text:
Contaminated land and other forms of environmental impairment: an approach to valuation
Paul Syms (Paul Syms is Visiting Research Fellow at Sheffield Hallam University, UK and Principal of Paul Syms Associates, Macclesfield, Cheshire)
Journal of Property Valuation and Investment
The valuation of properties affected by environmental impairment presents valuers with many problems, not least of which is the identification of those properties which may be so affected. Offers a list of land use activities, together with the types of contamination which may be found. Proposes a method of valuing impaired properties, taking account of present and future costs of treatment. Discusses the question of stigma and describes a method of assessing the valuation impact. Concludes with a proposal to establish an Environmental Impairment Databank, so as to assist valuers in the preparation of valuations and the advice which they provide to clients.
Environmental audit
Syms, P. (1996), "Contaminated land and other forms of environmental impairment: an approach to valuation", Journal of Property Valuation and Investment, Vol. 14 No. 2, pp. 38-47. https://doi.org/10.1108/14635789610112655
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Copyright © 1996, MCB UP Limited
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\section{Introduction}
\label{sec:introduction}

For any tamely embedded closed curve $\gamma$ in $\mathbb{R}^3$, the \emph{superbridge number} of $\gamma$, denoted $\operatorname{sb}(\gamma)$, is the maximum number of local maxima of the projection of $\gamma$ to any line. Compare to the bridge number $\operatorname{b}(\gamma)$, which is the \emph{minimum} number of local maxima, and the total curvature, which is $2\pi$ times the \emph{average} number of local maxima.

The superbridge number was introduced in Kuiper's 1987 paper~\cite{Kuiper:1987ki}, where he defined the \emph{superbridge index} to be the minimum of the superbridge number over all realizations of the knot; by construction this is a knot invariant, and the superbridge index of  a knot $K$ is denoted $\operatorname{sb}[K]$. While Kuiper computed it for all torus knots, the superbridge index is generally quite difficult to determine: for example, whereas the bridge index is known for all knots through 12 crossings~\cite{Blair:2020iz,knotinfo,Musick:2012uz}, the superbridge index is known for very few knots---as of September 1, 2020, the only knot invariant recorded by KnotInfo~\cite{knotinfo} which is known for fewer knots is the topological 4-dimensional crosscap number (a.k.a.\ nonorientable 4-ball genus)~\cite{Murakami:2000wb}.

The superbridge index often appears in conjunction with the \emph{stick number}---the minimum number of segments needed to construct a piecewise-linear realization of the knot---because of the bound ${\operatorname{sb}[K] \leq \frac{1}{2}\operatorname{stick}[K]}$ due to Jin~\cite{Jin:1997da}. Indeed, all determinations of superbridge indices of non-torus prime knots to date have come from showing this upper bound matches a lower bound on superbridge index.

However, this cannot be a winning strategy in general: as first observed by Furstenberg, Li, and Schneider~\cite{Furstenberg:1998fq}, the difference between the two sides of Jin's bound can be arbitrarily large.

\begin{figure}[t]
	\centering
		\subfloat[$8_{10}$]{\includegraphics[height=2in,valign=c]{8_10.pdf}} \qquad \qquad
		\subfloat[$9_7$]{\includegraphics[height=2in,valign=c]{9_7.pdf}}
	\caption{Polygonal realizations of the $8_{10}$ and $9_7$ knots. These curves both have superbridge number equal to 4: each has 4 local maxima when projected to the $x$-axis, so the superbridge number is at least 4, and \Cref{cor:gordan bound} will imply that superbridge number is $\leq 4$. Each knot is shown in orthographic perspective, viewed from the direction of the positive $z$-axis relative to the vertex coordinates given in \Cref{sec:coords}.}
	\label{fig:examples}
\end{figure}

The main contribution of the present paper is a new approach to giving upper bounds on superbridge index. In some sense the approach is obvious: find a realization of a knot $K$ whose superbridge number is no bigger than some $m$, and conclude that $\operatorname{sb}[K]\leq m$. The challenge comes in showing that, for a given realization of $K$, there is no direction so that projecting to a line in that direction has more than $m$ local maxima. The strategy in this paper is to use \emph{polygonal} realizations and to observe that the existence of a direction with $m+1$ local maxima is equivalent to the feasibility of a system of linear inequalities. In other words, ruling out the existence of such directions is equivalent to showing that the system has no solutions, which will be done using Gordan's Theorem~\cite{Gordan:1873dz}, a classical linear programming tool for certifying the non-existence of solutions.

See \Cref{fig:examples} for two such polygonal realizations of the knots $8_{10}$ and $9_7$. Both will be shown to have superbridge number $\leq 4$, and hence it will follow that $\operatorname{sb}[8_{10}],\operatorname{sb}[9_7]\leq 4$. More generally, the main theorem of this paper is:

\begin{theorem}\label{thm:main}
	The knots $8_{1-15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ have superbridge index $\leq 4$, and $10_{76}$, $13n_{226}$, $13n_{328}$, $13n_{342}$, $13n_{343}$, $13n_{350}$, $13n_{512}$, $13n_{973}$, $13n_{2641}$, $13n_{5018}$, and $14n_{1753}$ have superbridge index $\leq 5$.

	This implies that the knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ have superbridge index equal to 4, and that $13n_{226}$, $13n_{328}$, $13n_{342}$, $13n_{343}$, $13n_{350}$, $13n_{512}$, $13n_{973}$, $13n_{2641}$, $13n_{5018}$, and $14n_{1753}$ have superbridge index equal to 5.
\end{theorem}

The particular polygonal realizations providing these bounds were found by generating large ensembles of random equilateral polygons in tight confinement using the algorithm described in the paper~\cite{TomClay} and implemented in the open-source {\tt stick-knot-gen} project~\cite{stick-knot-gen}.

\Cref{sec:superbridge_index} below gives some background on superbridge index, including a survey of known bounds. The connection to Gordan's theorem is explained in \Cref{sec:a_new_approach}, which is where \Cref{thm:main} is proved, and \Cref{sec:conclusion} provides some discussion and open questions. \Cref{sec:table} consists of a table of values (or possible values, when the exact value is not known) of the superbridge index for all prime knots through 10 crossings, \Cref{sec:exact values} gives all prime knots through 16 crossings for which the exact value of superbridge index is known, \Cref{sec:coords} gives the coordinates for each of the polygonal knots discussed in this paper (which can also be downloaded from the {\tt stick-knot-gen} project~\cite{stick-knot-gen}), and \Cref{sec:snappy} gives diagrams for the 13- and 14-crossing knots and homomorphisms from each knot group to the symmetric group $S_5$ which will be used in the proof of \Cref{thm:main}.

\section{Background on the Superbridge Index}
\label{sec:superbridge_index}

The only infinite class of prime knots for which superbridge index is known is the class of torus knots:

\begin{theorem}[Kuiper~\cite{Kuiper:1987ki}]\label{torus}
	For relatively prime $2\leq p < q$, the superbridge index of the $(p,q)$-torus knot is
	\[
		\operatorname{sb}[T_{p,q}] = \min(2p,q).
	\]
\end{theorem}

Progress in determining the superbridge index of knots has been slow. Aside from torus knots, the superbridge index was known for only 41 prime knots prior to the present work. In particular, it was known for only 29 of the 249 nontrivial knots with 10 or fewer crossings. In virtually all cases, the strategy for computing superbridge index is to show that some upper bound matches a lower bound.

The most useful lower bounds on superbridge index are:

\begin{theorem}[Kuiper~\cite{Kuiper:1987ki}]\label{bridge bound}
	For any nontrivial knot $K$, the bridge index $\operatorname{b}[K] < \operatorname{sb}[K]$.
\end{theorem}

\begin{theorem}[Jeon--Jin~\cite{Jeon:2002gm}]\label{3 superbridge}
	Every knot except $3_1$ and $4_1$ and possibly $5_2$, $6_1$, $6_2$, $6_3$, $7_2$, $7_3$, $7_4$, $8_4$, and $8_9$ has superbridge index $\geq 4$.
\end{theorem}

This is slightly different than the statement in Jeon and Jin's paper, which included $8_7$ in the list of possible 3-superbridge knots. However, $8_7$ cannot have superbridge index equal to 3:

\begin{lemma}\label{lem:8_7}
	$\operatorname{sb}[8_7] \geq 4$.
\end{lemma}

\begin{proof}
	This result was first proved by Adams et al. in an early version of their paper~\cite{Adams:2020vm}; their proof, which is essentially the one given below, was also sketched in a talk given by Gyo Taek Jin in February, 2020~\cite{JinTalk}.

	Jeon and Jin's characterization of possible 3-superbridge knots begins by assuming that a given parametrization $\gamma: S^1 \to \mathbb{R}^3$ of a nontrivial knot has superbridge number 3, projecting to the orthogonal complement of a \emph{quadrisecant}---a line whose intersection with the image of $\gamma$ consists of at least 4 distinct components---and using the assumption $\operatorname{sb}(\gamma) = 3$ to constrain the resulting planar curve, and hence the possible knot types of~$\gamma$.

	\begin{figure}[t]
		\centering
			\subfloat[Simple]{\includegraphics[height=1.2in]{41simple.pdf}} \qquad \qquad
			\subfloat[Flipped]{\includegraphics[height=1.2in]{41flipped2.pdf}} \qquad \qquad
			\subfloat[Alternating]{\includegraphics[height=1.2in]{41alternating.pdf}}
			\put(-85,14){\small $a$}
			\put(-57,21){\small $b$}
			\put(-32,21){\small $c$}
			\put(-8,14){\small $d$}
			\put(-163,86){\small $a$}
			\put(-175,65){\small $b$}
			\put(-183,36){\small $c$}
			\put(-190,-5){\small $d$}
			\put(-335,49){\small $a$}
			\put(-316,41){\small $b$}
			\put(-282,41){\small $c$}
			\put(-263,49){\small $d$}
		\caption{The three types of quadrisecant illustrated on the figure eight knot from KnotPlot~\cite{knotplot}. Traversing the curve in the direction of the indicated orientation starting at point $a$ produces the word $acbd$ for the alternating quadrisecant, as opposed to $abcd$ for the simple quadrisecant and $abdc$ for the flipped quadrisecant.}
		\label{fig:quadrisecants}
	\end{figure}

	The fact that $\gamma$ has a quadrisecant was proved by Kuperberg~\cite{Kuperberg:1994et}, building on work of Pannwitz~\cite{Pannwitz:1933ks} and Morton and Mond~\cite{Morton:1982dn}, but Denne has proved~\cite{Denne:2004wq,Denne:2005wq} that every nontrivial knot has an \emph{alternating} quadrisecant; see \Cref{fig:quadrisecants}. Therefore, Jeon and Jin's argument can be modified to require the quadrisecant of $\gamma$ be alternating. However, all the $8_7$ knots in their catalog (one in their Table~T and two in Table~V) are associated with a non-alternating quadrisecant, contradicting this assumption and proving that $8_7$ cannot have superbridge index equal to~3.
\end{proof}

By far the most useful upper bound on superbridge index is given in terms of stick number.

\begin{theorem}[Jin~\cite{Jin:1997da}]\label{stick bound}
	For any knot $K$, $\operatorname{sb}[K] \leq \frac{1}{2}\operatorname{stick}[K]$.
\end{theorem}

This follows because the projection of a polygonal knot to a line cannot have more critical points than vertices. To date, all determinations of superbridge indices of non-torus knots have come from matching this upper bound with one of the lower bounds from Theorems~\ref{bridge bound} or~\ref{3 superbridge}. In particular, this accounts for all exact values of superbridge index in \Cref{sec:exact values} aside from those for torus knots and those attributable to \Cref{thm:main}.

