text:
\section{Introduction}
\label{intro}

A number of problems in machine learning, data mining and signal
processing require that one find a partition of a graph with small
boundary weight, but whose subsets have relatively large interior
weights. These graphs may directly represent the given data (as in the
case of social networks) or be derived from vector data via a
heuristic such as $k$--nearest neighbors. In either case, these graphs
often have millions of vertices, making a brute force search for
efficient partitions infeasible.

In the present paper, we introduce an approach we call \emph{Thin
Tree Position} (TTP) for finding efficient partitions of graphs by
carrying out a more targeted search, using intuition from
three dimensional geometry/topology. This approach builds on the
TILO/PRC (Topological Intrinsic Lexicographic Order / Pinch Ratio
Clustering)  algorithm introduced in~\cite{thingraphs}
and~\cite{sdm13prc}, but improves on a number of deficiencies of this
approach.
The TILO/PRC approach consists of two steps: First, the TILO algorithm
assigns a linear ordering to the vertices of the graph $G$ and then
progressively improves this ordering with respect to a carefully
chosen metric.  Next, the PRC algorithm picks out consecutive blocks
in the final TILO ordering that make up the partition.

Both TILO and TTP are algorithms that use ideas from low dimensional
topology, specifically the concept of \emph{thin position} of knots
and $3$--manifolds~\cite{Gabai,st:thin}.  As such, these methods rely
on the broad shape of the data set instead of its exact geometry.
Carlsson has argued that algorithms of this type should perform well
for certain problems and can give more information than existing
methods~\cite{Carlsson09}.  We expect that TTP will be useful
for clustering as well as other applications in topological data
analysis.

The present approach addresses two major issues with TILO/PRC: First,
the linear nature of TILO/PRC forces it to treat the subsets of the
partition at the front and back of the TILO  ordering differently from
the subsets in the middle. As we describe in Section~\ref{sect:tpts},
the TTP algorithm replaces the linear ordering with a trivalent tree
structure that allows all the subsets to be treated consistently. As
we discuss in Section~\ref{sect:compare}, this structure is a natural
generalization of TILO orderings.

Second, the gradient-like TILO step of the TILO/PRC is relatively
constrained. In order to make the search space reasonable, TILO only
looks for ways to shift a single vertex at a time forward or backward
in the ordering. Because TPT uses a tree structure, the equivalent
step in TPT is able to move entire branches, which would be equivalent
to allowing TILO to shift arbitrary subsets of consecutive vertices.
However, as we describe in Section~\ref{sect:reduction}, this tree
structure has a very simple reduction criterion analogous to that of
TILO/PRC.

\section{Thin position trees}
\label{sect:tpts}

Consider an undirected graph $G = (V,E)$ in which every edge $e \in E$
has two distinct vertices and every edge is assigned a positive real
number $w(e)$ called its \emph{edge weight}. In many cases, each edge
weight will simply be 1. The weight of a subset of edges $F \subset E$
will be the sum of the weights of those edges, $w(F) = \sum_{f \in F}
w(f)$. An edge is also specified by the two endpoints $f = (v_i, v_j)$
for $f \in E, v_i, v_j \in V$.  A short hand of the weight function is
used for vertices
\begin{gather*}
w(v_i,v_j) =
\begin{cases} w(\ (v_i,v_j)\ )\ \text{ if } (v_i,v_j) \in E\\
0 \text{ if } (v_i,v_j) \notin E\\
\end{cases}
\end{gather*}
and for sets $A \subset V, B \subset V$,
\begin{gather*}
w(A, B) = \sum_{v_i \in A, v_j \in B} w(v_i,v_j).
\end{gather*}

\begin{definition}
A \emph{$3$--Cayley tree\xspace} \cite{CayleyTrees2012} is  a connected acyclic graph in
which every non leaf vertex has  degree of exactly three.
We call the leaf vertices \emph{boundary vertices} and the
degree three vertices \emph{interior vertices}.
\end{definition}
A $3$--Cayley tree\xspace is also know as a \emph{boron} tree \cite{BoronTrees2013} in which the
interior vertices correspond to boron atoms and the boundary vertices
correspond to hydrogen atoms.
The number of interior vertices of a $3$--Cayley tree\xspace is always two less
than the number of boundary vertices; this can be proved, for example,
using an inductive argument, or from the fact that the Euler
characteristic of a tree is one.

We will use $3$--Cayley trees\xspace\ to study general graphs as follows:
\begin{definition}
Given a finite graph $G$, possibly with weighted edges, a \emph{tree
position} for $G$ is a pair $(C, \ensuremath{\phi})$ where $C$ is a $3$--Cayley tree\xspace
and $\ensuremath{\phi}$ is a one to one map from the vertices of $G$ onto the
boundary vertices of $C$.
\end{definition}

Figure~\ref{fig:treeposition} presents example tree positions
$(C,\ensuremath{\phi})$ and $(C',\ensuremath{\phi}')$ for the graph at the top of the
figure.  The outlined circles represent interior vertices, while the
filled in circles are boundary vertices of a tree position.  In this
figure, the relative positions of the vertices of the graph are
maintained in displaying the boundary vertices of the tree positions.

\begin{figure}[!tb]
  \begin{center}
\includegraphics{tpt-1.pdf}
\vspace*{4ex}

\includegraphics{tpt-2.pdf}
\hfill
\includegraphics{tpt-3.pdf}
\vspace*{2ex}
\caption{Graph $G$ with tree positions $(C,\ensuremath{\phi})$ and $(C',\ensuremath{\phi}')$.}
\label{fig:treeposition}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}

\includegraphics{tpt-4.pdf}
\hfill
\includegraphics{tpt-5.pdf}
\vspace*{2ex}
\caption{Edges of $G$ drawn through tree positions $(C,\ensuremath{\phi})$ and $(C',\ensuremath{\phi}')$.}
\end{center}
\label{fig:topomap}
\end{figure}

Given a tree position $(C, \ensuremath{\phi})$ for a graph $G$, let $v,w$ be
vertices of $G$ spanned by an edge $e$. Because $C$ is a tree, there
is a unique simple path in $C$ from $\ensuremath{\phi}(v)$ to $\ensuremath{\phi}(w)$. We
say that the edge $e$ \emph{passes through} each of these edges of
$C$. In other words, if $f$ is an edge of $C$ then for any topological
map from the realization of $G$ to the realization of $C$, as in the
picture of Figure~\ref{fig:topomap}, the image of $e$
would be forced to intersect $f$.   For an edge $f$ of $C$, let $R(f)$
be the set of edges of $G$ that pass through $f$.

\begin{definition}
Given a subtree $T$ of a tree position $(C, \ensuremath{\phi})$ of $G=(V,E)$, the
set of vertices of $V$ that are the image of the boundary vertices in
$T$ is $\posfInv{T} = \left\{ v \in V\ |\ \posf{v} \in T\right\}$.
\end{definition}
\begin{definition}
Let $f$ and $g$ be two distinct edges of a tree position $(C, \ensuremath{\phi})$.
Two subtrees of $C$ are defined by cutting the edge $f$.  The subtree
containing $g$ is denoted \cutT{f}{g}, and the subtree not containing $g$ is
denoted \cutTopp{f}{g}.
\end{definition}
Note that one endpoint of $f$ is in \cutT{f}{g} and the other is in
\cutTopp{f}{g}.  Thus cutting an edge $f$ of a tree position $(C,
\ensuremath{\phi})$ divides the vertices of $G$ into two disjoint sets:
\posfInv{\cutT{f}{g}} and \posfInv{\cutTopp{f}{g}}.
\begin{definition}
Given a tree position $(C, \ensuremath{\phi})$ for a graph $G$, the
\emph{width} $b(f)$ of an edge $f$ in $C$ is the sum of the weights of
the edges of $G$ that pass through $f$:
\begin{align*}
b(f) = w\left(R(f)\right) = w\left( \posfInv{\cutT{f}{g}} , \posfInv{\cutTopp{f}{g}}\right)
\end{align*}
or for some edge $g$ of $C$ with $g \neq f$.
\end{definition}

Note that we indicate the width of $f$ by $b(f)$ rather than $w(f)$
because it is equal to the area of the boundary of the subset of $G$
defined by the boundary vertices in either of the subtrees of $C$
defined by removing $f$. This notation is also in line with the notation
in~\cite{thingraphs}.
A
notion of width for a tree position, analogous to the width of a TILO
ordering, is created by placing the widths of all of the tree edges into a
multiset and sorting in nonincreasing order. Tree widths can then be
compared by a lexicographic ordering of the widths.
In Figure~\ref{fig:treeposition} the edges of graph $G$ have weight one and the edges
of the tree positions are denoted with their width.
The width of $(C,\ensuremath{\phi})$ is (3,3,3,3,2,2,2) and
the width of $(C',\ensuremath{\phi}')$ is (3,3,3,2,2,2,2).  Since the width of $(C',\ensuremath{\phi}')$ is less than
the width of $(C,\ensuremath{\phi})$, we say that tree position $(C',\ensuremath{\phi}')$ is \emph{thinner} than tree position
$(C,\ensuremath{\phi})$.
As with
linear TILO, we would like to find tree positions whose widths are as
low as possible. To do this, a \emph{branch shift} operation that is analogous to the
TILO shift is defined. A branch shift is described by two equivalent
operations: first an edit operation of removing and
reattaching a vertex (see Figure \ref{fig:bsEdit}),
and second a cyclic shifting of edges (see Figure \ref{fig:bsCycle}).

\begin{figure}[!t]
\begin{center}
\includegraphics{tpt-6.pdf}

\vspace*{2ex}

\includegraphics{tpt-7.pdf}

\vspace*{2ex}

\includegraphics{tpt-8.pdf}
\end{center}

\caption{Branch shift of $x$ to $y$ as an edit operation}
\label{fig:bsEdit}
\end{figure}

Given a $3$--Cayley tree\xspace $C$, let $x$ and $y$ be edges of $C$ that do
not share a common endpoint.  Since $C$ is a tree, there is a unique
simple path from $x$ to $y$.  Let vertex $u$ be the endpoint of $x$
that is shared with the first edge of the path, $e$.    Let edge $f$ be
other edge sharing vertex $u$.  Removing vertex $u$ from
$C$ is the operation of disconnecting edge $e$ from vertex $v$ (the
other endpoint of $e$), disconnecting edge $f$ from vertex $u$, and
attaching the free end of $f$ to vertex $v$.
Let vertex $s$ be the common endpoint of $y$ and the last edge of the
path.  Let edge $g$ be adjacent to $y$ and the last edge of the path.  Adding
vertex $u$ to $C$ is the operation of disconnecting edge $g$ from from
vertex $s$, attaching the free end of edge $e$ to vertex $s$, and attaching
the free end of edge $g$ to vertex $u$. See Figure \ref{fig:bsEdit}
for an example in which the original tree position is at the top, the middle is the tree position
after modifying the edges but not moving any vertices in the display, and the bottom is the same
tree position but with vertex $u$ moved in the display.

