text:
\section{Introduction}

Weak and strong $\eps$-nets were defined by Haussler and Welzl as a tool
for fast processing of geometric range queries~\cite{haussler1986epsilon}.
They have been consequently studied in many areas, including
computational geometry, combinatorics, and machine learning,
and they were used in many algorithmic applications,
including range searching and geometric optimization.

An $\eps$-net is a set that pierces all large sets in a given family of sets.
Formally, let $\mu$ be a probability distribution over a domain $X$
and let\footnote{Here and below
we assume that all sets considered are measurable.} $C \subseteq 2^X$
be a family of sets.
A subset $S$ of $X$ is called a {\em weak $\eps$-net} for $C$ over $\mu$
if $S \cap c \neq \emptyset$ for every $c\in C$ with $\mu(c) \geq \eps$.
A subset $S$ is called a {\em strong $\eps$-net} if in addition
$S$ is contained in the support of $\mu$.
We say that $C$ has weak/strong $\eps$-nets
of size $\beta=\beta(C,\eps)$
if for every distribution $\mu$
there is a weak/strong $\eps$-net for $C$ over $\mu$
of size at most $\beta$
(we stress that $\beta$ may depend on $\eps$, but not on $\mu$).

To illustrate the difference between weak and strong nets,
consider the uniform distribution on $n$ points
on the unit circle in the plane $X=\mathbb{R}^2$,
and the family $C$~to be all convex hulls of subsets of these $n$ points.
Any strong $\eps$-net must contain at least $(1-\eps)n$ points,
but there are weak $\eps$-nets of size $O(\alpha(\eps)/\eps)$,
where $\alpha(\cdot)$ is the inverse Ackermann function~\cite{Alon08weak};
using points inside the unit disc allows to use significantly less points.
In general, weak nets may be much smaller than strong nets.

The main question we address is under what conditions
do weak nets exist.
This question for strong $\eps$-nets is fairly well understood;
the fundamental theorem of statistical learning, which shows that
the VC dimension characterizes PAC learnability,
also shows that the VC dimension characterizes the existence of strong nets
(see~\cite{haussler1986epsilon,shalev2014understanding} and references within).
We show that the Radon number
characterizes the existence of weak nets in a pretty general setting (viz.~separable
convexity spaces).

\subsubsection*{Weak nets}

Weak $\eps$-nets were mostly studied in the context of discrete and convex geometry,
where they are related to several deep phenomena.
For example, Alon and Kleitman~\cite{alon1992piercing} used weak nets in their famous solution of the
$(p,q)$-conjecture by Hadwiger and Debrunner~\cite{hadwiger1957variante}.

\new{Barany, Furedi, and Lovasz~\cite{barany1990number} showed that
convex sets in the plane admit weak nets,
and Alon, Barany, Furedi, and Kleitman~\cite{alon1992point} established a bound of $O(1/\eps^2)$ on their size.}
Recently Rubin improved this bound to $O(1/\eps^{3/2 + \gamma})$,
where~$\gamma>0$ is arbitrarily small~\cite{Rubin18improved}.
Alon, Barany, Furedi, and Kleitman~\cite{alon1992point}
extended the existence of weak nets for convex sets to all dimensions $d$;
there are weak $\eps$-nets of size at most roughly $(1/\eps)^{d+1}$.
Their proof relies on several results from convex geometry, like
Tverberg's theorem and the colorful Caratheodory theorem.
Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl~\cite{chazelle1993improved} later improved the bound
to at most roughly $(1/\eps)^d$.
Overall, there are at least three different constructions
of weak nets for convex sets.
Matousek~\cite{matousek02lower} showed that any weak $\eps$-net for convex sets in $\mathbb{R}^d$
must \new{contain at least $\Omega(\exp(\sqrt{d/2}))$ points} for $\eps\leq1/50$.
Later, Alon~\cite{alon12lower} proved the first lower bound that is superlinear in $1/\eps$;
this was later improved to $\Omega\bigl((1/\eps)\log^d(1/\eps)\bigr)$ by Bukh, Matousek, and Nivasch~\cite{bukh2011lower}.
Ezra~\cite{Ezra10} constructed weak $\eps$-nets for the more restricted class of axis-parallel boxes in $\mathbb{R}^d$.
Alon, Kalai, Matousek, and Meshulam~\cite{alon2002transversal}
defined weak nets in an abstract setting,
and asked about combinatorial conditions that yield their existence.

We identify that the existence of weak nets
follows from a basic combinatorial property of convex sets,
Radon's theorem:

\begin{theorem}[Radon]
Any set of $d+2$ points in $\mathbb{R}^d$  can be partitioned into
two disjoint subsets whose convex hulls intersect.
\end{theorem}

We show that this property alone is sufficient and necessary
for the existence of weak nets in a general setting,
which we describe next.

It is worth mentioning that Radon's theorem
also plays a central role in the context of strong nets.
Indeed, it implies that the VC dimension of
half-spaces in $\mathbb{R}^d$ is at most $d+1$,
which consequently bounds the VC dimension
of many geometrically defined classes.

\subsubsection*{Convexity spaces}

We consider \new{an abstraction} of Euclidean convexity that originated
in a paper by Levi~\cite{levi1951helly}, and defined in the form \new{presented} here
by Kay and Womble~\cite{kay1971axiomatic}.
For a thorough introduction to this subject see
the survey by Danzer, Grunbaum, and Klee~\cite{danzer1963helly}
or the more recent book by van de Vel~\cite{van1993theory}.

A {\em convexity space} is a pair $(X,C)$ where $C \subseteq 2^X$ is a
family of subsets that satisfies\footnote{\new{Note that van de Vel~\cite{van1993theory} also requires that the union of an ascending chain of convex sets is convex.}}:
\begin{itemize}
\item $\emptyset,X\in C$.
\item $C$ is closed under intersections\footnote{We use the standard notation
$\cap C' = \bigcap_{c \in C'} c$.}: $\cap C' \in C$ for every $C'\subseteq C$.
\end{itemize}
The members of $C$ are called \emph{convex sets}.
The \emph{convex-hull} of a set $Y \subseteq X$, denoted by $conv(Y) = conv_C(Y)$, is
the intersection of all convex sets $c \in C$ that contain $Y$.
A convex set $b\in C$ is called a \emph{half-space} if its complement is also convex.

We next define the notion of separability, which is an abstraction of the
hyperplane separation theorem (and the more general Hahn-Banach theorem).
The convexity space $(X,C)$ is \emph{separable}, if for every $c\in C$ and $x \in X \setminus c$
there exists a half-space $b \in C$ so that $c \subseteq b$ and $x \not \in b$.
It can be verified that $(X,C)$ is separable if and only if
every convex set $c\in C$ is the intersection of all
half-spaces containing it.
This form of separability, as well as other forms,
have been extensively studied (e.g.~\cite{GrunbaumMotzkin61,hammer1955,hammer1961semispaces,Klee56,Chepoi1994}).

Convexity spaces appear in many contexts in mathematics.
For instance, the family of closed subsets in a topological space,
the subgroups of a given groups, and the subrings of a ring are all examples\footnote{One should sometimes add the empty set in order to satisfy all axioms of a convexity space.} of convexity spaces.
They are also closely related to the notion of $\pi$-systems in probability theory.
In Section~\ref{sec:Example} below we discuss a few examples of convexity spaces that arise in algebra and combinatorics.

\subsubsection*{Main results}

The combinatorial property that characterizes the existence
of weak nets is the Radon number~\cite{levi1951helly,kay1971axiomatic},
{which is an abstraction of Radon's theorem:
we say that $C$ \emph{Radon-shatters} a set $Y\subseteq X$ if for every partition of $Y$
into two parts $Y_1,Y_2$ it holds that $conv(Y_1)\cap conv(Y_2)=\emptyset$.
The {\em Radon number} of $(X,C)$ is
the minimum number~$r$ such that $C$ does not Radon-shatter \new{any} set of size~$r$.
Radon's theorem states that the Radon number of
the space of convex sets in $\mathbb{R}^d$ is at most~$d+2$.

\begin{theorem}\label{thm:main}
Let $(X,C)$ be a finite separable convexity space.
\begin{enumerate}
\item If the Radon number of $(X,C)$ is at most $r$ then
it has weak $\eps$-nets of size at most $(120 r^2/\eps)^{4r^2\ln(1/\eps)}$ for every $\eps > 0$.
\item
If the Radon number of $(X,C)$ is more than $r$
then there is a distribution $\mu$ over $X$ such
that every $\frac{1}{4}$-net for $C$ over $\mu$ has size at least $r/2$.
\end{enumerate}
\end{theorem}

We often refer to a construction of small weak $\eps$-nets
as an {\em upper bound}, and to a proof that no small weak $\eps$-nets
exists as a {\em lower bound}.
The upper bound in 1 above is quantitively worse than the one for Euclidean convex sets~\cite{chazelle1993improved},
but holds in a more general setting.
We do not know what is the optimal bound in this generality.

A possible interpretation of the upper bound is that the existence
of weak nets is not directly related to ``geometric'' properties of the underlying space
(as in the various constructions surveyed above).
This is somewhat surprising:
consider a family $C$ and a distribution $\mu$
such that $C$ has no strong $\eps$-net
with respect to $\mu$; in order to construct a weak $\eps$-net
we should have a mechanism that suggests ``good points''
outside the support of $\mu$. In Euclidean geometry
there are such natural choices, like ``center of mass''.
We notice that the Radon number provides such a mechanism
(see Section~\ref{sec:overview}).

We next discuss conditions
that are equivalent to the existence of small $\eps$-nets.
It is convenient to present these equivalences for infinite spaces.
Quantitative variants of these statements apply to finite spaces as well.

We use the following standard notion of compactness; a family $C$ is
\emph{compact} if for every $C' \subseteq C$
so that $\cap C' = \emptyset$ there is a finite $C'' \subseteq C'$
so that $\cap C'' = \emptyset$.
This condition is satisfied e.g.\ by closed sets in a compact topological space.

\begin{corollary}[Equivalences]
\label{cor:EQ}
The following are equivalent for a compact separable convexity space $(X,C)$:
\begin{enumerate}
\item $(X,C)$ has a finite Radon number.
\item $(X,C)$ has weak $\eps$-nets of finite size for some $0 < \eps <1/2$.
\item $(X,C)$ has weak $\eps$-nets of finite size for every $\eps >0$.
\end{enumerate}
\end{corollary}

The proof of Corollary~\ref{cor:EQ} appears in Section~\ref{sec:corEQ}.
It shows that the role of the Radon number in the existence of weak nets
is similar to the role of the VC dimension in the existence of strong nets,
at least for separable compact convexity spaces.

In Section~\ref{sec:NecAssum}, we provide an example
showing that the compactness assumption in Corollary~\ref{cor:EQ}
is necessary. We do not know if the separability assumption is necessary.