As mentioned in the introduction, this strategy cannot work in general: for any $n$ there are only finitely many knots with $\operatorname{stick}[K] \leq n$~\cite{Calvo:2001gv,Negami:1991gb}, but there are infinitely many 2-bridge knots and hence, by the next theorem, infinitely many knots with $\operatorname{sb}[K] \leq 5$.

\begin{theorem}[Adams et al.~\cite{Adams:2020vm}]\label{bridge upper bound}
	For any knot $K$, $\operatorname{sb}[K]\leq 3\operatorname{b}[K]-1$.
\end{theorem}

This bound is somewhat weaker than the one announced in~\cite{Adams:2009ha}, but it is still a significant improvement on the bound $\operatorname{sb}[K] \leq 5 \operatorname{b}[K]-3$ proved by Furstenberg, Li, and Schneider~\cite{Furstenberg:1998fq}. Notice, in particular, that this theorem implies $\operatorname{sb}[10_{37}] \leq 5$, even though the best extant bound on stick number is $\operatorname{stick}[10_{37}] \leq 12$~\cite{TomClay}.

Superbridge index is also bounded above by twice the braid index~\cite{Kuiper:1987ki} and by the harmonic index~\cite{Trautwein:1995wr,Trautwein:1998kn}, though these are less useful: the only knots through 11 crossings for which either of these bounds is better than those from Theorems~\ref{stick bound} and~\ref{bridge upper bound} are the 3-braid knots $11a_{240}$ and $11a_{338}$, both of which have bridge index 3~\cite{Musick:2012uz} and no known bound on stick number besides the general bound $\operatorname{stick}[K] \leq 18$ for all 11-crossing knots~\cite{Huh:2011co}.

\section{A New Approach}
\label{sec:a_new_approach}

One of the major challenges in computing superbridge index is that the inequalities go the wrong way: if $\gamma$ is a closed curve in $\mathbb{R}^3$ and the projection of $\gamma$ to a line has $m$ local maxima, this shows that $\operatorname{sb}(\gamma) \geq m$. But this gives no information about $\operatorname{sb}[\gamma]$, since there could well be some other line on which the projection of $\gamma$ has $m+1$ local maxima (contrast with bridge index: in the hypothetical, it follows that $\operatorname{b}[\gamma] \leq \operatorname{b}(\gamma) \leq m$). This means it is not so easy to use particular realizations of a knot to give bounds on superbridge index.

Working with polygonal knots---that is, piecewise linear embeddings $\gamma: S^1 \to \mathbb{R}^3$---discretizes the problem and will turn out to yield bounds coming from Gordan's theorem. Rather than thinking in terms of the parametrization, it will be more convenient to represent polygonal knots by a list $\vec{e}_1, \dots , \vec{e}_n$ of edge vectors.

In those terms,
\[
	\operatorname{b}_{\vec{v}}(\vec{e}_1, \dots , \vec{e}_n) = \#\{i \in [n] : \vec{e}_i \cdot \vec{v} >0 \text{ and } \vec{e}_{i+1} \cdot \vec{v} < 0\}
\]
with the convention that $\vec{e}_{n+1} = \vec{e}_1$. Here $\cdot$ is the usual dot product on $\mathbb{R}^3$.

The proof of \Cref{stick bound} implies that $\operatorname{sb}(\vec{e}_1, \dots , \vec{e}_n) \leq \frac{n}{2}$. Equality is achieved in this bound if and only if $n$ is even and the projection of each vertex to the line spanned by some $\vec{v} \in S^2$ is either a local minimum or a local maximum. In particular, this means that the list
\[
	\vec{v} \cdot \vec{e}_1, \vec{v} \cdot \vec{e}_2, \dots , \vec{v} \cdot \vec{e}_n
\]
must alternate signs. By replacing $\vec{v}$ with $-\vec{v}$ if necessary, it is no restriction to assume that $\vec{v} \cdot \vec{e}_1 > 0$, so that the sign pattern is $+1,-1,\dots ,-1$.

Equivalently, if $n$ is even and
\[
	E = \left[ \vec{e}_1 | -\vec{e}_2 | \cdots | -\vec{e}_n \right]
\]
is the $3 \times n$ matrix with the $(-1)^{i+1} \vec{e}_i$ as its columns, $\operatorname{sb}(\vec{e}_1, \dots , \vec{e}_n) < \frac{n}{2}$ if and only if there is no $\vec{v} \in \mathbb{R}^3$ so that $\vec{v}^T E$ has all positive entries. Contrapositively, the linear system $\vec{v}^T E > 0$ has a solution if and only if $\operatorname{sb}(\vec{e}_1, \dots , \vec{e}_n) = \frac{n}{2}$.

Gordan's theorem is a key tool for determining whether such systems of linear inequalities have solutions:

\begin{theorem}[Gordan~\cite{Gordan:1873dz}]\label{gordan}
	Suppose $A$ is a $k \times \ell$ real matrix. Then exactly one of the following is true:
	\begin{enumerate}
		\item \label{g1} There exists $\vec{v} \in \mathbb{R}^k$ so that $\vec{v}^T A$ has all positive entries.
		\item \label{g2} $A \vec{u} = \vec{0}$ for some nonzero vector $\vec{u} \in \mathbb{R}^\ell$ with nonnegative entries.
	\end{enumerate}
\end{theorem}

Among the class of theorems called theorems of the alternative (see, e.g.,~\cite{Tucker:1956ve} or~\cite[\S2.4]{Dantzig:2003ue}), Gordan's theorem was the first to appear, predating Farkas' Lemma~\cite{Farkas:1902bv} by almost 30 years. Here is an intuitive explanation of why Gordan's theorem is true: if \hyperref[g1]{(i)} is true, then the angles between $\vec{v}$ and each of the columns of $A$ are all less than $90^\circ$. But in this case the columns of $A$ all lie in the interior of a half-space, and no (nontrivial) conical combination produces the origin. On the other hand, \hyperref[g2]{(ii)} says exactly that the origin is a conical combination of the columns of $A$.

Gordan's theorem, combined with the preceding discussion, yields the following immediate corollary:

\begin{corollary}\label{cor:gordan bound}
	If $n$ is even and $\vec{e}_1, \dots , \vec{e}_n$ are the edge vectors of a closed polygonal curve in $\mathbb{R}^3$, then $\operatorname{sb}(\vec{e}_1, \dots , \vec{e}_n) < \frac{n}{2}$ if and only if there exists a nonzero vector $\vec{u} \in \mathbb{R}^n$ with nonnegative entries solving the matrix equation
	\begin{equation}\label{eq:gordan}
		\left[ \vec{e}_1 | -\vec{e}_2 | \cdots | -\vec{e}_n \right] \vec{u} = \vec{0}.
	\end{equation}
\end{corollary}

The vector $\vec{u}$ solving \eqref{eq:gordan} provides a certificate that there is no line onto which the polygonal curve can be projected with $\frac{n}{2}$ local maxima. This corollary is the essential result needed to prove \Cref{thm:main}.

\begin{mainthm}
	The knots $8_{1-15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ have superbridge index $\leq 4$, and $10_{76}$, $13n_{226}$, $13n_{328}$, $13n_{342}$, $13n_{343}$, $13n_{350}$, $13n_{512}$, $13n_{973}$, $13n_{2641}$, $13n_{5018}$, and $14n_{1753}$ have superbridge index $\leq 5$.

	This implies that the knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ have superbridge index equal to 4, and that $13n_{226}$, $13n_{328}$, $13n_{342}$, $13n_{343}$, $13n_{350}$, $13n_{512}$, $13n_{973}$, $13n_{2641}$, $13n_{5018}$, and $14n_{1753}$ have superbridge index equal to 5.
\end{mainthm}

\begin{proof}
	Coordinates of the vertices of polygonal realizations of each of these knots are given in \Cref{sec:coords} along with visualizations. For example, the entry for the $8_5$ knot is repeated in \Cref{tab:8_5}. The polygons were originally generated with coordinates given as double-precision floating point numbers, but to make it easier to verify the existence of exact solutions to \eqref{eq:gordan} these coordinates were rounded to three significant digits and converted to integers (while verifying that this did not change the knot type).

	\begin{table}
	\centering
	\begin{tabular*}{0.85\textwidth}{C{2.2in} R{.4in} R{.4in} R{.4in} | R{1in}}
	\multirow{10}{*}{\includegraphics[height=1.7in]{8_5.pdf}} & $0$ & $0$ & $0$ & $1$ \\
	& $1000$ & $0$ & $0$ & $1$ \\
	& $155$ & $535$ & $0$ & $8061667015$ \\
	& $57$ & $-456$ & $94$ & $1$ \\
	& $572$ & $183$ & $-478$ & $1$ \\
	& $842$ & $108$ & $482$ & $1$ \\
	& $-104$ & $233$ & $181$ & $496072961$ \\
	& $781$ & $398$ & $-254$ & $2237736971$ \\
	& $482$ & $-67$ & $579$ & $3514960071$ \\
	& $182$ & $877$ & $444$ & $4046282755$ \\
	\end{tabular*}
	\caption{Visualization of a 10-stick realization of the $8_5$ knot and the coordinates of its vertices. The rightmost column gives the entries of the vector $\vec{u}$ solving the equation in \Cref{cor:gordan bound}.}
	\label{tab:8_5}
	\end{table}

	In each case, the superbridge index bound will follow from \Cref{cor:gordan bound}, so the goal is to find an appropriate $\vec{u}$. In the case of $8_5$, it suffices to find a nonzero vector $\vec{u}$ with nonnegative entries so that
	\begin{equation}\label{eq:8_5}
		\begingroup
		\begin{bmatrix}
			1000 & 845 & -98 & -515 & 270 & 946 & 885 & 299 & -300 & 182 \\
			 0 & -535 & -991 & -639 & -75 & -125 & 165 & 465 & 944 & 877 \\
			 0 & 0 & 94 & 572 & 960 & 301 & -435 & -833 & -135 & 444
		\end{bmatrix}
		\endgroup
		\vec{u} =
		\begin{bmatrix}
			0 \\
			0 \\
			0
		\end{bmatrix}.
	\end{equation}
	The columns in the $3 \times 10$ matrix above are $(-1)^{i+1}\vec{e}_i$, where the $i$th edge vector $\vec{e}_i = \vec{v}_{i+1}-\vec{v}_i$, the vertices $\vec{v}_j$ have coordinates given in \Cref{tab:8_5}, and indices are computed cyclically so that $\vec{e}_{10} = \vec{v}_1-\vec{v}_{10}$.

