text:
\section{Introduction}
\label{intro}
Quantum mechanical many-body dynamics is essential  for the understanding of  wide range   phenomena in modern physics, including Bose-Einstein condensate and superfluidity  \cite{London1938Phenomenon,Landau:1941vsj,Bogolyubov:1947zz}. The   many-body dynamics usually rely on approximate approaches in the past, such as Hartree-Fock method \cite{Hartree1935}. In recent years, a lot progresses  have been made toward the study of few- and many-body dynamics from first principle, quantum chromodynamics (QCD)  \cite{Aoki:2007rd,Feng:2010es,Lang:2011mn,Aoki:2011yj,Dudek:2012gj,Dudek:2012xn,Wilson:2014cna,Wilson:2015dqa,Dudek:2016cru,Beane:2007es,Horz:2019rrn,Detmold:2008fn,Detmold:2008yn}. The calculation of lattice QCD   is usually performed  in Euclidean space with all particles   confined in a periodic cubic box, hence   the multihadron dynamics  is not directly accessible.  Instead dynamics is encoded in a discrete energy spectrum of   multihadron system in finite volume.
Therefore, establishing a method of  mapping out infinite-volume multihadron dynamics from discrete energy spectrum in finite volume has become an important subject in past few years.  Such a connection in two-body sector  is established   by L\"uscher formula in \cite{Luscher:1990ux} and its extensions \cite{Rummukainen:1995vs,Christ:2005gi,Bernard:2008ax,He:2005ey,Lage:2009zv,Doring:2011vk,Briceno:2012yi,Hansen:2012tf,Guo:2012hv,Guo:2013vsa}. Many promising developments along different approaches   have been made toward few- and many-body finite volume systems  recently \cite{Kreuzer:2008bi,Kreuzer:2009jp,Kreuzer:2012sr,Polejaeva:2012ut,Briceno:2012rv,Hansen:2014eka,Hansen:2015zga,Hansen:2016fzj,Hammer:2017uqm,Hammer:2017kms,Meissner:2014dea,Briceno:2017tce,Sharpe:2017jej,Mai:2017bge,Mai:2018djl, Doring:2018xxx, Romero-Lopez:2018rcb,Guo:2016fgl,Guo:2017ism,Guo:2017crd,Guo:2018ibd,Guo:2018xbv,Guo:2019hih,Blanton:2019igq,Romero-Lopez:2019qrt,Guo:2019ogp,Blanton:2019vdk,Mai:2019fba}.  One crucial thing to justify these  recent developments is to perform some tests and reproduce some known results, such as the threshold expansion formula that was originally derived by perturbation theory \cite{Huang:1957im,Lee:1957zzb,Beane:2007qr,Detmold:2008gh}.

Motivated exactly   by the purpose of  testing our formalism on finite volume $N$-body dynamics based on variational approach \cite{Guo:2018ibd,Guo:2019hih,Guo:2019ogp}, in this work, we illustrate how the well-known  threshold expansion formula for $N$-identical-boson system \cite{Huang:1957im,Lee:1957zzb,Beane:2007qr,Detmold:2008gh}  may be derived from  coupled dynamical equations. The exact value of eigen-energy of $N$-body system  are given by the eigen-solution of these Faddeev-type  coupled dynamical equations. Faddeev-type coupled dynamical equations is a non-perturbative approach, hence it applies in principle to both weakly and strongly coupled system. To reproduce  threshold expansion formula, the perturbation expansion in terms of weak coupling is carried out by iterations of coupled dynamical equations. A energy dependent  closed form    is thus obtained, and it ultimately yields the  threshold expansion formula by further expansion near threshold. The threshold expansion formula up to $\mathcal{O} (\eta^4/L^6)$ for pair-wise interaction and   $\mathcal{O} (\eta_3/L^6 )$ for three-body interaction  is already known \cite{Beane:2007qr,Detmold:2008gh}, where $\eta$ and $\eta_3$ are the two-body   and   three-body coupling strengths respectively.   The exact expression of $\mathcal{O} (\eta^4/L^6)$ expansion formula requires higher order terms by multiple iterations, which ultimately becomes a tedious task.  To simplify our presentation since the result is not new, in this work, we will only show  the derivation of the threshold expansion formula  up to  $\mathcal{O} (\eta^3/L^5)$ and  $\mathcal{O} (\eta_3/L^6 )$ by a single iteration, in terms of perturbation theory, they may be associated with   $\eta^2$ and $\eta \eta_3$ order diagrams respectively.

The paper is organized as follows.  The   formalism of finite volume  $N$-identical-boson systems is presented   in detail  in Section \ref{Nbdynamics}.  The derivation of threshold expansion formula  is illustrated       in Section \ref{ThresholdExpansion}.     Summary  is  given in Section \ref{summary}.

\section{ $N$-boson dynamics in finite volume }\label{Nbdynamics}

The dynamics of $N$ non-relativistic identical bosons in finite volume is described by Lippmann-Schwinger type integral equation, see Refs.~\cite{Guo:2019hih,Guo:2019ogp},
\begin{equation}
\Phi_E(\{\mathbf{ x}\})  = \int_{L^3} \prod_{i=1}^N d \mathbf{ x}'_i G_E (\{\mathbf{ x} - \mathbf{ x}'\} )  V(\{\mathbf{ x}'\})  \Phi_E(\{\mathbf{ x}'\}) , \label{LSeq}
\end{equation}
where the position of i-th particle is denoted by $\mathbf{ x}_i$, and $\{\mathbf{ x}\}  = \{\mathbf{ x}_1, \cdots , \mathbf{ x}_N \}$. The $N$-body finite volume Green's function is given by
\begin{equation}
G_E (\{\mathbf{ x}  \} )    = \frac{1}{L^{3 N}} \sum_{ \{ \mathbf{ p}\}} \frac{ e^{ i \sum_{i=1}^N  \mathbf{ p}_i \cdot \mathbf{ x}_i } }{E - \sum_{i=1}^N  \frac{\mathbf{ p}^2_i}{2m} },
\end{equation}
where $ \{ \mathbf{ p}\} =  \{ \mathbf{ p}_1, \cdots , \mathbf{ p}_N\}$, and $\mathbf{ p}_i = \frac{2\pi}{L} \mathbf{ n}_i$ with $\mathbf{ n}_i \in \mathbb{Z}^3$ stands for the free momentum of i-th particle. $L$ is the size of the cubic box. The finite volume Green's function is the solution of differential equation,
\begin{equation}
\left ( E + \sum_{i=1}^N  \frac{ \nabla^2_i}{2m} \right )G_E (\{\mathbf{ x}  \} ) =\prod_{i=1}^N \sum_{\mathbf{ n}_i \in \mathbb{Z}^3} \delta( \mathbf{ x}_i + \mathbf{ n}_i L),
\end{equation}
and satisfies periodic boundary condition,
\begin{equation}
G_E (\{\mathbf{ x}  + \mathbf{ n} L \} ) = G_E (\{\mathbf{ x}    \} ).
\end{equation}
Due to the periodic nature of Green's function, the periodicity of wave function, $\Phi_E(\{\mathbf{ x} + \mathbf{ n} L\}) =\Phi_E(\{\mathbf{ x}\}) $, is hence automatically warranted by Eq.(\ref{LSeq}). The interactions among particles is described by $V(\{\mathbf{ x}\})$,    the same form as given in  \cite{Beane:2007qr} is used in present work, {\it i.e.} only contact pair-wise and three-body interactions are considered,
\begin{equation}
V(\{\mathbf{ x}\}) = \eta \sum_{(i<j)=1}^N \delta (\mathbf{ r}_{i j}) +  \eta_3 \sum_{(i<j<k)=1}^N \delta (\mathbf{ r}_{i k})\delta (\mathbf{ r}_{j k}),
\end{equation}
where $ \mathbf{ r}_{i j} =\mathbf{ x}_i - \mathbf{ x}_j$ is relative coordinate between i-th and j-th particles, and $\eta$ and $\eta_3$ are the coupling strengths for pair-wise and three-body contact interactions respectively.

As illustrated in Refs.~\cite{Guo:2019hih,Guo:2019ogp}, two types of finite volume Faddeev amplitudes may be introduced by
\begin{equation}
T_{(i j)} ( \{ \mathbf{ k}\}) = -  \int_{L^3}  \left ( \prod_{l=1}^N  d \mathbf{ x}_l e^{-  i    \mathbf{ k}_l \cdot \mathbf{ x}_l } \right )  \eta \delta (\mathbf{ r}_{i j})\Phi_E(\{\mathbf{ x}\})  ,
\end{equation}
and
\begin{align}
& T_{(ij k )} ( \{ \mathbf{ k}\})  \nonumber \\
& = -  \int_{L^3}  \left ( \prod_{l=1}^N  d \mathbf{ x}_l e^{-  i    \mathbf{ k}_l \cdot \mathbf{ x}_l } \right )  \eta_3 \delta (\mathbf{ r}_{i k})\delta (\mathbf{ r}_{j k})\Phi_E(\{\mathbf{ x}\})  ,
\end{align}
where  $T_{(i j)}$'s and  $T_{(i j k )} $'s are associated with pair-wise and three-body contact interactions respectively. There are totally $\frac{N(N-1)}{2}$   $T_{(i j)}$'s and  $\frac{N(N-1) (N-2)}{6}$  $T_{(i j k )} $'s.  Eq.(\ref{LSeq}) is thus turned into $\frac{(N+1) N (N-1)}{6}$ coupled equations, for instance,
\begin{align}
 & T_{(12)}   ( \{ \mathbf{ k}\})  = \frac{1}{L^3} \sum_{   \mathbf{ p}_2  } \frac{ \eta  }{E - \frac{   \sum_{i=1}^N \mathbf{ p}^2_i}{2m}  } \nonumber \\
& \times \left [  \sum_{i<j} T_{(ij)} ( \{  \mathbf{ p} \} ) + \sum_{i<j<k} T_{(ij k)} ( \{ \mathbf{ p} \} )  \right ],   \nonumber \\
&    \mathbf{ p}_1=  \mathbf{ k}_1 +  \mathbf{ k}_2 -\mathbf{ p}_2 , \ \   \mathbf{ p}_l= \mathbf{ k}_l,  \ \  l=3, \cdots, N,
\end{align}
and
\begin{align}
 & T_{(123)}  ( \{ \mathbf{ k}\})    = \frac{1}{L^6} \sum_{   \mathbf{ p}_2,\mathbf{ p}_3 } \frac{  \eta_3  }{E - \frac{   \sum_{i=1}^N\mathbf{ p}^2_i  }{2m}   }  \nonumber \\
 & \times \left [   \sum_{i<j} T_{(ij)} ( \{  \mathbf{ p} \} ) + \sum_{i<j<k} T_{(ij k)} ( \{ \mathbf{ p} \} ) \right ],   \nonumber \\
 &     \mathbf{ p}_1=   \sum_{i=1}^3 \mathbf{ k}_i  -\mathbf{ p}_2 - \mathbf{ p}_3 , \ \   \mathbf{ p}_l= \mathbf{ k}_l,  \ \  l=4, \cdots, N.
\end{align}
The rest of equations for $T_{(i j)}$'s and  $T_{(i j k )} $'s are thus obtained by swapping particle indices: $ 1 \leftrightarrow i$, $ 2 \leftrightarrow j$ and $ 3 \leftrightarrow k$.