The implication $2 \Rightarrow 3$ in Corollary~\ref{cor:EQ} is an amplification
statement for the parameter~$\eps$  in weak nets.
In Section~\ref{sec:boost} we give an example showing that
the threshold $1/2$ in item 2 is sharp for amplification of weak nets:
\begin{example}
\label{ex:boost}
There is a compact separable {convexity} space that has
weak $\eps$-nets of finite size for every $\eps > 1/2$
but has no weak $\eps$-nets of finite size for $\eps<\frac{1}{2}$.
\end{example}

This demonstrates an interesting difference with
strong $\eps$-nets, for which there is no such threshold:
if $C$ has strong $\eps$-nets for some $\eps<1$
then it has finite VC dimension, which implies
{the existence of} strong $\eps$-net for all $\eps >0$.

\subsubsection*{Organization}

In Section~\ref{sec:Example} we provide some examples
of convexity spaces.
In Section~\ref{sec:overview} we outline
the construction that leads to the upper bound in Theorem~\ref{thm:main},
in Section~\ref{sec:LBintro} we outline the lower bound in Theorem~\ref{thm:main},
and in Section~\ref{sec:NecAssum} we provide some examples
that demonstrate the necessity of some of our assumptions.

In Section~\ref{sec:construction} we prove the upper bound,
and in Section~\ref{sec:RadonLB}  we prove the lower bound.
In Section~\ref{sec:cors} we prove the {characterizations (Corollary~\ref{cor:EQ}),}
and
in Section~\ref{sec:ext} we discuss an extension
to convexity spaces that are not necessarily separable or compact (like {bounded} convex sets in $\mathbb{R}^d$).
Finally, in Section~\ref{sec:discussion} we conclude the paper and offer some directions for future research.

\subsection{Some convexity spaces}
\label{sec:Example}
We now present a few examples of ``non Euclidean'' convexity spaces.
These examples will be used later on to show that some of our theorems/lemmas
are tight in the sense that each premise is necessary.
More examples can be found in the book~\cite{van1993theory}.

\vspace{0.3cm}
\noindent {\bf Example 1 (power set).}
Let $X$ be a set. Perhaps the simplest convexity space is~$(X,2^{X})$.
Here every convex set is a half-space and therefore this space is separable.
When $X$ is finite, the Radon number of this space is $\lvert X\rvert + 1$.
When $X$ is infinite, the Radon number is $\infty$.

\vspace{0.3cm}
\noindent {\bf Example 2 (subgroups).}
Let $G$ be a group with identity $e$.
The space $(G\setminus\{e\}, \{H\setminus\{e\} : H\leq G \})$
of all subgroups of $G$ (with the identity removed)
is a convexity space.
Here, $Y\subseteq G$ is Radon-shattered,
if every two disjoint subsets of~$Y$ generate groups
whose intersection is $\{e\}$.

\vspace{0.3cm}
\noindent {\bf Example 3 (cylinders).}
Let $X = \{0,1\}^n$,
and let the $C$ be the family of \emph{cylinders}:
a set $c\subseteq \{0,1\}^n$ is called a cylinder if there exists $Y\subseteq [n]$,
and $v\in\{0,1\}^Y$ such that
\[c = \bigl\{u\in\{0,1\}^n : u|_Y = v\bigr\}.\]
The size of $Y$ is called the co-dimension of the cylinder.
This is a separable convexity space.
Its half-spaces are the cylinders with co-dimension $1$,
and its Radon number is $\Theta(\log n)$.

\vspace{0.3cm}
\noindent {\bf Example 4 (subtrees).}
Let $T=(V,E)$ be a finite tree. Consider the convexity space $(V,C)$, where
\[C = \bigl\{U\subseteq V : \text{the induced subgraph on $U$ is connected}\bigr\}.\]
It is a separable convexity space and its Radon number is at most
$4$.
Theorem~\ref{thm:main} hence implies the existence of weak nets
of size depending only on $\eps$
(in this case there are elementary constructions of $\eps$-nets of size~$O(1/\eps)$).
This example is a special case of \emph{geodesic convexity}
in metric spaces (see~\cite{van1993theory}).

\vspace{0.3cm}
\noindent {\bf Example 5 (convex lattice sets).}
Consider the space $(\Z^d, C)$, where $C$ is the family of \emph{convex lattice sets} in $\mathbb{R}^d$;
these are sets of the form {$K \cap \Z^d$ for some convex~$K \subseteq \mathbb{R}^d$.}

This is a separable
convexity space. {Indeed,
let $c = K \cap \Z^d \in C$ and $x \in \Z^d \setminus c$.
Since $x \not \in c$ it follows that $x \not \in K$.
and therefore there is a half-space in $\mathbb{R}^d$ separating~$x$ and $K$.
This half-space induces a half-space in $(\Z^d, C)$.}

Onn~\cite{Onn91} proved that the Radon number of
this space is at most $O(d2^{d})$ {and at least $2^d$.}
Our results imply that weak $\eps$-nets
exist in this case as well.

Note that the family of half-spaces has VC dimension $d+1$,
which is much smaller than the Radon number.
Thus, Theorem~\ref{thm:weakeps} (which we state in the next subsection)
gives better bounds on the size of the $\eps$-net than Theorem~\ref{thm:main}.

\vspace{0.3cm}
\noindent {\bf Example 6 (linear extensions of posets).}
Let $\Omega$ be a set.
For a partial order $P$ on $\Omega$,
let $c(P)\subseteq X$ denote the set of all linear orders that
extend $P$.
Fix a partial order $P_0$, and
consider the family $C$ of
all sets of the form $c(P)$, where
$P$ is a partial order that extends $P_0$.
The space $\bigl(c(P_0),C\bigr)$ is a separable convexity space whose half-spaces
correspond to partial orders defined
by taking two $P_0$-incomparable elements
$x,y\in \Omega$ and extending $P_0$ by setting $x<y$.

\subsection{Upper bound}
\label{sec:overview}

\subsubsection*{{From Radon to Helly and VC}}
Let $(X,C)$ be a separable convexity space and let $B$ denote the family of half-spaces of $C$
(the notation $B$ is chosen to reflect that $B$ generates all convex sets in $C$ by taking intersection, and hence
can be seen as a \emph{basis}).

We first observe that the Radon number is an upper bound on the {Helly number}
and the {VC dimension} of $B$.
The {\em Helly number} of a family $B$ is the minimum number $h$ such that
every finite\footnote{The finiteness assumption can be removed when $B$ is compact.} $B'\subseteq B$
with $\cap B'=\emptyset$ \new{contains a subfamily $B''$ with
at most $h$ sets such that already the intersection of the sets in  $B''$ is empty}.
Helly theorem states that the Helly number of half-spaces in $\mathbb{R}^d$ is at most $d+1$.
The {\em VC dimension} of $B \subseteq 2^X$ is the supremum over~$v$ for which
there exists $Y \subseteq X$ of size $|Y|=v$ so that for every $Z\subseteq Y$
there is a member of the family that contains $Z$ and is disjoint
from $Y\setminus Z$.

\begin{lemma}
\label{lem:Radon}
Let $B$ the family of half-spaces of a separable convexity space $C$.
If the Radon number of $C$ is $r$
then the VC dimension and the Helly number of $B$ are less than $r$.
\end{lemma}

The bound on the Helly number follows from a result by Levi~\cite{levi1951helly},
and the bound on the VC dimension is straightforward.
The proof appears in Appendix~\ref{ap:Radon}.

Lemma~\ref{lem:Radon} reduces the construction of weak nets for separable convexity spaces
to the following, more general construction.

\begin{theorem}\label{thm:weakeps}
Let $X$ be a set, let $B\subseteq 2^X$ be a compact family
with $VC$ dimension~$v$ and Helly number $h$,
and let $B^\cap$ denote the family generated by taking arbitrary
intersections of members of $B$.
Then $B^{\cap}$ has weak $\eps$-nets of size at most $\beta<(120 h^2/\eps)^{4hv\ln(1/\eps)}$ for every $\eps>0$.
\end{theorem}

We next give an overview of
the proof of Theorem~\ref{thm:weakeps};
the complete proof appears in Section~\ref{sec:construction}.

\subsubsection*{Outline of construction}

The construction of weak nets underlying Theorem~\ref{thm:weakeps} is short and simple.
We want to pierce all convex sets $c \in C$ such that $\mu(c) \geq \eps$.
We distinguish between two cases.
The simpler case is when $c$ can be written as
the intersection of half-spaces, each of which has $\mu$-measure more than $1-1/h$.
By Helly's property, it follows that there is a single point
that pierces all such $c$'s.
In the complementary case, when~$c$ can not be written in this way,
we use Haussler's packing lemma~\cite{haussler1995sphere} and show that there is
a small collection $A \subseteq B$ such that conditioning $\mu$ on a single~$a \in A$
increases the measure of $c$ by a factor of at least $1+1/(2h)$.
The size of $A$ can be bounded from above in terms of the VC dimension of $B$.
So, constructing a set that pierces all $c$'s with measure at least $\eps$
is reduced to constructing a bounded number of nets for larger density $\eps' \geq (1+1/(2h))\eps$.

To summarize, the Helly number yields the theorem for $\eps$ close to $1$,
and the VC dimension allows to keep increasing the density until
the Helly number becomes relevant.

Going back to the discussion after Theorem~\ref{thm:main} concerning the mechanism that suggests ``good points'' we see that this mechanism is based on the Helly property
(roughly speaking, the Helly property is a mechanism that given a collection of sets
outputs a point).

\subsection{Lower bound}
\label{sec:LBintro}

The following lemma is a slight generalization of the lower bound in Theorem~\ref{thm:main}
(we do not assume separability or compactness, and replace $1/4$ by any $\eps>0$).

\begin{lemma}
\label{lem:LB}
Let $(X,C)$ be a convexity space.
If the Radon number of $C$ is greater than $r>0$
then there is a distribution $\mu$ on $X$
so that every weak $\eps$-net for $C$ over~$\mu$
has size at least $(1-2\eps)r$.
\end{lemma}

This gives a non-trivial lower bound as long as $\eps < 1/2$.
Example~\ref{ex:boost} shows that this is sharp in the sense that
when $\eps>1/2$ there is no lower bound that tends to infinity with the Radon number.

The proof of Lemma~\ref{lem:LB},
as well as a finer distribution dependent lower bound,
appear in Section~\ref{sec:RadonLB}.
The proof is essentially by reduction to
the chromatic number of Kneser graphs.

\subsection{The necessity of assumptions}
\label{sec:NecAssum}

\subsubsection*{Assumptions in Theorem~\ref{thm:weakeps}}
We first show that if either of the assumptions of having bounded VC dimension
or of having bounded Helly number is removed then Theorem~\ref{thm:weakeps}
ceases to hold.

To see why bounded Helly number is necessary,
let $X$ be any finite set and set $B=\{X\}\cup\{X\setminus\{x\} : x\in X\}$.
The VC dimension of $B$ is $1$,
but any subset of $X$
can be represented as an intersection of members of $B$.
Thus, $B^{\cap} = 2^X$,
which does not have weak $\eps$-nets of size which is
independent of $\lvert X\rvert$.

