text:
\section{Introduction}
Let $G = ([n],E)$ be a simple undirected graph on the vertex set
$[n]= \{1,\dots, n\}$.  Let $X = \left(
\begin{smallmatrix}
  x_{1}& \cdots & x_{n}\\
  y_{1}& \cdots & y_{n}
\end{smallmatrix}\right)
$ be a generic $2\times n$ matrix and $S = \Bbbk[\begin{smallmatrix}
  x_{1}& \cdots & x_{n}\\
  y_{1}& \cdots & y_{n}
\end{smallmatrix}
]$ the polynomial ring whose indeterminates are the entries of~$X$ and with
coefficients in a field~$\Bbbk$.  The \emph{binomial edge ideal of $G$} is
$\bei_{G} = \<x_{i}y_{j} - y_{i}x_{j} : \{i,j\}\in E\> \subset S$, the ideal of
$2\times 2$ minors indexed by the edges of the graph.  Since their inception in
\cite{herzogBEIorig,ohtani2011graphs}, connecting combinatorial properties of
$G$ with algebraic properties of $\bei_{G}$ or $S/\bei_{G}$ has been a popular
activity.  Particular attention has been paid to the minimal free resolution of
$S/\bei_{G}$ as a standard $\mathbb{N}$-graded $S$-module \cite{EneZarojanu2013,
kiani2016castelnuovo}.  The data of a minimal free resolution is encoded in its
graded Betti numbers
$\beta_{i,j}(S/\bei_{G}) = \dim_\Bbbk\text{Tor}_i(S/\bei_{G},\Bbbk)_j$.  An interesting
invariant is the highest degree appearing in the resolution, the
Castelnuovo--Mumford regularity
$\reg (S/\bei_{G})=\max\{j-i:\beta_{ij} (S/\bei_{G})\neq 0\}$.  It is a
complexity measure as low regularity implies favorable properties like vanishing
of local cohomology.  Binomial edge ideals have square-free initial ideals by
\cite[Theorem~2.1]{herzogBEIorig} and, using~\cite{conca2018square}, this
implies that the extremal Betti numbers and regularity can also be derived from
those initial ideals.  In this paper we rely on recursive constructions of
graphs rather than Gröbner deformations.

At the time of writing it is unknown if the regularity of $S/\bei_{G}$ depends
on the characteristic $\operatorname{char} (\Bbbk)$ of the coefficient field.  Indication that it
is indeed independent comes, for example, from a purely combinatorial
description of the linear strand of the determinantal facet ideals
in~\cite{linstrand}.  Here we use only combinatorial constructions that are
independent of $\operatorname{char} (\Bbbk)$, based on small graphs for which the minimal free
resolutions are independent of~$\Bbbk$.  Our starting point are the following bounds
due to Matsuda and Murai~\cite{matsuda2013regularity} which are valid
independent of~$\operatorname{char} (\Bbbk)$.
\begin{theorem}\label{theorem:matsudaMurai}
  Let $\ell$ be the maximum length of an induced path in a
  graph~$G$. Then
  \begin{equation*}
    \ell \leq \reg(S/\bei_{G}) \leq n-1.
  \end{equation*}
\end{theorem}
Our aim is to investigate families of graphs for which the lower bound is
constant.  We study the family of graphs with no induced path of length~$3$.
These are the \emph{complement reducible graphs (cographs)}.  They have been
characterized in \cite[Theorem~2]{CorneilLerchsBurlingham1981} as graphs such
that for every connected induced subgraph with at least two vertices, the
complement of that subgraph is disconnected.  Cographs are hereditary in the
sense that every induced subgraph of a cograph is a cograph
\cite[Lemma~1]{CorneilLerchsBurlingham1981}.  The graphs $G$ for which
$\reg(S/\bei_{G}) \le 2$ are all cographs by the ($\operatorname{char} (\Bbbk)$-independent)
characterization in \cite[Theorem~3.2]{madani2018binomial}.

One can quickly find that among (connected) cographs arbitrarily high regularity
is possible (Proposition~\ref{prop:cographsHighReg}), but when the regularity is
considered as a function of the number of vertices of the cograph, there is a
stricter upper bound than that in Theorem~\ref{theorem:matsudaMurai}.  Our
Theorem~\ref{theorem:cographs:maxReg} shows that for cographs the regularity is
essentially bounded by~$2n/3$.  The experimental results and also the proof
methods leading to the $2n/3$ bound lead us to study regularity bounds in terms
of other graph invariants.  Theorem~\ref{thm:boundByInvariants} bounds
$\reg(S/\bei_{G})$ by the independence number $\SizeOfMaxStabS(G)$ and the
number of maximal independent sets $\NumOfMaxStabS(G)$.  This stands in the
context of a general upper bound of $\reg(S/\bei_{G})$ by the number of maximal
cliques~$\NumOfMaxCl(G)$, shown by Malayeri, Madani and
Kiani~\cite{malayeri2021proof}.  Interestingly, for cographs the mentioned
invariants are computable in linear time, while in general they are hard to
compute.

The investigations for this paper started with a large experiment in which we
tabulated properties of binomial edge ideals and the corresponding graphs for
many small graphs.  To this end we developed a database of algebraic properties
of~$S/\bei_{G}$ and the necessary tools to extend the database.  Our source code
is available at
\begin{center}
  \url{https://github.com/kruesemann/graph_ideals}.
\end{center}
In the course of this work we found a counterexample to a conjecture of Hibi and
Matsuda.  We present this in Section~\ref{s:HibiMatsuda}, which leaves the class
of cographs.

\subsection*{Notation} All graphs in this paper are simple and undirected,
meaning that they have only undirected edges with no loops and no multiple edges
between two fixed vertices.  We often consider binomial edge ideals $\bei_{G}$
and $\bei_{H}$ of graphs on different vertex sets $V(G)\neq V(H)$.  In this case
we can consider both ideals in a bigger ring with variables corresponding to
$V(G) \cup V(H)$.  Embedding the ideals in a ring with extra variables has no
effect on the invariants under consideration and we still write $\bei_{G}$
and~$\bei_{H}$ independent of the ambient ring.  By \emph{the regularity of $G$}
we mean $\reg (S/\bei_{G})$.  The \emph{complement} $\compl{G}$ of a graph $G$
is a graph on the same vertex set, but with the edge set
$E(\compl{G}) = \{(i,j) : i\neq j, (i,j)\notin E(G)\}$.  The \emph{path of
length $\ell-1$} has $\ell$ vertices and is denoted by~$P_{\ell}$.

\section{Regularity for cographs}

Our first aim is to show that $\reg(S/\bei_{G})$ can take arbitrarily large
values, even if $G$ is restricted to connected cographs and the lower bound of
Theorem~\ref{theorem:matsudaMurai} does not apply.  If one allows disconnected
graphs, this follows from the simple observation that regularity is additive
under the disjoint union $G \sqcup H$ of two graphs:
\[
  \reg(S/\bei_{G \sqcup H}) = \reg(S/\bei_{G}) + \reg(S/\bei_{H}).
\]
As noted in \cite[proof of Theorem~2.2]{EneZarojanu2013}, this follows from the
fact that $\bei_{G}$ and $\bei_{H}$ use disjoint sets of variables.  In
particular this is true independent of~$\Bbbk$.
To see that arbitrary regularity is possible for connected cographs, we employ
the \emph{join} of two simple undirected graphs $G$ and $H$ which is
\begin{equation*}
  G \ast H = (V(G) \sqcup V(H), E(G) \cup E(H) \cup \{\{v,w\} : v \in V(G), w \in V(H)\}).
\end{equation*}
Here and in the following, $V(G)$ and $E(G)$ denote, respectively, the vertex
set and the edge set of an undirected simple graph~$G$.  The join also behaves
nicely with regularity as shown by Kiani and
Madani~\cite[Theorem~2.1]{madani2018binomial}.

\begin{theorem}\label{theorem:join:reg}
  Let $G$ and $H$ be simple, undirected graphs and not both complete. Then,
  independent of $\operatorname{char} (\Bbbk)$,
  \begin{equation*}
    \reg(S/\bei_{G \ast H}) = \max\{\reg(S/\bei_{G}), \reg(S/\bei_{H}), 2\}.
  \end{equation*}
\end{theorem}

The join of two cographs is a cograph as the path $P_{4}$ of length 3
is not a join itself.  It follows that the regularity can be made
arbitrarily high by forming \emph{cones}, that is joins with single
vertex graphs.

\begin{proposition}\label{prop:cographsHighReg}
  For any $r\ge 1$ there is a connected cograph with $\reg(S/\bei_{G})=r$.  If
  $r$ is even, then there is such a graph with $\frac{3}{2}r + 1$ vertices.  If
  $r$ is odd, then there is such a graph with $\frac{3}{2}(r+3)$ vertices.
\end{proposition}
\begin{proof}
  For $r=1$ a single edge suffices.  If $r = 2a$ is even, let $G$ be a disjoint
  union of $a$ copies of~$P_{3}$.  If $3 \le r=2a+1$ is odd, then let $G$ be a
  disjoint union of $a$ copies of $P_{3}$ and a single edge~$P_{2}$.  In both
  cases $\reg (S/\bei_{G}) = r$.  Let $\cone(G)$ be the join of $G$ with a
  single vertex graph.  Theorem~\ref{theorem:join:reg} implies that,
  $\reg(S/\bei_{\cone(G)}) = \reg(S/\bei_{G})$ and thus $\cone(G)$ is a
  connected cograph with regularity~$r$.
\end{proof}

With the lower bound settled, we aim for a stricter upper bound on the
regularity of cographs.  For this we employ the original definition of
complement reducible graphs: they are constructed recursively by
taking complements and disjoint unions of cographs, starting from the
single-vertex graph.  As a result, there is an unusual
one-to-one-relationship between connected and disconnected cographs of
the same order.

\begin{lemma}\label{lemma:cographs:complement}
  Let $G$ be a cograph with at least $2$ vertices. Then $G$ is
  connected if and only if $\compl{G}$ is disconnected.
\end{lemma}
\begin{proof}
  Let $\compl{G}$ be disconnected and $v,w \in V(G)$.  If $v$ and $w$ are in
  different connected components of $\compl{G}$, then $\{v,w\} \in E(G)$, so
  there is a path from $v$ to $w$ in~$G$.

  If $v$ and $w$ are in the same connected component of~$\compl{G}$, then there
  exists another vertex $u \in V(G)$ in a different connected component
  of~$\compl{G}$.  In particular, $\{v,u\}, \{w,u\} \notin E(\compl{G})$ and so
  $\{v,u\}, \{w,u\} \in E(G)$, so $(v,u,w)$ is a path from $v$ to $w$ in $G$.
  Thus $G$ is connected.

