text:
\section{\@startsection {section}{1}{\z@}{1.0ex plus

\begin{document}
\date{}

\title{The Gilbert Arborescence Problem\thanks{This research was supported by an ARC
Linkage Grant with Newmont Australia Limited.}}
\author{M.~G.~Volz\thanks{ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010,  Australia. CUBIN is an affiliated program of National ICT
Australia.} \and M.~Brazil\footnotemark[2] \and C.~J.~Ras\footnotemark[2] \and  K.~J.~Swanepoel\thanks{Fakult\"at f\"ur Mathematik,
    Technische Universit\"at Chemnitz,
    D-09107 Chemnitz, Germany. Parts of this paper was written while Swanepoel was visiting the Department of Mechanical Engineering of the University of Melbourne on a Tewkesbury Fellowship.} \and D.~A.~Thomas\footnotemark[2]}
\date{\today}

\maketitle

{\hspace{1pc} {\it{\small Abstract}}{\bf{\small---We investigate the problem of designing a minimum cost flow network interconnecting $n$
sources and a single sink, each with known locations and flows. The network may contain other unprescribed nodes, known as Steiner points. For
concave increasing cost functions, a minimum cost network of this sort has a tree topology, and hence can be called a Minimum Gilbert
Arborescence (MGA). We characterise the local topological structure of Steiner points in MGAs, showing, in particular, that for a wide range of metrics and cost-functions the degree of each Steiner point is $3$.

\em Keywords: Gilbert network, minimum cost networks, network flows, Steiner trees}}
 }

\section{Introduction}
The
\emph{Steiner Minimum Tree (SMT) problem} asks for a shortest
network spanning a given set of nodes (\emph{terminals}) in a given metric space. It differs from the minimum spanning tree problem in
that additional nodes, referred to as \emph{Steiner points}, can be included
to create a spanning network that is shorter than would
otherwise be possible. This is a fundamental problem in physical network design optimisation, and has numerous applications, including the design of telecommunications or transport networks for the problem in the
Euclidean plane (the $l_2$ metric), and the design of microchips for the problem in the rectilinear plane (the $l_1$ metric)~\cite{hwang-1992}.

Gilbert~\cite{gilbert-1967} proposed a generalisation of the SMT problem whereby symmetric non-negative \emph{flows} are assigned between each
pair of terminals. The aim is to find a least cost network interconnecting the terminals, where each edge has an associated total flow such that the
flow conditions between terminals are satisfied, and Steiner points satisfy Kirchhoff's rule (ie, the net incoming and outgoing flows at each
Steiner point are equal). The cost of an edge is its length multiplied by a non-negative \emph{weight}. The weight is determined by a given
function of the total flow being routed through that edge, where the function satisfies a number of conditions. The \emph{Gilbert network
problem} (GNP) asks for a minimum-cost network spanning a given set of terminals with given flow demands and a given weight function.

A variation on this problem that we will show to be a special case of the GNP occurs when the  terminals consist of $n$ sources and a
unique sink, and all flows not between a source and the sink are zero. This problem is of intrinsic interest as a natural restriction of the GNP; it is also of interest for its many applications to areas such as drainage networks~\cite{lee-1976}, gas
pipelines~\cite{bhaskaran-1979}, and underground mining networks~\cite{brazil-2000}.

If the weight function is concave and increasing, the resulting minimum network has a tree topology, and provides a directed path from each
source to the sink. Such a network can be called an \emph{arborescence}, and we refer to this special case of the GNP as the \emph{Gilbert
arborescence problem} (GAP). Traditionally, the term `arborescence' has been used to describe a rooted tree providing directed paths from the
unique root (source) to a given set of sinks. Here we are interested in the case where the flow directions are reversed, i.e. flow is from $n$
sources to a unique sink. It is clear, however, that the resulting weights for the two problems are equivalent, hence we will continue to use
the term `arborescence' for the latter case. Moreover, if we take the sum of these two cases, and rescale the flows (dividing flows in each
direction by $2$), then again the weights for the total flow on each edge are the same as in the previous two cases. This justifies our claim
that the GAP can be treated as a special case of the GNP. It will be convenient, however, for the remainder of this paper to think of
arborescences as networks with a unique sink.

A \emph{minimum Gilbert arborescence} (MGA) is a (global) minimum-cost arborescence for a given set of terminals and flows, and a given cost
function. All flows in the network are directed towards the unique sink. In this paper we investigate the local topological structure of Steiner points in MGAs, over a wide range of different metrics and cost-functions. Such an understanding of the topological structure is a key step towards designing efficient algorithms for constructing MGAs.

In Section~2 we specify the nature of the weight function that we consider in this paper, and formally define minimum Gilbert networks and
Gilbert arborescences in Minkowski spaces (which generalise Euclidean spaces). In Section~3 we give a general topological characterisation of
Steiner points in such networks, for smooth Minkowski spaces. We then apply this characterisation, in Section~4, to the smooth Minkowski plane
with a linear weight function to show that in this case all Steiner points have degree $3$. In Section~5 we derive a similar result in higher dimensional Euclidean spaces for a wide class of weight functions.

\section{Preliminaries}

\subsection{Minkowski spaces and Steiner trees}

The cost functions for the networks we consider in this paper make use of more general norms than simply the Euclidean norm. Hence, we introduce
a generalisation of Euclidean spaces, namely finite-dimensional normed spaces or Minkowski spaces. See \cite{Thompson} for an introduction to
Minkowski geometry.

A \emph{Minkowski space} (or \emph{finite-dimensional Banach space}) is $\numbersystem{R}^n$ endowed with a \emph{norm} $\norm{\cdot}$, which is a function
$\norm{\cdot}:\numbersystem{R}^n\to\numbersystem{R}$ that satisfies
\begin{itemize}
\item $\norm{x}\geq 0$ for all $x\in \numbersystem{R}^n$, $\norm{x}=0$ only if $x=0$,
\item $\norm{\alpha x}=\abs{\alpha}\norm{x}$ for all $\alpha\in\numbersystem{R}$ and $x\in \numbersystem{R}^n$, and
\item $\norm{x+y}\leq\norm{x}+\norm{y}$.
\end{itemize}

We use $\norm{\cdot}_2$ to denote the Euclidean ($l_2$) norm.

We now discuss some aspects of the SMT problem, since this is a special case of the GNP, where all flow are zero. Our terminology for the SMT
problem is based on that used in~\cite{hwang-1992}. Let $T$ be a network interconnecting a set $N = \{p_1,\ldots,p_n\}$ of points, called
\emph{terminals}, in a Minkowski space. A vertex in $T$ which is not a terminal is called a \emph{Steiner point}. Let $G(T)$ denote the
\emph{topology} of $T$, i.e. $G(T)$ represents the graph structure of $T$ but not the embedding of the Steiner points. Then $G(T)$ for a
shortest network $T$ is necessarily a tree, since if a cycle exists, the length of $T$ can be reduced by deleting an edge in the cycle. A
network with a tree topology is called a \emph{tree}, its links are called \emph{edges}, and its nodes are called \emph{vertices}. An edge
connecting two vertices $a,b$ in $T$ is denoted by $ab$, and its  length by $\norm{a-b}$.

The
\emph{splitting} of a vertex is the operation of disconnecting two edges $av,bv$ from a vertex $v$ and connecting $a,b,v$ to a newly created
Steiner point.
Furthermore, though the positions of terminals are fixed, Steiner points can be subjected to arbitrarily small movements provided the resulting
network is still connected. Such movements are called \emph{perturbations}, and are useful for examining whether the length of a network is
minimal.

A \emph{Steiner tree} (ST) is a tree whose length cannot be shortened by a small perturbation of its Steiner points, even when splitting is
allowed. By convexity, an ST is a minimum-length tree for its given topology. A \emph{Steiner minimum tree} (SMT) is a shortest tree among all
STs. For many Minkowski spaces bounds are known for the maximum possible degree of a Steiner point in an ST, giving useful restrictions on the
possible topology of an SMT. For example, in Euclidean space of any dimension every Steiner point in an ST has degree three. Given a set $N$ of
terminals, the \emph{Steiner problem} (or \emph{Steiner Minimum Tree problem}) asks for an SMT spanning $N$.

\subsection{Gilbert flows}

Gilbert~\cite{gilbert-1967} proposed the following generalisation of the Steiner problem in Euclidean space, which we now extend to Minkowski
space. Let $T$ be a network interconnecting a set $N = \{p_1,\ldots,p_n\}$ of $n$ terminals in a Minkowski space. For each pair $p_i,p_j,\;i\neq
j$ of terminals, a non-negative \emph{flow} $t_{ij} = t_{ji}$ is given. The cost of an edge $e$ in $T$ is $w(t_e)l_e$, where $l_e$ is the length
of $e$, $t_e$ is the total flow being routed through $e$, and $w(\cdot)$ is a unit cost \emph{weight function} defined on $[0,\infty)$ satisfying

\begin{align}
  w(0) &\geq 0\quad\text{and}\quad w(t) > 0  \text{ for all }t>0, \label{eq:cost-ftn-non-neg} \\
  w(t_2) &\geq w(t_1)\quad \text{for all}\quad t_2 > t_1\geq 0, \label{eq:cost-ftn-non-dec} \\
  w(t_1 + t_2) &\leq w(t_1) + w(t_2)\quad\text{for all}\quad t_1,t_2 > 0, \label{eq:cost-ftn-trnglr}\\
   w(\cdot)&\text{ is a concave function.} \label{eq:cost-ftn-concave}
\end{align}
That the function $w$ is concave means by definition that $-w$ is convex.
A network satisfying Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-trnglr} is called a \emph{Gilbert network}. For a given edge
$e$ in $T$, $w(t_e)$ is called the \emph{weight} of $e$, and is also denoted simply by $w_e$. The \emph{total cost} of a Gilbert network $T$ is
the sum of all edge costs, i.e.
\begin{eqnarray*}
  C(T) &=& \sum_{e\in E}w(t_e)l_e
\end{eqnarray*}
\noindent where $E$ is the set of all edges in $T$. A Gilbert network $T$ is a \emph{minimum Gilbert network} (MGN), if $T$ has the minimum cost
of all Gilbert networks spanning the same point set $N$, with the same flow demands $t_{ij}$ and the same cost function $w(\cdot)$. By the
arguments of~\cite{cox-1998}, an MGN always exists in a Minkowski space when Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave} are assumed for the weight function.

Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-trnglr} above ensure that the weight function is non-negative, non-decreasing and
triangular, respectively. These are natural conditions for most applications. Unfortunately, the first three conditions alone do not guarantee
that a minimum Gilbert network is a tree. To show this, we now give an example of a Gilbert network problem with two sources and one sink in the Euclidean plane, where there exists a split-route flow that has a lower cost than any arborescence.

For this example there are two sources $p_1, p_2$ and a sink $q$ which are the vertices of a triangle $\triangle p_1p_2q$ with edge lengths $\enorm{p_1-p_2}=1$ and $\enorm{p_1-q}=\enorm{p_2-q}=10$, as illustrated in Figure~\ref{figureEx1}.
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig1}
\end{center}
\caption{An example where split-routing is cheaper\label{figureEx1}}
\end{figure}
The tonnages at $p_1$ and $p_2$ are $2$ and $4$, respectively.
The weight function is $w(t)=\lceil (3t+1)/2\rceil$, i.e., $(3t+1)/2$ rounded up to the nearest integer.
This function is positive, non-decreasing and triangular, but not concave.
For the example we only need the following values:
\[
\begin{array}{c|ccccc} t & 1 & 2 & 3 & 4 & 6 \\ \hline w(t) & 2 & 4 & 5 & 7 & 10
\end{array}\]
Routing $1$ unit of the flow from $p_2$ via $p_1$ to $q$ gives a Gilbert network (Figure~\ref{figureEx1}) of total cost
$$w(1)\enorm{p_1-p_2} + w(3)\enorm{p_1-q} + w(3)\enorm{p_2-q} = 102.$$

For a Gilbert arborescence we  route the tonnages from $p_1$ and $p_2$ to $q$ via some point $s$ as in Figure~\ref{figureEx2}.
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig2}
\end{center}
\caption{The minimum Gilbert arborescence\label{figureEx2}}
\end{figure}
We calculate a minimum Gilbert arborescence by using the weighted Melzak algorithm as described in \cite{gilbert-1967}.
To construct the weighted Fermat-Torricelli point $s$, first construct the unique point $p$ outside $\triangle p_1p_2q$ such that $\enorm{p-p_1}=0.7$ and $\enorm{p-p_2}=0.4$ (Figure~\ref{figureEx3}).
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig3}
\end{center}
\caption{Constructing the weighted Fermat-Torricelli point\label{figureEx3}}
\end{figure}
Then construct the circumscribed circle of $\triangle pp_1p_2$, which will intersect the so-called \textit{weighted Simpson line} $pq$ in the required point $s$. The resulting total cost is
\begin{align*}
&\quad w(2)\enorm{p_1-s}+w(4)\enorm{p_2-s}+w(6)\enorm{s-q}\\
&=w(6)\enorm{p-q}=\sqrt{9982.5+7\sqrt{3890.25}}\\
&=102.074\dots,
\end{align*}
as some trigonometry will show.
  This shows that split routing can be necessary when the weight function is not concave.