To see why bounded VC dimension is necessary,
consider the convexity space $(X,C)$ of cylinders
(Example 3 in Section~\ref{sec:Example}).
Here, $X=\{0,1\}^n$, $C$ is the family of cylinders,
and the family of half-spaces $B$ consists of cylinders with co-dimension~1:
$B = B_0 \cup B_1$ with
\[B_t = \bigl\{\{x\in X : x_i=t\} : i\in[n]\bigr\}.\]
The Helly number of $B$ is $2$,
since an intersection of half-spaces
is empty if and only if two complementing
cylinders with co-dimension 1 participate in it.

The following claim gives a lower bound on weak $\frac{1}{4}$-nets
for $C$ over the uniform distribution $\mu$ over $X$.

\begin{claim}\label{claim:cylinders}
Every weak $(1/4)$-net for $C$ over $\mu$ has size at least $\log n$.
\end{claim}
\begin{proof}
$S\subseteq X$ pierces every cylinder with measure $1/4$
only if for every $i \neq j$ in $[n]$ there is $x\in S$ with $x_i=0$ and $x_j=1$.
Now, consider the mapping from $[n]$ to $\{0,1\}^S$, which maps $i\in[n]$
to $(x_i)_{x\in S}$. By the above, this mapping is one-to-one, and in particular
$2^{\lvert S\rvert}=\bigl\lvert \{0,1\}^S\bigr\rvert \geq n$.
\end{proof}

\subsubsection*{Assumptions in Corollary~\ref{cor:EQ}}

We now describe an example showing that
the compactness assumption in Corollary~\ref{cor:EQ} is necessary
(a related example appears in~\cite{BEHW89,wenocur1981some} in the context of strong $\eps$-nets).
Let $X=\omega_1$ be the first uncountable ordinal,
and $C$ be the family of all intervals in the well-ordering of $X$
(a set $I\subseteq X$ is an interval if whenever $a,b\in I$ and $a\leq x\leq b$ then also $x\in I$).
The space $(X,C)$ is separable with Radon number~$3$,
but is not compact.

We claim that it does not have finite weak $\eps$-nets, even for $\eps=1$.
Indeed, let~$\mu$ be the probability distribution
defined over the $\sigma$-algebra generated by countable subsets of $X$
and assigns every countable subset of $X$ measure $0$.
Every interval is either countable or has a countable complement, and is therefore measurable.

We now claim that there is no finite $S\subseteq X$ that pierces
all intervals of measure~$1$. Indeed, let $S$ be finite, and let $m$ be the maximum element in $S$.
The interval $\{x\in X : x > m\}$ has measure $1$
but is not pierced by $S$.

\section{Proof of upper bound}
\label{sec:construction}

Here we construct weak $\eps$-nets when the Helly number
and the VC dimension of the half-spaces are bounded (Theorem~\ref{thm:weakeps}).
The property of VC classes that we use is
the following packing lemma due to Haussler~\cite{haussler1995sphere}.

\begin{theorem}[Haussler]
\label{thm:packing}
Let $B\subseteq 2^X$ be a class of VC dimension $v$.
For every distribution $\mu$ on $X$ and for every $\delta > 0$,
there is $A \subseteq B$ of size $\lvert A\rvert \leq (4e^2/\delta)^v$
such that for every $b \in B$ there is $a\in A$
with $\mu(a\Delta b)\leq \delta$.
\end{theorem}

Haussler's stated the lemma
in a dual way; the number of disjoint balls of a given radius in a VC class
is small.
Haussler's proof is elaborate, but a weaker bound can be proved fairly easily.
Indeed, consider a finite set $A$ so that
$\mu (a \Delta a') > \delta$ for all $a \neq a'$ in $A$ .
Let $x_1,\ldots,x_m$ be $m$ independent samples from $\mu$ for \new{$m \geq 2 \log (|A|)/\delta$}.
Let $Y = \{x_1,\ldots,x_m\}$,
and let $A|_Y = \{a \cap Y : a \in A\}$.
On one hand, the Sauer-Shelah-Perles lemma implies that
$A|_Y$ is small:
$\bigl\lvert A|_Y\bigr\rvert \leq (e m / v)^v$.
On the other hand, by the union bound,
$\bigl\lvert A|_Y\bigr\rvert=|A|$ with positive probability.
This implies that $A$ is small.

\begin{proof}[Proof of Theorem~\ref{thm:weakeps}]
We start by focusing on the set
$B_0 = \{b\in B : \mu(b) > 1-1/h\}$.
By the union bound, every $h$ members of $B_0$ intersect.
Since $B_0\subseteq B$ is compact with Helly number $h$,
there is a single point $x_0 \in X$ that pierces all sets in~$B_0$.

Let $0 < \eps <1$.
We construct the  $\eps$-net by induction on $N(\eps)$,
which is defined to be the minimum integer $n$ such that
$\eps\bigl(1+1/(2h)\bigr)^n > 1-1/h$.

\subsubsection*{Induction base}
If $N(\eps)=0$, define the piercing set $S = S(\mu,\eps)$ as
$$S = \{x_0\},$$
where $x_0$ is the point that pierces all half-spaces in $B_0$.
Indeed, $S$ is an $\eps$-net as every $c\in C$ with $\mu(c)\geq \eps > 1-1/h$
{is the intersection of half-spaces} from $B_0$, so $x_0$ pierces $c$ as well.

\subsubsection*{Induction step}
Let $1 >\eps > 0$ such that $N(\eps) > 0$.
We construct the piercing set $S=S(\mu,\eps)$ as follows:
Set $\delta = \eps/(2h)^2$ and pick some $A\subseteq B$ as in Theorem~\ref{thm:packing}.
Also set $\eps'=(1+1/(2h))\eps$.
Note that $N(\eps')= N(\eps)-1$.
By induction,
for each $a \in A$ with $\mu(a) > 0$,
pick a piercing set $S_a = S(\mu|_{a},\eps')$, where
$\mu|_{a}$ denotes the distribution $\mu$ conditioned on $a$.
Finally, let
$$S=\{x_0\}\cup\bigcup_{a \in A : \mu(a)>0}S_a.$$
It remains to prove that $S$ satisfies the required properties.

\

\noindent
\underline{$S$ is piercing:}
Let $c \in C$ with $\mu(c) \geq \eps$.
If $c$ is generated\footnote{\new{I.e.\ $c$ can be presented as an intersection of sets from $B_0$.}} by $B_0$ then $x_0$ pierces $c$.
Otherwise, there is some $b\in B$ with $\mu(b) \leq 1-1/h$ that contains $c$.
Pick $a\in A$ such that $\mu(a\Delta b)\leq \delta$.
Since $\mu(b) \geq \eps$ and $\mu(a\Delta b)\leq\delta$,
$$\mu(a) \geq \mu(b) - \mu(b\setminus a) \geq
\eps- \delta > 0,$$
which means that $\mu|_a$ is well-defined.
We claim that
$S(\mu|_{a},\eps')$ pierces $c$. To this end, it suffices to show that $\mu|_{a}(c)\geq \eps'$:
\begin{align*}
\mu|_{a}(c) &= \frac{\mu(c\cap a)}{\mu(a)}\\
                   &= \frac{\mu(c)- \mu\bigl(c\cap(b\setminus a)\bigr)}{\mu(a)}\tag{$c\subseteq b$}\\
                   &\geq \frac{\mu(c)- \mu(b\setminus a)}{\mu(a)}\\
                   &\geq \frac{\eps- \delta}{1-1/h + \delta}
                   \tag{$\mu(c) \geq \eps, \mu(b) \leq 1-1/h,\mu(a\Delta b)\leq\delta$}\\
                 &\geq \frac{\eps(1-1/(2h)^2)}{1-1/(2h)} \tag{$\delta=\eps/(2h)^2$}\\
                   &=\eps'.
\end{align*}

\

\noindent
\underline{\em $S$ is small:}
Let $\beta(n)$ denote the maximum possible size of $S$ for $\eps$ with $N(\eps)=n$.
The argument above yields
\[ \beta(n) \leq 1 + (4e^2/\delta)^v \cdot\beta(n-1) =   1 + (16 e^2  h^2/\eps)^v\cdot\beta(n-1).\]
Since $\beta(0) = 1$, we get

$$\beta(n) \leq (120 h^2 / \eps)^{vn}.$$
Finally, since $N(\eps) \leq 4 h\ln(1/\eps))$, we get the bound
$$|S(\mu,\eps)| \leq (120 h^2/\eps)^{4hv\ln(1/\eps)}.$$

\end{proof}

\section{Proof of lower bound}
\label{sec:RadonLB}

The proof of Lemma~\ref{lem:LB} is based on the following distribution-dependent
lower bound on the size of weak nets:

\begin{lemma}\label{lem:LBDD}
Let $C$ be a family of subsets over a domain $X$ and let $\mu$ be a distribution on $X$.
For $\eps>0$ define a graph $G=G(\mu,\eps)$ whose vertices are the sets $c\in C$
such that $\mu(c)\geq \eps$, and two sets are connected by an edge if and only if they
are disjoint.
Then, every weak $\eps$-net for $C$ over $\mu$ has size at least the chromatic
number of $G$, which is denoted by $\chi(G)$.
\end{lemma}
\begin{proof}
Let $S$ be a set that pierces all $c\in C$ with $\mu(c)\geq\eps$.
Define a coloring of $G$ by assigning to every vertex $c$ an element $x\in S\cap c$.
This is a proper coloring of $G$,
since if $\{c,c'\}$ is an edge of $G$
then $c$ and $c'$ are disjoint and therefore can not be pierced by the same element of $S$.
\end{proof}

Lemma~\ref{lem:LBDD} is tight whenever the class $C$ has Helly number $2$
(like in examples~3 and~4 in Section~\ref{sec:Example}).
Indeed, consider an optimal coloring of $G(\mu,\eps)$.
Every color class is an independent set in $G$
(which means that every two sets in it have a non-empty intersection). So,
when the Helly number is $2$, each color class
can be pierced by a single element, and we get
a piercing set of size $\chi(G)$.

The proof Lemma~\ref{lem:LB} thus reduces to a lower bound on the chromatic
number of the relevant graph, which in our case contains a copy of the Kneser graph.
The {\em Kneser graph} $KG_{n,k}$ is the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint.

Lovasz~\cite{lovasz1978kneser} proved Kneser's conjecture on the chromatic
number of this graph (this proof is considered seminal
in the topological method in combinatorics):

\begin{theorem}[Lovasz]
\label{thm:Lovasz}
The chromatic number of $KG_{n,k}$ is $n-2k+2$.
\end{theorem}

We actually do not need the full strength of Lovasz's result.
A lower bound of the form $\chi(KG_{n,n/4}) \geq n/10$
suffices for deducing the equivalence between the existence
of weak nets and finite Radon number
(in fact, even much weaker bounds suffice).
Noga Alon informed us that for this range of the parameters
there is a short and elementary proof~\cite{AlonPrivate}.
Since this argument does not appear in the literature
and may be useful elsewhere, we include Alon's proof in Section~\ref{sec:Alon}.

\begin{proof}[Proof of Lemma~\ref{lem:LB}]
Since the Radon number is greater than $r$,
it follows that there is a set $Y\subseteq X$
of size $r$ that is Radon-shattered by $C$.
Pick $\mu$ to be the uniform distribution over $Y$.
By Lemma~\ref{lem:LBDD} it suffices to show that
$\chi\bigl(G(\mu,\eps)\bigr)\geq (1-{2\eps})r$.
This follows by noticing that since $Y$ is Radon-shattered, it follows that
the subgraph of $G(\mu,\eps)$ induced by the vertices
$conv(Z)$, for $Z\subseteq Y$ is of size $\lceil {\eps r}\rceil$,
is isomorphic to $KG_{r,\lceil {\eps r}\rceil}$.

