data10A, id = "conditon2", key = "Trial_type",
          leg = "Test",
          column1 = "NormalizedJerk_102",
          column2 = "NormalizedJerk_104",
          delt = 1, lab1 = "Pre-test", lab2 = "Post-test",
          p_col1 = "black", p_col2 = "grey40",
          textsize = 4, segsize = 1.5,
          pointsize = 2.5,
          title = "Change in Normalized jerk from Pre- to Post-test")  +
   facet_wrap( Trial_type ~ Session., scales="free", ncol=2) +
   theme(axis.text.x = element_text(size = 12, face = "bold"),
         axis.text.y = element_text(size = 11, face = "bold"),
         legend.position = "right",
         legend.text = element_text(size = 12),
         legend.title = element_text(size = 14),
         strip.text = element_text(face = "bold", size = 14, color = "black"))

A: The issue is that dumbbell::dumbbell does some data pre-processing before constructing the dumbbell chart via ggplot2. Among other things unnecessary columns get dropped during this step, i.e. only the columns specified in the arguments of dumbbell are kept while all other columns which in your case includes your Session column get dropped. As a consequence you are not able to facet by Session.
One workaround would be to add a new key column to your data which is simply the interaction of Trial_type and Session. This key column you could then passed to the key argument off dumbbell and could be used as faceting variable:
library(ggplot2)
library(dumbbell)

data10A$key <- paste(data10A$Trial_type, data10A$Session, sep = "\n")
dumbbell::dumbbell(data10A, id = "conditon2", key = "key",
                   leg = "Test",
                   column1 = "NormalizedJerk_102",
                   column2 = "NormalizedJerk_104",
                   delt = 1, lab1 = "Pre-test", lab2 = "Post-test",
                   p_col1 = "black", p_col2 = "grey40",
                   textsize = 4, segsize = 1.5,
                   pointsize = 2.5,
                   title = "Change in Normalized jerk from Pre- to Post-test")  +
  facet_wrap( ~ key, scales="free", ncol=2) +
  theme(axis.text.x = element_text(size = 12, face = "bold"),
        axis.text.y = element_text(size = 11, face = "bold"),
        legend.position = "right",
        legend.text = element_text(size = 12),
        legend.title = element_text(size = 14),
        strip.text = element_text(face = "bold", size = 14, color = "black"))

Created on 2021-11-14 by the reprex package (v2.0.1)


A stunning variety whose vibrant blooms are composed of masses of petals. The clump-forming plants perform well in the garden but make excellent cut flowers too! Orange - Flowers May-June. Height 20-30cm.
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A fabulous new ranunculus variety of this season's must-have plant! Amazing orange blooms with vivid green centres. Flowers May-June. Height 20-30cm (8-12").
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Person 3: I think I'll start with me, then it's you.
Person 1: *said threateningly* Make them go.


text:
Can your personality change after a TBI?
On Behalf of Winckler & Harvey, PLLC | Sep 29, 2022 | Personal Injury
A traumatic brain injury can leave you battling numerous repercussions that require ongoing treatment. You may feel and behave differently as a result of the trauma. Feeling foreign in your own body may trigger anxiety and feelings of uncertainty and hopelessness....
The long-lasting consequences of untreated preeclampsia
On Behalf of Winckler & Harvey, PLLC | Sep 12, 2022 | Medical Malpractice
Pregnancy naturally comes with certain risks, some minor, others serious. While for many women it remains a smooth process, others may suffer complications, including preeclampsia. According to the Centers for Disease Control and Prevention, in every 25 cases,...


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Crazy For Kids Books: Cybils Finalist: Go to Bed, Monster!
Cybils Finalist: Go to Bed, Monster!
Natasha Wing and Sylvie Kantorovitz have created a terrific bedtime story that will resonate with parents and kids alike. A very unsleepy Lucy gets out her crayons one evening and crayons herself into a series of adventures with a new friend named Monster. The two of them build castles and fly airplanes and march in a parade together until Lucy is tired and ready for sleep. But, Monster isn't.
Then the recognizable list of bedtime needs and complaints begins as Lucy takes on the parental role of putting Monster to bed and getting him to sleep. This role reversal will amuse children and parents both.Part of the charm of this story is the lean elegance of the text and illustrations. They effectively convey the power of a child's imagination while demonstrating to children the universality of bedtime games and the struggles parents have in getting their children to bed and to sleep.
Congratulations to Natasha and Sylvie on becoming a Cybils' Award Finalist.
I highly recommend this story for children ages 4-6.
Check out more blogger reviews for Go to Bed, Monster! at Seven Impossible Things Before Breakfast, Cheryl Rainfield's Book Blog, and World of Words.
ALSO check out this interview with Natasha Wing and Syvlie Kantorovitz at Becky's Book Reviews.
Welcoming 2008 with Cybils Award Finalists!