For the remainder of the paper we assume that the weight function $w$ is concave in addition to the other three conditions.
In this case it is known \cite{gilbert-1967, thomas-2006} that in the case where there is a single sink there always exists a minimum Gilbert network that is a Gilbert arborescence.
This means that we can (and will) without loss of generality only consider MGAs.
(Note that in~\cite{cox-1998}, Condition~(\ref{eq:cost-ftn-trnglr}), which we call the \emph{triangular condition}, was incorrectly
interpreted as concavity of the cost function.)

 The \emph{Gilbert network problem} (GNP) is to find an MGN for a
given terminal set $N$, flows $t_{ij}$ and cost function $w(\cdot)$. Since its introduction in~\cite{gilbert-1967}, various aspects of the GNP
have been studied, although the emphasis has been on discovering geometric properties of MGNs
(see~\cite{cox-1998},~\cite{thomas-2006},~\cite{trietsch-1985},~\cite{trietsch-1999}). As in the Steiner problem, additional vertices can be
added to create a Gilbert network whose cost is less than would otherwise be possible, and these additional points are again called
\emph{Steiner points}. A Steiner point $s$ in $T$ is called \emph{locally minimal} if a perturbation of $s$ does not reduce the cost of $T$. A
Gilbert network is called \emph{locally minimal} if no perturbation of the Steiner points reduces the cost of $T$.

The special case of the Gilbert model that is of interest in this work is when $N = \{p_1,\ldots,p_n, q\}$ is a set of terminals in a Minkowski
space, where $p_1,\ldots,p_{n}$ are \emph{sources} with respective positive flows $t_1,\ldots,t_{n}$, and $q$ is the \emph{sink}. All flows are
between the sources and the sink; there are no flows between sources. It has been shown in~\cite{thomas-2006} that concavity of the weight
function implies that an MGN of this sort is a tree. Hence we refer to an MGN with this flow structure as a \emph{minimum Gilbert arborescence}
(MGA), and, as mentioned in the introduction, we refer to the problem of constructing such an MGA as the \emph{Gilbert arborescence problem}
(GAP).

If $v_1$ and $v_2$ are two adjacent vertices in a Gilbert arborescence, and flow is from $v_1$ to $v_2$ then we denote the edge connecting the
two vertices by $v_1v_2$.

\section{Characterisation of Steiner Points}\label{section:characterisation}
In this section, we generalise a theorem of Lawlor and Morgan \cite{lawlor-1994} to give a local characterisation of Steiner points in an MGA.
The characterisation in \cite{lawlor-1994} holds for SMTs, which correspond to the case of MGAs with a constant weight function. Their theorem
is formulated for arbitrary Minkowski spaces with differentiable norm. Our proof is based on the proof of Lawlor and Morgan's theorem given in
\cite{swanepoel-1999}. A generalisation to non-smooth norms is contained in~\cite{swanepoel-2007} for SMTs and in \cite{thesis} for MGAs. Such a
generalisation is much more complicated and involves the use of the subdifferential calculus.

We first introduce some necessary definitions relating to Minkowski geometry, in particular with relation to dual spaces. For more details, see
\cite{Thompson}.

We denote the inner product of two vectors $x,y\in\numbersystem{R}^n$ by $\ipr{x}{y}$. For any given norm $\norm{\cdot}$, the dual norm $\norm{\cdot}^\ast$ is
defined as follows:
\[\norm{z}^\ast = \sup_{\norm{x}\leq 1}\ipr{z}{x}.\]

We say that a Minkowski space $(\numbersystem{R}^n,\norm{\cdot})$ is {\em smooth} if the norm is differentiable at any $x\neq o$, i.e., if
\[\lim_{t\to 0} \frac{\norm{x+th}-\norm{x}}{t} =: f_x(h)\]
exists for all $x,h\in \numbersystem{R}^n$ with $x\neq o$. It follows easily that $f_x$ is a linear operator $f_x:\numbersystem{R}^n\to\numbersystem{R}$ and so can be represented by a
vector $x^\ast\in\numbersystem{R}^n$, called the dual vector of $x$, such that $\ipr{x^\ast}{y}=f_x(y)$ for all $y\in\numbersystem{R}^n$, and $\norm{x^\ast}^\ast=1$. In
fact $x^\ast$ is just the gradient of the norm at $x$, i.e., $x^\ast=\nabla\norm{x}$.

More generally, even if the norm is not differentiable at $x$, a vector $x^\ast\in \numbersystem{R}^n$ is a {\em dual vector} of $x$ if $x^\ast$ satisfies
$\ipr{x^\ast}{x}=\norm{x}$ and $\norm{x^\ast}^\ast=1$. By the Hahn-Banach separation theorem, each non-zero vector in a Minkowski space has at
least one dual vector. A Minkowski space is then smooth if and only if each non-zero vector has a unique dual vector.

A norm is {\em strictly convex} if $\norm{x}=\norm{y}=1$ and $x\neq y$ imply that $\norm{\frac{1}{2}(x+y)}<1$, or equivalently, that the
\emph{unit sphere} \[ S(\norm{\cdot}) = \{x\in\numbersystem{R}^n: \norm{x}=1\}\]  does not contain any straight line segment. A norm $\norm{\cdot}$ is smooth
[strictly convex] if and only if the dual norm $\norm{\cdot}^\ast$ is strictly convex [smooth, respectively].

\begin{theorem}\label{theorem:gilb-arb-charac}
Suppose a smooth Minkowski space $(\numbersystem{R}^n,\norm{\cdot})$ is given together with  a weight function $w$ that satsfies Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave}, sources $p_1,\dots,p_n\in\numbersystem{R}^n$,
and a single sink $q\in\numbersystem{R}^n$, all different from the origin $o$. Let the flow associated with $p_i$ be $t_i$. (See Figure~\ref{Fig:th1}.)
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig4}
\end{center}
\caption{A Gilbert network with star topology.} \label{Fig:th1}
\end{figure}
For each $p_i$ let $p_i^\ast$ denote its dual vector, and let $q^\ast$ denote the dual vector of $q$. Then the Gilbert arborescence with edges
$op_i$, $i=1,\dots,n$ and $oq$, where all flows are routed via the Steiner point $o$, is a minimal Gilbert arborescence if and only if
\begin{equation}\label{balancing}
\sum_{i=1}^n w(t_i)p_i^\ast + w(\sum_{i=1}^n t_i)q^\ast = o
\end{equation}
and
\begin{equation}\label{collapsing}
\norm{\sum_{i\in I} w(t_i)p_i^\ast}^\ast\leq w(\sum_{i\in I} t_i)\text{ for all $I\subseteq\{1,\dots,n\}$.}
\end{equation}
\end{theorem}

Note: We think of Condition~\ref{balancing} as a flow-balancing condition at the Steiner point, and Condition~\ref{collapsing} as a condition
that ensures that the Steiner point does not split.

\begin{proof}
$(\Rightarrow)$ We are given that the star is not more expensive than any other Gilbert network with the same sources, sink, flows and weight
function.

In particular, $o$ is the so-called weighted Fermat-Torricelli point of the $n+1$ points $p_1,\dots,p_n,q$ with weights $t_1,\dots,t_n,\sum_{i=1}^n t_i$,
respectively, which implies the balancing condition \eqref{balancing}. We include a self-contained proof for completeness. If the Steiner point
$o$ is moved to $-te$, where $t\in\numbersystem{R}$ and $e\in\numbersystem{R}^n$ is a unit vector (in the norm), the resulting arborescence is not better, by the assumption
of minimality. Therefore, the function
\begin{align*}
 \varphi_e(t) &=\sum_{i=1}^n w(t_i)(\norm{p_i+te}-\norm{p_i})\\ &\quad+w(\sum_{i=1}^n t_i)(\norm{q+te}-\norm{q})\geq 0
 \end{align*}
attains its minimum at $t=0$. For $t$ in a sufficiently small neighbourhood of $0$, $p_i+te\neq o$ and $q+te\neq o$, hence $\varphi_e$ is
differentiable. Therefore,
\begin{align*}
0&=\varphi_e'(0) =\lim_{t\to 0} \left(\sum_{i=1}^n w(t_i)\frac{\norm{p_i+te}-\norm{p_i}}{t}\right.\\
& \qquad\qquad \qquad +\left.w(\sum_{i=1}^n t_i)\frac{\norm{q+te}-\norm{q}}{t}\right)\\
&= \sum_{i=1}^n w(t_i)\ipr{p_i^\ast}{e}+w(\sum_{i=1}^n t_i)\ipr{q^\ast}{e}\\
&= \ipr{\sum_{i=1}^n w(t_i)p_i^\ast+w(\sum_{i=1}^n t_i)q^\ast}{e}.
\end{align*}
Since this holds for all unit vectors $e$, \eqref{balancing} follows.

To show \eqref{collapsing} for each $I\subseteq\{1,\dots,n\}$, we may assume without loss of generality that $I\neq\emptyset$ and
$I\neq\{1,\dots,n\}$. Consider the Gilbert network obtained by splitting the Steiner point into two points $o$ and $+te$ ($t\in\numbersystem{R}$, $e$ a unit
vector) as follows. Each $p_i$, $i \notin I$, is still adjacent to $o$ with flow $t_i$, and $q$ is joined to $o$ with flow $\sum_{i=1}^n t_i$,
but now each $p_i$, $i\in I$, is adjacent to $te$ with flow $t_i$, and $te$ is adjacent to $o$ with flow $\sum_{i\in I} t_i$, as shown in
Figure~\ref{Fig:split}.
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig5}
\end{center}
\caption{The Gilbert network obtained by splitting the Steiner point $o$.} \label{Fig:split}
\end{figure}
Since the new network cannot be better than the original star, we obtain that for any unit vector $e$, the function
$$
 \psi_e(t)=\sum_{i\in I} w(t_i)(\norm{p_i-te}-\norm{p_i})+w(\sum_{i=1}^n t_i)\abs{t}\geq 0
$$
attained its minimum at $t=0$. Although $\psi_e$ is not differentiable at $0$, we can still calculate as follows:
\begin{align*}
0 &\leq \lim_{t\to 0+}\frac{\psi_e(t)}{t}\\
&= \lim_{t\to0+}\sum_{i\in I} w(t_i)\frac{\norm{p_i-te}-\norm{p_i}}{t}+w(\sum_{i=1}^n t_i)\\
&= \ipr{\sum_{i\in I} w(t_i)p_i^\ast}{-e}+w(\sum_{i=1}^n t_i).
\end{align*}
Therefore, $\ipr{\sum_{i\in I} w(t_i)p_i^\ast}{e}\leq w(\sum_{i=1}^n t_i)$ for all unit vectors $e$, and \eqref{collapsing} follows from the
definition of the dual norm.