	It is straightforward to verify that
	\[
		\vec{u} =(1,\, 1,\, 8061667015,\, 1,\, 1,\, 1,\, 496072961,\, 2237736971,\, 3514960071,\, 4046282755)
	\]
	is a vector with all positive entries solving~\eqref{eq:8_5}, so this vector in conjunction with \Cref{cor:gordan bound} certifies that the superbridge number of this realization of $8_5$ is less than 5, and hence that $\operatorname{sb}[8_5] \leq 4$. The entries in $\vec{u}$ are given in the rightmost column in \Cref{tab:8_5}.

	A similar argument works for each of the knots in the statement of the theorem: particular vectors $\vec{u}$ solving the equation in \Cref{cor:gordan bound} are given in the rightmost column of each table in \Cref{sec:coords}. The certificate vectors $\vec{u}$ were found using Mathematica's {\tt FindInstance} function.

The second sentence in the statement of the theorem follows from observing that the upper bounds just proved match lower bounds coming from Theorems~\ref{bridge bound} and~\ref{3 superbridge}. Specifically, the knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{32}$, and $9_{33}$ all have superbridge index $\geq 4$ by \Cref{3 superbridge}, and hence their superbridge index must be exactly 4.

As for the higher-crossing knots, the result will follow if their bridge indices are bounded below by 4, since then \Cref{bridge bound} implies that their superbridge indices are at least 5. Using \Cref{lem:homomorphism}---stated below---in each case it suffices to find a surjective homomorphism $\pi_1(S^3\backslash K)\! \twoheadrightarrow S_5$  from the knot group $\pi_1(S^3\backslash K)$ to the symmetric group $S_5$ sending meridians to transpositions.  \Cref{tab:K13n350} shows a diagram of $13n_{350}$ with strand labels defining such a homomorphism. Specifically, labeling the strand $(-7,-5)$ with the transposition $(1\,2)$, the strand $(-8,9,-10)$ with $(1\,3)$, the strand $(-6,7,11,-9)$ with $(1\,4)$, and the strand $(-2,3,-1)$ with $(2\,5)$ induces a complete labeling of strands when the labels are propagated to the other strands via the Wirtinger relations. This labeling satisfies the Wirtinger relations by construction and $\{(1\,2), (1\,3), (1\,4), (2\,5)\}$ is a generating set for $S_5$, so this induces the desired surjective homomorphism $\pi_1(S^3\backslash 13n_{350})\! \twoheadrightarrow S_5$.

\Cref{sec:snappy} gives similar homomorphisms for each of 13- and 14-crossing knots in the statement of the theorem, completing the proof. All of these homomorphisms were found using a preliminary version of {\tt Wirt\_Hm\_Suite}~\cite{Wirt-Hm-Suite}, which is being developed as part of forthcoming work by Blair, Kjuchukova, and Morrison~\cite{RyanNateSashka}.
\end{proof}

The following lemma giving a lower bound in bridge index is well-known. It is a classical example of a more general strategy that gives lower bounds on bridge index by finding a surjective homomorphism from the knot group to a group with nice properties~\cite{Baader:2019uf,Baader:2019vm}.

\begin{lemma}\label{lem:homomorphism}
		Let $K$ be a knot and $S_n$ the symmetric group on $n$ elements. If the knot group $\pi_1\!\left(S^3\, \backslash K\right)\!$ admits a surjective homomorphism to $S_n$ such that every meridian is sent to a transposition, then $\operatorname{b}[K] \geq n-1$.
\end{lemma}

\setlength{\tabcolsep}{2pt}

\begin{table}
	\begin{center}
	\begin{tabular*}{0.7\textwidth}{C{2.5in} R{1in} L{.45in}}
	\multicolumn{3}{c}{$13n_{350}$} \\
	\midrule
	\multirow{15}{*}{\includegraphics[scale=0.5]{K13n350SnapPy.pdf}
			\put(-47,50){$1$}
			\put(-46,70){$2$}
			\put(-25,70){$3$}
			\put(-66,50){$4$}
			\put(-66,70){$5$}
			\put(-66,153){$6$}
			\put(-66,112){$7$}
			\put(-129,71){$8$}
			\put(-129,111){$9$}
			\put(-129,132){$10$}
			\put(-109,111){$11$}
			\put(-150,111){$12$}
			\put(-150,71){$13$}
			\put(-88,87){$\mathbf{(1\,2)}$}
			\put(-149.5,90){$\mathbf{(1\,3)}$}
			\put(-49,115){$\mathbf{(1\,4)}$}
			\put(-30,48){$\mathbf{(2\,5)}$}
			\put(-115,35){$(1\,2)$}
			\put(-118,155){$(1\,2)$}
			\put(-160,131){$(2\,3)$}
			\put(-112,78){$(2\,3)$}
			\put(-18,164){$(2\,4)$}
			\put(-155,17){$(2\,4)$}
			\put(-189,75){$(3\,4)$}
			\put(-68,38){$(4\,5)$}
			\put(-108,51){$(4\,5)$}
			}
	& $(-7, -5)$ & $\mapsto (1\,2)$ \\
	& $(-8, 9, -10)$ & $\mapsto (1\,3)$ \\
	& $(-6, 7, 11, -9)$ & $\mapsto (1\,4)$ \\
	& $(-2, 3, -1)$ & $\mapsto (2\,5)$\\
	& & \\
	& & \\
	& & \\
	& & \\
	& & \\
	& & \\
	& & \\
	& & \\
	& & \\
	\end{tabular*}
	\end{center}
	\caption{The SnapPy~\cite{snappy} diagram of $13n_{350}$ with crossings labeled $1,2,\dots,13$. A surjective homomorphism $\pi_1(S^3 \backslash 13n_{350}) \twoheadrightarrow S_5$ is defined by mapping Wirtinger generators to transpositions as indicated to the right. The strands in the diagram, which correspond to Wirtinger generators, are specified by indicating the under crossings at which each strand begins and ends, and the over crossings (if any) along the way. For example, $(-6,7,11,-9)$ starts at the under strand of crossing 6, passes over crossings 7 and 11, and then ends as the under strand at crossing 9. The images of the remaining Wirtinger generators are determined by applying the Wirtinger relations at each crossing. In this diagram (though not in the diagrams in \Cref{sec:snappy}), all strands are labelled with their image in $S_5$; the generating labels are bolded.}
	\label{tab:K13n350}
\end{table}

\section{Conclusion}
\label{sec:conclusion}

The superbridge index remains frustratingly unknown for all the knots mentioned in \Cref{3 superbridge} except $3_1$ and $4_1$. \Cref{thm:main} eliminated the possibility that $\operatorname{sb}[8_4]$ or $\operatorname{sb}[8_9]$ is equal to 5, so each of the knots $5_2$, $6_1$, $6_2$, $6_3$, $7_2$, $7_3$, $7_4$, $8_4$, and $8_9$ has superbridge index equal to either 3 or 4.

\begin{conjecture}[Jeon--Jin~\cite{Jeon:2002gm}]\label{conj:jeon-jin}
	The only knots with superbridge index equal to 3 are $3_1$ and $4_1$.
\end{conjecture}

While the approach using Gordan's theorem described in \Cref{sec:a_new_approach} above cannot help prove this conjecture, it might be useful in finding a counterexample if one exists.

The specific polygonal realizations of knots used in the proof of \Cref{thm:main} were found by generating large ensembles of random equilateral polygons in tight confinement using the algorithm described in~\cite{TomClay}. Between \Cref{thm:main} and previous work~\cite{Blair:2020kv,TomClay}, upper bounds coming from random polygons generated in this way have increased the total number of prime, non-torus knots for which superbridge index is known from 23 to 71. While there are limits to this approach, especially in the absence of more refined lower bounds on superbridge index, its usefulness is probably not yet entirely exhausted. In the future, it might be helpful to explore algorithms for generating polygons with variable (rather than fixed) edgelengths in tight confinement.

Finally, there is a paucity of useful lower bounds on superbridge index beyond those given in Theorems~\ref{bridge bound} and~\ref{3 superbridge} (Adams et al.~\cite{Adams:2010do} notwithstanding). Lower bounds coming from algebraic information---analogous to the bridge index bound from \Cref{lem:homomorphism}---would be particularly welcome and might dramatically expand our ability to compute superbridge indices of particular knots.

\subsection*{Acknowledgments}
\label{sub:acknowledgments}

I am grateful to Jason Cantarella for introducing me to random polygons, to Tom Eddy for getting me serious about generating massive ensembles of them, and to Ryan Blair for sharing his knowledge about bridge index. As usual, thanks to Allison Moore and Chuck Livingston for maintaining KnotInfo~\cite{knotinfo}, which is an invaluable resource. This work was partially supported by grants from the Simons Foundation (\#354225 and \#709150).

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
Victoria University's College of Business MBA Available Fully Online
Ben Ready Online MBA News, Schools, Victoria University
Victoria University's (VU) College of Business will offer the first 100 per cent online version of its internationally recognised MBA program from September this year. The new offering responds to the demands of busy working professionals, offering cutting-edge career-based curricula developed with industry and unparalleled networking opportunities.
Students will enjoy greater success with VU Online, with VU tutors having more time allocated for individual student support than the industry standard, thus providing a higher level of one-on-one support.
The accelerated but flexible online MBA course structure enables students to complete the 12-unit course anywhere in Australia, while keeping in regular contact with classmates and academics. As with the on-campus program, VU's online MBA allows busy professionals to complete the program part-time in just two years.
Delivered by Victoria Graduate School of Business, VU's MBA was awarded a tier-one rating in CEO Magazine's 2018 Global Rankings.
Professor Chris Walsh
Professor Chris Walsh, Academic Director, Victoria University Online, said: "The VU MBA has been developed with industry input to ensure it is career-based and relevant to both employers and students. It is taught by some of Australia's most dedicated academics and facilitators, many of whom have corporate backgrounds and are trained to teach online.
"The higher levels of individual support will help to ensure higher pass rates with the online VU MBA and reduced drop-out rates. At the end of the course, our students will have developed commercial acumen and skills in strategic management, ethics, financial analysis and organisational change. Our graduates will emerge ready to succeed through promotion, salary increases and new business and leadership opportunities."
Independent research shows MBA graduates are more likely to be hired in a competitive job market and to benefit from the higher salaries that enhanced job opportunities bring.
The QS Top MBA Jobs & Salary Trends Report 2016/17, which collated data on salaries earned by MBA graduates around the world, found that Australian MBA holders earned an average of US$122,950 ($175,643), the fourth highest MBA salary level worldwide.
The VU online MBA is structured on six study periods a year, with intakes in September, October, January, March, May and July. This flexible structure helps students to fit coursework around work and family commitments.
Professor Walsh said: "The online MBA features flexible entry and exit points. Students can complete as many as six units in a year and complete all units consecutively to graduate faster."
"Alternatively, students may pause and take a break if needed, or achieve a Graduate Certificate in Business Administration after successfully completing four units. This feature allows students to receive recognition for their progress. It's up to them."
VU is among the world's top 2 per cent of universities according to the Times Higher Education World University Rankings 2018.
For more information on VU online click here.
Ben Ready
Ben Ready is the founder and Managing Editor of MBA News Australia (mbanews.com.au) and Online MBA (online-mba.com.au). He is a journalist, communications and digital marketing expert and also the founder of RGC Media & Mktng (rgcmm.com.au). He is a passionate advocate of MBAs and believes everyone who aspires to career advancement should get one.
Macquarie University Unveils $33,000 Online MBA
Revamped UTAS MBA — Building a Distinctive Community of Leaders

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Draw text in corners with graphviz I have a very simple graph:
digraph {
    node [shape=rect];
    rankdir=LR;
    A -> B
}

It outputs as I expect:

However, I need to place unique numbers in each corner of both A and B. I am currently only aware of xlabel, but from what I gather can only be used once and cannot be specified in a particular region. So how can I accomplish writing numbers in each corner?