\end{proof}

\section{Proof of equivalences}
\label{sec:cors}

\subsection{The existence of weak nets}
\label{sec:corEQ}

\begin{proof}[Proof of Corollary~\ref{cor:EQ}]
Let $(X,C)$ be a compact separable convexity space.

\paragraph{$1 \Rightarrow 3.$}
By Lemma~\ref{lem:Radon}, the family $B$ of half-spaces of $C$
has finite VC dimension and Helly number.
Theorem~\ref{thm:weakeps} now implies that $B^{\cap}=C$
has finite weak $\eps$-nets for every $\eps>0$.

\paragraph{$3\Rightarrow 2.$} Obvious.

\paragraph{$2 \Rightarrow 1.$}
Assume that $C$ has weak $\eps$-nets of size $\beta = \beta(\eps)<\infty$ for some $\eps < 1/2$.
By Lemma~\ref{lem:LB}, the Radon number of $(X,C)$
is at most $\frac{\beta}{1-2\eps}$.
\end{proof}

\subsection{The threshold $1/2$ is sharp}
\label{sec:boost}

Here we describe Example~\ref{ex:boost}.
Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space,
and let $C$ be the family of all cylinders;
recall that $c\subseteq X$
is a cylinder if there exist $Y\subseteq \mathbb{N}$
and $u\in\{0,1\}^Y$ such that $c=\{v\in X : v|_Y = u\}$.
The space $(X,C)$ is a separable convexity space.
It is also compact
(this follows e.g.\ from Tychonoff's theorem).

We claim that $(X,C)$ has $\eps$-nets of size $1$ for every $\eps > 1/2$.
This follows since for every distribution $\mu$ the family $\{c\in C: \mu(c) > 1/2\}$ is intersecting\footnote{A family of sets is {\em intersecting} if every two members of it intersect.},
and since $C$ has Helly number $2$.
Hence, $\cap \{c\in C: \mu(c) > 1/2\} \neq \emptyset$ for all $\mu$, as claimed.

It remains to show that $(X,C)$ has no finite weak $\eps$-nets
for $\eps < 1/2$. By Corollary~\ref{cor:EQ},
it suffices to consider the case $\eps=1/4$.
Let $\mu$ be the Bernoulli measure on the Cantor space;
namely the infinite product of uniform measure on $\{0,1\}$.
Now, $S\subseteq X$ is a weak $\eps$-net over $\mu$ if and only if
it intersects every cylinder with co-dimension $2$. In particular
for every $i \neq j$ in $\mathbb{N}$ there must be $x\in S$ with $x_i=0$ and $x_j=1$.
By the proof of Claim~\ref{claim:cylinders} it follows that such an $S$ must be infinite.

\section{An extension}
\label{sec:ext}

Consider the space of bounded closed
convex sets in $\mathbb{R}^d$.
The corresponding convexity space in not separable nor compact.
Nevertheless, our results extend to this space as well.

The following variants of separability and compactness suffice.
A convexity space is called {\em locally-separable} if
there is a set $B \subseteq C$ so that
$C = B^\cap$ and for every finite $Y \subseteq X$ and for every $b \in B$
there is $\bar b \in B$ so that $b \cap Y$ and $\bar b \cap Y$
form a partition of $Y$.
Every separable space is locally-separable,
but there are convexity spaces
which are locally separable but not separable
(like the space of bounded convex sets in $\mathbb{R}^d$).
A convex set $c\in C$ is called {\em compact}
if the restricted convexity space $(c,\{c'\cap c : c'\in C\})$ is compact.
The {\em Radon number} of $c$ is the Radon number of
the space $(c,\{c'\cap c : c'\in C\})$.

\begin{theorem}\label{thm:ext}
Let $(X,C)$ be a locally separable convexity space such that
there exists a chain $c_1\subseteq c_2\subseteq\ldots$
of compact convex sets each of which has Radon number at most $r$
such that $\bigcup_i c_i = X$.
Then $(X,C)$ has finite weak $\eps$-nets for every $\eps>0$.
\end{theorem}

Theorem~\ref{thm:main} and its proof apply
for locally-separable and compact convexity spaces.
Theorem~\ref{thm:ext} therefore follows
by applying it to a $c_i$ in the chain
such that $\mu(c_i)\geq 1-\eps/2$.

\section{Future research}
\label{sec:discussion}

We showed that for compact separable convexity spaces,
the existence of weak nets is equivalent to having a finite
Radon number. We now suggest several directions for future research.

One interesting direction is to find a characterization that is valid
even more generally (i.e.\ for families that are not necessarily separable convexity spaces).
It is worth noting in this context that the definition of
the Radon number can be extended to arbitrary families.
One may extend the definition of Radon-shattering as follows:
A family $C\subseteq 2^X$ \emph{Radon-shatters}
the set $Y\subseteq X$ if for every partition of $Y$
into two parts $Y_1,Y_2$ there are two disjoint sets
$c_1,c_2\in C$ such that $c_1 \cap Y = Y_1$ and $c_2 \cap Y = Y_2$.

This extension of the Radon number does not characterize the existence
of weak nets for arbitrary families.
For instance, let $C=\{Y\subseteq [n] : \lvert Y\rvert > n/2\}$.
The Radon number is $2$ since every two sets in $C$ intersect,
but every weak $\frac{1}{2}$-net over the uniform distribution
has size at least $n/2-1$.
However, this family $C$ is not convex (closed under intersection),
which is a crucial property in our work.
It may also be interesting to fully understand the role
of convexity in the context of weak nets.

Alon et al.~\cite{alon2002transversal}
studied weak nets and the $(p,q)$-property in an abstract
setting and described connections to fractional Helly properties.
It may further be interesting to investigate which other combinatorial
properties of convex sets apply in more general settings.

An additional question that comes to mind
is the dependence of the size of weak $\eps$-nets
on $\eps$.
In this direction, Bukh, Matousek and Nivasch
proved an $\Omega \left( \frac{1}{\eps} \log^{d-1} \frac{1}{\eps} \right)$
lower bound on the size of weak $\eps$-nets for convex sets in $\mathbb{R}^d$~\cite{bukh2011lower},
and recently Rubin~\cite{Rubin18improved} proved an upper bound of roughly $O(1/\eps^{3/2})$ in $\mathbb{R}^2$
(which improves upon the general bound in $\mathbb{R}^d$ of roughly $O(1/\eps^d)$ by~\cite{chazelle1993improved}).
Proving tight bounds on the size of weak $\eps$-nets
is a central open problem in this area.
The general framework developed here may be useful
in proving stronger lower bounds.

\subsection*{Acknowledgements}

We thank Noga Alon, Yuval Dagan, and Gil Kalai for helpful conversations.
We also thank the anonymous reviewers assigned by SoCG '19 for their helpful comments which
improved the presentation of this work.
\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


text:
The men's freestyle sprint cross-country skiing competition in the free technique at the 2014 Sochi Olympics took place on 11 February at Laura Biathlon & Ski Complex. Ola Vigen Hattestad won the gold medal.

Qualification

An athlete with a maximum of 100 FIS distance points (the A standard) will be allowed to compete in both or one of the events (sprint/distance). An athlete with a maximum 120 FIS sprint points will be allowed to compete in the sprint event and 10 km for women or 15 km for men provided their distance points do not exceed 300 FIS points. NOC's who do not have any athlete meeting the A standard can enter one competitor of each sex (known as the basic quota) in only 10 km classical event for women or 15 km classical event for men. They must have a maximum of 300 FIS distance points at the end of qualifying on January 20, 2014. The qualification period began in July 2012.

Results
 Q — qualified for next round
 LL — lucky loser
 PF — photo finish

In November 2017, Alexey Petukhov was disqualified from the event.

Qualifying

Quarterfinals
Quarterfinal 1

Quarterfinal 2

Quarterfinal 3

Quarterfinal 4

Quarterfinal 5

Semifinals
Semifinal 1

Semifinal 2

Final
The final was held at 17:40.

References

Men's cross-country skiing at the 2014 Winter Olympics
Men's individual sprint cross-country skiing at the Winter Olympics