\noindent $(\Leftarrow)$ Now assume that $p_1^\ast,p_n^\ast,q$ are dual unit vectors that satisfy \eqref{balancing} and \eqref{collapsing}.
Consider an arbitrary Gilbert arborescence $T$ for the given data. For each $i$, let $P_i$ be the path in $T$ from $p_i$ to $q$, i.e., $P_i =
x_{1}^{(i)}x_2^{(i)}\dots x_{k_i}^{(i)}$, where $x_1^{(i)}=p_i, x_{k_i}^{(i)}=q$, and $x_j^{(i)} x_{j+1}^{(i)}$ are distinct edges of $T$ for
$j=1,\dots,k_i-1$. For each edge $e$ of $T$, let $S_e=\{i:\text{$e$ is on path $P_i$}\}$. Then the flow on $e$ is $\sum_{i\in S_e} t_i$ and the
total cost of $T$ is
\[\sum_{\substack{e=xy\text{ is}\\ \text{an edge of $T$}}} w(\sum_{i\in S_e} t_i)\norm{x-y}.\]
The cost of the star is
\begin{align*}
&\quad\sum_{i=1}^n w(t_i)\norm{p_i}+w(\sum_{i=1}^n t_i)\norm{q} \\
&= \sum_{i=1}^n w(t_i)\ipr{p_i^\ast}{p_i}+w(\sum_{i=1}^n t_i)\ipr{q^\ast}{q} \\
&= \sum_{i=1}^n w(t_i)\ipr{p_i^\ast}{p_i-q}\qquad\text{by \eqref{balancing}}\\
&= \sum_{i=1}^n w(t_i) \sum_{j=1}^{k_i-1}\ipr{p_i^\ast}{x_j^{(i)}-x_{j+1}^{(i)}}
\end{align*}
\begin{align*}
&= \sum_{\substack{e=xy\text{ is}\\ \text{an edge of $T$}}} \ipr{\sum_{i\in S_e} w(t_i) p_i^\ast}{x-y}\\
&\leq \sum_{\substack{e=xy\text{ is}\\ \text{an edge of $T$}}}\norm{\sum_{i\in S_e} w(t_i) p_i^\ast}^\ast\norm{x-y}\\
&\leq \sum_{\substack{e=xy\text{ is}\\ \text{an edge of $T$}}} w(\sum_{i\in S_e} t_i)\norm{x-y} \quad\text{by \eqref{collapsing}}.\qedhere
\end{align*}
\end{proof}

Note that the necessity of the conditions \eqref{balancing} and \eqref{collapsing} holds even if the weight function is not concave. It is only
in the proof of the sufficiency that we need all minimal Gilbert networks with a single source to be arborescences.

\section{Degree of Steiner Points in a Minkowski plane with linear weight function
We now apply the characterisation of the previous section in the two\nobreakdash-\hspace{0pt}dimensional case, assuming further that the weight
function is linear: $w(t)=d+ht$, $d>0, h\geq 0$.

\begin{theorem}\label{thm2}
In a smooth Minkowski plane and assuming a linear weight function $w(t)=d+ht$, $d>0, h\geq 0$, a Steiner point in an MGA necessarily has degree
$3$.
\end{theorem}
\begin{proof}
By Theorem~\ref{theorem:gilb-arb-charac}, an MGA with a Steiner point of degree $n+1$ exists in $\numbersystem{R}^2$ with a smooth norm $\norm{\cdot}$ if and
only if there exist dual unit vectors $p_1^\ast,\dots,p_n^\ast,q^\ast\in\numbersystem{R}^2$ such that
\[ \sum_{i=1}^n (d+ht_i) p_i^\ast + (d+h\sum_{i=1}^n t_i) q^\ast = o \]
and
$$
 \norm{\sum_{i\in I} (d+ht_i)p_i^\ast}^\ast \leq d+h\sum_{i\in I} t_i \quad\text{for all } I\subseteq\{1,\dots,n\}.
$$
Label the $p_i^\ast$ so that they are in order around the dual unit circle. Let $v_i^\ast=(d+ht_i)p_i^\ast$ and $w^\ast=(d+h\sum_{i=1}^n
t_i)q^\ast$. Then the conditions become
\[ v_1^\ast+\dots+v_n^\ast+w^\ast=o,\]
and
\begin{equation}\label{collapsing2}
\norm{\sum_{i\in I} v_i^\ast}^\ast\leq d+h\sum_{i\in I} t_i \quad \text{for all } I\subseteq\{1,\dots,n\}.
\end{equation}
Thus we may think of the vectors $v_1^\ast,\dots,v_n^\ast,w^\ast$ as the edges of a convex polygon with vertices $a_j^\ast=\sum_{i=1}^j
v_i^\ast$, $j=0,\dots,n$ in this order (see Figure~\ref{Fig:polygon}).

\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig6}
\end{center}
\caption{A polygon with edges corresponding to $v_j^\ast$.}\label{Fig:polygon}
\end{figure}
Assume for the purpose of finding a contradiction that $n > 3$. Then the polygon has at least $4$ sides. Note that the diagonals $a_0^\ast
a_{j}^\ast$ and $a_{j-1}^\ast a_n^\ast$ intersect. Applying the triangle inequality to the two triangles formed by these diagonals and the two
edges $v_j^\ast$ and $w^\ast$ (as illustrated in Figure~\ref{Fig:polygon}), we obtain
\begin{align*}
 \norm{a_j^\ast}^\ast+\norm{a_n^\ast-a_{j-1}^\ast}^\ast
&\geq \norm{v_j^\ast}^\ast+\norm{w^\ast}^\ast\\
&= d+ht_j + d+h\sum_{i=1}^n t_i\\
&= d+h\sum_{i=1}^j t_i + d + h\sum_{i=j}^n t_i\\
&\geq \norm{\sum_{i=1}^j v_i^\ast}^\ast+\norm{\sum_{i=j}^n v_i^\ast}^\ast \qquad\text{by \eqref{collapsing2}}\\
&= \norm{a_j^\ast}^\ast+\norm{a_n^\ast-a_{j-1}^\ast}^\ast.
\end{align*}
Therefore, equality holds throughout, and we obtain equality in the triangle inequality. Since the dual norm is strictly convex, it follows that
$v_j^\ast$ and $w^\ast$ are parallel. This holds for all $j=2,\dots,n-1$. It follows that $p_1^\ast=\dots=p_n^\ast=-q^\ast$. Geometrically this
means that the unit vectors $\frac{1}{\norm{p_i}}p_i$ and $-\frac{1}{\norm{q}}q$ all have the same supporting line on the unit ball. We can
think of this condition on the vectors $p_i$ and $q$ as a generalisation of collinearity to Minkowski space.

Choose a point $s_2$ on the edge $op_i$ such that the line through $s_2$ parallel to $op_n$ intersects the edge $op_1$ in $s_1$, say, with
$s_1\neq o$. See Figure~\ref{Fig:unitball}.
\begin{figure}
\begin{center}
\includegraphics{gilbertflows-fig7}
\end{center}
\caption{Illustration of the proof of Theorem~\ref{thm2} for a unit ball with straight line segments on the boundary.}\label{Fig:unitball}
\end{figure}
Because of the straight line segments on the boundary of the unit ball,  $\norm{x+y}=\norm{x}+\norm{y}$ for any $x,y$ such that the unit vectors
$\frac{1}{\norm{x}}x$ and $\frac{1}{\norm{y}}y$ lie on this segment. In particular,
\begin{equation}\label{triangleineq}
\norm{s_2-s_1}+\norm{s_1-o}=\norm{s_2-o}.
\end{equation}
Now replace $p_2o$ by the edges $p_2s_2$ and $s_2 s_1$ , replace $p_1o$ by $p_1s_1$ and $s_1o$, and add the flow $t_2$ to $s_1o$. The change in
cost in the new Gilbert arborescence is
\begin{align*}
&\quad (w(t_1)\norm{p_1-s_1}+w(t_1+t_2)\norm{s_1-o} +w(t_2)\norm{p_2-s_2}+w(t_2)\norm{s_2-s_1}) \\
& \qquad - \left(w(t_1)\norm{p_1-o}-w(t_2)\norm{p_2-o}\right)\\
& = -w(t_1)\norm{s_1}-w(t_2)\norm{s_2}+w(t_1+t_2)\norm{s_1}+w(t_2)(\norm{s_2}-\norm{s_1}) \qquad\text{by \eqref{triangleineq}}\\
&= (w(t_1+t_2)-w(t_1)-w(t_2))\norm{s_1}\\
&= (d+h(t_1+t_2)-(d+ht_1)-(d+ht_2))\norm{s_1}\\
&= -d\norm{s_1} < 0.
\end{align*}
We have shown that a Steiner point of degree at least $4$ leads to a decrease in cost. In an MGA a Steiner point must then necessarily be of
degree $3$.
\end{proof}

\section{Degree of Steiner points in Euclidean space

We now consider Gilbert arborescences in Euclidean space (of arbitrary dimension) with the weight function $w(t)=d+ht^\alpha$, where $d, h>0$
and $0<\alpha\leq 1$.
Note that for all these values of $\alpha$, the weight function $w$ satisfies all of the conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave}.
We show that if $0<\alpha\leq 1/2$ or
$\alpha=1$, then the maximum degree of a Steiner point is $3$, while for each $\alpha\in(1/2,1)$ we provide an example of an MGA with a Steiner
point of degree $4$. This example is three-dimensional, and the tonnage goes to infinity as $\alpha$ approaches $1/2$ or $1$. When $\alpha=1$
then the previous section shows that Steiner points are necessarily of degree $3$ in the Euclidean plane. Although we have no examples of degree
$4$ Steiner points in the Euclidean plane for other values of $\alpha$, we consider their existence to be highly unlikely.

We show that there is a very general class of weight functions for which the Steiner points are necessarily of degree $3$ (Theorem~\ref{degree3} below). Our proof is
completely independent of the dimension.

As previously, we denote the Euclidean norm by $\enorm{\cdot}$, the inner product of two vectors $u$ and $v$ by $\ipr{u}{v}$, and the weight function by $w$.

We first reformulate Theorem~\ref{theorem:gilb-arb-charac} for Euclidean space.

\begin{corollary}\label{cor}
A Steiner point of degree $m+1$ is possible in some MGA in Euclidean space with a weight function $w(\cdot)$ that satisfies Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave}, if and only if there exist vectors
$v_1,\dots,v_m$ and tonnages $t_1,\dots, t_m>0$ such that
\begin{gather}
\label{a} \enorm{v_i}=w(t_i), \qquad i=1,\dots,m \\
\label{b} \enorm{\sum_{i=1}^m v_i}=w(\sum_{i=1}^m t_i), \\
\label{c} \forall I\subseteq\{1,\dots,m\}\text{ with } 2\leq\card{I}\leq m-2,\quad \enorm{\sum_{i\in I} v_i}\leq w(\sum_{i\in I} t_i).
\end{gather}
\end{corollary}

\begin{theorem}\label{degree3}
If the weight function $w(\cdot)$ satisfies Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave}, and is differentiable with $(w^2)'$ convex and $w(0)>0$, then all Steiner points in MGAs have degree $3$.
\end{theorem}

Note that the hypothesis is indeed satisfied for the weight function $w(t)=d+ht^\alpha$ for any $d,h>0$ and $\alpha\in[0,1/2]\cup\{1\}$, but not
when $\alpha\in(1/2,1)$.

For the proof of Theorem~\ref{degree3} we need the following inequality valid for functions with convex derivative.

\begin{lemma}\label{ineqlemma}
If $f:[0,\infty)\to\numbersystem{R}$ is differentiable with $f'$ convex, then for all $m\geq 2$ and all $t_i\geq 0$ \textup{(}$i=1,2,\dots,m$\textup{)},
\begin{multline}\label{ineq}
 \frac{(m-1)(m-2)}{2}f(0)+\sum_{1\leq i<j\leq m} f(t_i+t_j)\\ \leq (m-2)\sum_{i=1}^m f(t_i)+f\left(\sum_{i=1}^m t_i\right).
 \end{multline}
\end{lemma}
Note that in \eqref{ineq}, as well as in the sequel, the summation $\sum_{1\leq i<j\leq m}$ means that the sum is over all $m(m-1)/2$ pairs $(i,j)$ that satisfy $1\leq i<j\leq n$. We postpone the proof of the lemma.
\begin{proof}[Proof of Theorem~\ref{degree3}]
Suppose that a Steiner point of degree $m+1\geq 3$ exists. We intend to show that $m+1=3$. Note that we do not only consider the case $m+1=4$,
since it is \emph{a priori} possible that degree $4$ Steiner points don't exist, although degree $5$ points exist (although we don't have any
examples, and it seems highly unlikely).