A: Newest versions of Graphviz support HTML styling of nodes, including tables ("newer than mid-November 2003", that is). So you can make a 3x3 table like this:

Source:
digraph {
    node [shape=rect];
    rankdir=LR;
    A [shape=none label=<

1		2

A

3		4

    >];
    A -> B
}

Tested with http://sandbox.kidstrythisathome.com/erdos/; it also works with my local installed version (2.38.0).
See Graphviz: Node Shapes for the full set of supported HTML, and examples.

redpajama_set_name: RedPajamaStackExchange


We two girls read alot of blogs about flowers, weddings, brides, etc. Recently, we came across a few non traditional ways to use flowers at a wedding and we liked them so much we thought we'd share!
1. We live here in Florida and we LOVE our outdoor weddings. Sometimes though, the weather doesn't cooperate and the last thing you want is your guests to sit in a puddle of sweat. Hand out fans for your guests to use during the ceremony and attach some flowers to make a pretty statement.
2. Using greenery in a big way is very popular. This wedding idea uses a calligraphed leaf as a place card!
3. This wedding party in lieu of bouquets, carried a garland as a way to 'connect' the wedding party down the aisle.
And last but not least, a cool topper for a traditional centerpiece look: A chic looking lampshade on top of a submerged flower cylinder design.
Everyone expects flowers at a wedding, but there are plenty of ways to use flowers that will give YOUR big day a special twist!
Now go out and have yourself a special Wholesale Flower-Filled day!


text:
title: "Custom-Data-Type"
menu:
  main:
    name: "Custom-Data-Type"
    identifier: "sysadmin/configuration/easydb-server.yml/plugins/custom-data-type"
    weight: -930
    parent: "sysadmin/configuration/easydb-server.yml/plugins"
---

# Custom-Data-Type

redpajama_set_name: RedPajamaGithub


text:
Jan. 6 Committee Holds First Prime-Time Hearing; Migrant Caravan Grapples to Claim Visa Offers | NTD News Today
June 10, 2022NTD News Today
A House panel held its first hearing on the Jan. 6 Capitol breach, calling it an attempt to overthrow the government. But top Republicans say this is more about politics than facts.
President Joe Biden asks countries for shared responsibility on migration at the Summit of the Americas, despite the leaders of Mexico and other Latin American countries boycotting the event.
Alaska is running a nonpartisan, top four primary under a new election system. This year, 48 candidates are vying for the single open seat in the House.
* Click the "Save" button below the video to access it later on "My List."
Truth Social: https://truthsocial.com/@EpochTV


text:
Q: How do I remove redundant namespace in nested query when using FOR XML in SQL Function There is a solution for remove namespace here!, but I need it works in SQL function, I'm getting this error:

The FOR XML clause is invalid in views, inline functions, derived tables, and subqueries when they contain a set operator. To work around, wrap the SELECT containing a set operator using derived table syntax and apply FOR XML on top of it.

Can somebody help me?
Thanks Xtyan
From comment: This is the needed XML-output
<zoo xmlns:an="uri:animal">
  <an:animal species="Mouse">
    <an:legs>
      <an:leg>Front Right</an:leg>
      <an:leg>Front Left</an:leg>
      <an:leg>Back Right</an:leg>
      <an:leg>Back Left</an:leg>
    </an:legs>
  </an:animal>
  <an:animal species="Chicken">
    <an:legs>
      <an:leg>Right</an:leg>
      <an:leg>Left</an:leg>
    </an:legs>
  </an:animal>
  <an:animal species="Snake" />
</zoo>

A: You will not like this probably...
I did not find a generic solution for this... It is no problem to create this with repeated namespaces. This is not wrong, but annoying:
    declare @xml as xml;

    WITH XMLNAMESPACES('uri:animal' as an)
    select @xml = (
    select
        a.c2 as "@an:species"
        , (
           select l.c3 as "text()"
           from t2 l where l.c2 = a.c1
           for xml path('an:leg'), type
           ) as "an:legs"
    from t1 a
    for xml path('an:animal'),root('zoo'));

One solution: build this via string-concatenation
But this is super-ugly...
an (almost) working solution...
The following solution uses FLWOR to re-create the XML after its creation, but this is not generic. It is necessary to add one element of this namespaces to the <zoo>, otherwise the namespace is created on a deeper level repeatedly. I added the attribut an:Dummy.
create table t1 ( c1 int, c2 varchar(50))
create table t2 ( c1 int, c2 int, c3 varchar(50))
insert t1 values
(1, 'Mouse'),
(2, 'Chicken'),
(3, 'Snake');
insert t2 values
(1, 1, 'Front Right'),
(2, 1, 'Front Left'),
(3, 1, 'Back Right'),
(4, 1, 'Back Left'),
(5, 2, 'Right'),
(6, 2, 'Left');
GO
--the function

CREATE FUNCTION dbo.TestFunc()
RETURNS XML
AS
BEGIN
    declare @xml as xml;

    WITH XMLNAMESPACES('uri:animal' as an)
    select @xml = (
    select
        a.c2 as "@an:species"
        , (
           select l.c3 as "text()"
           from t2 l where l.c2 = a.c1
           for xml path('an:leg'), type
           ) as "an:legs"
    from t1 a
    for xml path('an:animal'));

    set @xml=@xml.query('declare namespace an="uri:animal";
                         <zoo an:Dummy="dummy">
                         {
                            for $a in /an:animal
                                return <an:animal an:species="{$a/@an:species}"><an:legs>
                                        {
                                        for $l in $a/an:legs/an:leg
                                        return <an:leg>{$l/text()}</an:leg>
                                        }
                                        </an:legs></an:animal>
                          }
                          </zoo>');

    return @xml;

END
GO
--the call

SELECT dbo.TestFunc();
GO
--clean up

drop function dbo.TestFunc;
drop table t1
drop table t2

The result
<zoo xmlns:an="uri:animal" an:Dummy="dummy">
  <an:animal an:species="Mouse">
    <an:legs>
      <an:leg>Front Right</an:leg>
      <an:leg>Front Left</an:leg>
      <an:leg>Back Right</an:leg>
      <an:leg>Back Left</an:leg>
    </an:legs>
  </an:animal>
  <an:animal an:species="Chicken">
    <an:legs>
      <an:leg>Right</an:leg>
      <an:leg>Left</an:leg>
    </an:legs>
  </an:animal>
  <an:animal an:species="Snake">
    <an:legs />
  </an:animal>
</zoo>

Previous answer
Okay, I think I got this completely wrong in my first attempt. The following is an example, where a function returns the XML without added namespaces.
I use one of the later answers which builds the inner XML in advance without a namespace and creates the full XML as a second call with a namespace. Please check this out:
create table t1 ( c1 int, c2 varchar(50))
create table t2 ( c1 int, c2 int, c3 varchar(50))
insert t1 values
(1, 'Mouse'),
(2, 'Chicken'),
(3, 'Snake');
insert t2 values
(1, 1, 'Front Right'),
(2, 1, 'Front Left'),
(3, 1, 'Back Right'),
(4, 1, 'Back Left'),
(5, 2, 'Right'),
(6, 2, 'Left');
GO

--the function
CREATE FUNCTION dbo.TestFunc()
RETURNS XML
AS
BEGIN
    declare @xml as xml;
    select @xml = (
    select
        a.c2 as "@species"
        , (select l.c3 as "text()"
           from t2 l where l.c2 = a.c1
           for xml path('leg'), type) as "legs"
    from t1 a
    for xml path('animal'));

    declare @resultXML XML;
    ;with XmlNamespaces( 'uri:animal' as an)
    select @ResultXML= (SELECT @xml for xml path('') , root('zoo'),TYPE);

    return @resultXML;
END
GO

--the call
SELECT dbo.TestFunc();
GO

--clean up
drop function dbo.TestFunc;
drop table t1
drop table t2

The result
<zoo xmlns:an="uri:animal">
  <animal species="Mouse">
    <legs>
      <leg>Front Right</leg>
      <leg>Front Left</leg>
      <leg>Back Right</leg>
      <leg>Back Left</leg>
    </legs>
  </animal>
  <animal species="Chicken">
    <legs>
      <leg>Right</leg>
      <leg>Left</leg>
    </legs>
  </animal>
  <animal species="Snake" />
</zoo>


^ 开展营营对抗活动. 人民网. 2011-09-27 [2011-09-27].
^ 下连当兵不走形式,走心!听九旬老兵讲白求恩. 凤凰网. 2016-06-13 [2017-10-14].
^ 第三軍醫大學正式更名陸軍軍醫大學. 多维新闻网. 2017-08-25 [2017-10-14].
^ 纪念白求恩逝世75周年论坛在白求恩医务士官学校成功举办. 人民网. 2014-10-13 [2016-09-04].
^ 白求恩医务士官学校扶贫助学温暖保定大台村. 中国日报网河北频道. 2016-01-22 [2016-09-04]. (原始内容存档于2016-09-16).
This page is based on the Wikipedia article 中国人民解放军白求恩医务士官学校; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.


text:
Q: Order statistics: Probability that a normal random variable is k-th out of n when ordered My question is almost the same as that of
Order statistics: probability random variable is k-th out of n when ordered
with the exception that the underlying distribution from which $X_1$ is drawn, having CDF $F$, is $N(\mu_1,\sigma_1^2)$. And the distribution from which $X_2,X_3,\ldots,X_n$ are drawn, having CDF $G$, is $N(\mu_2,\sigma_2^2)$. If that is the case then how to calculate the probability that $X_1$ is the $k^{th}$ out of the $n$ random variables when they are sorted, in order.
Even if the probability might not be expressible by any elementary formula for large $n$ could you please give hints as to how to numerically approximate. Also, in that case, could you please solve for the case of some small $k$ and $n$.

