text:
\section{Introduction}
\setcounter{equation}{0}

In previous work \cite{b01,b02,b03}, algorithmic methods
were presented to determine whether or not there exists an invertible
mapping of a given nonlinear partial differential equation (PDE) to
some linear PDE through computing the point or contact symmetries
admitted by the nonlinear PDE.
Moreover, it was also shown how to construct such an
invertible mapping, when one exists, from the admitted symmetries.

In this paper, we present an alternative algorithmic method for
mapping a nonlinear system of PDEs invertibly to some linear system of
PDEs by using admitted conservation laws of the nonlinear system.
Any linear system possesses an infinite set of conservation law
multipliers satisfying its adjoint system. This feature of linear
systems will be exploited to detect whether or not a given nonlinear
system can be linearized by an invertible transformation and, when
such a linearization mapping exists, to obtain the explicit form of
the linearizing transformation from an associated conservation law
identity.

In particular, using the determining equations for
conservation law multipliers,
one can find whether or not an invertible
linearization mapping will exist, and also find the adjoint of a target
linear system as well as the transformation of the independent variables
in the mapping whenever an invertible linearization is possible \cite{b04}.
Most importantly, the new
work presented in this paper will show that a conservation law
identity coming from the multiplier equations of an augmented system
consisting of the given nonlinear system and the adjoint of the linear
target system can be used to determine the
transformation of the dependent variables
(in addition to that of the independent variables)
in the invertible linearization mapping explicitly.
This work thus allows one to detect and construct
linearization mappings completely by the use of
algorithmic methods for obtaining multipliers of conservation laws
\cite{b05,b06,b07,b08,a}.

Our results generalize earlier computational work \cite{w}
in which a fully automatic algorithm was given using
the computer algebra program {\sc Crack/Conlaw} \cite{b14}
to detect and construct linearizations
in the case when the mapping only transforms the
dependent variables of the given nonlinear PDE system.
The work in \cite{w} also gave an extension of this algorithm to find
partial linearizations that consist of factoring out
a linear differential operator from all PDEs in a given system
(with the operator being of lower order than the highest order of derivatives
in the system).

To begin we summarize two well-known theorems
characterizing invertible mappings.
Our notation follows that in \cite{b11},
with the conventions that Latin indices $i,j,k$ will run $1$ to $n$
(labeling independent variables),
and Greek indices $\nu,\mu,\lambda$ and $\alpha,\beta,\gamma,\sigma,\tau$
will run $1$ to $M$ and $1$ to $m$
(labeling PDEs and dependent variables), respectively.
Throughout we use upper/lower case notation to distinguish
arbitrary functions $U,V$ from functions $u,v$ given by solutions of a PDE system.

Let ${\bf R}\{x,u\}$ denote a given $k$th-order nonlinear system of
$M$ PDEs with $n$ independent variables $x = (x_1 ,\ldots ,x_n )$ and
$m$ dependent variables $u = (u^1,\ldots ,u^m)$. In particular, let
the PDEs in ${\bf R}\{x,u\}$ be given by
\begin{equation}
\label{eq1}
G^\nu [u] = G^\nu (x,u,\partial_x u,\ldots ,\partial_x^ku) = 0,\quad \nu = 1,\ldots ,M,
\end{equation}
where $\partial_x^ju$ denotes $j$th-order partial derivatives of $u$ with respect to $x$.

Let ${\bf S}\{z,w\}$ denote a $k$th-order linear target system of $M$\ PDEs
with $n$ independent variables $z = (z_1 ,\ldots ,z_n )$ and $m$
dependent variables $w = (w^1,\ldots ,w^m)$,
\begin{equation}
\label{eq2}
H^\nu [w] = H^\nu (z,w,\partial_z w,\ldots ,\partial_z^k w) = 0,\quad \nu = 1,\ldots ,M,
\end{equation}
where $\partial_z^j w$ denotes $j$th-order partial derivatives of $w$ with respect to $z$.

Note that for common PDE systems such as
heat or diffusion equations and wave equations,
as well as more general parabolic, hyperbolic and Hamiltonian equations,
and elliptic equations,
we will have $m=M$.
However, for complete generality we allow $m\neq M$ hereafter,
as would occur for instance if one considers
a PDE system of dynamical evolution equations with differential constraints.

Now if one seeks an invertible mapping $\mu$ of
a given nonlinear PDE system (\ref{eq1}) to a linear target PDE system (\ref{eq2}),
then the following two theorems
show that the mapping is restricted to being a contact transformation
when (\ref{eq1}) and (\ref{eq2}) both contain a single dependent variable
($m = 1$; which we will call the {\it scalar case})
or to a point transformation
when (\ref{eq1}) and (\ref{eq2}) each contain two or more dependent variables
($m \ge 2$; the {\it multicomponent case}).

\begin{theorem}[B\"{a}cklund \cite{b09}]
If $u=u^1$ is a scalar $(m = 1)$, then a mapping $\mu $
defines an invertible transformation from
$(x, u, \partial_x u, \ldots , \partial_x^p u)$ -- space to
$(z,w,\partial_z w,\ldots ,\partial_z^pw)$ -- space for \emph{any}
fixed $p$ if and only if $\mu$ is a \emph{one-to-one contact transformation}
of the form
$z = \phi (x,u,\partial_xu)$, $w = \psi (x,u,\partial_xu)$,
$\partial_z w = \rho(x,u,\partial_xu)$,
which is subject to the contact condition $D_x \psi = \rho D_x \phi$ where $D_x$ denotes total derivatives with respect to $x$.
\end{theorem}

\begin{theorem}[M\"{u}ller and Matschat \cite{b10}]
If $u=(u^1,\ldots,u^m)$ has two or more components $(m \ge 2)$, then a mapping $\mu$ defines an invertible transformation from
$(x,u,\partial_x u,\ldots ,\partial_x^p u)$ -- space to
$(z,w,\partial_z w,\ldots ,\partial_z^p w)$ -- space for \emph{any}
fixed $p$ if and only if $\mu$ is a \emph{one-to-one point
transformation} of the form
$z = \phi (x,u)$, $w = \psi (x,u)$.
\end{theorem}

For formulating necessary and sufficient conditions
for the existence of such mappings between PDE systems,
we will consider only the natural situation where
the given nonlinear system (\ref{eq1}) and the linear target system (\ref{eq2})
contain the same number $M$ of PDEs with the same differential order $k$,
and no further assumptions will be required
(such as local solvability, or involutivity).
But for the computational implementation of our results
it is important that the given system of PDEs (\ref{eq1})
be consistent in the sense that all integrability conditions are
satisfied modulo these PDEs.
(To make this precise,
one would have to prescribe an elimination algorithm
which requires the definition of a total ordering of derivatives and of monomials built from them, as discussed in \cite{w}.)

The rest of this paper is organized as follows.
In section two, we review the method presented in \cite{b01,b02,b03}
to linearize nonlinear PDEs through admitted symmetries.
In section three, we present the new method to
linearize nonlinear PDEs through admitted conservation law multipliers
and outline the computational steps involved in this method.
As examples to illustrate and contrast both methods,
in section four we consider linearizations of
Burgers' equation, a pipeline flow equation, and a nonlinear telegraph system.
We make some closing remarks in section five.

\section{Use of Symmetries to Construct Linearizations}\setcounter{theorem}{0} \setcounter{equation}{0} \setcounter{remark}{0} \setcounter{definition}{0} \setcounter{prop}{0}

Necessary and sufficient symmetry conditions are stated in \cite{b01,b02,b03}
to determine whether or not a given nonlinear system ${\bf R}\{x,u\}$ of PDEs
(\ref{eq1}) can be transformed to some linear target system ${\bf
S}\{z,w\}$ of PDEs (\ref{eq2}) by an invertible mapping $\mu $.
These conditions provide a symmetry-based algorithmic procedure to determine
if they hold for a given nonlinear system of PDEs and
(when these conditions do hold)
a second symmetry-based algorithmic procedure to
construct such an invertible mapping $\mu $.
To apply these methods to a given nonlinear system of PDEs, it is
unnecessary to know a specific linear target system of PDEs. In
particular, the linear target system (when one exists) will arise from
computing the admitted point or contact symmetries
(Lie group of point or contact transformations)
of the given nonlinear system (first procedure).
Moreover, these admitted symmetries yield equations to
construct a specific mapping $\mu $ (second procedure).

The starting point is that a linear system ${\bf S}\{z,w\}$ of PDEs, given by
\begin{equation}
\label{eq5} L[z]w = g(z)
\end{equation}
as defined here in terms of a linear operator L[$z$], with any
inhomogeneous term $g(z)$, is completely characterized by its admitted
infinite-parameter set of point symmetries
\begin{equation}
\label{eq6} {\bf X} = f(z) \frac{\partial }{\partial w},
\end{equation}
where $f(z)$ is any function satisfying the related linear
homogeneous system
\begin{equation}
\label{eq7} L[z]f = 0.
\end{equation}

A point transformation is mapped to another point transformation under
any specific point transformation $(x,u) \rightarrow (z,w)$,
and a contact transformation is mapped to another contact transformation
(which could be a point transformation) under any specific contact
transformation $(x,u,\partial_xu) \rightarrow (z,w,\partial_zw)$
(which could be a point transformation). Hence an invertible point
transformation mapping a nonlinear system ${\bf R}\{x,u\}$ of PDEs to
some linear system ${\bf S}\{z,w\}$ must map each point symmetry
admitted by ${\bf R}\{x,u\}$ to a point symmetry admitted by ${\bf
S}\{z,w\};$ an invertible contact transformation mapping a nonlinear
scalar PDE system ${\bf R}\{x,u\}$ to some linear scalar PDE system
${\bf S}\{z,w\},$ must map each contact symmetry admitted by ${\bf
R}\{x,u\}$ to a contact symmetry (which could be a point symmetry)
admitted by ${\bf S}\{z,w\}$.
Moreover, this mapping must be an
isomorphism, i.e., there is a one-to-one correspondence between the
point (contact) symmetries admitted by ${\bf R}\{x,u\}$ and those
admitted by ${\bf S}\{z,w\}$.
Consequently, if there exists an
invertible transformation that maps the nonlinear PDE system ${\bf R}\{x,u\}$
to some linear PDE system ${\bf S}\{z,w\},$ then ${\bf R}\{x,u\}$
must necessarily admit an infinite set of point (contact) symmetries.