By Corollary~\ref{cor} there exist vectors $v_1,\dots,v_m$ and numbers $t_1,\dots,t_m>0$ that satisfy \eqref{a}, \eqref{b} and \eqref{c}. Square
\eqref{b}:
\begin{align}
\left(w(\sum_{i=1}^m t_i)\right)^2 &= \enorm{\sum_{i=1}^m v_i}^2 \notag\\
&= \sum_{i=1}^m\enorm{v_i}^2+2\sum_{1\leq i<j\leq m} \ipr{v_i}{v_j} \notag\\
&= \sum_{i=1}^m (w(t_i))^2 + 2\sum_{1\leq i<j\leq m} \ipr{v_i}{v_j}\quad\text{(by \eqref{a}).} \label{star}
\end{align}

Estimate $\ipr{v_i}{v_j}$ by applying \eqref{c} to $I=\{i,j\}$:
\begin{align*}
2\ipr{v_i}{v_j} &=\enorm{v_i+v_j}^2-\enorm{v_i}^2-\enorm{v_j}^2\\
&\leq w(t_i+t_j)^2-w(t_i)^2-w(t_j)^2 \quad\text{(again by \eqref{a}).}
\end{align*}
Sum this inequality over all pairs $(i,j)$ with $1\leq i<j\leq m$:
\begin{align*}
 2\sum_{1\leq i<j\leq m} \ipr{v_i}{v_j} &\leq \sum_{1\leq i<j\leq m} (w(t_i+t_j)^2-w(t_i)^2-w(t_j)^2) \\
 &= \sum_{1\leq i<j\leq m} (w(t_i+t_j))^2-(m-1)\sum_{i=1}^m(w(t_i))^2,
 \end{align*}
 since each $(w(t_i))^2$ is summed once for each of the $m-1$ pairs in which $i$ appears.
 Substitute this into \eqref{star}:
 \begin{equation}\label{A}
 \left(w(\sum_{i=1}^m t_i)\right)^2 \leq \sum_{1\leq i<j\leq m} (w(t_i+t_j))^2-(m-2)\sum_{i=1}^m(w(t_i))^2.
 \end{equation}
 Apply Lemma~\ref{ineqlemma} to $f=w^2$:
 \begin{multline}\label{B}
  \frac{(m-1)(m-2)}{2}(w(0))^2 + \sum_{1\leq i<j\leq m}(w(t_i+t_j))^2\\ \leq (m-2)\sum_{i=1}^m (w(t_i))^2+\left(w(\sum_{i=1}^m t_i)\right)^2.
 \end{multline}
 Combining \eqref{A} and \eqref{B}, we obtain $\frac{(m-1)(m-2)}{2}(w(0))^2\leq 0$, which implies $m+1\leq 3$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{ineqlemma}]
We use induction on $m\geq 2$. The base case $m=2$ is trivial.

Assume now that $m\geq 3$ and that the lemma holds for $m-1$, in particular that
\begin{multline}\label{indhyp}
 \frac{(m-2)(m-3)}{2}f(0)+\sum_{1\leq i<j\leq m-1} f(t_i+t_j)\\ \leq (m-3)\sum_{i=1}^{m-1} f(t_i)+f(\sum_{i=1}^{m-1} t_i).
\end{multline}
Consider $x:=t_m$ to be variable and $t_1,\dots,t_{m-1}$ fixed.
Set $T:=\sum_{i=1}^{m-1}t_i$, and define
\begin{align*}
 g(x) &:= (m-2)\sum_{i=1}^m f(t_i)+f(\sum_{i=1}^m t_i)-\sum_{1\leq i<j\leq m}f(t_i+t_j)\\
      &=  (m-2)\sum_{i=1}^{m-1} f(t_i) + (m-2)f(x)+f(T+x)\\
      & \qquad\qquad-\sum_{1\leq i<j\leq m-1}f(t_i+t_j)-\sum_{i=1}^{m-1} f(t_i+x).
\end{align*}
We have to show that $g(x)\geq \frac{(m-1)(m-2)}{2}f(0)$ for all $x>0$. First of all,
\begin{align*}
g(0) &= (m-2)\sum_{i=1}^{m-1}f(t_i)+(m-2)f(0)+f(T)\\
 & \qquad\qquad -\sum_{1\leq i<j\leq m-1}f(t_i+t_j)-\sum_{i=1}^{m-1}f(t_i)\\
&=(m-3)\sum_{i=1}^{m-1}f(t_i)+f(T)-\sum_{1\leq i<j\leq m-1}f(t_i+t_j)+(m-2)f(0)\\
&\geq \frac{(m-2)(m-3)}{2}f(0)+(m-2)f(0) \quad\text{(by \eqref{indhyp})}\\
&= \frac{(m-1)(m-2)}{2}f(0).
\end{align*}
It is therefore sufficient to show that $g'(x)\geq0$ for all $x>0$. We have
\[ g'(x)=(m-2)f'(x)+f'(T+x)-\sum_{i=1}^{m-1}f'(t_i+x).\]
If $T=0$ then $t_i=0$ for all $i=1,\dots m-1$, and then $g'(x)$ is identically $0$.
We may therefore assume without loss of generality that $T>0$.
Write each $x+t_j$ as a convex combination of $x$ and $x+T$:
\begin{equation*}\label{4}
t_j+x=\left(1-\frac{t_j}{T}\right) x+ \frac{t_j}{T}(x+T).
\end{equation*}
Since $f'$ is convex,
\begin{align*}
\sum_{j=1}^{m-1}f'(t_j+x) &\leq \sum_{j=1}^{m-1}\left(1-\frac{t_j}{T}\right)f'(x)+\frac{t_j}{T} f'(x+T))\\
&= (m-2)f'(x)+f'(x+T),
\end{align*}
which gives $g'(x)\geq0$. This finishes the induction step and the proof.
\end{proof}

\bigskip
We now show that for each $\alpha\in(1/2,1)$ there exist Gilbert arborescences with Steiner points of degree $4$ if the weight function is
$w(t)=d+ht^\alpha$, with $d,h>0$ chosen appropriately.
In the example all incoming tonnages will be equal.
We first use Corollary~\ref{cor} to formulate a result for general weight functions.
\begin{proposition}\label{degree4proposition}
Let $w(\cdot)$ be a weight function that satisfies Conditions~\eqref{eq:cost-ftn-non-neg}--\eqref{eq:cost-ftn-concave}.
There exists an MGA with degree $4$ in Euclidean $3$-space with equal tonnages $t_1=t_2=t_3=:t$ and with weight function $w(\cdot)$ if, and only if
\begin{equation}\label{degree4condition}
 3w(t)^2+w(3t)^2 \leq 3w(2t)^2.
\end{equation}
\end{proposition}
\begin{proof}
By Corollary~\ref{cor}, an MGA of degree $4$ exists in Euclidean space with equal tonnages  $t_1=t_2=t_3=:t$ if, and only if, there exist three Euclidean vectors $v_1, v_2, v_3$
such that
\begin{gather}
\norm{v_1}_2=\norm{v_2}_2=\norm{v_3}_2=w(t),\label{aa}\\
\norm{v_1+v_2}_2,\norm{v_2+v_3}_2,\norm{v_1+v_3}_2\leq w(2t),\label{bb}\\
\norm{v_1+v_2+v_3}_2=w(3t).\label{cc}
\end{gather}
(These vectors will of course span a space of dimension at most $3$.)

Square \eqref{cc} and use \eqref{aa} to obtain an expression for the sum of the three inner products:
\begin{equation}\label{iprsum}
 2(\ipr{v_1}{v_2}+\ipr{v_2}{v_3}+\ipr{v_1}{v_3})=w(3t)^2-3w(t)^2.
\end{equation}
Square \eqref{bb} and use \eqref{aa} to obtain an upper bound on each inner product $\ipr{v_i}{v_j}$ (geometrically this is a lower bound on
the angle between any two vectors):
\begin{equation*}\label{iprineq}
 2\ipr{v_i}{v_j}\leq w(2t)^2-2w(t)^2,
\end{equation*}
and substitute this into \eqref{iprsum} to obtain \eqref{degree4condition}.

Conversely, if we assume \eqref{degree4condition}, we have to find three vectors that satisfy \eqref{aa}--\eqref{cc}.
Note that for each $\lambda\in[-1/2,1]$ there exist three unit vectors $u_1,u_2,u_3\in\numbersystem{R}^3$ such that the inner product of each pair equals $\lambda$.
The one extreme $\lambda=1$ corresponds to three equal vectors, and the other extreme $\lambda=-1/2$ to three coplanar vectors such that any two are at an angle of $120^\circ$.
If we set
\[\lambda:=\frac{w(3t)^2}{6w(t)^2}-\frac12,\] then the vectors $v_i:=w(t)u_i$ satisfy \eqref{aa}--\eqref{cc}.
It remains to show that this value of $\lambda$ really lies in the interval $[-1/2,1]$.
The lower bound $\lambda\geq -1/2$ holds trivially, while the upper bound $\lambda\leq 1$ follows from  $0\leq w(3t)\leq 3w(t)$, which in turn follows from non-negativity (Condition~\eqref{eq:cost-ftn-non-neg}) and concavity (Condition~\eqref{eq:cost-ftn-concave}) of the cost function.
\end{proof}

\begin{corollary}
For each $\alpha\in(1/2,1)$ there exists $d,h,t>0$ and an MGA of degree $4$ in Euclidean $3$-space with cost function $w(t)=d+ht^\alpha$, and tonnages $t_1=t_2=t_3:=t$.
\end{corollary}
\begin{proof}
By choosing the unit of the weight function appropriately, we may assume without loss of generality that $w(t)=D+t^\alpha$.
We may similarly assume that $t=1$, and then by Proposition~\ref{degree4proposition}, we only have to show that $3w(1)^2+w(3)^2\leq 3w(2)^2$ will hold for some value of $D>0$, which is
\[ 3(D+1)^2+(D+3^\alpha)^2\leq 3(D+2^\alpha)^2,\]
or equivalently,
\[ D^2+(6+2\cdot 3^\alpha-6\cdot 2^\alpha)D+3+3^{2\alpha}-3\cdot2^{2\alpha}\leq 0.\]
A sufficient condition for this to hold for some $D>0$, is that the quadratic polynomial in $D$ on the left has a positive root.
For this to hold it is in turn sufficient that its constant coefficient is negative, i.e., that
\[f(\alpha)=3+3^{2\alpha}-3\cdot2^{2\alpha}< 0 \text{ for all }\alpha\in(1/2,1).\]
However, it is easily checked that $f(1/2)=f(1)=0$ and that $f''(\alpha)>0$ for all $\alpha\in(1/2,1)$, so that $f$ is convex on $(1/2,1)$.
It follows that $f$ is negative on $(1/2,1)$, which finishes the proof.
\end{proof}

\section{Conclusion}
In this paper we have studied the problem of designing a minimum cost flow network interconnecting $n$ sources and a single sink, each with
known locations and flows, in general finite-dimensional normed spaces. The network may contain other unprescribed nodes, known as Steiner
points. For concave increasing cost functions, a minimum cost network of this sort has a tree topology, and hence can be called a Minimum
Gilbert Arborescence (MGA). We have characterised the local topological structure of Steiner points in MGAs for linear weight functions,
specifically showing that Steiner points necessarily have degree $3$.