A: \begin{align}\varrho &= \binom{n-1}{k-1}\mathbb{E}^{X_1}[\mathbb{P}(X_2\le X_1|X_1)^{k-1}\mathbb{P}(X_2\le X_1|X_1)^{n-k}]\\
&=\binom{n-1}{k-1}\int \Phi(\{x_1-\mu_2\}/\sigma_2)^{k-1}\Phi(-\{x_1-\mu_2\}/\sigma_2)^{n-k}\sigma_1^{-1}\varphi(\{x_1-\mu_1\}/\sigma_1)\text{d}x_1\end{align}


What is (Mark status as "PL Listed") in the LST automated track? Where is this status at and what does it mean?
Automated processes are custom things and someone created that in your GoldMine.
I'm quite familiar with AP's and I'm using the Real Estate template very extensively but I've never been able to figure out what this other action pertains to?
Oh! Sorry, my mistake, I didn't notice which board this was on. Mark ought to chime in any time now!
One of the strategies I teach in my GoldMine workshops to keep separate owner ("people record") and property/listing ("non-people record"). The reference for that AP event, �Mark status as "PL Listed� which would be assigned to the listing (non-people) record in GoldMine refers to reminding the user to mark the property record as currently listed. The data field and actual field entry used for this is up to the user, but I recommend the key1 (Contact Type) field be used for this.
The �PL Listed� reference is actually a particular strategy which coded the record as �PL� for �property or listing� and the �Listed� for the status as �Listed� meaning currently listed. This would allow a double-click on the Key1 field label (Contact Type) to display all property records together (sorted by the �PL�� in that field and the listings would like up together as would the �PL Sold� (sold listings), �PL Hold� (listings on hold), �PL Off� (off market listings), etc or other coding classifications. And you could edit the F2 lookup table for that field to accommodate those field values.
Because you are not likely to have a �PL(space)something-else� as a �people code�, this would keep the property records sorted together and apart from the �people records� which you would identify with �people codes� in the Key1 (Contact Type) field.
There is no need to follow our �PL..� coding system if you have a different one, but was one for example purposes.
I teach many other strategies and tips for maintaining and tracking client records and data for building a ongoing lead/referral resource. The �PL Listed� reference is one that carried over into the Listing Automated Process.
Hope this clarifies the matter for you.
Thanks for the explanation. I'm using your templates and bought your book. I'm trying to use your model but make a few tweaks to separate my residential from commercial business yet make sure I don't drop the ball on anyone.
In my live webinar classes I teach and discuss multiple options for working with GoldMine and the Real Estate across multiple specialties. The good news is that GoldMine is very flexible and offers different alternatives to accomplishing your goals so you can find one you like the best.


text:
31st flowers, vegetables, bird show from April 5
Last Updated on Tuesday, 30 November 1999 05:00	Written by Shumaila Ahmed	Saturday, 30 March 2013 09:19
ISLAMABAD: Islamabad Horticultural Society in collaboration with Capital Development Authority (CDA) will arrange its 31st annual spring flowers, vegetables and bird show 2013 from April 5 to 7. Number of organisations from twin cities of Rawalpindi and Islamabad and other areas of the country are participating in the show, said a press release. Different varieties of spring flowers and birds will be exhibited in the show. Chairman CDA Syed Tahir Shahbaz will inaugurate the show.
Courtesy: Daily Times


If you're in need of tire repair in Orleans, you're in the right place. At your closest Mr. Lube, you'll find automotive specialists who are skilled, certified and understand that the right maintenance at the right time can go a long way to keeping your ride in top condition. That means you don't have to have an appointment to visit, and when you do come in, all you have to do is tell us about your tire repair needs and relax with complimentary coffee and newspapers while we take care of it.
In addition, you'll find that your vehicle's warranty will remain intact with Mr. Lube's warranty-approved service*. We pay close attention to service intervals and product specifications specified by the manufacturer, doing exactly what needs to be done at the right time. We also use the latest technology, including the exclusive Electronic Owner's Manual (EOM), which contains up-to-date information about every make and model on the road today — and enables us to access your personalized service interval requirements, keep your vehicle in peak condition and eliminate any unexpected surprises. In short, when it comes to tire repairs in Orleans, it's easy to see why we're the convenient alternative to your dealer. See for yourself how we make car care utterly effortless: get same-day service at your local store today.


text:
Q: En laravel ¿Es necesario que las tablas tengas el campo de "created_at" como el de "updated_at"? ¿Hay alguna forma de que pueda crear esos dos campos desde mi consola y como podria hacerlo?
Debo aclarar que la base de datos me la pasaron ya creada, ahora tengo que implementarla en este framework

A: Los campos created_at y updated_at son opcionales si no quieres usar por ejemplo la función de soft delete que tiene Laravel para hacer borrado lógico de datos, pero sobre todo depende de lo que estes resolviendo, tu propia lógica de negocio te va a indicar si requieres o no esos campos
Sin embargo si requieres que dichas columnas aparezcan en tu tabla, se puede hacer del modo siguiente, tendiendo en cuenta que:

*

*Son campos de tipo DATETIME

*Respeta la nomenclatura de los nombres de las columnas created_at y updated_at
Teniendo como base este ejemplo, te sugiero te guíes
Estructura original de la tabla de ejemplo
MariaDB [demon]> describe amigos;
+--------------+--------------+------+-----+---------+----------------+
| Field        | Type         | Null | Key | Default | Extra          |
+--------------+--------------+------+-----+---------+----------------+
| id           | bigint(20)   | NO   | PRI | NULL    | auto_increment |
| numero       | varchar(100) | NO   | UNI | NULL    |                |
| name         | varchar(100) | NO   | UNI | NULL    |                |
| username     | varchar(100) | NO   | UNI | NULL    |                |
| status_amigo | tinyint(1)   | NO   |     | NULL    |                |
| user_id      | bigint(20)   | NO   | MUL | NULL    |                |
+--------------+--------------+------+-----+---------+----------------+

Con la sentencia ALTER vamos a agregar la siguiente columna que sería created_at, usaremos AFTER para indicarle después de que columna la queremos
MariaDB [demon]> ALTER TABLE amigos ADD created_at DATETIME AFTER user_id;

Lo siguiente será repetir los pasos pero ahora para la columna updated_at igualmente con la sentencia AFTER para indicarle a donde la queremos
MariaDB [demon]> ALTER TABLE amigos ADD updated_at DATETIME AFTER created_at;

Teniendo como resultado final, lo siguiente
MariaDB [demo]> describe amigos;
+--------------+--------------+------+-----+---------+----------------+
| Field        | Type         | Null | Key | Default | Extra          |
+--------------+--------------+------+-----+---------+----------------+
| id           | bigint(20)   | NO   | PRI | NULL    | auto_increment |
| numero       | varchar(100) | NO   | UNI | NULL    |                |
| name         | varchar(100) | NO   | UNI | NULL    |                |
| username     | varchar(100) | NO   | UNI | NULL    |                |
| status_amigo | tinyint(1)   | NO   |     | NULL    |                |
| user_id      | bigint(20)   | NO   | MUL | NULL    |                |
| created_at   | datetime     | YES  |     | NULL    |                |
| updated_at   | datetime     | YES  |     | NULL    |                |
+--------------+--------------+------+-----+---------+----------------+


text:
Writer's Toolbox: Simile
5 Comments / Uncategorized / By De
I'm trying to explain to a member of my writing group how to write a good simile, which feels like the centipede trying to explain how it manages to walk with all those legs. This is the first time I've tried to work this out logically.
I ran into this:
…For fun, the next time someone corrects you and says "That's a simile, not a metaphor," you can respond by letting them know a simile is a type of metaphor, just like sarcasm is a type of irony. Resist the urge to be sarcastic in your delivery.
Okay, so most people can tell you that to write a good simile, you first have to know a simile starts with "like" or "as." But then what?
1) Cliches do not make effective similes, as such. If a cliche is exactly what you need*, then a simile with a cliche in it is appropriate. Otherwise, no.
The purpose of a simile (as with all metaphors) is to bring to light a hidden, unknown, or unappreciated quality. Because a cliche is an overused and therefore well-known phrase, it can't do the job, by definition.
"She fled like a bat out of hell" is meaningless. Try: "She disappeared like a good night's sleep after the birth of a first child."
2) Figure out exactly what aspect of the situation you're trying to bring to light.
If your heroine is fleeing a bad guy in a straightforward suspense story, "She disappeared like a good night's sleep after the birth of a first child" is inappropriate. "She trampled the old woman like a gazelle stampeding over its wounded dam, leaving her for the lions to devour."
A simile should impress the reader visually and/or viscerally (i.e., grabbing the guts). You don't want your reader to be thinking about their kids instead of the possibility your heroine isn't as nice a person as she would like everyone to think.
3) Your simile has to have an appropriate distance from its subject. How different is the thing itself from the thing you're comparing it to?
In "She trampled the old woman like a gazelle stampeding over its wounded dam, leaving her for the lions to devour," the subject, a woman, differs from the referent, a gazelle, significantly.
In "She ran like a woman in a nightmare," the subject, a woman, differs very little from the referent, a woman in a nightmare.
In "She ran like a shooting star," the subject, a woman, differs so significantly from the referent, a shooting star, that we have trouble seeing the connection.
The simile has to walk a middle ground between too much and too little difference between the subject and referent.
Similes both have to be different (i.e., not the exact same thing) and similar (it's called a simile!). –This may seem like a pretty obvious point, but it's something I run into a lot (although not necessarily in this case).
Another point to bring up here is extremely dissimilar similes can be used help build a surreal effect. "She shot across the highway like a shooting star, her silk dress burning white into my retinas long after my headlights passed her." A straightforward, plain-speaking, salt-of-the-earth character (or narrator) is never going to think something like this. A character with head trauma, a mental illness, or writerly tendencies? Sure.
4) Your simile has to have an appropriate distance from the reader. This distance is not the same thing as the distance between the subject and referent; this is how immediate (i.e., without anything intervening) or visceral versus how intellectual or ironic you make the simile. The appropriate distance depends on how much you want the reader to see the narrator behind the curtain.
"She trampled the old woman like a gazelle stampeding over its wounded dam, leaving her for the lions to devour" is more immediate/visceral than "She ran like a deflowered virgin in a horror movie."
To put the reader in the middle of the action, use a more immediate or visceral simile. To make the reader step back from the situation, use a more intellectual or ironic simile–possibly even a cliche.
Not only do you have to figure out exactly what aspect of the situation you want to reveal and how close you want the subject to the referent, you have to figure out whether you want the reader to question the narrator.
If you want the reader to say things like, "This narrator is a very close, faithful observer, and I can trust her," then make the similes more immediate and/or visceral. "She trampled the old woman like a gazelle stampeding over its wounded dam, leaving her for the lions to devour" belongs in a book with lots of adventure. –You don't trust the character, but you do have more trust for the narrator than with "She ran like a deflowered virgin in a horror movie."
In the first simile, the narrator is criticizing the character. The narrator seems to be speaking to the reader with the simple agenda of convincing the reader the character isn't all that nice. The distance between the reader and narrator is small, as if the two of you were sitting over a bistro table and talking. "No! She didn't!" says the reader. "Yep. And then…" says the narrator.
In the second, the narrator is making fun of the character's helplessness, and, if you think about it, criticizes the reader by implying the reader is someone who likes to laugh at women who are fleeing for their lives. A complicated agenda, with more distance. The narrator is up on stage, with the garish face paint of a commedia dell'arte player, walking down to the reader in the audience from time to time and totally creeping her out.
–Keep in mind, a simile is never perfectly immediate. Action is immediate; everything else has distance. Between similes and (non-simile) metaphors, I'd say the metaphor is more immediate. "She shot across the highway like a shooting star, her silk dress burning white into my retinas long after my headlights passed her" is further away than "She shot across the highway, a shooting star…" because you're not saying the woman is like a star, she is one. Also, "like a" is two extra words of separation between the reader and your point.
5) Sometimes, you should ditch the simile and go for something else.
Similes never advance the plot. As it turns out, plot is made up of action, and only action can advance the plot. Any time you do something that doesn't advance the plot, you cost your reader time and therefore interest.
Description (including similes), forshadowing, tone, mood, theme, and backstory are a waste of time, unless you do something with them that's at least as valuable as the action you just put off. Most of these elements, when used well, tell the reader something about the story that makes the plot an inevitable juggernaut. "She trampled the old woman like a gazelle stampeding over its wounded dam, leaving her for the lions to devour" gives the reader a hint about the character that affects the plot later on–without the simile, the reader might have thought the act was an accident. The simile sets up an expectation.
"She ran like a bat out of hell" sets up the expectation that the reader will put the book back on the shelf. "She ran," at least, gets to the point and lets the next thing happen.
*You need a cliche if the line is delivered by a character (or narrator) who is making fun of someone who uses cliches (possibly himself), trying to pretend he's dumb enough to speak in cliches, is so distressed he's unable to speak in anything but cliches, or generally lacks the ability to say what he means. You don't need a cliche if you're just writing something the reader could come up with just as easily as you can.
5 thoughts on "Writer's Toolbox: Simile"
Awesome post, DeAnna! I pimped you on PWB for it. Who's yer buddy? 😉
Glad you liked it. I've started reading Milkman, which was one of the things that got me thinking about the "distance from the reader" bit. None of the similes specifically, I just remember thinking, "this is an in-joke between the narrator and me, isn't it?"
Looking back, "…instead of making him look sickly it gave him a glow like a full moon" would be a good example.
1) I can imagine that, the kind of face with a little bit of blue to it. The kind of face that goths want to have and don't.
2) But wait, he said "full moon." Now I'm thinking about naked butts.
3) Heh. Lisa's really thinking, "He's pretty, but he's kind of a butt-head"!
Maybe not the kind of thing you meant there specifically, but the writing's such that by that point, I was looking at everything like that, if you see what I mean.
The writer group guy's someone from PPW. Since I don't know whether he minds, I won't mention his name here. But he writes thrillers, and I'm hoping he can help me shore up my weak spots, like putting too much crap in my story and not getting on with the plot, etc., etc.
Ah, yes! Ian was right, this is a great post. I love using similes in my writing, but it's only every blue moon that I come up with something that fits. Great tips here!
Bee Repartee
I arrived via Perpetual Writer, Ian.
Outstanding post. Bravo!
I'm reminded of Douglas Adams "Hitchhiker's Guide: Vogon "ships hung in the air in much the same way that bricks don't."
Great post, DeAnna! Ian is good at the pimpin'. 😉 That's how I found your post, too.
re: "writes thrillers, and I'm hoping he can help me… [with] not getting on with the plot, etc., etc."
Welcome to the dark side. 🙂 Story rules!