redpajama_set_name: RedPajamaWikipedia


This page does not claim to be a definitive compendium of BPA Members' Skydiving Records - it is a work in progress. The intention is that it will become more comprehensive over the years as it develops to become the BPA's online record book.
To claim a British National Record, please complete and return to BPA HQ Form 243 - UK National Skydiving Record, which may be downloaded by entering the form number in the Search Forms box in the header to this page. Claims are subject to verification before they appear on this page.
Records headed in orange have been broken but have been kept on this page for reference.
Aircraft: Cessna Caravan with Texas Turbine (900 hp) conversion.
BPA Members on record: Mikey Carpenter, Paul Cooper, Gary Crisp, Ally Milne, Jake Diliberto.
Congratulations to Mikey Lovemore who won double silver at the World Cup and European Championships in Speed Skydiving. His average speed of 498.23kph over four jumps is faster than the previous UK record for a single jump! He broke 500kph for the first time, one of only five people to ever do so in competition, and is now the 4th fastest speeder ever.
Skydivers: Jamie Arnold, Jacob Angier, Dunc Aspinal, Liz Boniface, Sarah Boyd, Nev Campbell, John Carter, Roy Castleman, Paul Cooper, Gary Crisp, Lee Cruddington, Oscar Daniel, Adam Dare, Peter Dennett, Chris Foster, Ted Foster, Daniel Guest, Steve Howes, Brian Jackson, Laura Kenyon, Karl Kissick, Greg Lucas, Rich Madley, Adam Mattacola, Chris Mayhew, Jimmy Meen, Mike McNulty, Ally Milne, Matty Mitchell, Daniel Parker, Glen Philips, Steve Smith, Weed Stoodley, Martin Roberson, Tiziana Rossetti, Glen Wainwright, Michael Wilson-Roberts, camera: Ewan Cowie.
This broke the 30-way head-down record of 8 September 2013.
Members kept their slots to the final attempt. This beat the 138-way set in 2012.
Skydivers: Liz Boniface, Leah Frost, Jill Hewitt, Laura Kenyon, Lorene Latour, Jenni Oakley, Naomi Maguire, Hannah Parker, Tizianna Rossetti, Emma Scrivener, Weed Stoodley.
Jackie Harper broke her own World, European and British Female Record for Time; and Spike Harper broke the British Record in Distance at on 12 September 2015 at the the British Nationals (see entry for that date).
Record broken by 14-way head-up British Record on 28 May 2017.
Skydivers: Duncan Aspinall, Liz Boniface, Mike Carpenter, John Carter, Nev Cartledge, Paul Cooper, Ben Cornick, Ewan Cowie, Gary Crisp, Adam Dare, James Davies, Jake Diliberto, Pete Dennett, Sean Freeman, Dan Guest, Steve Howes, Laura Kenyon, Ally Milne, Adam Morris, Anna Moxnes, Charlotte Newell, Matthew O'Riordan, Daniel Parker, Tim Parson-Smith, Glenn Phillips, David Reader, Martin Reynolds, Frazer Smith, Jason Southall, Weed Stoodley (= 30); Camera: Gary Wainwright & Andrew Ford.
Team members: Karen Bain, Leah Frost, Jill Hewitt, Laura Kenyon, Yolande Lee, Clara Melia, Weed Stoodley, Wendy Watkins, Karen Wilson-Roberts (organiser) and Naomi Yarrow.
Team Members: Martin Robiette, Gavin MacLeod, Douglas Preston, Colin Dickson and Craig Poxon (camera).
Matt O'Riordan, Jim Harris, Sean Freeman, Rob Silver, Dave Pacey, Matt Elliott, Alberto Mazzucalto, Ady Wild, Tiziana Rossetti, Duncan Aspinall, Darren Glover, Carl Baines, Charlotte Kemp, Gary Crisp, Steve Howes, Paul Cooper, Andy Newell, Ally Milne, Ted Foster, David Geffin, Ben Cornick, James Davies, Dan Parker, Mike Carpenter, Frazer Smith, Glenn Phillips, Jay Southall & Will Blackie.
Camera: Olly Burgin & Andrew Lovemore.
Anna Howerski, Catriona Adams, Kirsten McAndrew, Jane Henderson, Tiziana Rossetti, Yolande Lee, Karen Stryczek, and Karen Bain.
Record Score: Highest compulsory round score: 24 in time (score 9.9). British National record - 24, equalling world record set by USA on 3/8/2010 at same meet.
Brit Chicks 2007 formation jumpers: Alice Male, Ana Budjelan, Audrey Rowe, Beth King, Caroline Allen, Caroline Hughes, Charlie Cooke, Clare Smith, Clare Butcher, Conny Van Setten, Corinna Yap, Elaine Cherry, Elaine Curtis, Erica Richardson, Geetha Narayanan, Georgina Impey, Hannah Betts, Heather Little, Helen Samuelsson, Helen Taylor,Jan Gaynor, Jane Buckle, Jessica French, Jo Hawley, Jo Burns, Joanna Price, Julia Foxwell, Julie Woodrow, Kath Rybinski, Kate Stephens, Kate Shearer, Kate Burns, Kate Cooper, Katie Woods, Kris Young, Laura Brothers, Laura Mclelland, Leonie Needham, Lesley Gale, Lisa Christoffersen, Liz Matthews, Liz Danby, Liz Cass, Lynne Murray, Maria Allen, Maria Russell, Martine Howland, Maxine Hopkins, Maxine Tate, Mia Toftfelt, Michelle Garwood, Michelle Meakins, Rai Ahmed, Ros Pulham, Ruth Covell, Ruth Green, Sacha Chilton, Sarah Churchill, Sarah Smith, Siv Lindgren, Sophie Cook, Susan Farquhar, Tash Higman, Tina Wiseman, Vicki Score, Vicki Tomlinson, Vicky Scargill, Viki Bingham.
Record: Average Vertical speed over a vertical km of 442.73km/ph (275.09mph). Recorded on 2 red competition Pro-tracks and verified by the ISSA (International Speed Skydiving Association) board.
Skydivers (all British FAI International Sporting Licence Holders): Team Airkix - Julia Foxwell, Emma Hammersley, Amanda Kemp, Clare 'Sparky' Scott.
Congratulations to team Airkix (pictured above) - Claire 'Sparky' Scott, Amanda Kemp, Emma Hammersley, Julia Foxwell and their camera flyer Andy Wright. They completed a women's 4-way FS World Record sequence of 30 formations on one descent during Round 2 of the 4-way at the British Open Formation Skydiving Nationals on 31 July 2005 at Target Skysports, Hibaldstow.
Brit 100 formation jumpers: Andy Abbess, Chris Adams, Pete Allum, Paul Applegate, Steve Apps, David Bailey, Graeme Baxter, Marcus Beck, Robert Blake, Tony Bradbury, Mickey Brennan, Dave Brown, Ian Brown, Josh Burrill, Ian Cashman, Elaine Cherry, Ian Chick, Chopper, Deb Clarke, Andy Cliff, Knocker Cole, Rob Colpus, Steve Crabtree, Johnny Crewson. Phil Curtis, Eddie Davies, Steve Elvy, Rod Fairweather, Lesley Gale, Chris Gauge, Chris Good , Andy Goodall, Fran Gregory, Ken Gregory, Jason Grime, Andy Guest, Richard Hamblen, Graham Harris, Lee Harrison, Allan Hewitt, Dave Hickling, Craig Hill, Gordon Hodgkinson, Steve Houlker, Max Hurd, Nicky Johnston, Tony Jones, Jim Keery, Sally Kelvie, Dane Kenny, Pete Knight, Pete Lambsden, Ali Lawley, Rose Leggott, Dave Lewis, Lee Love, Paul Luffman, Scott Macdonald, Tim Mace, Pete Marsden, Robert Marshall, Brian Mason, Anne Maxwell, Ali Mayo, Gerry McAnally, Chris McCann, Kevin McCarthy, John McIver, Sandy McRobbie, Stuart Meacock, Ian Midgley, Steve Mikos, Robin Mills, Chris Mizzi, Darryl Moran, Dave Morris, Steve Murphy, Steve Mursell, Mick Nealis, Phil O' Connor, Sandy Oliver, Bob Parr, George Pilkington, Chris Place, Rhino, Andy Ring, Paul Rowe, Kev Sargent, Tom Scanlon, Carl Scarisbrick, Mark Scott, Chris Shaw, Tim Skinner, Eddie Smerdon, Dave Smith, Billy Somerville, Martin Soulsby, Toby Stafford, Dave Stephens, Pete Stone, Mike Strachan, William Strain, Derek Thomas, Steve Thomas, Wayne Thomas, Henry Thompson, Cathy Travers, Dave Tunks, Jan Tunks, Dave Turner, Jon Turner, Eno Van der Post, Gary Wainwright, Rich Ward, Steve Watson, Jim Weir, Richard Wheatley, Garry Willard.
Cameras: Simon Ward, Adrian Nicholas, Andy Ford, Tony Danbury.


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This is just a test post to try inserting photos, I'll need to know how to do that starting January 1st for my photo-a-day 2006 project.
Welcome to Bitstop's Blog, by me, Karen Chappell. I am hoping to start a "photo a day" project in 2006. Check back on January 1st for my first photo.


I hadn't made homemade gnocchi in at least seven years. I remember slaving away for hours last time and the fact that I had made my famous Gnocchi ai Quattro Formaggi that time meant that nobody was that keen to have something just as yummy but, unfortunately, equally fattening (we are all older and exercising less, unfortunately, so this time we stuck to a nice homemade red sauce). What got me to decide to try making them again was the fact that Trader Joe's recently started selling a pack of gnocchi imported from Italy. I tried it, and it was good. It was exactly like the gnocchi we buy at the supermarket in Italy, which means they are a bit hard and packed with preservatives. The thought of feeding my family preservatives is what pushed me over the edge. I want to avoid that whenever I can.
Chiara turned out to be an awesome potato peeler. I'll keep that in mind.
Since I remembered how long it took last time, I got an early start. The potatoes where cooked before noon, Chiara helped me peel them while hot (I did not ask her, she just insisted), then I mushed them really well. So by noon I had already made a few mistakes: too many potatoes (also I selected them without thinking about getting the driest kind) and mushing them well. After I was done I read that you don't want to do that at all. Some even suggest baking the potatoes for a while to dry them up.
I then started kneading this massive ball and I kept adding flour, then I used up the recommended amount of flour, then I kept adding flour because I was trying to get a dough. That was another big mistake. Most advice I found later says to start cutting the gnocchi as quickly as possible.
Chiara, again, came to help and she helped me roll the dough so that I could cut it up. She also tried to shape each gnocco against the cheese grater, but we still have to work on that technique.
The gnocchi turned out hard. Much harder than last time, when they wear soft and delicious. They were still good, we had them with a homemade red sauce that turned out really well, but they were chewy and filled us up quickly. Ten gnocchi were enough to make up a filling portion.
The end result. Good, but not as fluffy as I had envisioned.
This was a reminder, that handmade products require practice and passion. I was out of practice, but I resolve to trying again! What I liked most about the experience was that I got to spend time with Chiara and that she was thoroughly entertained. She did not ask to watch TV at all!
There is a new member of our key chain family. The Dragon key chain, that we have carried since the beginning of our existence, got a younger brother: the red dragon. After spending 10 years by himself and being referred to as simply the "dragon key chain", the veteran key chain will now be called Green Dragon Keychain.
Why the sudden addition you might ask? Well, we did not plan it. Last year we were contacted by a customer in Wales who told us that the Red Dragon is the symbol of Wales. After finding that out we immediately ordered it from Gianni and began reading about the association between Wales and the red dragon. It turns out that the red dragon has been officially part of the Welsh flag only since 1959, but the ties between dragons and Wales possibly go as far back as the second century when Roman cavalry stationed in Wales (and elsewhere) used the symbol. In 1485 Henry Tudor, of Welsh descent, flew the dragon of Cadwaladr during his invasion of England from France. He defeated Richard III and became king. It was then that he added the Tudor colors, green and white, to the dragon to complete the Welsh flag.
The Red Dragon Key Chain might be on the Welsh flag some day.
What was most interesting to find out was that there is no standard design for the dragon on the Welsh flag. So, technically, the design of Gianni's red dragon could, some day, make it to the flag. Until then, the Red Dragon Key Chain will be happy to represent Welsh pride and occasionally light a fire in your pocket.
House hunting for a business owner is a diversion. It does provide a diversion from my daily routine, but after not too long, I became so focused on this endeavor that I realize that this is going to be a massive diversion of my attention and of our funds. The market in Silicon Valley is moving up at roughly a 5% pace every month. This is the reason why we got beaten every time, even we offered $50,000 over asking, and possibly the reason why we will just decide to sit this one out.
Then, something very interesting came on the market. By interesting I don't mean something that will cause us to spend our bottom dollar: I mean something that we had to go see and bring Chiara and our cameras along.
The unit is entirely covered in tile. With the exception of the ceilings and a few, rare, surviving walls, it depicts Saint Francis and the Sun and what I think must be depictions of Assisi.
The unit is well laid out with two bedrooms and a den, an attached garage, two private patios, and an eye-popping asking price of $485,000. At 1102 square feet that is a whopping $440 per square foot. Far more than even the unit that just sold in the same complex and that was far more attractive. Take a look at it in detail on Redfin. It's worth it!
Zillow thinks it's worth less than $400,000. Let's not talk about e-appraisal, which hasn't yet figured out that the market is going in the other direction. On the other hand, it is located in a very attractive complex with a wonderful pool, and the schools are really good.
I think that it should fetch $468,000, but I would not be surprised if it sold for asking. In fact, given the craziness of the market today, I foresee 3 offers and a $495,000 winning bid. I will be curious to see how it pans out!
In the meantime we will hopefully be busy bidding on something else.
I don't like purses. Why? Well, the word "purse" ALWAYS implies that it is some sort of fashionable handbag that ladies carry. Any sort of bag a lady carries is automatically seen as a "purse". That makes no sense to me. With that said, I am definitely not the type of person to carry a purse, a handbag, or what have you. All I need are pockets to carry my phone, keys and wallet and I'm ready to go!
My first impression of Marcopoloni was how it is a company that perpetuates passion, hard work and genuineness. That definitely is what Marcopoloni embodies, no doubt. The products being offered here are so full of life and character that you can't help but get drawn to them. Well, let's fast forward to when I had access to the Campomaggi catalog. Found my first Campomaggi creation pretty quickly, I'd say. Not only is it my first Campomaggi bag in general, it is a custom order! Named after the inventor of the helicopter, Enrico Forlanini, it is every bit as logical and practical as its namesake.
After carrying the Forlanini with me on my adventures, the bag has become an appendage of me (Notice how I use the word "bag" to keep the ambiguity of what it is alive. It's a bag. Simple as that). What has changed since I have the Forlanini? Well, I still just carry my essentials, really. The typical phone, keys and wallet with the new addition of a sketchbook, a pencil, and mints (ginger mints, to be specific). The most important aspect of my bag is how it represents me. The ruggedness of Campomaggi's signature distressed leather coupled with its simplistic, utilitarian design just makes so much sense to me. It symbolizes my ideals of keeping things simple and straightforward. Lastly, I truly feel that Campomaggi embodies what it is like to be true to yourself–to your roots–in all of his products. As someone who has never envisioned herself seeing any sort of need to carry a bag, my first Campomaggi bag has truly become the companion I never expected.
My Forlanini as it is, scratches and all.
After owning my ALDO Leather Dress Belt for a few years, I found its quality so superior that I stopped wearing any other belt I owned. I soon realized that wearing a dress belt with jeans is not good for looks and it is not good for the wear and tear on the belt. So I set my eyes on the more casual LUCA Leather Belt but it was always close to going out of stock (I did not want to risk depriving our customers) as every time a new shipment would arrive from Italy the availability would appear to be just barely sufficient to meet demand. So I increased our order quantities and finally got enough belts to go around to feed my growing hunger for a LUCA. That's when I realized that I had another problem: our smallest size was for a range of waist sizes from 32 to 37. My pants size is 31 and I don't like the extremity of the belt to flap around too much.
Luckily I brushed my hesitation aside and ordered the 30-35, which has just arrived. I waited for Anita's stringent customs inspection to be complete (I'm referring to her quality control inspection) and I finally got to have my very own LUCA. This whole process took longer than a year!
I am thrilled with it so far, but I will write a report in a few weeks with all the details. For the time being enjoy the photo. Do you like my new Campomaggi bracelet?