An alternative view of a branch shift operation is that of a cyclic shift
of the edges adjacent to the path from $x$ to $y$.  In this view, $x$,
$y$, and all of the non-path edges sharing a vertex in the interior of the
path are disconnected.  Edge $x$ is reattached at edge $y$'s old
location.  The rest of the edges are reattached one vertex over in the
direction of $x$.  See Figure \ref{fig:bsCycle} for an example.

\begin{figure}[!t]
\begin{center}
\includegraphics{tpt-9.pdf}

\vspace*{5ex}

\includegraphics{tpt-10.pdf}

\end{center}

\caption{Branch shift of $x$ to $y$ as a cyclic shift of branches}
\label{fig:bsCycle}
\end{figure}

The difference between viewing the branch shift as a cyclic shift of
edges and the edit operation is just a relabelling of edges and
internal nodes.  The cyclic shift of edges will be used when making a
connection to shift operations on TILO linear orderings.  A typical
software implementation will use the edit operation.  Applying the
branch shift move on $C$ creates the new tree $C'$ and is described as
\emph{shifting $x$ to $y$}.  Note that shifting $y$ to $x$ produces a
different tree than shifting $x$ to $y$.

If we perform a branch shift on a $3$--Cayley tree\xspace $C$, the boundary vertices of $C$
are not touched. There is thus a canonical map from the boundary
vertices of $C$ to the boundary vertices of $C'$. If $(C, \ensuremath{\phi})$ is a
tree position for a graph $G$ then composing $\ensuremath{\phi}$ with this
canonical map defines a one to one map $\phi'$ from the vertices of
$G$ to the boundary vertices of $C'$, which in turn defines a new tree
position for $G$.  In this paper, $C$ and $C'$ will share the same set
of boundary vertices and the canonical map is just the identity map,
\ensuremath{\phi} = $\ensuremath{\phi}'$.

\begin{definition}
The tree position $(C', \phi)$ is a \emph{branch shift} of $(C, \phi)$
when $C'$ is a branch shift of $C$.
\end{definition}

The new tree position defined by a branch shift has its own width,
which may be higher or lower than the original.

\begin{definition}
A tree position $(C, \ensuremath{\phi})$ is \emph{weakly reducible} if
there is a branch shift that produces a new tree position with
strictly lower width. Otherwise,  $(C, \ensuremath{\phi})$ is
\emph{strongly irreducible} or a \emph{thin tree position}.
\end{definition}

\section{Comparing TILO tree position orderings to TILO linear orderings}
\label{sect:compare}

Initially, the idea of tree position may appear to be completely
different from the linear orderings used in the original TILO
algorithm. However, as we will see in this section, it is in fact a
very natural generalization.

The TILO algorithm as described in~\cite{thingraphs}
begins with a linear ordering of the vertex set $V$,
i.e.\ a one to one function from $V$ onto the set $[0,N-1] =
\{0,1,\ldots,N-1\}$, where $N$ is the number of vertices in $V$. We
will call this function a \emph{TILO ordering}. A TILO ordering of $V$
defines a sequence of subsets $A_i = \{v \in V\ |\ o(v) \leq i\}$.
This, in turn, defines a sequence of boundary widths $b_i =
w(A_i,\overline{A}_i)$ where $\overline{A}_i = \{ v \in V | v \notin
A_i\}$ is the complement of $A_i$.
Given a TILO ordering $o : V \rightarrow [0,N-1]$, we say that a
second TILO ordering $o'$ is the result of a \emph{shift} on $o$ from
$a$ to $b$ if $o'$ of the composition of $o$ with a cyclic permutation
of the block $[a,b]$ of consecutive numbers in $[0,N-1]$ that sends
$a$ to position $b$. For example, for $N = 8$, a shift of $2$ to $5$
takes the sequence $0,1,2,3,4,5,6,7$ to $0,1,3,4,5,2,6,7$. The shift
from $5$ to $2$ takes the initial sequence to $0,1,5,2,3,4,6,7$.
The width of a TILO ordering is the multiset of widths $b_i$ sorted in
nonincreasing order.  Lexicographic order is used to compare widths.

\begin{definition}
\label{def:stronglyirred}
A TILO linear ordering $o$ is called \emph{weakly reducible} if it satisfies
the reduction criteria in Lemma~\ref{lem:TILOreduction}.
A TILO linear ordering is called \emph{strongly
irreducible} if it is not weakly reducible.
\end{definition}

By definition, a weakly reducible ordering admits a shift that decreases
its width. The TILO algorithm performs the series of shifts determined by
Lemma~\ref{lem:TILOreduction} that reduce the width until it finds a strongly
irreducible ordering.  (Because there are a finite number of orderings, at
least one must be strongly irreducible.)
To present Lemma~\ref{lem:TILOreduction}, we first need to define
slope $s_{i,k}$ with respect to order $o$ is  as the sum of the edges from vertex
$v_{o(k)}$ to other vertices in $\overline{A}_i$ minus the sum of edges from
$v_{o(k)}$ to other vertices in $A_i$:
\begin{gather} \label{eqn:slopeTILO}
s_{i,k} =
w\left(v_{o(k)}, \overline{A}_i \setminus \{v_{o(k)}\}\right) -
w\left(v_{o(k)}, A_i \setminus \{v_{o(k)}\}\right)
\end{gather}
where $\setminus$ means set difference.  The adjacency $a_{i,j}$ with
respect to order $o$ is the edge weight between the $i$-th and $j$-th
vertices of order $o$: $a_{i,j} = w( v_{o(i)}, v_{o(j)})$.  The
following lemma is a slight rewording of \cite[Lemma 3]{thingraphs}
for the case of shifting a vertex earlier in a ordering with a vertex
later in the ordering.
\begin{lemma}
\label{lem:TILOreduction}
A TILO linear ordering $o$ is weakly reducible if for some $i$ and $k$
such that $k < i$ the following conditions hold:
\begin{gather*}
s_{i,k} > 0, \\
b_{t-1} \leq b_{t}\ \text{ for all }\ k < t \leq i, \text{ and}\\
s_{i,k} - s_{i,i+1} - 2 a_{k,i+1} > 0.
\end{gather*}
In this case, shifting $k$ to $i$ reduces the width of $o$.
\end{lemma}
The analogous result for when $k > i$ also holds but is not needed for
this paper.

To make a connection between a linear ordering and a path view of a
tree ordering, the following structures are defined.  Recall that a
\emph{partition} $\mathcal{P}$ of a set $A$ is a collection of
pairwise disjoint subsets $\mathcal{P} = \{P_0,\ldots,P_k\}$ of $A$
whose union is $A$. In other words, each element of $A$ is in exactly
one subset $P_j$.

\begin{definition} \label{def:quotientgraph}
Given a graph $G=(V,E)$ and a partition $\mathcal{P}$ of $V$,
the \textit{quotient graph} $Q = Q(G, \mathcal{P})$ is a weighted
graph whose vertices correspond to elements $P_j$ of $\mathcal{P}$ and
whose edges are defined as follows:
if $v_s \in P_i$, $v_t \in P_j$, $i \neq j$, and $(s,t) \in E$
then there is an edge $(i,j)$ in $Q$
with weight $w(P_i,P_j)$.
\end{definition}

Note that the quotient graph only accounts for edges that go between
different sets in the partition and ignores edges between vertices
that are in the same subset $P_j$.  The quotient graph corresponds to
a coarsening of the original graph in multilevel graph
algorithms~\cite{multilevelGraph,karypis1998parallel}.

A tree position $(C, \phi)$ for a graph $G$ can be used to define a
number of different partitions of $G$. This paper is interested in a
partitioning related to the branch shift operation with the following
definition.

\begin{definition} \label{def:bsqg}
Given a tree position $(C, \ensuremath{\phi})$ for a graph $G$ and any pair of nonadjacent
edges $x$ and $y$, we can construct a partition of the vertices of $G$ using
the following approach.  Denote the edges in the unique simple path from $x$ to $y$
as $p_t$ with $t=1$ for the edge adjacent to $x$ and $t=i$ for the
edge adjacent to $y$.  Label the edges sharing a common vertex with
adjacent path edge pairs as follows: $p_0$ for pair $x$ and $p_1$, $p_{i+1}$
for pair $y$ and $p_i$, and $f_t$ for pair $p_{t-1}$ and $p_{t}$ with
$2 \leq t \leq i$.  See Figure \ref{fig:bsqg} for an example.  Define the sets $P_t$ as
\begin{align*}
P_0 &= \posfInv{\cutTopp{p_0}{y}},\\
P_1 &= \posfInv{\cutTopp{x}{y}},\\
P_t &= \posfInv{\cutTopp{f_t}{y}}\ \text{ for } 2 \leq t \leq i,\\
P_{i+1} &= \posfInv{\cutTopp{y}{x}},\\
P_{i+2} &= \posfInv{\cutTopp{p_{i+1}}{y}}.
\end{align*}
We will call the quotient graph created using the partition
$\mathcal{P}=\set{P_0,\dots,P_{i+2}}$ the \emph{branch shift quotient
  graph} of $G$ induced from $x$, $y$, and $(C,\ensuremath{\phi})$.
\end{definition}

\begin{figure}[!t]
\includegraphics{tpt-11.pdf}

\caption{Before cyclic branch shift of $x$ to $y$}
\label{fig:bsqg}
\vspace*{5ex}
\includegraphics{tpt-12.pdf}
\caption{After cyclic branch shift of $x$ to $y$}
\label{fig:shifted}
\end{figure}

The partitions $P_j$ of a branch shift quotient graph are
disjoint. The only vertices not in the union of subtrees cut from $C$
are the vertices along the path from $x$ to $y$, which must be
interior vertices.  Hence all of the boundary vertices are covered by
the subtrees and the mapping back to $G$ covers all of the vertices of
$G$.  Thus $\mathcal{P}$ is indeed a partition of $V$.

The smallest number of partitions in a branch shift quotient graph of
G induced from $x$, $y$, and $(C,\ensuremath{\phi})$ is four.  This occurs
when the path from $x$ to $y$ is just a single edge.  The largest
possible number of partitions in a branch shift quotient graph is
$|V|$, the number of vertices in $G$.  This occurs when the path from
$x$ to $y$ contains every interior vertex of $C$.  Each partition is
then a singleton set.