  The other implication follows from the
  stronger~\cite[Theorem~2]{CorneilLerchsBurlingham1981}: every induced subgraph
  of a cograph with more than one vertex has disconnected complement.
\end{proof}

In Lemma~\ref{lemma:cographs:complement}, the implication $\compl{G}$
disconnected $\Rightarrow$ $G$ connected holds for any graph, not just
cographs.  In the same generality, the join of graphs can be expressed
using complement and disjoint union.

\begin{lemma}\label{lemma:join:complement}
  Let $G_1$ and $G_2$ be simple undirected graphs. Then
  $G_1 \ast G_2 = \compl{\compl{G_1} \sqcup \compl{G_2}}$.
\end{lemma}
\begin{proof}
  Let $V$ denote the common vertex set of both graphs.  Let $e = \{v,w\}$ where
  $v,w \in V$ are arbitrary vertices.  If $e \nsubseteq V(G_i)$ for
  $i \in \{1,2\}$, then $e$ is an edge in both
  $\compl{\compl{G_1} \sqcup \compl{G_2}}$ and $G_1 \ast G_2$.  If
  $e \subseteq V(G_i)$ for one $i \in \{1,2\}$, then
  $e \in E(\compl{\compl{G_1} \sqcup \compl{G_2}})$ if and only if $e$ is an
  edge in~$G_i$, which is the case if and only if $e \in E(G_1 \ast G_2)$.
\end{proof}

\begin{lemma}\label{lemma:cographs:join}
  A connected cograph $G$ is the join of induced subgraphs $G_1,\dots,G_m$,
  which are exactly the complements of the connected components of $\compl{G}$.
\end{lemma}
\begin{proof}
  By Lemma~\ref{lemma:cographs:complement}, $\compl{G}$ is disconnected.  Let
  $\compl{G_1},\dots,\compl{G_m}$ be the connected components of $\compl{G}$ and
  $G_1,\dots,G_m$ their complements.  The $G_{i}$ are induced subgraphs of $G$
  since their edge sets arise from that of $G$ by complementing twice.  Then, by
  Lemma~\ref{lemma:join:complement},
  $G = \compl{\compl{G_1} \sqcup \dots \sqcup \compl{G_m}} = G_1 \ast \dots \ast
  G_m$ as both $\sqcup$ and $\ast$ are associative.
\end{proof}
Using Lemma~\ref{lemma:cographs:join}, we can assume that any connected cograph
$G$ is written as a join of the complements of the connected components of its
complement.  Since any induced subgraph of a cograph is a cograph, the $G_{i}$
in the lemma are cographs too and by Lemma~\ref{lemma:cographs:complement} they
are disconnected or have only one vertex.

\begin{proposition}\label{proposition:cographs:regConnected}
  Let $G$ be a connected cograph that is not complete. Then
  \begin{equation*}
    \reg(S/\bei_{G}) = \max(\{2\} \cup \{\reg(S/\bei_{G_{i}}) : i \in \left[m\right]\}),
  \end{equation*}
  where $G = G_1 \ast \dots \ast G_m$ as in Lemma~\ref{lemma:cographs:join} and
  the $G_i$ are cographs that each are either disconnected or single vertices.
\end{proposition}
\begin{proof}
  Since $G$ is not complete, the $G_i$ cannot all be complete, so the equations
  follows from Lemma~\ref{lemma:cographs:join} and
  Theorem~\ref{theorem:join:reg}.
\end{proof}

We now have all the ingredients for a recursive computation of
$\reg(S/\bei_{G})$ for any cograph~$G$.  In the disconnected case, add the
regularities of all connected components.  In the connected case, compute the
maximum regularity of the complements $G_i$ of the connected components
$\compl{G_i}$ of~$\compl{G}$.  We use this to bound the maximum regularity.

\begin{theorem}\label{theorem:cographs:maxReg}
  Let $G$ be a cograph on $3k-a$ vertices, with $k \in \N$ and
  $a \in \{0,1,2\}$. Then
  \begin{equation*}
    \reg(S/\bei_{G}) \leq 2k - a.
  \end{equation*}
  If $G$ is connected, $k > 1$, and $a \in \{0,1\}$, then
  $\reg(S/\bei_{G}) \leq 2k - a - 1$.
\end{theorem}
\begin{proof}
  The proof is by induction over $k$.  The cographs with at most $3$ vertices
  are $K_1$, $K_2=P_1$, $\compl{K_2}$, $K_3$, $\compl{K_3}$, $P_2$ and
  $\compl{P_2}$.  Then $\reg(S/\bei_{G}) \leq 2 - a = \reg(S/\bei_{P_{3-a}})$.

  Now let $k>1$.  By Proposition~\ref{proposition:cographs:regConnected}, if $G$
  is connected, it either has regularity $2$ or there is a smaller, disconnected
  cograph with the same regularity.  So it can be assumed that $G$ is
  disconnected.  Let $H$ be a connected component of $G$ with $3k_H - a_H$
  vertices and $H' = G \setminus H$ have $3k_{H'} - a_{H'}$ vertices, where
  $k_H,k_{H'} \in \N$ and $a_H,a_{H'} \in \{0,1,2\}$.  Then both $H$ and $H'$
  have fewer vertices than $G$ and
  \begin{equation*}
      k_H + k_{H'} =
      \begin{cases}
	k &\text{if } a_H + a_{H'} = a, \\
      k + 1 &\text{if } a_H + a_{H'} = a + 3.
      \end{cases}
  \end{equation*}
  By induction $\reg(S/\bei_{H}) \leq 2k_H - a_H$ and $\reg(S/\bei_{H'})
  \leq 2k_{H'} - a_{H'}$, and
  \begin{align*}
    \reg(S/\bei_{G}) &\leq 2k_H - a_H + 2k_{H'} - a_{H'} \\
		     &= \begin{cases}
		       2k - a & \text{if } a_H + a_{H'} = a, \\
		       2k - a - 1 & \text{if } a_H + a_{H'} = a + 3 \\
		     \end{cases} \\
		     &\leq 2k - a.
  \end{align*}
  If $G$ is connected and $k > 1$, the regularity is either $2$ or at most that
  of a disconnected cograph with fewer vertices.  So if $G$ has $3k - a$
  vertices, with $a \in \{0,1\}$, then $\reg(S/\bei_{G}) \leq 2k - a - 1$.  If
  $G$ has $3k - 2$ vertices, then $\reg(S/\bei_{G}) \leq 2(k - 1) = 2k - 2$.
\end{proof}

The following example shows that the bounds for disconnected cographs in
Theorem~\ref{theorem:cographs:maxReg} can be realized.
\begin{example}\label{ex:maxreg}
  Let $n = 3k-a \in\mathbb{N}$ be a positive integer with $a\in\{0,1,2\}$.  If $a = 0$,
  let $G_{n}$ be a disjoint union of $k$ copies of $P_{3}$.  If $a = 1$, then
  let $G_{n}$ be a disjoint union of $k-1$ copies of $P_{3}$ and one of~$P_{2}$.
  If $k=2$, let $G_{n}$ be a disjoint union of two copies of $P_{2}$ and $k-2$
  copies of~$P_{3}$.  Since $P_{n}$ has regularity $n-1$ and regularity is
  additive under disjoint union, $G_{n}$ has regularity~$2k-a$.
\end{example}

To make connected examples, use the cone construction from
Proposition~\ref{prop:cographsHighReg}.  Even more, in two of three cases the
graphs from Example~\ref{ex:maxreg} are the only cographs of maximum regularity
as the following theorem shows.
\begin{theorem}\label{thm:Characterize}
  Let $G$ be a cograph with $3k - a$ vertices, where $k \in \N$ and
  $a \in \{0,1\}$. Then $\reg(S/\bei_{G}) = 2k - a$ if and only if $G$ is a
  disjoint union of $P_{3}$ and at most one~$P_{2}$.
\end{theorem}
\begin{proof}
  With Example~\ref{ex:maxreg} in place, just the only-if-direction remains.
  Let $G$ be a cograph.  By Theorem~\ref{theorem:cographs:maxReg}, there is
  nothing to prove if $G$ is connected, so we assume it is disconnected.  We
  first show that any connected component has at most $3$ vertices.  To this
  end, let $H$ be a connected component of $G$ with $3k_H - a_H$ vertices,
  $k_{H}>1$, and $a_H \in \{0,1,2\}$.  Let $H' = G \setminus H$ have
  $3k_{H'} - a_{H'}$ vertices with $a_{H'} \in \{0,1,2\}$.  Then
  \begin{equation*}
    k_H + k_{H'} =\begin{cases}
      k &\text{if } a_H + a_{H'} = a, \\
      k + 1 &\text{if } a_H + a_{H'} = a + 3.
    \end{cases}
  \end{equation*}
  If $a_{H}\neq 2$, then, by Theorem~\ref{theorem:cographs:maxReg},
  \begin{align*}
    \reg(S/\bei_{G}) &\leq 2k_H - a_H - 1 + 2k_{H'} - a_{H'} \\
		     &= \begin{cases}
		       2k - a - 1 & \text{if } a_H + a_{H'} = a, \\
		       2k - a - 2 & \text{if } a_H + a_{H'} = a + 3 \\
		     \end{cases} \\
		     &< 2k - a.
  \end{align*}
  If otherwise $a_H = 2$, then since $a \in \{0,1\}$, $G$ must have another
  connected component $H'$ with $3k_{H'} - a_{H'}$ vertices and
  $a_{H'} \in \{1,2\}$.  Let $H'' = G \setminus (H \sqcup H')$ with
  $3k_{H''} - a_{H''}$ vertices, where $k_{H''} \in \N_0$ and
  $a_{H''} \in \{0,1,2\}$.  Then
  \begin{gather*}
    k_H + k_{H'} + k_{H''} =
    \begin{cases}
      k + 1 & \text{if } a_H + a_{H'} + a_{H''} = a + 3, \\
      k + 2 & \text{if } a_H + a_{H'} + a_{H''} = a + 6.
    \end{cases}
  \end{gather*}
  Therefore Theorem~\ref{theorem:cographs:maxReg} implies
  \begin{align*}
    \reg(S/\bei_{G}) &\leq 2k_H - a_H + 2k_{H'} - a_{H'} + 2k_{H''} - a_{H''} \\
		     &= \begin{cases}
		       2k - a - 1 & \text{if } a_H + a_{H'} + a_{H''} = a + 3, \\
		       2k - a - 2 & \text{if } a_H + a_{H'} + a_{H''} = a + 6 \\
		     \end{cases} \\
		     &< 2k - a.
  \end{align*}
  We have thus shown that if $G$ has a connected component with more than $3$
  vertices, it cannot have maximum regularity. Since the only cographs of
  maximum regularity with $3$ and $2$ vertices are $P_3$ and $P_2$ respectively,
  it follows that if $G$ has maximum regularity, then it is a disjoint union of
  $2$-paths $P_{3}$, single edges $P_{2}$ and isolated vertices.  We now analyze
  these cases separately for two possible values of $a$.

  Suppose first that $a=0$.  If $G$ has a connected component $H$ with one or
  two vertices, that is $3 - a_H$ vertices and $a_H \in \{1,2\}$, then $G$ must
  have another connected component $H'$ with $3 - a_{H'}$ vertices, where
  $a_{H'} \in \{1,2\}$.  With a similar computation as above we find that
  $\reg(S/\bei_{G}) < 2k$.  Therefore if $G$ has $3k$ vertices and maximal
  regularity, each connected component must have exactly $3$ vertices and be
  equal to~$P_{3}$.