\subsection{Symmetry consideration}
Because of exchange symmetry of $N$ identical bosons system, only two independent amplitudes are required.  Let's define
\begin{equation}
T  ( \overline{\{ \mathbf{ k}\}}_{(12)}) = T_{(12)} ( \{ \mathbf{ k}\}) , \label{2bamp}
\end{equation}
where  $\overline{\{ \mathbf{ k}\}}_{(12)} = \{    \mathbf{ k}_3, \cdots, \mathbf{ k}_N  \}$ is a subset of $ \{ \mathbf{ k}\} = \{    \mathbf{ k}_1,  \mathbf{ k}_2,  \overline{\{ \mathbf{ k}\}}_{(12)}  \}$ by removing first two elements, and
\begin{equation}
 T_{ 3} (\overline{ \{ \mathbf{ k}\}}_{(123)} )  =  T_{(123 )} ( \{ \mathbf{ k}\}), \label{3bamp}
\end{equation}
where  $\overline{\{ \mathbf{ k}\}}_{(123)} = \{   \mathbf{ k}_4,  \cdots, \mathbf{ k}_N  \}$ is a subset of $ \{ \mathbf{ k}\} = \{    \mathbf{ k}_1,  \mathbf{ k}_2,   \mathbf{ k}_3,  \overline{\{ \mathbf{ k}\}}_{(123)}  \}$ by removing first three elements.   According to
    Eq.(\ref{2bamp}), $T_{(12)} ( \{ \mathbf{ k}\}) $    in fact depends on both $\mathbf{ k}_1 + \mathbf{ k}_2$ and $\overline{\{ \mathbf{ k}\}}_{(12)} $,    the $\mathbf{ k}_1 + \mathbf{ k}_2$ dependence has been dropped due to the fact that   all momenta are constrained by momentum conservation $\sum_{i=1}^N  \mathbf{ k}_i = \mathbf{ P}$, where $\mathbf{ P}$ stands for total momentum of N-particle.  Similarly,   the $\mathbf{ k}_1 + \mathbf{ k}_2 +  \mathbf{ k}_3 $ dependence in $ T_{(123 )} ( \{ \mathbf{ k}\})$ is dropped as well because of momentum conservation constraint. The rest of amplitudes are related to $T$ and $T_3$ defined in Eqs.(\ref{2bamp}) and (\ref{3bamp}) respectively by
\begin{equation}
T_{(i j)} ( \{ \mathbf{ k}\})  = T  ( \overline{\{ \mathbf{ k}\}}_{(ij)}), \ \   \ \ \ \  T_{(ij k )} ( \{ \mathbf{ k}\}) =   T_3  ( \overline{\{ \mathbf{ k}\}}_{(ij k)}),
\end{equation}
where $ \overline{\{ \mathbf{ k}\}}_{(ij)}$ and $ \overline{\{ \mathbf{ k}\}}_{(ij k)}$ can be obtained from
sets $ \overline{\{ \mathbf{ k}\}}_{(12)}$ and $ \overline{\{ \mathbf{ k}\}}_{(123)}$ by swapping particle momenta: $ \mathbf{ k}_1 \leftrightarrow \mathbf{ k}_i$, $ \mathbf{ k}_2 \leftrightarrow \mathbf{ k}_j$ and $ \mathbf{ k}_3 \leftrightarrow \mathbf{ k}_k$.  Two sets of coupled equations for $T_{(i j)}$'s and  $T_{(i j k )} $'s are hence reduced to two equations,
\begin{align}
 & T  ( \overline{\{ \mathbf{ k}\}}_{(12)})  \nonumber \\
 & = \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{ \sum_{i<j} T ( \overline{\{  \mathbf{ p} \}}_{(ij)}) + \sum_{i<j<k} T_{3} ( \overline{\{ \mathbf{ p} \}}_{(ij k)})   }{E - \frac{  ( \mathbf{ k}_1 +  \mathbf{ k}_2 - \mathbf{ p}_2 )^2  +   \mathbf{ p}_2^2 + \sum_{i=3}^N\mathbf{ k}^2_i }{2m}   }  ,   \nonumber \\
 &    \mathbf{ p}_1=  \mathbf{ k}_1 +  \mathbf{ k}_2 -\mathbf{ p}_2 , \ \   \mathbf{ p}_l= \mathbf{ k}_l,  \ \  l=3, \cdots, N,   \label{2bLSeq}
\end{align}
  and
\begin{align}
 & T_{3} (\overline{ \{ \mathbf{ k}\}}_{(123)} )   \nonumber \\
 &= \frac{\eta_3}{L^6} \sum_{   \mathbf{ p}_2,\mathbf{ p}_3 }  \frac{   \sum_{i<j} T  ( \overline{\{  \mathbf{ p} \}}_{(ij)}) + \sum_{i<j<k} T_{3} ( \overline{\{ \mathbf{ p} \}}_{(ijk)})    }{E - \frac{( \sum_{i=1}^3 \mathbf{ k}_i -   \mathbf{ p}_2- \mathbf{ p}_3  )^2 +    \mathbf{ p}_2^2 + \mathbf{ p}_3^2 +  \sum_{i=4}^N\mathbf{ k}^2_i  }{2m}   }  ,    \nonumber \\
 &     \mathbf{ p}_1=   \sum_{i=1}^3 \mathbf{ k}_i  -\mathbf{ p}_2 - \mathbf{ p}_3 , \ \   \mathbf{ p}_l= \mathbf{ k}_l,  \ \  l=4, \cdots, N,  \label{3bLSeq}
\end{align}
where   $\{ \mathbf{ p}\} = \{ \mathbf{ p}_1 ,\mathbf{ p}_2,  \overline{\{ \mathbf{ p}\}}_{(12)}\}  =  \{ \mathbf{ p}_1 ,\mathbf{ p}_2, \mathbf{ p}_3,  \overline{\{ \mathbf{ p}\}}_{(123)}\} $ in both Eqs.(\ref{2bLSeq}) and (\ref{3bLSeq}). The diagrammatic representation of  Eq.(\ref{2bLSeq}) and Eq.(\ref{3bLSeq}) is given  in Fig.~\ref{feynmanplot}.

  \begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{./faddeev_diag.pdf}
\caption{  Diagrammatic representation of Eq.(\ref{2bLSeq}) and Eq.(\ref{3bLSeq}), pair-wise and three-body interactions are represented by black solid circle and blue solid square.   }\label{feynmanplot}
\end{center}
\end{figure}

\subsection{Three-boson dynamical equations}
In the case of $N=3$, the dynamical equations   are  thus given by
\begin{equation}
  T  (   \mathbf{ k}_3 )   = \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{  T (  \mathbf{ k}_3 ) + 2 T (  \mathbf{ p}_2 )    + T_{3}  }{E - \frac{ ( \mathbf{ k}_1 +  \mathbf{ k}_2 - \mathbf{ p}_2 )^2  +  \mathbf{ p}_2^2 +  \mathbf{ k}^2_3 }{2m}   }  ,
\end{equation}
and
\begin{equation}
  T_{3}   = \frac{\eta_3}{L^6} \sum_{   \mathbf{ p}_2,\mathbf{ p}_3 }  \frac{   3  T (  \mathbf{ p}_2 )   +  T_{3}     }{E - \frac{  \mathbf{ p}_2^2 + \mathbf{ p}_3^2 + (   \mathbf{ P} -   \mathbf{ p}_2- \mathbf{ p}_3  )^2   }{2m}   }  .
\end{equation}
Eliminating $T_3$ amplitude, we find
\begin{align}
&  T  (   \mathbf{ k}_3 )
  = \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{  1     }{E - \frac{  \mathbf{ p}_2^2 + ( \mathbf{ k}_1 +  \mathbf{ k}_2 - \mathbf{ p}_2 )^2  +  \mathbf{ k}^2_3 }{2m}   }   \Bigg [  T  (   \mathbf{ k}_3 )  + 2 T (  \mathbf{ p}_2 )  \nonumber \\
&  \quad \quad \quad   + \frac{ \frac{\eta_3}{L^6} \sum_{   \mathbf{ q}_2,\mathbf{ q}_3 }  \frac{   3  T (  \mathbf{ q}_2 )      }{E - \frac{  \mathbf{ q}_2^2 + \mathbf{ q}_3^2 + (  \mathbf{ P} -   \mathbf{ q}_2- \mathbf{ q}_3  )^2   }{2m}   }  }{1- \frac{\eta_3}{L^6} \sum_{   \mathbf{ q}_2,\mathbf{ q}_3 }  \frac{   1    }{E - \frac{  \mathbf{ q}_2^2 + \mathbf{ q}_3^2 + (  \mathbf{ P} -   \mathbf{ q}_2- \mathbf{ q}_3  )^2   }{2m}   }} \Bigg ].   \label{N3Teq}
\end{align}

\section{Threshold Expansion}\label{ThresholdExpansion}
In this section, we illustrate that the threshold expansion formula may be derived from Eqs.(\ref{2bLSeq}) and (\ref{3bLSeq}) by iterations. Near the ground state energy threshold, all $N$ particles  are nearly at rest for weak interactions. Hence the dominant contribution comes from the zero momenta mode of amplitudes: $\{ \mathbf{ k}\} = \{ \mathbf{ 0}\}$. Thus, we find
\begin{align}
 & \left [ 1-  \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{1 }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  \right ] T  ( \overline{\{ \mathbf{ 0}\}}_{(12)})  \nonumber \\
 & = \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{ \sum_{i<j}^{ (ij) \neq (12) } T ( \overline{\{  \mathbf{ p} \}}_{(ij)}) + \sum_{i<j<k} T_{3} ( \overline{\{ \mathbf{ p} \}}_{(ijk)})   }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  ,      \nonumber \\
 &   \quad  \mathbf{ p}_1=   -\mathbf{ p}_2 , \ \   \mathbf{ p}_l= \mathbf{ 0},  \ \  l=3, \cdots, N,  \label{2bzeromode}
\end{align}
and
\begin{align}
 & T_{3} (\overline{ \{ \mathbf{ 0}\}}_{(123)} )   \nonumber \\
 &= \frac{\eta_3}{L^6} \sum_{   \mathbf{ p}_2,\mathbf{ p}_3 }  \frac{   \sum_{i<j} T  ( \overline{\{  \mathbf{ p} \}}_{(ij)}) + \sum_{i<j<k} T_{3} ( \overline{\{ \mathbf{ p} \}}_{(ijk)})    }{E - \frac{  \mathbf{ p}_2^2 + \mathbf{ p}_3^2 + (   \mathbf{ p}_2+ \mathbf{ p}_3  )^2   }{2m}   }  ,    \nonumber \\
 &   \quad   \mathbf{ p}_1=     -\mathbf{ p}_2 - \mathbf{ p}_3 , \ \   \mathbf{ p}_l= \mathbf{ 0},  \ \  l=4, \cdots, N  .
\end{align}