TILO uses a linear ordering of vertices.  A natural ordering of the
vertices of $Q$ is to use an identity map.  That is, if $u_j$ is the
vertex of $Q$ corresponding to subset $P_j$ then it is in the $j$th
position of the linear order.  We demonstrate a connection between
this TILO ordering and the tree position with a series of Lemmas.

\begin{lemma} \label{lem:widthEquiv}
Given a branch shift quotient graph $Q$ induced from edges $x$,
$y$, and tree position $(C,\ensuremath{\phi})$, the $t$-th width
$b_Q(t)$ using the identity map ordering on $Q$ is equal to $b(p_t)$, the
width of edge $p_t$ on the tree position $(C,\ensuremath{\phi})$ where path
edges $p_t$ are defined in Definition \ref{def:bsqg}.
\end{lemma}

\begin{proof}
The $t$-th width using the identity map linear ordering of vertices
is the sum of edge weights for edges $(j,k) \in Q$ such that $v_j$ is
in set $\{v_0,\ldots,v_t\}$ and $v_k$ is in set
$\{v_{t+1},\ldots,v_{i+2}\}$. Since edge weight $(j,k)$ is defined as
weight $w(P_j,P_k)$, this becomes
\begin{equation*}
b_Q(t) = w\left( \bigcup_{0 \leq j \leq t} P_j , \bigcup_{t < k \leq
i+2} P_k\right).
\end{equation*}
Recall that the boundary at an edge in $C$ can be defined in terms of
all the edges of $G$ that pass through it, $b(p_t) = \sum_{g \in
R(p_t)} w(g)$, or in terms of the edges between the sets of vertex images
of the subtrees created by cutting  at $p_t$,
\begin{equation*}
b(p_t)=
w\left(\posfInv{\cutT{p_t}{p_0}},\posfInv{\cutTopp{p_t}{p_0}}\right).
\end{equation*}
Edge $p_t$ is not contained in any subtree used to define the
partitions in Definition \ref{def:bsqg} therefore each subtree (and
corresponding partition) must be completely on one side of the cut of
edge $p_t$.
Thus
\begin{align*}
P_j &\subseteq \posfInv{\cutT{p_t}{p_0}} \ \text{ for } \   0\leq j \leq t \
\text{and}\\
P_k &\subseteq \posfInv{\cutTopp{p_t}{p_0}} \  \text{ for } \   t < k \leq i+2.
\end{align*}
Since subsets $P_j$, $P_k$ form a partition and
\begin{align*}
\posfInv{\cutT{p_t}{p_0}} \ \bigcap \ \posfInv{\cutTopp{p_t}{p_0}} = \emptyset
\end{align*}
then
\begin{align*}
\posfInv{\cutT{p_t}{p_0}} \subseteq  \bigcup_{0 \leq j \leq t} P_j
\quad \text{ and }   \quad
\posfInv{\cutTopp{p_t}{p_0}} \subseteq  \bigcup_{t < k \leq i+2} P_k.
\end{align*}
With the two pair of sets being equal, the weights are also equal:

\begin{align*}
b_Q(t) &= w( \bigcup_{0 \leq j \leq t} P_j , \bigcup_{t < k <= i+2} P_k)\\
 &= w(\posfInv{\cutT{p_t}{p_0}},\posfInv{\cutTopp{p_t}{p_0}}) \\
 &= b(p_t).
\end{align*}
\end{proof}

A branch shift on tree position $(C,\ensuremath{\phi})$ creates
a new tree position $(C',\ensuremath{\phi})$ for $G$. Let the
branch shift be the shifting of edge $x$
to edge $y$ for any pair $x$ and $y$ of nonadjacent edges of $C$.
Let $Q$ be the branch shift quotient graph of $G$ induced from $x$,
$y$, and $(C,\ensuremath{\phi})$.
Using the notation from Definition \ref{def:bsqg}, the before and after
views of a branch shift are presented in  Figures  \ref{fig:bsqg} and
 \ref{fig:shifted}.
Note that the edges $p_0$ to $p_{i+2}$ are not modified  nor are the
subtrees opposite the cut of edges $x'$, $y'$, $f_2'$ to $f_i'$.
Let $Q'$ be the branch shift quotient graph of $G$ induced from $f_2'$,
$x'$, and $(C',\ensuremath{\phi})$.

\begin{lemma}
\label{lem:branchmovequotient}
The branch shift quotient graph $Q'$ defined above is equal to the
original branch shift quotient graph $Q$.
\end{lemma}

\begin{proof}
The each $P_j$ of the partition used by the original branch shift
quotient graph is defined in Definition~\ref{def:bsqg} using the
subtree from the far side cut of an edge adjacent to, but not on,
the path between $x$ and $y$.  The branch shift operation cuts these
edges and then reattaches them to vertices on the path between $x$
and $y$.  This operation does not touch the subtree on the far side
cut of the edge.  Thus the subsets $P_k'$ used by $Q'$ are just a
relabelling of original subsets; explicitly, $P_0' = P_0$,
$P_{i+2}'=P_{i+2}$,
$P_{i+1}'=P_{1}$, and
$P_{t}'=P_{t+1}$ for $1\leq t \leq i$.
Therefore, the partition defined after the branch shift is the
same as the partition before the shift and hence the quotient graphs
are equal.
\end{proof}

Since the quotient graphs are equal, i.e.\ $Q' = Q$, we can compare
the linear TILO orderings on $Q$ defined by the tree positions $(C, \phi)$
and $(C', \phi)$.

\begin{lemma}
\label{lem:branchshiftTILO}
The TILO ordering on $Q$ defined by $(C',\phi)$ is the result of
starting with the ordering defined by $(C,\phi)$ and cyclicly shifting
the vertex at position $1$ to position $i+1$.
\end{lemma}

\begin{proof}
 Let $u_j$ and $u_j'$ be the vertices of $Q$ and $Q'$,
respectively.  As noted in the proof of Lemma~\ref{lem:branchmovequotient}, the
subsets $P_k'$ used by $Q'$ are relabelling of the subsets $P_j$ used
by $Q$. This results in the following relabelling of vertices:
$u_0' = u_0$, $u_{i+2}'=u_{i+2}$, $u_{i+1}'=u_{1}$, and
$u_{t}'=u_{t+1}$ for $1\leq t \leq i$.  The identity map linear
ordering of $Q'$ is
$(u_0',u_1',u_2',\ldots, u_i',u_{i+1}',u_{i+2}')$.
Substituting $Q$ vertices into this sequence yields the vertex order
$(u_0,u_2,u_3,\ldots, u_i,u_{i+1},u_1,u_{i+2})$.
This is precisely the result of applying a cyclic shift of vertex
$u_1$ to vertex $u_{i+1}$ of the identity map linear ordering of $Q$.
\end{proof}

\begin{corollary}
\label{coro:wrquotient}
A tree position $(C,\phi)$ of a graph $G$ is weakly reducible if
there is a pair of edges in $C$ such that the TILO ordering of
the induced branch shift quotient graph $Q$ of $G$ is weakly
reducible.
\end{corollary}

\begin{proof}
Note that if $(C',\phi)$ is the result of a branch shift of edge $x$
to edge $y$ on a tree position $(C, \phi)$ then the width of the
edges that are not along the path from $x$ to $y$ do not change. The
only widths that change are along the path from $x$ to $y$,
and by Lemma~\ref{lem:branchshiftTILO}, the way they change is
determined by the changes to the induced orderings of the quotient
graph.

Assume there is a path in which the induced ordering on the quotient
graph is weakly reducible, i.e.\ admits a shift that reduces its
width. This TILO shift determines a branch shift of the tree position.
Under this branch shift, the widths of the edges outside the path do
not change. Along the path, some of the widths may increase, but at
least one width strictly decreases and no width increases to a value
greater than or equal to the highest width that strictly decreases.
The same condition is true of the overall multiset of widths of the
tree position, so the width of the resulting tree position is strictly
less than the original width.

\end{proof}

\section{Reduction criteria}
\label{sect:reduction}

Corollary~\ref{coro:wrquotient} suggests a straightforward way to find
strongly irreducible tree positions of graphs: take all pairs of
edges, form the induced branch shift quotient graph, look for valid
TILO shifts, and apply the corresponding branch shifts on the tree
position.  Repeat this process until there are no valid TILO shifts,
and then perform a more comprehensive check to ensure that the tree
position is indeed strongly irreducible.  However, such an approach
involves a great deal of redundant computation. In particular, if two
paths overlap along a stretch of edges, then there will be TILO moves
in the two different quotients that correspond to the same branch
shifts.  In a naive implementation these will be checked multiple
times.  We therefore define criteria for weak reducibility of a tree
position that are determined by the quotient graphs but do not
require explicitly computing them.

\begin{definition} \label{def:adjwSlope}
Given a tree position $(C,\phi)$ and a pair of edges $e, g$ of $C$,
define the \emph{adjacency weight}
\begin{equation*}
a(e,g) = w\left(R(e) \cap R(g)\right) = w\left(\posfInv{\cutTopp{e}{g}},\posfInv{\cutTopp{g}{e}}\right)
\end{equation*}
to be the sum of the weights of the edges of $G$ that pass through both
$e$ and $g$ or equivalently the sum of the weights of edges of $G$ between
boundary vertices in \cutTopp{e}{g}  and boundary vertices in \cutTopp{g}{e}.

Define the \emph{slope}
\begin{align*}
s(e,g) &= w\left(R(g) \setminus R(e)\right) -  w\left(R(e) \cap R(g)\right)\\
       &= w\left(\posfInv{\cutT{g}{e}} \setminus \posfInv{\cutTopp{e}{g}}, \cutTopp{g}{e}\right) - w\left(\posfInv{\cutTopp{e}{g}},\posfInv{\cutTopp{g}{e}}\right)
\end{align*}
to be the sum of the weights of the edges of
$G$ that pass through $g$ but not through $e$  minus
the sum of the weights of edges of $G$ that
pass through both edges.
The slope $s(e,g)$ is the amount that the width of edge $e$ changes
if edge $g$ is disconnected from the tree and reattached on the far side
of $e$.
\end{definition}

 From the definitions, it follows
that $$a(e,g) + s(e,g) = w(R(g) \setminus R(e)).$$
From basic set theory,
$$R(g) = (R(g) \cap R(e)) \cup (R(g) \setminus R(e))$$
where the sets $R(g) \cap R(e)$ and $R(g) \setminus R(e)$ are
disjoint. Thus we have $b(g) = a(e,g) + a(e,g) + s(e,g)$. Rearranging
this gives us an alternate way to calculate $s(e,g)$:

\begin{lemma}
\label{lem:equivslopedef}
The slope $s(e,g)$ can also be calculated by the equation
\begin{align*}
s(e,g) = b(g) - 2a(e,g).
\end{align*}
\end{lemma}