Hello, everybody! Welcome to this week's edition of 7-Imp's 7 Kicks, our weekly tradition of listing Seven(ish) Exceptionally Fabulous, Beautiful, Interesting, Hilarious, or Otherwise Positive Noteworthy Things from the past week. We're continuing our recent habit of featuring the work of a different illustrator each week as our post image, with two works from the delightful and talented Jamie Hogan.
You may remember Jamie as the illustrator of Rickshaw Girl by Mitali Perkins, and the creator of the gorgeous portrait included in our interview with Mitali for the SBBT. She's also not the person one should sit behind when one is attending a drawing lesson given by Ralph Masiello at the Charlesbridge Open House, even if she is very charming and kind in person.
Thanks, Jamie, for the lovely summery images!
1* My mom fell and broke her wrist Friday night, which is definitely NOT a Kick. But she's okay – she's got this thingy on her arm (with PINS! UGH!) but she's not in a lot of pain, and she should be fine in a few weeks. It could have been worse. That's the Kick part, see?
2* Also, my brother is taking very good care of her. He stayed with her most of the night, but he finally had to leave the hospital at 4:00 a.m. because, as he put it, "the waiting room chairs have ARMS on them. What is up with that??" Good point.
3* I finished Michael Ondaatje's newest novel, Divisadero. It's not my very very favorite of his, but it's quite lovely.
4* I got to hang out with a friend and former co-worker who I hadn't really seen since she got her new job. And we went to Finale, which is #497 in the ever-growing list of Things Cambridge Has That Ithaca Doesn't And I'm Going To Miss A Lot.
Son of a Witch that I got at that Gregory Maguire/Daniel Handler thingy a few weeks ago.
7* A lot of my favorite patrons have heard that I'm moving, and keep telling me how much they'll miss me. Which is so totally rewarding, and utterly sweet of them. I'll miss them too.
Aw, man, am I glad Eisha's mama is okay! Also, I'm not surprised that Eisha's library patrons will miss her. She's not a "shush!"ing librarian for sure. She's actually friendly and kind and helpful and enjoys talking books, obviously, and would not ever give you a hard time if, say, you maxed out your library card and your daughter's library card on two huge stacks of picture books (sorry, but I've had too many bad experiences with the unhappy, "shush!"y kind).
#1). Those great images from Jamie Hogan!
#2). I got to go with my three-year-old daughter today to the baby shower of a friend from high school who has had a long, hard road to this joyous pregnancy. I'm so excited for her! And at this shower, I saw another high school friend, and she makes me laugh really, really hard and is a fabulous human being. I'm pretty bummed that the pictures taken of the three of us today did not come out (time for a new digital camera); otherwise, I'd share. Oh, and pictured here is my gift to the baby-to-come, 'cause no baby needs to be without that book (and now this book, too). BEST OF ALL, no one gave Love Your Forever as a gift. Whew.
#3). Cake at baby showers. And things like hand-made baby quilts and crocheted blankets that people like your aunts create for you. Awesome.
#4). The fact that the other bloggers in the Summer Blog Blast Tour have ended up being not unlike a Blogger Support Group. Also awesome.
#5). This post from Emily at …whimsy…. Go read it! Go read it! Great idea (thanks to Sara for pointing it out at a post over at Robin Brande's blog).
#6). Last week was the anniversary of the first date my husband and I had (hey, Jen Robinson has a similar anniversary around the same time, apparently!). On our first date nine years ago, we saw "The Planet of the Apes" ("Get your stinking hands off me, you damn dirty human!" — uh, that's a line from the movie, not anything I said to him on our date) in a summer classic-movie series at the historic Tennessee Theatre, but this time we saw "Knocked Up." Like Adrienne, I laughed so hard I thought I'd stop breathing or pass out or something. Very good movie. To boot, it seems like the soundtrack would be really great.
#7 — last but far from least). This will be cryptic, but a very (amazingly) kind offer was made to me, and it's all anonymous, so I don't know whom to thank, and I absolutely could not in good conscience — and for several reasons — take the unknown donor up on the ever-so generous offer, but it was touching nonetheless. Confused? Sorry. That's all I'm sayin'. But I had to mention it, 'cause such kindness is a definite kick.
I had some chest pain this week, which was not anything serious (just annoying). But Eisha called me tonight to make sure I was okay, 'cause I hadn't been online blogging or emailing all day. I was just busy all day, it turns out, but this tells me two things: 1) Eisha is a sweet friend, and 2) Damn, I must appear to be some kind of blogging addict! (But mostly it just tells me how thoughtful Eisha is) . . .
These lists really do make you find The Good in your week, 'cause I thought my list would be short and under seven things. Heh.
2 – I saw the season 3 finale of "Doctor Who" (The Last of the Time Lords) as it aired last night; it was a bit of a disappointment, but at least I saw it as it aired!
4 – I'm in Gloucestershire for the weekend with my parents and brother – four days off work too – Woo Hoo!
Hope it's drier where you are than it is here in soggy England!
Eisha, hope your mom is doing better and isn't hanging out with friends the best. I don't eny having to pack to move. Did that 2 years ago. Jules, I take it you are not a fan of Love you Forever? Um, not my fave either. Hope the rain stops, Michelle.
1. I actually, really left school for summer on Wednesday PM.
3. Am on CD # 5 of Harry Potter 4.
4. Author Party for Laini Taylor tomorrow at my house.
5. Leaving to see friends in Idaho on Tuesday.
6.I read Penny from Heaven by Jennifer Holm and cried.
7. I pulled weeds yeaterday ad deadheaded plants. Time in the garden, yes!!
And one more, a love box of goodies came from Michigan as part of the blogger's swap.
Have agreat week. I will try to post next week late but am gone the next two Sunday AM. Have a great week.
"Get your stinking hands off me, you damn dirty human!" — uh, that's a line from the movie, not anything I said to him on our date.
I love your first date story, too, jules! My hubbie and I saw many old movies at that theater. We were fans of the classic Humphrey Bogart/Lauren Bacall flicks. Eisha, how wonderful you can count on your brother, and that you have a sister to share books with!
1. My daughter's boyfriend stayed with us all week, and he is polite and charming. And he cooks.
2. Robin's party is real.
4. Having Éireann Lorsung comment on her poem yesterday.
6. Peaches now taste like peaches.
Sara, I fixed your link for you. I got your back. (I got rid of those messages that you said you wanted deleted, the paging-tech-support messages — I hope that's what you meant. I don't mean to be all delete-y fingers).
O MY HOW I LOVE PAUL SIMON and can't wait to see that show. Eisha and I also both happen to love Lyle Lovett. We both love that inside image in the CD insert of "The Road to Ensenada" — what is it? It's a picture of his hat, and the liner of the hat says "like hell it's yours." Did I get that right? I'll have to go find that CD and double-check.
And "Graceland" — ah, does it get any better? Thanks for telling us about that! And I'm going to go watch that video in a minute.
Awesome on your daughter's boyfriend. Five extra points for being a cook.
Jone, happy summer!! Have a safe trip to Idaho, and come back and report.
p.s. If either one of our girls had been a boy, we were likely going to name him Lyle!
and p.s. again — Yes, how KICKIN' was it to have that poet comment at our blog about her poem? So exciting.
Wow, guys, Hogan's illustrations are beautiful, the sort of thing I'd love to hang in my house.
Jules, So glad you enjoyed Knocked Up! It's so good to really, really laugh like that, isn't it? It's cleansing like a good cry, but so much better.
1. I finally got the new White Stripes album, and I'm totally loving it.
2. I also cannot stop listening to "Grace Kelly" by Mika. It makes me happy, I think, because I could also be purple.
3. Peter Rabbit visited my library this week, and the teen volunteer who wore the costume for three different hour-long programs made them the best costume character programs we've ever had. Staff, patrons, volunteers, me — everyone had a great time. That kid deserves a metal.
4. I finished Austenland by Shannon Hale, which was such a fun read.
5. This week, I climbed a ladder and WALKED on my roof to clean out the rest of the gutters. Talk about empowering (once I got over feeling like I was maybe going to throw up).
6. My second fifteen year high school reunion was totally fun.
7. I went to the drive-in to see 1408\ and Mr. Brooks, which are just exactly the kinds of movies one should watch at the drive-in.
Even now, I have an unplanned day and a cup of coffee. That's another Kick in itself, eh?
Sara, Oh, yes, I am in for games at Robin's party. Let it be a plan.
I hope the week has been good for everyone. So far, it looks like we have some excellent lists. Here are my kicks for the week.
1. Started teaching 2 online classes this week. It's craziness, but the teachers in them are awesome and we're all learning together. On most days I feel like they're teaching me.
2. Went to a very small pre-conference dinner with local school division heads and staff development folks. The food and company were good, as was the keynote speaker.
3. William and I stumbled on the Stink books by Megan Mcdonald and are having a terrific time reading through them.
4. Finally joined a writers group and had a fabulous first meeting with like-minded ladies. We talked about writing, our goals, how we want to function and did some free writing. Best of all, we all work at the same place, so we got to kvetch, which was a great opportunity for venting our frustrations.
5. Bought and finished Austenland on Thursday. Hey, I stayed up until 1 am, but hey, who needs sleep when a fine romantic novel is calling your name?
6. Also bought and finished Into the Wild this weekend. LOVED IT!
7. Was missing my Mom for some reason, so spent one wonderful hour on the phone with her this morning. And while I'm thinking about Adrienne's post about her high school reunion, let me give my Mom some props for attending her 60th this weekend. Even at 78 she still likes to party. I suppose that's another kick in itsefl.
Eisha-Hope your mom's wrist gets better soon.
Jules and Jen-I can't believe you remember the date of your first date with your husbands. Pretty impressive.
1. My parents are visiting. It makes me very happy.
2. Therefore, my house is the cleanest it has been in a very long time.
3. Therefore, Husband is happy.
4. Since my parents are here, my husband and I will be able to go out on a date.
5. The kids are totally in love with my parents and vice versa.
7. Warm Trader Joe's Garlic Naan with a spread of Chevre Cheese, Olive Tapenade and crushed red pepper. Delicious.
Freaky, Vivian. I bought Trader Joe's Garlic Naan this week, too. And I hardly ever shop at Trader Joe's. I went there for a Belgian chocolate fix, and the naan was an impulse buy.
Jules, I think it's cool that we celebrated first date anniversaries on the same week. We like to celebrate ours, especially this one b/c it's been 20 years. Scary thought, but cool, too. But I don't have a funny story like about the movie line.
Eisha, I hope that you Mom heals quickly.
1. I started reading The Mysterious Benedict Society while killing time in the bookstore yesterday, and I'm completely hooked, and have to get the book.
2. Perfect California weather (wishing you better weather soon Michele) all week. Cool enough to sleep at night, and walk in the mornings, but warm and sunny during the days.
3. I bought my ticket for Robin's Chicago party / kidlit conference, and made plans to spend a night with old friends from high school before the party. I hope that you all will come!!
4. I finished re-reading the third Harry Potter book. It's interesting to go back, and see what's different in the books from the movies, which I had seen more recently.
5. I got my to-be-reviewed books organized. It's still a daunting number of books (not helped at all by my compulsive need to re-read all six of the Harry Potter books before book 7 comes out), but at least I have better organization.
6. One of my reviews was blurbed in the Kane/Miller Fall Catalog (thanks to Jules for pointing that out!). It was for Jez Alborough's Fix-It Duck. And I've been loving the supportive SBBT team, too.
7. Yesterday afternoon we went to a tequila bar, where we sat outside on a comfortable couch, drinking simply amazing margaritas. It's nice to get away, even if it's only for a couple of hours.
Hey, everyone, that video Sara linked to is great.
Adrienne, I am so slow — I just figured out that there is even a new White Stripes album. I'm so behind. Can't wait to hear it. I gave a pregnant friend yesterday a copy of my rockin' children's-music-that-will-not-make-the-parents-wanna-gag mix, and it includes "We're Going to be Friends" by The White Stripes.
Yay, Tricia's mom!! Party down. Tricia, your writer's group sounds great.
Vivian, I was eating simple 'ol cheese and crackers when I read that about the garlic nan, and I think I might have drooled . . . I hope you have a great time with your parents.
Jen, I know how it feels to get review copies organized. I still need more space for them, but last week I actually — NERD ALERT! — made spreadsheets on my computer for my review copies, divided by publisher . . . and that's 'cause I've gotten so many. It's a good problem to have. Not a bad one at all. But my point is: I know how it feels to get that organized with them.
Jen, where did you all go on your first date, if I may boldly ask? Baseball game?!
Jules, when I said that I got my review books organized, I actually meant that I cleaned up the spreadsheet that I use to keep track of them, so that I know when each one was requested, received, and reviewed, and if I notified anyone about the review. Now it's sorted so that I can see the ones that I haven't reviewed, by date received. There are a couple that I feel very badly about, and hope to get to soon. Not that I'm 100% first in first out, but it's at least a guideline for things that slipped through the cracks. Guess I'm a nerd, too.
The first date… no I don't think I want to go into details about that right now. A baseball game, though. That would have been cool!
Jules, I keep telling the children's librarian I work with who also plays the guitar to learn "We're Going to Be Friends" for storytimes, but he stubbornly refuses to learn anything that isn't an action song. He gets all, "Why would the kids want to just sit there and here me play that?" And I get all, "Because the song is freaking cool." Stubborn, stubborn. One of these days I have to learn to play guitar so I can just do it myself.
Anyway, you should definitely listen to the new album. Good stuff. Incidentally, I have The Innocence Mission CD en route from amazon. Good stuff — thanks for turning me onto it.
Jules and Jen, you intimidate me. I just have three piles on the floor of my office: haven't read-read-library-books, haven't-read-review-copies, and have-read-but-haven't-reviewed.
And Jules, in my defence, I'm not always prone to panic if I don't get an email from you in a day – but I think my mom's accident had put me in hypervigilant mode.
Michele – I admire your discipline. There isn't much that could pull me out of bed at 5 a.m. besides a fire.
Jone – happy relaxing, and have fun in Idaho.
Sara – thanks for the heads-up! Paul Simon is one of my all-time fave singer-songwriters, and the thought of Lyle Lovett covering 50 Ways is giving me goosebumps. And yeah, I should have mentioned Eireann Lorsung commenting on the blog as a kick. That was totally awesome.
Adrienne – I did know there's a new White Stripes album out, but I've only heard the one song "Icky Thump" and it is fierce. I must get my hands on the CD soon.
Tricia – yay for Stink! And for Judy Moody! Those are really fun books.
Vivian, your lists always make me so HUNGRY. I may have to run out to Trader Joe's right now.
1. Reading all of these tasty, joyous, serendipitous, celebratory KICKS.
2. My daughters' swim meet when they both swam back stroke (even though they don't love it) because they were asked to.
4. I will be IN THE MOUNTAINS in less than a week. Home of my heart.
5. Date night last night with my husband and another couple who showed us their amazing photos from Turkey and Greece.
7. My husband's new job — making him very happy, which is making me very happy.
1. Cherries at $1.69 a pound.
2. Long afternoons at the pool. And with two seasoned swimmers, I can just sit back and read.
3. Some great books this week, like First Light and Story of a Girl.
4. Progress on one summer goal by joining Jacketflap.
5. Two pairs of my favorite baggy shorts for $3.99 at Rugged Warehouse.
6. Writing my Knuffle Bunny Too post and getting lots of comments on it.
Liz – ah, mountains. I love the mountains too. Have fun.
MotherReader – I also love cherries. Especially cheap ones. Actually, cheap is my favorite cuisine in general.
Baby, it's "Take your stinking paws off me, you damn dirty ape." Get the popcorn ready, you're sentenced to a remedial viewing of all five(*) movies.
Re. the rain – apparently we're due to have our summer on July 15th – St Swithin's Day !!! (If you're wondering "So what?" – there's a tradition that whatever the weather does on St Swithin's Day, it will then do for the next 40 days… You can bet lots of Brits will praying/hoping that it is a nice day on July 15th !
Eisha, my friend calls it "Sparrow fart" when talking of early morning… Makes me snigger !!
Hey, Michele, St. Swithin's Day factors into the recent (and wonderful) YA novel Billie Standish Was Here. That's where I first read about it.
(Just sharing) . . .
Here's to a nice day on July 15 . . .
Blaine, you're such a romantic.
I missed you yesterday — sorry!
Happy week all — enjoy the 4th!
Yay, Nancy! Thanks. There is no rule that says you *have* to post on Sundays. Thanks for doing that.
I'm painfully embarrassed that I got that "Planet of the Apes" quote wrong, 'cause that would be the Marky-Mark movie I quoted. Ew. Just . . . ew.
Blaine, I caught that, too, but I wasn't going to say anything because it was Jules' story and I was trying to be NICE. But you threw in that "baby" at the beginning, and that was awfully sweet, so you're forgiven.
But yeah, Jules, he's right.
Yeah, this is why I'm mortified. The Planet of the Apes series I love, the only sci-fi series of movies I can handle. And I somehow subconsciously quoted the MARKY-FREAKIN'-MARK VERSION? WHA???
(having to watch them again with my wonderful husband and a bunch of popcorn is not punishment, by the way) . . .
Jules, here's hoping July 15th is splendid !


