text:
\section{Introduction}
We consider quad equations defined in terms of a polynomial in four variables, $\mathcal{Q}$, that is equations of the form
\begin{equation}
\mathcal{Q}(u,\tilde{u},\hat{u},\th{u})=0,\label{ge}
\end{equation}
where in the simplest setting $u=u(n,m)$, $\tilde{u}=u(n+1,m)$, $\hat{u}=u(n,m+1)$ and $\th{u}=u(n+1,m+1)$ are values of a dependent variable taking values in $\mathbb{C}\cup\{\infty\}$ as a function of independent variables $n,m\in\mathbb{Z}$.
The quad equation is called multi-affine if the defining polynomial, $\mathcal{Q}$, is degree one in each variable, and we call it multi-quadratic if $\mathcal{Q}$ is degree two in each variable.

An important integrability feature that is possible for the quad equation is the multidimensional consistency \cite{FrankABS,BS}; it has proven to be a natural property of many integrable equations in the multi-affine class.
Significant work on this is the list of multi-affine quad equations with the consistency property obtained by Adler, Bobenko and Suris (ABS) in \cite{ABS,ABS2}.
Quad equations beyond the multi-affine class were considered in \cite{KaNie} where several multidimensionally consistent examples (also beyond the multi-quadratic) were obtained.
In fact these equations appeared naturally in relation to underlying models in a different class, namely the Yang-Baxter maps \cite{Ves,drin}, and emerge from a generalisation of results connecting the Yang-Baxter maps with the multi-affine quad equations \cite{ABS,PSTV,Tasos}.
One multidimensionally consistent multi-quadratic quad equation identified in \cite{KaNie} was known earlier due to observations of Adler and Veselov in \cite{AdVeQ}, it in fact arises as the superposition principle for B\"acklund transformations of the KdV equation.
Recently it has also come to light that the well-known discrete version of the KdV equation due to Hirota \cite{hirota-0}, which is multi-affine but absent from the ABS list, is naturally understood within the consistency framework as a special case of a multi-quadratic quad equation \cite{JamesQ}.
The few known examples therefore indicate that multi-quadratic quad equations are a quite natural and potentially rich class of integrable systems.

A feature of higher degree discrete models, like the multi-quadratic quad equations, is that they define multivalued evolution from initial data.
This adds richness, but also another level of difficulty in dealing with such systems.
This difficulty is however mitigated in the mentioned integrable cases because they can be reformulated as a single-valued system with augmented initial data.
For the models in \cite{KaNie} this is because the variables present in the associated Yang-Baxter map themselves satisfy a single-valued system.
But that situation can actually be viewed as a special case of a more general constructive approach to this kind of reformulation \cite{JamesQ}.
In this approach auxiliary variables are introduced on lattice edges, which are similar to variables of an associated Yang-Baxter map, however, rather than satisfying an independent system, they instead participate in a mixed system involving both vertex and edge variables.
The resulting model is of a type similar to the class introduced in \cite{hv11} but with the additional feature of preserving algebraic relations on the lattice edges.
The reformulation procedure relies completely on a discriminant factorisation property of the defining polynomial.
It is this special property that is the departure point and the main emphasis of the present work, we use it as a foundation to enable some systematic investigations of the multi-quadratic quad equations.

The main result presented here is a list of multi-quadratic quad equations with the discriminant factorisation property.
To construct the models we start from biquadratic polynomials (the factors of the discriminant) which are associated with the edges of the lattice, and this provides a natural correspondence with the multi-affine ABS equations whose construction also involves edge biquadratics.
Due to the factorised discriminant hypothesis the models we list admit reformulation as single-valued systems.
However the more important property of these equations, which is actually rather remarkable because it is not explicitly built into the construction, is the integrability feature of multidimensional consistency.

The second part of our paper is devoted to developing the transformation theory of the obtained models.
The most pressing question we seek to address relates to existence of transformations of B\"acklund or Miura type connecting these multi-quadratic models back to the better studied multi-affine ABS equations.
Such transformations are not inherent from our construction, rather we tackle the problem of obtaining transformations a posteriori using separate methods.
The main result is to connect all except the primary model, namely the multi-quadratic counterpart of Q4, back to equations in the multi-affine (ABS) class.

We proceed as follows.
In Section \ref{dfh} we explain the basic discriminant factorisation hypothesis for the multi-quadratic quad equations.
Additional assumptions are explained in Section \ref{assumptions} and the obtained models are listed in Section \ref{list}.
The method to reformulate these models as systems that define single-valued evolution from initial data is given in Section \ref{SVS} and their multidimensional consistency is described in Section \ref{MDC}.
Section \ref{transformations} is devoted to the transformation theory of the models, in particular we recall the multi-affine equations from the ABS list in Section \ref{ABSlist} and give transformations connecting the multi-quadratic models to them in Section \ref{BTex}.
The methods used to obtain these transformations are explained in Sections \ref{nsd}, \ref{idolons} and \ref{quadlin}.
Some questions raised by the results reported here are discussed in Section \ref{discussion}.

\section{The factorised-discriminant hypothesis}\label{dfh}
In the multi-quadratic case considered here, the defining polynomial $\mathcal{Q}$ of the quad equation (\ref{ge}) is of degree two in each of the four variables.
To calculate the evolution defined by (\ref{ge}) from some initial data requires solving this equation locally as a quadratic equation for one of the arguments, and in particular taking the square root of its discriminant with respect to that argument.
In general this discriminant is a polynomial of degree four in each of the remaining three arguments; the property of polynomial $\mathcal{Q}$ that we study here is the factorisation of this discriminant into a product of three factors:
\begin{equation}
\eqalign{
\Delta[\mathcal{Q}(u,\tilde{u},\hat{u},\th{u}),\th{u}] = H_1(u,\tilde{u})H_2(u,\hat{u})G_1(\tilde{u},\hat{u}),\\
\Delta[\mathcal{Q}(u,\tilde{u},\hat{u},\th{u}),\hat{u}] = H_1(u,\tilde{u})H_2(\tilde{u},\th{u})G_2(u,\th{u}),\\
\Delta[\mathcal{Q}(u,\tilde{u},\hat{u},\th{u}),\tilde{u}] = H_1(\hat{u},\th{u})H_2(u,\hat{u})G_3(u,\th{u}),\\
\Delta[\mathcal{Q}(u,\tilde{u},\hat{u},\th{u}),    u ] = H_1(\hat{u},\th{u})H_2(\tilde{u},\th{u})G_4(\tilde{u},\hat{u}),
}\label{df}
\end{equation}
where $H_1$ and $H_2$ are degree-two polynomials in each of two variables, i.e., biquadratics, and $G_1,\ldots,G_4$ are degenerate biquadratics of the form $G_i(\tilde{u},\hat{u})=(a_i+b_i\tilde{u}+c_i\hat{u}+d_i\tilde{u}\hat{u})^2$, $i\in\{1,2,3,4\}$, i.e., the square of a polynomial which is degree-one in each variable.
This is the most general hypothesis that allows replacement of (\ref{ge}) by an associated single-valued system through introduction of auxiliary edge variables $\sigma_1$ and $\sigma_2$ satisfying the edge relations
\begin{equation}
\sigma_1^2 = H_1(u,\tilde{u}), \quad \sigma_2^2 = H_2(u,\hat{u}).\label{er}
\end{equation}
The key features allowing this are that all discriminants in (\ref{df}) are squares which clearly leads locally to a rational model, and furthermore that the edge relations on opposite sides of the quad are the same, allowing the rational model to replace the quad equation globally (for instance in the simplest setting throughout $\mathbb{Z}^2$).
The procedure to obtain the single-valued system is straightforward, it was explained in detail in \cite{JamesQ} and an example will also be included later in this paper (Section \ref{SVS}).

We remark that the restriction to $G_1,\ldots, G_4$ being degenerate seems unnatural in some regards, in fact there exist sets of polynomials $\{\mathcal{Q},H_1,H_2,H_3\}$ satisfying system (\ref{df}) with $G_1=G_2=G_3=G_4=H_3$, and where $H_3$ is also non-degenerate.
These more symmetric solutions of the problem are interesting, however to understand the sense in which the resulting polynomial $\mathcal{Q}$ defines an integrable discrete model requires a substantial alteration of the setting, and this will be studied in detail elsewhere.
Also we remark that the assumed symmetry can be relaxed, for instance replacing $H_1$ appearing in the third and fourth equations of (\ref{df}) with $\hat{H}_1\neq H_1$.
Such systems are of interest too but the associated polynomial $\mathcal{Q}$ in that case is more naturally interpreted as defining a B\"acklund transformation between models.

\section{Additional assumptions}\label{assumptions}
The solution of (\ref{df}), in the sense of obtaining a set of polynomials \[\{\mathcal{Q},H_1,H_2,G_1,G_2,G_3,G_4\}\] for which system (\ref{df}) is identically satisfied, is a difficult classification problem in its full generality.
Further assumptions allow this problem to be solved using computer algebra, these come from looking at examples identified in \cite{JamesQ}.

We assume invariance of $\mathcal{Q}$ when the arguments are permuted by the Klein four-group (the Kleinian symmetry)
\begin{equation}
\mathcal{Q}(u,\tilde{u},\hat{u},\th{u})=\mathcal{Q}(\tilde{u},u,\th{u},\hat{u})=\mathcal{Q}(\hat{u},\th{u},u,\tilde{u}).
\end{equation}
This reduces the generic multi-quadratic polynomial $\mathcal{Q}$ to the form
\begin{equation}
\fl
\eqalign{
\mathcal{Q}(u,\tilde{u},\hat{u},\th{u})=
a_{{1}}  +a_{{2}} \left( u+\tilde{u}+\hat{u}+\th{u} \right)+a_{{3}} \left( {u}^{2}+{\tilde{u}}^{2}+{\hat{u}}^{2}+{\th{u}}^{2} \right)
+a_{{4}} \left( \hat{u}\th{u}+u\tilde{u} \right)
\\\quad
+a_{{5}} \left( \tilde{u}\hat{u}+u\th{u} \right) +a_{{6}} \left( \tilde{u}\th{u}+u\hat{u} \right)
+a_{{7}} \left( \tilde{u}\hat{u}\th{u}+u\hat{u}\th{u}+u\tilde{u}\th{u}+u\tilde{u}\hat{u} \right)
\\\quad
+a_{{8}} \left( u{\tilde{u}}^{2}+\hat{u}{\th{u}}^{2}+{u}^{2}\tilde{u}+{\hat{u}}^{2}\th{u} \right)
+a_{{9}} \left( u{\hat{u}}^{2}+{u}^{2}\hat{u}+\tilde{u}{\th{u}}^{2}+{\tilde{u}}^{2}\th{u} \right)
\\\quad
+a_{{10}} \left( u{\th{u}}^{2}+{u}^{2}\th{u}+\tilde{u}{\hat{u}}^{2}+{\tilde{u}}^{2}\hat{u} \right)
+a_{{11}}u\tilde{u}\hat{u}\th{u}
+a_{{12}} \left( \tilde{u}\th{u}+u\hat{u} \right)  \left( \tilde{u}\hat{u}+u\th{u} \right)
\\\quad
+a_{{13}} \left( \tilde{u}\hat{u}+u\th{u} \right)  \left( \hat{u}\th{u}+u\tilde{u} \right)
+a_{{14}} \left( \tilde{u}\th{u}+u\hat{u} \right)  \left( \hat{u}\th{u}+u\tilde{u} \right)
+a_{{15}} \left( {\hat{u}}^{2}{\th{u}}^{2}+{u}^{2}{\tilde{u}}^{2} \right)
\\\quad
+a_{{16}} \left( {\tilde{u}}^{2}{\hat{u}}^{2}+{u}^{2}{\th{u}}^{2} \right)
+a_{{17}} \left( {\tilde{u}}^{2}{\th{u}}^{2}+{u}^{2}{\hat{u}}^{2} \right)
+a_{{18}}u\tilde{u}\hat{u}\th{u} \left( u+\tilde{u}+\hat{u}+\th{u} \right)
\\\quad
+a_{{19}} \left( \tilde{u}{\hat{u}}^{2}{\th{u}}^{2}+{u}^{2}{\tilde{u}}^{2}\th{u}+u{\hat{u}}^{2}{\th{u}}^{2}+{u}^{2}{\tilde{u}}^{2}\hat{u} \right)
+a_{{20}} \left( {\tilde{u}}^{2}\hat{u}{\th{u}}^{2}+u{\tilde{u}}^{2}{\th{u}}^{2}+{u}^{2}{\hat{u}}^{2}\th{u}+{u}^{2}\tilde{u}{\hat{u}}^{2} \right)
\\\quad
+a_{{21}} \left( {\tilde{u}}^{2}{\hat{u}}^{2}\th{u}+u{\tilde{u}}^{2}{\hat{u}}^{2}+{u}^{2}\hat{u}{\th{u}}^{2}+{u}^{2}\tilde{u}{\th{u}}^{2} \right)
+a_{{22}}u\tilde{u}\hat{u}\th{u} \left( \hat{u}\th{u}+u\tilde{u} \right)
\\\quad
+a_{{23}}u\tilde{u}\hat{u}\th{u} \left( \tilde{u}\hat{u}+u\th{u} \right)
+a_{{24}}u\tilde{u}\hat{u}\th{u} \left( \tilde{u}\th{u}+u\hat{u} \right)
\\\quad
+a_{{25}} \left( {\tilde{u}}^{2}{\hat{u}}^{2}{\th{u}}^{2}+{u}^{2}{\hat{u}}^{2}{\th{u}}^{2}+{u}^{2}{\tilde{u}}^{2}{\th{u}}^{2}+{u}^{2}{\tilde{u}}^{2}{\hat{u}}^{2} \right)
\\\quad
+a_{{26}}u\tilde{u}\hat{u}\th{u} \left( \tilde{u}\hat{u}\th{u}+u\hat{u}\th{u}+u\tilde{u}\th{u}+u\tilde{u}\hat{u} \right)
+a_{{27}}{u}^{2}{\tilde{u}}^{2}{\hat{u}}^{2}{\th{u}}^{2},
}
\end{equation}
for some set of coefficients $\{a_1,\ldots,a_{27}\}$.
This symmetry is a strong additional assumption, for instance by itself it is actually sufficient for integrability of quad equations in the multi-affine class \cite{ABS2,via}.