\subsection{Perturbation expansion by iteration of $N$-body dynamical equations }\label{iteration}
As we can see from above equations, the leading order contributions of $T$ and $T_3$  start at the order of  $ \frac{\eta}{L^3} $  and $ \frac{\eta_3}{L^6} $ respectively. In present work, the aim is to just simply illustrate how the threshold expansion formula are derived from Eqs.(\ref{2bLSeq}) and (\ref{3bLSeq}).  For this purpose, we will only compute up to $  \eta^3/L^5 $ order in threshold expansion formula by iterating Eqs.(\ref{2bLSeq}) and (\ref{3bLSeq}) only once. The contributions from three-body force are only kept at the lowest order effect,  and also splitting  up zero momenta mode and non-zero momenta mode  contributions  in Eq.(\ref{2bzeromode}), so we obtain
\begin{align}
 & \left [ 1-  \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{1 }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  -  \frac{\eta}{L^3}  \frac{ \frac{N(N-1)}{2} }{E} \right ] T  ( \overline{\{ \mathbf{ 0}\}}_{(12)}) \nonumber \\
 & =  \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{ \sum_{i<j}^{ (ij) \neq (12) } T ( \overline{\{  \mathbf{ p} \}}_{(ij)})   }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  \nonumber \\
 &  +    \frac{\eta}{L^3} \frac{\frac{N (N-1)(N-2)}{6}}{E}  T_{3} ( \overline{\{ \mathbf{ 0} \}}_{(123)})    + \mathcal{O} (\frac{\eta^2 \eta_3}{L^{12}})  ,  \nonumber \\
 & \quad  \mathbf{ p}_1 = - \mathbf{ p}_2 , \ \ \ \ \mathbf{ p}_l =\mathbf{ 0}, \ \ l=3, \cdots, N,        \label{nonzeromode}
\end{align}
and
\begin{align}
 & T_{3} (\overline{ \{ \mathbf{ 0}\}}_{(123)} )  = \frac{\eta_3}{L^6}   \frac{     \frac{N(N-1)}{2}    }{E    } T  ( \overline{\{ \mathbf{ 0}\}}_{(12)}) + \mathcal{O} (\frac{\eta \eta_3}{L^{9}}) .
\end{align}
Eliminating $T_{3} (\overline{ \{ \mathbf{ 0}\}}_{(123)} ) $ term in Eq.(\ref{nonzeromode}), we find
\begin{align}
 & \left [ 1-  \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{1 }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  -  \frac{\eta}{L^3}  \frac{ \frac{N(N-1)}{2} }{E}     \right ] T  ( \overline{\{ \mathbf{ 0}\}}_{(12)}) \nonumber \\
 &  =     \frac{\eta \eta_3 }{L^9} \frac{\frac{N^2 (N-1)^2(N-2)}{12}}{E^2}     T  ( \overline{\{ \mathbf{ 0}\}}_{(12)})   \nonumber \\
 &+   \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{ \sum_{i<j}^{ (ij) \neq (12) } T ( \overline{\{  \mathbf{ p} \}}_{(ij)})   }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  + \mathcal{O} (\frac{\eta^2 \eta_3}{L^{12}})  ,   \nonumber \\
 & \quad  \mathbf{ p}_1 = - \mathbf{ p}_2 , \ \ \ \ \mathbf{ p}_l =\mathbf{ 0}, \ \ l=3, \cdots, N.        \label{T2nonzeromode}
\end{align}
Now, dominant zero momenta mode and sub-leading non-zero momenta mode are well separated in Eq.(\ref{T2nonzeromode}).
The terms that are given by non-zero momenta mode of $T$ amplitudes in Eq.(\ref{T2nonzeromode})  can be eliminated   and thus are related to zero momenta mode amplitudes by iterating Eq.(\ref{2bLSeq})  once.

Non-zero momenta mode of set $ \overline{\{  \mathbf{ p} \}}_{(ij)}$ in $T ( \overline{\{  \mathbf{ p} \}}_{(ij)})  $ can be split into two groups: (1)  $ \overline{\{  \mathbf{ p} \}}_{(1j )} = \{ \mathbf{ 0}, \cdots , \mathbf{ p}_2 , \cdots , \mathbf{ 0} \}$ with only a single non-zero momentum dependence at j-th position, $ \mathbf{ p}_j = \mathbf{ p}_2$  and  $j >2 $; (2) $ \overline{\{  \mathbf{ p} \}}_{(i j )} = \{ \mathbf{ 0}, \cdots, - \mathbf{ p}_2 ,  \cdots , \mathbf{ p}_2 , \cdots , \mathbf{ 0} \}$ with   two non-zero momenta  dependence at i-th and j-th positions,  $ \mathbf{ p}_i =- \mathbf{ p}_2$ and $ \mathbf{ p}_j = \mathbf{ p}_2$, where $ (i<j) = 3, \cdots, N$.

(1) For amplitudes in group one with only a single non-zero momentum dependence, using Eq.(\ref{2bLSeq}) again, we find  that each non-zero mode $ T ( \overline{\{  \mathbf{ p} \}}_{(1j)})$ is related to two   amplitudes that doesn't depend on $\mathbf{ p}_2$,
\begin{align}
& T ( \overline{\{  \mathbf{ p} \}}_{(1j)})   \nonumber \\
& =  \frac{\eta}{L^3}  \left [  \sum_{   \mathbf{ q}_2  } \frac{  T ( \overline{\{  \mathbf{ q} \}}_{(1j)})       }{E - \frac{   \mathbf{ q}_1^2  +   \mathbf{ q}_2^2 +  \mathbf{ p}^2_2 }{2m}   }  +  \sum_{   \mathbf{ q}_1  } \frac{    T ( \overline{\{  \mathbf{ q} \}}_{(2 j)})    }{E - \frac{   \mathbf{ q}_1^2  +   \mathbf{ q}_2^2 +  \mathbf{ p}^2_2 }{2m}   }   \right ] + \cdots \nonumber \\
& =  \frac{\eta}{L^3} \sum_{   \mathbf{ q}_2  } \frac{  2 T ( \overline{\{  \mathbf{ q} \}}_{(1j)})      }{E - \frac{  ( \mathbf{ p}_2 + \mathbf{ q}_2 )^2  +   \mathbf{ q}_2^2 +  \mathbf{ p}^2_2 }{2m}   }  + \cdots , \label{groupone}
\end{align}
where $ \overline{\{  \mathbf{ q} \}}_{(1j)} = \{ \mathbf{ 0}, \cdots , \mathbf{ q}_2, \cdots, \mathbf{ 0} \}$   and  $ \overline{\{  \mathbf{ q} \}}_{(2j)} = \{ \mathbf{ 0}, \cdots ,   \mathbf{ q}_1, \cdots, \mathbf{ 0} \}$   with $\mathbf{ q}_2$ and  $ \mathbf{ q}_1$ siting at j-th position.  Splitting sum of $\mathbf{ q}_2$ to zero momenta mode and non-zero momenta mode again in Eq.(\ref{groupone}), the dominant contribution for $ T ( \overline{\{  \mathbf{ p} \}}_{(1j)})$  comes from   zero momenta mode, sub-leading contribution from non-zero momenta mode may be eliminated by iteration again. Keeping only dominant zero momenta mode contribution,  we hence find
\begin{equation}
 T ( \overline{\{  \mathbf{ p} \}}_{(1j)})  = \frac{\eta}{L^3} \frac{1}{E - \frac{\mathbf{ p}_2^2}{m}} 2  T ( \overline{\{  \mathbf{ 0} \}}_{(12)})  +  \mathcal{O} (\frac{\eta \eta_3}{L^{6}}).
\end{equation}
There are $ (N-2)$  terms of $ T ( \overline{\{  \mathbf{ p} \}}_{(1j)})$  in group one, and another $(N-2)$  equivalent terms for     $ T ( \overline{\{  \mathbf{ p} \}}_{(2j)})$ amplitudes.  Therefore, the total dominant contribution from group one is  $ \frac{\eta}{L^3} \frac{4 (N-2)}{E - \frac{\mathbf{ p}_2^2}{m}}   T ( \overline{\{  \mathbf{ 0} \}}_{(12)})$.

(2) For amplitudes in group two with two non-zero momentum dependence, each  $ T ( \overline{\{  \mathbf{ p} \}}_{(i j)})$     is related to only one   amplitude that does not depend on $ \mathbf{ p}_i =- \mathbf{ p}_2$ and $ \mathbf{ p}_j = \mathbf{ p}_2$,
\begin{equation}
 T ( \overline{\{  \mathbf{ p} \}}_{(i j)})
 =  \frac{\eta}{L^3}    \sum_{   \mathbf{ q}_2  } \frac{  T ( \overline{\{  \mathbf{ q} \}}_{(i j)})       }{E - \frac{    \mathbf{ q}_2^2 +  \mathbf{ p}^2_2 }{m}   } + \cdots   ,
\end{equation}
where $ \overline{\{  \mathbf{ q} \}}_{(i j)} = \{ \mathbf{ 0}, \cdots ,  - \mathbf{ q}_2, \cdots,   \mathbf{ q}_2, \cdots, \mathbf{ 0} \}$     with $ - \mathbf{ q}_2$ and  $ \mathbf{ q}_2$ siting at i-th and j-th positions respectively.  Hence, the dominant zero momenta mode contribution from  $ T ( \overline{\{  \mathbf{ p} \}}_{(i j)})$ term is
\begin{equation}
 T ( \overline{\{  \mathbf{ p} \}}_{(i j)})    = \frac{\eta}{L^3} \frac{1}{E - \frac{\mathbf{ p}_2^2}{m}}   T ( \overline{\{  \mathbf{ 0} \}}_{(12)})  +  \mathcal{O} (\frac{\eta \eta_3}{L^{6}}).
\end{equation}
There are  $\frac{(N-2)(N-3)}{2}$ such terms,   the total   number of dominant contribution from group two is thus  $ \frac{\eta}{L^3} \frac{\frac{(N-2)(N-3)}{2}}{E - \frac{\mathbf{ p}_2^2}{m}}   T ( \overline{\{  \mathbf{ 0} \}}_{(12)})$.

\subsubsection{Zero momenta mode $N$-boson dynamical equation}
Combining all non-zero momenta mode terms in Eq.(\ref{T2nonzeromode}) from both group one and group two, we obtain,
\begin{align}
 &    \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{ \sum_{i<j}^{ (ij) \neq (12) } T ( \overline{\{  \mathbf{ p} \}}_{(ij)})   }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  \nonumber \\
 &  =  \frac{\eta^2}{L^6} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{ 4 (N-2) + \frac{(N-2) (N-3)}{2}  }{ \left ( E - \frac{  \mathbf{ p}_2^2     }{m}  \right )^2  }    + \mathcal{O} (\frac{\eta^2 \eta_3}{L^{12}})  .
\end{align}
Plugging  them back into Eq.(\ref{nonzeromode}),   we thus find
\begin{align}
 & \left [ 1-  \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{1 }{E - \frac{  \mathbf{ p}_2^2     }{m}   }  -  \frac{\eta}{L^3}  \frac{ \frac{N(N-1)}{2} }{E} \right ] T  ( \overline{\{ \mathbf{ 0}\}}_{(12)}) \nonumber \\
 &  =     \frac{\eta \eta_3}{L^9}  \frac{\frac{N^2 (N-1)^2(N-2)}{12}}{E^2}   T  ( \overline{\{ \mathbf{ 0}\}}_{(12)})     \nonumber \\
 &  + \frac{\eta^2}{L^6} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{   \frac{(N-2) (N+5)}{2}  }{ \left ( E - \frac{  \mathbf{ p}_2^2     }{m}  \right )^2  }      T  ( \overline{\{ \mathbf{ 0}\}}_{(12)})   + \mathcal{O} (\frac{\eta^2 \eta_3}{L^{12}})  .   \label{LO}
\end{align}
 Zero momenta mode amplitude $   T  ( \overline{\{ \mathbf{ 0}\}}_{(12)})  $  is thus cancelled out from both sides of equation, and Eq.(\ref{LO}) yields an analytic   form that depends on only $E$ and momentum sum.