  Finally, consider the case $a = 1$.  If $G$ has an isolated vertex, then $G$
  must have another connected component with fewer than $3$ vertices.  Then $G$
  cannot have maximal regularity as isolated vertices contribute no regularity
  and a disjoint union of an edge and a vertex has regularity 1.  If there are
  two isolated edges, then the subgraph on these $4$ vertices contributes
  regularity only $2$ and thus $G$ cannot have maximal regularity.
\end{proof}

\begin{remark}
  Graphs with $3k - 2$ vertices and maximal regularity, those in the $a=2$ case
  of Theorem~\ref{thm:Characterize}, do not have a simple characterization.  For
  example, the class contains cones over disjoint unions of $2$-paths as well as
  other types of joins and disjoint unions of joins, paths, and isolated
  vertices.
\end{remark}

The following corollary partially characterizes connected maximers of
regularity.
\begin{corollary}\label{cor:isCone}
  Let $G$ be a connected cograph with maximum regularity among connected
  cographs on $3k-a$ vertices with $k>1$ and $a \in \{0,2\}$, that is,
  $\reg(S/\bei_{G}) = 2k-1$ for $a=0$ and $\reg(S/\bei_{G}) = 2k-2$ for $a=2$.
  Then $G$ is a cone.
\end{corollary}
\begin{proof}
  Let $G$ be a connected cograph with $3k-a$ vertices and maximum regularity,
  where $k>1$ and $a\in\{0,2\}$.  By Lemma~\ref{lemma:cographs:join},
  $G = G_{1}\ast \dotsb \ast G_{m}$ where the $G_{i}$ are disconnected or single
  vertices.  Proposition~\ref{proposition:cographs:regConnected} shows that if
  $G$ is not a cone, that is, neither of the $G_{i}$ is a single vertex, then
  $\reg(S/\bei_{G}) = \reg(S/\bei_{G'})$ for some $G'$ which has at least two
  vertices fewer than~$G$.  So assume $G'$ has $3k-a-2$ vertices and apply
  Theorem~\ref{theorem:cographs:maxReg}.  If $a=0$, then $G'$ has $3k-2$
  vertices and thus regularity at most $2k-2$, but $G$ had maximum regularity
  $2k-1$, a contradiction.  If $a=2$, then $G'$ has $3(k-1)-1$ vertices and thus
  regularity at most $2(k-1)-1 = 2k-3$, but $G$ had maximum regularity $2k-2$,
  another contradiction.  By these contradictions, either $G$ is a cone after
  all, or its regularity was not maximal.
\end{proof}
Applying the reasoning in the proof to the $a=1$ case does not yield the
contradiction.  In this case, $G$ has $3k-1$ vertices and thus a regularity
bound of $2k-1-1 = 2k-2$ while $G'$ with its $3(k-1)$ vertices has regularity at
most~$2(k-1)$.

The results so far give a fairly clear picture of the asymptotic behaviour of
regularity in the class of cographs and individually for graphs with known
complement reducible decomposition.  If the exact structure of a cograph is
unknown, it can be useful to bound regularity using graph-theoretic invariants.
We consider here $\SizeOfMaxStabS(G)$ -- the size of the largest independent
set, $\NumOfMaxStabS(G)$ -- the number of maximal independent sets in $G$,
and~$\NumOfMaxCl(G)$ -- the number of maximal cliques of~$G$.  The recursive
construction of cographs yields bounds because these invariants satisfy simple
formulas under disjoint union and join:
\begin{align}
  \raggedright
  \NumOfMaxStabS(G \sqcup H) &= \NumOfMaxStabS(G)\NumOfMaxStabS(H), \label{equation:union:maxStable}\\
  \NumOfMaxStabS(G \ast H) &= \NumOfMaxStabS(G) + \NumOfMaxStabS(H), \label{equation:join:maxStable}\\
  \SizeOfMaxStabS(G \sqcup H) &= \SizeOfMaxStabS(G) + \SizeOfMaxStabS(H), \label{equation:union:stable}\\
  \SizeOfMaxStabS(G \ast H) &= \max\{\SizeOfMaxStabS(G), \SizeOfMaxStabS(H)\}. \label{equation:join:stable}
\end{align}

We find the following bounds which are independent of the number of vertices.
\begin{theorem}\label{thm:boundByInvariants}
  Let $G$ be a cograph. Then
  $\reg(S/\bei_{G}) \leq \min\{\NumOfMaxStabS(G), \SizeOfMaxStabS(G)\}$.
\end{theorem}
\begin{proof}

  The proof is by induction on the number of vertices of $G$.
  An isolated vertex has exactly one maximal independent set with one vertex and
  $\reg(S/\bei_{K_1}) = 0$, so the statement holds.
  Now let $G$ be any cograph.  If $G$ is connected, it is the join of smaller
  cographs $G_1,\dots,G_{m}$ and by induction
  \begin{equation*}
    \reg(S/\bei_{G_i}) \leq \min\{\NumOfMaxStabS(G_i), \SizeOfMaxStabS(G_i)\}
    \quad \text{ for all } i=1,\dots,m.
  \end{equation*}
  If $G$ is complete, both inequalities are trivial.  In the other case, it
  follows by Theorem~\ref{theorem:join:reg} and~\eqref{equation:join:maxStable}
  that
  \begin{equation*}
    \reg(S/\bei_{G}) = \max\{\{2\} \cup \{\reg(S/\bei_{G_i}) : i=1,\dots,m\}\}
    \leq \sum_{i=1}^{m} \NumOfMaxStabS(G_i) = \NumOfMaxStabS(G)
  \end{equation*}
  and, since $\SizeOfMaxStabS(G) = 1$ if and only if $G$ is complete,
  \eqref{equation:join:stable} leads to
  \begin{multline*}
    \reg(S/\bei_{G}) = \max\{\{2\} \cup \{\reg(S/\bei_{G_i}) : i=1,\dots,m\}\}
    \\
    \leq \max\{\SizeOfMaxStabS(G_i) : i=1,\dots,m\} = \SizeOfMaxStabS(G).
  \end{multline*}
  If $G$ is disconnected, it has connected components $G_1,\dots,G_{m}$ which by
  induction satisfy
  \begin{equation*}
    \reg(S/\bei_{G_i}) \leq \min\{\NumOfMaxStabS(G_i), \SizeOfMaxStabS(G_i)\}
    \text{ for all } i=1,\dots,m.
  \end{equation*}
  In this case, since $G$ is the disjoint union of its connected components, and
  since regularity is additive, with~\eqref{equation:union:maxStable} we have
  \begin{equation*}
    \reg(S/\bei_{G}) = \sum_{i} \reg(S/\bei_{G_i}) \leq \prod_i \NumOfMaxStabS(G_i) = \NumOfMaxStabS(G).
  \end{equation*}
  Finally, by ~\eqref{equation:union:stable} we have
  \begin{equation*}
    \reg(S/\bei_{G}) = \sum_{i} \reg(S/\bei_{G_i}) \leq \sum_{i} \SizeOfMaxStabS(G_i)
    = \SizeOfMaxStabS(G). \qedhere
  \end{equation*}
\end{proof}
The method of bounding by $\NumOfMaxStabS(G)$ seems coarse as the maximum and
sum over a set of intergers are, respectively, replaced by the sum and the
product over those integers.  Nevertheless $s(G)$ can be a good bound as
discussed in Remark~\ref{r:bestBound}.

Since independent vertices cannot be in the same clique,
$\SizeOfMaxStabS(G) \le \NumOfMaxCl(G)$.  Thus
$\reg(S/\bei_{G}) \leq \NumOfMaxCl(G)$ holds for cographs, but in fact it holds
for all graphs by~\cite{malayeri2021proof}.

Our last bound uses the maximum vertex degree $\delta(G)$ of a connected
cograph~$G$.
\begin{proposition}\label{proposition:cographs:regMaxdeg}
  Let $G$ be a connected cograph. Then
  \begin{equation*}
    \reg(S/\bei_{G}) \leq \delta(G) = \max\{\delta(v) : v \in V(G)\}.
  \end{equation*}
\end{proposition}
\begin{proof}
  If $G = K_{n}$, then $\reg(S/\bei_{G}) \le 1$, $\delta(G) = n-1$, and the
  inequality holds.  If $G$ is not complete but connected, it is the join of
  induced subgraphs $G_1,\dots,G_m$ as in Lemma~\ref{lemma:cographs:join}.  Then
  $\reg(S/\bei_{G}) = \max\{2, \reg(S/\bei_{G_{1}}), \dots, \reg(S/\bei_{G_{m}})\}$ by
  Proposition~\ref{proposition:cographs:regConnected}.  Writing $n_{i}$ for the
  number of vertices of $G_{i}$, Theorem~\ref{theorem:matsudaMurai} gives
  \begin{equation*}
    \reg(S/\bei_{G}) \leq \max\{2, n_1-1, \dots, n_m-1\}.
  \end{equation*}
  Let $n_{\text{max}} = \max\{n_1,\dots,n_m\}$.

  Then $\reg(S/\bei_{G}) \leq n_{\text{max}}$, since $G$ is not complete and
  thus $n_{\text{max}} \geq 2$.  Since $G$ is a join of the $G_{i}$, the maximum
  vertex degree satisfies $n_{\text{max}}\leq \max\{\delta(v) : v \in V(G)\}$
  and we conclude.
\end{proof}

\begin{remark}
  The bound in Proposition~\ref{proposition:cographs:regMaxdeg} does not give
  anything new for cone graphs since in this case it agrees with
  Theorem~\ref{theorem:matsudaMurai}.
\end{remark}

\begin{remark}\label{r:bestBound}
  One can ask if one of the bounds in this section is generally preferable over
  the other bounds.  Table~\ref{table:boundComparison} shows than any bound can
  beat any other bound, with the exception of
  $\SizeOfMaxStabS(G) \le \NumOfMaxCl(G)$.  On the other hand, if one asks for
  the best bound, it can be confirmed that among the $2341$ cographs in our
  database, for 505 $\NumOfMaxStabS(G)$ is strictly the best bound and for 724
  $\SizeOfMaxStabS(G)$ is strictly the best bound.  No other bound is ever
  strictly the best and for all remaining graphs there is a tie for the best
  bound.
\end{remark}

\begin{table}[htp]
  \centering
  \begin{tabular}{r | r r r r r}
    & order bound & $\NumOfMaxCl(G)$ & $\NumOfMaxStabS(G)$ & $\SizeOfMaxStabS(G)$ & max deg \\
    \hline
    order bound & $0$ & $968$ & $968$ & $146$ & $1090$ \\
    $\NumOfMaxCl(G)$ & $918$ & $0$ & $1049$ & $0$ & $724$ \\
    $\NumOfMaxStabS(G)$ & $920$ & $1050$ & $0$ & $514$ & $837$ \\
    $\SizeOfMaxStabS(G)$ & $1830$ & $1139$ & $1522$ & $0$ & $1150$ \\
    max deg & $5$ & $362$ & $201$ & $1$ & $0$
  \end{tabular}
  \caption{ \label{table:boundComparison} Comparisons of five regularity bounds
  for all $2341$ cographs in our database.  The numbers given are the numbers of
  cographs for which the bound in the left-most column is strictly better than
  the bound in the top row respectively. `Order bound' stands for the bound in
  Theorem~\ref{theorem:cographs:maxReg}, `max deg' denotes the maximum vertex
  degree in Proposition~\ref{proposition:cographs:regMaxdeg}. Comparisons with
  `max deg' are made only for the $1171$ connected cographs.}
\end{table}

\begin{remark}
  The questions about regularity in this paper can also be asked about the
  regularity of $S/\mathcal{I}_{G}$ where $\mathcal{I}_{G}$ is the \emph{parity
  binomial edge ideal} of~\cite{kahle2015parity}.  Using our database we
  observed the following inequality, slightly weaker than
  Theorem~\ref{theorem:matsudaMurai},
  \[
    \ell \le \reg(S/\mathcal{I}_{G}) \le n.
  \]
  Based on our computations we conjecture that the maximum regularity is
  achieved exactly for disjoint unions of odd cycles.  Minimal free resolutions
  of parity binomial edge ideals contain many interesting patterns that remain
  to be investigated.  At the time of the first posting of this paper,
  explaining even the minimal free resolution of $S/\mathcal{I}_{K_{n}}$ was
  open and we conjectured that $\reg(S/\mathcal{I}_{K_{n}}) = 3$.  In the
  meantime this has been confirmed in~\cite{hoang20:_hilber}.
\end{remark}

\section{Regularity versus \texorpdfstring{$h$}{h}-polynomials}\label{s:HibiMatsuda}