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text:
Contract for Processing
October 11, 2017 by KJ Hopt
Contract for Processing in Europe
Contract for Processing (in European Private Law)
In this context, this may be a concept of the term: A contract for processing is a contract under which one party, the processor, undertakes to perform a service on an existing movable or incorporeal thing or to an immovable structure for another party, the client (except where the service is construction work on an existing building or other immovable structure).
Conract for Design Conract for Design in Europe Conract for Design (in European Private Law) In this context, this may be a concept of the term: A contract for design is a contract under which one party, the designer, undertakes to design for another party, the client, an...
Obligation Obligation in Europe Obligation (in European Private Law) In this context, this may be a concept of the term: An obligation is a duty to perform which one party to a legal relationship, the debtor, owes to another party, the creditor. Resources See Also Private...
Dependent Personal Security Dependent Personal Security in Europe Dependent Personal Security (in European Private Law) In this context, this may be a concept of the term: A "dependent personal security" is an obligation by a security provider which is assumed in favour of a creditor in order to...
Independent Personal Security Independent Personal Security in Europe Independent Personal Security (in European Private Law) In this context, this may be a concept of the term: An "independent personal security" is an obligation by a security provider which is assumed in favour of a creditor for the purposes...
Withholding Performance Withholding Performance in Europe Withholding Performance (in European Private Law) In this context, this may be a concept of the term: "Withholding performance", as a remedy for non-performance of a contractual obligation, means that one party to a contract may decline to render due [...]...
Reciprocal Reciprocal in Europe Reciprocal (in European Private Law) In this context, this may be a concept of the term: An obligation is reciprocal in relation to another obligation if (a) performance of the obligation is due in exchange for performance of the other obligation; (b)...
Commercial Agency Commercial Agency in Europe Commercial Agency (in European Private Law) In this context, this may be a concept of the term: A "commercial agency" is the legal relationship arising from a contract under which one party, the commercial agent, agrees to act on a continuing...
Substitution of Debtor Substitution of Debtor in Europe Substitution of Debtor (in European Private Law) In this context, this may be a concept of the term: "Substitution" of a debtor is the process whereby, with the agreement of the creditor, a third party is substituted completely or incompletely...
Global Security Global Security in Europe Global Security (in European Private Law) In this context, this may be a concept of the term: A "global security" is a security which is assumed in order to secure all the debtor's obligations towards the creditor or the debit balance...
Proprietary Security Proprietary Security in Europe Proprietary Security (in European Private Law) In this context, this may be a concept of the term: A "proprietary security" covers security rights in all kinds of assets, whether movable or immovable, corporeal or incorporeal. Resources See Also Private Law...
Divided Obligation Divided Obligation in Europe Divided Obligation (in European Private Law) In this context, this may be a concept of the term: An obligation owed by two or more debtors is a "divided obligation" when each debtor is bound to render only part of the performance...
Proceeds Proceeds in Europe Proceeds (in European Private Law) In this context, this may be a concept of the term: "Proceeds", in relation to proprietary security, is every value derived from an encumbered asset, such as value realised by sale, collection or other disposition; damages or...
Confidential Information Confidential Information in Europe Confidential Information (in European Private Law) In this context, this may be a concept of the term: "Confidential information" means information which, either from its nature or the circumstances in which it was obtained, the party receiving the [...]...
Non-performance Non-performance in Europe Non-performance (in European Private Law) In this context, this may be a concept of the term: "Non-performance", in relation to an obligation, means any failure to perform the obligation, whether or not excused. It includes delayed performance and defective [...]...
Contractual Relationship Contractual Relationship in Europe Contractual Relationship (in European Private Law) In this context, this may be a concept of the term: A "contractual relationship" is a legal relationship resulting from a contract. Resources See Also Private Law...
Fundamental Non-performance Fundamental Non-performance in Europe Fundamental Non-performance (in European Private Law) In this context, this may be a concept of the term: A non-performance of a contractual obligation is fundamental if (a) it substantially deprives the creditor of what the creditor was entitled to expect [...]...
Transfer of Contractual Position Transfer of Contractual Position in Europe Transfer of Contractual Position (in European Private Law) In this context, this may be a concept of the term: "Transfer of contractual position" is the process whereby, with the agreement of all three parties, a new party replaces an...
Contract Contract in Europe Definition of Contract An agreement enforceable at law. An essential feature of contract is a promise by one party to another, or by two parties to one another, to do or forbear from doing certain specified acts. (according to the definition of...
Contract for Construction Contract for Construction in Europe Contract for Construction (in European Private Law) In this context, this may be a concept of the term: A contract for construction is a contract under which one party, the constructor, undertakes to construct something for another party, the client,...
Contract for Services Contract for Services in Europe Contract for Services (in European Private Law) In this context, this may be a concept of the term: A contract for services is a contract under which one party, the service provider, undertakes to supply a service to the other...
Contract for Storage Contract for Storage in Europe Contract for Storage (in European Private Law) In this context, this may be a concept of the term: A contract for storage is a contract under which one party, the storer, undertakes to store a movable or incorporeal thing for...
Contract for Barter Contract for Barter in Europe Contract for Barter (in European Private Law) In this context, this may be a concept of the term: A contract for the "barter" of goods is a contract under which each party undertakes to transfer the ownership of goods, either...
Contract for Sale Contract for Sale in Europe Contract for Sale (in European Private Law) In this context, this may be a concept of the term: A contract for the "sale" of goods or other assets is a contract under which one party, the seller, undertakes to another...
Contract for Donation Contract for Donation in Europe Contract for Donation (in European Private Law) In this context, this may be a concept of the term: A contract for the donation of goods is a contract under which one party, the donor, gratuitously undertakes to transfer the ownership...
Contract for Treatment Contract for Treatment in Europe Contract for Treatment (in European Private Law) In this context, this may be a concept of the term: A contract for treatment is a contract under which one party, the treatment provider, undertakes to provide medical treatment for another party,...
Consumer Contract for Sale Consumer Contract for Sale in Europe Consumer Contract for Sale (in European Private Law) In this context, this may be a concept of the term: A "consumer contract for sale" is a contract for sale in which the seller is a business and the buyer...
Contract for Proprietary Security Contract for Proprietary Security in Europe Contract for Proprietary Security (in European Private Law) In this context, this may be a concept of the term: A "contract for proprietary security" is a contract under which a security provider undertakes to grant a security right to...
Data Processing Data Processing in Europe Meaning of Deception in the Context of Data Processing in Germany Provisions relating to meaning of deception in the context of data processing in the German Criminal Code [1]: This criminal issue is covered by Chapter Twenty-three, under the heading [...]...
Principles of European Contract Law (Commision on European Contract Law) Principles of European Contract Law (Commision on European Contract Law) 1 The "Principles of Existing EC Contract Law" differ from to other initiatives relating to European private law, mainly in that they will be harvested from the existing Community law and not from domestic legal...
Common Frame of Reference for Common Principles of European Contract Law The Joint Network on European Private Law. By the end of 2008 this network will deliver a proposal for the so-called Common Frame of Reference containing the Common Principles of European Contract Law (CoPECL). The idea was that the Network will deliver a proposal for...
Distribution Contract Distribution Contract in Europe Distribution Contract (in European Private Law) In this context, this may be a concept of the term: A "distribution contract" is a contract under which one party, the supplier, agrees to supply the other party, the distributor, with products on a...
Loan Contract Loan Contract in Europe Loan Contract (in European Private Law) In this context, this may be a concept of the term: A loan contract is a contract by which one party, the lender, is obliged to provide the other party, the borrower, with credit of...
Aleatory Contract Definition of Aleatory Contract A wagering contract Browse You might be interested in these references tools: ResourceDescription Aleatory Contract in the Dictionaries, Aleatory Contract in our legal dictionaries, Related topics, Browse topics from the European Encyclopedia of Law, Browse [...]...
Apportionment Of Contract Definition of Apportionment Of Contract The division of a contract into several distinct acts, the performance of one or more of which may give the right to enforce the contract to that extent against the other party Browse You might be interested in these references...
Breach Of Contract Definition of Breach Of Contract The failure to carry out an undertaking, e.g., a breach of promise of marriage. See (-ontract; Damages; Lumley v. Gye; Hoehstcr v. Delatour Browse You might be interested in these references tools: ResourceDescription Breach Of Contract in the Dictionaries, [...]...
Categories C, Private Law, su1 Tags CO, Short Read
Benevolent Intervention
Emile Noël


First edition of this selection.
The copyright is assigned for the year 1973 to the Science Fiction Writers of America.
The copyright of the translations is assigned for the year 1978 to the publisher.
"Printed in Germany 1978" is stated on the copyright page.
It's 'Titan-8' on the cover, but 'Titan 8' on spine and title page.
Cordwainer Smith's novelette ends on p. 140, p. 139 is the last numbered page.
The tabe of contents appears on p. 5.
The cover design ("Umschlaggestaltung") is credited to Atelier Heinrichs.
Month of publication from "Science Fiction Times #145".