\subsubsection{Three-boson example}
Using three-body dynamical equation given in Eq.(\ref{N3Teq}) as a specific example, setting $\mathbf{ k}_1 = \mathbf{ k}_2 = \mathbf{ k}_3 =\mathbf{ 0}$, and keep only up to $ \frac{\eta \eta_3}{L^9}$ order, we obtain
\begin{equation}
  T  (   \mathbf{ 0} )
  \simeq \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  } \frac{   T  (   \mathbf{ 0} )  + 2 T (  \mathbf{ p}_2 )     }{E - \frac{  \mathbf{ p}_2^2  }{m}   }        + \frac{\eta \eta_3}{L^9}   \frac{   3  T (  \mathbf{ 0} )      }{E^2   }  .  \label{ThreebodyT2example}
\end{equation}
Splitting up to zero momenta and non-zero momenta mode in  Eq.(\ref{ThreebodyT2example}), and also use Eq.(\ref{N3Teq}) once to eliminate non-zero momenta mode,
\begin{equation}
  T  (   \mathbf{ p}_2 )
  = \frac{\eta}{L^3}  \frac{   2 T (  \mathbf{ 0} )      }{E - \frac{    \mathbf{ p}^2_2 }{m}   }    + \cdots,  \ \ \mathbf{ p}_2 \neq \mathbf{ 0},
\end{equation}
 hence we finally get
\begin{align}
& 1  - \frac{\eta}{L^3} \sum_{   \mathbf{ p}_2  \neq \mathbf{ 0} } \frac{   1  }{E - \frac{  \mathbf{ p}_2^2  }{m}   } -  \frac{\eta}{L^3} \frac{3}{E}   \nonumber \\
& \simeq   \frac{\eta^2}{L^6} \sum_{   \mathbf{ p}_2 \neq \mathbf{ 0}  } \frac{   4   }{ \left ( E - \frac{  \mathbf{ p}_2^2  }{m}  \right )^2  }    + \frac{\eta \eta_3}{L^9}   \frac{   3      }{E^2   }  .
\end{align}

\subsection{Near threshold   expansion and ground state energy}\label{thresexpand}
By assuming that energy shift near threshold is small due to weak interactions:  $E\sim 0$, Eq.(\ref{LO}) is thus turned into a polynomial equation by   near threshold expansion, keeping up to $\mathcal{O} (E^3)$, we   have
\begin{align}
 & \mathcal{O} (E^4) + \left (  \frac{1}{L^3}  \sum_{   \mathbf{ p} \neq \mathbf{ 0}  } \frac{1}{ \frac{  \mathbf{ p}^4     }{m^2}   } \right )  E^3  \nonumber \\
 & +  \left ( \frac{1}{\eta}+  \frac{1}{L^3} \sum_{   \mathbf{ p} \neq \mathbf{ 0}  } \frac{1}{ \frac{  \mathbf{ p}^2     }{m}   } -  \frac{\eta}{L^6} \sum_{   \mathbf{ p} \neq \mathbf{ 0}  } \frac{\frac{(N+5) (N-2)}{2}}{ \frac{  \mathbf{ p}^4     }{m^2}   }   \right ) E^2  \nonumber \\
 &  -  \frac{1}{L^3}    \frac{N(N-1)}{2}  E   =     \frac{ \eta_3}{L^9}  \frac{N^2 (N-1)^2(N-2)}{12}      .    \label{threshold}
\end{align}
Introducing renormalized two-body coupling constant
\begin{equation}
\frac{1}{\eta} = \frac{1}{\eta_R} - \frac{ m \Lambda}{\pi L},
\end{equation}
where $\Lambda$ is related to the cutoff on momentum sum,  and also using  relations given in Refs. \cite{Beane:2007qr},
\begin{align}
& \frac{1}{L^3} \sum_{   \mathbf{ p}_1 \neq \mathbf{ 0}  } \frac{1}{ \frac{  \mathbf{ p}_1^2     }{m}   } - \frac{ m \Lambda}{\pi L} =  \frac{ m \mathcal{I}}{(2\pi)^2 L} , \ \ \ \ \mathcal{I} =   \sum_{ \mathbf{ n} \neq \mathbf{ 0}  }^{| \mathbf{ n} |  \leqslant \Lambda} \frac{1}{   \mathbf{ n}^2     } -  4\pi  \Lambda , \nonumber \\
&  \frac{1}{L^3} \sum_{   \mathbf{ p}_1 \neq \mathbf{ 0}  } \frac{1}{ \frac{  \mathbf{ p}_1^4     }{m^2}   }  =  \frac{ m^2 L \mathcal{J}}{(2\pi)^4 } , \ \  \ \ \mathcal{J} =   \sum_{ \mathbf{ n} \neq \mathbf{ 0}  }  \frac{1}{   \mathbf{ n}^4     } ,
\end{align}
we can rewrite Eq.(\ref{threshold}) to
\begin{align}
 &  \frac{   L \mathcal{J}}{(2\pi)^4 }    (mE)^3  \nonumber \\
 & +  \left ( \frac{1}{ m \eta_R}+   \frac{  \mathcal{I}}{(2\pi)^2 L}  - \frac{(N+5) (N-2)}{2}   \frac{  m \eta_R   \mathcal{J}}{(2\pi)^4 L^2 }   \right ) (m E)^2  \nonumber \\
 &  -  \frac{1}{L^3}    \frac{N(N-1)}{2}  (m E)   \simeq    \frac{ m \eta_3}{L^9}  \frac{N^2 (N-1)^2(N-2)}{12}      .    \label{cubiceq}
\end{align}
The cubic equation, Eq.(\ref{cubiceq}), can be easily solved by perturbation theory
\begin{equation}
m E= \frac{N(N-1)}{2} \frac{4\pi a_0}{L^3}  \left [ 1 +\sum_{n=1}^3  \left (\frac{a_0}{ \pi L}  \right )^n c_n   \right ],
\end{equation}
where $a_0$ is two-body scattering length and is related to coupling constant of pair-wise contact  interaction  by
\begin{equation}
 m \eta_R = 4\pi a_0 .
 \end{equation}
    The solution of cubic equation, Eq.(\ref{cubiceq}) is thus given by
\begin{align}
 E &= \frac{N(N-1)}{2} \frac{ 4\pi a_0 }{ m L^3} \bigg [ 1 - \left  ( \frac{   a_0 }{ \pi L}  \right  ) \mathcal{I}   \nonumber \\
 &+  \left ( \frac{ a_0 }{\pi L} \right   )^2 \left ( \mathcal{I}^2 +(2 N-5) \mathcal{J}  \right ) +  \mathcal{O} (  \frac{   a^3_0 }{L^3}      )  \bigg]  \nonumber \\
 & + \frac{N(N-1) (N-2)}{6} \frac{ \eta_3}{L^6}  ,
\end{align}
 which is consistent with well-known results in Refs.~\cite{Huang:1957im,Lee:1957zzb,Beane:2007qr,Detmold:2008gh}.

\begin{figure*}
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{./E3bplot.pdf}
\caption{ Three-boson energy spectrum   in 1D.}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{./E5bplot.pdf}
\caption{ Five-boson energy spectrum   in 1D.}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{./E7bplot.pdf}
\caption{ Seven-boson energy spectrum   in 1D.}
\end{subfigure}
\caption{ Plot of  $m E$ as the function of $L$: red dashed curves are  exact solutions given by Eq.(\ref{exact1D}), and black solid curves are approximate solution given by threshold expansion in Eq.(\ref{approxE1D}),  $m \eta = 0.1$.  }
\label{ENbplot}
\end{figure*}

\subsection{Threshold expansion formula in 1D and comparison to exact solutions}
 The iteration of coupled finite volume $N$-body dynamical equation approach  and results presented in section \ref{iteration} and  \ref{thresexpand} can be applied to $N$-boson interaction in 1D with little   changes. $N$ identical bosons interacting with pair-wise contact potentials in 1D is in fact exactly solvable, see Refs.~\cite{Yang:1967bm,Lieb:1963rt,Guo:2016fgl}. The exact analytic solutions are given by $E = \frac{1}{2m} \sum_{i=1}^N p_i^2$, where $p_i$'s satisfies  coupled equations
\begin{equation}
\frac{p_i L}{2} = \sum_{j =1 (j \neq i)}^N \cot^{-1}  \left ( \frac{p_i - p_j}{m \eta} \right ), \ \ \sum_{i=1}^N p_i=0. \label{exact1D}
\end{equation}

 Keeping only pair-wise contact interaction and expanding up to $\mathcal{O} (E^3)$, the  threshold   expansion equation in 1D can be obtained by replacing 3D momentum sum $\frac{1}{L^3} \sum_{\mathbf{ p}} $ in Eq.(\ref{threshold}) by 1D counterpart $\frac{1}{L} \sum_{p} $, hence, we obtain
  \begin{align}
 & \frac{1}{L}    \frac{N(N-1)}{2}     \simeq \left (  \frac{1}{L}  \sum_{  p \neq  0  } \frac{1}{  p^4        } \right )  (mE)^2  \nonumber \\
 & +  \left ( \frac{1}{ m \eta}+  \frac{1}{L } \sum_{  p \neq  0 } \frac{1}{  p^2      } -  \frac{\eta}{L^2} \sum_{   p \neq  0 } \frac{\frac{(N+5) (N-2)}{2}}{   p^4       }   \right ) (mE)   .    \label{threshold1D}
\end{align}
The infinite momentum sum in 1D can be carried out rather easily,
\begin{equation}
 \sum_{  p \neq  0  } \frac{1}{  p^2        }= \frac{L^2}{12}, \ \ \ \ \sum_{  p \neq  0  } \frac{1}{  p^4        }= \frac{L^4}{720},
\end{equation}
the solution of Eq.(\ref{threshold1D}) is thus given by
\begin{equation}
m E= \frac{ \sqrt{ b^2  + 4 c  }   -  b}{2}  , \label{approxE1D}
\end{equation}
where
\begin{align}
b &= \frac{720}{L^3}  \left ( \frac{1}{ m \eta}+  \frac{L}{12} -   \frac{(N+5) (N-2)}{2}  \frac{m \eta  L^2}{720}  \right )  , \nonumber \\
  c &= \frac{N(N-1)}{2}  \frac{720}{L^4} .
\end{align}
The comparison of $m E$ as the function of $L$ between exact solutions given by Eq.(\ref{exact1D}) and approximate solution by threshold expansion in Eq.(\ref{approxE1D}) is illustrated in  Fig.~\ref{ENbplot}.