We also assume that both biquadratic polynomials $H_1$ and $H_2$ are symmetric, and furthermore that they are taken from one of the following one-parameter families (the parameter being denoted by $p$)
\begin{equation}
\label{q4}
\fl\textrm{q4}^*:\quad\frac{c}{2}\left(1+u^2\tilde{u}^2+\tilde{u}^2p^2+p^2u^2\right)-\frac{1}{2c}\left(u^2\tilde{u}^2p^2+u^2+\tilde{u}^2+p^2\right)-\left(c^2-\frac{1}{c^2}\right)u\tilde{u}p,
\end{equation}
\begin{equation}
\label{q3}
\fl\textrm{q3}^*:\quad\frac{1}{2}(u^2+\tilde{u}^2)+\frac{\delta^2}{2}(p^2-1)-u\tilde{u}p,
\end{equation}
\begin{equation}
\label{q2}
\fl\textrm{q2}^*:\quad\frac{1}{4}\left(u^2+\tilde{u}^2+p^2\right)-\frac{1}{2}\left(u\tilde{u}+\tilde{u}p+pu\right),
\end{equation}
\begin{equation}
\label{q1}
\fl\textrm{q1}^*:\quad (\tilde{u}-u)^2-p,
\end{equation}
\begin{equation}
\label{a2}
\fl\textrm{a2}^*:\quad\frac{1}{2}(1+u^2\tilde{u}^2)-pu\tilde{u},
\end{equation}
\begin{equation}
\fl\textrm{a1}^*:\quad\label{a1}(\tilde{u}+u)^2-p,
\end{equation}
\begin{equation}
\label{h3}
\fl\textrm{h3}^*:\quad u\tilde{u}p+\delta^2,
\end{equation}
\begin{equation}
\label{h2}
\fl\textrm{h2}^*:\quad u+\tilde{u}+p.
\end{equation}
Where it appears $\delta \in\{0,1\}$, whilst in (\ref{q4}) $c\in\mathbb{C}\setminus\{0,1,-1,i,-i\}$ is a fixed constant.
Up to a point transformation of the parameters these biquadratic polynomials coincide with those associated with the ABS equations \cite{ABS,ABS2} (precise parameter associations will be given later in Section \ref{ABSlist}).
Note that (\ref{h3}) and (\ref{h2}) are considered to be biquadratics here also, but this is in a projective sense, so loosely speaking we view for instance (\ref{h2}) as the polynomial $(u+\tilde{u}+p)(u-\infty)(\tilde{u}-\infty)$.
We remark that by a M\"obius change of variables a symmetric biquadratic polynomial can always be brought to one of the forms (\ref{q4})--(\ref{h2}) for some choice of parameter $p$, or else is the square of a multi-affine polynomial (like $G_1,\ldots,G_4$ above), a possibility which we therefore exclude by restricting to (\ref{q4})--(\ref{h2}).
The discriminant of biquadratics (\ref{q4})--(\ref{h2}) with respect to $\tilde{u}$ is a polynomial of degree at most four in $u$, the main characterising feature of the biquadratic families is that roots of this discriminant polynomial do not change upon altering parameter $p$.

Finally, the assumed symmetry of $\mathcal{Q}$ together with (\ref{df}) implies already that \[G_1=G_2=G_3=G_4=:G,\] in fact we will go further and make the assumption that
\begin{equation}
G(\tilde{u},\hat{u}) = (\tilde{u}-\hat{u})^2,
\end{equation}
which is our last additional assumption.

\section{The obtained multi-quadratic quad equations}\label{list}
The additional assumptions enable solution of the discriminant factorisation hypothesis directly by computer algebra, which gives the main result of our paper as follows.
\begin{proposition}\label{listprop}
Let $\mathcal{Q}$ be a polynomial of degree two in each of four variables with the Kleinian symmetry
\begin{equation}
\mathcal{Q}(u,\tilde{u},\hat{u},\th{u})=\mathcal{Q}(\tilde{u},u,\th{u},\hat{u})=\mathcal{Q}(\hat{u},\th{u},u,\tilde{u}),
\end{equation}
such that the discriminants factorise as follows
\begin{equation}
\Delta[\mathcal{Q}(u,\tilde{u},\hat{u},\th{u}),\th{u}]\propto (\tilde{u}-\hat{u})^2H_1(u,\tilde{u})H_2(u,\hat{u}),\label{disc}
\end{equation}
where $H_1$ and $H_2$ are biquadratic polynomials.
If $H_1$ and $H_2$ are taken from one of the biquadratic families (\ref{q4})--(\ref{h2}) with generic parameter choices denoted $p$ and $q$ respectively, then $\mathcal{Q}$ is determined up to an overall constant and defines respectively the quad equations (\ref{QQ4})--(\ref{QH2}) below.
\end{proposition}
Q4$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[(c^{-2}p-c^2q)(u\tilde{u}-\hat{u}\th{u})^2-(c^{-2}q-c^2p)(u\hat{u}-\tilde{u}\th{u})^2]\\
-(p-q)^2[(u+\th{u})^2(1+\tilde{u}^2\hat{u}^2)+(\tilde{u}+\hat{u})^2(1+u^2\th{u}^2)]\\
+[(u-\th{u})(\tilde{u}-\hat{u})(c^{-1}-cpq)-2(p-q)(1+u\tilde{u}\hat{u}\th{u})]\\
\times[(u-\th{u})(\tilde{u}-\hat{u})(c^{-1}pq-c)-2(p-q)(u\th{u}+\tilde{u}\hat{u})]=0,\\
}\label{QQ4}
\end{equation}
Q3$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u\tilde{u}-\hat{u}\th{u})^2-q(u\hat{u}-\tilde{u}\th{u})^2]-\delta^2(p-q)^2[(u+\th{u})^2+(\tilde{u}+\hat{u})^2]\\
\fl\quad +[(u-\th{u})(\tilde{u}-\hat{u})-2\delta^2(p-q)][(u-\th{u})(\tilde{u}-\hat{u})(pq-1)-2(p-q)(u\th{u}+\tilde{u}\hat{u})]=0,
}\label{QQ3}
\end{equation}
Q2$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u\tilde{u}-\hat{u}\th{u})(u+\tilde{u}-\hat{u}-\th{u})-q(u\hat{u}-\tilde{u}\th{u})(u+\hat{u}-\tilde{u}-\th{u})] \\
+(u-\th{u})(\tilde{u}-\hat{u})[p(u-\hat{u})(\tilde{u}-\th{u})-q(u-\tilde{u})(\hat{u}-\th{u})-pq(p-q)]=0,
}\label{QQ2}
\end{equation}
Q1$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u+\tilde{u}-\hat{u}-\th{u})^2-q(u+\hat{u}-\tilde{u}-\th{u})^2]\\+4(u-\th{u})(\tilde{u}-\hat{u})[p(u-\hat{u})(\tilde{u}-\th{u})-q(u-\tilde{u})(\hat{u}-\th{u})]=0,
}\label{QQ1}
\end{equation}
A2$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u\hat{u}-\tilde{u}\th{u})^2-q(u\tilde{u}-\hat{u}\th{u})^2]\\
+(u-\th{u})(\tilde{u}-\hat{u})[(u-\th{u})(\tilde{u}-\hat{u})(pq-1)+2(p-q)(1+u\tilde{u}\hat{u}\th{u})]=0,
}\label{QA2}
\end{equation}
A1$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u-\tilde{u}+\hat{u}-\th{u})^2-q(u-\hat{u}+\tilde{u}-\th{u})^2]\\+4(u-\th{u})(\tilde{u}-\hat{u})[p(u+\hat{u})(\tilde{u}+\th{u})-q(u+\tilde{u})(\hat{u}+\th{u})]=0,
}\label{QA1}
\end{equation}
H3$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u\hat{u}-\tilde{u}\th{u})^2-q(u\tilde{u}-\hat{u}\th{u})^2]\\
+(u-\th{u})(\tilde{u}-\hat{u})[(u-\th{u})(\tilde{u}-\hat{u})pq-4\delta^2(p-q)]=0,
}\label{QH3}
\end{equation}
H2$^*$:
\begin{equation}
\eqalign{
\fl\quad (p-q)[p(u-\tilde{u}+\hat{u}-\th{u})^2-q(u-\hat{u}+\tilde{u}-\th{u})^2]\\+(u-\th{u})(\tilde{u}-\hat{u})[(u-\th{u})(\tilde{u}-\hat{u})-2(p-q)(u+\tilde{u}+\hat{u}+\th{u})]=0.
}\label{QH2}
\end{equation}

We have denoted the list of models here $Q4^*,\ldots,H2^*$ due to their natural correspondence with equations from the ABS list, which are denoted $Q4,\ldots,H1$ \cite{ABS}. (The ABS list and its precise relationship with (\ref{QQ4})--(\ref{QH2}) will be given in Section \ref{ABSlist}.)
The biquadratic polynomials associated with ABS equations $Q1_{\delta=0}$, $A1_{\delta=0}$ and $H1$ are already squares, which means the multi-quadratic counterparts of those models factorise into products of multi-affine polynomials, cases we exclude here as degenerate.

Similar to the situation for ABS equations, all of (\ref{QQ3})--(\ref{QH2}) can be obtained from the primary model (\ref{QQ4}) by limiting procedures.
All listed models are non-equivalent by autonomous M\"obius changes of variables and point transformations of the parameters.
However, the models $A2^*$ and $Q3^*_{\delta=0}$ are related by the non-autonomous point transformation $u\rightarrow u^{(-1)^{n+m}}$, whilst $A1^*$ and $Q1^*$ are related by $u\rightarrow (-1)^{n+m}u$.

To our knowledge the equations in the list (\ref{QQ4})--(\ref{QH2}) are new except for the following cases.
The model H2$^*$ (\ref{QH2}) appeared as the superposition principle for solutions of the KdV equation in \cite{AdVeQ} and in relation to Yang-Baxter maps in \cite{KaNie}.
The model A1$^*$ (\ref{QA1}) was obtained, although not in explicit form, in \cite{KaNie3}.
The model A2$^*$ (\ref{QA2}) was obtained originally in \cite{JamesQ} as the superposition principle for B\"acklund transformations of Hirota's discrete KdV equation \cite{hirota-0}.

\section{Reformulation as single-valued systems}\label{SVS}
One of the most salient features of quad equations from the multi-quadratic class is that they define multivalued evolution from initial data.
However, built into the construction of models (\ref{QQ4})--(\ref{QH2}) is the discriminant factorisation property, and this allows the quad equation to be reformulated as a system that defines single-valued evolution from augmented initial data \cite{JamesQ}.
Here we perform this reformulation on the primary model Q4$^*$ (\ref{QQ4}), and describe the relationship between the quad equation and its reformulation.