As a standard graded $\Bbbk$-algebra, the Hilbert series of $S/\bei_{G}$ takes the
form $\frac{h_{G}(t)}{(1-t)^{d}}$ where $d$ is the Krull dimension and
$h_{G}(t) \in \mathbb{Z}[t]$.  The numerator $h_{G}$ is known as the
\emph{h-polynomial}.  In the first arXiv version of \cite{HibiMatsuda2018}, Hibi
and Matsuda conjectured that for binomial edge ideals its degree bounds the
regularity from above.  The conjecture was removed from a subsequent version of
their paper after we informed them of the following minimal counterexample on
eight vertices:

\begin{example}
  Let $G$ be the graph in Figure~\ref{figure:regHPolDeg:counterexample}, that is
  the graph on the vertex set $\{1,\dots, 8\}$ with edges $\{1,8\}$, $\{2,6\}$,
  $\{3,7\}$, $\{3,8\}$, $\{4,5\}$, $\{4,8\}$, $\{5,6\}$, $\{5,7\}$, $\{6,7\}$,
  $\{6,8\}$, $\{7,8\}$.  Then $\reg(S/\bei_{G}) = 4$ and $\deg(h_{G}) = 3$.
\end{example}

\begin{figure}[ht]
	\centering
	\begin{tikzpicture}
	[
	scale=0.02,mynode/.style={draw,fill=white,circle,outer sep=4pt,inner sep=2pt},myedge/.style={line width=1.5,black}
	]

	\coordinate(p1) at(100,0);
	\coordinate(p2) at(70,70);
	\coordinate(p3) at(0,100);
	\coordinate(p4) at(-70,70);
	\coordinate(p5) at(-100,0);
	\coordinate(p6) at(-70,-70);
	\coordinate(p7) at(0,-100);
	\coordinate(p8) at(70,-70);

	\draw[myedge] (p1) -- (p8);
	\draw[myedge] (p2) -- (p6);
	\draw[myedge] (p3) -- (p7);
	\draw[myedge] (p3) -- (p8);
	\draw[myedge] (p4) -- (p5);
	\draw[myedge] (p4) -- (p8);
	\draw[myedge] (p5) -- (p6);
	\draw[myedge] (p5) -- (p7);
	\draw[myedge] (p6) -- (p7);
	\draw[myedge] (p6) -- (p8);
	\draw[myedge] (p7) -- (p8);

	\node[mynode] at (p1) {1};
	\node[mynode] at (p2) {2};
	\node[mynode] at (p3) {3};
	\node[mynode] at (p4) {4};
	\node[mynode] at (p5) {5};
	\node[mynode] at (p6) {6};
	\node[mynode] at (p7) {7};
	\node[mynode] at (p8) {8};
	\end{tikzpicture}
	\caption{\label{figure:regHPolDeg:counterexample} A graph with
	$\reg (S/\bei_{G}) > \deg (h_{G})$.}
\end{figure}

At the time of writing, our database contains $39$ counterexamples and none
shows a difference greater than 1 between $\reg(S/\bei_{G})$ and $\deg(h_{G})$.
However, gluing two copies of the counterexample in
Figure~\ref{figure:regHPolDeg:counterexample} at vertex $1$ yields a graph $G$
(visible in Figure~\ref{figure:regHPolDeg:counterexample2}) which satisfies
$\reg(S/\bei_{G})=8$ and $\deg(h_{G}) = 6$.  We now show that the difference can
be made arbitrarily large.  To this end we employ the following two theorems
that explain the behaviour of the regularity and the Hilbert series upon gluing
two graphs $G_{1}$ and $G_{2}$ over a vertex which is a free vertex in both
graphs.  If $G$ is a gluing like this, then $G_{1}$ and $G_{2}$ are a
\emph{split} of~$G$.

\begin{theorem}[{\cite[Theorem~3.1]{JayanthanNarayananRao2016}}]\label{theorem:glue:reg}
  Let $G_1$ and $G_2$ be a split of a graph $G$ at a vertex~$v$. If $v$ is a
  free vertex in both $G_1$ and $G_2$, then
  $\reg(S/\bei_{G}) = \reg(S/\bei_{G_1}) + \reg(S/\bei_{G_2})$.
\end{theorem}

\begin{theorem}[{\cite[Theorem~3.2]{KumarSarkar2018}}]\label{theorem:glue:hilbert}
  Let $G_1$ and $G_2$ be the decomposition of a graph~$G$ at a vertex~$v$. If
  $v$ is a free vertex in both $G_1$ and $G_2$, then
  \begin{equation*}
    \Hilbert_{S/\bei_G}(t) = (1-t)^2 \Hilbert_{S/\bei_{G_1}}(t) \Hilbert_{S/\bei_{G_2}}(t).
  \end{equation*}
\end{theorem}

\begin{theorem}\label{thm:differenceLarge}
  Let $k \in \N$. Then there exists a graph $G$ such that
  \begin{equation*}
    \reg(S/\bei_{G}) = \deg(h_{G}) + k.
  \end{equation*}
\end{theorem}
\begin{proof}
  Let $G_1$ be the graph in Figure~\ref{figure:regHPolDeg:counterexample}.  The
  reduced Hilbert series of $S/\bei_{G_1}$ can be computed with
  \textsc{Macaulay2}~\cite{Macaulay2} as
  \begin{equation*}
    \Hilbert_{S/\bei_{G_1}}(t) = \frac{1+7t+17t^2+13t^3}{(1-t)^9}
  \end{equation*}
  and its regularity as $\reg(S/\bei_{G_1}) = 4$.  Since $G_{1}$ has two free
  vertices $1,2$, we can glue a chain of $k$ copies of $G_{1}$ along free
  vertices (see Figure~\ref{figure:regHPolDeg:counterexample2} for the case
  $k=2$ in which the vertices $2$ and $9$ are available for further gluing).  By
  Theorem~\ref{theorem:glue:hilbert}, the Hilbert series of the resulting graph
  is
  \begin{equation*}
    \Hilbert_{S/\bei_{G_2}}(t) = \dfrac{(1+7t+17t^2+13t^3)^k}{(1-t)^{7k+2}}
  \end{equation*}
  and, by Theorem~\ref{theorem:glue:reg}, $\reg(S/\bei_{G_2}) = 4k$.  Thus
  $\reg(S/\bei_{G}) - \deg(h_{G}) = k$.
\end{proof}

\begin{figure}[ht]
\centering
\begin{tikzpicture}
[
scale=0.02,mynode/.style={draw,fill=white,circle,outer sep=4pt,inner sep=0pt, minimum size=14pt},myedge/.style={line width=1.5,black}
]

\coordinate(p1) at(100,0);
\coordinate(p2) at(70,70);
\coordinate(p3) at(0,100);
\coordinate(p4) at(-70,70);
\coordinate(p5) at(-100,0);
\coordinate(p6) at(-70,-70);
\coordinate(p7) at(0,-100);
\coordinate(p8) at(70,-70);
\coordinate(p9) at(130,70);
\coordinate(p10) at(200,100);
\coordinate(p11) at(270,70);
\coordinate(p12) at(300,0);
\coordinate(p13) at(270,-70);
\coordinate(p14) at(200,-100);
\coordinate(p15) at(130,-70);

\draw[myedge] (p1) -- (p8);
\draw[myedge] (p2) -- (p6);
\draw[myedge] (p3) -- (p7);
\draw[myedge] (p3) -- (p8);
\draw[myedge] (p4) -- (p5);
\draw[myedge] (p4) -- (p8);
\draw[myedge] (p5) -- (p6);
\draw[myedge] (p5) -- (p7);
\draw[myedge] (p6) -- (p7);
\draw[myedge] (p6) -- (p8);
\draw[myedge] (p7) -- (p8);
\draw[myedge] (p1) -- (p15);
\draw[myedge] (p9) -- (p13);
\draw[myedge] (p10) -- (p14);
\draw[myedge] (p10) -- (p15);
\draw[myedge] (p11) -- (p12);
\draw[myedge] (p11) -- (p15);
\draw[myedge] (p12) -- (p13);
\draw[myedge] (p12) -- (p14);
\draw[myedge] (p13) -- (p14);
\draw[myedge] (p13) -- (p15);
\draw[myedge] (p14) -- (p15);

\node[mynode] at (p1) {1};
\node[mynode] at (p2) {2};
\node[mynode] at (p3) {3};
\node[mynode] at (p4) {4};
\node[mynode] at (p5) {5};
\node[mynode] at (p6) {6};
\node[mynode] at (p7) {7};
\node[mynode] at (p8) {8};
\node[mynode] at (p9) {9};
\node[mynode] at (p10) {10};
\node[mynode] at (p11) {11};
\node[mynode] at (p12) {12};
\node[mynode] at (p13) {13};
\node[mynode] at (p14) {14};
\node[mynode] at (p15) {15};
\end{tikzpicture}
\caption{\label{figure:regHPolDeg:counterexample2} A graph with
$\reg (S/\bei_{G}) > \deg (h_{G}) + 1$.}
\end{figure}

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


They say there are no limits to marketing. The latest on the list of retailers is contests. Despite freebies being given to customers retail experts believe that contests make for an excellent marketing tactic. Let us see how.
Modern retailing has brought in the existence of organised retail and many new strategies for marketing products, services and stores. Marketing does not cease at the advertising level and brand promotion and can take place in the form of retail contests which retailers hold on their store floor or at a particular venue. Contests are also a good way to generate foot traffic for the store and gather consumer data which can be used for direct marketing and e-mail marketing advertising.
It does not take a humungous effort for a retailer to organise a contest on his retail floor. Nevertheless, what he seeks to achieve out of it should be clearly delineated. Educating customers about the brand, its USP and why it is the best option for the consumer is something that can be a part of the contest, banners, promotional flyers, pamphlets and so on. Customers walking into the store may not always participate in the contest, but that does not mean the retailer should alienate this prospect. Making them feel a part of the brand by just acknowledging their presence in the store can work wonders for the store and the brand.
The types of contests a retailer can plan are as limitless as his imagination. The objective of any contest should be to create the best possible incentive to get customers into the store, and motivate them to provide their contact information. Incentives can be a valuable grand prize, but can also be in the form of a small gift given just for participating in the contest. Naming the brand ambassador or mascot/ character associated with the business is a good example of a contest. It educates the consumer, promotes the brand and in turn, you have a ready-to-use contest at your disposal. If the retailer has a website, customers can enter the contest via an online entry form as well. The retailer can set dates associated with the contest such as an entry deadline, a first round selection of three finalists, and finally the date to announce the grand prize winner. The contest rules and progress can be included in all advertising and marketing campaigns within the contest timeframe. National players such as Shoppers Stop, Pantaloons, Big Bazaar etc are known to hold such contests to attract foot traffic and make winners out of customers. Another popular kind of contest is a lucky draw. This ensures that the retailer gets customer details for his database and the preferences mentioned in contest forms help with future merchandise selection decisions. In addition to being a fun idea, this type of contest can create great word-of-mouth advertising and a good deal of shopper traffic.
The grand prize for contests can be an opportunity for the winner to appear in a series of local advertisements with the mascot and to have her name and photo appear on the air, in the press, and on the website. Gift cards, a prize basket, a travel package are also other examples of prizes that could be offered to the winner and to the semi-finalists.
Contests with media tie-ups are particularly good traffic builders, as the reach for potential entrants is greater than usual. The publication or station printing or airing your ads for the contest benefits from the tie-up contest, as it receives an increase in listeners and newspaper sales on a particular day. A radio station may provide a discounted advertising rate to motivate a retailer to direct listeners to a show in need of a ratings boost, or to a particular section of the newspaper. Serving multiple purposes, contests should be seen more than just events for dishing out freebies. A discerning retailer will understand it is more than just that.
Contests are great opportunities to obtain contact information, generate foot traffic, and inspire word-of-mouth advertising for a retail store. Include a series of contests in your annual retail marketing plan.


text:
Christening may refer to:

 Baptism, a Christian sacrament of admission and adoption
 Infant baptism, the practice of baptising infants or young children
 Christening, an example of a naming ceremony, practiced by Christians and often involving a baptism
 Christening, naming of a vessel, as in ceremonial ship launching
 Anointing, the ritual act of putting aromatic oil on a person

Other uses
 "Christening" (The Office), an episode of the US version of The Office
 The Christening, a 2010 Polish drama film