If we think of the vertices $u_e$ and $u_g$ in a quotient
graph that correspond to the subtrees \cutTopp{e}{g} and
\cutTopp{g}{e}, respectively, then the adjacency weight $a(e,g)$
is the weight of edge $(e,g)$ of the quotient graph.
If $u_e$ and $u_g$ are vertices in a branch shift quotient graph
then the slope $s(e,g)$ is $\pm s_{e,g}$, the slope (as defined
in~\cite{thingraphs}) of a  linear ordering of the vertices of
the induced quotient graph.
In particular, we have the following:

\begin{lemma}
\label{lem:slopetranslation}
Given $Q$, a branch shift quotient graph of graph $G$ induced by edges
$x$, $y$, and tree position $(C,\ensuremath{\phi})$, let $e=p_i$, an edge
on the path from $x$ to $y$ and $g=f_k$, a non-path edge adjacent
to a path edge from $x$ to $y$.  Using the identity map induced linear
TILO ordering on $Q$, we find that
\begin{align*}
s(e,g)= s(p_i,f_k) = \begin{cases}
-s_{i,k} & \text{ if } k \leq i,\\
s_{i,k} & \text{ if } k > i
\end{cases}
\end{align*}
where the slope $s_{i,k}$ of a linear order is defined in
(\ref{eqn:slopeTILO}).
\end{lemma}

\begin{proof}
Using the notation from Definition \ref{def:bsqg} and Lemma \ref{lem:widthEquiv}, the
path edge $p_i$ corresponds to
the location between quotient graph vertices $u_i$ and $u_{i+1}$ in the identity map linear ordering.
The edge $f_k$ corresponds to vertex $u_k$ of the quotient graph.
Recall that $s_{i,k}$ is defined as the sum of the weights of edges from vertex
$v_k$ to vertices with indices strictly greater than $i$ minus the sum of the
edge weights from vertex $v_k$ to vertices with indices less than or equal to
$i$.
\begin{align*}
s_{i,k} &= \sum_{\substack{i < j\\j \neq k}} w(k,j) - \sum_{\substack{0 \leq j \leq i \\j \neq k}} w(k,j)\\
        &= w \Bigl( P_k,\bigcup_{\substack{i < j\\j \neq k}} P_j \Bigr) - w\Bigl(P_k, \bigcup_{\substack{0 \leq j \leq i \\j \neq k}} P_j\Bigr)
\end{align*}

Cutting at edge $p_i$ and set subtracting $P_k$ creates the sets corresponding to the two unions in the above equation.
When $k>i$ this is
$\bigcup_{\substack{i < j\\j \neq k}} P_j =
\posfInv{\cutT{p_i}{f_k}} \setminus \posfInv{\cutTopp{f_k}{p_i}}$
and $\bigcup_{\substack{0 \leq j \leq i \\j \neq k}} = \posfInv{\cutTopp{p_i}{f_k}} \setminus \posfInv{\cutTopp{f_k}{p_i}}
=\posfInv{\cutTopp{p_i}{f_k}}$
with
$P_k = \cutTopp{f_k}{p_i}$.  Using
$\posfInv{\cutT{p_i}{f_k}} \setminus \posfInv{\cutTopp{f_k}{p_i}}
= \posfInv{\cutT{f_k}{p_i}} \setminus \posfInv{\cutTopp{p_i}{f_k}} $ and substituting into the above equation yields
\begin{align*}
s_{i,k} &= w\left(\cutTopp{f_k}{p_i}  ,\posfInv{\cutT{f_k}{p_i}} \setminus \posfInv{\cutTopp{p_i}{f_k}} \right) -
w\left( \cutTopp{f_k}{p_i}, \posfInv{\cutTopp{p_i}{f_k}} \right),\\
 &= w\left(\cutTopp{g}{e}  ,\posfInv{\cutT{g}{e}} \setminus \posfInv{\cutTopp{e}{g}} \right) -
w\left( \cutTopp{g}{e}, \posfInv{\cutTopp{e}{g}} \right)
\end{align*}
which is the definition of slope $s(e,g)$.  When $k\leq i$, then the unions swap their correspondence to \cutT{p_i}{f_k}
and \cutTopp{p_i}{f_k} yielding $s_{i,k} = -s(e,g)$.
\end{proof}

\begin{lemma} \label{lem:treeReducible}
A tree position $(C, \phi)$ is weakly reducible if for
some pair of nonadjacent edges $x$ and $y$ the following conditions hold:
\begin{gather}
s(p_i,x)<0,   \\
b(p_{t-1}) \leq b(p_t) \ \text{ for all }\  1<t \leq i, \text{ and }\\
s(p_i,x) + s(p_i,y) +2a(x,y) < 0  \label{eqn:treduceB}
\end{gather}
where the edges of the simple path connecting $x$ to $y$ is denote by $p_t$ for $t$ from 1 to $i$.
Moreover, if this is the case then shifting $x$ to $y$ strictly
reduces the width of the tree position.
\end{lemma}

\begin{proof}
Let $Q$ be the branch shift quotient graph of $G$ induced by $x$, $y$,
and $(C,\ensuremath{\phi})$ with an identity map linear TILO ordering of $Q$'s
vertices $u_t$.  In terms of Definition \ref{def:bsqg}, $x$ is
associated with $u_1$ and $y$ is associated with $u_{i+1}$.  In terms
of Lemma~\ref{lem:TILOreduction}, $k=1$ as $u_1$ is being shifted to
$u_{i+1}$.  The resulting weakly reducible conditions are a
nonincreasing sequence of boundaries from $u_1$ to $u_i$, $s_{i,1}>0$,
and $s_{i,1} - s_{i,i+1} - 2a_{1,i+1}>0$.  By
Lemma~\ref{lem:slopetranslation}, we have $s_{i,1} = -s(p_i,x)$ and
$s_{i,i+1} = s(p_i,y)$.  By Definitions \ref{def:quotientgraph} and
\ref{def:adjwSlope}, we have $a_{1,i+1} = a(x,y)$.  By
Lemma~\ref{lem:widthEquiv}, we have $b_Q(t)$ = $b(p_t)$.  Therefore by
substituting, we find that the TILO ordering of the quotient graph $Q$
is weakly reducible if
\begin{gather*}
-s(p_i,x) > 0,\\
b(p_{t-1}) \leq  b(p_t), \ \text{ and }\\
-s(p_i,x) - s(p_i,y) - 2a(x,y) > 0.
\end{gather*}
By Corollary~\ref{coro:wrquotient}, the tree position is weakly
reducible if this condition holds.  Multiplying the slope inequalities
by $-1$ gives us the condition stated above.
\end{proof}

Condition (\ref{eqn:treduceB}) of the lemma gives the amount by which
the width of edge $p_i$ changes due to shifting $x$ to $y$,
\begin{equation} \label{eqn:widthChange}
 b^{'}(p_i) = b(p_i) +s(p_i,x) + s(p_i,y) +2a(x,y)
\end{equation}
where $b^{'}(p_i)$ is the width after the branch shift operation.

\section{Properties of thin tree positions useful for clustering and
  implementation}

The properties of a thin tree position of a graph $G=(V,E)$ can be
used to find \emph{pinch clusters} of $G$ (see~\cite[Definition
  1]{thingraphs}).  A subset $A \subset V$ is a pinch cluster if for
any sequence of vertices $v_1,\dots, v_m$, if adding $v_1,\dots, v_m$
to $A$ or removing $v_1,\dots, v_m$ from $A$ creates a new set with
smaller boundary, then for some $k<m$, adding or removing $v_1,\dots,
v_k$ to or from $A$ creates a set with strictly larger boundary.
Recall that, given a subset $A\subseteq V$, we say that the
\emph{boundary} of $A$ is $w(A, V\setminus A)$.

In linear TILO, pinch clusters are determined by local
minima in the width values defined by the TILO ordering.
Any path in a thin tree position with local minimums of edge widths
determines pinch clusters of the quotient graph by cutting the graph at the
local minimum. These clusters can be refined to create pinch clusters
of the original graph by creating a linear ordering of vertices
compatible with the tree position and checking for any TILO
shifts. Limiting the search for refinement shifts and approximating
the pinch ratios by effective bounds calculated from a tree position
will be explored in future papers.  Some pinch clusters of $G$ can be determined
without futher refinement.  These occur at locations defined as follows.

\begin{definition}
Given a graph $G$ and a tree position $(C, \ensuremath{\phi})$ for $G$, we say that
an edge $e$ of $C$ is a \emph{local minimum} if each endpoint of $e$
is shared with an edge of $C$ that has strictly greater width than $e$
and whose second endpoint is not a boundary vertex.
\end{definition}

When an edge is a local minimum of a tree position then it is a local minimum of every path
through it in the tree.  Thus the edge determines pinch clusters in every quotient graph induced
by a path through the edge.  The next theorem shows that a local minimum edge finds a pinch cluster
of the original graph.

\begin{theorem}
If $e$ is a local minimum of a strongly irreducible tree position
$(C,\ensuremath{\phi})$ of a graph $G$ then the two subsets of $G$ defined by the
subtrees that result from cutting $C$ at $e$ are pinch clusters.
\end{theorem}

\begin{proof}
Assume $e$ is a local minimum of a tree position $(C, \ensuremath{\phi})$.
We will prove that $e$ is a local minimum in a strongly irreducible
TILO ordering on $G$.

Let $\{P_1, P_2\}$ be the partition of the vertices of $G$ defined by the
local minimum $e$, i.e., $P_1 = \posfInv{\cutT{e}{g}}, P_2 =
\posfInv{\cutTopp{e}{g}}$ for some other edge $g$ in $C$.  Let $o$ be a TILO
ordering of $G$ that has minimal width among all orderings for which every
vertex of $P_1$ appears before every vertex of $P_2$. In other words, $P_1$
consists of the vertices $v_0,\ldots,v_k$, while $P_2$ consists of vertices
$v_{k+1},\ldots,v_{N-1}$ (where the subscript indicates the TILO ordering).
Note the the width $b_k$ between $v_{k}$ and $v_{k+1}$ is precisely
$b(e)$.

If $b_{k-1}>b_k$ and $b_{k+1}>b_k$ then by the second condition of Lemma~\ref{lem:TILOreduction}
it is not possible for a valid shift of a vertex between $P_1$ and $P_2$ of the ordering.  With
the initial assumption that ordering $o$ has minimal width within the partitions, this means that
$o$ is a strongly irreducible TILO ordering on $G$.

Let $x$ be the edge in $C$ that has $\posf{v_k}$ as an endpoint.
If $x$ and $e$ share an endpoint then $b(x)=b_{k-1}$.  Since $e$ is a local minimum,
$b(x)>b(e)$ and thus $b_{k-1}>b_{k}$.
Assume that $x$ and $e$ do not share an endpoint and
denote the edges of the simple path from $x$ to $e$ as $p_1,\ldots,p_i$.  Let $y$ be the edge
adjacent to $e$ and $p_i$.
Assume $b_{k-1} < b_k$.
Then $s_{k-1,k} = b_k - b_{k-1} > 0$
and by Lemma~\ref{lem:slopetranslation}, $s(e,x) = -s_{k-1,k} < 0$.
For path edges $k > j$, $\cutTopp{p_k}{x} \subset \cutTopp{p_j}{x}$ and
$a(p_k,x)\leq a(p_j,x)$. Using the slope definition in Lemma~\ref{lem:equivslopedef},
$s(p_j,x)=b(x)-2a(p_j,x)\leq b(x)-2a(p_k,x) = s(p_k,x)$. Since $e$ is at the end of the path,
$s(p_j,x) \leq s(e,x)$.
This is also the same path from $x$ to $y$.
Consider a branch shift of $x$ to $y$.
After shifting $x$ to $y$, $b^{'}(p_i)$ is $b_{k-1}$ as
cutting at $p_i$ induces the partitions of $G$ as $P_1 \setminus \set{v_k}$ and $P_2 \cup \set{v_k}$.
Shifting $x$ to $y$ strictly reduces the width $b(p_j)$
for every path edge $p_j$ since $s(p_j,x) < 0$.  No other widths are changed.
Thus shifting $x$ to $y$ reduces the width
of the tree position $(C,\ensuremath{\phi})$.  But this contradicts the original assumption that
$(C,\ensuremath{\phi})$ is strongly irreducible.  Thus $b_{k-1}$ can not be less than $b_k$.