The following two theorems on mappings of nonlinear PDEs to linear PDEs
were presented and proved in \cite{b01,b02,b03}.

\begin{theorem}[Necessary conditions for the existence of
an invertible linearization mapping]
If there exists an invertible mapping $\mu$ of a given nonlinear
system ${\bf R}\{x,u\}$ of PDEs \eqref{eq1}
to some linear system ${\bf S}\{z,w\}$ of PDEs \eqref{eq2}, then
in the multicomponent case (i.e. $m \ge 2$):
\begin{enumerate}
\item $\mu$ is a point transformation of the form
\begin{equation} \label{eq8}
z_i = \phi_i (x,u),\quad i = 1,\ldots ,n, \qquad
w^\sigma = \psi ^\sigma (x,u),\quad \sigma = 1,\ldots ,m;
\end{equation}
\item ${\bf R}\{x,u\}$ admits an infinite-parameter Lie group of point symmetries given by an infinitesimal generator
\begin{equation}
\label{eq9}
{\bf X} = \xi_i (x,u)\frac{\partial }{\partial x_i } + \eta ^\tau(x,u)\frac{\partial }{\partial u^\tau}
\end{equation}
{with infinitesimals of the form}
\begin{equation} \label{eq10}
\xi_i =
\alpha _{i\sigma }(x,u) F^\sigma (x,u), \quad
\eta^\tau = \beta_\sigma^\tau(x,u) F^\sigma (x,u),
\end{equation}
where
$\alpha_{i\sigma}$, $\beta_\sigma^\tau$
($i = 1,\ldots ,n;\sigma,\tau = 1,\ldots ,m$)
are specific functions of $x$ and $u$,
and where $F = (F^1,\ldots ,F^m)$
is an arbitrary solution of some linear system of PDEs
\begin{equation}
\label{eq11}
L[X]F = 0
\end{equation}
in terms of some linear differential operator $L[X]$ and specific
independent variables
$X = (X_1 (x,u),\ldots ,X_n (x,u))=(\phi_1,\dots,\phi_n)$.
\end{enumerate}
\end{theorem}

\begin{theorem}[Sufficient conditions for the existence of an invertible
linearization mapping]
Suppose a given nonlinear system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1})
admits infinitesimal point symmetries \eqref{eq9} of the form \eqref{eq10}
involving an arbitrary solution $F(X)$ of a linear system
\eqref{eq11} with specific independent variables
$X = (X_1(x,u),\ldots ,X_n (x,u))$.
If the $m$ first order scalar linear homogeneous PDEs
\begin{equation}
\label{eq12}
\alpha _{i\sigma}(x,u)\frac{\partial \phi }{\partial x_i }
+ \beta_\sigma^\tau(x,u) \frac{\partial \phi}{\partial u^\tau} = 0,
\quad \sigma = 1,\ldots ,m,
\end{equation}
whose coefficients are formed from \eqref{eq10}
have $\phi_1=X_1 (x,u),\ldots ,\phi_n=X_n (x,u)$ as $n$ functionally independent solutions,
and if the $m^2$ first order linear inhomogeneous PDEs
\begin{equation}
\label{eq13}
\alpha _{i\sigma}(x,u)\frac{\partial \psi^\tau}{\partial x_i}
+ \beta_\sigma^\gamma(x,u) \frac{\partial \psi^\tau}{\partial u^\gamma}
= \delta_\sigma^\tau ,
\quad \tau,\sigma = 1,\ldots ,m,
\end{equation}
(where $\delta _\sigma^\tau$ is the Kronecker symbol)
have a particular solution
$\psi = (\psi ^1(x,u),\ldots ,\psi ^m(x,u)),$ then the mapping $\mu$ defined by
\begin{equation} \label{eq14}
z_i = X_i (x,u),\quad i = 1,\ldots ,n,\qquad
w^\sigma = \psi^\sigma(x,u),\quad \sigma = 1,\ldots,m ,
\end{equation}
is invertible and transforms ${\bf R}\{x,u\}$ to the linear system ${\bf S}\{z,w\}$ of PDEs given by
$L[z]w = g(z),$
for some inhomogeneous term $g(z)$.
\end{theorem}

It is easy to see that Theorems 2.1 and 2.2 also hold
in the scalar case (i.e. when $m=1$)
if one seeks to transform nonlinear scalar PDEs
to linear scalar PDEs only by a point transformation.
Furthermore, these theorems in the scalar case
can be extended to the more general possibility of
seeking invertible mappings of
nonlinear scalar PDEs to linear scalar PDEs by contact transformations.
For details, see \cite{b01,b02}.

\section{Use of Conservation Law Multipliers to Construct Linearizations}\setcounter{theorem}{0} \setcounter{equation}{0} \setcounter{remark}{0} \setcounter{definition}{0} \setcounter{prop}{0}

Now we present a new method for finding mappings that transform
a nonlinear system of PDEs invertibly to some linear system of PDEs.
Our method uses admitted conservation law multipliers of the nonlinear system
and is based on two theorems that are the
conservation law multiplier-based counterparts of the symmetry-based Theorems~2.1 and~2.2.

The first new theorem gives necessary conditions that conservation law multipliers must satisfy in order for
a given nonlinear system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1})
to have an invertible mapping to some linear target system ${\bf S}\{z,w\}$
of PDEs (\ref{eq2}) by a point transformation.
As in the case of Theorem~2.1,
the necessary conditions yield an algorithmic procedure to
find a specific linear target system as well as its independent variables.

The second new theorem gives sufficient conditions for the existence of the mapping. In particular, as in the case of Theorem 2.2,
these conditions yield an algorithmic procedure to find
the components of an explicit point transformation giving
an invertible mapping to the linear target system.

The extension of these two theorems to include contact transformations
will be discussed in section 3.2.

\subsection{Necessary and Sufficient Linearization Conditions}
Recall, a conservation law of a system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1}) is a divergence expression
$D\Upsilon_i[u]/Dx_i$ that is required to vanish for all solutions $u(x)=(u^1(x),\ldots,u^m(x))$ of the given PDE system.
Conservation laws are nontrivial if and only if the conserved fluxes $\Upsilon_i[u]$ do not have the form of curls
$D\Theta_{ij}[u]/Dx_j$, with $\Theta_{ij}[u]=-\Theta_{ji}[u]$, for every solution $u(x)$.

\begin{definition}
\textit{Conservation law multipliers} for a system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1}) are a set of factors
$\{\Lambda_\nu(x,U,\partial_x U,\ldots,\partial_x^\ell U)\}$
$(\nu=1,\ldots,M)$ such that,
for arbitrary functions $U = (U^1(x),\ldots ,U^m(x))$,
\begin{equation}
\label{eq26}
\Lambda_\nu[U] G^\nu[U] \equiv D\Upsilon_i[U] / Dx_i
\end{equation}
holds as an identity for some functions $\Upsilon = (\Upsilon_1[U],\ldots,\Upsilon_n[U])$,
where each factor $\Lambda_\nu[U]$ is non-singular for all solutions $U=u(x)$
of the given PDE system (\ref{eq1}).
\end{definition}

It is well known that multipliers will yield all nontrivial conservation laws (to within the
addition of curls to the conserved fluxes) admitted by a
given PDE system ${\bf R}\{x,u\}$ if it is of Cauchy-Kovalevskaya type.

For any PDE system ${\bf R}\{x,u\}$, its admitted conservation law multipliers satisfy the determining equations
\begin{equation}
\label{deteq}
E_{U^\sigma}(\Lambda_\nu[U]G^\nu[U]) = 0, \quad \sigma= 1,\ldots,m,
\end{equation}
given by the Euler operators
\[
E_{U^\sigma} = \frac{\partial}{\partial
U^\sigma} - \frac{D}{Dx_i}\frac{\partial}{\partial U^\sigma_{x_i}}
+\frac{D^2}{Dx_iDx_j}\frac{\partial}{\partial U^\sigma_{x_ix_j}} +\ldots
+(-1)^k\frac{D^k}{Dx_i\ldots Dx_j}\frac{\partial}{\partial U^\sigma_{x_i\ldots x_j}}
+\ldots
\]
which annihilate divergence expressions.
One can solve these determining equations by an algorithmic
procedure \cite{b08,w}
analogous to that for solving the determining equations
for point or contact symmetries in evolutionary form.
(The similarity is made explicit in the work in \cite{b06,b07}
which shows how to reformulate (\ref{deteq}) as a kind of adjoint of
the symmetry determining equations for Cauchy-Kovalevskaya PDE systems.)

The starting point, leading to the formulation of an invertible mapping between
nonlinear and linear PDE systems through conservation law
multipliers, is that any linear operator $L$ and its adjoint
operator $L^*$ satisfy a conservation law identity.
Specifically, consider a $k$th-order linear PDE system $L[z]w = 0$,
denoted by ${\bf S}\{z,w\}$ with dependent variables $w=(w^1,\ldots,w^m)$
and independent variables $z=(z^1,\ldots,z^n)$.
In particular, let the linear PDEs be given by
\begin{equation}
\label{eq27}
L_\alpha ^\nu [z]w^\alpha = 0, \quad \nu = 1,\ldots ,M
\end{equation}
in terms of linear operators
\footnote{We freely raise and lower indices for convenience here in displaying
the adjoint relation between $L$ and $L^*$.}
\begin{equation}
\label{eq28}
L_\alpha ^\nu [z] =
b_{\nu \alpha}(z) + b_{\nu \alpha i}(z)\frac{\partial }{\partial z_i }
+ \cdots
+ b_{\nu \alpha i_1 \cdots i_k}(z)
\frac{\partial^k}{\partial z_{i_1 } \cdots \partial z_{i_k } }.
\end{equation}
The corresponding adjoint linear system, $L^*[z]v = 0,$ is given by
\begin{equation} \label{eq29}
\begin{array}{ll}
L_\nu^{*\alpha}[z]v^\nu
= \displaystyle b_{\mu\alpha}(z)v^\mu
-\frac{\partial}{\partial z_i}\Big(b_{\mu\alpha i}(z)v^\mu\Big) + \cdots
+ ( - 1)^k\frac{\partial ^k}{\partial z_{i_1 } \cdots \partial z_{i_k } }
\Big(b_{\mu\alpha i_1 \cdots i_k}(z)v^\mu \Big)
\\\qquad\qquad
= 0,  \ \ \alpha = 1,\ldots,m,
\end{array}\end{equation}
where $L^{*\alpha}_\nu[z]$ is the linear operator defined through formal
integration by parts of the operator $L^\nu_\alpha[z]$ with respect to $z$.
Then for arbitrary functions $V(z)=(V^1(z),\ldots,V^M(z))$
and $W(z) = (W^1(z),\ldots ,W^m(z)),$ which can be viewed as multipliers $\{V^\mu(z)\}$
and $\{W^\alpha (z)\}$ for the augmented linear system consisting of the linear system (\ref{eq27}) and the adjoint system
(\ref{eq29}), there is a conservation law identity
\begin{equation}
\label{eq31}
D\Upsilon_i / Dz_i =
\delta_{\nu\mu} V^\mu L_\alpha^\nu [z]W^\alpha -
\delta_{\alpha\beta}W^\beta L_\nu^{*\alpha} [z]V^\nu
\end{equation}
holding for some specific functions $\Upsilon_i[V,W]$
that have a bilinear dependence on the multipliers and their derivatives.
(An explicit expression for the fluxes $\Upsilon_i[V,W]$ is given in \cite{b}.)
Here both $\delta_{\nu\mu}$ and $\delta_{\alpha\beta}$ are Kronecker symbols.