See also
 Christen (disambiguation)

Christian terminology

redpajama_set_name: RedPajamaWikipedia


text:
How to Write GPS Coordinates on Paper
What do you mean by GPS? How To Read GPS Coordinates from Google Maps? How do you write latitude and longitude coordinates? What is GPS format converter?
byJack Williams
You would have wondered how the locations are mentioned on the maps. Are the map locations universal? How to write GPS coordinates on paper? Let's first discuss what GPS actually means. You should also know how to read GPS coordinates. For this, you must know how to write longitude and latitude. Let's learn it all.
The potential of Global Positioning System (GPS) technology is only limited by your creativity. Since they are so adaptable, GPS devices are used in practically every industry. Today, GPS is used to monitor forests, assist farmers in harvesting their crops, and aid ground and aerial pilots in navigating. Systems for locating people in need of aid are essential for both military applications and rescue teams. Numerous applications of GPS technology exist beyond your imagination.
1. What do you mean by GPS?
GPS stands for Global Positioning System. It is a system designed by the USA that helps in navigation and locating places. It is a very essential system for providing positioning, timing, and navigation. (See How to find North Direction in Room?)
The system has three segments:
Space segment
Control segment
User segment
2. What are the Different GPS formats?
The three most commonly used GPS formats are:
Degrees, Minutes and Seconds
Degrees and Decimal Minutes
Decimal Degrees
These formats can also help us to know how to write GPS Coordinates on paper. (See What is a Circle Degree Chart?)
3. How to Read GPS Coordinates from Google Maps?
Learning to write latitude and longitude coordinates is very important for locating a place. To read GPS coordinates from Google Maps, you need to learn certain symbols like degrees, minutes, and seconds.
For example, 41°24'12.2″N, 2°10'26.5″E will be read as forty-one degrees, twenty-four minutes, twelve point two seconds north and two degrees, ten minutes, twenty-six point five seconds east. This will help you to understand how to read GPS coordinates to look for the universal placement of a location. (See How to find House Facing Direction in Google Maps?)
4. How to Find Location by Latitude and Longitude?
You need to follow certain steps to find a location with the help of longitude and latitude:
1. Open Google Maps on your device.
2. Enter the coordinates in the search box. There are certain formats for it:
3. Click on the Search button to get the result.
5. How do you Write Latitude and Longitude Coordinates? How to Write Coordinates of a Location?
To write the longitude and latitude coordinates, you need to learn the formatting of coordinate writing. The degrees, minutes, and seconds are mentioned along with the direction to give latitude and longitude coordinates. For example, example 41°24'12.2″N and 2°10'26.5″E.
To find the coordinates of a location, you need to search for the location on Google Maps. The location, along with its universal coordinates, will appear. You can copy the coordinates from there. (See 13 Different Types of Compasses)
6. How to Write Latitude and Longitude in Degrees, Minutes and Seconds?
To know about degrees, minutes, and seconds, you must learn how to read GPS coordinates. You can write the coordinates by using the DMS format. The degrees, minutes, and seconds format is one of the most commonly used formats for writing coordinates. For example, 40:26:46.302N and 079:58:55.903W. (See What is the Difference between Latitude and Longitude?)
7. How do you Enter GPS Coordinates in Google Maps?
You need to follow certain steps to enter the GPS coordinates in Google Maps:
2. Type the address into the search box.
3. Copy the coordinates mentioned in the tab.
4. Search coordinates to get the location with the help of coordinates.
8. How are Map Coordinates Written?
In order to write the map coordinates, you need to write the longitude and latitude. You have to mention the exact degrees, minutes, and seconds to provide the exact location. The map coordinates are written by recording the longitude and latitude of the location on earth. (See What is the Imaginary Line that Runs from the Geographic North To South Pole?)
9. How to Write GPS Coordinates on Paper?
It is very important to learn how to write GPS coordinates on paper in a certain format. You can represent coordinates with one latitude and one longitude line. You can use the degrees, minutes, and seconds method. This method provides you with more accuracy in the map coordinates. (See Why Do We Have Different Time Zones?)
10. List 5 Geographic Coordinates' Examples.
Examples of geographical coordinates are:
41.40338, 2.17403.
41°24'12.2″N, 2°10'26.5″E
41 24.2028, 2 10.4418
27.2046° N, 77.4977° E
77° 13′ 46.0056" E (See What is the Absolute Location of the Philippines?)
11. What is GPS Format Converter?
The GPS format converter is a tool which is used to convert the GPS coordinates to an address and the address to longitude lines. It has several coordinate formats, for example, decimal degrees, degrees, minutes, seconds, degrees and decimal minutes. (See How Long does it Take to Drive 1 Mile?)
We hope you learned what is GPS, how to read or how to write GPS coordinates on paper as well as how to copy GPS coordinates to Google Maps to find a particular location. Keep reading for more such cool tips!
Degrees Minutes and Seconds
different GPS formats
read GPS coordinates
read latitude and longitude
write GPS coordinates on paper
write latitude and longitude
Jack Williams
How does it feel like to fly on a plane for the first time?
What to Expect Being on a Plane for the First Time? What does it feel like to fly a plane for the first time?
10 Features of Physical Environment
What is Physical Environment? What are the Most Important Parts of the Physical Environment? What is an Example of Physical Environment?
What is the Importance of Transportation in your Life?
What is Transportation? What does Transport mean? What are different Types of Transportation? What is the Importance of Transport in your life? What is the Importance of Transportation in Marketing?
45 Delicious Sandwich Bread Wrap Alternative
What are the Different Sandwich Bread Wrap Alternatives? What Flavors are these available in? Are there Vegan & Vegetarian options too?

redpajama_set_name: RedPajamaCommonCrawl


text:
'I've a lot to learn and quickly' says MotoGP rookie Binder
Red Bull KTM's Brad Binder pretty-much dismissed his top-ten position at the end of the Qatar MotoGP test, saying doing one fast lap means very little in the blue riband class.
Binder was surprised by the 1'54.283 he put in on the final day and almost immediately said that putting together 22 laps like that was where he needed to be. But it would be very difficult.
"Race distance is hard on this bike; on the MotoGP bike generally it's a different story. The tyres drop, it starts to spin a lot more, when you lose the fuel how different the bike is crazy. I've a lot to learn and very quickly," said Binder.
"We made a change and the bike got way better, It felt more comfortable to get the bike stopped and also to get it turned quickly and on the throttle earlier. Once we figured that out, it was easier to put a fast lap together.
"I put on a set of new tyres. I didn't really expect it. I came across the line and was like, 'jeez'. But it's one thing to do a fast lap and it's another thing to do 22 of them."


I'm converting to a powered bathroom fan too but I'm letting Reace install it in a new camper.
Thought the powered bathroom fan was standard equipment on new units?
It is, I was joking.
I imagine this upgrade in the bath vent to a powered model will be quite the costly modification.
I did mine the less expensive way, although it was still an expensive way to fix a leak. One year later next October, I'll do it the more expensive way with a new 21.
Congratulations Jon, this is the first notice on my part that a 21 awaits.
Sometimes I'm a little slow on the uptake (too much Bookers while in Kentucky).
Continuing off-topic, I finished the bathroom vent fan installation today following a couple of excellent posts here by other members (jamman and Vermilye). I don't really have anything to add to what's already been posted, but I'm really glad that's finished! I was a little disappointed with the kit, but it's in and working now, so on to other things!


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Home » Lifestyle » Conan O'Brien Riffs On Burglary Of 'Conan' Set: 'We Got Robbed!'
Conan O'Brien Riffs On Burglary Of 'Conan' Set: 'We Got Robbed!'
Conan O'Brien was upstaged by a scene-stealer on Monday.
The old Los Angeles theater where O'Brien has set up shop for "Conan" during the coronavirus pandemic was burglarized.
"We got robbed!" the host declared. (See the clip below.)
"That's the lowest," he said. "I can't think of anything lower."
But he found the funny side of the theft, in which laptops used for the remote interviews and a slate clapperboard were taken.
O'Brien imagined the burglar addressing the cardboard cutouts used as audience stand-ins: "Hey, don't mind me. I'm just gonna steal some shit."
"This kind of shit isn't happening to other big-time late night shows," O'Brien said. "Nobody breaks into 'The Tonight Show' and steals all the equipment." Watch below:
The incredible true story behind Freida Pinto's new spy series
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The Hidden Heart Problem That's Claiming Men's Lives Now
Whole Foods Is Selling Huge Bags Of Dill Pickle-Flavored Almonds
See Hilaria Baldwin's Baby Bump Ahead of 5th Child: Pregnancy Pics
« Jacqui Smith opens up on 'secret advice' from Strictly Come Dancing partner Anton du Beke
Burger King's Halloween stunt in Sweden has Ronald McDonald-like clown creepily appearing in bathroom mirror »


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Eduard VII., král Spojeného království Velké Británie a Irska a britských dominií a císař Indie, obdržel během svého života řadu britských i zahraničních titulů a vyznamenání. Během své vlády v letech 1901 až 1910 byl také velmistrem britských řádů.

Tituly

 9. listopadu 1841 – 8. prosince 1841: Jeho královská Výsost vévoda z Cornwallu
 8. prosince 1841 – 22. ledna 1901: Jeho královská Výsost princ z Walesu
 22. ledna 1901 – 6. května 1910: Jeho Veličenstvo král

Erb

Vojenské hodnosti

Čestné vojenské hodnosti

 čestný plukovník Gardehusarregimentet – Dánsko, 1870
polní maršál Německé armády – Německé císařství, 1883
čestný plukovník 27. (krále Eduarda) dragounský pluk z Kyjeva – 5. února 1901
admirál loďstva à la suite Císařského námořnictva – Německé císařství, 26. června 1902
čestný generál kapitán španělské armády
čestný admirál španělského námořnictva
čestný admirál švédského námořnictva – 1905
čestný generál švédské armády – 1906

Vyznamenání

Britská vyznamenání

Velmistr (1901–1910)

  velmistr a zakladatel Řádu za zásluhy
  velmistr a zakladate Imperiálního řádu Za zásluhy
  zakladatel Královského Viktoriina řetězu

Řády

  rytíř Podvazkového řádu – 9. listopadu 1858
  rytíř společník Řádu indické hvězdy – 25. června 1861
  velkokomandér Řádu indické hvězdy – 24. května 1866
  rytíř velkokříže Řádu lázně, vojenská divize – 10. února 1865
  rytíř Řádu bodláku – 24. května 1867
  rytíř Řádu svatého Patrika – 18. března 1868
  rytíř spravedlnosti Nejctihodnějšího řádu sv. Jana Jeruzalémského – 1876
  rytíř velkokříže Řádu svatého Michala a svatého Jiří – 31. května 1877
  velkokomtur Řádu Indické říše – 21. června 1887
  rytíř velkokříže Královského Viktoriina řádu – 6. května 1896

Medaile

  Medaile zlatého výročí královny Viktorie – 21. června 1887
  Medaile diamantového výročí královny Viktorie – 1897