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text:
Home | Journeys | Beaches | Nature | Adventure | Destinations | Wildlife | Heritage | Spritiual | Leisure | Foodie
ADVENTURE TOP 10
Top 10 zipping sites
Mohanchatti, Rishikesh, Uttarakhand
This is indeed the most exhilarating zipping stuff in India- Flying Fox at Jumpin Heights near Rishikesh in Uttarakhand. One is harnessed to the wire and on release by gravity roll down the wire from 113 mt to just 7 mt above the river level and then goes up. In this process one accelerates upto speeds of 160 kmph. Your body moving in free air with this speed is just mind-blowing. Only difference with traditional zip lines is that you don't sit on a seat harness but are harnessed into a special flying suit with zip line and go down in position of diving in air just like Superman does. Enjoy breath-taking views as you fly down a series of ultra-strong steel zip lines, securely attached by a harness and pulley propelled simply by gravity. Most trained instructors from the industry accompany you to ensure your safety.
Neemrana, Rajasthan
It was first name and place to introduce zipping to Indian adventurers and so far is the most popular one. It was first permanent multi zip course in India. Located at a distance of 100 km from Delhi International Airport in Neemrana fort and its adjoining hills, Flying Fox is a perfect destination for those looking for an adventure in Neemrana. You can enjoy a unique, exhilarating aerial journey between hill forts and ridge tops located at Neemrana Fort-Palace one of India's oldest heritage hotels, built almost a thousand years ago. It has six sessions daily from Monday to Friday at 10:00, 11:00, 12:15, 14:00, 15:30, 16:30 and eight sessions on weekends i.e. Saturday and Sunday 09:30, 10:30, 11:30, 12:30, 14:00, 15:00, 16:00 and 16:30. For a group of 12 persons total tour takes about 2 hours. Neemrana is 115 kms from Delhi in Alwar district of Rajasthan. It has total five ziplines on the whole tour, named adventurously. First one, 'To Qila Slammer' has a length of 350 m. Second one is named 'Where Eagles Dare' and it has length of 400 m. It is the longest zip on the course. Third one is smallest and named 'Pussy Galore'. It is just 90 m long. Fourth one, 'Goodbye Mr Bond' is 250 m long. Fifth and the last zip is 'The Big B' and it has length of 175 m.
Clouds End, Mussoorie, Uttarakhand
There is another zip line at Mussoorie where you can fly over the dense forest of deodar, oak, rhododendron and a very beautiful valley in Mussoorie wild life sanctuary. The India's first wild life Zip Line and also the highest zip line in the India (600 ft. high) at Clouds End. This is the most beautiful area of the Mussoorie. Here is 800 ft long zip line ride at a height of 600 ft where you can fly up to 90 km/hr. speed. Located at an altitude of 7300 ft Mussoorie Adventure Park is six kilometers from Mussoorie at Clouds End Estate. Zip lines India is a special body of The Real Adventure Sports who manages, operates and constructs all the zip lines here with assistance of the Zip line experts from Colorado USA. There are many activities associated with this zipline here including Sky walk, Sky ride, zip swing, besides normal other adventure activities. Arround Mussoorie Adventure Park you can wander to many of the points like – Eco point , sunset point , sunrise point , Molty tibba , lovers point , Dun view point , Valley view point , Himalaya view point , thunder point and Tipri.
Kerwa Dam, Bhopal, Madhya Pradesh
At 520 metres long, Flying Fox Bhopal is the longest twin zip in Asia, and the first twin zipline course to be constructed in India. The twin zipline experience includes personal equipment fitting at the Flying Fox office inside the Piknik complex at Kerwa, a short trek across the dam and up to the hilltop start point, a safety briefing for all participants by trained instructors, and a short practice zipline. Participants then race down the twin ziplines in pairs, starting from a 20 ft high hill-top look out platform, crossing the Kerwa Lake and finishing back at the Piknik Café. The launch platform has extraordinary panoramic views of the surrounding landscape, and offers an excellent opportunity to observe the flora and fauna of the site. Instructors accompany participants throughout the tour. Kerwa Dam is located just 15 kms from the city of Bhopal at Mendora. The safety equipment used by Flying Fox is imported from France and checked on a daily basis. The zipline installation is checked according to a rigorous daily, monthly, quarterly and annual inspection programme. The ziplines are British designed and managed and all the instructors are trained to European standards.
Kikar Lodge, Ropar, Punjab
Flying Fox Kikar is the longest zip tour in South Asia and India's first forest based aerial adventure zip line experience. Soar over the canopy and forested valleys in this 5 zipline tour, with views of the Himalaya. A two hour aerial adventure, Flying Fox essentially involves zipping through 5 zip wires, usually 180 to 400 meters in length, separated by short walks. The length of the zipline totals 1.5 km and is considered as the longest zipline in south Asia. The entire experience takes place away from the city at Kikar Lodge where a number of other fun-filled activities are provided as well such as paintball, quad-biking and night safari. Take an early morning tour of the chance to see wildlife such as Sambar and Cheetal – then either head back to the Lodge for a delicious meal or embark on Kikar's extensive range of exciting adventure activities. Located at Kikar Lodge 1 hour 45 minutes form Chandigarh in the village of Nurpur Bedi. This zipline operates six days in week except for Wednesday. There are four zipping session 9:30AM, 11:30 AM, 14:00 and 16:00. Flying Fox customers are also eligible for 15% discount on all food and beverage items at The Kikar Lodge. Flying Fox Kikar is situated 80 km from Chandigarh. It takes place at Kikar Lodge in Ropar district.
Mehrangarh Fort, Jodhpur, Rajasthan
If you are more adventurous, you can do Zipline ride from one of the highest fort in India like Mehrangarh Fort which is situated 400 feet (122 m) above the city. It is the most impressive as well as the biggest fort of Rajasthan. For those of you looking for adventure trips in Jodhpur, Flying Fox is just the right place. It is located at Mehrangarh Fort in the Blue City. There are total seven tours daily at 0930, 1030, 1130, 1530, 1630 and 1700. It takes 1.5 hours for a full group of 12 persons to do the tour. There are total six ziplines on the whole tour, named traditionally. First one, 'Chokelao Challenge' has a length of 110 m. Second one is named 'Ranisar Rollercoaster' and it has length of 170 m. Third one is smallest and hence named 'Chhota Wallah'. It is just 70 m long. Fourth one, 'Jai Jodha' is second longest with a length of 270 m. Fifth zip is 'Rajput's Revenge' and it has length of 160 m. Last one is the longest one- 'The Magnificent Marwar' and it has a length of 300 m.
Buranshkanda, Mussoorie-Dhanaulti, Uttarakhand
Zipping in backdrop of snow clad mountains can be very thrilling. Snow adventure zone gives you this opportunity. It is located at Buranskhanda, 18 km from Mussoorie and 6 km before Dhanaulti on Mussoorie - Chamba state highway road. Having a perfect blend of Nature & Adventure "Snow Adventure Zone" is a perfect outdoor adventure destination nearby Mussoorie and Dhanaulti. Surrounded by beautiful alpine meadows and trees of deodar, pine, wall nut, apricot, oak and apple having spectacular and breathtaking views of Himalayan peaks rewarded as the best location for views of Himalayas. Zip line is the famous adventure activity for fun and thrill. You are harnessed and attached on wire rope with the help of pulley at the higher end and then slide free fall on that wire rope with speed of 50 kmph to the lower end of 330 ft long. Claimed as the Highest Adventure Park in India (8600 ft), Snow Adventure Zone has the brand new adventure activities including some traditional adventure activities like rock climbing, rappelling, and trekking etc. That's the best part that alongwith zipping you get to do here a lot of other activities. You also have camping opportunities here.
Orsang, Vadodra, Gujarat
The Orsang Resort, is a local getaway resort for adventure centric guests at Chanod. Chanod is a riverside area at 58 kms from Vadodara and 15 kms from Rajpipla in Gujarat.This zipzone has six ziplines. Longest zipline is of 1400 ft which passes over the forest & river. This adventure park also has a high rope adventure zone. Located at Orsang Eco Resort at Orsang Gamdi, Chanod Tal, Dabhoi in district Vadodra of Gujarat. This is an adventure centric getaway offering activities besides zipping, such as ATV rides, jungle trekking, Burma bridge, river crossing etc. This place is in close proximity to Chanod, a pilgrim destination at the confluence of the Narmada, Orsang and Saraswati rivers. Rajpipla is a nature centric heritage getaway at 40 kms from Bharuch and about 100 kms from Vadodara. Rajpipla is known for its palaces, several forest areas and renowned Sardar Sarovar Dam nearby.
Dawki, Meghalaya
One of the most popular adventure site of the north-east is situated in Dawki (which borders Bangladesh) and provides adventure activities like scuba diving, trekking, river rafting, cliff jumping, zip lining, caving, rock climbing, canoeing and camping. Intent is to have people stay on site for a couple of days and take part in the various adventure activities on offer. This adventure zone is run by Pioneer Adventure group from Jarman Villa, Lower Lachumiere in Shillong, capital of Meghalaya. Zipping, also known as a flying fox, zip wire, aerial runway, aerial ropeslide, consists of a pulley suspended on a cable. It is designed to enable a user propelled by gravity to travel from the top to the bottom of the inclined cable, usually made of stainless steel, by holding on or attaching to the freely moving pulley. And of course, a safety line is also attached.
Shivpuri, Rishikesh, Uttarakhand
Just thought of zipping over Ganges can pump up your adrenaline. Snowleopard adventures provide this thrilling stuff at Shivpuri in Rishikesh. There are two zip lines going across the river starting from one point but landing at two different spots, with a Leopard beach camp on the banks of the river. Thrill is so much that one may have done it hundreds of times, but each time on the zip is amazing! It's a top class facility with international standard and all safety and contingency measures and a partnership with Flying Fox. There are numerous riverside beach camping accommodations in Shivpuri. Shivpuri is small village and popular for river rafting and beach camping. Shivpuri village is 18 Kms from Rishikesh City. There are lots of adventure activities in this area.
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Manchester fashion retailer boohoo.com has acquired 66% of Ancoats-based PrettyLittleThing for £3.3 million.
The remaining 34% will be used to incentivise PrettyLittleThing CEO Umar Kamani and his senior management team.
PLT's revenue grew by more than 400% to £17.0m in the financial year ended 29 February 2016 (FY 15: £3.1m), and revenue was £19m in the six months to 31 August 2016 (H1 16: £6.4m).
Boohoo.com says the transaction is expected to complement the group's presence both in the UK and internationally. It has continued to perform well since reporting its interim results on 27 September.
Trading was strong across the Black Friday weekend and peak season trading continues to be encouraging. Reflecting this strong trading performance, the Board now expects boohoo.com to deliver revenue growth of between 38% and 42% in the next financial year, against previous guidance of between 30% and 35%.
Boohoo continues to benefit from improved operating leverage in the business, and the board anticipates an EBITDA margin of between 11% and 12% in FY 17, against previous guidance of around 11%.
In addition, the boohoo Group will benefit from the acquisition of 21 Three Clothing Company Ltd which is expected to be consolidated from 3 January 2017. PLT was founded by Boohoo boss Mahmud Kamani's sons Umar and Adam in 2012 and is popular with celebrities including Hollywood actress Lindsay Lohan and The X Factor winners Little Mix.