\section{Summary }\label{summary}

As a sanity check and a test on the formalism of finite volume $N$-body system developed in  \cite{Guo:2018ibd,Guo:2019hih,Guo:2019ogp}, we illustrate   how the well-known threshold expansion formula of   $N$-identical-boson system   may be derived by   iterations of Faddeev-type coupled dynamical equations. The ground state energy of $N$-boson system   near threshold  is dominated by zero momenta mode of  $N$-body amplitudes, non-zero momenta mode amplitudes are associated with sub-leading order contributions and are related to leading order zero momenta mode     through  Faddeev-type coupled dynamical equations. Eliminating non-zero momenta modes by iterations ultimately yields an   analytic expression that   depends on only system energy and free momentum sum,  thus it can be turned into a polynomial equation by treating energy shift near threshold as a small parameter.  With only a single iteration, we are  able to compute  threshold expansion formula up to  $\mathcal{O} (\eta^3/L^5)$ for pair-wise interaction and   $\mathcal{O} (\eta_3/L^6 )$ for three-body interaction.

\bigskip

\begin{acknowledgements}
 We   acknowledge support from the Department of Physics and Engineering, California State University, Bakersfield, CA.   This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.   We also thank M.~D\"oring for suggesting such an investigation.
  \end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
There's a big discussion on an hp-internal newsgroup as to what to name your pets. People are suggesting "Fluffy & [...]
Hah! A coupla months ago a notice went up on the general notice board at the office asking for suggestions for names for the company's new barcode gun and cradle. Apparently, the company always had product names which include 'zen'. For example, Gunzen and Morzen.
Me and Frank [waves] thought up the name Fluffy. This was taken from Barcode READER --> fast breeder --> rabbit --> Fluffy. Other suggestions which were ignored were 'IBM PC AT' and 'The Barcode Gun and Cradle'.
Joe "I have a dog named Pestilence"
Hmmph. I'm gonna call me dog 'Woof!'. I just hope that I don't lose it and have to call for it.
Alternatively, I have considered 'my mother's coat' as a name for a pet.
"Ohmygod, my mother's coat has been run over!"
"It's time to feed my mother's coat"
"My mother's coat has just taken a fancy to a kitchen table leg"
Talking of mcdonald's burgers. Since I became a vegetable I have had to make do with Wimpy's beanburgers. I used to adore BigMacs because of all the gunge (pickles, mayonnaise, salad etc etc) but Spicy Beanburgers come with cheese and tomatoes [gak]. No one will make me up a burger with the gunge because "it's a set menu' [cry].
Has anyone else tried the strawberry Ribena? It's wonderful.

redpajama_set_name: RedPajamaCommonCrawl


"The Believers," a 31-voice gospel choir from Sweden, will present a concert at 6:30 p.m. on Monday, April 6, at Ascension Lutheran Church in Fond du Lac. The Believers' music includes classic choral music, hymns and spirituals.
The choir's name in Sweden is Troendegruppen, which literally translated means "The group that believes." In 2013 The Believers performed G.F. Handel's "Messiah" and in the spring of 2014, they presented the "St. John Passion" by J. S. Bach. Their music can be heard at www.Troendegruppen.com.
The Believers was first formed in 1968 under the direction of Stig Sandlund, who still serves as the choir's director. In 2012, Stig was designated as "The Choir-Leader of the Year" in Sweden. The choir comes from a music institute that is affiliated with the Lulea University of Technology, a research and education school located about 190 miles south of the Arctic Circle.
The Believers have toured in America, Israel, Poland, Hungary and throughout Scandinavia. The choir members, who range in age from 20 to 25, will be hosted by members of Ascension.
The Believers' 2015 tour in Wisconsin and Illinois is sponsored by Shepherd's Canyon Retreat, Inc., an Arizona ministry that provides counseling retreats for men and women in full-time ministry who are in the midst of burnout, stress, depression and conflicts of all kinds. A free-will offering will be received to benefit this ministry.


From the grain of dust lying in Earth's orbital plane, to interstellar clouds and vast plumes of vaporized debris, the music of Dave Luxton portrays the magnificence of the cosmos - and attempts to connect us to our own inner universe. Strange Environs (51'43") contains nine tracks, each a unique spell cast by Luxton's slow motion synthesized fields of harmony and grandeur. His music is serene and calm, and meant to liberate these states which already exist within each listener. Atmosphere, texture and mood all unite in gentle converging psychological currents. Ethereal melodies play out across a building electronic choir, until darker notes remind each of a coming wrath. Secret tones emit warm glows amid rushing forms, as abandoned solo lines echo and hover above undulating sonic designs. Strange Environs bestows upon us a feeling somewhere between the dreamy realms of ritualistic portent and galactic comfort. This work is in no particular hurry to get to its destination - we shall arrive whenever we get there. Luxton knows he cannot provide us with meaning, but does offer a path towards it - striking deep chords that words cannot approach.


Happy Throwback Thursday! SquadUP is too young of a company to be reminiscing on how it all started, but for #TBT we'll make an exception. Today, let's go back to our first event in #NYC. Sure, we already had some traction on college campuses at this point last year. But the all-you-can-eat and all-you-can-drink sushi night at Zest was our soft NYC beta opening.
Looking back, it's one of the best deals we've ever had on an event we curated ourselves. We host very few events these days because we want to it to be clear that SquadUP is an event planning platform, not a company of event planners. That being said, we got started partly by using the platform to show people what an improvement it is for event hosts of all kinds. Our logo may have changed but the loyal supporters have been there all along. This #TBT is for you!
We had three and a half hours of as much lychee infused saki, sapporo, and amazing food as we could consume… all for a mere $30 per person. If you consider the quality of the food (this was NOT a sushi boat filled with california rolls) and typical NYC prices, we made out like bandits.
We ended up forming a great relationship with the owner who had just opened up in the neighborhood and was thrilled to have 50 twenty-somethings pack the place on a Wednesday. You would regularly find co-founder, Willie Litvack laughing with him over some late night sake. Good times.
Can you think of an early stage SquadUP event you'd like to see revived? Let us know by commenting or tweeting @squadup.
I truly wanted to write down a brief note so as to appreciate you for these lovely guidelines you are giving out on this site. My extensive internet search has at the end been paid with good facts and strategies to go over with my friends. I 'd express that most of us website visitors actually are undoubtedly endowed to exist in a decent site with very many perfect individuals with very helpful principles. I feel somewhat grateful to have discovered your web page and look forward to tons of more awesome times reading here. Thanks once more for a lot of things.


text:
Oceans North Celebrates Progress on Indigenous-led Protection of Seal River Watershed
MONTREAL—Today, the Canadian government announced an important step towards advancing Indigenous-led protection of the Seal River watershed in partnership with the province of Manitoba and four First Nations.
The announcement of a feasibility assessment for a new Indigenous Protected Area comes during the United Nations Biodiversity Conference in Montreal, where countries are gathered to draft new goals that will help stop and reverse the loss of nature globally. Last week, Canada made a historic commitment of up to $800 million to advance four other Indigenous-led conservation initiatives.
The Seal River watershed—a 50,000-square-kilometre expanse of tundra, forest and wetland located in Northern Manitoba—is an extraordinary repository of northern biodiversity, including culturally important caribou herds. Through the Seal River Watershed Alliance, Sayisi Dene First Nation has been working in partnership with the Northlands Denesuline Nation, Barren Lands First Nation and O-Pipon-Na-Piwin Cree Nation to conserve their ancestral lands and waters.
During the pandemic, Oceans North helped support harvesting programs in Tadoule Lake, the Sayisi Dene community in the heart of the watershed. The programs were aimed at improving local food security and passing down knowledge and skills. "The people who know the land and depend on it for their wellbeing are the ones best placed to protect it," says Chris Debicki, vice-president of policy and counsel at Oceans North. "We're happy to see the province and federal government working with Cree and Dene Nations, as well as the expansion of solutions that benefit both communities and nature."
The Seal River watershed is connected to the marine ecosystem of Western Hudson Bay. The estuary of the Seal River itself is home to some of the 57,000 belugas that migrate to Western Hudson Bay each summer—and recent research suggests it could be an important nursery for beluga calves. Yet this marine habitat remains unprotected. "We've seen at this conference what's possible when multiple levels of government, Indigenous communities and other groups come together to protect species and places that matter to them," Debicki says. "We're hopeful that spirit of collaboration will continue as Canada continues its work to protect global biodiversity and reach our conservation targets."
Alex Tesar