As described in Section \ref{dfh}, the auxiliary variables on lattice edges (see Figure \ref{quadpic}) are introduced through the relations
\begin{equation}
\eqalign{
\fl \sigma_1^2 = \frac{c}{2}\left(1+u^2\tilde{u}^2+\tilde{u}^2p^2+p^2u^2\right)-\frac{1}{2c}\left(u^2\tilde{u}^2p^2+u^2+\tilde{u}^2+p^2\right)-\left(c^2-\frac{1}{c^2}\right)u\tilde{u}p,\\
\fl \sigma_2^2 = \frac{c}{2}\left(1+u^2\hat{u}^2+\hat{u}^2q^2+q^2u^2\right)-\frac{1}{2c}\left(u^2\hat{u}^2q^2+u^2+\hat{u}^2+q^2\right)-\left(c^2-\frac{1}{c^2}\right)u\hat{u}q,
}\label{dvdef}
\end{equation}
which are in terms of the associated biquadratic polynomial given earlier in (\ref{q4}).
\begin{figure}[t]
\begin{center}
\begin{picture}(100,100)(0,0)
\multiput(21,12)(60,0){2}{{\line(0,1){60}}}
\multiput(21,12)(0,60){2}{{\line(1,0){60}}}
\put(7,39){{$\sigma_2$}}
\put(48,0){{$\sigma_1$}}
\put(87,39){{$\tilde{\sigma}_2$}}
\put(48,78){{$\hat{\sigma}_1$}}
\put(21,12){\circle*{3}}
\put(81,12){\circle*{3}}
\put(21,72){\circle*{3}}
\put(81,72){\circle*{3}}
\put(12,4){$u$}
\put(84,4){$\tilde{u}$}
\put(12,72){$\hat{u}$}
\put(84,72){$\th{u}$}
\end{picture}
\end{center}
\caption{Variables assigned to the vertices of a quadrilateral, and auxiliary variables to the edges.}
\label{quadpic}
\end{figure}
By solving equation (\ref{QQ4}) as a quadratic equation for $\th{u}$ and exploiting these edge variables we obtain an equation of the form
\begin{equation}
\mathcal{F}(u,\tilde{u},\hat{u},\th{u},\sigma_1\sigma_2)=0
\end{equation}
which is by construction of polynomial degree one in $\th{u}$ and $\sigma_1\sigma_2$, for instance:
\begin{equation}
\eqalign{
\fl\quad\mathcal{F}(u,\tilde{u},\hat{u},\th{u},\sigma_1\sigma_2):=u[(c^{-2}p-c^2q)(u\tilde{u}-\hat{u}\th{u})+(c^{-2}q-c^2p)(u\hat{u}-\tilde{u}\th{u})]\\
-(u-\th{u})[(c^{-1}-cpq)(u^2+\tilde{u}\hat{u})+(c^{-1}pq-c)(1+u^2\tilde{u}\hat{u})+2\sigma_1\sigma_2]\\
-(p-q)(\tilde{u}-\hat{u})(1+u^3\th{u})
}\label{Fdef}
\end{equation}
(the precise form of this expression is chosen because it is simple, but it is not unique).
Sequentially solving for each of the quad variables of (\ref{QQ4}) in the same way leads to the following system
\begin{equation}
\eqalign{
\mathcal{F}(u,\tilde{u},\hat{u},\th{u},\sigma_1\sigma_2)=0,\\
\mathcal{F}(\tilde{u},u,\th{u},\hat{u},\sigma_1\tilde{\sigma}_2)=0,\\
\mathcal{F}(\hat{u},\th{u},u,\tilde{u},\hat{\sigma}_1\sigma_2)=0,\\
\mathcal{F}(\th{u},\hat{u},\tilde{u},u,\hat{\sigma}_1\tilde{\sigma}_2)=0,
}
\label{quadsys}
\end{equation}
up to the choice of sign of the discriminant terms appearing in the last argument.

The system (\ref{quadsys}), (\ref{dvdef}) is the single-valued model associated with (\ref{QQ4}).
It is easily verifiable that each equation in (\ref{quadsys}) implies (\ref{QQ4}) is satisfied modulo the relations (\ref{dvdef}).
The signs of the discriminant terms in (\ref{quadsys}) have been chosen for self-consistency (ensuring that any one equation in (\ref{quadsys}) is a consequence of the other three), to preserve the Kleinian symmetry, and to preserve the consistency property described in the following section.

The usual initial value problem for a quad-equation involves specifying the dependent variable on vertices along some admissible lattice path or collection of paths.
The modification required for system (\ref{quadsys}) is that variables on path edges, subject to the constraint (\ref{dvdef}), also need to be specified.
The system (\ref{quadsys}), (\ref{dvdef}) is therefore actually a model of vertex-bond type as described in \cite{hv11}, but it has the additional feature of preserving algebraic relations (\ref{dvdef}) on lattice edges.
In this setting the multivalued evolution defined by equation (\ref{QQ4}) is reflected in the non-uniqueness of initial edge variables when they are defined in terms of the initial vertex variables through (\ref{dvdef}).

In Table \ref{svms} the polynomial $\mathcal{F}$ that appears in the single-valued system associated with each equation from the list (\ref{QQ4})--(\ref{QH2}) is also given.
The auxiliary variables present in the table are introduced through the edge relations which can be written generically as
\begin{equation}
\sigma_1^2 = H_1(u,\tilde{u}), \quad \sigma_2^2 = H_2(u,\hat{u}),\label{erer}
\end{equation}
in terms of the associated biquadratic polynomials (cf. Proposition \ref{listprop}).

\begin{table}
\begin{tabular}{rl}
\hline
Q4$^*$
& $u[(c^{-2}p-c^2q)(u\tilde{u}-\hat{u}\th{u})+(c^{-2}q-c^2p)(u\hat{u}-\tilde{u}\th{u})]$\\
& \quad $-(u-\th{u})[(c^{-1}-cpq)(u^2+\tilde{u}\hat{u})+(c^{-1}pq-c)(1+u^2\tilde{u}\hat{u})+2\sigma_1\sigma_2]$\\
& \quad $-(p-q)(\tilde{u}-\hat{u})(1+u^3\th{u})$\\
Q3$^*$
& $u[p(u\tilde{u}-\hat{u}\th{u})+q(u\hat{u}-\tilde{u}\th{u})]-(u-\th{u})[u^2+\tilde{u}\hat{u}-\delta^2(1-pq)-2\sigma_1\sigma_2]$\\
& \quad $-\delta^2(p-q)(\tilde{u}-\hat{u})$ \\
Q2$^*$
& $u[p(u+\tilde{u}-\hat{u}-\th{u})+q(u-\tilde{u}+\hat{u}-\th{u})]+p(u\tilde{u}-\hat{u}\th{u})+q(u\hat{u}-\tilde{u}\th{u})$\\
& \quad $-(u-\th{u})[(u-\tilde{u})(u-\hat{u})+pq+4\sigma_1\sigma_2]$\\
Q1$^*$
& $p(u+\tilde{u}-\hat{u}-\th{u})+q(u-\tilde{u}+\hat{u}-\th{u})-2(u-\th{u})[(u-\tilde{u})(u-\hat{u})-\sigma_1\sigma_2]$\\
A2$^*$
& $u[p(u\hat{u}-\tilde{u}\th{u})+q(u\tilde{u}-\hat{u}\th{u})]-(u-\th{u})(1+u^2\tilde{u}\hat{u}-2\sigma_1\sigma_2)$\\
A1$^*$
& $p(u+\hat{u}-\tilde{u}-\th{u})+q(u-\hat{u}+\tilde{u}-\th{u})-2(u-\th{u})[(u+\tilde{u})(u+\hat{u})-\sigma_1\sigma_2]$\\
H3$^*$
& $u[p(u\hat{u}-\tilde{u}\th{u})+q(u\tilde{u}-\hat{u}\th{u})]+2(u-\th{u})(\delta-\sigma_1\sigma_2)$\\
H2$^*$
& $p(u+\hat{u}-\tilde{u}-\th{u})+q(u-\hat{u}+\tilde{u}-\th{u})+(u-\th{u})(\tilde{u}+\hat{u}+2u-2\sigma_1\sigma_2)$\\
\hline
\end{tabular}
\caption{
Polynomials $\mathcal{F}(u,\tilde{u},\hat{u},\th{u},\sigma_1\sigma_2)$ that define the single-valued system (\ref{quadsys}) associated with the multi-quadratic models (\ref{QQ4})--(\ref{QH2}) of Proposition \ref{listprop}.
The auxiliary variables $\sigma_1$ and $\sigma_2$ satisfy edge relations $\sigma_1^2=H_1(u,\tilde{u})$ and $\sigma_2^2=H_2(u,\hat{u})$ in terms of the associated biquadratic polynomials listed in (\ref{q4})--(\ref{h2}).
}\label{svms}
\end{table}

The single-valued reformulation (\ref{quadsys}), (\ref{erer}) that we have proposed for the quad equation (\ref{ge}) is autonomous, but there is the possibility to make a non-autonomous reformulation of the model in which signs of discriminant terms in (\ref{quadsys}) change from quad to quad.
Such non-autonomous reformulation is more difficult to study, however in the autonomous case the quad equation and its reformulation are not globally equivalent.
They are of course locally equivalent, by which we mean that on a single quad the equation (\ref{ge}) is sufficient for consistency of system (\ref{quadsys}) in $\sigma_1$, $\sigma_2$, allowing the edge variables to be constructed from vertex variables on a single quad.
The simplest setting in which the quad equation and its autonomous reformulation are not equivalent is a square domain involving nine points (four quads).
On this larger domain we can write the system
\begin{equation}
\eqalign{
\mathcal{F}(u,\tilde{u},\hat{u},\th{u},\sigma_1\sigma_2)=0,\\
\mathcal{F}(u,\ut{u},\hat{u},\ut{\hat{u}},\ut{\sigma}_1\sigma_2)=0,\\
\mathcal{F}(u,\tilde{u},\uh{u},\uh{\tilde{u}},\sigma_1\uh{\sigma}_2)=0,\\
\mathcal{F}(u,\ut{u},\uh{u},\ut{\uh{u}},\ut{\sigma}_1\uh{\sigma}_2)=0.
}
\label{crossys}
\end{equation}
where $\ut{u}=u(n-1,m)$, $\uh{u}=u(n,m-1)$ etc, this system being a consequence of imposing (\ref{quadsys}).
The equation (\ref{ge}) imposed on each quad is {\it not} sufficient for consistency of (\ref{crossys}), specifically elimination of $\sigma_1,\sigma_2,\ut{\sigma}_1$ and $\uh{\sigma}_2$ from (\ref{crossys}) yields a constraint on the nine vertex-variables which is not a consequence of (\ref{ge}).
In particular (\ref{crossys}) can be used to obtain $\th{u}$ uniquely from data $\{\ut{\uh{u}},\ut{u},\uh{u},u,\ut{\hat{u}},\uh{\tilde{u}},\tilde{u},\hat{u}\}$, whereas if (\ref{ge}) alone were imposed this data would yield two possible values of $\th{u}$.
This example therefore demonstrates that global equivalence of the quad equation and its associated single-valued system is not possible unless a non-autonomous reformulation would be considered.

More generally than the example above, if initial data is specified on verticies along some admissable lattice path, then the autonomous single-valued system (\ref{quadsys}), (\ref{erer}) generically determines $2^{(x-1)}$ solutions where $x$ is the number of edges along the path (corresponding to the number of auxiliary edge variables), whereas using the equation (\ref{ge}) alone the number of solutions determined would be $2^y$ where $y$ is the number of quads in the domain.
Because (on a finite domain) $y\ge x-1$, the autonomous single-valued reformulation of the system can be said to limit the multivaluedness from initial data.

\section{Multidimensional consistency}\label{MDC}
The purpose of this section is to explain a key integrability feature of models (\ref{QQ4})--(\ref{QH2}), namely their {\it multidimensional consistency} \cite{FrankABS,BS,ABS}.
This emerges quite naturally, although it has not been explicitly built into the construction of these equations.