Taken from here and nixCraft page (click there to see larger text).
Python: my formatting is my syntax!
PHP: can I GET guys anything?
C/C++: If you're not allocating memory, you're not living!
C#: I am nothing like my father.
Rust: I'll be designated driver tonight!
Fortran: I am unparalled in numerical ^ scientific computation!
Ah there are some updates!
Master programming assignment help for C, C++, Java and other dialect Ask inquiries concerning homework assignments or other C/C++ related problems.
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Elevation on Central is located at 4650 N. Central Avenue Phoenix, AZ and is managed by Pillar Communities, a reputable property management company. Elevation on Central offers Studio to two-bedroom apartments. Amenities include BBQ/Picnic Area, Bike Racks, Business Center, Ceiling Fan, Clubhouse and more. This rental community is pet friendly, welcoming both cats and dogs. Property is located in the 85012 ZIP code. For more details, contact our leasing office today!


text:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading;
using CIAPI.IntegrationTests.Streaming;
using CIAPI.Rpc;

using NUnit.Framework;
using Salient.ReliableHttpClient;
using Client = CIAPI.Rpc.Client;

namespace CIAPI.IntegrationTests.Rpc
{
    [TestFixture]
    public class ErrorHandlingFixture : RpcFixtureBase
    {
        [Test]
        public void ShouldGiveGuidanceWhenSpecifyingInvalidServerName()
        {
            try
            {
                var rpcClient = new Client(new Uri("http://invalidservername.asiuhd8h38hsh.wam/TradingApi"), new Uri("http://foo.bar.com"), AppKey);
                rpcClient.LogIn("username", "password");
                Assert.Fail("Expected exception");
            }
            catch (Exception ex)
            {
                Assert.That(ex.Message, Is.StringContaining("Invalid response received.  Are you connecting to the correct server Url?"), "Expecting some info explaining that the server Url is invalid");
            }
        }

        [Test]
        public void Http200ErrorsShouldThrow()
        {
            try
            {
                var rpcClient = BuildRpcClient();
                rpcClient.Http200ErrorsOnly = true;

                const int moreThanMaxHeadlines = 1000;
                rpcClient.News.ListNewsHeadlinesWithSource(source: "dj", category: "UK", maxResults: moreThanMaxHeadlines);
            }
            catch (Exception ex)
            {
                Assert.That(ex.Message, Is.StringContaining("You cannot request more than 500 news headlines"), "Expecting some info explaining that the server Url is invalid");
            }
        }

        [Test]
        public void LogOutShouldInvalidateSession()
        {
            var rpcClient = BuildRpcClient();

            var headlines = rpcClient.News.ListNewsHeadlinesWithSource("dj", "UK", 3);
            Assert.That(headlines.Headlines.Length, Is.GreaterThan(0), "you should have a set of headlines");

            rpcClient.LogOut();

            try
            {
                rpcClient.News.ListNewsHeadlinesWithSource("dj", "AUS", 4);
                Assert.Fail("the previous line should have thrown an 'Session is null. Have you created a session? (logged in)' exception");
            }
            catch (ReliableHttpException e)
            {
                Assert.That(e.Message, Is.StringContaining("Session is null. Have you created a session? (logged on)"), "The client should have rejected the request");
            }
        }

        [Test]
        public void TheSameErrorShouldhaveTheSameErrorMessage()
        {
            var rpcClient = BuildRpcClient();

            var error1 = GetBadRequestErrorMessage(rpcClient);
            var error2 = GetBadRequestErrorMessage(rpcClient);
            var error3 = GetBadRequestErrorMessage(rpcClient);

            Console.WriteLine("Error1:{0}\nError2:{1}\nError3:{2}", error1, error2, error3);
            Assert.That(error1, Is.EqualTo(error2), "errors should be the same");
            Assert.That(error2, Is.EqualTo(error3), "errors should be the same");
        }

        private string GetBadRequestErrorMessage(Client rpcClient)
        {
            var errorMessage = "No error thrown";
            try
            {
                const int moreThanMaxHeadlines = 1000;
                rpcClient.News.ListNewsHeadlinesWithSource(source: "dj", category: "UK", maxResults: moreThanMaxHeadlines);
            }
            catch (ReliableHttpException ex)
            {
                errorMessage = "Message: " + ex.Message;
            }
            return errorMessage;
        }

        [Test]
        public void ErrorMessageShouldContainDetailsOfErrorResponseDTO()
        {
            var rpcClient = BuildRpcClient();

            var error1 = GetBadRequestErrorMessage(rpcClient);

            Assert.That(error1, Is.StringContaining("You cannot request more than 500 news headlines"));
        }

    }
}

redpajama_set_name: RedPajamaGithub


text:
Florida vs. Charleston Southern: Seven Gators suspended for opener
Florida will be missing several players in its debut under Dan Mullen.
By Andy Hutchins Sep 1, 2018, 6:30pm EDT
Share All sharing options for: Florida vs. Charleston Southern: Seven Gators suspended for opener
Photo by Logan Bowles/Getty Images
Seven Florida Gators players will miss Florida's season opener against Charleston Southern on Saturday, the program announced on Saturday. Nick de la Torre of Gator Country reported the news prior to the program's confirmation; both de la Torre's report and Scott Carter's confirmation of it include a source saying the players are suspended for "not living up to the Gator standard."
(I'll let you guess what sort of source might repeat the same phrase Dan Mullen has repeated for months while revealing suspensions to the media.)
Senior pass-rusher CeCe Jefferson — who attended SEC Media Days as part of Florida's traveling party this summer — is perhaps the biggest name of the players suspended, with sophomore wide receiver Kadarius Toney serving as the biggest name among offensive players. They are joined in purgatory by sophomore defensive tackle Kyree Campbell, redshirt junior defensive lineman Luke Ancrum, sophomore cornerback Brian Edwards, sophomore running back Adarius Lemons, and walk-on freshman wideout James Washington.
But while Ancrum, Campbell, and Toney were all part of the well-publicized summer encounter between Florida players and locals including a man familiarly known as "Tay Bang" — Ancrum was named by "Tay Bang" as an instigator of a simmering feud between players and local, Campbell and Toney both brandished AirSoft guns in the encounter, and Toney later being stopped by Gainesville police with an AR-15 in his car — Edwards, Jefferson, Lemons, and Washington were not known to have had any run-ins with law enforcement or other rule-breaking.
So Jacqui Franculli of Gators Territory reporting that, yes, Campbell and Toney's suspensions are related to that, but that Jefferson's is related to academics? That makes sense.
It would also make sense if any or all of these suspensions related to Florida's drug-use policies, as many Gators have served season-opening suspensions for violations of that policy. But this round of announced suspensions does not feature any explanatory verbiage about Florida players being suspended "due to University of Florida policy," long understood as code for violations of the program's drug-use policies.
Then again, that code was largely not in use during Mullen's first stint in Gainesville as offensive coordinator under Urban Meyer, as Meyer often held players out of games without explicitly addressing their suspensions.
Campbell, Edwards, Jefferson, Lemons, and Toney all appeared on Florida's depth chart for this week, but none was listed as a starter.

redpajama_set_name: RedPajamaCommonCrawl


text:
The 2006 Victoria Park state by-election was a by-election held on 11 March 2006 for the Western Australian Legislative Assembly seat of Victoria Park in the inner southeastern suburbs of Perth.

Background
The by-election was triggered by the resignation of Labor member Dr Geoff Gallop on 16 January 2006. Gallop had represented the area since a by-election on 7 June 1986, and had served as a minister, shadow minister, Opposition Leader (1996–2001) and, finally, as Premier of Western Australia since 2001. Gallop announced his resignation as Premier and from State Parliament effective immediately in order to aid his recovery from depression, and that "in the interests of my health and my family, I've decided to rethink my career."