text:
Pakistan on war with Pakistan
Marvi Sirmed
The tragic happenings of last week provide a glimpse into the malaise that infests Pakistan's society and state. A young girl who could not get justice after being gang raped immolated herself in broad daylight under the prying eyes of TV cameras. The Council of Islamic Ideology (CII), the state's body to oversee the Islamic-ness of the law, made a mockery of it and made two ridiculous recommendations. One, to repeal the provision of Pakistan's Family Laws that make it incumbent upon the husband to seek the first wife's permission before re-marrying. The other gross recommendation of CII was about allowing the under-age marriages of minor girls.
As if it was not enough, the state went overboard in dealing with a peaceful protest demonstration of female paramedical staff at Lahore. The nurses were asking to be regularized in their jobs. What a nonsensical demand it was, from a government too busy with more important issues like kneeling before the holy Taliban for piece negotiations. Spell check: peace I meant.
Not enough yet. Women had it from us but how could we forget religious minorities? That too the worst of them all; Hindus. After all, our 'ideology' calls for a complete social breakdown with them. We invoke blasphemy accusations, that handy tool for dealing with not only the minorities but anyone lacking enough weight in society.
It was holi celebrations in the offing over the weekend. Holi, the spring festival that symbolizes the victory of good over evil, is considered to be a festival of colors, celebrated with music, dance, laughter, forgiveness and repairing ruptured relationships. It signifies the South Asian way of celebrating life.
As the myth goes, the demon King Hiranyakashipu ordered everyone to worship him but his own son Prahlada instead worshipped Lord Vishnu. The King then asked his evil sister Holika, who was immune to fire, to sit on pyre with little Prahlada in her lap. As the divine intervention went, the pyre burnt Holika instead of Prahlada.
READ MORE: Philippines: Over 50,000 affected by volcano eruption
To symbolize this defeat of evil, every year Holika is burnt on the pyre on the night of the full moon during the Holi festival. Holika Dahan, as the tradition is called, was about to happen in Larkana last week when an enraged crowd appeared and set on fire to the Dharma Shala and the adjoining Mandir. The evil Holika, in this case, finally was able to burn the innocent Prahlada's abode. Welcome to Pakistan.
The tragic incident followed an accusation of burning the pages of the Quran by a member of the local Hindu community. However unfounded these kinds of accusations may be, they normally lead to a similar turn of events: burning their homes or places of worship, and in the worst cases, the mass killings of minority members. These vigilante actions which always go unpunished and unaddressed by the state, have brought death and destruction upon the minority communities quite frequently in the past too. This, in fact has become an easy recourse to settling personal scores turned shrewdly into collective crime, so everyone goes scot-free.
Here we are not talking about the Islamic radical extremists or militants? This is a routine course of this society followed by ordinary citizens. The agent provocateurs in such cases may be miscreants, but the action is always completed with the active participation of the common man in the street.
For a long time I have been a strong holder of the view that we have a 'Sufi Sindh' and a 'secular Balochistan' as opposed to a 'religiously inclined KP' and 'radical Punjab'. The first two labels now increasingly appear as myths. Even a couple of decades ago, what happened in Larkana this weekend, could not have been imagined in Sindh. However adamant our complacency may be, however strong our delusions may be, the fact remains that Sindh's soul has been captured and Balochistan's heart has been assaulted. So has been that of Pakistan.
READ MORE: AC sends Ahsan Iqbal to jail on judicial remand in NSC case
Besides the debate of whether Jinnah was secular or a fundamentalist, one thing needs to be appreciated as a hard glaring fact. That is, the diverse ethnic make-up and plural cultural roots were and are central to the pulse of Pakistan. And that Pakistan is home to one of the most learned and ancient civilizations – the Indus Valley. We are home to the world's first ever university founded around a thousand years BC, i.e., Takshashila. People indigenous to this country are known by historians to have pioneered the banking and accounting system centuries ago. Indeed, we are the world's ancient seat of learning, culture and intellect. What happened to us?
The indigenous people of this proud civilization, the Hindus, may have turned into a 'minority' today, but they continue to contribute to the social and intellectual health of this country. Ever since their native land turned into the 'Land of the Pure', there has been no refuge for them from the perpetual persecution they have had to face. Their own progeny that sought to get 'purer' over the centuries, entangled itself in a vicious trap of identity, struggling with which it not only disassociated itself from, but also brutally cut every route that led to its original roots.
In theory, Pakistan's Hindus enjoy all kinds of freedom to lead their life according to their beliefs. There is no bar on any community (save Ahmadis) in Pakistan to openly and freely observe their religious rites. It however, gets more complex than a simple statement of granting the rights of observing the religion freely in a state that adopts the majority's religion to be its own, rendering all others as peripheral ones albeit with 'equal rights'.
Desecration of ancient Hindu temples by Muslim rulers throughout history is routinely eulogized with pride in textbooks and the media. In addition to having an adverse psychological effect on Hindu students, this factor could be blamed for general apathy on frequent desecrations and demolitions of Hindu temples in Pakistan.
READ MORE: Pak-China friendship to reach greater heights in 2020: Ambassador Hashmi
Whereas it is impossible to demolish a mosque to occupy the land it is built on, it is many times easier rather, in most cases kosher, to demolish the places of worship of other religious communities, especially if they belong to Hindus or Ahmadis. The 'pagan' status that Hindus enjoy in Pakistani textbooks for almost all levels, earns them this 'special' status where it is nearly impossible for an average child to have in their imagination any justification for the existence of religious symbols of Hindus in Pakistan, let alone respecting them.
To make things worse in Balochistan and Sindh, homes to the majority of Pakistani Hindus, a well-planned establishment of Madrassa networks was undertaken in late 1990s and early 2000s. This patronage of and pandering to extremist elements by the state has emboldened the radicals in the last few decades.
Never has any law-enforcing agency opened a blasphemy case against those who rampaged the temples. Never have any arrests been made for vigilante actions on minorities and women. Does the state even want to stop these atrocities on our minorities?
This war is ironically being waged on Pakistan by Pakistan itself. It remains to be seen who will 'negotiate' with whom to stop this war.
READ MORE: US envoy Alice Wells to meet Pakistani officials today in Islamabad
The writer is an Islamabad based campaigner for human rights and works on parliamentary strengthening and democratic governance.
Email:marvi@marvisirmed.com
Tweets at:@marvisirmed
Pakistan, US to discuss vital issues today
Pakistan brokering ME, regional peace: FM
Pakistan -Morocco revamping ties, agree to enhance bilateral cooperation at different levels
Tax payers' money Rs7m spent on ex-CJ's temporary residence: News
Harris Bin Munawar
The art of dying
Javid Husain
Pakistan's blunders in Afghanistan
The Lawn Anthropologist
Mob mentality
Right under your nose
MH730 and internet frenzy
Law of the Land
Sickly mothers and malnourished children
Goodbye local cars?


Romantic comedy anime My Teen Romantic Comedy SNAFU is getting a third season!
Originally based on a light novel series of the same name by Watari Wataru, it follows two isolated high schoolers named Hikigaya Hachiman and Yukinoshita Yukino who are members of the "Service Club", which offers help to others.
My Teen Romantic Comedy SNAFU has been adapted into several manga series, OVAs and a game. The anime series began in 2013, with Season 2 airing in 2015.
Details on the broadcast and cast are yet to be released.


There are lots of excellent distillers in the world, and lots of reasons to get excited about the products they make. Still, the ones that we chime with the most are the ones who remind us of the books we read as children. You know, the mad, eccentric Grandmother who had pots and pans piled high with strange concoctions. The inventors who made breakfast machines, the scientists who brought dreams to life. These were people who had made the adventure into adulthood without losing any of their wonder; they were the people to aspire to.
There seems to be something Trouvaille Spirits founder Tim James that hums with that same energy, so we were keen to get talking to him as soon as we heard of his Blossom & Hops Gin.
Chaos then, is an attraction. When Trouvaille Spirits was born, the distilling scene in its home country, South Africa, was still very young. "There were probably no more than 30 local gins on the market at the time," James explains, "but two years later there are in excess of 100." At the time, it seemed clear that South Africa needed a lot more representation, and with a mad world of flavours at his disposal out there, he was going to be able to satisfy his boundless wonder.
In total, it took 10 months to perfect two recipes, with a lot of trial and a great deal of luck involved. Luck, in fact, became a key part of the brand. 'Trouvaille' translates to lucky find or valuable discovery in French, and it was the mixing of their separate gins that helped them to crack the final recipe. Lucky indeed.
The initial plan was to launch a Hops Gin and a Blossom Gin, but when James went for a weekend away with his partner Teresa, the idea to mix his two products struck. He added one to the other and struck gold; Blossom & Hops Gin was born.
The recipe's key success lies in its traditions. So many South African distillers cannot resist ripping all of the Fynbos out of the land and cramming it into their stills. The end result can be great or it can be a little vulgar, and it can all too rarely be classed as gin either, as juniper hovers around in the background, transparent and ghostly.
In our opinion, Gins from the region are, by and large, heavily medicinal botanical vodkas, so when one comes out wearing traditional colours with a sense of local flair, we can't help but root for it. The local angle here comes from coriander seeds, citrus (lime leaves, lime blossom, lemon rind and orange rind) and hops. The rest of the ingredients – juniper, angelica, cardamom, liquorice root, cassia, orris and grains of paradise are in imported from Europe, the East and other parts of Africa.
The Southern Aroma Hops, incidentally, come form the Outeniqua mountains in the South Eastern Cape, known locally as The Garden Route. The particular strain used in Blossom & Hops Gin is low in acid and much less bitter than usual, bringing a citrus, grapefruit-like finish to the gin.
The inclusion of hops into a gin recipe was an obvious choice given James' brewing background. Still, there was research that went into it as well. "We found that hops was used in very old Genever recipes dating back to the 1600s. In these old recipes we also found the presence of lime blossom, a botanical that we were already experimenting with in our lime blossom gin".
The majority of the Blossom & Hops Gin recipe was created in Trouvaille Spirits' basement, although help to bring it to completion came from the gin's current maker, Hope on Hopkins.
Gin Foundry regulars will know just how much we admire Hope & Hopkins Distillery, based not far from Trouvaille Spirits' home. Co-founder Lucy and Leigh are insanely dedicated to their craft and everything they produce is made with integrity and heart. The duo have an intrinsic knowledge of botanicals, so when James hit a flavour wall, Lucy stepped in to suggest lime leaf. It was the final piece in the jigsaw, and gave James great faith in their knowhow.
The barriers to entering the market in South Africa are, arguably, more stringent than in England; a distillery needs to be built and fully functional before the licensing process can get started. This is a process that can take up to a year, so it's an incredibly costly waiting time for an idea that may not even get approved in the end anyway. James knew the time was now to move into the market – the wave was there to catch and he'd have been foolish to pass it up – so following their first positive interactions, he decided that collaboration with Hope on Hopkins was the way forward.
At first, Lucy and Leigh were quite hesitant to take on the project. They're busy as it is (they seem to be the go to contact distiller in the region now) and the duo had their concerns about the impact hops residue has on distilling equipment. After a sip, though, they couldn't resist, so the combined team worked on a plan for the hops that wouldn't affect the overall equipment.
Gin geeks, place your bets now: what was the alternative? Why, it was a vacuum still, of course. By cold distilling the hops, the distilling team were able to extract the fresher flavours and aromas, keeping the bitterness in the leftover products, rather than taking it over into the spirit. The process is simple and quick, with the hops put through a mere three hour maceration ahead of distillation.
The rest of the botanicals are placed in Hope of Hopkins' pot still. A cane neutral spirit is added to the pot with juniper, coriander and the citrus, which sit in amongst the liquid as the still is gently heated. The lime blossom and lime leaves (along with a pinch of juniper and coriander) are added to the vapour basket above the pot. The distillation takes around six hours, with the gin coming off the still at around 78% ABV. This is then blended down to 43% and left to sit overnight, before the hops distillate is blended in.
On the nose, juniper and earthy hops combine with the heady aromas of of lime blossom. Distinct and captivating, there are many of the same cues to taste, although there's also the addition of a notable citric warmth that joins the chorus as coriander seed warms towards the mid journey. The grains of paradise add a real depth to the flavour, but it's the core duo of aromatic hop and lime blossom that hold true throughout, with the latter that drags onto the smooth, slightly sweet finish.
In a G&T, the tonic further accentuates the lime blossom and lime leaves on the front end while the more assertively spiced and earthy elements, that are so rewarding when sampled straight, disappear all together on the finish. It's an accessible, pleasant melange of favours, if a little underwhelming given the promising sequence of botanical high notes (and the way they are integrated one into the next) that's evident when served up neat. For some, that adaptable, slightly ubiquitous and delicate profile will be this gin's biggest selling point as it can be paired with many a garnish (try rosemary or grapefruit peel as both work, and both bring out completely different aspects of the gin). Others may find the gin's transience a little forgettable come G&T time, a mid ground that makes sense as the best of both worlds on paper and when served neat on a tasting panel, yet somehow, as in our case, leaves us wondering if the assimilation is in fact a compromise that's left it incapable of surviving a mixer. For the record, we tried it with four tonic brands at different ratios before writing the above.
While the collaboration with Hope on Hopkins began as a workaround, the team has become a close one, so it's very unlikely indeed that James will look to produce his Gin at home. "If we look at the original strategy, it was very much to grow the brand and, once established, to build our own production facility. At the journey has progressed, however, so has our partnership with Hope on Hopkins, who have added tremendous value, not only to our product but also to our knowledge and understanding of the industry.
Interestingly enough, given the beery background of James, the only other edition in the Trouvaille Spirits lineup is a Lime Blossom Gin, which is said to be much sweeter than Blossom & Hops (which we felt was already quite sweet, with the cane spirit setting the tone for proceedings). It's not quite ready yet, but it's next on the production schedule. After that, a Hops Vodka, perhaps but the reality is that Blossom & Hops Gin is still a very small player in a big market, and like many in their stage of development, almost every plan they have largely involves finding routes for growth.
So far, the brand work has been great. The gin's bottle is pretty and the box it comes in is beautiful, with stunning illustration work. It works, certainly, but in an age where custom glass is becoming order of the day, it might be a good move for James to start looking in that direction. Yes, it's expensive to do, but the amount of beautiful, crimped glass on offer in Europe at the moment is mind blowing, and even though our role as critics is to take a rational approach to choosing our gins, you can bet your house on the fact that we just can't stop buying the tall, ripples, glorious glass.
That trend will make its way to South Africa in no time and for those with long term aspirations and such promise, it may be a clever move that will help them jump in ahead of the crowd. Still, this is a brilliant start, and the contents of the bottle are the most important part (once you've convinced someone to try it, that is).
Blossom & Hops Gin is a good drink. It's familiar, yet unique, with whispers of strange, exotic botanicals rising up alongside a gin core. We are enamoured by this storytelling; one sip, and you haven't got the wild planes of Africa in your mouth, but the subtle flavours of invention, curiosity and of following an idea down the rabbit hole and return clutching something interesting in both hands. In a world where everything's trying to grab your attention, a little bit of subtlety is nice to see, and we hope it makes its way to our home turf so that more can get to try it for them selves.