The multidimensional consistency involves not just one equation in isolation, and in fact for the equations listed it involves the whole family of equations obtained by varying parameters $p$ and $q$.
The consistency is between members of this family with different choices of the parameters.
Due to the central role they play it is therefore convenient to explicitly include the dependence on parameters $p$ and $q$ when writing a generic equation from (\ref{QQ4})--(\ref{QH2}):
\begin{equation}
\mathcal{Q}_{p,q}(u,\tilde{u},\hat{u},\th{u})=0.\label{pqs}
\end{equation}

The key properties (involving the parameter dependence) of the generic defining polynomial are first the symmetry
\begin{equation}
\mathcal{Q}_{p,q}(u,\tilde{u},\hat{u},\th{u}) = \mathcal{Q}_{q,p}(u,\hat{u},\tilde{u},\th{u}), \label{covar}
\end{equation}
and second the consistency of the system
\begin{equation}
\eqalign{
\mathcal{Q}_{p,q}(u,\tilde{u},\hat{u},\th{u})=0, \quad& \mathcal{Q}_{p,q}(\bar{u},\bt{u},\hb{u},\thb{u})=0,\\
\mathcal{Q}_{q,r}(u,\hat{u},\bar{u},\hb{u})=0, \quad& \mathcal{Q}_{q,r}(\tilde{u},\th{u},\bt{u},\thb{u})=0,\\
\mathcal{Q}_{r,p}(u,\bar{u},\tilde{u},\bt{u})=0, \quad& \mathcal{Q}_{r,p}(\hat{u},\hb{u},\th{u},\thb{u})=0,
}\label{cubesys}
\end{equation}
for any choice of the parameters $p$, $q$ and $r$, which is usually visualised by assigning variables to vertices of a cube as in Figure \ref{cubepic}, and equations to faces.
\begin{figure}[t]
\begin{center}
\begin{picture}(200,140)
\input{dat.tex}
\end{picture}
\end{center}
\caption{\label{cubepic}Variables assigned to the vertices of a cube. In the case of the multi-quadratic quad equations (\ref{QQ4})--(\ref{QH2}) the system (\ref{cubesys}) determines four possible values of $\thb{u}$ from initial data $u$, $\tilde{u}$, $\hat{u}$ and $\bar{u}$.}
\end{figure}
By consistency we mean that for generic initial data $\{u,\tilde{u},\hat{u},\bar{u}\}$ the system (\ref{cubesys}) has at least one solution for the remaining variables $\{\th{u},\hb{u},\bt{u},\thb{u}\}$.
Directly by polynomial manipulation it can be confirmed that for each quad-equation (\ref{QQ4})--(\ref{QH2}) the system (\ref{cubesys}) is consistent in this sense, and in fact determines four possible solutions $\{\th{u},\hb{u},\bt{u},\thb{u}\}$.

Alternatively, and more explicitly, the consistency property can be verified by reformulating the equation as a single-valued system as described in Section \ref{SVS}.
For instance in the case of the primary model (\ref{QQ4}) a direct calculation yields the following expression that determines $\thb{u}$ in terms of the initial data on the cube:
\begin{equation}
\fl
\eqalign{
u(c^2-c^{-2})[p(q-r)^2(\hat{u}\bar{u}-\tilde{u}\thb{u})+q(r-p)^2(\bar{u}\tilde{u}-\hat{u}\thb{u})+r(p-q)^2(\tilde{u}\hat{u}-\bar{u}\thb{u})]\\
-(q-r)(r-p)(\bar{u}-\thb{u})[(c^{-1}-cpq)(u^2+\tilde{u}\hat{u})+(c^{-1}pq-c)(1+u^2\tilde{u}\hat{u})+2\sigma_1\sigma_2]\\
-(r-p)(p-q)(\tilde{u}-\thb{u})[(c^{-1}-cqr)(u^2+\hat{u}\bar{u})+(c^{-1}qr-c)(1+u^2\hat{u}\bar{u})+2\sigma_2\sigma_3]\\
-(p-q)(q-r)(\hat{u}-\thb{u})[(c^{-1}-crp)(u^2+\bar{u}\tilde{u})+(c^{-1}rp-c)(1+u^2\bar{u}\tilde{u})+2\sigma_3\sigma_1]=0,\\
}\label{cubesol}
\end{equation}
where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are determined from the initial data through equations (\ref{dvdef}) and
\begin{equation}
\fl \sigma_3^2 = \frac{c}{2}\left(1+u^2\bar{u}^2+\bar{u}^2r^2+r^2u^2\right)-\frac{1}{2c}\left(u^2\bar{u}^2r^2+u^2+\bar{u}^2+r^2\right)-\left(c^2-\frac{1}{c^2}\right)u\bar{u}r.\label{s3}
\end{equation}
Although the transformation group $(\sigma_1,\sigma_2,\sigma_3)\mapsto(\pm\sigma_1,\pm\sigma_2,\pm\sigma_3)$ leaving (\ref{dvdef}), (\ref{s3}) invariant contains eight elements, the initial data $\{u,\tilde{u},\hat{u},\bar{u}\}$ leads to only four distinct values of $\thb{u}$ due to the symmetry $(\sigma_1,\sigma_2,\sigma_3)\mapsto(-\sigma_1,-\sigma_2,-\sigma_3)$ of (\ref{cubesol}).

It is important to note that the consistency property of the associated single-valued system can be broken by reversing sign of all discriminant terms in (\ref{quadsys}).
This subtlety reflects the fact that polynomial system (\ref{cubesys}) determines only four possible values of $\thb{u}$ from the initial data, and not eight as might be expected.
Other polynomials given in Table \ref{svms} are also chosen with the consistency property in mind, so that in particular the sign chosen for the discriminant terms is important.

The consistency property of models (\ref{QQ4})--(\ref{QH2}) means that their more general setting is for a dependent variable $u$ defined on $\mathbb{Z}^d$, where $d>1$ is the dimension, determined by the system
\begin{equation}
\mathcal{Q}_{p_i,p_j}(u,\mathsf{T}_iu,\mathsf{T}_ju,\mathsf{T}_i\mathsf{T}_ju)=0, \quad i,j\in\{1\ldots d \}.\label{mds}
\end{equation}
Here $p_1,\ldots,p_d$ are some set of parameters, while $\mathsf{T}_1,\ldots,\mathsf{T}_d$ are shift operators in each dimension.
Probably the simplest Cauchy problem for (\ref{mds}) is to specify initial data on coordinate axes, but actually this multidimensional setting is the departure point for an extremely rich variety of initial value problems and lattice configurations \cite{BS,AdVeQ,AS}.
This also aligns with the notion of solvability for models in statistical mechanics \cite{baxter}.
From a slightly different perspective the consistency property yields a great deal of control over the solution structure of these models by providing immediately a natural auto-B\"acklund transformation.
This allows for example to construct exact solutions as periodic B\"acklund chains (the discrete analogue of finite-gap solutions), as well as soliton-type solutions from B\"acklund iteration.

We remark that on the domain of a hypercube, $\{0,1\}^d$, associated edge variables can be obtained (as described in Section \ref{SVS}) for {\it any} solution of (\ref{mds}), this domain is therefore singled out as one on which the quad equation and its autonomous single-valued reformulation are globally equivalent.

\section{Transformations to ABS equations}\label{transformations}
All equations listed by ABS in \cite{ABS}, with the exception of Q4, are mutually related via B\"acklund or Miura type transformations \cite{NC,James,NAH-Sol,James2}.
Table \ref{bts} here lists similar transformations connecting all except the primary multi-quadratic model Q4$^*$, specifically equations (\ref{QQ3})--(\ref{QH2}), back to multi-affine equations from the ABS list.
Before describing these transformations in more detail we first recall the ABS list in (\ref{Q4})--(\ref{H1}) below.
\subsection{The ABS list}\label{ABSlist}
Q4:
\begin{equation}
\eqalign{
\fl\quad \mathop{\mathrm{sn}}\nolimits(\alpha)(v\tilde{v}+\hat{v}\hat{\tilde{v}})-\mathop{\mathrm{sn}}\nolimits(\beta)(v\hat{v}+\tilde{v}\hat{\tilde{v}})\\
-\mathop{\mathrm{sn}}\nolimits(\alpha-\beta)[\tilde{v}\hat{v}+v\hat{\tilde{v}}-k\mathop{\mathrm{sn}}\nolimits(\alpha)\mathop{\mathrm{sn}}\nolimits(\beta)(1+v\tilde{v}\hat{v}\hat{\tilde{v}})]=0,
}
\label{Q4}
\end{equation}
Q3:
\begin{equation}
\eqalign{
\fl\quad(\alpha-1/\alpha)(v\tilde{v}+\hat{v}\th{v})-(\beta-1/\beta)(v\hat{v}+\tilde{v}\th{v})\\
-(\alpha/\beta-\beta/\alpha)[\tilde{v}\hat{v}+v\th{v}+\delta^2(\alpha-1/\alpha)(\beta-1/\beta)/4]=0,
}\label{Q3}
\end{equation}
Q2:
\begin{equation}
\eqalign{
\fl\quad \alpha(v-\hat{v})(\tilde{v}-\th{v})-\beta(v-\tilde{v})(\hat{v}-\th{v})\\
+\alpha\beta(\alpha-\beta)(v+\tilde{v}+\hat{v}+\th{v}-\alpha^2+\alpha\beta-\beta^2)=0,
}\label{Q2}
\end{equation}
Q1:
\begin{equation}
\alpha(v-\hat{v})(\tilde{v}-\th{v})-\beta(v-\tilde{v})(\hat{v}-\th{v})+\delta^2 \alpha\beta(\alpha-\beta)=0,
\label{Q1}
\end{equation}
A2:
\begin{equation}
\fl\quad (\alpha-1/\alpha)(v\hat{v}+\tilde{v}\th{v})-(\beta-1/\beta)(v\tilde{v}+\hat{v}\th{v}) - (\alpha/\beta-\beta/\alpha)(1+v\tilde{v}\hat{v}\th{v})=0,
\label{A2}
\end{equation}
A1:
\begin{equation}
\alpha(v+\hat{v})(\tilde{v}+\th{v})-\beta(v+\tilde{v})(\hat{v}+\th{v})- \delta^2\alpha\beta(\alpha-\beta)=0,
\label{A1}
\end{equation}
H3:
\begin{equation}
\alpha(v\tilde{v}+\hat{v}\th{v})-\beta(v\hat{v}+\tilde{v}\th{v})+\delta(\alpha^2-\beta^2)=0,
\label{H3}
\end{equation}
H2:
\begin{equation}
(v-\th{v})(\tilde{v}-\hat{v})-(\alpha-\beta)(v+\tilde{v}+\hat{v}+\th{v}+\alpha+\beta)=0,
\label{H2}
\end{equation}
H1:
\begin{equation}
(v-\th{v})(\tilde{v}-\hat{v})-\alpha+\beta=0.
\label{H1}
\end{equation}

In (\ref{Q4})--(\ref{H1}) we have reproduced the ABS list.
Where it appears $\delta\in\{0,1\}$ and for Q4 (\ref{Q4}) $k\in \mathbb{C}\setminus\{0,1,-1\}$ is the modulus of the Jacobi elliptic function $\mathop{\mathrm{sn}}\nolimits$.
The parametrisations coincide with the ones given by ABS in \cite{ABS} except for the case of Q4, this canonical form (the Jacobi form) was obtained by Hietarinta in \cite{Hie}.
Each ABS equation is defined by a polynomial $\mathcal{Q}=\mathcal{Q}(v,\tilde{v},\hat{v},\th{v})$ that is degree one in four variables.
It was shown in \cite{ABS} that this polynomial is characterised in terms of biquadratics through a generalised discriminant formula as follows
\begin{equation}
\fl\eqalign{
(\partial_{\hat{v}}\mathcal{Q})(\partial_{\th{v}}\mathcal{Q})-(\partial_{\hat{v}}\partial_{\th{v}}\mathcal{Q})\mathcal{Q} \propto H_1(v,\tilde{v}), \quad (\partial_v\mathcal{Q})(\partial_{\tilde{v}}\mathcal{Q})-(\partial_v\partial_{\tilde{v}}\mathcal{Q})\mathcal{Q}  \propto H_1(\hat{v},\th{v}),\\
(\partial_{\tilde{v}}\mathcal{Q})(\partial_{\th{v}}\mathcal{Q})-(\partial_{\tilde{v}}\partial_{\th{v}}\mathcal{Q})\mathcal{Q} \propto H_2(v,\hat{v}), \quad (\partial_v\mathcal{Q})(\partial_{\hat{v}}\mathcal{Q})-(\partial_v\partial_{\hat{v}}\mathcal{Q})\mathcal{Q}  \propto H_2(\tilde{v},\th{v}).
}\label{gendisc}
\end{equation}
The polynomials $H_1$ and $H_2$ appearing in (\ref{gendisc}) coincide with those in (\ref{disc}) up to an overall constant, provided we make the following associations between parameters $\alpha$ and $\beta$ appearing in the multi-affine equations (\ref{Q4})--(\ref{H2}) and parameters $p$ and $q$ appearing in their multi-quadratic counterparts (\ref{QQ4})--(\ref{QH2}),
\begin{equation}
\begin{array}{rlll}
Q4:& p=\sqrt{k}\mathop{\mathrm{sn}}\nolimits(\alpha+K),& q=\sqrt{k}\mathop{\mathrm{sn}}\nolimits(\beta+K),& c=\sqrt{k},\\
Q3,A2:& p=(\alpha+1/\alpha)/2,&q=(\beta+1/\beta)/2,\\
Q2:& p=\alpha^2,&q=\beta^2,\\
Q1,A1:& p=\delta^2\alpha^2,&q=\delta^2\beta^2,\\
H3:& p=1/\alpha,&q=1/\beta,& \delta^2\rightarrow \delta,\\
H2:& p=\alpha,&q=\beta,
\end{array}
\label{pa}
\end{equation}
where in the case of Q4 the parameter $K$ is standard notation for the quarter period of the $\mathop{\mathrm{sn}}\nolimits$ function (with modulus $k$) satisfying $\mathop{\mathrm{sn}}\nolimits(K)=1$, $\mathop{\mathrm{sn}}\nolimits'(K)=0$.