http://www.flickr.com/photos/36918746@N00/sets/72157594362534970/

A Flickr user by the name of "fatherfork" managed to snap a few shots of the Zune (okay, it's a non-functional retail display unit, but it's not like it matters in this case) next to the 5G iPod, iPod nano, Zen Vision:M and Sansa e200 series. While it does appear a little on the large side, Engadget assures us that it's "deceptively small and still quite pocketable". Now that's what we like to hear!
Personally, from this picture it doesn't look too bad. However, its hard to tell how thick it is compared to the iPod. Also, what is it going to be like when you put a case on it?
However, its hard to tell how thick it is compared to the iPod. Also, what is it going to be like when you put a case on it?
It will be interesting to see how big of an issue the size continues to be after it is released.
That's the 80GB iPod though, isn't it?
Yeah, the screen is 0.5" bigger than the iPod, but it seems much bigger, or perhaps it just uses the space better. It seems more "intense" to me in terms of a user experience, which is cool.
Yes, it is, so you're right in thinking that it's not exactly a fair comparison size-wise. I believe the Zune is nearly twice as thick as a 30GB iPod.


Walmart's shopping spree continues with the latest purchase of Art.com, a startup founded in 1998, and now the largest online art and wall decor retailer with more than two million curated images. Walmart announced the pending acquisition Thursday, (Dec. 6), but provided no financial details. The acquisition is expected to close early next year.
Art.com has annual sales of roughly $300 million and is similar to the other recent acquisitions by Walmart like Bare Necessities.
The retail giant said this acquisition fits within its strategy to enhance the experience through increased assortment (like Moosejaw and Bare Necessities). Walmart said the addition of Art.com will allow the retailer to augment its assortment and accelerate the growth in categories like home decor.
"This news builds upon the recent Walmart.com site re-design for introductions to new experiences and brands, further positioning Walmart.com as a specialty retail destination for home," said Anthony Soohoo, senior vice president, group general manager for home, at Walmart U.S. eCommerce.
He said in the past 10 months since Walmart revamped its home category website to provide an enhanced customer experience the retailer has learned more about how customers want to shop, and the process they undergo to create a space they love. Soohoo said a big factor in creating that personal space starts with the wall, and that's a $10 billion market in the U.S. for wall art and decor.
Walmart said it will operate Art.com as a standalone and complementary site. Over time, the retailer will add the wide-ranging assortment of art to Walmart.com, Jet.com and Hayneedle.com.
"This will enhance the customer experience with millions of additional choices for art, wall décor, and personalized print-on-demand capabilities while creating a richer, deeper shopping experience across the Home category," Soohoo said.
Walmart saw Art.com as a value given its deep understanding of the art and wall decor category allowing for a more personalized shopping experience overall from affordable posters to limited edition prints and loft art pieces that consumers want to add to their homes.
Art.com has earned a great reputation for its on-demand and customization capabilities with the majority of its inventory being printed on demand at the time of customer ordering. The company also offers custom framing and mounting on canvas which is done on-site by artisans.
Walmart U.S. eCommerce CEO Marc Lore has said Walmart could own as many as 40 digital native brands before its acquisition spree is done. Walmart's digital native brand acquisitions since the big $3 billion purchase of Jet.com in 2016 includes ModCloth, Moosejaw, Shoes.com, Bonobos, Eloquii and Bare Necessities.
The retailer's strategy is aimed at targeting a younger audience that isn't necessarily shopping in its physical stores and are also likely Amazon Prime members.


Norway's 5G pilot will have driverless buses, drones, and real-time medical diagnoses When Telenor's 5G pilot begins near Oslo later this year, it will be putting a string of applications to real-life testing.
Late last year telecoms provider Telenor and the municipality of Kongsberg signed a contract for 5G pilot testing in the town. The choice of location is not random.
Although Kongsberg is an hour's drive from Oslo and small even by Norwegian standards, with just over 27,000 inhabitants, it is home to many of Norway's most innovative high-technology companies.
"The 5G technology can fundamentally change many critical operations for society, like traffic management, health services and important communication services as emergency communications. Therefore, we're very happy to announce this collaboration with the exciting innovation environment here on Kongsberg," said Ove Fredheim from Telenor.
Telenor tells ZDNet that it is in the process of planning deployment of 5G base stations in and around the industrial park where the Kongsberg tech companies are concentrated.
The company will be building 5G coverage between the industrial park and the local railway station, to support a test of an autonomous bus service on the 3km (1.9-mile) stretch of road between the locations.
In this pilot project, Telenor is combining 4G and 5G communications. For Internet of Things applications, 4G may be as applicable as 5G, in terms of separation of traffic into service classes. This technique is referred to as 'network slicing', where logical segments of the network can have different service characteristics, such as bandwidth priority, without affecting one other.
In October 2018, Telenor will enable the IoT technologies NB-IoT (Narrowband IoT) and LTE-M (LTE for Machines) on all their 4G base stations in Norway.
Network slicing is also part of the 5G specification where, for instance, emergency communication can be allocated to a separate slice. This approach ensures such traffic gets through the network even if the ordinary telephony part is congested due to extreme loads after a catastrophe of some sort.
5G also offers shorter response times and higher capacity. Telenor did a 5G demo earlier in 2017 and achieved more than 70Gbps throughput in that test. That capacity is approximately 700 times greater than what 4G can offer. Both speed and low latency will be tested in the Kongsberg pilot test, Telenor tells ZDNet.
Autonomous buses, cars and drones that need to be operated in real time.
Applications within e-health that are in particular need of real-time high bandwidth. An example use case could be remote medical diagnostics.
Networked use of virtual and augmented reality (VR/AR). This application needs high bitrate and low response time to work well enough.
Separate emergency communications with new applications, like video calls. This use case calls for sliced networking to give the communication highest priority in the network, as well as high capacity.
The project is in the planning stage, with Telenor and the current pilot test partners working on which applications and tests to implement during the pilot. More organizations will be welcome to enter later in the project, according to Telenor.
During Telenor's 5G technology demo in March 2017, it said it plans to open its 5G mobile network in Norway "around 2020".
Telenor used the 5G demo to showcase possible applications for this technology, such as autonomous shuttle buses and industrial drones. From left: Telenor CEO and president Sigve Brekke, minister of Transport and Communications Ketil Solvik-Olsen, and Telenor Norway CEO Berit Svendsen.
Qualcomm's future Snapdragon 5G mobile chipsets will be manufactured using Samsung's 7-nanometer Low Power Plus (LPP) extreme ultra violet (EUV) process.
Starting next week, startups and researchers will be able to test out their ideas for new Internet of Things apps on a free pilot service in Norway.


text:
Q: Error in application when trying to convert date I have this method which gets data from the db and sends it to the client. on one stage I am trying to convert the date from Gregorian to Persian date,but for some reason I get this error which I have attached below.
private DataProvider getSupplierOrderData(Long supplierorderid)
{
    SupplierOrder supplierOrder = SupplierOrderBussinessLogic.getInstance().getSupplierOrderById(supplierorderid);
    List<SupplierOrderDetails> list = SupplierOrderDetailBussinessLogic.getInstance().getSupplierOrderDetailsFromsupplierOrder(supplierOrder);

    DataProviderBuilder dpb = new DataProviderBuilder();

    // add heading data
    dpb.add("so", supplierOrder.getSupplierOrderNo());
    dpb.add("sn", supplierOrder.getSupplier().getPerName());
    dpb.add("sec", supplierOrder.getSection().getAlternateName());

// I GET THE ERROR ON THIS BELOW LINE
    String date = CommonUtil.gregorianToJalali(supplierOrder.getSupplierOrderCreated()).toString();

    System.out.println("Document render date      !!!!      "+date);

    dpb.add("od", supplierOrder.getSupplierOrderCreated().toString());