text:
Star Wars' Hyperspace Explained
One of the Star Wars galaxy's most iconic visuals is more fascinating than ever right now.
James Whitbrook
Punch it!
Screenshot: Lucasfilm
Lucasfilm's version of faster-than-light travel—hyperspace—is as fundamental to the worldbuilding of Star Wars as the Force. It's the backbone of making its galaxy an interconnected universe of potential, a concept that has become more and more integral to some of the franchise's biggest storytelling pushes. But how it works, and how people use it, has become more complex and fascinating than ever.
Recreating Star Wars' "Jump to Hyperspace"
So What Is Hyperspace?
Hyperspace is one of the oldest elements of Star Wars we know, having first glimpsed it in action when the Millennium Falcon raced on its way from Tatooine towards Alderaan (or, what was left of it) in 1977's A New Hope. Although the concept of faster-than-light travel is no stranger to science fiction, how Star Wars has started playing with it in recent years has made it one of its most quietly compelling worldbuilding mechanics. It can be charted, traveled, manipulated, even weaponized—and the very ramifications of using it outside of the expected point-a-to-point-b mechanics can upend what we thought we knew about the galaxy far, far away in an instant.
On its most simple level, hyperspace—not unlike Star Trek's warp speed or myriad other sci-fi series' take on interstellar travel—is an alternate dimension of reality that vessels (among other things) can slip in and out of at speeds unbound by the rules of physics in realspace. Accessible only by breaking the speed of light (jumping to lightspeed, as it's known in Star Wars lingo), time and velocity operate differently in hyperspace, vastly reducing the time it takes to get from one point to another. Its discovery formed the possibility for galactic civilization as we know it in Star Wars to actually exist.
It's not all that easy though. Navigating it is still not fundamentally unlike traveling in realspace, it just happens at impossible-to-comprehend speeds. Things that exist in realspace cast adjacent shadows in hyperspace, meaning navigation can be difficult without proper knowledge or, as we will see become common in the Star Wars galaxy, pre-charted safe routes. Traveling in hyperspace can be incredibly dangerous both during the process and coming out of it.
As Han tells Luke in A New Hope, an unplanned exit (intentional or otherwise) could lead you to come out of hyperspace in the middle of a star, or within the gravity well of a planet or spatial anomaly. Hitting another ship or debris in hyperspace can have massive ramifications—in hyperspace and realspace—even beyond the fatal consequence of hitting something else at FTL speeds. We've witnessed this in the Great Distaster of the High Republic era, when debris from a ship destroyed mid-hyperspace travel was shot across the known galaxy and accelerated to speeds capable of wiping out entire planets. Hyperspace might be fundamental to how the Star Wars galaxy works, but it's also something that, even by the time of the movies, people generally just don't know all that much about.
How Do You Travel Through Hyperspace?
Most hyperspace travel we see in Star Wars media is technological; starships, everything from as small as an X-Wing to as large as a Star Destroyer, as long as it had a hyperdrive, could enter and exit the dimension. Fuelled by hypermatter, the engines shoot vessels to the speed that allows them to break into hyperspace in the first place, maintaining their mass and velocity (which is how Admiral Holdo's maneuver in The Last Jedi proved to be so devastating). This travel takes place along hyperspace lanes, safe passageways in the dimension that had been charted out hundreds and hundreds of years beforehand by the earliest explorers of the Old Republic, marked by beacons which provide route data and computational power as well as physical guide points to ships as they made jumps. In "contemporary" Star Wars—for want of a better word about a story that takes place a "long time ago"—the computation power and layouts of those routes required to travel was now handled by onboard navigational systems known as navicomputers, or, to enhance limited computers in smaller vessels, an astromech droid's internal systems.
Gif: Lucasfilm
Vessels could also fall out of hyperspace, intentionally or otherwise, at other points. Failure of a hyperdrive, for example, would cause a ship to jerk back into realspace. But they could also be forced: a gravitational pull could affect the velocity needed to breach hyperspace (which is why attempting to jump to lightspeed within the atmosphere of a planet was considered dangerous enough to be close to impossible) and a strong enough pull could block or terminate the process. The Empire built specific cruisers, known as the Interdictor class, that generated powerful gravity wells that could both stop nearby ships from entering hyperspace and pull them out mid-jump. This technology was vital to the Empire as, for much of Star Wars' history, it was deemed impossible to track a vessel once it made the jump to lightspeed. Just decades later, however, Imperial successor the First Order cracked the ability to do so. It broke a generation of conventional wisdom by isolating a computer system in a localized hyperspace field and accelerating its programming capabilities.
Are There Other Ways to Travel in Hyperspace?
Yes, there are other ways to navigate hyperspace. The Chiss Ascendancy existed in the "Unknown Regions" of the galaxy which had not been charted by hyperspace prospectors for the Galactic Republic or the Empire. They employed a secretive corps of hyperspace navigators (known in their language as ozyly-esehembo) to help facilitate the computations required to navigate the volatile spaces in the region. Translated into basic as "Sky-walkers," these Chiss were primarily very young girls between the ages of eight and 13. Sensitive to the Force, they had what was known in Chiss culture as "Third Sight," the precognitive ability to help ships navigate through realspace and to map limited hyperspace jumps. Kept secret from all except the upper echelons of Chiss military and public society, each ship in the Ascendancy navy would be assigned a Sky-walker, and they would rotate out frequently. Why? Because Sky-walkers, no matter what training or preventative measures were taken, largely lost their ability to use Third Sight by the time they were teenagers.
The Chiss were not the only Force-sensitive species that could use those abilities in such a manner, however incredibly rare such a manifestation of power actually was. In the time of the High Republic—roughly 200 years before the events of the main Star Wars movies—a force-sensitive human named Mari San Tekka could use her abilities to detect safe, but unconventional passageways through hyperspace known simply as "Paths." Deemed too risky or even impossible by traditional prospectors' standards, San Tekka's paths could defy previously held conventional wisdom—like the ability to enter and exit hyperspace near spatial anomalies, or within and near the gravitational pull of a planet.
Due to the rarity of her abilities, however, San Tekka was an extremely valuable asset. She was captured and utilized exclusively by the pirate force known as the Nihil and sustained well beyond her natural lifespan by being hooked into medical systems aboard the flagship of the Nihil's leader Marchion Ro (and his father before him, Asgar), the Gaze Electric. As they accelerated their operation outside of the Outer Rim territories they originated in and became a galactic-scale threat to the Republic, every ship was equipped with a "Path Engine." It was a secondary hyperdrive that translated the data provided by San Tekka's abilities into more standardized navigational information.
But humanoids weren't the only sapient beings that could tap into this biological method of navigation. Introduced in Star Wars Rebels, the Purrgil were a species of whale-like semi-sentients that lived in deep space and had the unique natural ability to manifest "simu-tunnels" in hyperspace. While rare in and of themselves, Purrgils have taken on an almost folkloric status among space pilots and navigators. They're ancient enough that it's believed the study of their travel is what inspired galactic society to develop their own means of traveling through hyperspace in the first place, thousands of years before the Star Wars films. Additionally, their biological ways of travel and thought-processing formed the basis for the technology behind everything from the Wayfinders of ancient Force orders like the Jedi and Sith, to the hyperspace beacons and drives that made early intergalactic scouting possible.
Wondering where our RSS feed went? You can pick the new up one here.
io9MoviesStar Wars


I love this Chrome extension! Really improves your desktop experience of the game. Includes features such as showing the difficulty of each team's upcoming fixtures, extra columns in your leagues with more data.
A handy chart to get your head around the season's blank and double game week fixtures.
In this video Jay discusses FPL Chips, blank gameweeks, potential double gameweeks, teams with good fixtures, captaincy options & Alonso, Aguero, De Gea.
When you've gotta get extra cash, getting cheap defenders is the best way to make savings. Who are the reliable options?


text:
Jeffrey Peel
Evidence Based Consulting
TECH | The New Era
Big Data and Government
Northern Ireland's "Pandemic"
Post author By Jeffrey Peel
1 Comment on Northern Ireland's "Pandemic"
Every day people die. It's an unfortunate fact. But people die. We'll all die. And, when we do, our deaths – little data-points in the aggregate – will get added to the mix. Northern Ireland, like every administrative region in every part of the developed world, has a statistics agency that counts up all the deaths every day and collates them by month. And this year, 2020, is no different to any other year. Deaths have been tallied. There's no denying the data. Or is there?
In this, a 'pandemic' year, the month by month death data are more interesting than most. I've been looking at the numbers. And one month really jumps out. The month? January 2018. Why is this month particularly note-worthy? Well in most months, most Winter months, around 1,200 or 1,300 or even 1,400 people die. But in January 2018 2,101 people died. In fact, more than 500 more people died in January 2018 than the average number of deaths for the five years previously. This is the excess deaths number. 506 to be precise.
But, we're told, 2020 is a pandemic year? So presumably we've knocked that previous record out of the park? Well, no. In not one month of 2020 have we seen that number of deaths. In April we came close with 1,933 deaths. In January this year we had much fewer deaths than in January 2018.
In 2018 it was well publicised that the health service came close to being overrun. There was a crisis. Seasonal deaths were very high. But, of course, no convoluted tests were being used to determine what people were dying of. No doubt, respiratory diseases played a big part in causing end of life, especially among older people. But no new test was conceived. After all, if people are dying from chronic respiratory failure the symptoms are obvious, the diagnosis easy. No fancy tests are needed to test for what is normally described as Winter flu. Or perhaps a particularly virulent form of cold.
Now let's focus on April 2020 again. This is the month this year that was nearly as bad as January 2018 in terms of death count. If the health service was completely overrun in 2018, surely that was the case again in April, in a pandemic year? Well, no. That's not the case. Because in 2020 – in March and April and May – the chronically ill patients, mostly elderly, were sent to care homes to die – for fear of hospitals being overwhelmed. So there was no real crisis in the hospitals. And, of course, elderly people with co-morbidities aren't eligible for the limited ICU beds (there are less than 100 of them in Northern Ireland). So ICU beds never got to capacity in April.
So what about the so-called "second wave" in 2020 in Northern Ireland? Well, we're told, hospitals, many of them, are at more than 100% capacity. But in September deaths were pretty average for time of year at 1,384. This number isn't significantly worse than the death number in September 2015. And yet, in 2015, most of the economy hadn't been closed down. People weren't on state-funded furlough. We could all still get out for a meal or a pint, and still get our hair cut. But in 2015 we didn't have the PCR test – a test contrived to define people as sick who clearly aren't. And one result of this PCR test is that perfectly healthy medical staff are being sent home to "self-isolate" – meaning that they can't help the spike in Winter patients – spikes that have occurred frequently in the past.
And remember, the most severe spike was in January 2018, not in any month, so far, in 2020.
By Jeffrey Peel
← Lockdown Folly → Check out this BIG data dump…
One reply on "Northern Ireland's "Pandemic""
Weevysays:
As of today Dec 20, we've been told in Northern Ireland that we are going into another lockdown FOR 6 WEEKS on Boxing Day!! The reason…protect the bloody NHS…again….why haven't our Executive boosted the NHS resources, staff and facilities in the last 9 months?!! What have they been doing? When we all know hospitals get busy at this time every.single.year. Scandalous
(c) Jeffrey Peel
© 2021	Jeffrey Peel