A symmetric argument implies that $b_{k+1}$ can not be less than
$b_k$.  In the case that $b_{k-1}=b_k$, the argument can be extended
to show that the closest nonequal boundary in each direction can not
be less than $b_k$.  So $b_k$ must be a local minimum of the TILO
ordering.  By~\cite[Theorem 4]{thingraphs}, this implies that $T_1$
and $T_2$ are pinch clusters.
\end{proof}

In addition to the conceptual advantages of thin tree position noted
in the introduction, this method is more amenable to efficient
implementation.  Width, slope, and adjacency weight are defined both
in terms of a weight of sets of edges and in terms of a weight between
sets of vertices. The edge based definition is better for accumulating
and propagating the tree position reduction calculations over sparse
matrices. The vertex based definition is better for accumulating and
propagating these calculations over dense matrices.

The following relationships between the width, slope, and adjacency
weight of edges sharing a common vertex are useful when accumulating
and propagation calculations across a tree position.  Given edges $e$,
$f$, and $g$ sharing a common interior vertex of a tree position, we have
\begin{align*}
b(e)  &= a(e,f)+a(e,g), \\
b(e) &= b(f) +b(g) -2a(f,g), \text{and}\\
  s(f,g) &= a(g,e)-a(f,g)=b(e)-b(f). \\
\end{align*}

The reduction checks of Lemma \ref{lem:treeReducible} can be done in
parallel. Since a branch shift of $x$ to $y$ does not modify the
subtrees off the path from $x$ to $y$, sets of branch shift
operations can be applied in parallel if the paths are independent
(do not intersect).

\bibliographystyle{spmpscinat}

redpajama_set_name: RedPajamaArXiv


text:
What is a bluestocking?
The cover for To Charm a Bluestocking has been released.
I'm pleased that Escape Publishing kept my title, even though my husband said "A what? No-one will know what a bluestocking is."
"Except every historic romance reader in the history of the genre," I said.
The term 'bluestocking' to denote a clever woman was first used in literature in 1638. The concept is much older. Back in the 1400s, the Venetians had a group of intellectual men nicknamed 'della calza' (of the stocking). A century later, a selection of literary women in Paris were named the 'bas bleu'. The idea gained prominence in English society in 1750 when The Blue Stockings Society was founded by coal heiress Elizabeth Montagu and three friends. Blue stockings were cheap men's wear made of wool (what we would call working class socks today). Important (rich) men wore white silk stockings. Montagu's group were initially seen as an elegant cultural innovation – a balance between fashion, feminine sophistication and virtue. In 1778, Richard Samuel painted the group as female celebrities, and the artwork featured on a pocket diary, to inspire women to write down their little musings.
Unfortunately, the idea of clever women was quickly seen as dangerous – especially considering the American Revolutionary War (1775-1783) and the French Revolution (1789-1799). Women might also forget their place in society and want something more. Bluestocking became an insult. In an 1811 play, a character says "The bluestocking is the most odious of creatures." Lord Byron wrote in 1821 that fellow poet Felicia Dorothea Hemans should 'knit bluestockings instead of wearing them'. Bluestockings had gone from a cute gathering to a group of radical political thinkers who spoke out for equality, liberty and social justice. Those who didn't want change painted them as mannish, frumpy creatures – grotesque harridans with no graces who attacked each other over the tea-table.
The term fell out of fashion. Hardly surprising when it came with such negativity. But romance writers have reclaimed it for the clever women. Those women who could think for themselves. Who were not the vacant gossiping society girls. Who were slighted by society. But ultimately, who find true love because they are clever and interesting, not despite that fact.
The rise of anti-intellectualism
We've boundless plains to share.
I adore it when folks come together and share opinions, great site, keep it up.
S M Spencer says:
Thanks for the history lesson! I'd heard the term but wasn't sure what it meant … now I do! I've had a real brush with history the last few days, having been to Port Arthur and the the Cascades Women's Factory (where the women who were transported to the Colonies were sent). I may become a history buff yet!

redpajama_set_name: RedPajamaCommonCrawl


An easy, but elegant, brunch idea. This Puff Pastry Cream Cheese Danishes are delicate, pretty and tasty. Make them any flavor you want. I did chocolate chip, raspberry and blueberry, but use whatever filling sounds good to you!


This video provides a description of advancements starting in vSphere 6.0 that improved the scalability of compute resources on the host.
- [Instructor] In regard to Compute Enhancements in vSphere 6 and vSphere 6.5, there are a tremendous number of them. And they are sweeping in their impact. Most of them actually happened at 6.0, but then there are also some advances at 6.5. There isn't anything really for me to demonstrate here, it's just a bunch of exciting facts. The first thing is we went from 32 hosts per cluster all of a sudden to 64 hosts per cluster, which meant that large companies who were managing hundreds and hundreds of hosts, or thousands and thousands of hosts maybe, could create larger clusters, larger DRS clusters.
Less to manage in each cluster. Larger resources to work with. So the number of hosts per cluster increasing changed things for some companies that can take advantage it. Also we have larger hosts. We can hosts with virtual CPUs up to 4,096, in other words, the virtual CPUs that the virtual machines are using that are on the hosts can go up to 4,096. That's higher than it's ever been.
As a matter of fact, with the overhead reduction that was part of 6.0 and 6.5, we could potentially have 1024 virtual machines on the same host sharing the same resources. We've got physical RAM support all the way up to 12 terabytes. We talked about that earlier, but it bears mentioning again because physical RAM? We're not talking about disk, we're talking about RAM.
Physical RAM support up to 12 terabytes of RAM, that's a whole lot of RAM. And again, would you need that for every host? No, would you need it maybe for any host? Maybe not, but if you do, then the capability is there. We talked about that earlier, that VMware does that because they don't like to say No. If somebody can do it in the physical world then they want to make that possible to do with a virtual machine as well. And physical CPU support up to 480.
So, we've got a tremendous amount of resource available on each host, but wait, there's more. We also have USB 3.0 support, kind of had it 5.5 but really began to support it for CAC, for Common Access Cards for the government and for the military, fully supported in vSphere 6. We have NVM Express support.
These are cards that can be added to physical devices that are automatically recognized. The host can recognize those. Also we have SMART card support for SSDs, self-monitoring cards that will tell you when they're getting sick. Used to be that when we just spinning disks we had smart support for those, and they can tell you when they were getting sick but solid straight drives really couldn't do that, we didn't have the tools to get information.
Now we have commands to retrieve data about SSDs so we can get an idea of when it's going to all of the sudden go out on us. We don't want that to necessarily be all of a sudden, we want to have the fault tolerant support with that. And speaking of fault tolerance, beginning with vSphere 6.0, we have Fault Tolerance support for 4 virtual CPU virtual machines. That takes that into a different realm as well. Whereas, some companies would have said Well, we'd love to use that fault tolerance because that actually allows the virtual server to run in two different physical environments basically at the same time within milliseconds of each other so if one fails then the other one can take over.
So we'd like to use that for our very important machines, like our SQL and our Exchange and our Java. And then VMware had to say Well, those are multi-threaded applications and you're going to get a whole lot better performance if you have multiple virtual CPUs, and at this time, before vSphere 6, we do not have support for multiple, virtual CPUs. And now we do, we can support up to four virtual CPUs so that takes that into a whole new realm. As I said, I'm saying 6.x here because most of this stuff actually happened at 6.0.
But there are some things that happened right at 6.5. As the technology changes VMware keeps up with it. Raw Device Mapping support has been increased. Now, do you always use Raw Device Maps? No, you should use a data store. And with all the new storage technologies coming up, Raw Device Mapping should be a one-off kind of thing anyway, where I want to have a virtual machine that connects to a disk that's not actually a data store but it's actually on the SAN. And if I want to go through ethernet support to get to it rather than some type of a fiber channel support then I've got that capability on 6.5.
All the latest Input/Output drivers, support for the latest Intel chipset and, back to fault tolerance for a second, now there's support for multi-NIC. If I'm going to set up Fault Tolerance then what I want to do is eliminate every single point of failure. So, being able to have multiple physical NICs associated with the VMkernal ports that are associated with fault tolerance, certainly make sense.
That's one less single point of failure, and I don't want any single points of failure. I want to have separate NICs, separate switches, the whole path be separate from everything else or have two separate paths at least, so that I actually have that fault tolerance in place. So these Compute Enhancements in vSphere 6 and 6.5 allow us to do those kinds of things. So as you can see, there are tremendous number of compute enhancements that actually put things in different realm for some companies.
What that means to your company remains to be seen, but it's there if you need it.


text:
Home Research > Members > 1558-1603 > NORRIS, Sir Henry II (c.1554-99)
NORRIS, Sir Henry II (c.1554-99), of Rycote, Oxon.
Published in The History of Parliament: the House of Commons 1558-1603, ed. P.W. Hasler, 1981
b. c.1554, 4th s. of Sir Henry Norris I and bro. of Edward, Sir John and William. educ. Magdalen Coll. Oxf. 1571. unm. Kntd. Sept. 1586; KG Apr. 1592.1
Capt. English volunteers at Antwerp June 1583; capt. Berks. contingent for defence of England 1588, on Portugal expedition 1589, in Brittany 1591; col.-gen. of infantry in Ireland 1595.2
Norris and his five brothers were all soldiers, and comprised the most distinguished military family of their time. Norris followed the second-born, 'General' Sir John Norris, to the Netherlands, Portugal, Brittany and Ireland. He was knighted by Sir Robert Dudley, Earl of Leicester at Zutphen in 1586, and returned to England a year later to help in the preparation of defences against the Armada and lead the Berkshire contingent in Leicester's army at Tilbury. In April 1588 Norris received an honorary MA at Oxford. That autumn he was chosen knight of the shire for Berkshire and Sir John was elected for Oxfordshire. Although he was not mentioned by name in the journals of the 1589 Parliament he may have attended the subsidy committee to which the first knights of the shires were appointed on 11 Feb. 1589. Soon after the end of the Parliament he took part in the Portugal venture, and he was not, in fact, in England for any length of time until 1596. In December of that year he was ordered by the Privy Council to bring to London the ringleaders of the recent insurrection against enclosures in Oxfordshire, whom his father had examined. In 1597 he was again elected for Berkshire and was appointed to committees on bills concerned with the gentry's obligation to maintain armour and weapons (8 Nov.), the punishment of rogues disguised as soldiers and mariners (20 Dec.), maltsters (12 Jan. 1598) and the relief of mariners and soldiers (26 Jan. 1598). As knight for Berkshire he was entitled to attend the committees concerning enclosures (5 Nov.), the poor law (5, 22 Nov.), the penal laws (8 Nov.), monopolies (10 Nov.), the town of Wantage (10 Nov.) and the subsidy (15 Nov.). He was apparently served with a subpoena while in Parliament, a breach of privilege which Sir Edward Hoby brought to the attention of the House on 16 Jan. 1598. He was sent to Deptford in December 1597 to receive and escort to London the governor of Dunkirk, captured by Sir Edward Norris.3
By the summer of 1599 he was back in Ireland. Serving under Essex in Munster, he and his brother, Sir Thomas Norris, who had become president of Munster on Sir John Norris's death two years before, both received mortal wounds. The deaths of Sir Thomas on 16 Aug. and of Sir Henry, after the amputation of his leg, on 21 Aug. were reported as an 'infinite disaster'. The Queen wrote to console Lord and Lady Norris 'in this bitter accident', in which she proposed herself as an example of fortitude, 'our loss being no less than yours'; and promised to recall from the Netherlands Sir Edward, the last survivor of the six martial brothers. No will or inquisition post mortem survives for Norris.4
Author: Alan Harding
1. DNB (Sir Henry, Baron Norris); Lansd. 94, f. 110.
2. HMC 2nd Rep. 73; HMC Foljambe, 47, 49, 51; CSP For. 1589 (Jan.-July), 119; APC, xxi. 25; CSP Dom. 1591-4, p. 397; Cal. Carew Pprs. iii. 113.
3. VCH Berks. iii. 105; HMC Hatfield, iv. 202; CSP Dom. 1595-7, pp. 13, 318; D'Ewes, 431, 552, 553, 555, 557, 561, 575, 578, 581, 588; APC, xxviii. 11, 165.
4. CSP Ire. 1599-1600, pp. 66, 128; Chamberlain Letters ed. McClure, i. 82, 84; CSP Dom. 1598-1601, p. 319.