\begin{prop}
Suppose a given nonlinear system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1})
can be mapped into some linear system ${\bf S}\{z,w\}$ of PDEs (\ref{eq27})
by some point transformation (\ref{eq8}).
Then the nonlinear and linear PDEs will be related through
a set of factors $\{Q_\nu^\mu(x,U,\partial_x U)\}$ such that
\begin{equation}
\label{eq32}
Q_\nu^\mu [U]G^\nu [U] = L_\alpha^\mu [z]W^\alpha ,\quad \mu = 1,\ldots ,M,
\end{equation}
for arbitrary functions $U(x) = (U^1(x),\ldots ,U^m(x))$,
where the functions $W(z) = (W^1(z),\ldots ,W^m(z))$ are
given by the linearization mapping
\[
z = \phi (x,U(x)),\quad W(z) = \psi (x,U(x)) .
\]
(Explicit expressions for $Q_\nu^\mu$ can be readily derived
from the mapping formulae given in \cite{f}.)
Moreover :
\begin{enumerate}
\item
This mapping will define an invertible point transformation iff
$\{\phi_i(x,U(x))\}$ ($i=1,\ldots,n$)
are functionally independent
and $\{Q_\nu^\mu(x,U(x),\partial_x U(x))$ ($\nu,\mu=1,\ldots,M$)
are non-degenerate,
namely,
$\det(D\phi_i(x,U)/Dx_j)\neq 0$
and also $\det( Q_\nu^\mu(x,U,\partial_x U) )\neq 0$.
\item
Under such a point transformation
the conservation law identity (\ref{eq31}) becomes
\begin{equation}
\label{eq33} D\Gamma_i / Dx_i = (\delta_{\lambda\mu}V^\lambda Q_\nu^\mu [U]G^\nu [U]
- \delta_{\alpha\beta}W^\beta L_\nu^{*\alpha} [z]V^\nu )J[U],
\end{equation}
where $D\Gamma_i / Dx_i$ is related to $D\Upsilon_i / Dz_i$
by the non-vanishing Jacobian factor
\begin{equation}
\label{eq34}
J(x,U,\partial_x U)
= \left| {\frac{Dz}{Dx}} \right|
= \det \left( {\frac{D\phi_i(x,U(x))}{Dx_j }} \right).
\end{equation}
(See \cite{b11} for the explicit expression for $\Gamma_i$
in terms of $\Upsilon_i$.)
\end{enumerate}
\end{prop}

This proposition leads to the following two theorems for linearization from
admitted conservation law multipliers,
which are the counterparts to Theorems~2.1 and 2.2 for linearization
through admitted point symmetries.

\begin{theorem}[Necessary conditions for the existence of an invertible
linearization mapping]
If there exists an invertible point transformation \eqref{eq8}
for $m\geq 1$ under which
a given $k$th-order nonlinear system ${\bf R}\{x,u\}$ of PDEs \eqref{eq1}
is mapped to some linear system ${\bf S}\{z,w\}$ of PDEs (\ref{eq27}),
then ${\bf R}\{x,u\}$ must admit conservation law multipliers of the form
\begin{equation}
\label{eq35}
\Lambda_\nu [U] = \delta_{\lambda\mu}v^\lambda(X) Q_\nu^\mu [U] J[U],
\end{equation}
where
$Q_\nu^\mu$ ($\mu,\nu = 1,\ldots ,M$)
are specific functions of $x$, $U$ and $\partial_x U$;
$v = (v^1,\ldots ,v^M)$ is an arbitrary solution of some linear system of PDEs
\begin{equation}
\label{eq3_11}
\tilde {L}^\alpha_\mu[X] v^\mu
=
\tilde{b}_{\alpha\mu}(X)v^\mu
+ \tilde{b}_{\alpha\mu i}(X)v^\mu_{X_i} + \cdots
+ \tilde{b}_{\alpha\mu i_1 \cdots i_k}(X)v^\mu_{X_{i_1}\cdots X_{i_k}}
= 0,\quad \alpha = 1,\ldots ,m,
\end{equation}
in terms of specific independent variables $X = (X_1 (x,U),\ldots ,X_n (x,U))$; and $J[U] = \vert DX / Dx\vert =\det(DX_i(x,U)/Dx_j)$
which is a function of $x$, $U$ and $\partial_x U$.
\end{theorem}

\begin{proof}
The existence of multipliers (\ref{eq35}) follows from equation (\ref{eq33})
with $X$ playing the role of $z$,
$\tilde {L}_\nu^\alpha [X]$ playing the role of $L_\nu^{*\alpha} [z]$
and with $V^\mu = v^\mu $ satisfying system \eqref{eq3_11}.
\end{proof}

\begin{theorem}[Sufficient conditions for the existence of an invertible
linearization mapping]
Suppose a given nonlinear system ${\bf R}\{x,u\}$ of PDEs \eqref{eq1} admits
conservation law multipliers of the form \eqref{eq35}
where the components of $v$ are dependent variables of some linear system \eqref{eq3_11}
with specific independent variables $X = (X_1 (x,u),\ldots ,X_n (x,u))$.
Let $\tilde {L}^* [X]$ be the adjoint of the linear operator $\tilde {L}[X]$
in (\ref{eq3_11})
and consider the augmented system of PDEs
consisting of the given nonlinear system \eqref{eq1} and the linear system \eqref{eq3_11}.
Then, from identity \eqref{eq33},
there exists an infinite set of multipliers
\[
\{\Lambda_\mu[U,V], \tilde\Lambda_\alpha[U]\}
= \{
Q_\mu^\nu(x,U,\partial_x U) \delta_{\lambda\nu}V^\lambda(X(x,U)) J[U],
-\delta_{\alpha\beta} W^\beta(x,U)J[U]
\}
\]
$(\mu=1,\ldots,M;\alpha=1,\ldots,m)$
yielding a conservation law
\begin{equation}
\label{eq36}
\Lambda_\nu[U,V] G^\nu [U]
- \tilde\Lambda_\alpha[U] \tilde{L}_\mu ^\alpha [X(x,U)]V^\mu
= D\Gamma_i / Dx_i
\end{equation}
for some specific functions $\Gamma_i$,
in terms of the Jacobian
\begin{equation}
\label{eq37}
J[U] = \vert DX / Dx\vert  = \det (DX_i(x,U)/Dx_j ).
\end{equation}
This conservation law \eqref{eq36} is equivalent to the identity
\begin{equation}
\label{eq38}
\delta_{\lambda\mu}V^\lambda Q_\nu^\mu(x,U,\partial_x U) G^\nu [U]
- \delta_{\alpha\beta}W^\alpha (x,U)\tilde{L}_\mu^\beta [X(x,U)]V^\mu
= D\Upsilon_i / DX_i ,
\end{equation}
holding for some functions $\Upsilon_i$.
If the variables $\{X_i\}$ ($i=1,\ldots,n$) are functionally independent
and if the factors $\{Q_\nu^\mu\}$ ($\mu,\nu=1,\ldots,M$) are non-degenerate,
then the point transformation given by
\begin{equation}
\label{eq39}
z = X(x,u),\quad
w = W(x,u)
\end{equation}
maps the nonlinear system of PDEs \eqref{eq1} invertibly into the linear system
${\bf S}\{z,w\}$ given by
\begin{equation}\label{eq40}
\begin{array}{ll}
\tilde {L}_\alpha ^{*\nu} [z]w^\alpha
= \displaystyle \tilde b_{\alpha\nu}(z)w^\alpha
-\frac{\partial}{\partial z_i}\Big(\tilde b_{\alpha\nu i}(z)w^\alpha\Big) + \cdots
+ ( - 1)^k\frac{\partial ^k}{\partial z_{i_1 } \cdots \partial z_{i_k } }
\Big(\tilde b_{\alpha\nu i_1 \cdots i_k}(z)w^\alpha \Big)
\\\qquad\qquad
= 0,\quad \nu=1,\ldots,M .
\end{array}\end{equation}
\end{theorem}

\begin{proof}Since $\tilde {L}[X]$ is a linear operator, the identity (\ref{eq31}) gives
\begin{equation}
\label{eq41}
\delta_{\alpha\beta}W^\alpha (x,U)\tilde {L}_\mu^\beta [X(x,U)]V^\mu =
\delta_{\mu\nu}V^\mu \tilde {L}_\alpha^{*\nu}[X(x,U)]W^\alpha (x,U) + D\Gamma_i / DX_i
\end{equation}
for some specific functions $\Gamma_i [U,V,W]$.
Consequently, the identity (\ref{eq38}) becomes
\begin{equation}
\label{eq42}
\delta_{\lambda\mu}V^\lambda (Q_\nu^\mu [U]G^\nu [U]
- \tilde {L}_\alpha^{*\mu} [X(x,U)]W^\alpha (x,U))
= D(\Gamma _i + \Upsilon_i ) / DX_i .
\end{equation}
Now apply the Euler operators with respect to $V^\mu,$ i.e.,
\[
E_{V^\mu } = \frac{\partial }{\partial V^\mu } - \frac{D}{DX_i }\frac{\partial }{\partial V^\mu_{X_i}} + \cdots ,
\quad \mu = 1,\ldots,M,
\]
to each side of equation (\ref{eq42}).
Each of these Euler operators annihilates the right-hand side of (\ref{eq42}) and hence we obtain the identity
\begin{equation}
\label{eq43} Q_\nu ^\mu [U]G^\nu [U] = \tilde {L}_\alpha^{*\mu} [X(x,U)]W^\alpha (x,U)
\end{equation}
holding for arbitrary functions $U$.
Now suppose $U = u(x)$ solves the given nonlinear system of PDEs (\ref{eq1}),
then it follows that $w = W(x,u(x))$ solves the linear system given by (\ref{eq40}).
Consequently, one obtains the point transformation (\ref{eq39})
which maps the nonlinear system of PDEs (\ref{eq1})
into the linear system (\ref{eq40}).
Invertibility of this transformation is established by Proposition~3.1.
\end{proof}

\subsection{Computational Steps}
We now outline the computational steps involved in applying Theorems 3.1 and 3.2 to linearize a given nonlinear system of
PDEs through admitted conservation law multipliers.

\textbf{Step 1:}
For a given nonlinear system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1})
with highest derivative of order $k$,
solve the determining equations (\ref{deteq})
for admitted sets of multipliers $\{\Lambda_\nu[U]\}$
depending on the independent variables $x = (x_1 ,\ldots, x_n )$,
the dependent variables $U = (U^1,\ldots,U^m)$
and their first derivatives $\partial_x U = (U^1_{x_i},\ldots,U^m_{x_i})$ .
Two cases arise, depending on whether ${\bf R}\{x,u\}$ admits any
set of multipliers of the form (\ref{eq35}) with functions $v=(v^1(X), \ldots, v^M(X))$
satisfying some linear system of PDEs of the form \eqref{eq3_11}
in terms of specific independent variables $X = (X_1(x,U),\ldots ,X_n (x,U))$.