Zahraniční vyznamenání

 rytíř Domácího řádu věrnosti – 1861
 velkokříž Řádu zähringenského lva – 1861

  rytíř Řádu svatého Huberta – 19. března 1901

  velkokříž Řádu Jižního kříže – 11. července 1871
  Černohorské knížectví
  velkokříž Řádu knížete Danila I. – 1902

 rytíř Řádu slona – 16. listopadu 1863
 Čestný kříž Řádu Dannebrog – 14. října 1864
Pamětní medaile zlaté svatby krále Kristiána IX. a královny Luise – 1892
 velkokomtur Řádu Dannebrog – září 1901
 Ernestinská vévodství
 velkokříž Vévodského sasko-ernestinského domácího řádu – 1859

  velkokříž Řádu Šalomounovy pečeti – 1874
  velkokříž Řádu etiopské hvězdy – 9. října 1901

 velkokříž Řádu čestné legie – 15. března 1863

 velkokříž Řádu Guelfů – 1864
 rytíř Řádu svatého Jiří – 1865
  Havajské království
  velkokříž s řetězem Řádu Kalākaua – červenec 1881
  Hesenské velkovévodství
 velkokříž Řádu Ludvíka – 8. října 1862
 velkokříž Velkovévodského hessenského řádu za zásluhy – 18. února 1878

  velkostuha Řádu chryzantémy – 1886
  velkostuha s řetězem Řádu chryzantémy – 13. června 1902
  Meklenbursko
 velkokříž se zlatou korunou Domácího řádu vendické koruny – 1865
  Nasavsko
 rytíř Nassavského domácího řádu zlatého lva – srpen 1865

  velkokříž Řádu nizozemského lva – 1849

  velkokříž Řádu svatého Olafa – 3. října 1874
  velkokříž s řetězem Řádu svatého Olafa – 1906

 Řád Osmanie I. třídy – 25. května 1862
Hanedan-i-Ali-Osman – červen 1902

 velkokříž Řádu věže a meče – 25. listopadu 1858
 velkokříž Stuhy tří řádů – 8. února 1901

  rytíř Řádu černé orlice s řetězem – 1869
  velkokříž Řádu červené orlice – 17. ledna 1869
  rytíř spravedlnosti Řádu johanitů – 1884

  velkokříž Královského uherského řádu svatého Štěpána – 13. června 1867

  velkokříž Řádu rumunské hvězdy – 1882
  velkokříž s řetězem Řádu Karla I. – 1906

  rytíř Řádu svatého Ondřeje – 1844
  rytíř Řádu svatého Alexandra Něvského – 1844
  rytíř Řádu bílého orla – 1844
  rytíř I. třídy Řádu svaté Anny – 1844
  Řád svatého Stanislava I. třídy – 1844
  Řád svatého Vladimíra III. třídy – 1881

 velkokříž Řádu Spasitele – 29. května 1862

  velkokříž Rytířského řádu San Marina – srpen 1902

 rytíř Řádu zvěstování – 20. února 1859

  rytíř Řádu routové koruny – 1844
  Sasko-výmarsko-eisenašské vévodství
 velkokříž Řádu bílého sokola – 17. dubna 1860

  rytíř Řádu Mahá Čakrí – 1880
  velkokříž Řádu bílého slona – 1887

  rytíř Řádu zlatého rouna – 7. května 1852
  velkokříž s řetězem Řádu Karla III. – 6. května 1876

 rytíř Řádu Serafínů – 27. září 1864
 rytíř Řádu Karla XIII. – 21. prosince 1868
 komtur velkokříže s řetězem Řádu Vasova – 26. dubna 1908

  velkokříž Řádu württemberské koruny – 1883

Odkazy

Reference

Související články

 Tituly a vyznamenání Alžběty II.
 Tituly a vyznamenání Eduarda VIII.
 Tituly a vyznamenání prince Charlese
 Tituly a vyznamenání Jiřího V.
 Tituly a vyznamenání Jiřího VI.

Tituly a vyznamenání osob ze Spojeného království

Rytíři Podvazkového řádu
Nositelé Řádu indické hvězdy
Rytíři velkokříže Řádu lázně
Rytíři Řádu bodláku
Rytíři Řádu svatého Patrika
Nositelé Nejctihodnějšího řádu sv. Jana Jeruzalémského
Nositelé Řádu Indické říše
Rytíři velkokříže Řádu sv. Michala a sv. Jiří
Rytíři velkokříže Královského Viktoriina řádu
Nositelé Řádu routové koruny
Rytíři Řádu svatého Ondřeje
Nositelé Řádu svatého Alexandra Něvského
Rytíři Řádu bílého orla
Nositelé Řádu svaté Anny
Nositelé Řádu svatého Stanislava
Nositelé Řádu svatého Vladimíra
Nositelé velkokříže Řádu nizozemského lva
Rytíři španělského Řádu zlatého rouna
Rytíři velkokříže s řetězem Řádu Karla III.
Rytíři velkokříže Řádu věže a meče
Nositelé Stuhy tří řádů
Nositelé Řádu zvěstování
Nositelé Vévodského sasko-ernestinského domácího řádu
Nositelé Řádu bílého sokola
Nositelé Domácího řádu věrnosti
Nositelé Řádu zähringenského lva
Nositelé Řádu Osmanie
Nositelé velkokříže Řádu Spasitele
Nositelé Řádu Ludvíkova
Nositelé velkokříže Řádu čestné legie
Rytíři Řádu slona
Nositelé Čestného kříže Řádu Dannebrog
Velkokomtuři Řádu Dannebrog
Nositelé Řádu Serafínů
Nositelé Řádu Karla XIII.
Komandéři velkokříže Řádu Vasova
Nositelé Řádu Guelfů
Nositelé Řádu svatého Jiří (Hannoversko)
Nositelé Domácího řádu vendické koruny
Nositelé Královského řádu svatého Štěpána
Rytíři Řádu černé orlice
Nositelé Řádu červené orlice
Nositelé velkokříže Řádu Jižního kříže
Rytíři velkokříže Řádu svatého Olafa
Rytíři velkokříže s řetězem Řádu svatého Olafa
Nositelé Řádu Šalomounovy pečeti
Nositelé Řádu etiopské hvězdy
Nositelé Řádu Mahá Čakrí
Nositelé Řádu bílého slona
Nositelé velkokříže Řádu rumunské hvězdy
Nositelé Řádu Karla I.
Nositelé Řádu württemberské koruny
Nositelé velkokříže Řádu chryzantémy
Nositelé velkokříže s řetězem Řádu chryzantémy
Nositelé Řádu svatého Huberta
Nositelé Řádu San Marina
Nositelé Řádu knížete Danila I.
Nositelé Medaile zlatého výročí královny Viktorie
Nositelé Medaile diamantového výročí královny Viktorie


text:
Capstone Property Recruitment
Bristol 0117 287 2008
Capstone News
Construction and Development
£65000 - £75000.00 per annum
Kevin Boakye
k.boakye@capstone-recruitment.com
Commercial Manager - Developer
£65k - £70k basic p.a. plus excellent benefits and guaranteed bonus.
West End, London
A specialist Development and Project Management platform based in London with investor roots in Asia are contributing to London's most luxurious residential, leisure and hospitality projects. With a track record of delivering landmark schemes across the world, my client is growing a team to be involved in the delivery of a major hotel project; the first of its kind in Europe, which needs a Commercial Manager to support the Commercial Director and lead the project.
- Reporting directly into the Commercial Director, overseeing the main contractor and cost consultant parties, ensuring that they work closely together to deliver this iconic project on time and within budget.
- The role will encompass a range of cost management processes as part of the procurement team, which includes the monitoring of all financial models from forecasting to variations in cost.
- You will collaborate with external insurance adviser and assist in identifying and recommending appropriate strategies
The successful candidate must have a relevant degree in Quantity Surveying or Commercial Management, having worked previously at a construction consultancy or developer.
You will ideally have good JCT contract knowledge with the aspiration to work on major real estate developments.
If you'd like to be a part of a hugely successful young and dynamic company then please check the requirements and contact Kevin Boakye on 020 3757 5000 or email a copy of your CV to k.boakye@capstone-recruitment.com.
Transparency Notice to Prospective Candidates
We process certain personal information about you for our legitimate business interests in order to identify and contact suitable candidates about positions that may be relevant to them. At Capstone Property Recruitment Ltd we respect your Privacy details all of which are set out in our Privacy Policy on our website at www.capstone-recruitment.com .If you don't want to be contacted by us about potential positions that may be of interest to you in the future, then please email our Data Protection Officer at dpo@capstone-recruitment.com
Senior Consultant - Construction and Development UK
020 37575000 / 07494368464
US$100000.00 - US$120000 per annum + + bonus + 20 days holiday
Chartered Surveyors have always been Leaders......
Stephen Langton
Qaunity Surveying, Working in Property, Quantity Surveyor, Building Surveying, Building surveyor, Real Estate, Construction and Development
Working in Construction - A definition of a Civil Engineering Surveyor
Civil Engineering Surveyor
Insight into a Senior Project Manager Role
All rights reserved 2018 Capstone


text:
Q: How to limit the number of digits in a calculator program for the windows phone? I'm having a hard time limiting the number of digits allowed on the text box. I have this code right here:
if (inputPanelTextBox.Text.Length >= 16)

I just don't know what to do under the if statement. What I want to happen is the same thing as the e.handled = true. When the text reaches 15 digits, don't do anything to it if the user attempts to add more digits.
Any help would be very helpful. Thank you so much in advance.

A: The way I would do it handle the OnKeyDown event.
inside of the handler, check the length of the string (as you have done), then set the e.Handled property to true, this will stop the character from being added to the textbox.

redpajama_set_name: RedPajamaStackExchange


Member of the European Association of Agricultural Economists, the Balkan Scientific Association of Agricultural Economists, the Scientific Association of Serbian Economists, the Serbian Association of Agricultural Economists, and the Belgrade Association of Economists.
Member of the EU Food Unique network (the international expert group).
Contributed definition of the Draft Law on Public Warehouses within the high level working group (Project Current State and Modeling of Wheat Public Warehousing System in Serbia, financed by The Republic of Serbia Ministry of Agriculture, Forestry and Water management 2003-2009). Participated the Ministry of Agriculture, Trade, Forestry and Water management working group focused on The Commodity Exchange Act (2011-2012).
University of Kiel, Germany (2004).
Teaches Agricultural Economics at the Undergraduate level and agro-rural courses at Master level (Study program – Economic Policy and Development); a lecturer of Developmental and Regional Economics at the PhD level.
(EAG) Agricultural Economics, (MAEP) Agricultural Economy and Policy, (MFTB) Financial Market and Exchange, (MGRP) Globalization and Agricultural Development, (MRUR) Rural Development.
FP7 – Food Consumer Science in the Balkans: Frameworks, Protocols and Networks for a better knowledge of food behaviors, European Commission, 2008-2011.
The support system for cooperatives in Serbia and Lessons Learned, UNDP 2009.
Business Enabling Project: Financial Market Development, USAID 2011-2012.
FP7 – International comparison of product supply chains in the agri-food sector: determinants of their competitiveness and performance on EU and international markets, 2012-2015.
FP7 – Global and local food assessment: a multidimensional performance-based approach, 2013-2016.
H2020 – Sustainable finance for sustainable agriculture and fisheries, 2015-2019.
Žaklina Stojanović, Gnjatović Dragana (2015): Cooperatives in Serbia (in Serbian), Hisperia Edu, Belgrade.
Žaklina Stojanović, Rakonjac Antić Tatjana, Popović Petrović Ivana (2015), The impact of catastrophic risks on agricultural sector in Serbia, in: Jelena Kočović, Biljana Jovanović Gavrilović, Vesna Đukić (red.), Catastrophic Risks and Sustainable Development, Belgrade: Faculty of Economics, University of Belgrade.
Žaklina Stojanović, Dragutinović-Mitrović Radmila, Laniau Matine (2013): Products with nutrition and health claims in the Western Balkans: labelling behaviour, regulation and policy implications, European Journal of Law and Economics, DOI: 10.1007/s10657-013-9386-z.
Dominique Barjolle, Gorton Matthew, Milosević-Djordjević Jasna, Stojanović Žaklina (ed.) (2013): Food Consumer Science: Theories, Methods and Application to the Western Balkans, Springer, The Netherlands.
Žaklina Stojanović, Filipovic Jelena, Mugoša Boban (2013): Consumer acceptance of functional foods in Montenegro, Montenegrin Journal of Economics, Vol. 9, No. 3.
Iris Zezelj, Jasna Milosevic-Djordjevic, Zaklina Stojanovic, Galjina Ognjanov (2012): The motivational and informational basis of attitudes toward foods with health claims, Appetite, Vol. 59, Issue , pp. 960-967.
Zaklina Stojanovic, Svetlana Popovic (2012): Labour Supply and Demand Mismatches in the Rural Serbia – The current state and consequences, print in ed. Cvijanovic D., Subic J., Andrei J., Sustainable Agriculture and Rural Development in Terms of The Republic of Serbia Strategic Goals Realization within The Danube Region, IEP, pp. 975-990.