text:
Israel adopts national plan for reducing green house gases
by bayanihan on November 30, 2010
JERUSALEM, Nov. 29 — The Israeli cabinet on Sunday approved a national project to reduce green house gas emissions, allocating 2.2 billion shekels (about 598 million U.S. dollars) over the upcoming decade.
The plan includes reducing home electricity consumption, renovation of public and residential buildings for energy- efficiency and information campaigns to increase public awareness, said a statement issued by the prime minister's office.
The plan's implementation carries long-term significance for the state, said the statement.
"The recent dry months, including the driest November in the history of the state, are a warning light to us all that the threat of climate change is no less menacing than the security threats that we face. In a country that suffers from a severe water shortage, this is an existential struggle," Israeli Prime Minister Benjamin Netanyahu was quoted as saying in the statement.
Meteorologists say that rainfall in the area this year is the lowest since 1962, and that temperatures are the warmest they've seen in decades.
The Israeli Water Authority has said, according to local Ynet news site, that this November was the driest in almost half a century. (PNA/Xinhua) ALM
Next post: Mexico president calls for responsible response to climate change
Previous post: Today in History


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text:
Jamie Rose
Jamie Rose is a co-owner of Momenta Group LLC. She has worked as an international photojournalist on five continents and has been a contract photographer and communication strategist with some of the world's largest nonprofit organizations.
In 2011, Jamie was awarded the United Nations' IPC's Photographers Leadership Award. She has also won awards and grants for her documentary and nonprofit work including from The Alexia Foundation, and the prestigious White House News Photographers' Association Project Grant for her portfolio of nonprofit work from Africa. Jamie received her Master's Degree in photojournalism from the Newhouse School at Syracuse University and her bachelor's degree from American University. She completed her Master's thesis photographing volunteer health care workers in the West Bank and Israel. This project established her passion for working with nonprofits and covering humanitarian issues, while earning her a place in the prestigious agency team at Aurora Novus.
Her work has been published in National Geographic Books, The New York Times Magazine, The Washington Post Magazine, O Magazine, Rolling Stone, Better Homes and Gardens, Newsweek, TIME Magazine and others. Her nonprofit clients include The Global Fund, Doctors Without Borders, The Calvert Foundation and Physicians for Human Rights among many other local and domestic organizations. Her photo editing skills were rewarded when she was asked to be the lead photo editor National Geographic's "American Heroes" book.
Jamie is considered an expert in nonprofit photography business development. She has been interviewed twice by Photo District News (PDN), News Photographer Magazine, as well as online publications and podcasts about the importance of nonprofit documentary photography. She has been a guest lecturer at Georgetown University, American University, Hiram College, Syracuse University, and Columbia University's Journalism schools about her work with international nonprofits.
Jamie lives in the heart of the mountain in ski country with her husband and their awesome pound puppy, Cody, in Redmond, Oregon.


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text:
package com.microsoft.store.partnercenter.samples.analytics;

import com.microsoft.store.partnercenter.IPartner;
import com.microsoft.store.partnercenter.analytics.IPartnerLicensesAnalyticsCollection;
import com.microsoft.store.partnercenter.models.ResourceCollection;
import com.microsoft.store.partnercenter.models.analytics.PartnerLicensesDeploymentInsights;
import com.microsoft.store.partnercenter.samples.BasePartnerScenario;
import com.microsoft.store.partnercenter.samples.IScenarioContext;

public class GetPartnerLicenseDeploymentInsights extends BasePartnerScenario {

	public GetPartnerLicenseDeploymentInsights(IScenarioContext context) {
		super("Show partner license deployments", context);
	}

	@Override
	protected void runScenario() {
		IPartner partnerOperations = this.getContext()
				.getUserPartnerOperations();
		IPartnerLicensesAnalyticsCollection analyticsOperations = partnerOperations
				.getAnalytics().getLicenses();
		this.getContext().getConsoleHelper()
				.startProgress("Getting the partner license deployments");
		ResourceCollection<PartnerLicensesDeploymentInsights> deployment = analyticsOperations
				.getDeployment().get();
		this.getContext().getConsoleHelper().stopProgress();
		this.getContext().getConsoleHelper()
				.writeObject(deployment, "Partner license deployments");
	}
}