\textit{Case I. No set of multipliers of the required form are admitted.}
Then one concludes from Theorem 3.1 that
${\bf R}\{x,u\}$ cannot be mapped invertibly by any point transformation
to a linear system.

\textit{Case II. There is an infinite set of multipliers with the required form.}
Typically in this case, the variables $X_i(x,U)$ are found directly from
the specific form of the multiplier determining equations,
either by inspection (see examples 4.1.2 and 4.2.2)
or, more generally, through integration of characteristic first-order linear PDEs
contained in the system of multiplier determining equations (see example 4.3.2).
Then from the admitted multipliers (\ref{eq35}) one reads off (by Theorem 3.1)
\begin{enumerate}
\item[(a)]
the independent variable part of the linearizing point transformation \eqref{eq39},
i.e. $x_i\rightarrow z_i = X_i(x,u)$;
\item[(b)]
the specific linear target system which is given by the adjoint of the linear system \eqref{eq3_11}.
\end{enumerate}

\textbf{Step 2:}
Assuming that the necessary conditions of Theorem 3.1 hold (\textit{Case II}),
one proceeds by replacing $v(X)$ by an arbitrary function $V(X)$
in the conservation law (\ref{eq26}) arising from
the multipliers (\ref{eq35}) for the nonlinear system ${\bf R}\{x,u\}$ of PDEs (\ref{eq1}).
This will automatically yield ---
after integration by parts if necessary
(with the use of the transformation formula for divergence expressions
derived in \cite{f}) ---
a conservation law identity of the form (\ref{eq38})
for the augmented system consisting of the given nonlinear system (\ref{eq1})
and the linear system (\ref{eq3_11}).
 From the components of the multipliers for the linear system (\ref{eq3_11})
in this identity, one directly obtains
\begin{enumerate}
\item[(c)]
the dependent variable part
$u^\alpha \rightarrow w^\alpha=W^\alpha(x,u)$
of the linearization mapping to the linear target system.
\end{enumerate}

In summary,
when there exists a linearization of a given nonlinear PDE system
${\bf R}\{x,u\}$ $(m\geq 1)$
by an invertible point transformation then
the necessary conditions stated in Theorem~3.1
yield a linear target system along with the independent variables of this system,
while the sufficient conditions stated in Theorem~3.2
yield the dependent variables of the linear target system,
which completes the transformation.
(Examples are illustrated in sections~4.1.2 and 4.3.2.)
Furthermore, if the necessary conditions do not hold then
no linearizing point transformation exists.

It is easy to extend these results to include
the linearization of scalar PDEs ($m=1$) by contact transformations,
where $X_i$ and $W^\alpha$ can now depend on first derivatives of $U$.
Since in this case it follows that
the Jacobian $J[U]$ and the factors $Q_\mu^\nu[U]$
may depend on second derivatives of $U$,
one must consequently seek sets of multipliers $\{\Lambda_\nu[U]\}$
depending on $x$, $U$, $\partial_x U$ and $\partial_x^2 U$.
Then in Step 1 any admitted multipliers of the form (\ref{eq35})
will yield the independent variable part of
a linearizing contact transformation,
i.e., $x_i \rightarrow z_i = X_i(x,u,\partial_x u)$;
similarly, in Step 2 the resulting conservation law identity (\ref{eq38})
will yield the dependent variable part of this mapping,
i.e., $u^\alpha \rightarrow w^\alpha=W^\alpha(x,u,\partial_x u)$.
(An example is illustrated in section~4.2.2.)
If no such multipliers are admitted then
no linearizing contact transformation exists.

We remark that in certain special cases
a lower differential order is possible in the required form of
conservation law multipliers (\ref{eq35})
for the existence of a linearizing transformation.
Such situations are important computationally
because the multiplier determining equations will then become
more over-determined and hence easier to solve.

(1)
For point or contact transformations that involve
{\it no change in the independent variables},
$x_i \rightarrow z_i = x_i= X_i$,
one can easily show from the results in \cite{f} that
the Jacobian (\ref{eq34}) will be trivial,
i.e. $J=\det(DX_i/Dx_j)=\det(\delta_i^j)=1$,
while the factors $Q_\nu^\mu[U]$ in (\ref{eq32}) will have
no dependence on $\partial_x U$ in the multicomponent case ($m\geq 2$)
and no dependence on $\partial_x^2 U$ in the scalar case ($m=1$).
Hence, in this situation,
conservation law multipliers can be sought in the form
\[
\Lambda_\nu [U] = \delta_{\lambda\mu}v^\lambda(x) Q_\nu^\mu [U]
\]
depending at most on $x$, $U$ and $\partial_x U$ if $m=1$
or only on $x$ and $U$ if $m\geq 2$.
(A fully algorithmic version of the linearization steps
applied to this situation was given in \cite{w},
which allows one to detect and explicitly find all linearizing mappings
that involve only a transformation of dependent variables.)
See example 4.1.2.

(2)
Suppose a given nonlinear PDE system \eqref{eq1} of order $k$ admits
{\it integrating factors}
$\{\lambda_\nu^\mu(x,U)\}$ for its highest derivative terms,
so that the system has the form
$\lambda_\nu^\mu[U] G^\nu[U] = D\tilde G^\nu_i[U]/Dx_i + \tilde G^\nu_0[U]$
where $\tilde G_0^\nu$ and $\tilde G^\nu_i$ ($i=1,\ldots,n$)
are of order at most $k-1$ in derivatives of $U$.
Then it is straightforward to show that the factors relating
the nonlinear and linear PDE systems in (\ref{eq32})
will be given by $Q_\nu^\mu [U] = \lambda_\nu^\mu(x,U) J[U]^{-1}$,
and consequently one can seek conservation law multipliers of the form
\[
\Lambda_\nu [U] = \delta_{\lambda\mu}v^\lambda(X) \lambda_\nu^\mu(x,U)
\]
where $X_i[U]$ depends at most on $x$, $U$ and $\partial_x U$ if $m=1$
or only on $x$ and $U$ if $m\geq 2$
(corresponding to linearizing contact transformations or point transformations,
respectively).
See examples 4.2.2 and 4.3.2.

\section{Examples}\setcounter{theorem}{0} \setcounter{equation}{0} \setcounter{remark}{0} \setcounter{definition}{0} \setcounter{prop}{0}

To illustrate the conservation law approach
for obtaining linearizations and contrast it with the symmetry approach, we now consider three examples.

\subsection{Linearization of Burgers' Equation}
Our first example is a nonlinear PDE system
with independent variables $(x_1 ,x_2 ) = (x,t)$
and dependent variables $(u^1,u^2)$ as given by
\begin{equation}\label{eq44}
G^1[u] = \displaystyle \frac{\partial u^2}{\partial x} - 2u^1 = 0,\qquad
G^2[u] = \displaystyle \frac{\partial u^2}{\partial t} - 2\frac{\partial u^1}{\partial x} + (u^1)^2 = 0,
\end{equation}
where $u^1$ satisfies Burgers' equation
\begin{equation}
\label{eq45} u^1_{xx} - u^1u^1_x - u^1_t = 0.
\end{equation}

\subsubsection{Linearization Through Admitted Point Symmetries}

By a direct computation (solving the symmetry determining equations),
one finds that equation (\ref{eq45}) admits at most a finite number of contact symmetries.
Hence there does not exist any contact (or point) transformation
that linearizes Burgers' equation.

On the other hand the nonlinear system (\ref{eq44}) is found to admit
an infinite set of point symmetries
given by the infinitesimal generator \cite{b12,b13}
\begin{equation}
\label{eq46}
{\bf X} = e^{u^2 / 4}\left( (2g_x(x,t) + g(x,t)u^1)\frac{\partial }{\partial u^1}
+ 4g(x,t)\frac{\partial }{\partial u^2} \right)
\end{equation}
where $g(x,t)$ is an arbitrary solution of the linear PDE
\begin{equation}\label{eq47}
g_{xx} = g_t .
\end{equation}
Consequently, one can apply Theorems 2.1 and 2.2 to linearize the system (\ref{eq44}) as follows.
First we see from a comparison of (\ref{eq46}) with expressions (\ref{eq10}) that
one has
$F^1 = g_x(x,t)$, $F^2 = g(x,t)$,
with
$\alpha_{ij} = 0$,
$\beta _1^1 = 2e^{u^2 /4}$, $\beta _2^1 = u^1e^{u^2 / 4}$,
$\beta _1^2 = 0$, $\beta _2^2 = 4e^{u^2 / 4}$;
the associated linear homogeneous system (\ref{eq12})
has $X_1 = x$, $X_2 = t$
as functionally independent solutions
and the corresponding linear inhomogeneous system (\ref{eq13})
has a particular solution
$(\psi ^1,\psi ^2) = (\frac{1}{2}u^1e^{ - u^2 / 4}, - e^{ - u^2 /4})$.
Then from (\ref{eq47}) we have that $F=(F^1,F^2)$ satisfies the linear system
\begin{equation}
F^2_x = F^1 ,\quad F^1_x=F^2_t .
\end{equation}
Hence one obtains the invertible point transformation
\begin{equation}
\label{eq48}
z_1 = x,\quad
z_2 = t, \quad
w^1 = \textstyle\frac{1}{2} u^1e^{ - u^2 / 4},\quad
w^2 = - e^{ - u^2 / 4}
\end{equation}
mapping the given nonlinear system (\ref{eq44}) into the linear system
\begin{equation}\label{eq49}
\frac{\partial w^2}{\partial x}= w^1 ,\quad
\frac{\partial w^1}{\partial x} = \frac{\partial w^2}{\partial t},
\end{equation}
where both $w^1$ and $w^2$ consequently satisfy the heat equation
\begin{equation}
\label{eq50}
\frac{\partial ^2w^1}{\partial x^2} - \frac{\partial w^1}{\partial t} = 0 ,\quad
\frac{\partial ^2w^2}{\partial x^2} - \frac{\partial w^2}{\partial t} = 0 .
\end{equation}

We thus note that the mapping (\ref{eq48}) and the linear system (\ref{eq49})
together yield
\begin{equation}
\label{eq51}
u^1 = - \frac{2}{w^2}\frac{\partial w^2}{\partial x}
\end{equation}
which is the well-known Hopf-Cole transformation
that maps any solution of the heat equation (\ref{eq50})
into a solution of Burgers' equation (\ref{eq45}).