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Savin, the Juniper Savin (Sabina), or Saffern, was known of old as the "Devil's Tree," and the "Magician's Cypress," because much affected by witches and sorcerers when working their spells.
Added Dec 1, 2010 | Healing Shrubs | 3,395 Reads
It is a herb which grows freely in our bed of garden Herbal Simples, if properly cared for, and which possesses medicinal virtues of a potential nature. The shrub is a native of southern Europe, being a small evergreen plant, the twigs of which are densely covered with little leaves in four rows, having a strong, peculiar, unpleasant odour of turpentine, with a bitter, acrid, resinous taste. The young branchlets are collected for medicinal use. They contain tannin, resin, a special volatile oil, and extractive matters.
A medicinal tincture (H.) is made from the fresh leaves, and the points of the shoots of the cultivated Savin. But this is a powerful medicine, and must be used with caution. In small doses of two or three drops with a tablespoonful of cold water it is of singular efficacy for arresting an active florid flux from the womb at the monthly times of women when occurring too profusely, the remedy being given every two, three, or four hours. Or from one to four grains of powdered Savin may be taken instead of each dose of the tincture.
The stimulating virtues of Savin befit it for cleansing carbuncles, and for benefiting baldness. When mixed with honey it has removed freckles with success; the leaves, dried and powdered, serve, when applied, to dispel obstinate warty excrescences about the genitals.
Rubbed together with cerate, or lard, powdered Savin is used for maintaining the sores of blisters, and of issues, open when it is desired to keep up their derivative action.
The essential oil will stimulate the womb to functional activity when it is passively congested and torpid. As to its elementary composition this oil closely resembles the spirit of turpentine; and when given in small well diluted doses as a tincture (made of the oil mixed with spirit of wine), such medicine does good service in relieving rheumatic pains and swellings connected with impaired health of the womb. For these purposes the ordinary tincture (H.) of Savin should be mixed, one part thereof with nine parts of spirit of wine, and given in doses of from six to ten drops with a tablespoonful of water. Dr. Pereira says about the herb: "According to my own observation, Savin is the most certain and powerful stimulator of the monthly courses in the whole of our Materia Medica; and I never saw any ill effects result from its administration." The essential oil may be preferred in a dose of from one to four drops on sugar, or in milk, when this functional activity is sought.


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Gender-neutral bathrooms now legal in Minneapolis
The change aims to accommodate the transgender community in the cities
Barry Lytton
Due to a recently passed resolution, Minneapolis restaurants and bars now have the option to offer gender-neutral, single-user bathrooms as a substitute for gender-segregated restrooms.
City Council members unanimously passed the resolution at a meeting Friday. The change, City Council members said, encourages gender-neutral restrooms throughout the city's businesses, educational facilities and other buildings, in part to better accommodate the transgender community.
"Being able to have gender-neutral bathrooms is one way to reduce the questioning, the harassment … [and people] having to face potential discriminatory treatment," said Elizabeth Glidden, who represents Ward 8 on the City Council and co-authored the resolution.
Transgender people have long grappled with gendered bathrooms, City Council members said at Friday's meeting, and no longer requiring restrooms to be classified by sex is a first step toward equality.
On the University of Minnesota's campus, there are 52 known gender-neutral bathrooms, according to the University of Minnesota's Gay, Lesbian, Bisexual, Transgender, Ally Programs Office.
A few blocks from the edge of the University's West Bank campus — which alone provides four gender-neutral bathrooms — the Hard Times Cafe has had two for at least 15 years, employee and co-owner Erica Johnson said.
She said transgender employees and customers alike have expressed gratitude to the business for providing the spaces, and to her knowledge, no one has ever complained to management about the gender-neutral bathrooms.
Andrea Jenkins, a transgender female, worked on the measure alongside City Council members as part of the city's recently established Transgender Issues Work Group. Since March, the group has teamed up with city officials to examine issues facing the transgender community and promote policy changes in hopes of fostering greater equality.
"The bathrooms [were] a really good first step, but we still have more issues that we intend to address, and hopefully [we can] make some substantial changes," Jenkins said.
The resolution also intends to help others, like families with small children and people with disabilities.
"You often have a reason to go into the restroom with your child, and having that gender-neutral bathroom makes that situation a whole lot easier," Glidden said. "If the child is a different sex than the parent, there isn't an issue created."
For businesses with existing single-occupancy bathrooms that wish to make the switch, there is little cost involved, said Andrew Johnson, who represents Ward 12 on the City Council and co-authored the resolution. The change entails replacing "men's" and "women's" bathroom placards with gender-neutral signing, like "unisex" or "family," he said.
Johnson said the resolution also benefits local companies and establishments.
"The biggest impetus for doing this is to have greater equality and equity in our city — but at the end of the day, this is also a good change for small businesses," he said, "[by] giving reasonable control to our businesses."
Johnson said the former bathroom ordinance was one of many "antiquated" city laws.
"I've been going through our ordinances and really working to clean up [ones] that don't make a lot of sense," he said.


text:
COVID-19 Health Protection Guidelines
According to the Centers for Disease Control, our current transmission level in the community is high. As such, here are the health protection guidelines you need to be aware of before you enter one of our facilities:
Masks required
Self-screening. Contact your primary care team before visiting any of our locations if you have COVID-19 symptoms.
No more than 2 visitors allowed per inpatient
To reduce the number of people in our facilities, if you are able, please come alone to your outpatient medical appointments.
Physical distancing required.
View health protection level for all Houston VA locations
Houston health care
Introduction to Mindfulness Workshop
Tue. Feb 7, 2023, 11:00 am – 12:00 pm CT
This is an online event.
Introduction to Mindfulness Workshop: 60-minute workshop offering an overview of mindfulness and its benefits, opportunities to try out mindfulness exercises, and resources on ways to begin a mindfulness practice or to go deeper after the workshop. No appointment needed.
When: 1st Tuesday of every month, 11:00 a.m. - 12:00 p.m.
Where: Join WebEx meeting
Passcode: umGjgeu@325
Dial in: Join by video system. Dial 27624509480@veteransaffairs.webex.com. You can also dial 207.182.190.20 and enter the meeting number.
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The master Prevention and Public Health provides tools for planning and designing theory- and evidence based health intervention programmes. The master also provides tools for evaluation research of these interventions.
An intervention at worksites to prevent overweight: nutrition education for employees and changes in the food supply of the canteen.


Can you focus on more than one book at a time?
Well, not literally. That would be impossible.
But, you can switch between books. Which is what I do.
At any given time, I'm reading at least three books. Always thought it was weird until I read a recent post. Now I know I'm not the only person who keeps several books in the air.
It's quite a balancing act.
There is always at least one or two books about writing, a mystery, and a comedy/love story open on my Kindle. Which one I read when depends on my mood. And a really odd thing is, no matter how long it's been since I started the book, it's as if I began reading it yesterday. I have no problem picking up the story thread. No matter the genre.
But, from time to time, there are books I can't put aside.
So, what will keep me read a book from cover to cover in one sitting?
A story that grabs me by the throat and I must find the answers.
A book on writing that is filled with real meat on the craft.
Why do I read several books?
Here are the books I'm reading now. Click on image to read about the book.
Do you read several books at once?
Can you pick up the thread in them?
Do you read different genres? Or stick to one?
Or do you prefer to finish one book before starting another?
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Usually I have a non-fiction or self-help going; a novel or two, but they cannot be set in the same time or place; and a sci fi or "literature"/classic or both. So far, I haven't been confused and like you, I am able to pick up without having to backtrack-read to start.
The comments on this post were interesting as well. Thanks for opening up this discussion.
Check out this helpful post from Jean's Writing blog on whether you can focus on more than one book at a time.
I currently have three books on the go and have been known to have four. For manuscripts that you have written yourself, what I find works is working on one project after completing it and then putting it aside for a year or so. That way you can come back to it completely afresh.
I almost always read 3-4 books at a time, shifting around based on mood, mental energy, goal for reading. Currently reading books on Buddhist teachings, physics, and historical fiction/romance.
Reading: I keep at least 3 books on my nightstand usually of two different genres. One title usually takes priority and occasionally I find the third book is not to my taste and eventually makes it back to the library shelves.
Writing: It's easy for me to shift from blog to memoir writing. One is a sprint, the other a marathon.
Yes. At the moment there is "Snowchild", and a book about a baltic-German family (kind of family biography), a fantasy book, a high-brow literature book about a cruise, Dickens' Great Expectations, Döblin's Berlin Alexanderplatz, a reread of Jane Austen's Mansfield Park, a Barbara Metzger regency romance and Robin Hobb's third part of the farseer-trilogy on the go – and I might have left out those of which I only read the first chapter or so … Oh, I forgot two more fantasy books – one is "high fantasy" (completely new world) the other is a horror-fantasy-mixture taking place in this world.
Can you pick up the thread in them? Mostly. Last year was the first time I had to completely reread a book to get it going again. (Christopher Moore, the second of his vampire-books). Sometimes I can lapse a whole decade and start reading again as if I dropped the book yesterday.
Do you read different genres? Or stick to one? See answer above. I think that is what keeps me able to remember the plots. If there were too many books of a genre that is usually prone to fulfill a pattern (like regency romances) I would defo mix things up.
Sorry for the long comment, but you asked.
Thanks so much. Love long answers. But wow! You keep a lot of books in the air. I'm impressed. I agree that different genres make it easier to remember the story thread. Also, I too don't count the ones I never finish. Time is too precious to spend time reading something I don't enjoy.
I will only be able to say if I finished them after I did or after I gave up – these books here are all books right now I want to finish.
Always; a paperback on the go, because I have so many good ones waiting to be read and I like to read them on the bus or train and beach hut where I daren't take my precious Kindle. On Kindle mostly Indie authors and the occasional classic downloaded for free when it has come to my attention; a novel, a non fiction and always a short story collection to read in between books.
My Kindle White fits in pockets or purse and is almost always with me. Wow you read more than I do and that's quite an accomplishment.
I'm opposite. I can have several books going at the same time, but find writing more than one WIP is too hard. Even using a style-sheet.
I can but I don't like to read more than one book at a time. If I do, it's normally a how-to non-fiction and a fiction. I definitely can't write more than one book at a time. As it is, I sometimes get the character names mixed up when I finish writing one and start another right away.
Sounds like you need a style-sheet. Since I started one for my WIP, I've kept things straight.
I'm always reading several books at a time. Literature (those 800-words tomes) can get a bit much if I don't take a break from time to time. Same goes for my obsession with history books and autobiographies. And then there are those books I promise to review but can't get into. Oh boy. I may have 5 or 6 of them going right now. Eek! I need help!
I can't imagine! My longest novel was 90,000 words and that was enough!
A book has to really grab me for me to stick to one.
Reading or writing? I have tried reading more than one at a time but prefer to stick to one. I tent to lose the polt a bit if I swap around. Writing; I have three on the go but they are all in the same series. I'm mainly focused on the first but add bits to the other two as I think of them.
Good idea to add bits to the other WIP manuscripts.
I'm hoping it will connect the stories.