TP Mechanical Contractors will be holding informative Open Enrollment Benefit Meetings to provide important information about our benefit programs. During this time, we will be offering flu shots to all employees and their spouses. Please be sure to bring your current medical card. The meeting schedule is listed below.
Open enrollment will run from October 2, 2017 through October 13, 2017.
If you elected not to participate in any or all benefit plans during your initial eligibility period or if you have changes that you need to make to your current benefits, you may enroll or make those changes during this open enrollment period.
We are having a passive enrollment This means if you don't want to make any changes, you don't need to do anything.
FSA contributions DO NOT roll-over so you will need to re-elect the contribution you want for this upcoming year.
Any questions you have about open enrollment or the company's benefits, please contact Jen Madden at (513) 851-8881 x1244 or jen.madden@tpmechanical.com.
Congratulations estimating and sales for securing work totaling $3,778,856 and total man hours of 17,674 for the months of August.
Like or Share and You Could Win!
TP Mechanical has been building its presence on social media over the past few years. Our social media sites can be accessed from our website, where you can find up-to-date information about our company or even what in going on in the industry.
Please follow us on your favorite social media site. Over the next few months we will hold one contest a month; by "Liking" or "Sharing" a post your name will be put into a drawing to win a gift from the marketing closet. Contest starting October 1, 2017 ending December 22, 2017.
Thank you to everyone who took the time to participate in the newsletter survey over the past month. Email was the unanimous winner.
Prizes winners will be contacted this week.
Ticket sales for our upcoming Monte Carlo are now on sale. To purchase your tickets please click here to complete the order form. Once your order form is completed, Tara Teepe will email over an invoice with your payment options. If you have questions about our fundraising event please contact Tara either by email tara.teepe@tpmechanical.com or directly 513-898-8908.
Evening proceeds will benefit A Kid Again, who gives families with a child experiencing a life-threatening illness the chance to have fun and feel hope. The Ohio charity offers adventures and practical support to help family members cope with the uncertainty during the most difficult time of their lives.


Peaceful wooded lot in popular Steeplechase Subdivision. Lots of privacy on this desirable cul-de-sac surrounded by nice homes.Community water included.
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Beautiful elevated lot with trees, clearings and mountain views. Well share agreement in place. Horses welcome in this quiet, serene and lovely neighborhood with nice homes. Power pole at lot line. Excellent location with easy access to I-40, services and shopping.


text:
Q: Is the complex symmetric part of a $2\times2$ normal matrix always normal? I want to prove (or disprove) that, for a normal matrix $A\in\mathbb{C}^{2x2}$ (so that $AA^\ast=A^\ast A$), the sum $A+A^T$ is (or isn't) normal.
I know that for $3\times3$ normal matrices, $A+A^T$ isn't normal, but for the $2\times2$ case, when I try to come with a counterexample I do not manage to make it.

A: Yes, $A+A^T$ is normal if $A$ is $2\times2$ normal. The reason is that all $2\times2$ complex skew-symmetric matrices commute (because they are scalar multiples of the rotation matrix for angle $\frac\pi2$).

*

*Let $A=S+K$, where $S=\frac12(A+A^T)$ is complex symmetric and $K=\frac12(A-A^T)$ is complex skew-symmetric.

*Using the fact that $K\bar{K}=\bar{K}K$, rearrange the equality $AA^\ast=A^\ast A$ to $[S,\bar{S}] = [S,\bar{K}]+[\bar{S},K]$, where $[X,Y]$ denotes the commutator $XY-YX$.

*Since $[S,\bar{S}]$ is purely imaginary (because $\overline{[S,\bar{S}]}=[\bar{S},S]=-[S,\bar{S}]$) but $[S,\bar{K}]+[\bar{S},K]$ is real, the rearranged equality in last step implies that $[S,\bar{S}]=0$. As $\bar{S}=S^\ast$, we conclude that $S$ is normal.

redpajama_set_name: RedPajamaStackExchange


The governor of the state of Ohio, John Kasich, just plain hates teachers. The teaching profession is made up mostly of women. I've often felt that his dislike of teachers has to do with the fact that he doesn't like smart, outspoken women. Now he has gone too far.
Shame on Kasich. He has never and will never get my vote.


The job description, sent out on 20 March to EU states' embassies in Brussels, says their task will be "correction and fact-checking of misinformation/myths".
It will also involve "development and regular updating of EU 'narrative' via key messages/lines to take, articles, op-eds, factsheets and infographics, with an emphasis on communicating the benefits of the EaP".
The EaP, or Eastern Partnership, is an EU policy on closer trade and political ties with six former Soviet countries.
The new staff are to be Russian speakers and to do "analysis/monitoring of reporting on EU policies" in Russian language media.
The group - between two and five people - will also prepare press "materials" in Russian.
They'll be initially hired for one year and paid at least €4,350 a month. But they'll be paid by whichever EU countries put them forward because EU institutions have not earmarked any money for the project.
The job description notes they should be former journalists or press officers and know their way around "social media".
The move comes after EU leaders, in March, tasked the foreign service "to challenge Russia's ongoing disinformation campaign".
They said the EU should "prepare, by June, an action plan on strategic communication in support of media freedom. The establishment of a communication team is a first step in this regard".
Russia's media machine was on show on Thursday (16 April) when president Vladimir Putin held his annual Q&A session with the general public.
Most questions focused on Russia's economy, the rising cost of food and medicine, or problems with housing and social services.
Putin said in Moscow that EU economic sanctions "have an impact" and that "it's highly unlikely that [they] will be lifted anytime soon".
He claimed the EU and US imposed them "to restrain our growth" and that they "have nothing to do" with the conflict in Ukraine, because "the Russian army is not in Ukraine".
He predicted the Russian economy will return to growth in two years.
He also claimed the US forced EU leaders to boycott his 9 May military parade in Moscow, a WWII memorial event.
The TV show, which included glamorous female presenters and questions from starry-eyed children, was broadcast, with live translation, by RT.
The Kremlin's media flagship has a budget of $300 million a year, employs 2,000 people in 16 countries, and airs shows in English, French, German, and Spanish.
It is reaches millions of households worldwide and it makes the EU counter-project look tiny by comparison.
But the new group of EU foreign service specialists is just one of several measures.
The job description notes they'll work together with the EU's foreign embassies and relevant departments in the European Commission.
The EU Council, which represents member states, is also co-ordinating groups of diplomats who will liaise with the EU foreign service cell.
"We're thinking how best to make use of whatever this team of myth-busters produces. One idea is a virtual network, anchored in the Council's communications department, to distribute materials to member states", an EU official said.
A second EU official noted the EU foreign service cell "is only one element".
He said there will be "a wide range of EU activities aimed at communicating on EU policies and programmes, supporting independent media and media diversity … and promoting freedom of the media throughout the neighbourhood".
The only effective way to address the region's challenges is to invest in balanced, trustworthy, independent and diverse media, working for Russian-language audiences.


text:
Q: R: setwd delay when using file.choose So what im doing is making my browse button so that when i click on them bring the user straight to the directory that i want them to save their file in or look for their file.
For example
     setwd("C:\\Users\\Eric\\Desktop\\Program\\graphs") #set directory
     file.choose()

However in the earlier script i have already set my work directory at
    setwd("C:\\Users\\Eric\\Desktop\\Proram") #set directory

so when i ran the first example it brought me to directory Program instead of graphs .
but when i ran file.choose() on the second time, it then brought me to the graphs directory
why is this happening? any idea how to fix this ?

A: Here's a quick and dirty solution to your problem:
 dirPath <- "C:\\Users\\Eric\\Desktop\\Program\\graphs"
 setwd(dirPath)
 # Tell R to sleep until the current directory matches the expected directory
 while(getwd() != normalizePath(dirPath)) {
   Sys.sleep(0.02)
 }
 file.choose()


1:00 p.m. Lunch is provided and childcare is available by request. Please contact the church office if you would like more information. You may let us know in the RSVP section if you require childcare.
Come to learn about church of the Woods or become a member!


You will look fashionably pretty in this two-piece set gown, Rachel Allan Princess 2111. V-neckline with contrasting laced sweetheart lining formed the crop top. Wide band cinched the waist with side pockets secure your small essentials. The gathered skirt falls down the ground that balances the look with an edge.


text:
/**
 * The global state selectors
 */

import { createSelector } from 'reselect';

const selectGlobal = () => (state) => state.get('global');

const selectError = () => createSelector(
  selectGlobal(),
  (globalState) => globalState.get('error')
);

const selectIdToken = () => createSelector(
  selectGlobal(),
  (globalState) => globalState.get('idToken')
);

const selectIsLoggingIn = () => createSelector(
  selectGlobal(),
  (globalState) => globalState.get('isLoggingIn')
);

const selectProfile = () => createSelector(
  selectGlobal(),
  (globalState) => globalState.get('profile')
);

// selectLocationState expects a plain JS object for the routing state
const selectLocationState = () => {
  let prevRoutingState;
  let prevRoutingStateJS;

  return (state) => {
    const routingState = state.get('route'); // or state.route

    if (!routingState.equals(prevRoutingState)) {
      prevRoutingState = routingState;
      prevRoutingStateJS = routingState.toJS();
    }

    return prevRoutingStateJS;
  };
};

export {
  selectGlobal,
  selectError,
  selectIdToken,
  selectIsLoggingIn,
  selectProfile,
  selectLocationState,
};

redpajama_set_name: RedPajamaGithub


text:
La cinquième circonscription du Finistère est l'une des huit circonscriptions législatives françaises que compte le département du Finistère.