The Speaker of the Legislative Assembly advised Parliament on 7 March 2006 that he had received a letter from Gallop on 25 January announcing his resignation as the member for Victoria Park, and announced that under section 67(2) of the Electoral Act 1907, he had on 2 February 2006 directed the Acting Electoral Commissioner to proceed with an election in the district. The battle for Labor party preselection was hard fought, with 31-year-old DPP lawyer Ben Wyatt defeating former national president of the Liquor Hospitality and Miscellaneous Workers' Union, Helen Creed, at a vote on 1 February 2006 of the 14-member Administrative Committee. The by-election, called for 11 March 2006, attracted a total of eleven candidates, one of the largest in Western Australian electoral history.

The seat of Victoria Park, which elects a single member to the Western Australian Legislative Assembly using a preferential method, was established in 1929 by the Government of Western Australia under the Redistribution of Seats Act (No 1 of 1929), which was given assent by the Governor on 15 April 1929. The seat is considered a safe Labor Party seat, and has been held by the party between 1930 and 1945, and since the 1953 state election.

Timeline

Candidates
The by-election attracted eleven candidates, significantly more than the previous Victoria Park election, which attracted only five candidates. Both major parties fielded a candidate, with the Greens, One Nation, Daylight Saving Party, Family First, the Christian Democrats and the unregistered Socialist Alliance also represented, as well as three independents.

One Nation candidate Sue Bateman made national headlines when it emerged that she was running for the seat of Victoria Park. The former One Nation president was accused of being a prolific online racist, using the forum Stormfront. She was also dismissive of Gallop's battle with depression, saying that average Joe did not have the luxury of becoming depressed.

Results
As largely predicted, Ben Wyatt retained the seat for the Labor party, with a swing against him of 4.86%. Wyatt claimed victory within two hours and became one of three Aboriginal parliamentarians in Western Australia.

Voter turnout was low, given that Victoria Park is a safe Labor seat, with only 64% of voters casting their vote: the lowest for a by-election in the state since 1936.

References

2006 elections in Australia
Western Australian state by-elections
2000s in Perth, Western Australia


text:
Christian Heights UMC
The blog of Christian Heights United Methodist Church. Where Everybody is Somebody Special!
feedback@christianheightsumc.org
Sunday School: 9:30 AM
Morning worship: 10:30 AM
All are welcome to join us for all publicized events.
Map to CHUMC
Cross Fundraiser
Sermons for Shut-ins
Scripture quoted by permission. Quotations designated (NET) are from the NET Bible® copyright ©1996-2006 by Biblical Studies Press, L.L.C. www.bible.org All rights reserved.
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Help Us Add A Methodist Flame To Our Cross
Model for the new cross
Christian Heights wishes to upgrade the cross on the outside of our building. We are asking our extended church family to help us do this. Since the beginning Christian Heights has had the simple white cross on the side of the building. Then late last year a sum of money was donated to jump start the process of adding a flame.
Based on a quote from a metal shop, it will cost $2,948.00 to add a flame to the existing cross. In addition, it was proposed that we could upgrade the lighting shining on the cross to new high-intensity bulbs. If the funds were available this would be an extra $500.
The church wishes for funds to be donated as memorials to someone. When you send the check please include a note designating the donation as for adding the Methodist flame to the cross and to whom the memorial is made in the name of. Please pray on this while you consider participating. Thank You!


text:
Bust, Eleanor Roosevelt, Presidential First Lady Series, Famous American Woman, 13", White Stone, Custom Casting by American Artists, Museum Quality Detail, Democrat Woman
Product Code: 737W
PRESIDENTIAL FIRST LADY SERIES
13" CAST WHITE STONE, MUSEUM QUALITY PIECE
Eleanor Roosevelt, the wife of President Franklin D. Roosevelt, was a distinguished public figure in her own right. She served as first lady from 1933-1945, supporting the New Deal policies of her husband, as well as being a strong advocate for civil rights and human welfare. Much of her life was spent in service to others, from working as a social worker in the East Side slums of New York at age seventeen, to chairing the United Nations committee responsible for getting the Universal Declaration of Human Rights drafted and approved. At her memorial service, Adlai Stevenson said of her, "What other single human being has touched and transformed the existence of so many?"
President Obama Oath of Office Pen Souvenir White House Pen- No Box Army Lapel Pin, 7/8" Diameter, Enamel, Brass, Polybag Vietnam Veteran Lapel Pin, White House Gift Shop Gift Box Options, Thank you for your Service! American Flag Lapel Pins, Classic Size, 3/4" Across, 24K gold, Premium Clutch, Velvet Jewel Case and Custom Formal Black White House 2-Piece Outer Presentation Box, Select Package Option
1600-Department of Commerce "BO" The First Family Portuguese Water Dog is 5" x 7" x 7", Huggable and Sweet! Presidential Guest Style Eagle Seal Cufflinks, 24K Gold Finish, Elegant Design, with Custom Case and Outer White House Seal Box, Coins, Washington D.C. Coin, Panorama Commemorative, White House, U.S. Capitol, Lincoln & Jefferson Memorial, Washington Monument, Gold Finish, 1.5" Diameter, Two Piece Protective Capsule
Excellent product; welcomed addition to my home January 24, 2018
Reviewer: Steven James Beto from Northfield, MN United States
For years I have looked for a likeness of Eleanor Roosevelt in my price range. (She looks great amongst the orchids.) For future needs, I will first check your store.
Products > Statues
Products > Statues > Presidential & American Leaders


Born and raised in Tucson, Arizona, Danny chose to attend Pima Community College on a golf scholarship, he graduated with his Associates Degrees in Business and Art. Immediately following graduation, Danny joined the PGA of America in 2003 where he has had a successful career. His teaching philosophy is to simplify the swing and to educate his students so they can better enjoy their rounds, he enjoys working will all caliber of golfers, especially our youth! Danny is available for private lesson sessions, golf schools, and group/corporate clinics.
A native Tucsonan, Rich was a baseball player growing up, not picking up golf until age 17. He fell in love with the game while competing on the team and earning an Associates Degree in Psychology at Pima Community College, where he now serves as the Head Men's Golf Coach. He turned pro in 1997 and has had a successful career as a player and coach. Rich approaches teaching from a players' perspective and has a passion for short game and course management. He is available for private lesson sessions, golf schools, and group/corporate clinics.
Rich Elias, PGA - Head Golf Professional.
Born and raised in Seattle, Washington, Glen chose to attend the University of Montana where he taught golf classes and graduated with a Bachelor's Degree in business. Immediately following graduation, he joined the PGA of America in 1993 and moved to Tucson, Arizona where he has enjoyed a successful career as a player, coach, and instructor. His philosophy is simple and encourages a self-coaching approach utilizing balance and self-awareness. He specializes in all aspects of the short game and on course coaching. Glen is available for private lesson sessions, golf schools, and group/corporate clinics.
Glen Griffith, PGA - Director of Membership.
Born and raised in Minnesota, Joe Attended the University of Minnesota-Duluth, where he played on the golf team while also teaching golf and assisting with coaching duties. Joe's Golf instruction career started at Braemar Golf Course in Edina, MN. A PGA member for over 20 years, Joe gained experience in teaching at facilities from South Carolina to California. Joe's teaching philosophy is to keep things as simple as possible and focusing on his best to communicate with each student. Joe enjoys teaching golf at all levels, from juniors to low handicappers. Joe Is certified in teaching adaptive golf to people with disabilities, and is the lead instructor for PGA Hope Tucson, a program teaching golf to veteran's with disabilities. Joe is available for private lessons, group clinics and golf schools.
Joe Riekena, PGA - Assistant Golf Professional.
Born and raised in beautiful western Montana. Trevor left Gods country to move down to Arizona so He could attend The Golf Academy of America, where he earned an Associates Degree in Golf Complex Operations. Trevor is certified in club fitting and instruction. He is also Omni's on-site club repair specialist, handling any club repair or fitting needs for members and guests. Furthering his education he joined the PGA in May of 2015, and is working toward his PGA membership. Trevor specializes in full swing instruction with a philosophy of teaching timing and tempo in order to bring the club into the proper position at impact. Trevor enjoys giving one on one lessons to break down exactly what is occurring during the swing. He can work with multiple styles and abilities of golfers, and is available for private lessons and fittings.
Trevor McLean - Assistant Golf Professional.
Born and Raised in Tucson, Arizona, Dillon grew up playing basketball and baseball, and started playing competitive golf at 15 years old. Currently in level 3 of PGA PGM program, Dillon has been teaching golf for over 5 years. He loves working with the youth, running many junior camps and coaching the PGA Jr. League team. Having both parents, aunts and uncles as school teachers, Dillon has earned a passion for teaching. Dillon's teaching philosophy focuses on bettering the students flexibility, strength and mental abilities. Dillon is available for private lesson sessions, golf schools, and group/corporate clinics.
Dillon Molinar - Assistant Golf Professional.
An Arizona native splitting time between Bullhead City and Tucson, Daulton started golfing as a freshman in high school. He began working in the golf industry at a local driving range while continuing his education at Pima Community College. In 2016 Daulton joined the PGA as an Associate and continues to work towards his Class A membership. He specializes in junior golf instruction and also enjoys teaching short game fundamentals. He is available for private lessons, golf schools, and group/corporate clinics.
Daulton Altamirano - Assistant Golf Professional.
Born and raised in Midland, Texas, Bevan attended Texas A&M University-Corpus Christi, where he earned his Bachelor's Degree in Music Performance. Following graduation, Bevan moved to Tucson to begin work on his Master's Degree in Music Performance at the University of Arizona. While there, he began his golf career in 2018 and was promoted to Assistant Golf Professional with Omni Tucson National Resort. Strengths include: on course game management, mental focus, and properly striking the golf ball. Bevan is available for private lesson sessions and group/corporate clinics.
Bevan Moody - Assistant Golf Professional.


text:
Sponkit Celebrity Blog
Celebrity Couples, Royal Family News, Celebrity News, Hollywood Stars, Music Celebrities.
Brad Pitt: Green Day Concert With His Sons
Brad Pitt was spotted as he gets in some father-son bonding time with his two oldest sons — Maddox, 8, and Pax, 5 — at the Green Day concert in the first row of section 114 at the New Orleans Arena on Friday (August 7).
Band members Billie Joe Armstrong, Mike Dirnt and Tre Cool rocked out for a straight 2.5 hours.
At one point during the concert, Bille Joe commanded. "Everybody off your as*es!"
Meanwhile, Brad Pitt thinks his new movie is "outrageous".
The actor loved working with director Quentin Tarantino on second world war film 'Inglourious Basterds' and says it was an even better experience than he had expected.
He said: "I did the movie because I'm a Tarantino fan. It's an outrageous movie and it's hilarious. From my perspective, making it was like a reward for decades of hard work. I've never had a nicer experience making a movie. It's a revenge fantasy. It kind of stomps and tap dances all over the WWII stigma. I expect I will get some flack for it. But it's Quentin Tarantino, man."
While Brad enjoyed working on the movie, he insists his own enthusiasm failed to match that of his German co-stars.
He added to Parade magazine: "You know who enjoyed making the movie the most? The German actors! I realize that they, being a younger generation, have to carry the stigma from World War II. And, in this film, they got to stomp all over that thing. Everyone's born of a particular nationality and it seems you have to carry the sins of the fathers even if you had nothing to do with their crimes. And you really feel the frustration with that and the need to shake that off."
Categories Brad Pitt Post navigation
Source: Lindsay Lohan Tearful At Dinner With Samantha Ronson
Robert Pattinson & Kristen Stewart's Low-Key Concert Night
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Description: The garbage disposal is mounted to the underside of a sink and is designed to store waste food in a hopper chamber (just beneath the sink drain and the upper part of the disposal). When turned on, the motor spins the flywheel and attached impellers at almost 2,000 RPM.... 17/09/2011�� Dr Fix It and Bob Cat Billy show you how to save money fixing a broken garbage disposal.
With garbage disposals, there are some distinct signs that may give you clues about what to do. You might consider replacing your kitchen garbage disposal if you notice any of these situations. 1.
I have found that grinding up glass in a garbage disposal is actually good for the disposal. If I ever break a regular class I take the broken pieces and put in my disposal. It seems to sharpen the blades and I don't have any problems After.