\subsubsection{Linearization Through Admitted Conservation Law Multipliers}
A straightforward computation of conservation law multipliers
$\{\Lambda_1(x,t,U), \Lambda_2(x,t,U)\}$
shows that the nonlinear system (\ref{eq44}) admits
an infinite set of multipliers given by
\begin{equation}
\label{eq52}
\Lambda_1 = \textstyle\frac{1}{2}U^1e^{- U^2/4}f(x,t) + e^{- U^2/4}f_x(x,t) ,\quad
\Lambda_2 = e^{ - U^2/4}f(x,t),
\end{equation}
where $f(x,t)$ is an arbitrary solution of the linear scalar PDE
$f_{xx}+f_t=0$
(namely, the backward heat equation).
Hence, the necessary conditions of Theorem~3.1
for the existence of an invertible mapping of the nonlinear system (\ref{eq44})
to a linear target system are satisfied, where,
through a comparison of (\ref{eq52}) with (\ref{eq35}),
the adjoint target system is given by
\begin{equation}\label{eq53}
\frac{\partial v^1}{\partial x} - v^2 = 0, \quad
\frac{\partial v^2}{\partial x} + \frac{\partial v^1}{\partial t} = 0
\end{equation}
with dependent variables
\begin{equation}\label{eq53b}
v=(v^1,v^2)=(f,f_x)
\end{equation}
and the same independent variables $(x,t)$ as the given system (\ref{eq44}).

In the conservation law (\ref{eq26}) arising from
the multipliers (\ref{eq52}), (\ref{eq53}) and (\ref{eq53b}),
we replace $(v^1(x,t),v^2(x,t))$ by arbitrary functions $(V^1(x,t),V^2(x,t))$.
This leads to the following conservation law identity for the augmented system consisting of the given nonlinear system
(\ref{eq44}) and the linear system (\ref{eq53}):
\begin{equation}\label{eq54} \begin{array}{ll}
\displaystyle
\left(V^1(\textstyle\frac{1}{2}U^1e^{ - U^2 / 4}) + V^2e^{ - U^2 / 4}\right)G^1[U] + V^1e^{ - U^2 / 4}G^2[U]
\\[2ex]
\displaystyle
- 2U^1e^{ - U^2 / 4}\left(\frac{\partial V^1}{\partial x} - V^2 \right)
- 4e^{ - U^2 / 4}\left(\frac{\partial V^2}{\partial x} + \frac{\partial V^1}{\partial t} \right)
\\[2ex]
 = \displaystyle
\frac{D}{Dx}\left(e^{ - U^2 / 4}( - 4V^2 - 2U^1V^1) \right) +
\frac{D}{Dt}\left(- 4V^1e^{ - U^2 / 4} \right).
\end{array}\end{equation}
Consequently,
directly comparing (\ref{eq54}) with the identity (\ref{eq38}) in Theorem 3.2,
we obtain an invertible mapping
$(u^1,u^2) \rightarrow (w^1=2u^1e^{-u^2/4}, w^2=4e^{-u^2/4})$
of the nonlinear system (\ref{eq44})
to a linear target system given by the adjoint of the linear system (\ref{eq53}).
In particular, this yields the point transformation (\ref{eq48})
that maps the nonlinear system (\ref{eq44}) into the linear system (\ref{eq49})
up to a suitable rescaling of the dependent variables $w=(w^1,w^2)$.

\subsection{Linearization of a pipeline flow equation}
As a second example, we consider the pipeline flow equation \cite{b11}
\begin{equation}
\label{eq55}
G[u] = u_t u_{xx} + u_x{}^p = 0, \quad p={\rm const}.
\end{equation}

\subsubsection{Linearization Through Admitted Contact Symmetries}
The nonlinear scalar PDE (\ref{eq55}) admits an infinite set of contact symmetries
whose infinitesimal generator is given by
\begin{equation}
\label{eq56}
{\bf X} = - \frac{\partial F}{\partial u_x }\frac{\partial }{\partial x} + \left( {F - u_x \frac{\partial F}{\partial
u_x }} \right)\frac{\partial }{\partial u} + \frac{\partial F}{\partial t}\frac{\partial }{\partial u_t }
\end{equation}
where $F = F(t,u_x )$ is an arbitrary solution of the second-order linear PDE
\begin{equation}
\label{eq57}
u_x{}^p \frac{\partial ^2F}{\partial u_x{}^2 } - \frac{\partial F}{\partial t} = 0.
\end{equation}
Consequently, one can linearize the PDE (\ref{eq55})
by applying the generalization of Theorems~2.1 and~2.2
for seeking a linearizing contact transformation.
In particular,
$X_1 = u_x$, $X_2 = t$ are found to be functionally independent solutions
of the appropriate generalization (cf. \cite{b01,b02})
of the linear homogeneous system (\ref{eq12}),
while the corresponding generalization of the linear inhomogeneous system (\ref{eq13})
has the particular solution
$\psi = u - xu_x$
(from which one deduces $(\rho^1,\rho^2) = (u_t,-x)$ through
the contact condition in Theorem~1.2).
Consequently, one obtains an invertible mapping $\mu$
given by the contact transformation
\begin{equation}
\label{eq58} z_1 = t,\quad z_2 = u_x ,\quad w = u - xu_x ,\quad w_{z_1} = u_t ,\quad w_{z_2} = - x,
\end{equation}
which maps the nonlinear PDE (\ref{eq55}) to the linear PDE given by
\begin{equation}
\label{eq59} z_2^{\ p} \frac{\partial ^2w}{\partial z_2^{\ 2}} - \frac{\partial w}{\partial z_1} = 0.
\end{equation}

\subsubsection{Linearization Through Admitted Conservation Law Multipliers}

Through a computation of conservation law multipliers $\Lambda(x,t,U,U_x,U_t)$,
one finds that the given nonlinear PDE (\ref{eq55}) admits
an infinite set of multipliers
\begin{equation}
\label{eq60}
\Lambda = v(U_x,t),
\end{equation}
where this function satisfies the linear scalar PDE
\[
v_t + (U_x{}^p v)_{U_x U_x} = 0.
\]
By inspection, since $v$ depends on two variables
(i.e., the same number of independent variables as $U$ has),
we see that the necessary conditions of Theorem~3.1
for the existence of an invertible contact transformation that linearizes
the nonlinear scalar PDE (\ref{eq55}) are satisfied
with the adjoint target system given by
\begin{equation}
\label{eq61}
\frac{\partial v}{\partial T} + \frac{\partial ^2(X^pv)}{\partial X^2} = 0
\end{equation}
for $v=v(X,T)$ in terms of the independent variables
\begin{equation}
\label{eq61b}
X = U_x,\quad T = t .
\end{equation}

In the conservation law (\ref{eq26}) arising from the multipliers (\ref{eq60}),
we replace $v(X,T)$ by an arbitrary function $V(X,T)$.
This leads to the following conservation law identity for the augmented system
consisting of the given nonlinear PDE (\ref{eq55}) and the linear PDE (\ref{eq61}):
\begin{equation}\label{eq62} \begin{array}{ll}
\displaystyle
VG[U] - (xU_x - U)J[U]\left(\frac{\partial V}{\partial T} +
\frac{\partial ^2(X^pV)}{\partial X^2}\right)
\\[2ex]
 = \displaystyle
\frac{D}{Dx}\Big((xU_x - U)(U_{tx} V + U_x ^pV_X ) + \big((1 - p)xU_x
+ pU\big)U_x ^{p - 1}V \Big)
+ \frac{D}{Dt}\Big(U_{xx} (U - xU_x )V \Big),
\end{array}\end{equation}
where
\begin{equation*}
J[U] = \left| {\frac{D(X,T)}{D(x,t)}} \right|
= U_{xx}
\end{equation*}
is the Jacobian of (\ref{eq61b}).
(For verifying the identity (\ref{eq62}), we note that
$V_x = V_X U_{xx}$, $V_t = V_X U_{xt} + V_T $.)
Consequently, the sufficiency conditions of Theorem 3.2 hold
for the existence of an invertible mapping by a contact transformation
of the nonlinear PDE (\ref{eq55}) to a linear target PDE
which is the adjoint of the linear PDE (\ref{eq61}).
In particular,
from a comparison of (\ref{eq62}) with the identity (\ref{eq38}),
it follows that the contact transformation
determined by
\begin{equation}
\label{eq64}
z_1 = X = u_x ,\quad
z_2 = T = t,\quad
w = xu_x - u
\end{equation}
maps the nonlinear pipeline equation (\ref{eq55}) invertibly into the linear PDE
\begin{equation}
\label{eq65}
X^p\frac{\partial ^2w}{\partial X^2} - \frac{\partial w}{\partial T} = 0.
\end{equation}

\subsection{Linearization of a Nonlinear Telegraph Equation}

For a final example, we consider a nonlinear telegraph system given by
\begin{equation}\label{tele1}
G^1[u] = \displaystyle
\frac{\partial u^2}{\partial t}-\frac{\partial u^1}{\partial x} = 0,\qquad
G^2[u] = \displaystyle
\frac{\partial u^1}{\partial t}+u^1(u^1-1)-(u^1)^2\frac{\partial u^2}{\partial x} = 0
\end{equation}
with dependent variables $(u^1,u^2)$ and independent variables $(x_1,x_2)=(x,t)$.
Note that $u^1$ satisfies the nonlinear telegraph equation
\[ \frac{\partial^2 u^1}{\partial x^{\,2}}-
\frac{\partial}{\partial t}\left((u^1)^{-2}\frac{\partial u^1}{\partial t}+1-(u^1)^{-1}\right) = 0.\]

\subsubsection{Linearization Through Admitted Point Symmetries}

The nonlinear system (\ref{tele1}) admits an infinite set of point symmetries
given by the infinitesimal generator
\begin{equation}
{\bf X}=
F^1(X,T)\frac{\partial}{\partial x}+e^{-t}F^2(X,T)\frac{\partial}{\partial t}
+e^{-t}u^1 F^2(X,T)\frac{\partial}{\partial u^1}+F^1(X,T)\frac{\partial}{\partial u^2} \label{tele2}
\end{equation}
where
\begin{equation}
X_1=X=x-u^2,\quad X_2=T=t-\log u^1, \label{tele3}
\end{equation}
and $(F^1(X,T), F^2(X,T))$ is an arbitrary solution of the linear system of PDEs
\begin{equation}
\frac{\partial F^2}{\partial T} - e^T\frac{\partial F^1}{\partial X} = 0 ,\quad
\frac{\partial F^2}{\partial X} - e^T\frac{\partial F^1}{\partial T} = 0 .
\label{tele4}
\end{equation}
Consequently, one can apply Theorems 2.1 and 2.2 to linearize (\ref{tele1}).
In particular,
we see that by comparing (\ref{tele2}) with expressions (\ref{eq10}),
one has
$\alpha_{11}=\beta_1^2=1$, $\alpha_{12}=\alpha_{21}=\beta_1^1=\beta_2^2=0$,
$\alpha_{22}=e^{-t}, \beta_2^1=e^{-t}u$;
the associated linear homogeneous system (\ref{eq11})
has $X_1=x-u^2, X_2=t-\log u^1$ as functionally independent solutions
and the corresponding linear inhomogeneous system (\ref{eq12})
has a particular solution  $(\psi^1,\psi^2)=(x,e^t)$.
Hence the mapping given by
\begin{equation}
\label{tele5}
z_1=x-u^2 ,\quad
z_2=t-\log u^1 ,\quad
w^1=x ,\quad
w^2=e^t
\end{equation}
is an invertible point transformation under which
the nonlinear telegraph system (\ref{tele1}) is mapped to the linear PDE system
given by
\begin{equation}
\frac{\partial w^2}{\partial z_2} - e^{z_2}\frac{\partial w^1}{\partial z_1}
= 0 ,\quad
\frac{\partial w^2}{\partial z_1} - e^{z_2}\frac{\partial w^1}{\partial z_2}
= 0.
\label{tele6}
\end{equation}