> Is my child too active?
Young children may surprise you with the amount of energy they have! We offer tips to help you cope with active children, creating a calm time and space at home, and making sure they get to sleep at night. Featuring some information on spotting the signs of hyperactivity.
Part of the series: Ready To Learn?


text:
Q: Equivalence Class Let $A:= \{1,2,3,\dots\}$ and $P := A \times A$. Now define a relation on $R$ on $P\,$ by
$$(x,y)R(a,b) \iff x^y = a^b$$
1) Determine the equivalence class $[(9,2)]$ of $(9,2)$
Note that I already verify it is indeed an equivalence relation.
But what to do with the points?
$9^2 = 81 = a^b$
Not sure what to do
2) Determine an equivalence class with exactly four elements
Need to do on question 1 first

A: How many integer solutions are there to $a^b = 81 = 3^4$? Think about the Fundamental Theorem of Arithmetic (unique factorization into primes).
For (2), again, think about the Fundamental Theorem of Arithmetic, and what you need to be able to write $a^b$ in exactly four different ways.


text:
/* globals module */
/**
 * @module userEducationRoute
 * @description Baasic User Education Route Definition provides Baasic route templates which can be expanded to Baasic REST URIs. Various services can use Baasic User Education Route Definition to obtain needed routes while other routes will be obtained through HAL. By convention, all route services use the same function names as their corresponding services.
 */

import { injectable, inject } from "inversify";
import { BaseRoute, TYPES as commonTypes } from '../../common';
import { IGetRequestOptions, IOptions } from '../../common/contracts';;
import { IUserEducation } from './contracts';
import { IAppOptions, TYPES as coreTypes } from '../../core/contracts';

export class UserEducationRoute extends BaseRoute {

    public readonly findRoute: string = 'profiles/{userId}/educations/{?searchQuery,page,rpp,sort,embed,fields}';

    public readonly getRoute: string = 'profiles/{userId}/educations/{id}/{?embed,fields}';

    public readonly createRoute: string = 'profiles/{userId}/educations';

    public readonly updateRoute: string = 'profiles/{userId}/educations/{id}';

    public readonly deleteRoute: string = 'profiles/{userId}/educations/{id}';

    constructor(@inject(coreTypes.IAppOptions) protected appOptions: IAppOptions) { super(appOptions); }

    /**
     * Parses find route which can be expanded with additional options. Supported items are:
     * - `searchQuery` - A string referencing user education properties using the phrase or BQL (Baasic Query Language) search.
     * - `page` - A value used to set the page number, i.e. to retrieve certain user education subset from the storage.
     * - `rpp` - A value used to limit the size of result set per page.
     * - `sort` - A string used to set the user education property to sort the result collection by.
     * - `embed` - Comma separated list of resources to be contained within the current representation.
     * @method
     * @param options Query resource options object.
     * @example userEducationRoute.find({searchQuery: '<search-phrase>'});
     **/
    find(options?: IOptions): any {
        return super.baseFind(this.findRoute, options);
    }

    /**
     * Parses get route; this route doesn't expose any properties.
     * @method
     * @param id User education id which uniquely identifies user education resource that needs to be retrieved.
     * @param options Query resource options object.
     * @example userEducationRoute.get(id);
     **/
    get(id: string, options?: IGetRequestOptions): any {
        return super.baseGet(this.getRoute, id, options);
    }

    /**
     * Parses create route; this URI template does not expose any additional options.
     * @method
     * @param data An user education object that needs to be inserted into the system.
     * @example userEducationRoute.create(data);
     **/
    create(data: IUserEducation): any {
        return super.baseCreate(this.createRoute, data);
    }

    /**
     * Parses update route; this URI template does not expose any additional options.
     * @method
     * @param data An user education object used to update specified skill resource.
     * @example userEducationRoute.update(data);
     **/
    update(data: IUserEducation): any {
        return super.baseUpdate(this.updateRoute, data);
    }

    /**
     * Parses delte route; this URI template does not expose any additional options.
     * @method
     * @param data An user education object used to delete specified skill resource.
     * @example userEducationRoute.delete(data);
     **/
    delete(data: IUserEducation): any {
        return super.baseDelete(this.deleteRoute, data);
    }
}

/**
 * @overview
 ***Notes:**
 - Refer to the [REST API documentation](https://github.com/Baasic/baasic-rest-api/wiki) for detailed information about available Baasic REST API end-points.
 - [URI Template](https://github.com/Baasic/uritemplate-js) syntax enables expanding the Baasic route templates to Baasic REST URIs providing it with an object that contains URI parameters.
 - All end-point objects are transformed by the associated route service.
 */


text:
The Camp – A Home for Refugees
May 8, 2013 by Open Doors in General
Hundreds of thousands of refugees fleeing the violence in Syria are staying in camps just across the border inside Turkey. In light of that, the April 10 news item from Reuters, announcing the creation of two more camps in Turkish town of Midyat, just beyond Syria's northeastern border, did not garner much notice – at first.
The announcement of construction on a camp specifically for Syrian Christians, however, was new.
Of the thousands of Syrian refugees in Turkey, only a few hundred are Christian. Nowhere in the Islamic world has a refugee camp for Christians from one country been built across the border in a neighboring country. Now Turkey is building a camp that will hold between 3 and 30 times the number of Syrian Christians currently taking refuge in the country.
But why? Why is Turkey creating a small city to handle a flood of Syrian Christians?
Jamil Diarbakerli, a local representative of the Assyrian Democratic Organization, maintains that the persecution of Christians in Syria is exaggerated. "They are just looking for an excuse to leave Syria and escape to Europe," he says. Sargon, a Christian from the Syrian town of Qamishly, just across the Turkey border, strongly disagrees. "What are you talking about? The body of my cousin Alexi Skandar has been lying on a street in Aleppo for more than two weeks. His lacerated corpse has rotted. No one dares to go and get him; if you do you'll be killed. And he is not alone; many other bodies of Christians are rotting on the streets because Islamists threaten to kill people who want to bury them."
While the civil war continues to cause suffering for Muslims and Christians alike, Christians and churches are often specifically targeted. "We were doing fine under Bashar's regime, before the so-called revolution," points out Sano, a Syrian refugee. "Certainly, everything wasn't okay, but at least we weren't oppressed because of our religion. It was much better before. Now, al Qaeda and Salafists have taken over certain parts of Syria and are about to occupy more territory. It is really scary."
Journalist Nuri Kino, of Syriac Orthodox background, traveled to Midyat, Turkey, to observe construction of the camp. He met with Turkish Christian leaders who are divided on the idea of opening a camp. He also met with Syrian Christians; some believe the camp is needed, while others it will become a Christian ghetto. Some see the potential for sectarian trouble arising from the camp's proximity to a separate camp for Muslim refugees, also under construction; other see the potential to empty Syria of Christians entirely.
"It's not time for a camp yet, not for Christians," Diarbakerli contends. "There are only around 300 of them in all of Turkey. If the situation becomes acute, you can always pitch tents; it'd only take a few hours. I don't understand why they are taking these drastic steps. They should wait." He suggests an alternative plan to providing shelter to Assyrians and other Christians in a humanitarian zone within Syria.
The few hundred Syrian Christians taking refuge in Turkey do not relish the prospect of being relocated to a tent city, but construction of the camp is already in full swing. "I'm convinced that letting land for the camp was the right thing to do," says the vice president of the Midyat church board. "It's indeed needed for the Christian refugees that are already in Turkey. And it could be needed for many more."
Some of the refugees are currently staying at the Syriac Orthodox monastery; others are living in apartments in the center of Midyat, in a Catholic church and in villages around the city. Saliba men of the monastery explains, "We have been taking care of them for a year and a half. They have lived and eaten for free in the monastery. We fear that they will arrive in large numbers. That is why we turned to the Turkish government and asked for help." He confesses, though, that things have not gone as they hoped. "We asked the government for barracks, not tents," he says. "The whole thing has turned out so wrong. Anyway, the refugees that are here, and are coming."
The refugees say they have each received 150 Turkish Liras, about 80 US dollars, from the Turkish government. This is the first time any of them have received government support. While appreciative, their main hope is to be granted work permits. The board vice president, Yusuf Tker, says, "Many Christians in Midyat will be able to find work; we are going to need interpreters and people with other professional skills in the camp." The members of the refugee committee are pleased that many other Christian associations have promised their help in meeting needs. Tker adds proudly, "The Turkish Ministry of Foreign Affairs has promised that it's going to be a state-of-the-art refugee camp with sports grounds, food stores, health care and many other facilities lacking in other camps."
One concern is the proximity of the camp to another being built for Muslims. "They are going to build not one but two camps, one for 6,000 Muslims and one for 4,000 Christians," says a Turkish journalist, herself a Muslim. "What will happen if Christians can't flee from Syria? Then both camps will be used by Muslims and 10,000 Syrian Muslims will be located in close vicinity to one of the world's oldest monasteries." Tker counters that this situation will not be allowed to occur. "Our agreement with the government is that only Christians will be allowed to stay in the camp located near the monastery. Should anyone commit a crime or create problems, that person will be sent back to Syria." Tker adds, "We hope and pray that there will be peace in Syria and the situation for Christians will improve. The camp will be there if the situation gets worse, in other words, just in case."
Father, we pray for Christians in Syria who are enduring persecution in their homeland. We pray for the many who remain, hoping to reach out with the gospel to their Muslim neighbors and offer comfort and help to those who suffer. And we pray for those who have fled to Turkey, those who are homeless, that You will be their refuge. Protect them, Father. Provide shelter and food. Comfort them with Your presence. Thank You for the churches in Turkey that have provided shelter for their fellow Christians during the past year-and-a-half. Grant them wisdom for the future as they prepare and maintain this refugee camp and as they seek other ways to aid these Syrian believers. And we pray for an end soon to the conflict in Syria. In the name of Jesus, our shelter from all the storms of life, Amen.
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