    // add table data
    dpb.addJavaObject(list, "data");

    return dpb.getDataProvider();
}

ERROR I get is
java.lang.ExceptionInInitializerError
      at com.kaizen.erp.client.utill.CommonUtil.<clinit>(CommonUtil.java:17)
      at com.kaizen.erp.server.docmosis.SupplierOrderPrintController.getSupplierOrderData(SupplierOrderPrintController.java:138)
      at com.kaizen.erp.server.docmosis.SupplierOrderPrintController.handleRequest(SupplierOrderPrintController.java:64)
      at org.springframework.web.servlet.mvc.SimpleControllerHandlerAdapter.handle(SimpleControllerHandlerAdapter.java:48)
      at org.springframework.web.servlet.DispatcherServlet.doDispatch(DispatcherServlet.java:859)
      at org.springframework.web.servlet.DispatcherServlet.doService(DispatcherServlet.java:793)
      at org.springframework.web.servlet.FrameworkServlet.processRequest(FrameworkServlet.java:476)
      at org.springframework.web.servlet.FrameworkServlet.doGet(FrameworkServlet.java:431)
      at javax.servlet.http.HttpServlet.service(HttpServlet.java:617)
      at javax.servlet.http.HttpServlet.service(HttpServlet.java:717)
      at org.mortbay.jetty.servlet.ServletHolder.handle(ServletHolder.java:487)
      at org.mortbay.jetty.servlet.ServletHandler$CachedChain.doFilter(ServletHandler.java:1097)
      at com.isomorphic.servlet.CompressionFilter.doFilter(CompressionFilter.java:260)
      at org.mortbay.jetty.HttpParser.parseAvailable(HttpParser.java:211)
      at org.mortbay.jetty.HttpConnection.handle(HttpConnection.java:380)
      at org.mortbay.io.nio.SelectChannelEndPoint.run(SelectChannelEndPoint.java:395)
      at org.mortbay.thread.QueuedThreadPool$PoolThread.run(QueuedThreadPool.java:488)
Caused by: java.lang.UnsupportedOperationException: ERROR: GWT.create() is only usable in client code!  It cannot be called, for example, from server code.  If you are running a unit test, check that your test case extends GWTTestCase and that GWT.create() is not called from within an initializer or constructor.
      at com.google.gwt.core.shared.GWT.create(GWT.java:51)
      at com.google.gwt.core.client.GWT.create(GWT.java:85)
      at com.kaizen.erp.client.Kaizenerpweb.<clinit>(Kaizenerpweb.java:25)
      ... 30 more
[WARN] Nested in java.lang.ExceptionInInitializerError:
java.lang.UnsupportedOperationException: ERROR: GWT.create() is only usable in client code!  It cannot be called, for example, from server code.  If you are running a unit test, check that your test case extends GWTTestCase and that GWT.create() is not called from within an initializer or constructor.
      at com.google.gwt.core.shared.GWT.create(GWT.java:51)
      at com.google.gwt.core.client.GWT.create(GWT.java:85)
      at com.kaizen.erp.client.Kaizenerpweb.<clinit>(Kaizenerpweb.java:25)
      at com.kaizen.erp.client.utill.CommonUtil.<clinit>(CommonUtil.java:17)
      at com.kaizen.erp.server.docmosis.SupplierOrderPrintController.getSupplierOrderData(SupplierOrderPrintController.java:138)
      at com.kaizen.erp.server.docmosis.SupplierOrderPrintController.handleRequest(SupplierOrderPrintController.java:64)
      at org.springframework.web.servlet.mvc.SimpleControllerHandlerAdapter.handle(SimpleControllerHandlerAdapter.java:48)
      at org.springframework.web.servlet.DispatcherServlet.doDispatch(DispatcherServlet.java:859)
      at org.springframework.web.servlet.DispatcherServlet.doService(DispatcherServlet.java:793)
      at org.springframework.web.servlet.FrameworkServlet.processRequest(FrameworkServlet.java:476)
      at org.springframework.web.servlet.FrameworkServlet.doGet(FrameworkServlet.java:431)
      at javax.servlet.http.HttpServlet.service(HttpServlet.java:617)
      at javax.servlet.http.HttpServlet.service(HttpServlet.java:717)
      at org.mortbay.jetty.servlet.ServletHolder.handle(ServletHolder.java:487)

redpajama_set_name: RedPajamaStackExchange


The liver is significant to your pet's well being. Data is energy, so the extra you realize about your pet's health, the higher off your pet can be. Furthermore, it's necessary you know the way to choose an excellent veterinary hospital, earlier than you can take your pet to be handled there.
As you get into the habit of enjoyable in front of your tropical aquarium pet fish after feeding them each day, keep in mind to breathe deeply and to immerse yourself into the sensation of their watery, peaceful aquarium world. Taking care of animals is an expensive business.
Nevertheless, if the animal arrives without well being or vaccination certificates, or is showing signs of dangerous communicable animal disease, it will likely be quarantined. Contemporary meals that's fed to your birds will do them an entire lot of excellent; nevertheless, meals like espresso, chocolate and avocado are deadly for birds and they need to never be given such meals.
Once per week can actually make a difference and SURE Cats will get used to their weekly bath and even take pleasure in them in time. In case you have other animals at home it might be best to introduce your pigmy goat to this group carefully, as a result of they could be harmed by them.
How Do You Deal With Pet Illness?
The liver is outstanding in its means to regenerate itself and so many animals go on to stay normal, healthy lives after experiencing liver failure.
Dog grooming is a profession that's good for the animal lover that has decided to devote their life's work to animals, but does not see themselves working in a veterinarian hospital. Also, when remedy is placed in the food or water it is rather troublesome to determine how a lot of the remedy the pet has actually ingested. He additionally makes sure the canines and cats are always separated so there's no added tension. Many owners don't wish to ask about the prognosis of their pet's illness.
Or you can volunteer time on the native animal shelter as an alternative of adopting a pet. It consists of medicines which might be used to treat serious pet health issues, however they may also consist of simple nutritional supplements that handle the general health of pet animals.
A lot of petlovers care for their pets as though they had been their very own youngsters. Chihuahua dogs have reached the head of recognition as 'Bruiser', as 'Legally Blonde' lady Reese Witherspoon's pet canine. Except the cat refuses to be trained to a litter box, they're very clear pets. You probably have already chosen to think about buying a pet wheelchair and if your dog is apt for it, a number of types of pet wheelchairs are available available in the market.
By setting apart a room specifically for pet homeowners, they permit different lodge guest to be able to spend their stay within the resort in a room free from pet dander, as well as the hassle of running into, or hearing other individuals's pets.
Pets are fun and they have been confirmed to scale back stress. Wanting slightly extra to the longer term, and I mean to the surroundings of all the animals, pets crops and people that can come after we're pushing up the daisies may not be a nasty idea. Every year more than 6 million cats and canines are taken in by animal shelters and more than half of those have to be ultimately euthanized.


text:
Home Blogs Micro-shares and the democratization of horseracing
Micro-shares and the democratization of horseracing
The sport of kings today is accessible to ordinary mortals allowing them to co-own top-quality racing horses with extremely affordable micro-investments. Without losing anything of the charms of being part of the competitive equine universe. Racing clubs are multiplying, and the era of micro-shares is here. The emotion of participating in these noble competitions is also available thanks to horse racing betting sites 2021, but now you can make the experience even more immersive. While syndications have been around for more than half a century, they still require a relatively deep pocket, in the order of several thousand pounds of dollars. Racing clubs, on the other hand, offer shares that can be as low as $50.
Let's take a step back in history, to 1969, when the president of Dogwood Stable, W. Cothran, launched the first public partnership to co-own for a horse named Social Arrest. In the following years, the initiative expanded and recruited 1200 co-owners who invested more than $100 million in pedigreed horses, also dubbed "bloodstock." Among them, there have been more than 70 stakes winners who competed in 10 Triple Crown races. By 2011, they were earning a median purse of $150.000.
The expansion of the syndicate model brought abundant and much-needed fresh blood into horseracing. It also opened up a world that appeared very secretive and arcane to a multitude of very willing investors. In times when attendance of the racecourses was shrinking and a crisis was looming, this injection of new actors meant a new leash of life for the whole sector. A quarter of a century ago, perhaps 20 percent of horses were owned by syndicates. Today the estimate is that they account for no less than 50-60 percent, though precise data is not really available since thoroughbred racing is very decentralized.
Horses syndicates have very varied entry levels. Some allow buying a share for as little as $1,000, though a more serious investment to get at least 10-15 percent ownership of a quality horse will not be below $10,000. A yearling will be even more expensive, in the range of at least $15,000. Still, for champions whose full price runs in the hundreds of thousand dollars, these are great deals. Sometimes the financial return, what with earned prizes and breeding rights, can be extremely rewarding.
Nevertheless, "what you are buying is a lifestyle", say the ones who have joined the movement. The next step, the appearance of micro-shares, was a logical consequence. Today the racing clubs offering this opportunity have thousands of members. And their horses are co-owned by hundreds and thousands of fans. They get a number of perks in return for their membership. The best part is obviously all that creates a real-life experience for members, from visits to the stables to enjoying the races from the box and accessing the winner's circle when their horse wins a race.
Check out what a racing club offers, if you are planning to join one. Each of them has a specific list of the rights acquired, together with a share of ownership, when subscribing. The best ones make the most of the social aspect of crowd-ownership and organize social events to bind co-owners with a team spirit. Cheerleading for your champion at the racecourse is more fun with friends!
Previous articleBroder and Gorin-Byrne Finish WEF 2021 On a High Note in the Bruno Delgrange Palm Beach Adult Medal
Next articleWill the public be allowed back to Royal Ascot?
Marissa Carl
Learn How To Win More Bets in the Upcoming Horse Races...

redpajama_set_name: RedPajamaCommonCrawl


GUILTY is No1 on Amazon!
TOPIC: GUILTY is No1 on Amazon!
In their Pop Culture chart. You too can buy it now.
576 paperback copies shipped out in a day (pre orders) - and, illustrating how few people actually read, a further 11 copies ordered by Amazon for stock. So they don't expect to sell more than 600 in total. Fascinating. We sold 500 copies of the hardback 65 My Life So Far (the total print run) in a week but that was at £20 a copy. Still haven't reprinted that in paperback (it's available in E Book/Kindle). Wonder whether we should.
oh really? I rather like Kindle and e-books and I'm awaiting the arrival of my new Samsung tablet (so annoying in Australia- stores never order enough) and my first treat in honor of JK's innocence will be Guilty.
I love the way you can carry 50 books on your tablet which makes waiting in doctor's waiting rooms (something I spend a lot of time doing now) almost a treat and you never have to bother with all those out-of-date magazines.


text:
James "Jake" Warren Foster
June 30, 1936 - April 23, 2021
James "Jake" Warren Foster, 84, of Albemarle, passed away Friday, April 23, 2021 at Atrium Health Stanly. James will have a visitation on Thursday, April 29, 2021 from 2pm until 4pm at Stanly Funeral and Cremation Care of Albemarle. Mr. Foster was born June 30, 1936 in Stanly County, NC son of the late John Mansfield Foster and Effie Jane Harward Foster. He retired from the United States Air Force and from the United States Postal Service. He is survived by a son, Michael Foster and wife Angela of Spokane WA; a daughter, Kathy Foster of Albemarle, NC; two grandchildren, Clinton Foster and Danielle Foster both of Spokane, WA and two great-grandchildren, Colby Foster and Aaron Foster. Stanly Funeral and Cremation Care of Albemarle is serving the Foster family.
James "Jake" Warren Foster, 84, of Albemarle, passed away Friday, April 23, 2021 at Atrium Health Stanly. James will have a visitation on Thursday, April 29, 2021 from 2pm until 4pm at Stanly Funeral and Cremation Care of Albemarle. Mr.... View Obituary & Service Information
The family of James "Jake" Warren Foster created this Life Tributes page to make it easy to share your memories.
James "Jake" Warren Foster, 84, of Albemarle, passed away Friday,...
Send flowers to the Foster family.


text:
is a manga series written by Leiji Matsumoto and illustrated by Kōichi Shimahoshi. It was the final manga by Leiji Matsumoto before his death in February 2023.

The series was launched in August 2014 in the pages of Akita Shoten's Champion Red magazine, to celebrate the 60th anniversary of Matsumoto's debut.

The series is a retelling of the original 1978 Space Pirate Captain Harlock manga, with some significant plot differences, and illustrated by Shimahoshi in a modernized version of Matsumoto's classic style. In January 2017, it was licensed in the U.S. by Seven Seas Entertainment. It's also been published in Spain by Norma Editorial, and in France by Kana.

Plot
In the future, Earth is a decadent planet. With everything provided by the state, the Earth Federation, nobody works and a few rich people control everything. The willful people had left the planet, leaving behind only the decadent and the ones who don't have money to leave. The Gaia Fleet, composed by Earth's best warriors, had not been seen in many years, and is now dedicated to protect humankind's new home planet, Big Earth. When the threat of the Mazon looms over Earth, only one man can protect it: Captain Harlock. While the government sees Harlock as an outlaw, others see him as Earth's last hope.

About a year before the beginning of the story, a strange spherical object arrived from space. Dr. Daiba has repeatedly tried to warn Earth's Prime Minister of an incoming threat, but the government is more interested in riches and pleasure than in protecting the planet. After he's killed by the Mazon, his son Tadashi becomes frustrated by the government's flippant attitude, landing himself in prison. Tadashi is invited aboard the battleship Arcadia, under the command of the legendary Captain Harlock. Tadashi joins its crew in their quest to protect Earth against the Mazon.

Characters

The crew of the Arcadia
 Captain Harlock: Captain of the Arcadia and former officer of the Gaia Fleet. He owns one of the legendary Cosmo Dragoon pistols.
 Tadashi Daiba: Dr. Daiba's son who joins the Arcadia's crew to fight the Mazon. He later goes back to Earth to investigate and joins Mayu and Zero.
 Yattaran: Arcadia's second-in-command.
 Dr. Zero: Arcadia's Chief Medical Officer. He has a cat named Mii, whom he has trained to steal sake from Masu's kitchen.
 Masu: Arcadia's kitchen steward. She wields a really big knife and has zero tolerance for Mii's raids on her kitchen.
 Maji: Arcadia's Chief Engineer.
 Yuki: A young woman who's a bridge officer on the Arcadia.
 Miime: An alien woman who describes herself as "a woman who has given everything to Harlock" She's also the last of the Nibelungs.
 Zero: A young man who loathes both humankind and the Machine People, but who would give up his life to protect his chosen family, the Arcadia and its crew. That's why he goes to Earth to protect Mayu. They later rescue Tadashi and join him.