La circonscription de 1958 à 1986

Description géographique
Dans le découpage électoral de 1958, la cinquième circonscription du Finistère était composée des cantons suivants :
 Canton de Daoulas
 Canton de Landivisiau
 Canton de Lesneven
 Canton de Ploudiry
 Canton de Plouescat
 Canton de Plouzévédé
 Canton de Saint-Thégonnec
 Canton de Sizun.

Historique des députations

Historique des élections

Élections de 1958

Élections de 1962

Élections de 1967

Le suppléant d'Antoine Caill était Yves Michel, agriculteur, conseiller municipal de Plouescat.

Élections de 1968

Le suppléant d'Antoine Caill était Albert Crenn, de Landivisiau.

Élections de 1973

Le suppléant d'Antoine Caill était Yves Michel. Yves Michel remplaça Antoine Caill après son décès, du 27 avril 1976 au 2 avril 1978.

Élections de 1978

La suppléante de Charles Miossec était Marguerite Caill, maire de Plouzévédé.

Élections de 1981

La suppléante de Charles Miossec était Marguerite Caill.

La circonscription depuis 1986

Description géographique et démographique
Dans le découpage électoral de la loi n°86-1197 du 24 novembre 1986
, la circonscription regroupe les cantons suivants :
 Canton de Guipavas
 Canton de Landerneau
 Canton de Landivisiau
 Canton de Lannilis
 Canton de Lesneven
 Canton de Plouescat.

D'après le recensement général de la population en 1999, réalisé par l'Institut national de la statistique et des études économiques (INSEE), la population totale de cette circonscription est estimée à .

Le découpage de la circonscription n'a pas été modifié par le redécoupage des circonscriptions législatives françaises de 2010.

Historique des députations

Historique des élections

Élections de 1988

Le suppléant de Charles Miossec était Jean-Pierre Cuiec, avocat au barreau de Brest, adjoint au maire de Landerneau.

Élections de 1993

Le suppléant de Charles Miossec était Emmanuel Cuiec, avocat au barreau de Brest.

Élections de 1997

Élections de 2002

Élections de 2007

Élections de juin 2012

Les élections législatives françaises de 2012 ont lieu les dimanches 10 et 17 juin 2012.

 Le taux d'abstention est de 38,87 % au premier tour, et de 37,65 % au second tour.

Présidentielle 2012 ( tour) : François Hollande (PS) 58,86 %, Nicolas Sarkozy (UMP) 41,14 %

Législatives 2007 ( tour) : Jacques Le Guen (UMP) 54,80 %, Chantal Guittet (PS) 45,20 %

Élections de juin 2017

Députée sortante : Chantal Guittet (PS)

Élections de 2022

Notes et références

Voir aussi

Articles connexes

 Liste des députés du Finistère
 Liste des circonscriptions électorales de France (par ordre croissant d'habitants)
 Découpage électoral
 Élections législatives en France

Liens externes

Département du Finistère

 La fiche de l'INSEE de cette circonscription :

Circonscriptions en France

 Description et Atlas des circonscriptions électorales de France sur http://www.atlaspol.com, Atlaspol, site de cartographie géopolitique, consulté le 13 juin 2007.

5

redpajama_set_name: RedPajamaWikipedia


text:
Niels Rasmus Trads Rasmussen (født den 30. januar 1945 i Viborg) er tidligere administrerende direktør i PFA Byg da han i januar 1999 blev varetægtsfængslet og sad isolationsfængslet i mere end 40 dage, sigtet for dokumentfalsk.

PFA-sagen

Rasmus Trads havde sammen med Kurt Thorsen forfalsket PFA-direktør André Lublins underskrift på økonomiske garantier til Ejendomsselskabet Norden og finansieringsselskabet Gefion – begge selskaber ejet af Kurt Thorsen.

I alt havde Rasmus Trads og Kurt Thorsen udstedt garantier for 1,8 milliarder kroner, mente man på dette tidspunkt. Beløbet voksede siden til det dobbelte – 3,7 milliarder kroner.

To dage senere blev også byggematadoren Kurt Thorsen varetægtsfængslet.

11. januar 2002 fik Kurt Thorsen 6 års fængsel og Rasmus Trads 4 års fængsel. Under afsoningen tog han 2003-2004 en journalistisk diplomuddannelse fra Danmarks Journalisthøjskole. Han blev løsladt i juni 2004, hvorefter han afsonede samfundstjeneste og var ansat som mediekonsulent i mediekoncernen Eniro indtil maj 2006. Ved siden af sit fuldtidsjob freelancede Rasmus Trads blandt andet for talkshowet Clement Direkte på DR2.

Kilder og henvisninger

Erhvervsfolk fra Danmark
Bedrageridømte fra Danmark
Personer fra Viborg


A2: rec. 1942-05-26 (live radio show).
A5: rec. 1939-04-04 (live radio show from the Meadowbrook Ballroom, Cedar Grove, New Jersey.
A7: rec. 1940-07-31 (live radio show from the Chicago Civic Theater).
A9: rec. 1943; Army Force Band.
A10: rec. 1943; Army Force Band.
B2: rec. 1942-06-02 (live radio show).
B3: rec. 1939-04-10 (Arrangement unknown).
B6: rec. 1942-04-23 (live radio show).
B9: rec. 1940-06-04 (live radio show from the Wardman Park Hotel, Washington).
B10: rec. 1939-11-26 (live radio show from the Meadowbrook Ballroom, Cedar Grove, New Jersey).


text:
Jaguar is celebrating the 60th anniversary of the iconic E-Type with the debut of the new limited-edition F-Type Heritage 60 Edition. Only 60 of the special F-Type Heritage 60 Edition models will be produced and they are all based on the supercharged F-Type R.
Since each F-Type Heritage 60 Edition is based on the F-Type R, that means there's the nice supercharged 5.0-liter V8 with 575 horsepower under the hood. Each one is painted in Sherwood Green, just like what was offered on the E-Type back in the 1960s. Inside there's a two-tone Caraway and Ebony Windsor interior, which is basically a fancy way of saying it's tan and black. Lastly there's also a unique aluminum console finisher, inspired by the rear-view mirror casing of E-type
You'll also like the 60th anniversary logo that's embossed on the headrests and the E-Type 60 badges that are shared with the limited-edition E-Type 60 Collection vehicles
"Celebrating 60 years of the iconic Jaguar E-type is the perfect moment to create our first-ever SV Bespoke limited edition – and the rarest F-Type, with just 60 cars available globally," said Mark Turner, Commercial Director, Jaguar SV Bespoke. "We've worked closely with Jaguar Design to develop a theme for the F-Type Heritage 60 Edition that pays homage to the E-Type in a contemporary way. It's testament to Jaguar's sports car design lineage that the 1960s Sherwood Green color looks as though it was designed for today's F-Type."
Jaguar hasn't announced the pricing or availability.
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Jaguar Land Rover offers zero percent financing for 72 months and no payments for 90 days


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Step 2 download YUMI – Multiboot USB Creator Download: YUMI – Multiboot USB Creator This is an easy way of making a Bootalble USB FlashDrive for Hirens Boot eegal Downloadable Music. Freegal is a downloadable music service that the Westchester Library System subscribes to.
All you need to access the service is your Westchester Library card.


Our OBDII Breakout Box gives you a practical and easy to use in-line breakout box to make an electrical connection to the vehicle On-Board Diagnostic connector (SAE J1962 or OBD connectors) without disturbing the connection between the vehicle and the tool making it possible to execute a test while simultaneously monitoring the testing process.
• Two OBDII Standard J1962 Female (Car-Side) Connectors.
• One J1962 Male Connector on a 3' extension cable.
• All 16 pins broken out to Banana Plug connections.


Gone are the days when skinny models set the standards for good looks. Curvy girls are making a big comeback. No one knows this better than plus-size dating sites that are full of people looking to meet and have fun with big beautiful women.
One of these sites is FatFlirt – a plus-size dating platform that is quickly gaining popularity in this segment of the online dating world. Read our FatFlirt review to find out all you need to know about this exciting dating service that may help you find a perfect chubby match in your area.
FatFlirt has a simple and straightforward registration procedure. It literally takes only a minute and you're ready to upload your profile photo. If you choose, you may also add a video and details about your ethnic origin, hair color, body type, orientation, height, weight, education, smoking and drinking habits, and more. Once you are done, your search for the perfect chubby companion can begin.
FatFlirt features a very nice and user-friendly interface. All the elements are arranged in a logical fashion, which means that even the most inexperienced user won't have trouble using the site.
It is worth noting that this dating site is very well optimized so there are no annoying delays when you're opening pages and searching through the member base. The color theme is pleasing to the eye and notifications are clearly visible so you'll never miss them. Overall, the creators of this dating site have done a really good job and their efforts result in first-class user experience.
When it comes to features, FatFlirt offers all you can expect from a premium dating platform and it has some special options that you can use to make your plus-size dating even more enjoyable. We are glad to see that even non-paying members can use almost all the features, which makes this dating site highly attractive.
First of all, FatFlirt allows its users to search through the member base using various criteria such as age, location, orientation, ethnicity, body type, and even hair and eye color. You can limit your search to new members or those who are online at the moment. Alternatively, you can choose to see only those profiles that contain a photo and/or video.
Like Gallery is a useful dating tool that allows you to store profiles of all the members that you find interesting and to see who's liked your profile. Of course, you have the option to communicate with potential matches via instant messages. In your profile settings, you can choose which notifications you want to see. FatFlirt also offers you the possibility to promote your account so that you can be seen by more people and get more interaction.
Some of the communication options, like the ability to get read receipts of all your messages, are reserved for paying members. Paying members also have the option to activate incognito mode so that they can browse profiles anonymously. Users who choose to upgrade to a paid membership also get the benefit of extra security.
It is important to mention that FatFlirt offers a very nice mobile version of its site, which is great news for all singles who prefer to use their cellphones for online dating. Lastly, this dating platform also offers some useful tips for safe dating.
FatFlirt is undoubtedly a great place to meet big beautiful women. The member base is filled with open-minded individuals who are ready to flirt and have fun. Whether you prefer casual interactions or want to make lasting connections, FatFlirt is an excellent platform that will allow you to get the best out of plus-size dating.
Finding interesting chubby singles in your area has never been so easy. FatFlirt is a user-friendly platform that will make your pursuit of a perfect chubby companion highly enjoyable. We strongly recommend that you join this plus-size dating site and take advantage of the numerous exciting features it offers. Try it! After all, you have nothing to lose and chances are you'll have great fun finding that special someone.
Great place to be if you want to have some fun. I met some very cute ladies.
I enjoy using this site. It has some good features.