We remark that for the multi-affine equations H1 (the lattice potential KdV equation \cite{we,NC}), Q1$_{\delta=0}$ (the lattice Schwarzian KdV equation \cite{NC}) and A1$_{\delta=0}$ the associated biquadratic is the square of a polynomial which is degree one in each variable, and as mentioned before their multi-quadratic counterparts factorise into products of multi-affine polynomials which we have excluded as degenerate cases here.

\subsection{An example transformation}\label{BTex}
\begin{table}[t]
\begin{tabular}{lll}
Eq. in $u$ & B\"acklund transformation & Eq. in $v$ \\
\hline
Q3$^*_{p=2\alpha^2-1}$ & $\alpha[(\tilde{u}+\delta)\tilde{v}^2+(u+\delta)v^2]=(\tilde{u}+u+2\delta\alpha^2)\tilde{v}v$ & Q3$_{\delta=0}$ \\
Q2$^*_{p=\alpha}$ & $(\tilde{v}-v)(\tilde{u}\tilde{v}-uv)+\alpha \tilde{v}v=0$ & Q1$_{\delta=0}$\\
A2$^*_{p=2(\alpha^2+1)/(\alpha^2-1)}$ &$\alpha (\tilde{v}^2v^2+1)=\tilde{v}v[\tilde{u}u(\alpha^2-1)+(\alpha^2+1)]$ & A2\\
H3$^*_{p=4/\alpha^2}$ & $\tilde{u}u=\tilde{v}v(\tilde{v}v+\delta\alpha)$ & H3\\
H3$^*_{p= 4/\alpha}$ & $4\alpha\tilde{u}u=(\tilde{v}+v)^2-\delta^2\alpha^2$ & A1 \\
A1$^*_{p=\alpha^2}$ & $2 \tilde{v} v(\tilde{u}+u)=\alpha(\tilde{v}^2 v^2+1)$ & H3$_{\delta=0}$\\
A1$^*_{p=\alpha}$  & $2 (\tilde{v}+v)(\tilde{u}+u)= (\tilde{v}+v)^2+\alpha$ & A1$_{\delta=0}$\\
H2$^*_{p=-\alpha}$ & $\tilde{u}+u-\alpha=(\tilde{v}+v)^2$ & H1 \\
\hline
A2$^*_{p=2\alpha^2-1}$ & $\alpha(\tilde{u}u\tilde{v}^2v^2+1)=(\tilde{u}u+1)\tilde{v}v$ & A2\\
H3$^*_{p=4/\alpha}$ & $4\alpha \tilde{u}u=(\tilde{v}-v)^2-\delta^2\alpha^2$ & Q1 \\
Q1$^*_{p=\alpha^2}$ & $2\tilde{v}v(\tilde{u}-u)=\alpha( \tilde{v}^2 v^2+1)$ & H3$_{\delta=0}$\\
Q1$^*_{p=\alpha}$ & $ 2(\tilde{v}-v)(\tilde{u}-u)=(\tilde{v}-v)^2+\alpha$ & Q1$_{\delta=0}$\\
\hline
\end{tabular}
\caption{B\"acklund transformations connecting equations from the multi-quadratic class to the multi-affine (ABS) class.
Details of implementing the transformations are explained in Section \ref{BTex}.
Transformations are given up to composition with point symmetries of the equations in $u$ and $v$.
For completeness we include in the second part of the table some transformations that can be obtained from those in the first part by composition with (non-autonomous) point transformations.
The transformation connecting H2$^*$ to H1 was given originally in \cite{AdVeQ} up to composition with point symmetries.
Methods to obtain the transformations are described in the text, first by non-symmetric degeneration of the auto-B\"acklund transformation in Section \ref{nsd}, second by taking advantage of a natural connection with Yang-Baxter maps in Section \ref{idolons}, and third in Section \ref{quadlin} by exploiting the fact that the transformations, like the equations themselves, can be characterised by discriminant properties of the defining polynomial.
}\label{bts}
\end{table}
To explain precisely the meaning of the entries in Table \ref{bts} we give here an explicit example, specifically the second entry of the table.
Consider the coupled system of equations
\begin{equation}
(v-\tilde{v})(uv-\tilde{u}\tilde{v})+\alpha v \tilde{v} = 0, \quad (v-\hat{v})(uv-\hat{u}\hat{v})+\beta v\hat{v} = 0,\label{bt}
\end{equation}
which involve two functions $u=u(n,m)$ and $v=v(n,m)$, where as usual $n,m\in\mathbb{Z}$, $\tilde{u}=u(n+1,m)$ and $\hat{u}=u(n,m+1)$, etc.

For a fixed function $v=v(n,m)$ the equations (\ref{bt}) are coupled discrete Riccati equations for $u$, which are compatible if $v$ satisfies Q1$_{\delta=0}$.
Due this choice of $v$ the function $u=u(n,m)$ that emerges as the solution of (\ref{bt}) then satisfies equation Q2$^*$, that is (\ref{QQ2}), with parameter associations $p=\alpha$ and $q=\beta$ (notice that this parameter association is different from the one in (\ref{pa})).
For brevity the second equation in (\ref{bt}) and the parameter association between $q$ and $\beta$ are omitted from Table \ref{bts} because they can be inferred from the first equation in (\ref{bt}) and the association between $p$ and $\alpha$ that have been listed.

The transformation defined by (\ref{bt}) from $v$ to $u$ is therefore a quite standard discrete-Riccati type of B\"acklund transformation, see for instance \cite{NC,BS,James}.
However the system (\ref{bt}) also defines an inverse transformation from $u$ to $v$ which is less standard because the equations for $v$ are not of Riccati type.
Similar to verifying the consistency property of the multi-quadratic models (\ref{QQ4})--(\ref{QH2}) (cf. Section \ref{MDC}), it is convenient to handle this system by introducing auxiliary variables.
Specifically for the example (\ref{bt}) the auxiliary variables enter through the edge relations
\begin{equation}
\eqalign{
\sigma_1^2 = \frac{1}{4}\left(u^2+\tilde{u}^2+\alpha^2\right)-\frac{1}{2}\left(u\tilde{u}+\tilde{u}\alpha+\alpha u\right),\\
\sigma_2^2 = \frac{1}{4}\left(u^2+\hat{u}^2+\beta^2\right)-\frac{1}{2}\left(u\hat{u}+\hat{u}\beta+\beta u\right),
}
\end{equation}
so in fact they are the Q2$^*$ auxiliary variables.
They allow (\ref{bt}) to be re-written as
\begin{equation}
2\tilde{u}\tilde{v}=(u+\tilde{u}-\alpha-2\sigma_1)v, \quad 2\hat{u}\hat{v}=(u+\hat{u}-\beta-2\sigma_2)v, \label{bt2}
\end{equation}
which have degree-one polynomial dependence in $v$, $\tilde{v}$ and $\hat{v}$.
Implementation of the transformation from $u$ to $v$ therefore requires a solution $(u,\sigma_1,\sigma_2)$ of the single-valued reformulation of Q2$^*$ (cf. Table \ref{svms}).
From such solution the system (\ref{bt2}) determines the function $v$ satisfying Q1$_{\delta=0}$.

Similar to the example we have focused on in this section, all transformations listed in Table \ref{bts} are standard Riccati-type systems for $u$.
The inverse transformation to obtain $v$ is always more involved, reducing to a Riccati type system for $v$ only after the introduction of auxiliary variables associated with the solution of the multi-quadratic quad equation for $u$.

\subsection{Obtaining the transformations}\label{nsd}
It is rare to have a systematic method to obtain non-local transformations of B\"acklund or Miura type between a given pair of equations.
The transformations obtained here do however fit into a general framework, specifically they are defined in terms of polynomials which are characterised by their discriminants, a framework which is therefore consistent with the main theme of this paper.
We exploit this general point of view in Section \ref{quadlin} however, we instead use more direct methods to obtain most of the transformations listed in Table \ref{bts}.
A method involving {\it non-symmetric degeneration} of the auto-B\"acklund transformation is explained in this section, transformations obtained through a connection with Yang-Baxter maps will be explained in detail in the following section (Section \ref{idolons}).

There is a method to obtain B\"acklund transformations between distinct quad equations that was developed in \cite{James}, it is constructive insofar as it takes as a starting point the already obtained equations.
It relies on the natural auto-B\"acklund transformation, which for the equations in question is inherent from the multidimensional consistency, and exploits this in conjunction with the hierarchical relationships between equations.
Specifically we seek to connect an equation to its degenerate counterpart by making a non-symmetric degeneration of the auto-B\"acklund transformation.

The principal example where the non-symmetric degeneration technique has been used is to obtain the second transformation in Table \ref{bts} (which was also the example considered in Section \ref{BTex}).
We exploit the fact that the substitution
\begin{equation}
u=\frac{v}{\epsilon(1+v)}
\end{equation}
into equation Q2$^*$ yields equation Q1$_{\delta=0}$ to first order as $\epsilon \longrightarrow 0$.
The natural auto-B\"acklund transformation for Q2$^*$ (\ref{QQ2}) is as follows
\begin{equation}
\mathcal{Q}_{p,r}(u,\tilde{u},v,\tilde{v})=0, \quad \mathcal{Q}_{q,r}(u,\hat{u},v,\hat{v})=0,
\end{equation}
where $\mathcal{Q}_{p,q}$ is the defining polynomial of this equation, and the transformation connects a solution $u=u(n,m)$ to another solution of the same equation $v=v(n,m)$.
Applying the degeneration procedure to solution $v$ and judiciously choosing the B\"acklund parameter we are led to write
\begin{equation}
\fl \mathcal{Q}_{p,r}(u,\tilde{u},rv/(1+v),r\tilde{v}/(1+\tilde{v}))=0, \quad \mathcal{Q}_{q,r}(u,\hat{u},rv/(1+v),r\hat{v}/(1+\hat{v}))=0,
\end{equation}
which at leading order as $r=1/\epsilon\longrightarrow \infty$ yields exactly the desired non-auto B\"acklund transformation (\ref{bt}).

\subsection{From Yang-Baxter maps to B\"acklund transformations}
\label{idolons}
The Yang-Baxter maps given in \cite{ABSf}, when suitably interpreted as a system of equations for functions defined on edges of the lattice, are naturally connected with multi-affine quad equations from the ABS list \cite{ABS}, in particular a potential function for the edge variables is governed by a quad equation.
A systematic method to obtain the Yang-Baxter system on edge variables starting from the multi-affine equations from the ABS list was given in \cite{Tasos}.
Developments reported in \cite{KaNie,KaNie3} go further, but in the opposite direction, showing that also non-multi-affine multidimensionally consistent quad equations can emerge as potential for the Yang-Baxter systems.
And furthermore, that two different quad equations emerging in this way from the same Yang-Baxter system, as we shall see in this section, can immediately be connected through a B\"acklund-type transformation.

Of particular relevance here are the models that have been considered in \cite{KaNie5} alongside the present work.
This section is devoted to recalling the relevant models from \cite{ABSf} and giving details of the theory developed in \cite{KaNie,KaNie3,KaNie5} which, combined with the methods developed in this paper, can be used to obtain many of the transformations in Table \ref{bts}.

The starting point is a system of equations for two variables, say $s$ and $t$, assigned to the edges of $\mathbb{Z}^2$ oriented in the $n$ and $m$ directions respectively (similar to $\sigma_1$ and $\sigma_2$ introduced in Section \ref{SVS}, cf. Figure \ref{quadpic}).
The particular systems relevant here are as follows.