Welcome to the first in our new series of "Featured Resource" posts! As we continue to build the Tool Library, we thought it would be helpful to introduce you to some of these tools with a little more detail and an invitation for you to provide your feedback. The featured resource this month is The Free Mindfulness Project.
Guided Mindfulness Exercises – Whether you are looking for a meditation guide to take you through breathing awareness, imagery, or a body scan you'll find multiple options here. You'll also find a range of timed self-guided options available. Try the "5 minutes just bells" download for a quick mindfulness break at your desk. More than 25 files are available for you to download with an open, Creative Commons license (CC: BY-NC-SA).
How is my mind right now? – "We are living in an increasingly busy world. Often we are wrapped up in thoughts and feelings, constantly moving from one thing to the next without a moment's peace." Sound familiar? This brief page provides the place to begin your mindfulness journey.
Discussion Forum – Are you interested in talking to other people about mindfulness? This site provides a link to Everyday-mindfulness.org, an active online community in which you can join conversations and groups to learn more.
While this site's blog and Twitter account have not been active recently, the resources listed above remain available and relevant.
P.S. You may not be surprised that I would pick a mindfulness tool first. I have been a fan of the Stop, Breathe & Think app for years now, and continue to be amazed at the growing focus on mindfulness in education. If you are interested in learning more, please take a look at my recent interview with Coach and Counselor Tiffany Guske – Mindful Online Teaching and Learning.
We had an amazing time at the NCDA Global Conference this year! Thanks to all of the organizers and attendees for the many effective learning and community-building opportunities. And a special thanks to all who joined our session on site! If you were not able to attend the conference, you can catch up on some of the event's highlights by searching Twitter, Instagram, and Facebook for the event hashtag: #NCDAPhoenix.
In this post, we are excited to share our Snapchat presentation. It includes some Snapchat basics, along with a description of how we used Snapchat this past year in our courses – both campus-based (Deb) and online (Melissa). We each shared our goals for integrating social media, some lessons learned, tips for getting started, a list of suggested activities, and additional resources (also provided below).
Are you using Snapchat with your students? If so, we'd love to hear more about your experience! If you aren't using Snapchat, but are thinking about it, please add your questions to the comments here, so we can respond and continue the conversation.
This week I (Melissa) had the privilege of not only attending NCDA's Career Practitioner Institute in Charleston, SC, but also presenting. When I was approached about developing a session related to technology, I quickly identified this topic – it's one I field questions about fairly frequently.
Are you interested in presenting a webinar? This presentation offers an overview to get you started. Two tools are outlined, as well as a checklist to make sure you cover key tasks before, during, and after your live session.
Thanks to all who attended in person for sharing their questions, stories, and feedback! Please continue the conversation in the comments area of this post. What are your questions about webinar delivery? What are your tips for those just getting started?
If you aren't already aware of the "Top Tools for Learning" project from Jane Hart at the Centre for Learning & Performance Technologies, this post is for you!
This year's list includes the top 200 tools for learning this year, as reported by 2,174 learning professionals from 52 countries. This is the 11th year of the study, which tracks the movement of popular tools, such as YouTube and Dropbox, up and down the list.
What can you do with this list as a career practitioner?
Search the top 100 tools for workplace learning to find out what your students and clients may be using in their current or future employment settings.
Search the top 100 tools for personal and professional learning for ideas to inspire your own professional development or professional development opportunities you would like to offer your students and clients.
Search the 200 tools by category to find possible solutions for problems you are trying to solve, (e.g., forms and survey tools, web and video meeting tools, mind mapping tools, screencapture tools, online course platforms, and many more).
Also … new on the list this year – more than 50 tools that haven't appeared before. Typeform for surveys and forms, Microsoft Teams for collaboration, and ActEXS a productivity app are just a few of these to explore.
Which tech tools are you currently using in your career office or practice? How do they compare to those in this study? Share your favorites!


Appliance Repair By Red repaired our clothes washer and dryer unit.
Appliance Repair By Red came out promptly and got straight to work. I appreciated how they showed us the part that needed to be replaced. I think they're an honest and hardworking company, and I would recommend them to anyone.


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Inequality>
Donald Trump Tax Reform Speech in Springfield, Missouri 8/30/2017 (Photo: screenshot/trump tv network)
Trump's Tax Plan Is a Confidence Scam
Donald Trump traveled to Springfield, Missouri to kick off the administration's next big initiative: cutting taxes. Here was the essential Trump, the confidence man, peddling a plan that does not yet exist on paper. Populist bluster was deployed to cover for what will be a one-percenter's elixir.
Each of the four principles Trump claims will inform what he calls "tax reform" is actually mocked in the plans the administration is cooking up behind closed doors.
Robert Borosage
Aug 31, 2017Campaign for America's Future
The only question is whether Americans are gullible enough to buy what Trump, House Speaker Paul Ryan and Republicans are selling. And whether Democrats will unite to expose the con and demand progressive tax reforms that are long overdue.
Simplicity?
Trump's first principle is simplicity: the tax code should be "simple, fair and easy to understand." But Republicans aren't interested in simplicity. They are interested in cutting taxes. particularly for the wealthy and the corporations, and starving the government of revenue.
This sets up their argument for the necessity of slashing popular government programs - from Medicare, Medicaid and Social Security to disaster relief, investment in education and environmental protections.
Were simplicity the actual goal, Trump would promise a plan that was revenue neutral, or given the current deficits, produced more revenue, by closing loopholes and shutting down tax dodges. The words never crossed his lips.
Cutting Corporate Taxes
The second "principle" is to cut corporate taxes. With corporate profits a record percentage of the GDP and revenue from corporate taxes at record lows, this isn't exactly a compelling cause. So, this goal is spun as a way to make corporations competitive, creating more jobs and higher wages.
Every part of this argument is a lie. Corporations pay far lower effective tax rates than the nominal rate. And as a new study by the Institute for Policy Studies shows, the corporations that already pay at the lower 20-percent rate Republicans are peddling have cut, not created, jobs over the last nine years.
And while cutting corporate taxes lines the pockets of shareholders and CEOs, workers suffer job loss and pay stagnation.
Tax Relief?
The third principle is "tax relief for middle class families." Whenever politicians start talking about cutting taxes on the middle class, hold on to your wallet.
In every version of Trump tax plans that have been made public, the richest 1 percent pocket nearly half or more of the tax breaks. In the most recent plan, the top one-tenth of 1 percent would enjoy a tax break of over one million a year.
The average taxpayer in the middle would pocket about $1,000. Trump argues that these "forgotten people" will get the pay raise they've been waiting for, and "do lost of things with their paychecks." Yeah, like making a nominal payment on the record credit card debts, and auto and school loans that burden them. This is a heist. The Trump tax cuts will overwhelmingly benefit the few and the corporations. The middle class is offered a pittance simply to make the dish palatable.
"The American Model"
The fourth principle is the most outrageous: Trump promises to create "the American model" by bringing back trillions in wealth parked overseas. Large corporations have indeed booked trillions of profits overseas to avoid paying U.S. taxes. The numbers are staggering: an estimated $2.5 trillion, with $700 billion in owed taxes.
But does Trump's "America Model" require these corporations to pay the taxes that they owe?
No, he offers them a massive tax break - taxing the money at 5-10 percent -- to reward them for their tax avoidance. This, he claims, would "return trillions" to invest in our country and rebuild our communities.
All this is simply nonsense. The money isn't stashed off shore; it is a bookkeeping entry. The corporations already have use of it if they so choose, simply by borrowing against it and writing off the interest rates.
The last time the U.S. offered"repatriation" of overseas profits, U.S. companies used the tax giveaway to accelerate mergers, raise dividends, and buy back their own stock.
This pumped up stock prices, and CEO and executive bonuses. There was no jobs boom. And of course, the success of the scam encouraged far more companies to start rigging their books to report profits overseas to line up for the next sting.
If Trump were serious about his "American Model," he would simply make corporations pay the American taxes that they owe, not lavish rewards on multinationals gaming the tax code.
Real Tax Reform
No one likes taxes. The subject is boring, painful and complex. But there are a few simple realities that shouldn't be forgotten. The U.S. pays among the lowest taxes in the industrialized world. One result is that our infrastructure is crumbling; our schools are aged and inadequate; our shared security - from health care to retirement security - inadequate.
In a time of obscene inequality, tax reform should raise rates on the wealthy. It should make the corporations pay their fair share. It should tax activities we want to discourage -like financial speculation and fossil fuels. It should tax the income of investors at the same rates as the income of workers. It should crack down on tax dodges, close loopholes and tax havens.
And yes, it should allow the IRS to automatically fill out the tax forms for the 60 million Americans who do not itemize deductions. That sensible reform is blocked by the lawyers and tax preparation companies who profit at tax time.
Trump's plan will do none of these.
The Pied-Piper of Plutocracy
Senate Democratic leader Chuck Schumer and 45 of 48 Democratic Senators have put forth principles for bipartisan cooperation: no tax breaks for the richest 1 percent, no increased burden on the middle class, taxes should be done in "regular order" requiring bipartisan cooperation and reform must result in the revenue needed to fund the government adequately. Trump's tax plans will violate each of these.
Trump says he wants bipartisan support. But his plan is being cooked up behind closed doors in meetings only with Republican congressional leaders. He offers not an outstretched hand, but a clenched fist. In Missouri, he urged supporters to demand that Democratic Senator Claire McKaskill vote for his tax plan or be voted out of office.
Peddling a plan that doesn't yet exist on paper. Promising jobs and raises while pushing tax cuts for the rich and corporations. Puffing an "American Model" that rewards global corporations for their tax dodges. Trump, the pied piper of the plutocracy, would lead us into another round of privileged pillage. He won't change; he must be stopped.
Robert Borosage is the founder and president of the Institute for America's Future and co-director of its sister organization, the Campaign for America's Future.
InequalityPlutocracyTaxationTrumpism


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דף הבית > "Unity in Diversity" Reloaded? The European Court of Human Rights' New Subsidiarity Doctrine and its Consequences
Mikael Rask Madsen, Danish National Research Foundation's Centre of Excellence for International Courts (iCourts) "Unity in Diversity" Reloaded? The European Court of Human Rights' New Subsidiarity Doctrine and its Consequences
When the European Convention of Human Rights (ECHR) was drafted in the late 1940s and early 1950 the project was driven by geopolitical interests and was mainly put in place to sound the alarm if genuine democracy was threatened in the member states (Bates 2011; Madsen 2007; Simpson 2004). The real fear was Soviet expansionism into Western Europe and a robust institutional bulwark consisting of the European Court of Human Rights (ECtHR), the European Commission of Human Rights (EComHR) and the Committee of Ministers of the Council of Europe was put in place to defend the free world. While this set-up worked reasonably well, notably at the intergovernmental political level, the system itself eventually developed into a different kind of regime with a focus on providing individual justice in a growing number of member states. This raised fundamental questions as to the level of difference allowed within the common European human rights space. Basically, was the system to rest on minimum standards with room for domestic differences or was it to create uniform standards. In other words, how much diversity was allowed within the unified space? These questions came up already in the early 1990s but have received new attention in the context of the ECtHR's recent turn towards a new procedural subsidiarity doctrine (Spano 2014; Spano 2018). Scholars have started investigating both the new subsidiarity standard (Arnardóttir 2017; Arnardóttir 2014) and its consequences (Cali 2018; Madsen 2018). Based on limited data analysis, these initial studies suggest that the new approaches are possibly creating divergences within the space of European human rights where the established democracies tend to come out ahead. This paper picks up where these initial studies ended and explore more systematically and with a much larger dataset the hypothesis that the new subsidiarity doctrine is indeed introducing new forms of difference within European human rights. It proceeds as follows. In an initial section, the overriding question of unity in diversity – and its accompanying ideas of unity with uniformity and diversity without disintegration – is revisited. It then uses a big data approach to explore how this articulates in the actual case law of the ECtHR, seeking to identify patterns of change based on a set of variables of difference. In a third section, it discusses and concludes the findings of the paper with a view to its consequences for European human rights more generally.


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