Harlock's allies
 Emeraldas: Captain of the Queen Emeraldas. A wanted outlaw like Harlock, she possesses one of the Cosmo Dragoon pistols. She's also Mayu's mother.
 Tochirô Ooyama: Harlock's friend and a genius engineer who reconverted a Yukikaze-kai type ship into the powerful Arcadia and gave it to Harlock to protect Earth while he took care of his own enemy, the Machine Empire. He also build Deathshadow Island, a hollowed out asteroid that serves and Arcadia's base, following the ship at a distance, as well as the Cosmo Dragoon pistols.
 Mayu: The 14 years-old daughter of Tochirô and Emeraldas, who lives in an orphanage in Earth. The mechanized police tries to arrest her, but she's saved by Zero.
 Tetsuro Hoshino: Owner of another of the Cosmo Dragoon pistols.

Earth
Dr. Daiba: A scientist who tried to warn Earth about the Mazon threat before being killed by them.
Isora Kirita: Vicecommander of the Gaia Fleet. Kirita hates Harlock, whom he considers a coward. One of the few willful men left on Earth, he's decided to defend Earth at any cost.
 Prime Minister: Prime Minister of Earth Federation. His only interests are in women, golf and fame, and believes that divine providence would protect Earth.
 Commander Tôdô: Commander of the Gaia Fleet on Big Earth and Kirita's superior, who believes one day Kirita and Harlock's ideals will be the same.

Mazon
Raflesia: Queen of the Mazon. She considers Harlock the only threat of their plans to conquer Earth.
Paphio aka Shizuka Namino: A Mazon who masquerades as the Prime Minister's secretary.
Jogibel: Captain of the Mazon's special fleet against Harlock.

Other
 Faust: President of the Galactic Railway company. Under his true name, Gôzô Hoshino, he was an officer of the Gaia Fleet until he resigned. Also known as the Black Warrior.
 Bolgazanda: President of Earth's autonomous region of Megalopolis. He plans to run in the next elections to Prime Minister opposing the incumbent.

References

Captain Harlock
2014 manga
Adventure anime and manga
Akita Shoten manga
Leiji Matsumoto
Seven Seas Entertainment titles
Seinen manga
Space opera anime and manga

redpajama_set_name: RedPajamaWikipedia


text:
Welcome from the Head
Our Founder and History
Why Aiglon?
Meet Aiglon Worldwide
Visiting Admissions
Senior Curriculum
Junior Curriculum
Learning Enrichment
University Advising
Discovery Programme
Aiglon Expeditions
Sport Programmes
Activities Programme
Ski Programmes
Boarding & Houses
SAT Courses
Facility Lettings
Dates & Downloads
Aiglon Magazine
The Aiglon Chapel in a new light
Posted on 9th September 2022 in Spirit, Alumni
Now that the first renovation phase of the Aiglon College Chapel is complete, discover how fresh stonework, re-wiring, and innovative lighting have rejuvenated this valued community space.
White as snow, our unassuming chapel, situated next to the Villars Palace, harmoniously blends with its surroundings in winter. Inside, however, following recent optimisations this summer, the small space now holds even greater year-round appeal, and, with further improvements in the pipeline, promises for a an even brighter future.
First established in 1883 by the Intercontinental Church Society, the English chapel came into the hands of Aiglon College just over a century later in 1996. Fascinatingly though, the pasts of Aiglon College and the chapel have long been intertwined. The chapel also commemorates British skiers lost during the WWII, including the son of Alpine College's founder. Alpine College is where John Corlette - founder of Aiglon College – was educated.
Aiglon alumni have preached here. Students and parents have attended services and ceremonies here. College staff families have celebrated marriages and christenings here. Its four beautiful stained-glass windows were conceived by our teachers and funded by parents of Aiglon alumni.
Indeed, when the sun shines through these windows it casts a kaleidoscope of colours across the pews and, framed by the new decorative stone wall, creates a magical atmosphere. Adding to this, the recently installed light features bring a touch of modernity to our Chapel, which is of course available to the broader school family for a variety of functions throughout the year.
While the Chapel cannot accommodate whole school events, the next phase of renovations will further increase the Chapel's adaptability and potential. Plans for the construction of a partial second floor with a large meeting room will allow for increased meeting possibilities. Guests will be better catered for thanks to the installation of a bar, and, vitally, flexible seating in the main nave area will open up the Chapel for more unconventional events such as cinema projections and exhibitions. Consequently, these advances will enable extended engagement with our village.
We are delighted that you are considering Aiglon College. We are always pleased to introduce prospective students and their families to life at the school, and we welcome all enquiries.
School Reception
Avenue Centrale 61
1885 Chesières
Les Collonges 4
Media/Press Enquiries
Online School Shop


text:
Wisconsin governor to veto anti-infanticide bill that's 'redundant' and 'not a productive use of time'
Gov. Tony Evers says the state 'has all sorts of issues to deal with' and protecting the unborn from failed abortion attempts is not one of them.
Wed Apr 24, 2019 - 7:20 pm EST
Wisconsin Gov. Tony Evers
By Martin M. Barillas
Follow Martin
MADISON, Wisconsin, April 24, 2019 (LifeSiteNews) — Right to Life of Wisconsin deplored Democrat Gov. Tony Evers' vow to veto legislation that would save babies who survive abortion.
"Thankfully, Wisconsin did pass a bill to protect unborn babies born alive from failed abortion attempts that was signed into law" in 2013, said Heather Weininger, executive director of Wisconsin Right to Life.
Weininger encouraged Evers to study how Wisconsin's current law differs from the proposed bill that she said would assist health care providers in understanding the degree of care they must offer to babies who survive abortion.
The bill would also call on medical professionals to notify law enforcement officials in cases where survivors of abortion had not received proper care.
"When a child is born prematurely via natural labor, induced labor or via cesarean section, doctors have a clear understanding of the steps that must be taken to ensure they are giving the child every opportunity at life," Weininger said. "Unfortunately, that isn't the case when a baby is born alive during a failed abortion."
The proposed "Born Alive Protection Act" (AB 79) requires abortion doctors to "exercise the same degree of professional skill, care, and diligence to preserve the life and health of the child as a reasonably diligent and conscientious health care provider would render to any other child born alive at the same gestational age."
According to the legislation, doctors also must ensure that a survivor of abortion is immediately taken and admitted to a hospital. Providers who do not comply could be found guilty of a felony and face as many as six years in prison and up to a $10,000 fine. In February, a similar bill failed in the U.S. Senate.
According to the Milwaukee Journal Sentinel, Evers said he would veto the bill.
"I think those protections already exist," he said. "We have all sorts of issues to deal with in the state of Wisconsin and to pass a bill that is redundant seems to be not a productive use of time. And clearly I ran on the belief — and I still believe — that women should be able to make choices about their health care. But this deals with a specific issue that's already been resolved."
The bill currently resides in the House health committee and must be reported out before being considered by the legislature as a whole.
Wisconsin state Senate President Roger Roth, a Republican co-author of the bill, said when he introduced the legislation, "I am introducing the Born Alive Protection Act because, unfortunately, some people including the Governor of Virginia, are unclear on this, so we need to reaffirm these protections."
"We are providing explicit outlines to health care practitioners that babies who survive an abortion need to be given the same, equal rights as any other person," he added.
The Milwaukee Journal Sentinel quoted retired professor Donald Downs, who said that once a baby is born and outside the womb, the state has an interest in providing protections accorded to all citizens.
"If indeed this is redundant, then there's no need for it, but I don't know what the previous protection is," he said. "The law protects you when you're born — you're a person."
In Evers' gubernatorial campaign, he pledged to veto legislation that places restrictions on abortions in Wisconsin. In a 2018 tweet, he wrote, "Wisconsin is 1 of 4 states that has a criminal abortion ban on the books. Government shouldn't be making personal health decisions for women and we shouldn't be treating physicians like criminals. Women's lives will be jeopardized. We must remove this ban from our statutes."
Nearly 80% of voters support protecting babies born alive during an abortion procedure. Yet, Tony Evers will veto 'born alive' abortion bill advanced by GOP lawmakers. https://t.co/b0t6guglin via @journalsentinel
— Rep. Barbara Dittrich (@RepDittrich) April 23, 2019
When the bill was introduced, Republican co-authors Sen. Patrick Testin and Rep. Barbara Dittrich said in a memo seeking co-sponsors, "No unborn child should be deprived of life due to any quality that they have inherited during any point during their development." The memo went on to say, "This legislation corrects that and adds legal protections for the unborn against a selective abortion stemming from discrimination."
abortion, born-alive protection act, right to life of wisconsin, tony evers, veto, wisconsin house of representatives
Wisconsin governor to veto anti-infanticide bill that's 'redundant'… Wisconsin governor to veto anti-infanticide bill that's 'redundant' and 'not a productive use of time' News By Martin M. Barillas


Since making its debut on the Fall '16 runway, Prada's 'Cahier' bag has been updated in various new sizes and color-ways. This two-tone version was expertly crafted from smooth leather in a structured silhouette inspired by antique books with a sectioned interior to keep you organized. Adjust the strap to find your desired drop.


Pat is currently responsible for logistics and administration. This includes dealing with incoming orders, ordering supplies and dealing with general enquiries.
Graham is responsible for project management, design and development of low-power solutions, vehicle telemetric and microcontroller PIC firmware.
Graham is an experienced engineer with over 30 years experience of computer based systems from inception through specification, tendering, design, documentation and test to on-site commissioning and final system hand over. Graham graduated from Salford University, in 1975, with a B.Sc. Honours degree in Electrical Engineering.
David is responsible for project management, hardware design and development, FPGA design.
David is a design engineer with over 30 years experience in the concept development, system design, hardware design and test of industrial products and systems. David graduated from Salford University in 1984, with a B.Sc. Honours degree in Electrical and Electronic Engineering. He is a chartered member of the Institute of Electrical Engineers M.I.E.T.
Helen is responsible for project management of Software projects, software and FPGA design and development.
Helen is a design / software engineer with over 25 years experience in both hardware and software engineering disciplines, system design and test of industrial systems. Helen graduated from Manchester University, in 1987, with a first class B.Sc. Honours degree in Electrical and Electronic Engineering. She is a chartered member of the Institute of Electrical Engineers M.I.E.T.
Ian is responsible for Windows and Linux driver design and development.
Ian is an experienced software engineer with over 30 years experience of software engineering specialising in low-level interfaces, protocol handlers and device drivers for Linux and Windows. Ian helps maintain the Comedi Linux control and measurement device interface. Ian graduated from Manchester University, in 1984, with a B.Sc. Honours degree in Computer Science.
Keith is responsible for production test software, commissioning software, microprocessor firmware and back office desktop solutions.
Keith is a software engineer with over 20 years experience in software development. After leaving British Army (R.E.M.E.) as a Telecommunications Technician Class 2, he first worked in the games software industry before joining MEV.
Nick joined MEV shortly after we formed and has become a key member of the team responsible for in electronic PCB capture, testing and documentation tasks.
Connor joined MEV as an apprentice in 2016 and has quickly became part of the team in electronic PCB capture, hardware commissioning and testing.
Dave has joined MEV where he continues to provide consultancy on secure solutions with partner organisations. Already at MEV he has been involved in GDPR considerations for a vehicle tracking system, interfacing to a USB Ultrasonic Testing solution and performance enhancements to a customer SQL database.
Dave was the Architect for HP's ProtectTools Products for many years and he is recognised as the worldwide expert on this range of solutions. He also provides consultancy on security standards and requirements, particularly those of governments.


For our magazine, he reached for a family recipe—one his grandmother and mother traditionally serve during the holidays. "I've changed it up by using different vinegars," he said, "which makes the greens a lot brighter due to the acidity of the vinegar. It has a little different feel—not quite so heavy." He likes serving it with bacon Cheddar cheese grits and smoked pork chops with ham hock maple glaze sauce.
METHOD: Combine all ingredients into a large stock pot. Simmer (don't boil) over medium heat till greens are tender, 2 to 2 1/2 hours. Drain before serving.