{\noindent $F_I$:}
\begin{equation}
\label{A2I}
\eqalign{
\hat{s} = t\frac{\alpha (1-{\beta}^2) s - \beta (1-{\alpha}^2) t-{\alpha}^2+{\beta}^2}{\beta (1-{\alpha}^2) s - \alpha (1-{\beta}^2) t+({\alpha}^2-{\beta}^2)st}, \\
\tilde{t} = s\frac{\beta (1-{\alpha}^2) t - \alpha (1-{\beta}^2) s-{\beta}^2+{\alpha}^2}{\alpha (1-{\beta}^2) t - \beta (1-{\alpha}^2) s+({\beta}^2-{\alpha}^2)st},
}
\end{equation}
$F_{II}$:
\begin{equation}
\label{II}
\eqalign{
\hat{s}= t\frac{\alpha s-\beta t-\delta(\alpha^2-\beta^2)}{\beta s-\alpha t},\\
\tilde{t}= s\frac{\alpha s-\beta t-\delta(\alpha^2-\beta^2)}{\beta s-\alpha t},
}
\end{equation}
$F_{III}$:
\begin{equation}
\label{III}
\eqalign{
\hat{s}= t \frac{\alpha s-\beta t}{\beta s-\alpha t},\\
\tilde{t}= s \frac{\alpha s-\beta t}{\beta s-\alpha t},
}
\end{equation}
$F_{V}$:
\begin{equation}
\label{V}
\eqalign{
\hat{s}=t+\frac{\alpha-\beta}{s-t},\\
\tilde{t}=s+\frac{\alpha-\beta}{s-t},
}
\end{equation}
which are nothing but the quadrirational Yang-Baxter maps presented in \cite{ABSf} suitably interpreted as an equation on the lattice (we have omitted model $F_{IV}$ because it is not used here).
The relevant feature of systems (\ref{A2I})--(\ref{V}) is that they admit a three-parameter family of potentials, we denote the potential here by $f$.
The potentials corresponding to these models, which are derived in \cite{KaNie5}, are as follows:

\begin{equation}
\label{A2p}
\fl (F_{I})\quad
\begin{array}{l}
\tilde{f}+f = A\ln(s)+B\ln(s-\alpha)+C\ln(\alpha s-1)+\frac{1}{2}(B+C)\ln({\alpha}^2-1),\\
\hat{f}+f = A\ln(t)+B\ln(t-\beta)+C\ln(\beta t-1)+\frac{1}{2}(B+C)\ln({\beta}^2-1),
\end{array}
\end{equation}
\begin{equation}
\fl (F_{II})\quad
\begin{array}{l}
\tilde{f}+f=A\alpha (2s-\delta\alpha)+B \ln(s) +C \ln (s-\delta\alpha),\\
\hat{f}+f=A\beta (2t-\delta\beta)+B \ln(t) +C \ln (t-\delta\beta),
\end{array}
\end{equation}
\begin{equation}
\fl (F_{III})\quad
\begin{array}{l}
\tilde{f}+f=A\ln(s)+\alpha B s+\displaystyle {C}/{s},\\
\hat{f}+f=A\ln(t)+\beta B t+\displaystyle {C}/{t},
\end{array}
\end{equation}
\begin{equation}
\fl (F_{V})\quad
\begin{array}{l}
\tilde{f}+f=As+B(s^2+\alpha)+C(s^3+3 \alpha s),\\
\hat{f}+f=At+B(t^2+\beta)+C(t^3+3 \beta t),
\end{array}
\label{Vp}
\end{equation}
The parameters $A$, $B$ and $C$ may be chosen freely in each case.

For the purpose of illustration we focus on the system (\ref{A2I}).
In particular writing $f=\ln(v)$ for the potential corresponding to the choice of the parameters $A=1$, $B=C=0$ in (\ref{A2p}) we find
\begin{equation}
\label{vv}
\tilde{v}v = s, \quad \hat{v}v = t,
\end{equation}
whilst writing $f=\ln(u)$ for the potential corresponding to the choice of the parameters $A=-1$, $B=C=1$, (\ref{A2p}) becomes
\begin{equation}
\label{uu}
\tilde{u}u =\frac{(s-\alpha)(\alpha s-1)}{s({\alpha}^2-1)}, \quad
\hat{u}u =\frac{(t-\beta)(\beta t-1)}{t({\beta}^2-1)}.
\end{equation}
Eliminating $s$ and $t$ from (\ref{A2I}) by means of (\ref{vv}) we find that $v$ satisfies equation A2 (\ref{A2}), whilst eliminating $s$ and $t$ from (\ref{A2I}) using (\ref{uu}) we find that variable $u$ satisfies equation A2$^*$ (\ref{QA2}) with the parameter associations
\[p =2\frac{\alpha^2+1}{\alpha^2-1},\quad q =2\frac{\beta^2+1}{\beta^2-1}.\]
Equations governing the potential $f$ for different choices of the parameters $A$, $B$ and $C$ are referred to in \cite{KaNie} as {\em idolons} of the underlying system governing $s$ and $t$, in particular we have shown that equations A2 and A2$^*$ are idolons of (\ref{A2I}).

A B\"acklund transformation can be obtained by composing these relations, that is by eliminating $s$ and $t$ from formulas (\ref{vv}) and (\ref{uu}),
\begin{equation}
\label{BTI}
\tilde{u}u =\frac{(\tilde{v}v-\alpha)(\alpha \tilde{v}v-1)}{\tilde{v}v({\alpha}^2-1)}, \quad
\hat{u}u =\frac{(\hat{v}v-\beta)(\beta \hat{v}v-1)}{\hat{v}v({\beta}^2-1)}.
\end{equation}
This is the B\"acklund transformation between $A2$ and $A2^*$ that appears in the first part of Table \ref{bts}.
The second transformation between these models can be obtained similarly after observing that the potential corresponding to choice of parameters $A=C=-1$, $B=1$ is also governed by model $A2^*$.

Table \ref{mtab} contains the data required to construct, by the same procedure, all but the first two transformations in the first part of Table \ref{bts}.
\begin{table}
\begin{tabular}{llll}
System & Parameters ($A$,$B$,$C$) & Potential & Equation\\
\hline
$F_{I}$ & (-1,1,1) & $f=\ln(u)$ & A2$^*_{p=2(\alpha^2+1)/(\alpha^2-1)}$\\
        & (-1,1,-1) & $f=\ln(u)$ & A2$^*_{p=2\alpha^2-1}$\\
        & (1,0,0) & $f=\ln(v)$ & A2\\
\hline
$F_{II}$ & (0,1,1) & $f=\ln(u)$ & H3$^*_{p=4/\alpha^2}$\\
         & (0,0,1) & $f=\ln(v)$ & H3\\
         & (1,0,0) & $f=v$ & A1$_{\alpha\rightarrow \alpha^2}$\\
\hline
$F_{III}$ & (0,$\textstyle \frac{1}{2}$,$\textstyle \frac{1}{2}$) & $f=u$ & A1$^*_{p=\alpha^2}$\\
          & (1,0,0) & $f=\ln(v)$ & H3$_{\delta=0}$\\
          & (0,1,0) & $f=v$ & A1$_{\delta=0,\alpha\rightarrow \alpha^2}$\\
\hline
$F_{V}$ & (0,1,0) & $f=u$ & H2$^*_{p=-\alpha}$\\
        & (1,0,0) & $f=v$ & H1\\
\end{tabular}
\caption{
Equations governing different potentials of the systems (\ref{A2I})--(\ref{V}).
The potentials are introduced through equations (\ref{A2p})--(\ref{Vp}) with the indicated choice of parameters $(A,B,C)$.
It is explained in the text how to construct a transformation between equations (idolons) that govern different potentials of the same system.
Parameter associations in the listed equations are given connecting $\alpha$ and $p$ or transforming $\alpha$, similar associations connecting $\beta$ and $q$ or transforming $\beta$ are implicit, they are omitted from the table for brevity.
}
\label{mtab}
\end{table}

\subsection{Discriminant properties of the transformations}
\label{quadlin}
All transformations listed in Table \ref{bts} are in the same class, they involve equations on lattice edges and the defining polynomial $\mathcal{B}=\mathcal{B}(u,\tilde{u},v,\tilde{v})$ is degree-one in each of $u$ and $\tilde{u}$, and degree two in each of $v$ and $\tilde{v}$.
This defining polynomial also has the following discriminant properties,
\begin{equation}
\eqalign{
(\partial_{u}\mathcal{B})(\partial_{\tilde{u}}\mathcal{B})-(\partial_u\partial_{\tilde{u}} \mathcal{B})\mathcal{B} \propto \mu{\tilde{\mu}} h(v,\tilde{v}),\\
(\partial_{\tilde{v}}\mathcal{B})^2-2({\partial^2_{\tilde{v}}} \mathcal{B})\mathcal{B} \propto \eta^2 h^*(u,\tilde{u}),
}\label{ddd}
\end{equation}
where $\mu$ and $\eta$ are polynomials in $v$, and $h$, $h^*$ are the edge biquadratics associated with the two equations connected by the B\"acklund transformation.
Specifically $h$ coincides with $H_1$ associated with the multi-affine equation in $v$ through the generalised discriminant (\ref{gendisc}), while $h^*$ coincides with $H_1$ associated with the multi-quadratic equation in $u$ through the discriminant formula (\ref{disc}).
This discriminant property is an important feature of the transformations that combines asymmetrically the underlying discriminant characterisations of the multi-quadratic models (\ref{QQ3})--(\ref{QH2}) and the multi-affine ABS equations (\ref{Q3})--(\ref{H1}).
In particular it may be used as a basis for their construction.

One approach to such construction is offered by recognising the biquadratics themselves (\ref{q4})--(\ref{h2}) can take natural discriminant forms, as the primary example we recognise that (\ref{q3}) is proportional to
\[ (\tilde{u}+u+2\delta \alpha^2)^2-4[\alpha(\tilde{u}+\delta)][\alpha(u+\delta)], \qquad 2\alpha^2=p+1. \]
It is easily seen that this expression emerges as the discriminant, with respect to $\tilde{v}$ or $v$, of the expression
\[ [\alpha(\tilde{u}+\delta)]\tilde{v}^2 - [\tilde{u}+u+2\delta\alpha^2]\tilde{v}v + [\alpha(u+\delta)]v^2, \]
which itself coincides with the polynomial defining the first transformation in Table \ref{bts}.
The precise choice of this expression is not unique, and in particular the biquadratic obtained from the generalised discriminant in $u$ and $\tilde{u}$ emerges a posteriori after choosing the expression on the basis that the required discriminant property in $v$ and $\tilde{v}$ is obtained.

We remark that the classification of polynomials $\mathcal{B}$ with the discriminant properties (\ref{ddd}) is interesting from the point of view of exhausting transformations in the same class as those listed in Table \ref{bts} (such task is similar to the one solved for multi-affine quad equations in \cite{boll}).

\section{Discussion}\label{discussion}
The discriminant factorisation property, as laid out in Section \ref{dfh}, allows reformulation of the multi-quadratic quad equation as a system that defines single-valued evolution from initial data.
Our main result is the list of models constructed on the basis of this hypothesis in Section \ref{list}, it contains all previously known integrable multi-quadratic quad equations as well as a substantial number of new models that exhibit the same integrability features.
The nature of the relationship between the discriminant factorisation property and the integrability is therefore an important question.
Here we have made additional assumptions beyond the discriminant factorisation hypothesis, therefore although our investigations are suggestive, they leave open the problem of determining whether this property is sufficient for integrability in this class of models.

Beyond elliptic (and hyperelliptic) functions, degree-two equations studied systematically from the point of view of integrability seem to be relatively rare.
An isolated precedent exists within the framework of Painlev\'e analysis for ordinary differential equations.
A list of degree-two counterparts of the Painlev\'e equations was obtained by Chazy \cite{chsy}, loosely speaking these are second-order ordinary differential equations with the Painlev\'e property that are quadratic in the second derivative term.
The difficult step of rigorously classifying this class of equations was made by Cosgrove and Scoufis \cite{CoSc}.
In that setting one of the principal features is that the higher degree models do not define new transcendents (see \cite{Cosg}), in particular they are solvable in terms of the degree-one Painlev\'e equations.
Here the primary model, namely the equation that we identify as a multi-quadratic counterpart of Q4, is the only new model we have not yet connected back to an equation from the multi-affine class.
The parallels between integrable quad-equations and the Painlev\'e-type equations suggest that a B\"acklund-type transformation establishing such connection should exist.
Verifying or falsifying this is therefore an important open problem.

\ack
J.A is supported by the Australian Research Council, Discovery Grant DP 110104151.
M.N is grateful for financial support from Australian Research Council Discovery Grant DP 0985615, which enabled his stay at the University of Sydney, during which the present surveys started.