\subsubsection{Linearization Through Admitted Conservation Law Multipliers}

Through a computation of conservation law multipliers of the form
$\{\Lambda_1(x,t,U^1,U^2)$, $\Lambda_2(x,t,U^1,U^2)\}$,
one can show that the nonlinear telegraph system of PDEs (\ref{tele1})
admits an infinite set of multipliers yielding a linearization as follows.
After some integrability analysis of the multiplier determining equations
(e.g. using {\sc Crack} \cite{b14}), one first obtains
\begin{equation}
\Lambda_1=f_{U^2},\quad \Lambda_2=f_{U^1},
\label{tele7}
\end{equation}
in terms of a function $f(x,t,U^1,U^2)$ satisfying the three linear PDEs
\begin{eqnarray}
&&
f_x+f_{U^2}=0,\quad
f_t+U^1f_{U^1}=0,
\label{tele8a} \\
&&
(U^1)^2f_{U^1U^1}+2U^1f_{U^1}-f_{U^2U^2}=0.
\label{tele8b}
\end{eqnarray}
To proceed we integrate the pair of first-order PDEs (\ref{tele8a}),
which yields a reduction of the independent variables such that
$f=f(X,T)$ where
\begin{equation}
\label{tele8c}
X=x-U^2,\quad T=t-\log U^1.
\end{equation}
Then the second-order PDE (\ref{tele8b}) combined with (\ref{tele7})
gives the infinite set of multipliers
\begin{equation}
\Lambda_1=-f_X(X,T),\quad \Lambda_2=-f_T(X,T)/U^1,\quad
f_{XX} -f_{TT} +f_T =0 .
\label{tele9}
\end{equation}
Finally, comparing (\ref{tele9}) with (\ref{eq35}) in Theorem~3.1,
we see that the necessary conditions for the existence of an invertible mapping of
the nonlinear system (\ref{tele1}) to a linear target system are satisfied,
with the adjoint target system being given by
\begin{equation}
v=(v^1,v^2) =(-f_X,-f_T) .
\label{tele10a}
\end{equation}
and
\begin{equation}
\frac{\partial v^1}{\partial X} - \frac{\partial v^2}{\partial T} + v^2 = 0 ,
\quad
\frac{\partial v^2}{\partial X} - \frac{\partial v^1}{\partial T}   = 0.
\label{tele10}
\end{equation}

In the conservation law (\ref{eq26}) arising from the multipliers (\ref{tele9}),
(\ref{tele10a}) and (\ref{tele10}),
we replace $(v^1(X,T),v^2(X,T))$ by arbitrary functions $(V^1(X,T),V^2(X,T))$.
This leads to the following conservation law identity for the augmented system
consisting of the given nonlinear system (\ref{tele1})
and the linear system (\ref{tele10}):
\begin{equation}\label{tele11}
\begin{array}{ll}
V^1G^1[U]+V^2(U^1)^{-1}G^2[U]
\displaystyle
- U^1J[U]\left(\frac{\partial V^1}{\partial X}
-\frac{\partial V^2}{\partial T}+V^2\right)
-xJ[U]\left(\frac{\partial V^2}{\partial X}-\frac{\partial V^1}{\partial T}\right)\\\displaystyle
=
\frac{D}{Dx}\left(-V^1\left(x\frac{\partial U^2}{\partial t}+\frac{\partial U^1}{\partial t}-U^1\right)
+V^2\left(x-x(U^1)^{-1}\frac{\partial U^1}{\partial t}
-U^1\frac{\partial U^2}{\partial t}\right)
\right)
\\\qquad\displaystyle
+ \frac{D}{Dt}\left(-V^1\left(x-x\frac{\partial U^2}{\partial x}+\frac{\partial U^1}{\partial t}\right)
+V^2\left(x(U^1)^{-1}\frac{\partial U^1}{\partial x}+U^1\frac{\partial U^2}{\partial x}-U^1\right)\right),
\end{array}\end{equation}
where, from (\ref{tele8c}),
\[
J[U]= \left| {\frac{D(X,T)}{D(x,t)}} \right|
= (U^1)^{-1}\left( \left(1-\frac{\partial U^2}{\partial x}\right)\left(U^1-\frac{\partial U^1}{\partial t}\right)
-\frac{\partial U^2}{\partial t}\frac{\partial U^1}{\partial x}\right).
\]
Consequently,
by comparing (\ref{tele11}) with the identity (\ref{eq38}) in Theorem 3.2,
we obtain an invertible mapping of the nonlinear system (\ref{tele1})
to a linear target system
which is given by the adjoint of the linear system (\ref{tele10}).
In particular, the point transformation
\begin{equation}
z_1 = X = x-u^2,\quad
z_2 = T = t-\log u^1,\quad
w^1 = x,\quad
w^2 = u^1
\label{tele12}
\end{equation}
maps the nonlinear telegraph system (\ref{tele1}) invertibly into the linear system
\begin{equation}
\frac{\partial w^1}{\partial X} - \frac{\partial w^2}{\partial T} - w^2 = 0 ,\quad
\frac{\partial w^2}{\partial X} - \frac{\partial w^1}{\partial T} = 0 .
\label{tele13}
\end{equation}
Note that the further point transformation
$\tilde{w}^1 = w^1$, $\tilde{w}^2 = e^Tw^2$
maps this linear system (\ref{tele13}) into
the equivalent linear system (\ref{tele6}) obtained from
linearization of the nonlinear telegraph system (\ref{tele1}) through
its admitted infinite set of point symmetries (\ref{tele2}).

\section{Concluding Remarks}

An important question is which of the two methods presented in this paper
is better for linearization?
It would appear likely that the approach using
admitted conservation law multipliers
will be better computationally
since, in general,
the solution space for the determining equations for conservation law multipliers
is smaller than that for point (or contact) symmetries.
A way of seeing this is to consider
the case when a given system of PDEs is variational,
i.e., its linearization operator (Frechet derivative) is self-adjoint,
so then Noether's theorem is applicable.
For any such system
its conservation law multipliers correspond to a variational subset of
its admitted point (or contact) symmetries
and hence, in general,
the determining equations for conservation law multipliers are then
more over-determined than those for point (or contact) symmetries
since they include the symmetry determining equations as a subset
\cite{b05,b06,b07}.

An interesting computational way for checking for the possibility of linearization of a given system of nonlinear PDEs,
especially for classification problems,
is to apply the work in \cite{c,d}
to find the size of the solution space of a given system of
determining equations
without the need for obtaining any of its actual solutions.
In particular, if the size of the solution space of the determining systems
either for conservation law multipliers
or for point (or contact) symmetries is finite dimensional,
then no linearization by an invertible mapping is possible.
But the converse (that the solution space is infinite dimensional)
is not sufficient to imply the existence of a linearization by
a point or contact transformation.
Specifically,
the solution space must have a sufficiently large number of parameters
such that the number of functions and independent variables,
as well as the number of linear PDEs,
matches the corresponding number in the given nonlinear system of PDEs.
This counting can be performed fully algorithmically \cite{c,d}
provided a differential Gr\"{o}bner basis is available.
However, it is important to recognize that in practice
the computational complexity of finding a differential Gr\"{o}bner basis
will be very high.

It would also be interesting to develop a fully algorithmic version
of the two linearization methods themselves.
This would entail, firstly,
the use of a differential Gr\"{o}bner basis computation
to reduce the determining equations into a consistent over-determined linear system
satisfying all integrability conditions
and having a sufficiently large solution space.
Secondly, a strictly computational procedure must be specified to extract
from this linear system the form of the linearizing transformation for
the dependent and independent variables in the given nonlinear system of PDEs.

The algorithmic steps we have used in our present work
(e.g. for linearizing Burgers' equation, the pipeline equation,
and the nonlinear telegraph system)
are less automatic than finding a differential Gr\"{o}bner basis
but have much less computational complexity in practice.
In particular, they take advantage of the integration capabilities of {\sc Crack} \cite{b14},
including algorithms for direct and indirect separation of equations,
for dropping redundant functions, for checking integrability conditions,
and others.
Furthermore, {\sc Crack} also handles solving the first-order linear PDEs
as needed in finding the transformations of the independent variables
in the linearization mapping.

Finally, we mention that a hodograph-type approach for linearization
has been given some time ago in work \cite{e} of Varley and Seymour
on linearizing the nonlinear telegraph system (example 4.3).
Their clever procedure for finding a linearization is not symmetry-based;
it only applies to specific types of nonlinear PDEs
and cannot determine, in general,
whether a given system of nonlinear PDEs admits a linearization
by an invertible mapping.

\section*{Acknowledgements}
The authors are each supported by an NSERC research grant.

redpajama_set_name: RedPajamaArXiv


Besides its low cost of living, Iowa also has a large variety of senior housing options in large cities such as Des Moines and in smaller cities and towns. Choose from independent living, assisted living, memory care, and more. Active seniors can take part in the state's yearly Iowa Senior Games.


A miniature model of bustling Bangkok, the pavilion offers heaps of glittering accessories, hair and hand bands, clothing, children's wear, cosmetics, massage shops and food. Exhibitors loudly call out enthusiastic visitors to buy from them at knock-off prices, giving you a first-hand authentic Thai street shopping experience. Thai fruits are available in all colours and tastes and include mangoes, bananas and others.


text:
Q: Existence of zeroes of holomorphic Functions I really am struggling with this kind of easy question, i hope you can help me out here.
Let $\Omega \subset \mathbb C$ be a domain, $f : \Omega \rightarrow \mathbb C$ a holomorphic function and $z_0 \in \Omega$ be a zero of $f$.
1) Show that if f is not the constant nullfunction there exists a $k \in \mathbb N$, so that $f^{(n)}(z_0) = 0 \quad \forall n \lt k$ and $f^{(k)}(z_0)  \neq 0$
2) Furthermore show that there always exists a $\epsilon \gt 0$ so that for all $0 \lt r \lt \epsilon$ the equation$$\frac{1}{2 \pi i} \int_{\gamma_{B_r(z_0)}} \frac{f´(z)}{f(z)} $$ returns the order of zeroes of f.