In a revealing interview, the newly grounded young star opens up about her love life, family drama, and relaunching her acting career. Read the full-length article here.
Lindsay opens up. Jacket, $1,195, Balenciaga by Nicolas Ghesquière. balenciaga.com. Bangle (worn throughout), $3,850, Cartier. 800-CARTIER.
Jacket, $1,300, D&G. dolcegabbana.it. Jeans, $750, Diesel Black Gold. 877-433-4373. Pumps, $825, Christian Louboutin.
Jacket, $6,010, Chanel. 800-550-0005. Tank, $14.50, Gap. gap.com.
Lindsay Lohan's fashion party. Jacket, $6,010, Chanel. 800-550-0005. Tank, $14.50, Gap. gap.com. Shorts, $69, Guess by Marciano. 800-39-GUESS. Necklaces, $259-$265, Iosselliani.
One-of-a-kind jumpsuit, Michael Kors with Crystallized Swarovski Elements for Saks Fifth Avenue. On display at Saks Fifth Avenue, NYC; 212-940-4145.


text:
Cerodontha nitidiventris är en tvåvingeart som beskrevs av Malloch 1934. Cerodontha nitidiventris ingår i släktet Cerodontha och familjen minerarflugor. Inga underarter finns listade i Catalogue of Life.

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nitidiventris


THE search for the genetic causes of autism has identified another likely candidate.
Dan Arking at Johns Hopkins University in Baltimore, Maryland, and his colleagues studied the DNA of 1295 autistic children and their parents. They found that a common variant of a gene called CNTNAP2, which helps coordinate interactions between cells in the nervous system, was often associated with autism.
While many genes are believed to contribute towards the disorder, CNTNAP2 could be a good target for drugs because it is active throughout life, rather than just during development as other such genes are. The results were presented at a meeting at Cold Spring Harbor Laboratory, New York, earlier this month.


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Beware the Cresting Wave of Emerging Market Debt
Investors buying a record volume of risky bonds from borrowers in developing economies may become filled with regret.
By Dan Weil
Illustration by Andrea Manzati
Risky bonds are being sold at an unprecedented pace in developing economies as borrowers rush to do favorable deals with yield-hungry investors. The surge may not end well.
Governments and companies in emerging markets issued $252.7 billion of speculative-grade bonds this year through November 14, smashing the record $172.1 billion sold in all of 2016, according to Dealogic. The 47 percent jump from last year's level may be a boon to borrowers, but it raises concern that investors will be harmed by their reach for yield.
"With the high-yield markets facing headwinds after a very impressive run, an onslaught of new issuance runs a notably higher risk of being the straw that breaks the back of elevated valuations," says Mohamed El-Erian, chief economic adviser at Allianz. "This is a possibility that investors in the asset class would regret."
Tajikistan — not exactly a bastion of wealth — raised $500 million in September from its first international bond sale, a sum more than 7 percent of its gross domestic product. The Central Asian country will pay an annual interest rate of 7.125 percent for the below-investment-grade debt, according to the Financial Times. In June, Argentina, which has defaulted on its debt eight times since gaining independence from Spain in 1816, sold $2.75 billion of 100-year bonds, also with a 7.125 percent coupon.
It's difficult to blame borrowers for extracting favorable deals when markets are wide open. Eager to put their money in emerging-markets bonds on the belief that U.S. interest rates will remain very low, investors may be underestimating how much the Federal Reserve will tighten its monetary policy and the impact that will have on emerging-markets debt, according to analysts.
"This is the tail end of the credit cycle," says Win Thin, global head of emerging-markets strategy at Brown Brothers Harriman & Co. "Borrowers are trying to push through the door as fast as possible, given that the Fed is raising interest rates."
If the forecasts of Federal Reserve officials are correct, by the end of next year the fed funds rate will have at least tripled from a range of 0.5 percent to 0.75 percent at the beginning of 2017. The central bank — which last raised its benchmark rate in June to a range of 1 percent to 1.25 percent — signaled in September that it would likely hike rates again in December and then three times in 2018. Still, on November 14, the fed funds futures market priced in a 55.4 percent probability that the central bank will lift rates no more than once in the first 11 months of next year, meaning many investors may be in for a rude awakening if the Fed's projections turn out to be correct.
"We already saw the impact [of Fed tightening] during the taper tantrum of 2013," says Ali Hassan, senior fixed-income analyst at Thornburg Investment Management in Santa Fe, New Mexico. He's referring to the surge in bond yields globally after former Fed chair Ben Bernanke shook markets in May 2013 with his comments that the central bank was considering tapering its bond-buying program, which it had implemented after the 2008 financial crisis. Now, as the Fed continues steps toward tightening its monetary policy, Hassan warns that higher rates will hurt countries in the early stages of economic recovery, such as Brazil.
Rate hikes tend to push the U.S. dollar higher against foreign currencies, making it more expensive for emerging-markets borrowers to repay their dollar debt. Meanwhile, the fragile state of many developing countries' finances is problematic because of their large budget and current-account deficits. Other risks include volatile commodity prices — rising crude prices can hurt oil-importing nations, for example, while falling prices can squeeze oil exporters — and geopolitical instability in areas like the Middle East and North Korea.
Many emerging-markets bonds are illiquid, which means they can be difficult for bondholders to sell when others are trying to do the same. "People could be running for a small and shrinking door," says Thin, warning that investors may become stuck holding debt that's dropping in value as buyers vanish.
Knowing the risks, why would investors accept the 7.125 percent rate offered by Argentina and Tajikistan? "It's an indication of frothiness and a reach for yield," Hassan says. By comparison, the Bank of America Merrill Lynch U.S. High Yield Index had an effective yield of 5.84 percent on November 17.
Thin is a bit surprised by investors' enthusiasm for emerging-markets debt, saying that "credit quality has really declined" and interest rates on junk bonds "aren't high enough in many cases to compensate for the risk." He points to Venezuela defaulting on some of its bonds in November, catching many debt holders off guard.
"A lot of investors are talking themselves into the idea that this is not such bad debt," Thin says of emerging-markets deals. Even with more cynical investors, he adds, "you get the sense that people are holding their nose as they buy."
This doesn't mean every bond is toxic, with analysts still seeing investment opportunities in countries with stronger fundamentals, such as China, South Africa, Russia, and Thailand. But the burst of junk bond deals has made Arvind Rajan, head of global and macro at Prudential Financial's PGIM Fixed Income division, more cautious. "We're participating in fewer than one in ten issues that come to market," he says.
High Yield: Not Armageddon, but Hardly Paradise Either
Dan Weil
This content is from: Research
Investors Are Keeping a Close Eye on the Fed
Thomas W. Johnson


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The Peppermint Kicks - self titled
So you like the pop side of rock and roll, but you still need it to really rock? Well then The Peppermint Kicks are the band for you! Co-fronted by Dan Kopko (Watts, The Shang Hi Los) and Sal Baglio (The Stompers, The Amplifier Heads), The Peppermint Kicks are a throwback to the days of CREEM magazine, Don Kirshner's Rock Concert, $7 arena shows, and band posters all over your bedroom wall. The band's self-titled debut album releases today on the mighty Rum Bar Records. This ten-song long player would not have sounded out of place on your teenage turntable alongside T. Rex, Cheap Trick, the Ramones, and The Who. Equal parts glam rock, power pop, first wave punk, and big hooky classic rock, The Peppermint Kicks sound has been likened to "a cool drink of strawberry lemonade, with a sprig of mint and a gut-punch chaser". Yum! Kopko and Baglio are both longtime fixtures of the Boston rock and roll scene. Combining the musical visions of both of these artists (much of it shared), you get a killer rock and roll band steeped in the timeless traditions of pop songwriting. What could be better?
I love that The Peppermint Kicks have come out big! The band kicks off its debut album (and entire discography) with a full-on rock anthem in "When Rock & Roll Met Your Dad". An epic love letter to rock and roll, this is the perfect introduction to The Peppermint Kicks and what they're all about. It's immediately followed by "Hey Fanzine!", which takes us back to the glory days of the rocking power pop radio hit. Building off such an auspicious start, the album proceeds to take you on a whirlwind tour of the coolest sounds of rock and roll's heyday. "Strawberry Girls" is a snappy, melodic tune that sounds like it could have sprung from the later '60s U.K. "I Don't Hear A Single" is a timeless pop song that ironically sounds...like a single! "Johnny D's (Play It Again)" is a vintage Danny The K fists-in-the-air sing-along rocker. "Shag '72" is an old style rock and roll toe-tapper with hooks for days. "Stooge" channels peak-era Rolling Stones. To my absolute delight, "Rock & Roll Rampage" embraces nearly every arena rock cliché in the book.
It is 100% safe to say that any fan of Watts or The Amplifier Heads will love The Peppermint Kicks. You get the best of both bands here. Kopko and Baglio both sing, play guitar, and contribute some piano and percussion. Lead songwriting duties are split 50/50. Legend has it that the ghost of Little Richard commanded these two individuals to join forces. I, for one, do not doubt that narrative. This band and this album are a celebration of a moment in time when rock was king. Every song is a hit -- unless it's an anthem. And while no one is getting rich these days writing rock and roll songs with great hooks, you know you've made it when the likes of Palmyra Delran and Genya Ravan are singing your praises. Whether you were a teenager in the '70s or simply wish you could have been, this pop-rock cocktail ought to hit the spot!
The Peppermint Kicks by The Peppermint Kicks The Peppermint Kicks by The Peppermint Kicks
https://rumbarrecords.bandcamp.com/album/the-peppermint-kicks
https://thepeppermintkicks.bandcamp.com/releases
https://www.facebook.com/thepeppermintkicks
https://www.facebook.com/RumBarRecords
Posted by Lord Rutledge at 8:26 AM
Labels: Boston, Glam, Hard rock, Power Pop, Rock n' Roll, Rum Bar
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