redpajama_set_name: RedPajamaGithub


Ivara opened the door slowly. She hesitated then, wondering if her task was just. Her Superior had his elbows on the 机, デスク and appeared to be staring intently over the 上, ページのトップへ of his open laptop at the 画像 taped to the wall. He wore the same outfit as he had worn that morning at practice: dark red sweats and nothing more.
Ivara could see the marks that were etched in Nic's back, light scars on his pale skin. She was reminded of the markings that decorated her own body, the バイオレット tendrils swirling around her limbs and torso. Ivara felt a connection with her Superior in this way: they both were cursed to carry around the scars of their pasts.
"Are あなた going to just stand there, または did あなた come for a reason?" Nic asked.
Ivara took a spilt 秒 to collect herself before beginning, "I have something I need to speak with あなた about." The leader let out a groan before swiveling around in the chair to face her, arms crossed over his chest. "What is it now? もっと見る trouble for the team, I suppose with our luck."
"Actually, it's about Superior Titanium."
This got Nic's full attention; he sat forward. "I'm listening."
"I-I think he's been stealing Reverse from Dr. Xian's lab."
"BLAZE!" Jace ran into the room, nearly knocking into Ivara. "Seeker- he- went after- Ti," he informed between pants.
Nic was on his feet and headed toward the door in a スプリット, 分割 second, pulling on a coat, and shouting an order over his shoulder. "Fliers, with me. Jace, get the others and prepare a cot."
"In your dreams," Twan's daughter responded. She let up on the gas the slightest. Chance's car pulled off the ramp just in front of hers. He drifted with the momentum, and Tessa took her opportunity. She sped up and rammed into the race car, sending him spiraling into the railing.
"Ha!" she cheered as her car crossed the finish line, first place flashing on the screen.
Chance stated at the screen as he made fifth place. "Who taught あなた to play like that?"
Tessa shrugged, a grin on her face. "Who's next?"
"Me!" Blade volunteered from the arm chair.
"You wanna play?" Chance asked, offering his controller to Alek. He shook his head, reluctant to take his arm from around his girlfriend's shoulder. Tara accepted the controller and snuggled closer to Alek. He grinned and kissed the 上, ページのトップへ of her head as the race started.
It was the first time in awhile he had joined the group while they relaxed, playing games, and chatting about nothing of great importance. It was gratifying, and though he would miss regularly sparring with Nic, taking Jace and Ivy to swim in the ベイ, 湾 while Sounder barked at them from the pier, the ボクサー occasionally leaping into the water to 登録する the fun, Alek was glad he would get to be around the other heroes-in-training. He felt it was where he belonged.
Chance glanced up from his place on the floor where he sat cross-legged watching the race as Alek accidentally kicked him as he coughed into his sleeve.
"You okay?" the tawny-haired boy asked. Blade suddenly tensed and turned he red eyes toward the Russian. "I smell blood," she announced, already standing to make her retreat as the overwhelming scent taunted her.
Alek doubled over coughing and all eyes turned to him. When he pulled his arm back, it was spotted with blood.
Tara's eyes widened. "Alek!" she cried as he slipped off the ソファー, ソファ onto the floor on his knees. His shoulders shuddered with each cough, blood staining the carpet. Tessa and Chance kneeled beside him, unsure of what to do.
In the chaos and shock no one noticed when the hooded man suddenly materialized out of thin air until he had already jabbed the syringe into Alek's shoulder. "Catch him," Seeker instructed as the sick boy let out a final wheeze and collapsed into his teammate's arms.
Nic suddenly charged into the room, Emily and Ivara on his heels.
"What did あなた do?" The fire-caster demanded, suddenly five inches from the assassin, steam rising off his clothes.
"Injected him with an anesthetic before he coughed his lungs out," Declan answered infuriatingly calmly. "I suggest we get him back to base before he wakes up."
Nic's nostrils flared as he suppressed his rage. "Help me get him up," he ordered. Chance and Tessa stepped back out of the way as Emily, Nic, and Seeker knelt to lift the limp boy.
"My car's outside the Gotham zeta," Declan informed. They led him toward the tube. Ivy made a 移動する to go after them, but back peddled as a fiery eyed brunette stepped in her path.
"I'm going with," she stated. Ivara opened her mouth to argue, but realized she would not be able to dissuade her, and nodded. She turned and sprinted after the others, Tara right on her heels.
Ahhhhh! D: smerghakfroofmrdfgrprktreofrepdgr Declan maybe I have to do that to you so I can catch you and you can be locked in my closet forever....( eyes widen) I mean what.........pffft. Chance weooooooo wait that's my Chancey right? Smergader I'm suprised Rowan wouldn't be there he's a video game addict. But again lovely work I hope to see another chapter soon! Another cliff hanger agh!
((HEY I'M IN THE COMMENTS!!!!!!!!!!!!! Yeah R_L is busy with something so I'm here!!!!!
I try to put an many other people's characters in articles, and since I use Tara a lot... Plus I don't think Id portray you very well.
And I'm not sure if anyone caught it, but Nic told Jace to prepare a cot web he went after Declan, BEFORE he knew that Alek was sick.
Basically he planned on putting Declan on that cot for attacking Alek.
There's not enough violence in this series, just a lot of heartbreaking shiz... Hmm..
*tosses a cookie at becca* You can have a cookie too, oh scary one.
Most enthusiastic character in the entire club.
Yeah Cat doesn't get sarcasm.
*pats his head* I'll explain later, darling.