A: Hints: for (1), use the identity principle to prove that if all the derivatives at $z_0$ are zero, then $f=0$ (look at the Taylor expansion in some ball around $z_0$). For (2) use that the zeros of any holomorphic function are isolated: write $f(z)=(z-z_0)^{{\rm ord}(f,z_0)}g(z)$, with $g$ holomorphic and $g(z_0)\neq 0$ and then compute that integral.

redpajama_set_name: RedPajamaStackExchange


text:
CWL - YTO
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Pros: "The entertainment options were good."
Pros: "He missed it due to having a 32 minute layover, which he ran all the way to board."
Cons: "he had no choice but to look at the closed door of his flight, and wait an additional 7 hours and they didn't even give him a voucher for food. I'm his mother, I'm currently visiting family in Scotland, so couldn't help him much. I'm pretty disgusting and disappointed by the whole experience."
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Cons: "I paid €60 extra to sit in the emergency door seat to get more leg room but once.i was in the plane learned for more leg room I accepted a smaller seat next to a bathroom and no entertainment. I could not store my belongings near me. No power outlet to charge my devices for my own entertainment on a 7 hour flight. Dealing with constant traffic for the bathroom. I sat in the middle seat between two hours other men. It was a constant struggle arm space. After 7hours in the air I could not wait to get off the plane. If I didn't actually pay an extra €60 for the seat I may not feel bad but I feel like I might have been better off without paying for the "upgrade""
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Cons: "Lost our bags, little sympathy from staff"
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Pros: "Professional, welcoming and very friendly crew, well-maintained aircraft, comfortable seat and good food. Keep up the good work, KLM."
Cons: "Not much to comment on here."
Cons: "That I was forced to check my baggage. I never received a chain ticket for my luggage. The flight has a 32 hour layover and KLM lost my luggage and it was not found until a week after my arrival back home. I still have not received my bag at this time. The boarding crew was manually tracking who boarded and who didn't. It was a cluster."
Pros: "The crew were just amazing, such lovely male stewards! They were helpful and friendly and generally wonderful."
Cons: "The entertainment was quite dated and didn't have new movies etc"
Cons: "Why do the crew have to slam the overhead and catering compartments to get them to shut?"
Cons: "Only one queue for boarding. Flight late on arrival."
Pros: "The staff were friendly and all smiles. Seats were comfortable and we had a quick, smooth flight from Edinburgh to Amsterdam. The breakfast supplied was an egg and mayonnaise sandwich plus a drink, which seemed about right and very Dutch in character."
Cons: "It was rainy and necessary to walk outside to access the plane; the staff should give their customers a heads-up so that jackets can be donned. Additionally, the check-in queue seemed rather sluggish."
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Cons: "Nothing to improve"
Pros: "Efficiency and speed of boarding was acceptable. I have no complaints about that aspect. The Comfort+ seats were worth the upgrade, pretty comfortable and reasonable legroom unless you have a big carry on stowed under the seat in front of you."
Cons: "The first announcement they made to all of us in the economy section was thay they did not have any headphones for anyone but business class. I was told when I booked the seat that the headphones would be complementary. So I did not bother to bring any assuming they were going to keep their word; I figure I'd make it work and watch a movie without sound. If that isn't bad enough midway through my nine hour flight my monitor went out and would not turn back on."
Pros: "."
Pros: "Food and staff were great"
Cons: "Entertainment system old and not as many options as other KLM flight. No power on aircraft for charging"
Pros: "Very good flight, excellent service and good food."
Pros: "Food and drink service"
Cons: "Boarding pass facility"
Cons: "Very slow service. I wanted to by several goods from duty free. But crew served only first 20 rows of aircraft."
Pros: "KLM Is an impressive flying experience. We enjoyed the comfortable seats, the in-flight entertainment, and the food. The flight attendants are top notch as well."
Cons: "The only complaint I have regarding my most recent flight was the incredibly rude service staff whom we experienced at the front counter at check-in. After a harrowing cab ride to the airport in which our driver continuously ignored my request to follow the GPS insisting that he knew faster shortcuts, we arrived 40 minutes later than intended. Additionally he dropped us off at the wrong terminal such that we finally arrived at the KLM gate an hour later than we had planned. This was especially frustrating as we are traveling with two small children and made every effort to be punctual and leave early to be sure we got there with time to spare. Not the case today. All of that I could deal with, but when we arrived at the KLM counter, and I apologized to the attendant staff member that we were late and explained that our cab driver got lost, she responded by chastising us first in Spanish and then in broken English. She made sure to make it clear to us that it was our fault and we should not blame the Argentinian cabdriver. The cabdriver was not actually Argentinian, but that is totally besides the point. I was totally taken aback to be treated so rudely as a greeting at the airline counter. My wife share the same sentiment. Nothing like this has ever happened to us before. Thank God, every single airport personnel and of course KLM staff member on board was incredibly kind and accommodating of our children subsequent to this. It saved what otherwise could've been a terrible experience."
Pros: "Smooth boarding. Good food. Decent leg room at my seat. Good crew. Flight was on time. Good magazine on the flight for entertainment."
Cons: "They didn't give any napkins out with the food which was weird."
Pros: "Even for this short flight, drinks and sandwich was provided staff were efficient and friendly"
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Tankless water heaters have been in use throughout Europe and Asia for decades as people have used them as domestic hot water heaters. The technology behind tankless water heaters has been adapted by North American manufacturers in particular in the last decade. These devices are able to handle the more demanding rates of water flow found in the majority of whole-house residential U.S. and Canadian applications. Tankless water heaters are especially popular in apartments, condominiums, single-family homes, mobile homes and manufactured homes, vacation properties and guest houses. Hotel/motel operators, builders and even real estate developers have benefitted from the added value that tankless water heaters have brought to their projects in recent years.
For more information on tankless or any other kinds of water heaters, please call Approved Comfort today!


ƒProgramming principles and concepts including block-based coding, flowcharts, sequence, loops, pseudocode, algorithms and desk checking.
ICAS is developed annually by a team of highly experienced assessment developers and psychometricians, with rigorous reviews at each stage of construction. Subject matter experts develop new ICAS questions each year, drawing on their in-depth understanding of the way students learn. All ICAS assessments are reviewed by experienced teachers to ensure that they accurately assess students' skills and are relevant to what they are learning at school.
ICAS is a product of UNSW Global Pty Limited, a world leader in the provision of education, assessment and expertise services and a wholly owned enterprise of UNSW Australia. UNSW Global Pty Limited ABN 62 086 418 582. ICAS is distributed exclusively in [Country] by Macmillan Education [Country].


New Celebrity Bikinis: Rachel Bilson, Hayden Panettiere, and More!
Summer's just around the corner, and celebrities know it! Rachel Bilson was spotted splashing around in the Barbadian waves wearing a black and silver two-piece, and Hayden Panettiere worked on her backhand in a brown string bikini while on vacation in Hawaii. Click through to see more of our favorite celebrity bikinis of 2012—so far!


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Ratcliffe exchange
On Thursday, March 29th, 15 students and two teachers left Invercargill on the fifth biennial Ratcliffe Exchange. Excitement mixed with parental anxiety was the mood in the terminal as farewells were made.
First stop – seven hours in Singapore airport (more precisely, two hours in the rooftop swimming pool) before touching down in Rome, Italy – about 30 hours after leaving New Zealand. Three action packed days followed as sights were 'ticked off': the Colosseum, Santa Maria Church, Trevi Fountain, Spanish Steps, Pantheon, Vatican Museum and many more wonderful sights.
Unfortunately, with Easter we were unable to get an audience with the Pope, however when we did 'pop into' St Peter's Square we managed to be there at the right time for the Papal blessing on Easter Monday. Though being some distance away we still managed to see the Pope.
From Rome we made our way to Domodossola via Milan. Arriving in Domodossola to a cool and showery evening, high in the Italian Alps. This unfortunately was typical of weather for the next few days. For many students, having the opportunity to stay at Calvario was one of the highlights of the trip. Here we walked past Antonio Rosmini's room every day on the way to the dining room. A day in Stresa, beside Lake Maggiore, was filled with learning more of the history of this humble and forward thinking man. We toured the first Rosmini College and inside its beautiful chapel viewed the tomb of Rosmini. Along the foreshore, we toured Villa Bolongaro where Rosmini spent his final years and where many of his writings are stored. His writings were so extensive that they are still being translated. We also went into the room where he worked and eventually died.
From Domodossola we travelled by train to the Milam Malpensa airport where we stayed the night before making our way to Dublin, Ireland. The next day was a long days travel to Ireland via Luton, but finally arriving at the Hostel round 11pm.
Ireland was an added addition to this trip compared to previous exchanges, as we wanted to include the Mercy Sisters as part of this exchange. After visiting McAuley House and getting a comprehensive history of Catherine McAuley's life and an understanding of what the Sisters of Mercy do around the world, this was definitely a worthy inclusion. The delicious tea and scones did help. We also managed to fit in quite a few sights in our time in Dublin. These included, Natural History Museum, a walking tour chronicling the fables and folklore of Dublin, Epic Museum – the migration story of the Irish ancestors.
On Monday 9th April we travelled to London for the rest of the week. This allowed us a good snapshot of such a large city. During this time we were able to visit the Tower of London, Tower Bridge, Westminster Abbey, St Paul's Cathedral, Trafalgar Square, Piccadilly Circus, the London Zoo, Kensington Palace, some of St James' and Hyde Parks and see the "Lion King' stage show. The visit to Windsor Castle will be remembered for our meeting up with past teacher Susan Winters. Riding the underground and counting the stations before getting off became the norm for the students.
Come Friday 13th, it was off to the long anticipated reunion at Ratcliffe College with the host families and the students who had visited SPC in 2017. During the 14 day stay here, besides attending classes, students visited Warwick Castle and the theme park, Alton Towers, along with other activities organised by the individual host families. Tears flowed as farewells were said – some of these 'connections' will continue for years to come, I am sure.
Returning to London for the last few days meant a few more sights were 'ticked off'; visiting the New Zealand War Memorial, the Imperial War Museum, Buckingham Palace, London Eye, Madam Tussauds, the Natural History Museum as well as the all-important shopping.
Finally it was time to board the plane for home – the complete spectrum of emotions from the group were felt, with some students ready to return to family while others wanted longer!! However, there was definitely an air of excitement as parents greeted their children back in Invercargill on April 30.
What an action packed four weeks. Thousands of kilometres travelled, hundreds of photos taken and many, many stories to be told.
Written by Ronnie Kotkamp


Active in both the American Indian Movement and in the U.S. Civil Rights Movement, Jimmie Durham fuses his artistic practice with a keen political activism. Durham's incisive works draw on his Cherokee roots, probing and questioning Western hegemonic values and perspectives in order to offer alternative narratives and modes of reflection. Stone plays a central role in his art, used often as a destructive force. Boulders are used to crush cars and rocks are used to quite literally "stone" a refrigerator, acts of nature triumphing over the manmade technology of the West as well as alluding to—and questioning the validity of—ancient Christian rituals.
Born in Arkansas in 1940, Jimmie Durham has exhibited widely in internationally renowned institutions including the Serpentine Gallery, London, Whitney Museum of American Art, New York, and the Musée d'Art de la ville de Paris, amongst many others. He has also shown work at the Venice Biennale, documenta, and the Whitney Biennial. Durham lives and works in Berlin.