\section*{References}
\bibliographystyle{unsrt}

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TUCSON INTL is the most important airport of Tucson, Arizona, United_States. It is modern and one of the largest airport of the North America. TUCSON INTL is important for people and goverment of United States.


text:
The 1972 Old Dominion 500 was a NASCAR Winston Cup Series racing event that was held on September 24, 1972, at Martinsville Speedway in Martinsville, Virginia.

The race car drivers still had to commute to the races using the same stock cars that competed in a typical weekend's race through a policy of homologation (and under their own power). This policy was in effect until roughly 1975. By 1980, NASCAR had completely stopped tracking the year model of all the vehicles and most teams did not take stock cars to the track under their own power anymore.

Background
Martinsville Speedway is one of five short tracks to hold NASCAR races. The standard track at Martinsville Speedway is a four-turn short track oval that is  long. The track's turns are banked at eleven degrees, while the front stretch, the location of the finish line, is banked at zero degrees. The back stretch also has a zero degree banking.

Race report
Five hundred laps were done on a paved track spanning  in a time of three hours and forty-five minutes. Richard Petty would defeat Bobby Allison by a time of six seconds in front of 31,000 live audience members; making this race the closest Bobby Allison ever came to winning at Martinsville. Although Cecil Gordon had a competitive finish of 10th place during this event, his lackluster performance during the 1972 NASCAR Winston Cup Series season would eventually cost him a shot at winning the championship.

Exactly ten minutes from the race along with a flyover by the local military jets were used in the 1973 American film The Last American Hero. While Richard Petty and Bobby Allison got filmed most of the time, a moment involving Ray Hendrick's #2 vehicle crashing on lap 311 was also a notable event during the filming.

Allison would earn the pole position with a speed of . The average speed of the race, however, would be a mere . Eight cautions slowed the race for 58 laps in the entire race. All thirty-six of the qualifying drivers on the grid were male and born somewhere in the United States of America. Originally, 56 drivers auditioned for this race; including Joe Frasson, who was one of the ones who failed to qualify. Bill Shirey would be the unfortunate last-place finisher due to an overheating issue on lap 19. Fred Lorenzen would retire from NASCAR after this race. This was also LeeRoy Yarbrough's last race, never to be heard from again until his death. He ran most of the last two seasons in non-competitive cars yet he was one of the most fearless racers ever.

Richard Petty would go on to win the championship shortly after this race was over while Bobby Allison would find a new employer named Ralph Moody for the 1973 NASCAR Winston Cup Series season. The total purse of this racing event was $46,650 ($ when considering inflation); the first-place finisher received $7,350 ($ when considering inflation) while the last-place finisher received $425 ($ when considering inflation).

Six of NASCAR's most notable crew chiefs were on attendance for this race; including Harry Hyde, Dale Inman, Vic Ballard and Lee Gordon.

Technological concerns
Improvements in tire and engine technology in the early 1970s had made NASCAR Cup Series vehicles overpowered compared to the limited space that Martinsville Speedway had for breathing space. Further technological advancements by 2013 have caused further concern for driver safety at Martinsville; indicating that the Monster Energy NASCAR Cup Series may have to use restrictor plates in Martinsville if tire/engine technology keeps its current pace of development.

Qualifying

Finishing order
Section reference:

Timeline
Section reference:
 Start: Bobby Allison was leading the starting grid as the green flag was waved.
 Lap 19: Bill Shirey and Bobby Isaac jointly overheated their vehicles.
 Lap 77: Larry Smith blew his vehicle's engine.
 Lap 88: Richard Childress caused the rear end of his vehicle to become dangerously loose.
 Lap 95: Frank Warren caused the rear end of his vehicle to become dangerously loose.
 Lap 105: Bill Dennis caused terminal vehicle damage.
 Lap 108: LeeRoy Yarbrough caused terminal vehicle damage; Dave Marcis caused the rear end of his vehicle to become dangerously loose.
 Lap 132: Richard Petty took over the lead from Bobby Allison.
 Lap 134: Dean Dalton overheated his vehicle.
 Lap 148: Bobby Allison took over the lead from Richard Petty.
 Lap 216: Fred Lorenzen blew his vehicle's engine.
 Lap 221: David Pearson took over the lead from Bobby Allison.
 Lap 222: Bobby Allison took over the lead from David Pearson.
 Lap 223: David Pearson took over the lead from Bobby Allison.
 Lap 225: Bobby Allison took over the lead from David Pearson.
 Lap 272: Tiny Lund caused the rear end of his vehicle to become dangerously loose.
 Lap 279: David Pearson took over the lead from Bobby Allison.
 Lap 280: Bobby Allison took over the lead from David Pearson.
 Lap 311: Ray Hedrick caused terminal vehicle damage.
 Lap 328: David Ray Boggs blew his vehicle's engine.
 Lap 369: Henley Gray overheated his vehicle.
 Lap 436: Richard Petty took over the lead from Bobby Allison.
 Lap 441: Bobby Allison took over the lead from Richard Petty.
 Lap 442: Richard Petty took over the lead from Bobby Allison.
 Lap 446: Bobby Allison took over the lead from Richard Petty.
 Lap 459: John Sears' vehicle developed problems with its transmission.
 Lap 462: Richard Petty took over the lead from Bobby Allison.
 Finish: Richard Petty was officially declared the winner of the race.

References

Old Dominion 500
Old Dominion 500
NASCAR races at Martinsville Speedway

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text:
Q: Generate PDF from rails in rest way I want to generate PDF in rails application. I had saw different options for doing this but
I want all data needed for PDF should be collected in a format (mostly XML) such that it can be utilized by my rails application for generating the reports as well as be consumed by other applications( Web Service)
 The data I have use for generating PDf it comes from different object and basically it contains images and graphs. Data volume is also big
I want to generate PDF with proper style and layout. Can anybody suggest me the best option I can use in my rails application to achieve above problem

A: Prawn or PDFKit seem to be the two most popular right now. Each has their ups and downs depending on what you are trying to do and the styles they want to apply. For PDFKit here is an article talking about how it integrates with Rails

A: If using an online/cloud service is an option, you might want to look at Docmosis.  It can generate PDFs using a rest interface and has at least one Ruby example.  Please note I work for the company that created Docmosis.

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Trump lies to military spouses.
Trump will lie to anybody, anywhere — even military spouses, during Military Spouse Appreciation Day, at the White House.
It takes a special kind of craven person to lie to the those married to U.S. military members about giving their spouses their first pay raise "in 10 years." But that's exactly what Trump did this week.
On Friday, MSNBC's Stephanie Ruhle denounced Trump's pathological behavior.
"For those who say we don't call the White House out, this is 100 percent a lie told by our president," Ruhle announced.
The ugly incident unfolded inside the White House during a reception that featured a group of virtually all-white women, even though nearly half of today's U.S. military is made up of non-white members. There, Trump lied repeatedly about how he had single-handedly improved pay for service members.
Referencing his recent signing of the National Defense Authorization Act for fiscal year 2018, Trump bragged, "I know you don't care about this, but that also includes raises for our military — first time in 10 years."
Moments later he insisted, "That is why, earlier this year, I was proud to sign that big pay raise that I've already spoken about. And I am proud of it. And I guess there will be others, too. Would you like one sooner, or do you want to wait another 10 years?"
The clear implication of the hideous lie was that during President Barack Obama's two terms in office, U.S. military members were ignored and had to suffer through a decade of static pay.
But that's not how any of this works.
By law, the military receives at least a 1 percent pay increase every year. And that has been true for the last three decades.
The most recent pay increase will be 2.4 percent. During 2009 and 2010, when Obama was president, military members received 3.9 and 3.4 percent raises, respectively.
Trump has already logged more than 3,000 lies since taking office, according to the Washington Post Fact Checker, and he clearly couldn't care less about adding to that tally. Even if that means manufacturing falsehoods to try to impress military spouses.
As Ruhle said, "No, thank you, Mr. President, for the lie."