text:
12 Types Of Computer Games Every Gamer Should Know About | Thought Catalog All kinds of games to play
🎰 List of types of games - Wikipedia
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All kinds of games to play
But, many games can be considered to be more than one genre.
For instance, a all kinds of games to play game could be considered a sports game, as well as a simulation game.
Here is some information that will help you to better understand the various computer game genres.
Massively Multiplayer Online MMO These games are played over a LAN local area network or via the Internet.
Players use a network and interact with other players in the virtual game room.
You can play against people from all over the world.
The creation of these games involves thousands of hours of programming, giving you, the player, the most amazing gaming experience possible.
Simulations These games involve taking control of real-world vehicles, including tanks, ships, and aircraft.
You learn how to control these vehicles, and use simulation games that can also be used to train professionals.
In fact, many pilots are trained using before they actually take flight.
Adventure These are usually single player games, and are often set in fantasy or adventure worlds.
You complete puzzles to advance levels.
The game generally starts with a back story of your character, and let you know what your mission is.
You have to figure out how to complete the mission.
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Similar to a strategy game, RTS games move in real-time, and players can play at once without taking turns so you can play together at the same time.
Puzzle These games appeal to those who love to solve.
There are many levels, from beginner to expert, and games usually have coloured shapes and simple actions.
These are brain games, with no action involved.
Action You need to be fast to enjoy these fast-paced games, and you need to have excellent reflexes.
Complete challenges by fighting with enemies, and use a character of your choice to represent yourself and jump into the action.
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Combat Fight one on one with opponents, up close and personal.
You need good reflexes, and the ability to use the controls for all kinds of fighting moves.
First Person All kinds of games to play FPS You are the protagonist, and the.
You can really get into these games.
The only downfall is that you are not able to see how you look in the game, because you are seeing things through your own eyes.
Sports Play real-world sports like baseball, basketball, soccer, and more.
As you work up through the various skill levels, your game will involve mimicking real professional athletes and how they move.
The most popular sports games are usually based around specific popular sporting events.
Role-Playing RPG If you love fantasy, you will love role-playing games.
You get to act out the part of the main character, be the hero, etc.
Many of all kinds of games to play games have narrative guides.
Educational Learn more here many parents and teachers complain about video games, there are some great out there that can help with the learning process.
Train in a variety of subjects, using games to make learning fun instead of boring.
There are testing functions, where you can answer multiple choice questions.
The most common types of education games are for math, science, and ICT.
These games are perfect for anyone looking for intense competition and plenty of depth.
Today, games of all sorts can be played smoothly and easily virtually anywhere WiFi or cellular connectivity is available.
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Because each card game has a different setup and desired outcome, the rules vary depending on the game, but almost all card games involve at least one deck of cards, a particular dealing pattern and a definitive winning outcome. There are thousands of card games, and many basic games also have variations designed to make play more interesting.
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text:
Q: Working example of datatables with lazy loading of images I'm looking for a working example of lazy-loading of images within datatables that continues to work after clicking on a column heading to change the sorting.
The method I've had the most success with uses the jquery.lazyload plug:
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery.lazyload/1.9.1/jquery.lazyload.min.js"></script>
<script type="text/javascript">
    $(function() {
    $("img.lazy").lazyload();
    });
</script>

Here's how I reference the image:
<img class="lazy" data-original="https://example.com/image.jpg" width="50" height="50" />

The lazy loading works fine if I never change the sorting. However, if I decide to change the sorting of the table by clicking on one of the column headings, the lazy loading stops working and any images that were not downloaded at this point will remain blank when I scroll them into view.
I wasn't born a javascript or jQuery expert, so had to learn from examples here on SO and elsewhere. But this problem stumps me. From my research, I see a lot of comments and snippets about how it is possible, but no actual working example that proves this is supported.
jquery-lazyload images in jquery-databables
https://datatables.net/forums/discussion/1959/image-thumbs-in-table-column-lazy-loading
I wont rattle off the numerous other links I found here and elsewhere, I just need some kind soul to help.

A: Use drawCallback option to define a function that will be called every time DataTables performs a draw.
For example:
$('.table-data').DataTable({
   drawCallback: function(){
        $("img.lazy").lazyload();
   }
});

See this jsFiddle for code and demonstration.

A: I'm sort of stuck with this one seeing as I don't have access to your backend or more information about your circumstances but I think what will work will be to call your lazy load code in the draw event like this:
$(document).ready(function() {
    $("img.lazy").lazyload();
    $('.table-data').DataTable({
        "drawCallback": function(){
            $("img.lazy").lazyload();
            setTimeout(function(){
                $(window).trigger("scroll")
            }, 1000);
        }
    });
});

I've got an example up here, but I think you might be using server-side processing?