Several production companies put nonsense and funny warning labels on their products just to give their customers a smile. This list contains 10 of the funniest warning labels we've come across.
Found on an American clothing label, in French.
Found on a bottle of dog's pills.
Found on a jet-ski user manual.
Found on a vending machine.


text:
Invest in Canada promotion glosses over the real opportunities
Did you know that on the streets of Davos, they are handing special sponsored editions of the Financial Times? It's not actually the Financial Times newspaper, though, it's a paid infomercial to invest in Canada. So Canadian taxpayers are paying to have people wandering around handing out an "Invest in Canada" glossy brochure that was paid for.
There must be thousands of copies that they're handing out here, but I'm not sure if handing out an infomercial like that on the streets of Davos is going to generate investment in Canada on its own.
And I don't know if it's going to generate investment in Canada when the prime minister himself says there is no business case. For example, for LNG liquefied natural gas, not one but two national leaders have come to Canada begging Justin Trudeau to let them buy Canadian oil and gas.
And they're handing out these promotions right outside "Project Ukraine". Do you know what the main weapon Russia is using to attack Ukraine is? It's not a tank. It's not a drone. It's the oil weapon. It's the natural gas weapon. The reason that Russia is so economically strong with foreign currency reserves is because they're one of the world's largest producers of oil and gas.
They're tied with Saudi Arabia and the United States producing about 10 million barrels of oil a day. But their real stranglehold on Europe, especially Germany, is in natural gas. You can't actually put sanctions on Russia if you're Europe, because that would be tantamount to putting sanctions on yourself. Is Germany simply going to stop using a third of its energy?
So both the new chancellor of Germany and the new prime minister of Japan, they both came to Canada and said, please liberate us from this conflict energy, liberate us from the conflict natural gas that comes from the Russian company Gazprom and please let us buy Canadian ethical oil and gas. And in the case of both those countries, they really wanted the natural gas.
And Justin Trudeau said no both times. In fact, he didn't even have the courage to say no like a normal person. He said "Oh, well, we're going to have the green energy revolution". But they need natural gas to cook on a stove, to heat a furnace.
And he gave this weird, goofy answer about hydrogen or ammonia or whatever. So my point is, you've got these nice young ladies earning a few thousand bucks over the course of Davos. And there are all these glossy brochures everywhere. But that's not going to convince someone to invest millions, let alone billions of dollars in the country.
But Canada's prime minister says there's no business case to sell oil and gas. Well, meanwhile, the Germans since signed a deal with Qatar.
Canada has one of the stupidest prime ministers in the world, and that's a damn big contest.
GUEST: Noor Bin Ladin
Canada News Analysis


Many Betta breeders, as well as some pet owners, do not think about the Nitrogen Cycle because most of these people are changing 90-100% of their fishes water every few days. Even if you are one of those doing the refular water changes, it is not a bad idea to understand what the nitrogen cycle because all water containing fish will embark on that journey. The regular water changes just keep the cycle at the initial stage.
The nitrogen cycle is not difficult to understand, but it may take reading over this article or actually having your tanks go through it for it to really make sense. So do not feel bad, or feel you will never get it. Just be persistent and read this over a few times and see where you end up. Honestly, I read many articles and NEVER got it untill I watched it go through its progression in my barracks. Suddenly the light bulb went on and it made sense. So again, keep at it untill you understand it because most of the health and spawning issues we face can be traced back to a water quality issue.
The fact that we call this a cycle should tell you there are going to be several stages that must be progressed through before we it can be complete and we can say a tank is cycled. So lets start at the beginnig and work our way through it. You are going to need a good Master Test kit that will test ammonia, nitrite and nitrate. The Master Kit will also have tests for your pH. Get one that uses drops as they are more accurate than test strips. You will not know where you are in the cycle without a kit, and the first thing anyone will ask you if you have fish problems is what are your water parameters. So best to get this now and understand what you are using the test for.
Everything that enters a closed system, stays in the system unless removed. Typically it just changes form. With your fish, any food you give them will be used by the body, and any unused food will leave the fish as waste. This waste will change and ammonia is created. Any ammount of ammonia is toxic to most aquatic life. So if your test kit gives a reading of anything but zero, you now have an issue that needs addressed.
With the production of ammonia in a system, bacteria that feed on this ammonia grow in numbers untill they eventually have a population that will convert ammonia produced into nitrite. It takes a litte time for this population to reach optimum levels, so until that happens, the ammonia levels will continue to build. If the rising levels of ammonia are not removed, it will eventually it kill the fish. Even small amounts are harmful and you will often get gill burning and health problems before the fish finally dies. Ammonia as low as .5 will also wipe out fry, and is the bigest killer of new spawns. Untill the bacteria build their numbers you control excess ammonia with water changes.
It is difficult to give a percentage of water to change as it depends on the current level of ammonia in your system. All over the internet you will have people tell you to do a 10% all the way up to 100% water change. How do you know what to do? Lets assume your tank has an ammonia level of 1.0. If you do a 50% water change, although your tank will look clean, it still has a level of .5, and it still toxic. In that case 90% would be better and another change the following day to get the level barely above zero. Do you see now where that test kit is going to be a big help in your fish room?
So you now are goimg to spend the next few weeks testing the water dai;y and doing water changes as needed to keep it healthy for your fish. As time progresses you will see the ammonia levels start to level off and decline. Bacteria that like the ammonia will have built in numbers and ans should be keeping the ammonia stabilize close to zero. What is happening is those little bacteria are taking in the amonia and turning it into nitrite, the second stage of the cycle.
When you first test your water you need to just test the ammonia. In the beginning you will see the ammonia level climb. As the ammonia starts to build less quickly, it is time to test for nitrites as well. When large numbers of the ammonia eating bacteria start building, the ammoia levels climb more slowly and you start to see the nitrite produced as a result, and you are in entering the second part of the cycle. Nitrite is also toxic to fish, so you do not want levels other than zero in this too.
Water changes will keep the ammonia levels under control untill the bacteria grow in numbers to handle what your tank typically produces. When you see the nitrite levels start to climb you once again will do water changes to get them back to zero. And, just like what you saw with the ammonia, in time you will start to see this level start to stay steady close to zero. Now a new type of bacteria are growing in numbers that will take the nitrite and consumer it and produce nitrate.
How long will it take you to get to this point? Depends on the water volumne and bioload which is the numbers of fish and how often fed. It is not uncommon to take a few months to get to this point. There will be a lot of water changes along the way, but to keep your fish healthy or alive, it will be necessary.
Just as you saw with the ammonia, the levels of the nitrites will start to grow more slowly until it too levels off close to zero. During this time your ammonia will also be staying close to zero. As the nitrites start to level off it is now time to test for the nitrates. With the nitrite eating bacteria growing in numbers you will see the nitrate level rise. Now, nitrate is not as harmful as ammonia or the nitrite, and a low level here is generally acceptable. Plants in the water will consume it as fertilizer, but you will also control this level with water changes. By this point you will not have to do daily or even every other day water changes. You should be getting to a point where you can do a once a week water change and keep the system at healthy levels. When you do you water changes this water can be used to water house plants and they will thrive with the nutrients in the water. I have found bamboo to be especially found of this water and will show its happiness with a deep rich green coloring.
Once the tank is cycled you should be able to preform a weekly water change and keep all the levels at acceptable levels. Feed your fish moderate ammounts and strive for a regular schedule so your water can adapt to a consistend level of waste that will be produced. At this time you can test every few days until you get to know your tank, then a once a week test just to make sure is my recommendation.
If you add any additional fish or decide to feed more heavy, make sure you run some additional tests so you can see if it makes a difference in your water quality. By testing you are in a position where you can do a water change before the system gets unhealthy or critical.
Hopefully you were able to follow and understand the nirtogen cycle, and if you are working through one in a tank of your own, you have gotten to the end and have a happy and healthy tank for your fish. It probably took a few months, but it is worth it because your fish are happy and healthy. Should you have any additional questions please feel free to ask them on our forum.
So here you are, enjoying your new tank and thinking maybe you want another tank. Or what if you have to break down and start over the one you just did somewhere a long the line. Are there no shortcuts?
As you just saw, the cycle is all about certain types of bacteria building to numbers that are able to handle the waste production of your system. The cycle I just took you through allowed nature to prodce and build the bacteria numbers up to aceptable levels for a particular tank. It takes time for the bacteria to develop and grow in numbers to handle the amount of ammonia your tank will regularly produce. If you want to speed this process up, you can get a products consisting of nitrifying bacteria to seed your tank. Even with a jump start of nitrifying bacteria, you will want to regularly test your water to see where you are in the cycle and maintain the water quality until the cycle completes. I completly cycled a barrack system of 48 fish in about 60 gallons of water in two weeks. using nitrifying bacteria I purchased.
A proven product that works well is the Dr Tim's One and Only shown to the left. Dr Tim has a Ph.D. in nitrifying bacteria and has 16 years of experience in the field. He uses these bacteria to set up huge public aquariums in about a weeks time. For more information on cycling a tank and Dr Tim's products and research you can see his web page . He also has several information videos like the one above if you want to know more about the nitrogen cycle in a new tank.
Once you get your tanks cycled, you want to make sure you do not accidently reduce the numbers of bacteria, or you will end up with a mini cycle until it gets reestablished. Your bacteria will have built up along many surfaces, but the prime place to find them will be your filter and gravel beds. As time goes on most filter media will need to be rinsed to continue to work properly. Always use old tank water or tap water that has been conditioned to removed chlorine and chlorimines. Not only will chlorine kill your fish, but your bacteria as well. A typical gravel siphon should remove organic patter while leaving bacteria pretty much intact.
Use care if you decide to treat for fish diseases. Most medications to treat fish diseases will kill bacteria, good and bad. If you are adding meds to treat something like fin rot, you are adding a bacteria killing agent and it will wipe out your biological filtration along with the disease you are treating. Massive die off of these bacteria will cause unhealthy water conditions as they decompose, and the fish waste they were taking care of now is building. It can be a matter of hours to go from healthy to toxic conditions so always use another bare botromed tank as your treatment tank.