Chinua Achebe Interview about "Things Fall Apart"
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Although there are many prescription medications for anxiety there are many people who wish to find other options to treat their anxiety. One of the potential issues with taking prescription medications and especially anti-anxiety meds is that there may be undesirable side effects or withdrawal symptoms. Another problem is that there are some anti-anxiety medications which may be habit forming especially those medications which fall under the category of benzodiazepines such as Xanax or Valium. In addition, if you use some of these prescription anti-anxiety medications, they may not work as well over time and you end up having to keep increasing the dosage. These reasons and more make some people look to "natural" treatments for their anxiety.
There are some precautions, however, that you need to think about when looking to supplements and herbal remedies to treat the symptoms of your anxiety disorder. Here are just a few things to think about.
Just because it is natural doesn't mean there won't be any side effects. And one of the problems with the natural supplements is that you do not get a list on the bottle of any possible side effects primarily because they may be unknown.
Most natural supplements are not FDA approved and this means that sometimes fillers are used which may be useless or even harmful additives. Unlike prescription medications that go through a rigorous safety and effectiveness evaluation, herbal remedies and supplements have no such evaluation. The safety and effectiveness is left up to the manufacturer.
Due to the lack of scientific studies on many of these supplements, the possibility of drug interactions may be unknown. If you are taking multiple prescription drugs as well as supplements you may be brewing a dangerous cocktail of substances without knowing.
The lack of regulations for supplements also might mean that the herbal product does not contain enough of the active ingredient to make it effective.
Natural supplements may not be strong enough to help with moderate to severe anxiety symptoms and you could be losing valuable time when you could be taking something more effective for you.
The bottom line is that you need to do your homework and research any supplement you are thinking about taking. Not only that, you need to tell your doctor what vitamins and supplements you are taking in addition to any prescription medications so that they can tell you (if they know) about any possible adverse interactions.
Can Kava Help Social Anxiety Disorder?


Open year round, weather permitting. During the winter, reservations will be accepted no more than one week before your requested stay, to determine weather conditions.
Water at the sites is typically turned on around the beginning of May, and turned off no later than the end of October. This is weather driven, and extremely cold temperatures could make us alter the schedule. There is a water spigot on the side of the store for you to fill your tanks during the off-season, but you must bring your own water hose.
Heated shower house open year round!
We have plenty of campsites available for your stay with us. If you require water and electric, or if you prefer a full hook-up site, we have a site that is perfect for you in either a wooded or open area. Clean, modern restrooms and showers are also available for your convenience.
Sites Offered: W/E and Full Hook-ups — Larger units V.I.P. pull-throughs with charcoal grills.
Check-in Times: Monday – Saturday 3:00 pm – 9:00 pm; Sunday 3:00 pm – 5:00 pm.
Please plan your travel time accordingly as we cannot offer early check-ins. (We have a very small staff, and we are busy getting ready for you!) We do not offer late night check-ins, as a courtesy to our other guests.
10% Active Military Discount (excluding special or holiday weekends). Please advise at the time of reservation; we will need proper I.D. at check-in to honor the discount.
$5 extra per night during all holiday weekends.
PLEASE NOTE THAT "DAY GUESTS" MAY NOT USE THE POOL, DUE TO INSURANCE AND LICENSING RESTRICTIONS.
All daily and overnight guests must register upon entering the campground. Daily guests must leave by 9:00 pm or be charged the overnight rate.
$1560 plus metered electric. Seasonal guests have access to the campground all year. We currently have three seasonal sites available!
ATVs are welcome on a trailer or in the back of your truck bed. However, our insurance will not permit ATVs to be placed on the ground unless you need to take your ATV out of the trailer so that you can sleep in the trailer. Please talk to us first so that we know why the ATV is off the trailer. Sorry, but we do not have trail access from the campground, and ATVs may not be ridden in the campground.
Your "quiet", well behaved dogs are welcome on a leash. Please clean up after your dogs. All pets must be current on all vaccines, and are not to be left at the campsite unattended. Dogs bark when their people are gone, so please take your dogs with you. Our insurance prohibits aggressive dogs in the campground. Please help us maintain the tranquility of the campground by keeping your dogs from barking. Thank you.
*Pets are NOT permitted in the cabins (with the exception of the Rocky Mountain Cabin and the Covered Bridge)!
Payment: All reservations must be paid in full at the time of reservation by Visa, MasterCard, or Discover. All credit card transactions will be processed at the time of reservation.
Cancellations & Refunds: No refunds for early departure, illnesses or other reasons, including inclement weather.
Camping: If cancellation is received 14 days prior to the arrival date, a one year rain check will be issued. Sorry, but we are not able to issue or accept rain checks for holiday weekends (including Tour De Forest).


text:
Biomass of Forests
Biomass gasification employing low-temperature carbonization pretreatment for tar reduction
Umeda et al. in: Biomass & Bioenergy (2019), 126, pp 142-149
Harvest interval and row spacing of SRC willow influence yield and nutrient content
Larsen, Jorgensen, Laerke in: Biomass & Bioenergy (2019), 126, pp 181-189
Simulation of a tilted-slide reactor for the fast pyrolysis of biomass
Choi et al. in: Biomass & Bioenergy (2019), 126, pp 94-105
Effects of bedrock type and soil chemistry on the fine roots of European beech – A study on the belowground plasticity of trees
Kirfel et al. in: Forest Ecology and Management (2019), 444, pp 256-268
Exceptionally high mangrove root production rates in the Kelantan Delta, Malaysia; An experimental and comparative study
Muhammad-Nor et al. in: Forest Ecology and Management (2019), 444, pp 214-224
Global patterns of tree stem growth and stand aboveground wood production in mangrove forests
Xiong et al. in: Forest Ecology and Management (2019), 444, pp 382-392
Nutrient cycling in forests: A historical look and newer developments
Johnson, Turner in: Forest Ecology and Management (2019), 444, pp 344-373
Optimizing stand structure for tradeoffs between overstory and understory vegetation biomass in a larch plantation of Liupan Mountains, Northwest China
Ahmad et al. in: Forest Ecology and Management (2019), 443, pp 43-50
Quantifying tree and volume mortality in Italian forests
Bertini et al. in: Forest Ecology and Management (2019), 444, pp 42-49
FOREST POLICY AND ECONOMICS
Innovation in the use of wood energy in the Ukrainian Carpathians: Opportunities and threats for rural communities
Soloviy et al. in: Forest Policy and Economics (2019), 104, pp 160-169
Estimation of the volume, biomass and carbon content of coarse woody debris within two forest types in the State of São Paulo, Brazil
Moreira, Gregoire, do Couto in: Forestry (2019), 92:3, pp 278-286
Arbuscular Mycorrhizal Fungi Effectively Enhances the Growth of Gleditsia sinensis Lam. Seedlings under Greenhouse Conditions
Wang et al. in: Forests (2019), 10:7, Article 567
Changes in Carbon Balance of Harvested Wood Products Resulting from DifferentWood Utilization Scenarios
Parobek et al. in: Forests (2019), 10:7, Article 590
Comparison of Three Algorithms to Estimate Tree Stem Diameter from Terrestrial Laser Scanner Data
Ravaglia et al. in: Forests (2019), 10:7, Article 599
Contrasting Development of Canopy Structure and Primary Production in Planted and Naturally Regenerated Red Pine Forests
Hickey et al. in: Forests (2019), 10:7, Article 566
Development of a Tree Growth Difference Equation and Its Application in Forecasting the Biomass Carbon Stocks of Chinese Forests in 2050
Zhang et al. in: Forests (2019), 10:7, Article 582
Forest Structure, Wood Standing Stock, and Tree Biomass in Different Restoration Systems in the Brazilian Atlantic Forest
Pontes, Engel, Parrotta in: Forests (2019), 10:7, Article 588
Growth and Tree Water Deficit of Mixed Norway Spruce and European Beech at Different Heights in a Tree and under Heavy Drought
Schafer et al. in: Forests (2019), 10:7, Article 577
Land Use Changes, Disturbances, and Their Interactions on Future Forest Aboveground Biomass Dynamics in the Northern US
Ma et al. in: Forests (2019), 10:7, Article 606
Linking Dendrometry and Dendrochronology in the Dominant Azorean Tree Laurus azorica (Seub.) Franco
Matos et al. in: Forests (2019), 10:7, Article 538
Measuring the Labile and Recalcitrant Pools of Carbon and Nitrogen in Forested and Agricultural Soils: A Study under Tropical Conditions
de Almeida et al. in: Forests (2019), 10:7, Article 544
Optimizing Quality of Wood Pellets Made of Hardwood Processing Residues
Thiffault et al. in: Forests (2019), 10:7, Article 607
Removing the Scaling Error Caused by Allometric Modelling in Forest Biomass Estimation at Large Scales
Zhou, Zhou in: Forests (2019), 10:7, Article 602
Vertical Distribution of Leaf Area of European Larch (Larix decidua Mill.) and Norway Spruce (Picea abies (L.) Karst.) in Pure and Mixed Stands
Sterba, Dirnberger, Ritter in: Forests (2019), 10:7, Article 570
Warming Alters Plant Chemical and Nutrient Compositions by Affecting Metabolites in Cunninghamia lanceolata (Lamb.) Hook
JOURNAL OF FORESTRY
Estimating Biomass Availability and Cost When Implementing Forest Restoration with Tethered Harvest Systems
Petitmermet, Fried, Sessions in: Journal of Forestry (2019), 117:4, pp 323-339
JOURNAL OF GEOPHYSICAL RESEARCH (JGR) – BIOGEOSCIENCES
Global Patterns in Net Primary Production Allocation Regulated by Environmental Conditions and Forest Stand Age: A Model-Data Comparison
Xia et al. in: Journal of Geophysical Research – Biogeosciences (2019), 124:7, pp 2039-2059
Two Decades of Experimental Manipulation Reveal Potential for Enhanced Biomass Accumulation and Water Use Efficiency in Ponderosa Pine Plantations Across Climate Gradients
Liles et al. in: Journal of Geophysical Research – Biogeosciences (2019), 124:7, pp 2321-2334
The effect of tree planting density on the relative development of weeds and hybrid poplars on revegetated mine slopes vulnerable to erosion
Remaury, Guittonny, Rickson in: New Forests (2019), 50:4, pp 555-572
Standing volume yield, canopy structure and allometric relationships of dominant urban trees in Melbourne, Australia
Orozco-Aguilar et al. in: Urban Forestry & Urban Greening (2019), 43, Article 126363
Soil carbon and nitrogen availability are key determinants of soil microbial biomass and respiration in forests along urbanized rivers of southern China
Mgelwa et al. in: Urban Forestry & Urban Greening (2019), 43, Article 126351
Category: BiomassTags: Australia, Biomass & Bioenergy, Forest Ecology and Management, Forest Policy and Economics, Forestry, Forests, JGR Biogeosciences, Journal of Forestry, New Forests, Publication Date: July 2019, Urban Forestry & Urban Greening
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