These silver Horn Tusk Stud Earrings are symbolic of protection, making the perfect gift for a loved one.
The Horn is an amulet recognised for it's powers of protection and qualities of repelling evil.
These gorgeous studs come complete with luxury gift packaging and a meaning card.


C-5 is a 32.5 kilometers long circumferential artery road that connects the north and the south of Metro Manila, which makes it an important alternative to the always-congested EDSA. C-5 is actually a network of roads and bridges that is still being extended.
C-5 is short for Circumferential Road 5. Its construction began in 1986, as part of a bigger plan to build 10 circumferential roads in the Megacity. In 1996, all the road segments that form the Circumferential Road 5 together were renamed Carlos P. Garcia Avenue, after the 8th President of the Republic of the Philippines. Different controversies regarding the relocation of poor residents and favoritism for a property-owning senator have delayed a further extension of the road.
C-5 roughly runs parallel to EDSA, just a few kilometers east of it. It begins in Valenzuela City in the north were North Luzon Express Way and Mindanao Avenue meet. The road merges into the South Luzon Expressway near NAIA Airport, at the city borders of Taguig and Parañaque. On its way, C-5 also passes the cities of Quezon, Marikina, Pasig, and Makati.
The street allows access to the HotSpots Ortigas and Fort Bonifacio from the east. It passes by the malls Market Market and SM Aura and Eastwood. The universities Ateneo and UP Diliman are also accessible via C-5.
The character of C-5 ranges from a winding two-lane street with many intersections and traffic lights in the northern parts to a fully-fledged 14-lane expressway in the south. Thus, the availability of public transport and the traffic situation vary substantially depending on the stretch. In general, C-5 experiences heavy traffic during rush hour, near the adjacent business districts, and at important intersections. Being an alternative to EDSA therefore does not mean that C-5 is necessarily faster or less congested. For updates on the current traffic situation at C-5, you can refer to this link.
C-5 is the alternative route of EDSA.


Hezbollah's Image Campaign: Theme Parks, Anyone?
In Doha 12, Hezbollah launches its effort to eliminate the Doha 12 (including Our Heroes) as part of their worldwide publicity/propaganda campaign. Hezbollah is one of the most sophisticated users of media, image and messaging in the terrorist world, with their own newspapers and satellite TV channel.
VICE (an online news/documentary producer) recently explored Hezbollah's museum/theme park/propaganda center of Mleeta. There's also a Part 2 (not as well-done) that looks at how Hezbollah fits in the greater Lebanese political scene and how the Lebanese public views them. Interesting viewing, and Part 2 partly explains why the on-camera host doesn't challenge his interviewees more.


Having a famous doppelgänger can have its perks. But it can also be incredibly annoying.
Maybe that's why a Bernie Sanders lookalike was spotted at an unknown FedEx location wearing a white shirt with blue and red lettering (very patriotic) that read, "I am not Bernie."
Although the man's face isn't fully visible, the resemblance to the Vermont Senator is so uncanny that we're not completely convinced this isn't Sanders himself.
The photo, which was posted to Reddit's r/pics by Dasbeardog, quickly hit the site's front page. In the comments section, another Redditor posted a behind shot of what we can only assume is the same man, rockin' the same outfit.
Comment from discussionsometimes your doppelganger gets too famous.
Finally, another Redditor claims to have seen Not Bernie at a rally for the real Bernie Sanders in Santa Monica in May.
The thread also spawned a very important question: Could Not Bernie be the internet's next Grate Guy? Only time will tell.


text:
The 2008 Beni Amrane bombings were two bombings on June 9, 2008 that killed 13 people in the town of Beni Amrane in the Boumerdès Province,  from Algiers, the capital of Algeria. The first bomb killed a French citizen and his Algerian driver as they were leaving the town's railway station. The second device exploded about five minutes later as rescue workers arrived. Eight soldiers and three firefighters died in the second blast while an unconfirmed number of people suffered injuries.  Both devices appeared to have been detonated remotely. No group has claimed the bombings, which follow attacks blamed on the al-Qaeda Organization in the Islamic Maghreb group. The Frenchman was an engineer working for a French firm on a renovation project at the station.

International reactions

Countries
 – The French Foreign Minister Bernard Kouchner condemned the attack. "I want to express my feeling of disgust and my absolute condemnation of this blind terrorist violence that nothing can justify,".

See also
 Terrorist bombings in Algeria
 Insurgency in the Maghreb (2002–present)
List of terrorist incidents, 2008

References

External links
BBC News

Mass murder in 2008
Terrorist incidents in Algeria in 2008
Improvised explosive device bombings in Africa
Terrorist incidents in Algeria
Islamic terrorism in Algeria
2008 murders in Algeria


text:
Apple: No Decision on Encrypting Cloud Info
Hill explores encryption, public safety issues
The House Energy & Commerce Committee Tuesday took a deep dive into the tension between encryption and law enforcement. Among the takeaways were that Apple has made no decision about whether to encrypt information in the cloud and the company's assertion that it did not share source codes with China, though that government had asked for them.
The three-plus-hour hearing featured two panels, including witnesses from the FBI and Apple—though not on the same panel.
The poster case for the issue was FBI v. Apple over access to a San Bernardino shooter's encrypted cell phone information, but as House Energy & Commerce Committee chairman Fred Upton (R-Mich.) put it in his opening statement, "[i]t is important that we move beyond the 'us versus them' mentality."
Upton said there are valid concerns on both sides, and no ready answers, but that was no reason "to put our head in the sand or resort to default positions."
But the tensions were still obvious. Thomas Galati, chief of intelligence with the NYPD, said encryption leaves "the police, prosecutors, and the people we are sworn to protect in a very precarious position" and laid the blame at the feet of tech companies.
"What is even more alarming is that this position is not dictated by our elected officials, our judiciary system, or our laws. Instead, it is created and controlled by corporations like Apple and Google. These corporations have taken it upon themselves to decide who can access critical information in criminal investigations."
For his part, Apple general counsel Bruce Sewell said that strong encryption is "a good thing" and "a necessary thing" to protect consumers information. He also suggested that helping the government circumvent those protections would do more harm than good. "[T]he people subject to law enforcement inquiries represent far less than one-tenth of one percent of our hundreds of millions of users," he said. "But all of those users — 100% of our users would be made more vulnerable if we were forced to build a back door."
The FBI has said it does not want Apple to create a back door, just to call off the dogs so it can try to pick the lock on the front door if there are exigent circumstances, like getting info about terrorists or criminals.
Sewell suggested that there was a fundamental difference in perspective that needed resolving.
"As you heard from our colleagues in law enforcement, they have the perception that encryption walls off information to them," he said. "But technologists and national security experts don't see the world that way. We see a data-rich world that seems to be full of information. Information that law enforcement can use to solve—and—crimes."
The Information Technology Industry Council saw the hearing as progress on a tough issue.
"Unlike most Washington debates, where people go into their separate corners and repeat 'us versus them' positions, a serious discussion is emerging between lawmakers and tech on encryption," said ITIF SVP for government affairs Andy Halataei. "Members of Congress on both sides of the aisle are asking thoughtful questions to understand the critical role encryption plays in securing our digital world. These hearings are also giving lawmakers a better sense of the trade-offs involved for the security and safety of Americans that would occur by placing restrictions or limitations on encryption that would do more harm than good."
Activist Group: FBI Should Share Hack Info With Apple
Says that is necessary to protect public safety
Apple Balks at 'Hacking' iPhones
Fights government request to create encryption work-around
Encryption Commission Legislation Introduced
Some groups less than enthusiastic
FCC's Pai Calls on Apple to Activate FM Chips
Says it is matter of public safety
House Leaders Create Encryption Working Group
Will look at key legal and policy issues
Encryption-Protection Bill Introduced
Would preempt state and local laws
Justice: We Want Un-Secret, Front Door to Encrypted Info
Rosen raises specter of warrant-proof encryption abetting child abuse
Tech Groups Tout Hill Shout-Out for Encryption
Welcome report, say weakened protections would help bad actors


Hi, I am using the latest version of phpBB (3.1.8). Unfortunately this version seems to be incompatible with php7, where mysql_connect is deprecated. Can we soon expect a version of phpBB that is fully compatible with php7? There are ISPs out there who begin charging an extended services of php5 on their servers!
phpBB 3.2 is currently is in beta, so it probably will be released rather soonish, and it will support PHP 7.
Also it's using mysql_connect because you're using the MySQL connection method in your installation instead of MySQLi.


text:
Carl Gottlieb Strauß (* 6. April 1743 in Danzig; † 11. Juli 1790 ebenda) war ein deutscher evangelischer Geistlicher und Pädagoge.

Leben
Carl Gottlieb Strauß wurde als Sohn des Benjamin Strauß, Ratsherr in Danzig, geboren. Er erhielt anfangs Unterricht durch Hauslehrer und besuchte ab 1760 das Akademische Gymnasium Danzig. Dort hörte er unter anderem Vorlesungen bei Gottfried Less, außerordentlicher Professor der Theologie; er entwickelte bereits damals eine Neigung zur Philosophie, die durch Michael Christoph Hanow noch gefördert und unterstützt wurde.

1764 begann er sein Theologie- und Philosophie-Studium an der Universität Jena. Er trat dort in die philosophische Gesellschaft unter dem Vorsitz von Professor Johann Gottlieb Waldin (1728–1795) ein, bei der Gelegenheit wurde seine Abhandlung Von der nöthigen Behutsamkeit beym Tadel der Libellatiker. Er war auch ein Mitglied des Disputatoriums des Professor Johann Ernst Immanuel Walch und wurde durch seine gedruckte Rede De commodis et incommodis studii singularitatis, quod in societatibus regnare solet in die Lateinische Gesellschaft aufgenommen. Nach einem vierjährigen Studium beendete er 1768 seinen Aufenthalt in Jena als Dr. phil.

Nach seiner Rückkehr nach Danzig predigte er in der Folge fünf Jahre lang, bis ihm 1774 nach dem Tod von Professor Hanow durch den Magistrat der Stadt dessen Stelle als Professor der Philosophie am akademischen Gymnasium angeboten wurde; gleichzeitig wurde er Aufseher der öffentlichen Ratsbibliothek. Nach dem Tod des Benjamin Groddeck (1723–1776) übernahm er für drei Jahre dessen griechischen und hebräischen Vorlesungen.

Carl Gottlieb Strauß war seit 1774 verheiratet. Die Ehe blieb kinderlos.

Schriften (Auswahl)

 Abhandlung betreffend die nöthige Behutsamkeit beym Tadel der Libellatiker. Jena Schill 1766.
 De commodis et incommodis quae in societatibus ex singularitatis studio oriuntur. Jenae: Strauss, 1766.

Literatur

Pädagoge (18. Jahrhundert)
Hochschullehrer (Akademisches Gymnasium Danzig)
Lutherischer Geistlicher (18. Jahrhundert)
Christlicher Geistlicher (Danzig)
Bibliothekar (Deutschland)
Deutscher
Geboren 1743
Gestorben 1790
Mann