text:
Dave Ursillo
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A Note and a Thought on Tragedy, Human Nature, and Staying 'Boston Strong'
Many things have changed since 9/11, the tragedy of all tragedies for us Americans, which brought out both the best and the worst in us: pride and jingoism, strength and anger, determination and discrimination, unity and bloodlust.
We've learned much since then–more about the world, more about ourselves. We have seen the sins of our own ways, and seem to recall it more quickly than what we've done right. We do still understand that things are not so black and white. Evil is not just evil and good is not just good. The good can do evil. And evil is just human, like us.
Life, we see, is truly a frustrating thing for the ignorant, for the ideologues, for the fundamentalists.
If so much has changed since 9/11, one lingering consequence is the game of public opinion. At first, there is perfect unity: those affected are shown compassion and understanding by those unaffected, because those unaffected feel themselves in those harmed. But with time, those unaffected, quite naturally, see their compassion wane. A tragedy that could have claimed anyone has instead become a simple fact of history–and one, at that, that did not claim them.
But those affected by the tragedy are still ripe with emotion, because they are human and that is our human nature. The affected still recall the victims whose lives have been shattered. These are their neighbors, their friends, the community members who share the same grocer and root on the same ball clubs. We are them, they are us, and we won't soon forget it. This tragedy is not a simple fact of history that spared our lives, but an unchosen event that lives on still. And, our show of "strength" is not a show at all, but an undying testament to them.
We are the ones who don t-shirts with slogans, and bear affirmations from faith in human resilience. We stand a flag in the sand that reads "Pride" to remember what happened so that it may not happen again.
Then there are those indirectly affected by the same tragedy, or who were not affected at all.
Without the direct emotional ties to the tragedy, the others resort to logic, to forethought, to the memory of our collective wrongdoing from a time gone by when unwieldy emotion blinded our collective moral compass and our good judgment–when we thought we were good and they were evil, and found that our "goodness" could lead us to more wrong than right.
And in the space between these two parties, who are not enemies but themselves countrymen, the affected who still bleed and emote are dubbed "exploiters" of tragedy–weak-minded ones who linger in their tragedy, in spite of the fact that the day becomes more ancient by the second as the clock ticks ever forward.
And the affected, in turn, feeling offended and scolded, call those non-affected–who do not bleed or cry still for the loss and for the pain and for the suffering experienced–high-minded and callous, dark-hearted and unaware, insensitive and inhumane.
Neither party is right. Neither is wrong. I write this piece to say this to both groups, with a hope of uniting them:
Suffering is not a competition.
An affirmation for one does not mean the exclusion of another.
Rightness or wrongness is not at stake–what we witness when parties split into warring camps of opinion reflects the proximity to tragedy, and when there is a tragedy, all ought to side with compassion and understanding, not angry fists formed from insensitive words.
Let those unaffected, who ponder why the affected still "linger" upon the tragedy, remember that when any group of our brethren bleed, those by them will huddle up to protect them. That is our nature. Strike at the bleeding with words or fists, and that huddling pack will be incited to violent response.
Let those affected, who may now feel threatened or abandoned by the unaffected, remember that we are all fighting our own battles and our own tragedies. Let us remember, as well, to show the compassion and care for those affected by their own tragedies, when we suddenly become "the unaffected" in that game of public opinion.
We need feel neither shame nor fear for meeting the suffering of others with compassion. This emotion is, in fact, a most logical choice: a steadfast and strong one that is a testament to the free will of humans and the ability to choose their thoughts and their words with deep care, consideration, and mindfulness.
We tend to think and speak kindly, especially to neighbors and friends, when we recall what pain feels like.
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Space: the debris frontier
To mark the International Day of Human Space Flight, we celebrate the significance of mankind's space era, reflect on the potential risks and how standards can help mitigate them.
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Since the launch of the first satellite in 1957, the number of orbital objects has skyrocketed. Around the world, amateur radio enthusiasts tuned in to hear Sputnik 1's plaintive beeps – a sound that heralded a new technological era. Mankind's space era and the space race had begun. In 1961, the first manned flight breached the confines of our atmosphere. Since then, private companies have joined the space race.
But in the 65 years since Sputnik took its first lonely spin around Earth, the region of space closest to us has become rather more crowded. Today, on International Day of Human Space Flight, it is important to remember that space is no longer inaccessible. The great void of our stratosphere and beyond have never been closer – and that comes with risks.
A crowded space
Today, around 7 500 satellites are operating at altitudes below 2 000 km – the upper limit for low-Earth orbit (LEO). More than a third of those were launched in the past two years, and there are many more to come. Proposed launches by private companies such as SpaceX and Amazon would swell the number of satellites in LEO by more than 45 000.
But satellites represent only a small fraction of the objects whirling around our planet. Orbital debris, or "space junk", is becoming an increasingly urgent problem. NASA is tracking 27 000 pieces of orbital debris larger than a tennis ball, from discarded launch vehicles and parts of old spacecraft to tools and jettisoned bags of garbage. Added to those are an estimated one million pieces of debris larger than a marble, and a further 330 million between 1 mm and 1 cm in size. All these objects are travelling at 25 000 km/h, or 7 km/s. At that speed, even tiny pieces of debris can do a significant amount of damage.
Any collisions or explosions that do occur will only add to the problem. When Russia successfully tested an anti-satellite missile in November 2021, it created 1 500 more pieces of trackable debris as well as hundreds of thousands of smaller pieces. In the immediate aftermath, crew members aboard the International Space Station (ISS) had to take refuge in their docking capsules in case they needed to make a quick getaway. The ISS itself has had to move out of the way of space debris 25 times since 1999.
The Kessler effect
What happens if we keep on adding to the pile of space junk? In 1978 – when the space race was in full swing – a NASA scientist called Donald Kessler came up with an answer. He proposed that once the number of objects in LEO reaches a tipping point, the debris created by a collision could start a chain reaction of further collisions. Kessler argued that this would eventually create a cloud of debris encircling Earth, rendering the region both unusable and impassable.
Why is this a problem?
The greatest threat from orbital overcrowding is the damage it could cause to operational satellites. Those in LEO are used for imaging, surveying, and – increasingly – communications. This region is also becoming progressively more militarized. It hosts satellites used for surveillance and battlefield communications, which means any collision could have unpredictable geopolitical consequences.
Orbital debris also poses a threat to missions beyond LEO. This risk is small at present, but – like the volume of debris itself – it's growing. In the worst-case scenario, space junk could eventually box us in entirely and close off any future exploration of our solar system (see Box).
Standards to help
Last year, the G7 countries pledged to tackle the growing problem of space debris. In a joint statement, leaders said they "recognize the importance of developing common standards, best practices and guidelines related to sustainable space operations", and called on organizations, including ISO, to help "preserve the space environment for future generations".
The work being done by ISO in this field – in particular, by technical committee ISO/TC 20, which focuses on aircraft and space vehicles – contributes in two main ways; firstly by reducing the amount of debris that finds its way into orbit, and secondly by mitigating the risks posed by the increasing congestion of near-Earth space.
ISO 24113 focuses on the former. It sets out requirements for the design and operation of all spacecraft and launch vehicles that are either launched into, or passing through, near-Earth orbit that ensure they leave as little debris as possible behind during – and after – their useful lives. When a satellite runs out of fuel, for example, it becomes just another big chunk of debris. ISO 24113 requires satellite designers and manufacturers to take this into account – for example by building satellites that can either be refuelled or removed from orbit at the end of their life to burn up safely in the upper atmosphere.
A safe space
Standards also play an important role in reducing the risk of collisions. LEO isn't just becoming more crowded; it's being populated by an increasing number of players. As well as the world's space agencies, more than a dozen countries have successfully launched satellites and a growing number of private companies are also now jostling for space. To minimize the risk of collisions between spacecraft, and to ensure safe passage for rockets bound for, or through, LEO, a standardized communication protocol is essential.
Standards play an important role in reducing the risk of collisions.
That's where ISO 26900 comes in. It specifies message formats that spacecraft operators can use to communicate quickly and clearly with each other – whether that's sharing pre-flight plans or alerting other operators about potential collisions in good time so they can manoeuvre their spacecraft out of the way.
Last year saw a record 145 orbital launches take off around the world. Near-Earth space is only going to become more important to the commercial, scientific and military interests of countries and corporations – and standards like these are one way we can ensure that, despite the growing congestion and the risks that poses, everyone has the space they need to operate safely.
Este artículo ha sido traducido por nuestro miembro español y se puede descargar en versión PDF.
本文已由我们的中国成员翻译,可下载中文PDF版本。
The great space clean-up
Space Sustainable Rating references ISO standards to help reduce the amount of debris in our galaxy.
Space debris mitigation requirements
Space data and information transfer systems
Orbit data messages
Aircraft and space vehicles
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The Border NewsVictoria Hits New Covid Record Amid Concerns Over Testing Changes
Victoria Hits New Covid Record Amid Concerns Over Testing Changes
'The horse has bolted'
Victoria reported a rise in cases with 21,997 new infections on Thursday and six more Covid-related deaths.
It marks a new state record as Victoria's infections exceed 20,000 for the first time since the pandemic started.
There are currently 61,120 active coronavirus cases, with 631 people in hospital, while 51 of those are in ICU and 22 are on ventilator.
Another 49 people are in intensive care with Covid, but the health Department have confirmed their infections are no longer considered active.
The rapid surge in presentations and hospitalisations is an increase up from 591 Covid patients in hospital the previous day.
The new cases were detected from 64,861 swabs collected on Wednesday, while 20,078 vaccine doses were administered at state-run hubs.
Currently, more than 93 per cent of Victorians aged over 12 are now fully vaccinated, while 12 per cent have received their third dose.
It comes as health experts warn case numbers will less likely reflect the true number of Covid community cases after Wednesday's national cabinet meeting, where the federal government slashed testing requirements in a bid to ease pressure on testing sites and procedures.
The new rules, now stipulate that people who deliver a positive rapid antigen test (RAT) will no longer need to confirm their diagnosis with a PCR test.
The critical change from PC to rapid testing does however present a major concern for many medical experts, who say cases numbers may increasingly become less representative of state and territory's infection rates.
University of Melbourne epidemiologist Professor Tony Blakely said surveillance will no longer be accurate.
"The horse has bolted, this is the biggest policy failure so far in Australia," he told the Seven Network on Thursday.
"We also haven't thought about how you can load up that data to the surveillance system, so we won't get that in place in the next couple of weeks."
Prof Blakely hopes rapid tests sold in Australia would be equipped with a QR code for patients to upload their results.
Experts are calling for the government to roll out an online booking system for home tests sold in Australia to be equipped with a QR code for patients to upload their results, to more accurately reflect infection figures.
In the meantime, Prime Minster Scott Morrison has urged anybody who tests positive with an at-home kit to contact their GP so their result can be included in official daily tallies.
"That is always your first point of contact when it comes to managing your illness, if you are not in hospital. And your GP would be able to assist you through telehealth if you need to go and get further treatment in hospital," he said.
There is currently no way to register a positive rapid antigen test with health authorities, meaning those infections will not be included in daily case numbers.
Testing Changes
Fox FM


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Private sector to step in at Parks sites
Parks Canada will allow private contractors to take over tours this summer at the S.S. Klondike and Dredge No. 4. Yukon MP Ryan Leef made the announcement today in Whitehorse.
Jesse Winter
Mar. 8, 2013 9:00 a.m.
Parks Canada will allow private contractors to take over tours this summer at the S.S. Klondike and Dredge No. 4.
Yukon MP Ryan Leef made the announcement today in Whitehorse. It comes at the 11th hour for the two popular historic sites, with tourist season mere months away.
Parks will begin contacting 18 local businesses that expressed interest in running the tours, as well as seeking other proposals. The deadline for applications for a licence to run the tours is April 2. Parks plans to have the applications reviewed by April 15; the licences will be valid starting June 1.
"We've been working on this for just over a year and at every step of the way (federal environment minister) Peter Kent was available and responsive to us. Support from the territorial government helped us keep this issue alive in the federal mind and is what ultimately led to the solution," Leef said.
Leef also credited the Tourism Industry Association of the Yukon for its efforts earlier in the winter to contact businesses that might be interested in operating tours. This will likely help speed up the process, he said.
"TIA was an active and involved partner right from the get go, and the minister of tourism, Mike Nixon, was fantastic as well. I don't think a week went by that he wasn't communicating directly with us about what solutions we were looking for," Leef said.
"Ultimately, this is a Yukon solution to a Yukon challenge. Yukoners will finally get to decide how things are run," he said.
He acknowledged that timelines are now tight, with just over a month for a successful applicant to prepare to run the tours.
"The length of time and challenge has been directly related to the fact that this is national in scope and scale. I think we all would have liked that flick-of-the-switch response and immediate return in action when this occurred, but because it was national in scope and scale, that took some time," Leef said.
While Parks won't be running the tours itself anymore, Leef said it will work closely with the private operators to ensure that all the safety and visitor experience requirements are met. That also leaves private operators flexibility in how they meet those requirements, he said.
"They could go all out with costumes and everything, or they could just provide that safety from a simple guided tour," Leef said.
As well as establishing guided tours for the summer, Leef also said that Parks has committed to maintaining all the historical artifacts from the Dredge No. 4 in the Yukon.
"None of those artifacts will be leaving the Yukon. I know that was a big concern for some people, especially folks in Dawson who donated artifacts from their personal collections. They would not have been happy if their family items ended up being stored somewhere in Ottawa," Leef said.
In April last year, Parks Canada announced it was cutting guided tours from the S.S. Klondike and Dredge No. 4 as part of the federal government's 2012 budget cutbacks.
Local businesses worried that losing the popular tours would cause a drop in tourism revenue, especially for tour companies wanting to include the historic sites in their tour packages.
The tours had been run by Parks Canada staff for the last couple of years, but were managed by a private contractor up until 2005.
Contact Jesse Winter at
jessew@yukon-news.com
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Kaska condemn Supreme Court action


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Home News You don't know what you've got till it's gone
You don't know what you've got till it's gone
With internet speeds down to a crawl, students confront an essential question: Are we too reliant on the internet?
Jason Keller
When construction on I-25 and Highway 34 damaged fiber lines feeding internet to UNC, the university was forced to fall back on their secondary service to supply the campus with connectivity. While usable, the internet at UNC is still struggling to keep up with campus demands.
In an email blast sent out by the Office of Information and Management Technology, Chief Information Officer Bret Naber said that they are still exploring ways to add bandwidth to their backup solution, a process which has left many UNC students like undeclared freshman Maura Calahan frustrated. Calahan is a resident in Snyder Hall, and says that she's worried how the lack of internet will affect her academics.
"It'll definitely hurt my grades a little bit if I can't access the internet from my dorm," she said. "That could be an issue if I have a lot to get done."
In an era where technology is the backbone of society, so much so that the United Nations considers internet access a human right, students like Calahan have to confront life without being plugged in, a process she says is something she's still not used to.
"I think it's become the way that the world is. We're in a technological age where we all use it [the internet] to interconnect ourselves. It creates a more efficient society. I believe that it is very helpful in connecting us all," she said.
Calahan was not alone in her sentiments. As the days drag on since the initial damage on Aug. 20, other students chimed in, such as Larissa Daniels, a senior working towards a double major in anthropology and criminal justice. Daniels, like Calahan, is living in one of the residence halls on campus. In her case, Turner.
She's making do with her unlimited mobile data plan, but she says it's not enough to be able to follow through with her academic obligations.
"There's a book that I don't actually have, that my teacher uploaded the first two chapters on the internet. It took me about two hours to actually load both chapters up," Daniels said.
As a self-proclaimed 90s kid, Daniels remembers a time when the internet wasn't as available as it is today. She says that we're too reliant on the internet, but that's just something we're going to have to accept.
"I think we're too reliant, but at the same time it's such a big part of our lives now that we really can't do much about it, except maybe be more patient with it. It's something we're just going to have to get used to," she said.
Unlike Daniels or Calahan, Griffin Myers, an undeclared sophomore, had different thoughts on the matter.
"I never had a phone, so I couldn't really relate to people who were married to their phones. But when I got one, I fell victim to the same thing – I became married to the phone. So I think it's [the internet] a problem, but everyone has some sort of problem," Myers said.
Myers, who is thinking of pursuing a major in geology or computer science, said that while it's a problem for people who cannot connect to class resources, those who can't connect to Netflix need to "get over it."
"The problem of internet addiction is not black and white, because obviously we need it for some things, but people mostly use it as a distraction," he said. "Students take it for granted, professors take it for granted. The internet going down could definitely present a problem for classwork, but for what people use it for, entertainment, distraction, if it goes down, yeah, get over it."
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Though a lingering nip is still hanging around, there's something else in the air now too. Signs of spring abound— in the receding snow, the morning songs of birds, the longer days and the warming heat of the sun.
Though I posted a few blogs about how to get great winter photos, I most definitely didn't practise what I preached. Like the bear, I hibernate in the cold months, limiting my amateur shutterbug tendencies to loved ones and social events.
But now that spring is mightily pushing winter out of the way, I think it's time to start thinking about taking my camera outdoors. The rebirth and rejuvenation is quite frankly a blooming opportunity for some lovely pictures. The first signs of returning flora and fauna, the colours are the images of the season that beg for attention. The trick is in how to make the pictures you take as fresh as the subjects on which you're focussing.